Economic Evaluation and Investment Decision Methods

Economic Evaluation and Investment Decision Methods Tenth Edition

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FRANKLIN J. STERMOLE PRopEssoR EuERrrus,

Colonaoo Scsoor_ op MrNrs CHarRrrtnN, INvnsrnr ENr EveLuarroNs CoRpoRetrorv JOHN M. STERIVIOLE

IxslirucroR. ColoRaoo Scuoor_ or. N{rNas AoruNcr [,Ror,ESSon, UrurvERsny op DeNvER

Col_lecr op Law

PRnsroeNt INvEsrlrpNr EvaluertoNs CoRpoRartoN

INVESTMETTIT EVALUATIONS CORPORATION 3070 South Newcombe Way Lakewood, Colorado 90227

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Franklin J. Stermole,8.S., M.S., Ph.D., Chemical Engineering, Iowa State University, is Professor Emeritus of Mineral Economics and Chemical and Petroleum Refining at Colorado School of Mines where he has taught since 1963. Frank also serves as Chairman of Investment Evaluations Corporation. He has taught economic evaluation techniques for over 30 years tri undergraduate and graduate students and has done economic evaluation consulting for numerous mineral and non-mineral companies. Since 1970 when the first short course was prcsented, Frank has taught more than 650 "Economic Evaluation" short courses to over 16,000 persons from mineral and nonmineral industry companies and govemment agencies. In addition to the United States these courses have been presented in Armenia, Australia, Canada, Colombia, Egypt, France, Germany, Great Britain, Guyana, Indonesia, Kazakhstan, Kuwait, Mexico,

Norway, Philippines, Saudi Arabia, S-outh Africa, Trinidad and Venezuela. This domestic and foreign industrial consulting and teaching experience has had a direct eft'ect on the applications-oriented content and organization of the text. John N{. Stermole, B.S.B.A., Finarce, University of Denver, and M.S., Mineral Economics, Colorado School of Mines, is President of Investment Evaluations Corporation. Since 1988 John has taught as an Instructor at Colorado School of Mines for the Departments of Mineral Economics, Chemical And Petroleum Refining Engineering and Environmental Sciences. John has also served as a Fellow to the Institute for Globai Resources Policy at Colorado School of Mines, and was co-author in the 1st Edition of the Global Mining Taxation Comparative Study. Since 1997 John has also taught as an Adjunct Professor at The University of Denver, College of Law in the Natural Resources and Environmental Law Program. John has presented more than 250 "Economic Evaluation" short courses for mineral, petroleum and non-mineral companies and govemment agencies. In addition to the United States, these courses have been presented in AustrrLlia, Canada, Chile, Colombia, Indonesia, South Africa, Srvitzerland and the United Arab Emirates. Prior to joining Investment Evaluations Corporation on a full tirne basis, John gained three years of industry experience with

Loq,dermilk Construction of Englewood, Colorado, applying economic evaluation techniques to heavy construction projects related to mine site development and highwiiy construction, and in replacement analysis.

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This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understantling that neither the authors nor the publisirer is engaged in rendering legal, accounting, tax, futures/securities tnding, or other professional services. If legal advice or other expert assistance is required, the services of a competent professional person should be sought.

From a Declaration of Principles jointly adopted by a Committee of tlte American Bar Association and a Committee of Publishers.

Trademarks:

Microsoft Excel is a registered trademark of Microsoft Corporation. I{Pl0B, HP12C, HPITBII are registered trademarks of Hewlett Packard Company. E\A is a registered trademark of G. Bennett Stewan IIL

Copyright @ 2000 by Investment Evaluations Corporation 3070 South Newcombe Way, Lakewood, CO 80227 USA Ph: 303-984-9954 Fx: 303-984-9895 Web site: www.sterrnole.com Earlier Editions Copydght @ 1996,1993,1990,1987, 1984,1982,1980, 1977 md 1974

By Investment Evaluations Corporation

All rights reserved. No Part of This Text May be Reproduced in Any Form Without Permission in Writing from Investment Evaluations Corporation ISBN r-878740-09-1 Library of Congress Control Number 00-134613 Printed in the United States of America

TABLE OF CONTENTS

t Page

CHAPTER 1: INVESTMENT DECISION MAKING

l.l 1.2 1.3 1.4 1.5 .6 1.7 1

Introduction to Investment Analysis "Engineering Economy" and "Economic Evaluation" Making Decisions Definition of Discounted Cash FIow Analysis Example of Discounted Cash Flow Minimum Rate of Return/Opportunity Cost of CapitallDiscount Rate InvestmentAnalysis

I 3

4 6 9

t2 t3

CHAPTER 2: COMPOUND INTEREST FORMULAS

2.1 2.2 2.3

2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.1

1

Introduction to Equivalence Compound Interest Formula Derivations and Illustrations Nominal, Period and Effective Interest Rates Based on Discrete Compounding of Interest Nominal, Period and Effective Interest Rates Based on Continuous Compounding of Interest Applications of Compound Interest Formulas 'Add-On" or "Flat" Interest (Applied to the Rule of 78) "Loan Points" and Buying Down Interest Arithmetic Gradient Series Alternative Time Line Diagrarn and the Concept of Cash Flow Introduction to Rate of Return Analysis Summary

15 L6

26 29 36

4t 44 46 49

5l 52

CHAPTER 3: PRESENT, ANNUAL, AND FUTURE VALUE, RATE OF RETURN AND BREAK-EVEN ANALYSIS

3.1 3.2 3.3 3.4

3.5 :1.6 3.7 3.8 3.9

3.10 3.I

I

lntroduction

62

Break-even and Rate of Return (ROR) Calculations Using Present, Annual and Future Worth Equations Rate of Return and Cumulative Cash Position Alternative Methods to Obtain Annual Value From Initial Cost, C

63 72

and Salvage, L Rate of Return on Bond Investments Rate of Return Related to T-Bill Discount Rates Financial Cost of Capital vs Opportunity Cost of Capital Rate of Return and the Revenue Reinvestment Question Growth Rate of Return Net Present Value, Net Annual Value, and Net Future Value lMethods of Analysis Benefit-Cost Ratio and Present Value Ratio

3.12 Etfect of Income Producing Project Life on Project Economics 3.13 Mineral and Petroleum Project Analysis 3.14 ROR, NPV and PVR Analysis of Service-Producing Investments 3. I

5

3.1

6

With Equal Lives Present, Annual and Future Cost Analysis of Service-Producing Investments With Equal Lives Comparison of Unequal Life Alternatives that Provide the Same Service tv

80 82 86 8'7

93 98

r03 112 122 124

r32 t36

t39

117 i.18

Comparison of Service-Producing Alternatives that Provide Diflerent Service Summary

CHAPTER 4: NIUTUAI-L\. EXCLUSIVE AND NON.MLiT{;ALLY EXCLUSIVE PROJECT ANALYSIS

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1? :t.-3

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Anall sis lvlutuaiiy Exclusive Irrct,nre pr.or]gci1g Aftcrnatiyes 'ri Using Rrle oi Return. Net Value antj Ratios

Life Nlutuarly Excrusive rncome-protJucing Arternative Analysis r\lutually Excrusive Inr.esrment Analysis using Gr.ivth nî. nrir." erid Fulure \\brth profit Methods "r Llnequal

.+.-i Changing rhe Minimum Rare of Return With Time -i 5 Diftbrences Between Net Value Analysis and Cost Analysis .1.6 Efl'ect of Evaluation Lit'e on Economic Analt sis Results Investment Analysis When lncome or Savings precedes Cost 11 Alternating Investmenr, Income, Investrnent: The problem 1I 1.9 Alternating Income, Investment, Income SituationsDuar Rate of Return

4.10

4.ll

144 148

Evaluation of Non-lVlutually Exclusive Investments Summary of Mutually Exciusive and Non_Mutually Exclusive

,\lternativc Anall sis

164 170

t82 185 189 193

t94 201 219

221 234

CHAP,IER 5: ESCAL,\IED AND CONSTANT DOLLARS

5.1 1? 5.3

Inflation and Escalariou in Econornic Anulvsis Exchange Rate Et{ecrs on Escalarion and iash Flow Analysis Summary

246 271 276

CHAPTER 6: UNCERTAINTY AT{D RISK ANALYSIS

I Introduction Anail sis to Analyze Efl'ects ol.Uncerrainty I : Scnsiriviry n I ]hg Ralrre Approach to Sensiriviry AnalS.sis 5.,1 Prob:rbilisricScnsitivityAnalvsis 6 5, Expected \hlue Analyiis (Economic Risk Analysis) Expecred NPV, Expected pVR, and Expected ROR 99 6-7 Probability of Survival (Financial RiskAnalysis) Analysis 6.8 Risk Due to Natural Disaster 6.9 Option Pricing Theory Related to ENpV 6.

6.10

Summary

282 284 287 288

296 299 311

Jl-1 315

320

CHAPTER 7: DEPRECIAIION, DEPLETION, AMORTIZATION AND CASH FLOW

7.1 7.2 7.3 7.,1 75

Introduction

After-Tax Cash Flow in Equation Form Business Costs That May be Expensed Depreciation L)epreciation N{ethotls 7.5a Straight Line Depreciation 7.5b Declining Balance Depreciation !2, S-witching liom Declining Balance to Straight Line Depreciation 7 .5d Units of Production Depriciation 7'-5e Modified Accelerated cosr Recoverv System (ACRS) Depreciation Election to Expense Limired Depreciabie Costs 79 7.7 Depletion Methods 7.7a Cost Depletion 7.7h PercenrageDepletion

326 328

330 335 337

340 a^1

343 344 350 3s0 351

352

.8 '.9 '.10

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Amortization

357

Royalties, Production Payments, Severance and Property Taxes Four Investor Financial Situatlons'That Affept Caqh Flow Calculatig.ns

361

Introduction Forms of Business Organizations and Tax Considerations Corporate and Individual Federal Income Tax Rates Corporate and Individual Capital Gains Tax Treatment Tax Treatment of Investment Terminal (Salvage) Value Alternative Minimum Tax Effective Tax Rates for Combined State and Federal Income Tax

8.10 8.1

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8.12

Tax Credits Discounted Cash Flow Rate of Return (DCFROR), Net Present Value, (NPV) and Ratio AnalYsis Working Capital Intemational Project Evaluation Considerations Mining and Petroleum Project After-Tax Analysis

CHAPTER 9: AFTER-TAX INVESTMENT DECISION METHOT'S AND APPLICATIONS

9.1 9.2 9.3 9.4 9.5 9.6 9.1

Introduction Payback Perioo Analysis Savings are Analogous to Income Sunk Costs and Opportunity Costs in Evaluations

Break-evenAnalysis Three Methods of Investment Valuation NPV Use For Break-even Acquisition Cost Valuation 9;7a NPV Use for Break-even Sale Value Analysis 9.8 Valuation of Pubtic Projects and Investments 9.9 Tax Analysis Versus Financial (Shareholder Report) Analysis 9.10 Net Income Analysis Cornpared to Cash Flow Analysis 9.10a "Economic Value Added" Net Income Analysis 9. l1 "Regulated" Company Investment Analysis

CHAPTER 10: AFTER-TAX SERVICE ANALYSIS

10.1 10.2 10.3

lO.4 10.5

l0-6

General ReplacementPhilosophy Leasing Compared to Purchasing Sunk Costs and Opportunity Costs Related to Replacement Evaluation of Alternatives That Provide Different Service Unequal Life Service-Producing Alternatives Optimum Replacement Life for Equipment

CHAPIER 11: EVALUAIIONS INVOLVING BORROWED MONEY I

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366

Summary

IH.{PTER 8: INCOME TAX, CASH FLOW WORKING CAPTIAL AND DISCOUNTED CASH FLOW ANALYSIS

i. I 1.2 t.3 1.4 1.5 3.6 3.1 3.8 8.9

359

Introduction ofLeverage Applications

I I .l a Joint Venture Analysis Considerations ll.2 Considerations Related to Leveraged Investment Analysis' vt

? 378

379 380 383 384

391 394 395 398

106 415 419

436 437 443 445 451

460 462 467

469

4't0 478 483 490

505

513 529

534 540 543

553

562 567

1.3 Current U.S. Tax Lau, Regarding Interest Deductions 1.4 Minimum Rate of Return and Leverage 1.5 Capitalization of Interest in Certain I_'everaged Investments 1.6 Leveraged Purchase Versus Lease Analvsis1.7 Summary r

s69 573 576 578

584

CHAPTER 12: PERSONAL INVESTMENTS AND HEDGING

12.l

Itrroducrion

12.2 12.3

Common Stock lnr.estments Put and Call Option Investmenrs 12.3a Writing Put and Call Option Conrracts 12.3b Index Options 12.-jc Foreign Currency and Debt Options 12.1 Futures Contract Transactions I 3.-1a Options on Furures 12.5 Net Wbrth, Stock Equity, Bonds and Debentures l?9 Placing o'ders to Buy or Seil stocks, Bonds, Debenrures, options and Fulures 12.7 Comparison of Alternative personal Investments I2.7a Life Insurance Aiternatives 12.7b Home Purchase Versus Renting I 2.7c Personal Auto purchase u".suiLeasc I1.8 Summary of Selected Investment Terminology

,_i89

:r)3 600 604 607

609 611

614

6r6 625 627 633 635

bJv 642

APPENDIX A: Discrete Inrerest, Discrete lhlue Facrors

649

APPENDIX B: Continuous Interest, Discrete Value Factors

6'10

APPENDIX C: continuous Interest, continuous Flowing value APPENDIX D: Production Cost Variations and Break_even APPENDIX E: Arithrnetic Gradient Series Facror Equivalence and Conversion

Information

SELECTED REFERENCES

Factors

Analysis

Development

6E2

696 7O"l

709 710

INDEX

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PREFACE

This textbook represents the ongoing efforts and interests of a father and son who have worked together at various times and in varying amounts for over twenty years. The original versions of this text were the sole efforts of Frank Stermole. John began making some contributions as early as the Fourth Edition, but more significant contributions began with the Sixth Edition. While some might view writing a book as a difficult task, we have

truly enjoyed the opportunity to work together and try to improve our understanding of economic evaluation concepts through the textbook and its examples and problems. This text is an introduction to the concepts of the time value of money and the application of time value of mbney decision criteria to the beforetax and after-tax evaluation of virtually all types of investment situations. We would like to emphasize that the concepts to be developed throughout the text are used by investors in all investment situations. Other than the obvious engineering differences, whether you are considering development or expansion of an existing ore body, coal deposit, oil and gas field, or considering refining projects or a real estate investment, the same economic

evaluation tools are utilized. From an economic evaluation viewpoint, the primary difference between oil and gas, refining, mining and real estate

evalnations

will relate to the relevant tax considerations which

are

addressed in this textbook beginning in Chapter Seven. The {irst six chapters present decision criteria on a before-tax basis to sim-

plify the understanding in developing each of the criteria such as rate of return, net present value and ratios and how they are applied in different investment situations. Chapters Seven through Eleven address the same issues on an after-tax basis. This involves developing an understanding of cash flow, and other subtle aspects of proper after-tax evaluations. Chapter Twelve addresses personal investment considerations which can also be tied directly to any other type of project evaluation. For example, investing in futures is discussed in the final chapter. The use of futures, options, options on futures, etc., are all known mechanisms to reduce the level of uncertainty in some economic evaluation project parameters early on in the project evaluation life. vill

Ti;e material covered in the text is applicable for students in all engineering drsciplines, geology, geophysics, business, accounting, finance, management,

operations research, and anvone interested in economic er.aluation issue

ihis tcxtbook

s.

iras been designed tor use in threp basic ways. First, it SrriYcS as a university textbook for unilergraduate or graduaie students. on thc Coloi-atlo School of Mines campus ,,i.e have used the textbook for a one

seilester course in which all the material from chapters one throush

Twelve is addressed. In a quarterry s1,stem. you might pief'er to break it irrto two components with one quarter on a before-tax basis addressing chapters one through six and a second addressing after-tax applications in cnapters Seven through Eleven. chapter Twelve addresses perional investment considerations and could be addressed in either or both courses. Second, we make extensive use of the text in continuing education courses for industry and government personnel with interests and backgrounds in the aforemen-

tioned categories. The examples and problems throughout the text are

designed with this in mind as they address specific e'aluation consideratiols ibr a variety of industry applications. Third, the text may also be used for self-study to teach one's self economic evaluation techniques and their proper application. To suppiement this later use, we have also written a "Self reaching Manual" for this textbook which specifically addresses in a more step-by-step process, the material in chapters one through Four of the textbook. If you find yourself stru_sgling with the firsr three oitbu, chapters you might seriously consider this g0 page manual as supplemental reading. w'e mzrke use of the Self reaching I\{anual as pre-course ieading when presenting the textbook material in a one week short course format. As you go through the textbook, you'll find we have a common theme centered on "consistency" throughout the evaluation process. Expanding on this simply implies that evaluators must compare projects and alternatives on the same basis. This means making a proper analysis for the evaluation situation and being consistent in terms of discount rates, timing, the type of

doliars irvolved, whether borrowed money is being consiJered and of

course, properly considering the relevant tax issues. You'll also find that evaluation work is onlv as good as the information provided for the analysis. The old saying "garbage in, garbage out,' could not be more accurate than for economic evaluation work. one big advantage with personal computers today is the ability to consider a wide range of sen-

sitivity analyses to uncertainty concerning input parameters. This helps immeasurably in establishing the most sensitive criteria which then can be emphasized through the engineering or cost esrimating process. tx

In most investment decision making situations, you will find that proper application of the concepts and techniques presented in the text together with a little common sense and good management judgment will enable you to do a better job of economic. investment dec,ision-making"than you can achieve without using these methods. Finally, we would like to offer our thanks to Pattie Stermole (John's wife) for her eftbrts in research and editing for this latest edition of the textbook. Frank Stermole and John Stermole

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CHAPTER

1

INVESTMENT DECISION MAKING

1.I

Introduction to Inyestment Analysis

Economic evaruation of investment alternatives rerates to systematicaily e,aluating the rerative profit potenrial of i,r,estment utt.*utir"r. If sy:,tematic, quantitative methods arg not used to compare the econcrnic considera_ tions of investment alternatives, it seems evident that in cedain investment decision making situations the wrong choices may be made frcrn an economic viewpoint. For example, in thJanalysis of investment alternatives for a given investment situation, the artematives under consideration may have differences with respec.r to costs and profits or savings unJ irr" timing of costs and profits or savings. Differences may also exisi in project lives, tax considerations, and the effects of escalation and inflation on projected costs and revenues. If a systematic approach is not used to quantify the economic effects of these factors, it is very difficult to correctry assess which arterna_ tives have the best economic potential. Since the days of the writings of economist Adam smith it has been rec_ ognized that capital accumuration has been the primary investment objective of capitalistic individuars, companies and societies to enable them to improve their standard of.riving. it rs emptrasized Iater in this chapter that factors other than economic consideration, into most i,vestment decisio,s, but from an economic viewpoint it is"n,". assumed that maximizing capi_ tal accumulation (or the value of aisets that courd be converted to capital) is During.rhe ren year period between the late r9g0,s and rate ll"^39;".rt'e' 1990's, it is estimated that **. .upitul investment dolars wil be spent in the united states than were spent cumulatively in the past 200years of u.S. history' The importance of proper evaluation techniques in deter_ ".onorni. mining the most economicaily effective way to spend this money seems evi_

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Economic Evaluation and lnvestment Decision Methods

dent whether you analyze it from an individual, corporate, or government viewpoint. This text presents the development and applicatiofl of economic"evaluation techniques that can be used to enhance your ability to make correct investment decisions from an economic viewpoint. Note thafit is not purported that the use of these techniques will enable you to make correct economic decisions all the time. Because of the effects of risk and uncertainty, including escalation and inflation of costs and revenues, it is not possible to develop evaluation techniques that guarantee investment decision making ,u.."ri. However, by using one or more of the economic evaluation techniques presented and recommended in this text, you should be able to do a consistintly better job of economic decision making than you can do without using these techniques. Obviously a given analysis is only as good as the inpui cost and revenue data that go into it. Risk and uncertainty effects make it impossible to know for certain that a given set of data for a proposed investment situation is correct. This, of course, means we cannot be rertain of the economic analysis results based on the data. Even when probirhilities c'rf success and failure are incorporated into the analyses, aS is introduced in Chapter 6, we clo not have analysis results that aIe certain for any given investment situation. However, even under evaluation conditions of great uncertainty, the use of the evaluation techniques presented in this text *itt giu" the decision maker a much better feeling for the relative risks and uncertainties between alternatives. This information together with the numerical economic evaluation results usually will put the decision maker in a better position to make a correct decision than he would be in if systematic evaluation procedures were not used. Evaluation of investment alternatives to select project investments that per dollar invested is a key goal of every successful corporate manager or individual investor. To fully achieve this goal, manageis or individuals should be familiar with the principles of economic evaluation and investment decision methods which provide the basis for quar'tified economic evaluation of alternative engineering projects and general

will maximize profit

investment opportunities. Most business decisions are made by choosing what is believed to be the best alternative out of several courses of action. Problems in this area are therefore called alternative choice problems. In many business situations, decisions are made intuitively because systematic, quantified decision making methods are not available to weigh the alternatives. This should not be the case for weighing the economic consideratiom related to most invest-

Chaoter 1: lnvestment Decision Makin.:

ment decisions. systematic erronomic decision methods are available for evaluating individual investment projects and for comparing alternative investment projects. The "u,hims of management" should not be the basis for reaching decisions concerning economic dift'erenceg between inr.estment alternatives. In this age of increasingll, complex iinestmeill situatii::rs, to be successful over the long run it is imperative that a primary economil evaluirtion criterion be selected and applied to compare alternative inr,cstmont choices. This text presents economic evaluation criteria which are brrsed tt;: the premise that profit maximization is the investment objective; that is, maximization of the future worth of available investment dollars. Ir{ethorls are developed and illustrated in this text to enable a person to determine the courses of action that will make best economic use of lirnited resources. In general, this involves answering the question, "Is it better to invest cash in a given investment situation or will the cash earn more if it is invested in an alternative situ ation?"

1.2 "Engineering Economy" and,,Bconomic Evaluation,, Engineering and science technology in one way or another provide the basis for most of the investment opportunities iir this world today. Even the economic desirability of investrnents in land often relates to engineering technology that may make the Iand more valuable several years fiom now for apartrnents, a park or some in,Justrial plant utilization. In a capitalistic society it is imperative that engineering proposals as well as all other types of investment proposals be evaluated in terms of worth and cost before they are undertaken. Even in public activitres, benefits must be greater than costs before expenditures normally are approved. Thus, the term "engineering economy" which is used widely in literature and texts applies in general to the economic evaluation of all types of investment situations. The terms "economic ev?rluation" arrd "engineering economy" are considered to have the same meaning in this text. A perscln does not need to be an engineer to be proficient in the application of engineering economy principres to evaluate investment alternatives. The well known prerequisite of successful engineering ventures is economic feasibility. This prerequisite applies to both engineering and non-engineering investment situations, so the terms "economic evaluation" and "engineering economy" have valid meaning and importance not only to engineers, but also to bankers, accountants, business managers and other personnel in a wide variety of job descriptions where

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they are concerned with economic evaluation of investment alternatives. This text is written for peop[with th3s-1krnd; of backqrounds or in1e1est,

,

1.3 Making

Decisions

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Peter Drucker, in his management texts, has stated that decision making has five distinct phases:

1. Defining the problem the problem 3. Developing alternate solutions 4. Deciding upon the best solution 5. Converting the decision into effective action

2. Analyzing

:

These decision making phases apply to economic evaluation decision making as well as general managerial decision making. Defining economic evaluation problems clearly is as important in economic analysis as any other situation that requires a decision. In any situation requiring decision making it is necessary to ask the right questions before one can expect to get the answers that are needed. Analysis of the problem or questions is the next step in the decision making process for economic analysis as well as general managerial decisions. This leads to the third phase of decision making concerning whether alternative approaches or investments might not be better. Analysis of these alternative investments then leaves us in a position to decide upon thti best economic choice and to take action to implement the best choice. This text, and the concept of economic decision making, is primarily con-

cerned with the three middle phases defined by Drucker. Again, this includes presenting and illustrating methods that can be used to analyze correctly various investment situations, develop alternative solutions and the economic analysis of these solutions. Emphasis is directed toward the fact that econornic analysis always involves comparison of alternatives, and determining the best way to invest available capital. From an economic viewpoint this means we want to maximize the future profit that can be accumulated from the available investment dollars. Economic evaluation decision making relates to two basic classifications of projects or investments: 1. Revenue producing investments 2. Service producing investments

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Sometimes people think a third investment classification might be "savings producing projects," but it will be illustrated in Chapter 3 that looking at differences between the costs of providing a service by alternative methods gives the savrngs that incremental investments in the more costly iniriai investment alternatir.es will generate. Analvsis of thes6 savings and incremental costs is just one of severai valid ways oi evaluating generai service producing projects. Ir4anv analvsis techniques are presented and iliustrated in this text. However, emphasis is placed on compound interest rarc of return analysis and net present value analysis, properly applied on an after tax basis. A large rnajority of individuals, companies and government organizations that use tormal evaluation techniques use rate of return analysis as their primary decision making criterion with net present value the second most used technique. There are other correct techniques for evaluating various investment situations including future and annual value, and several ratios. These techniques are presented in the text. It is necessary in economic evaluation work to be familiar with many dift'erent approaches to economic analysis because eventually you will interact with people that have a wide variety of evaluation backgrounds who use or advocate widely varying economic evaluation techniques. Familiarity with different evaluation techniques enhances communication with these people. Also, it will be shown in Chapter 4 that in ceriain evaluation situations, methods such as net present value have significant advantages over rate of return analysis. To communicate effectively u,ith diff-erent evaluation people you must be farniliar with the principles and advantages or disadvantages associated with different evaluation techniques. Also, beware that when you discuss rate of return analysis with different people the chances are that the term, "rate of return", may mean something very different to the other person than it does to you. Many different rates of return are defined in the literature, some of which follow: return on initial investment (ROI), which may be defined as being based on initial investment, average investment or solne other investment; return on assets (ROA), which is also called accounting rate of return and generally is based on the non-depreciated asset value, return on equity (ROE), which refers to return on individual or stockholder equity capital as the basis of the calculation; return on sales (ROS), which is not an investment rate of return at all; and the compound interest rate of return (ROR), or discounted cash flow rate of return on an after-tax basis (DCFROR) which is analogous to a bank account or mortgage interest rate and is the interest rate that makes project costs and revenues equivalent at a given point in time. Only this lat-

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ter rate of return, DCFROR, is valid consistently for anaiyzing alternative investments. The other rates of return may have use in certain analysis or accoirnting'sitUations but they should not, in general, be used to evaluate the relative economic merits of alternative investments becaule they do not account for the time value of money properly over the project evaluation iife. In general these other rates of return look at project rate of return at a specific point in time, or for some kind of average profit and cost considerations.

that the time value of money be handled correctly in all valid economic evaluation methods. Also, Since taxes are a cost relevant to most evaluation situations, economic analyses must be done after-tax. To omit a major project cost such as taxes may be more important than omitting operating costs and few people would think we should leave operating costs out of an analysis. In certain government project evaluations where tares do not apply. it is of course proper to neglect taxes. Irt general you -should think in terms of doing all analyses after tax, omitting tax considerations only when appropriate. [n Chapters 2 through 6 of the text, evaluation techniques and illustrations are presented primarily on a before tax basis to avoid confusing the reader with significant tax considerations, at the same time various evaluation techniques and the time value of money are being introduced. Starting in Chapter 7 everything is presented on an after-tax analysis basis and this is the way all evaluations should be done for decision making purposes.

It is imperative

1.4 Definition of Discounted Cash Flow Analysis In all industries, tvhether for corporations or individuals, economic analysis of potential investment projects is done to select the investment project or projects that will give maximum value from the investment of available capital. Investors usually use economic anal1'sis techniques based on either rate of return, present value, annual value, future value, or various breakeven analyses to reach economic analysis decisions. When the techniques just mentioned are based on handling the time value of money with a compound interest rate, these techniques are all referred to as "Discounted Cash Flow Anal-vsis Techniques". Understanding this concept requires definition of terms "discounted" and "cash flow". The term "discount" is generally considered to be synonvmous tt'ith "present v,orth" in econonic evaluation work. ln handling the time I'alue of money, investors want to account for the fact that a dollar in hand todav has

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Cnapter 1: lnvestment Decision Making

greater value than a doiiar at some future tin-ie because a doliar in hand trrday can be put to work now in a bank account or other investments to accrue interest, or refurn on investment. Compound interest is the generally iicccpted approach today for calcuiaring accrucd interest, or returil on il".'ritment. in time ."'alue of money calculatiors. TIle fut'-:re value thal is prolected to be accrued trom the investurent of dollars toei:r1' at a speciiied compounci interest rate is equal to the sum of the accrued interest and the initial doilars lprincipal.l invested. The concept oi present worth is just the o1-''posite of compounding. The present worth of a future value is the sum of rironey that invested today, at a specified compound interest rate, would grow to the given future value. when you are working with positive interest rates, present values are always less than future values. Since the term "discounting" implies reducing the value of something, the use of the terms "discounting" and "present worth" have equivalent meaning because they botli relate to reducing the value of assets or dollars. The term "cct,.rh flovv" is used to re.fer to the net inflow or outflow of zlstne), that occtrrs durirry a specified csyserating Ttcriod such as a month or r"C0r.

Gross Revenue or Savings

- Operating Expenses - Tax Costs - Capital Costs = Cash

Flow

1-t

Inflows of money from revenues and savings, minus outflows of money for expenditures such as operating costs, income taxes and capital expenditures, equal the project cash flow for a given period. If outflows exceed inflows of money, then cash flow is negative for that period. Of course, it follows that if inflows of money exceed outflows, then cash flow will be positive. Sometimes investors look at project evaluations on a before-tax basis, so they omit income iax costs and savings from economic analyses and define cash flow on a before tax basis. Generally, it is undesirable to evaluate investments on a before-tax basis, uniess the investor is not subject to income taxation. As previously mentioned, Chapters 2 through 6 do not directly address "after-tax" cash flow calculations. The reader can use Equation l-1 to help visualize the after-tax cash flow values that are illustrated in more detail in Chapter 7 through 12 examples.

The term "discounted cash flow" evolved from the fact that investors most often handle the time value of money uging present value calculations

,Dli

'.iiil,: i':ill:]}

.ri

ti rt

.r,;:,iiijl::1fi.

rllti:r:i|Njiiiljilliilfi$S$'.

,l.,,iti.:ii..l,i.lii:ililll{

ilirriti{iiirf


so they "present worth" or "discount" positive and negative "cash flow" anticipated from an investment to evaluate the project economic potential. Discounted cash flow analysis forces an investor to think systematically and quantitatively about all the relevant economic factors that may affect the economic poiential of investments. In the past, successfuT entrepreneurs ''intuitively" took into account investment economic analysis factors such as the magnitude and relative timing of investment costs and revenues, the effects that inflation and escalation may have on costs and revenues, the risk of failure involved with the overall investment, the uncertainty associated with projection of specific investment analysis parameters, the tax effects relevant to a proper after-tax evaluation for the financial situation of the investor, and finally, how to assimilate these considerations in a manner that enabled fair, consistent comparison of alternative investments. As investors have become more diversified, it has become more difJicult to use entrepreneurial judgments consistently and corectly in analyzing the economic potential of different investments. Discounted cash flow analysis has provided a systematic approach to quantitatively take into account the fac-

tors that are relevant in all industries for the proper economic analysis of investments. Examples of the use of discounted cash flow analysis today are innumerable. Income and service producing project investments of all types are analyzed using discounted cash flow analysis. Investors in minerals, petroleum, timber, real estate, manufacturing, leasing, etc., use discounted cash flow

analysis to determine the upper"limit that they could be willing to pay for mineral rights, land or assets to generate projected negative and positive cash flows over future years that yield a specified return on invested capital. Major companies use discounted cash flow analysis to evaluate the economic value of other companies. In a simplified form, by evaluating the value of individual properties and businesses that make up a company, the overall company value may be considered to be the cumulative sum of the value of individual properties or businesses that make up the cornpany. The acquisition bids for companies in recent years are situations where discounted cash flow analysis by major investors has indicated the value of companies to be considerably different than the value common stock shareholders had been placing on the companies utilizing net income approaches. Sometimes the discounted cash flow analysis will give a higher indicated value of a project or company than other evaluation approaches might give. Sometimes the discounted cash flow value may be less. The advantage of discounted cash flow analysis is that in all cases the assumptions on which


the analysis is based can be explicitly stated and understood by all. If you do not like the input assumptions they can be changed to what you consider more realistic. The non-discounted chsh flow, older econornic analysis methods have various implicit assumptions built in may or may not be (orrect in different analysis situations and, therefore,$at often lead to inccrr:ct economic evaluations. In particular, the older evaluation techniqr.:es d: not properly account for the time value of money. This is the single most inipor_ t:int consideration that has caused companies and investors in most in,Justries to shift to discounted cash flow analysis since the mid 1960,s. The real utiiity of discounted cash flow analysis is that it puts all invest_ rnents on a common evaluation basis of handling the time value of money with compound interest rate of return. In all industries, we are concerned rvith analyzing inflows of money (such as revenues and savings) and outflows of money (such as operating costs, capital expenditures and income tax costs). Discounted cash flow analysis enables investors to fairly and properly account for the magnitude and timing of these dollar value consid_ erations regardless of the type of investmeirt.

1.5 Example of Discounted Cash Flow Remembering that investment cash flow in any year represents the net difference between inflows of money from all sources, *lro, investrnent outflows of money from ari sources, consider the cash flow diagram preserited in thousanCs of dollars: Year Revenue

-

Operating Cost Capital

Cosrs -200 -100

Thx Costs

Project cash

23456 170 200 230 260 290 -40 -50 -60 _70 _80 -30

Flow -20a -100 +100

_40

+l

-50

-60

-70

l0 +120 +130 +140

The negative cash flows incurred during years 0 and 1 will be paid off by the positive cash flows in years 2 through 6, very much like loans of $200 and $100 thousand today and one year from today respectively, would be paid off by mortgage payments in amounts equal to the positive cash flow in years 2 through 6. what after-tax discounted cash flow rate of return (DCFROR) would an investor be receiving if he incurs the negative cash flows in years 0 and 1 to generate the positive cash flow from revenue in

,rui

rii;lHlirii}ilhif,

10


years 2 through 6? The compound interest rate that makes the present worth positive cash flow plus the present worth negative cash flow equal to zero.is the desired rate of return, compound interest rate, or DCFROR. using those terms interchangeably. This value is 20.8Vo for this stream;of positive and negative cash flows. Net present value (NPV), is the cumulative present worth of positive and negative investment cash flow using a specified discount rate to handle the time value of money. In general, the discount rate represents the minimum acceptable investment DCFROR. For this example a discount rate of l57a is used. Positive net present value represents the present worth positive cash flow that is above what is needed to cover the present worth negative cash

flow for the discount rate used. In other words, positive NPV represents additional costs that could be incurred in the year NPV is calculated, and allow the project to still have a DCFROR equal to the discount rate. Remember that rate of return (or DCFROR) is the discr:unt rate that makes NPV equal to zero. For the l5Vo discount rate. the NPV for the above values is +$54.75 thousand. This represents the additional negative cash flow that could be incurred in year 0 (in addition to the -$200 thousand cash flow in year zero) and have the project yield a 157o DCFROR. Sensitivity analyses can be made to see how the acquisition cost of $54.75 thousand is affected by changing the relative timing of when the costs and revenues are to be incurred. First, instead of incurring the cumulative positive investment cash flow of $600 thousand over years 2 through 6, assume the same cumulative positive cash flow will be realized over years 2 and 3 with +$280 thousand in year 2 and +$320 rhousand in year 3. Project Cash Flow

-200 -100 +280

+320

Year

For this case the DCFROR will increase to 37.1% and rhe NPV grows to +$135.2 thousand. Accounting for the rime value of money, and realizing positive cash flow much quicker, enhances the economics of a project significantly. Second, if we slow down the receipt of positive cumulative cash flow of $600 thousand so that the cash flow is realized more slowly over years 2 through 9 with +$55, +$60, +$65, +$70, +$75, +$85, +$90, and +$100 thousand per year respecrively, what is the effect on the project economics?

Project Cash

Flow -200 -100 +55 +60 +65 +70 +75 +85 +90 +100

t2

l-

4'56789


11

Def'erring positive cash flow into the future drops the Npv to -$11.7

thousand and the DCFROR to l4vo. Both values indicate that the project is

unsatisfactory compared to other opportunities thought to exist at a 15vo

rate of return and, in f'act, ttie Npv inclicates that we would have to be paid s11.7 thorrsancl to takr this project ancl receive a fsq, return on invested

ciipihl.

Suinmary of Findings lnvestment Life

3 Years

6 Years

9 Years

DCFROR

37Vo ZtVa I4Vo +$135.2 +$ 54.7 _$ 11.7 Cumulative +CF, Thousands $600.0 $600.0 $600.0 Cumulative -CF,'Ihousands $300.0 $300.0 $300.0 Project

ProjectNPV @ 157o

In this example we have looked at three different evaluations o1. the same cumulative negative cash flow (investment dollars) of $j00 thousand and cumulative positive cash flow of $600 thousand. If we neglect rhe tin.)e value of money, we would consistently determine that the project yieliis $300 thousand in profits. Yet the economic conclusions that aicount fbr the tiine value of money indicate a rarlge of Npv's for these three cases ftom -S11.7 thousand to +$135.2 thousand. Obviously, project economics properly accounting for the time value of money can be very sensitive to the relative timing of investment capital costs and revenues over the expected proj-

ect 1ife.

The discount rate selected can arso have a very significant effect on economic evaluation results. To illustrate this concept we will analyze the Npv of the six year life analysis for discount rates of l0 and 20 perbent, as well as l5 percent. The results are presented below: Discount Rate

NPV

107o

+$1 I 6.1

157o

207o

+$

54.8

+$

6.8

NPV results vary by a factor of l7 from +$l16.1 to +$6.g thousand as the discount rate is increased by a factor of two from 10 to 20 percent. In the following section, discussion is related to determining the appropriate dis-

count rate.


12

1.6 Minimum Rate of Return/Opportunity Cost of CapitaMDiscount Rate

It is widely

accepted in industry and government practigq for private government organizations, and regulated utilities alike that the <:ompanies, desired or allowed investment rate of return should equal the "cost of capital". Proper application of this concept is based on defining the "cost of capital" as the accepted rate of return that could be realized on similar altenrative investments of equivalenr rist. Many investors refer to this rate more explicitly as the "opportunity cost of capital" since it reflects the rate of return that the investor feels represents other opportunities in which to invest available capital with a similar level of risk. If these other investment opportunities are passed up, then the investor forgoes realizing the potential rate of return and thereby incurs an "opportunity cost of capital" equal to the foregone rate of return. The terms "minimum rate of return," "hurdle rate," "discount rate," "minimum discount rate," and "opportunity cost of capital" are all interchangeable with the term "cost of capital" as used in this text and in common practice. These interchangeable terms which repre' sent "opportunity cost of capital" must not be confused with the "financial cost of capital" which is the cost of raising money by borrowing or issuing

new bond, debenture, common stock or related debt/equity offerings. Regardless of the source of investment dollars, the objective of discounted cash flow analysis is to evaluate the economic potential of alternative income-producing and service-producing investrnents to select the optimum investments that will maximize the future value that can be generated from available investment capital. To achieve this objective, "opportunity cost of capital" rather than "financial cost of capital" must be used in discounted cash flow calculations and economic decision-making. Many people are confused by the differences in "opportunity cost of capital" and "financial cost of capital". It is not uncommon for people to mistakenly either physically use or think of a comparry "hurdle rate" or "minimum discount rate" as a "financial cost of capital" rate rather than aq "opportunity cost of capital" rate. It is shown in Chapters 3 and 4 that when the usual situation of capital rationing exists, the "opportunity cost of capital" generally is larger than the "financial cost of capital" and you will not achieve optimum economic investment decisions if you use "frnancial cost of capital" rather than "opportunity cost of capital" in your analyses. If bonowed money is unlimited so capital is not rationed, then and only then will the "opportunity cost of capital" equal the "financial cost of capital." These "cost of capital" dis-

.

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sMi

ll*r'ffiiq#rHuM,ril&$Bffiiiii::Sidt$ir,rilllliiJi:ll$rilirlitllltril}lElll*Hffi.:.illllrrlllii:

Chapter '1 - lnvestment Decision Making

13

count rate considerations are extremely important and fundamental to all discounted cash flow analyses calculations and decisions. Therefore, these "cost of capital" concepts and applications are explained and illustrated in greater detail in section 3.7 of Chapter 3 and in several sections and examr ples in Chapter.l.

1.7 Inl'estment Analysis Beiore proceeding to the development of the compound interest formulas in the next chaprer, it shoutd be pointed out that this text is concerned primarily with decision methods for "economic analysis" of alternative investment opportunities. An overall investment analysis should, and usually does, involve three analyses: 1. Economic Analysis

2. Financial Analysis 3. Intangible Analysis Economic analysis involves evaluation of the relative rnerits of investment situations fiom a profit and cost (or economic) viewpoint. Financial anal\,sis refers to where the investment funds for proposed investments will be obtained. Some alternate methods of financing investments include use of personal or corporate funds. borrowing from a bank, having a corporate funded debt offering of bonds or debentures, or going to the pubric with a new common stock offering. Intangible analysis involves consideration of factors that affect investments but which cannot be quantified easily in economic terms. Ty'pical intangible factors are legal and safety considerations, public opinion or goodwill, political considerations in foreign ventures, ecological and environmental factors, uncertain regulatory or tax law conditions, and many others. often times an alternative that looks best economically may be rejected for financial or intangible reasons. For instance, attractive projects may have to be rejected for financial reasons if internal funds are not available to finance the projects and outside financing cannot be obtained at attractive interest rates. Intangible factors that may cause rejection of economically sound projects are innumerable, but high on the list of importance are potential public opinion and legal problems from possible air, land, or water pollution. The importance of financial and intangible analysis factors in relation to economic factors must never be underestimated in management investment decision making. They often are as important as economic considerations.

'14


There is a tendency in the literature and practice to interchange the use of the terms "economic analysis" and "financial analysis". This often leads to confusion and improper use of investment decision methods' It is important to recognize that, as used in this text, "economic anallsis" relates to evaluation of the profitability of a proposed project and "financial analysis" relates to how the project will be financed. This text primarily is concerned with the development and illustration of economic analysis techniques;

L-

CHAPTER 2

COMPOUND INTEREST FORMULAS

2.1 Introduction to Equivalence Economic evaluation of investment alternatil,es requires that the alterna_ tives be evaluated on the same basis and that the time value of money be accounted for properly. when ahernate sources cf lcan rnoney are al.ailable with different paymenr schedules that make it difficult or irnpossible to determine intuitively the source that is least expensive, it is nccessaiy to convert the alternati\res to an "equi.,,alent" basis that permits conparison cf the alternatives. This necessitates correct accounting for the time value of money. For example, would you rather have $100 now or $102 a year from now? A majority of peopre would take gr00 now because they intuitively_ know that putting the $r00 in a bank at 4vo or 5vo interestwill give them more than $102 a year from now. Would you rather have $100 now or $I50 a year from now? A_rnajority of people would probably take $150 a year

from now because they intuitively know that tley proiabry do not have

other places ro invest $100 where ir will earn $5opiofit or'a 50vo rare of return in one year. Now if you are asked whether you would rather have $100 now or $150 tive years from now, the problem is more difficult to evaluate intuitively. What compound interest rate will cause $i00 now to gro$'to $150 in 5 years? Do you have alternative places to invest the $100 where it will grow to more than $150 in 5 years? Th"r" questions must be answered by using one of several possible "equivalence', -methods to com_ pare the relative economic merits of $100 today with $150 five years from today. In the following section the compound interest formulas that provide the basis for equivalence calculations ari developed and ilustrated.

15


15

2,2

Compound Interest Formula Derivations and Illustrations

This section presents the derivation of the six basic compound interest formulas commonly needed to apply engineering economy @cision methorJs for proper comparison of investment alternatives. To develop general formulas, the following letter symbols will be used throughout the remainder of this text:

P: Present single sum of money. Normally, "P" refers to a Sum of money at time zero, but may represent a sum of money at any point from which we choose to measure time.

F: A future single sum of money at some designated future date. A: The amount of each payment in a uniform series of equal payments made at each period. When the periods are years, 'A" refers to annual payments or value.

n: The number of interest compounding periods in the project evaluation life.

i: The period compound interest rate. Depending on the situatiol, "i" may iefer to either the cost of borrowed money, the rate of return on invested capital, or the minimum rate of return, in which case this value will be designated as "i*." To assist in understanding these letter symbols and their relationship to investment evaluation problems, refer to the horizontal time line diagram shown in Figure 2- 1.

AAAAA

..... . n-1

Figure 2-1 ATime Diagrarrt lllustrated compounding periods are designated 1,2,3, ' ' "1, below

the The interest horizontal line. At time zero, "P" designates a present single sum of money; 'A" designates a uniform series of equal payments at each compounding period; urd "F" designates a future sum of money at the end of period "n"' it usually is desirable to put the monetary numbers from economic evaluation problems on this type of time diagram to reduce confusion that creeps into problem statement; before attempting to calculate the desired quantities. Once the given monetary values are determined, it is necessary to

,

E

Chapter 2: Compound lnterest Formulas

17

establish where they occur on the time diagram, what increment

of time

cicsignates a period, what the interest rate per period is, and what you want

to calcularc. At this point, the probrem is essentialry solved. It is then just a i]larier of purring the information into the appropriqle equation that wiil cal_

culate the desired quantitv.

oa the lbllowing p.g., tt,. compound interest formula factors are devel_

oped to maihematically relate "p", "F", an,J ,A," In deveioping and dircrsring the appiication of these factors, the terms ,.u,orth,, and ,'value,, are used inter_ changeabiy to refer to either cost or income quautities and

calculations.

There are six different two variabre rerationships that can be developed

betwegn "P", "F",

and,A",

as

follows:

Calculated Quantir), F

F P P

A A

Given

= =PX

Quantity

Appropriate

X

F/Pi.n

=Ax =FX =Ax =FX =Px

Table 2-1 Variable Relationships Between ,,p,,, the Appropriate Factors 'ihe

Factor

F/A;,n

PFi,, P/A1,,

AFi,n

Mi,n ,,F,,,

and "A" and

factor sl,mbolism given here ctnd used throughout tr.te text is based on .first letter in each fa.ctor desigriating the quan"tity that the factor calcu_ lates, while the second retter designot"i the quantrty that is girrr. The two subscripts on eachfactor are the period intirest rate, ,,i,,, ioilorrra bt,the number of interest compounding period.s, "n". rJse of this ,v*u"iirnates the confusion of trying to memorize the name of each factor and "iir"i when to use each. For the student initiaily learning time value of money calcuia_ tions, this is a great help in becoming familiar with the apptication of com_ pound interest formulas. However, the common names of the different fac_ tors should be learned eventually since riterature commonlf us", ,u,r," terminology rather than symbolism. There are only three basic types of time varue of money calcurations: .(1) calculation of future varue, "F", from either ..p,, or ,,,q-ie) calculation of uniform and equal period values, .A,,, from either ,,F,, or .ip,,; (3) calcu_ lation of present value, "p", from eitirer ,.F,, or ,A',. All time value of money calculations invorve writing an equation or equations to calculate the


18

,,F", ,.P", or ,A" so familiarity with the development and application of the factors needed to make these calculations is very important'

either

SINGI,EPAYMENTCoMPOUND-AMoUNTFACTOR.Whena

futnre value, "F", "n" periods frOm nOw, is tO be CalCulated ffom a present the calcusum of money, "P", With cOmpOunded interest At"'i%o" per period, lations are made as shown on the time diagram inFigure2-2' Principal Interest

f,

).,'*', ililil,) P(1+i)2 i[i]l]l.1, ]

P(1+i)n F = P(1+i)n

Figure 2,2 Time Diagram lllustration of calculation of the single PaYment ComPound-Amount Factor' Interest is paid each year on the principai in the account at the beginning of the y"0.. Fo, year one, the principal is "P" and interest is. "Pi" which glves eit+i; accumulated at the end of year one. Therefore, in year two, the principal P(i+i) draws interest P(l+i)i which gives a total value of P(1+i)2 accumulated at the end of year two. The final value accrued at the end of year

"n" is given in Equation 2-1.

F=

2-l

P(l+i)n

The mathematical expression (l+i)n is called the "single payment comand is designated as F/Pi,n because a future single pound-amount factor" -sum ,,F", is to be calculatedfroru a present single sum of money, of money, ,,P" at a given interest rate "i" for a git,en nuntber of compouttding periods "n ". Example 2-I illustrates the use of the F/Pi,n factor'

EXAMPLE 2-1 Single Payment Compound-Amount Factor lllustration

Calculate the future worth that $1,000 today will have six years {rom now if interest is 10% per year compounded annually'

Solution: 1.7716

P=$1,000

1

2.............6

F = $1 ,000(F/P10%,0) = $1 ,771.6

factor is found in the tables in Appendix A, or it can be calculatdii'mathematically to be (1 +'10)o = 1 '7716'

The F/Pi

,.,


SINGLE PAYMENT PRESENT-WORTH FACTOR. [f rhe value of a future sum, "F", is given and the present value, "p", is desired, solve for .,p,, using Equation 2-1 as follows:

P=F[1/(1+i)o]

r'

z_z

factor l/(1+i)n is called the "single pa,,.rnent present-w,crth factt.tr;" and is designated br- P/Fi,r. Thisfactor is uset! t,t calculate a present sirigle sttrtt, "P", that is eqtdt'alem rc afutLtre single strrn, ,,F',. Note that P/Fi,n = t/(F/Pi,n). Aithough one factor can be calculated from the other. the rables of compound interest formulas (Appendix A) give both The

factors for convenience.

EXAMPLE 2-2 Single Payment present-worth Factor lttustration

calculate the present value of a g1,000 payment to be received six years from now if interest is 10% per year compounded annually.

Solution: P=

0.564s $'1 ,000(P/F10%,6) = $SG4.S0 F = 91,000

The P/F167.6 factor is found in Appendix A. The result shows that $564.50 invested today at 10"/. interest per year would grow to $1,000.00 in six years.

uNrFoRM SERIES coMpouND-At\{ouNT FACTOR. Uniform

series of equal investments are encountered frequently in economic evaluation problems, and it often is necessary to calculate the future worth, .,F,,, of these payrnents. Derivation of the equation to do this follows. Each equal

single investment, 'A", draws compound interest for a different number of periods as illustrated in Figure 2-3. The investment, 'A", at the end of period "n" draws no interest. The investment, 'A", at the end of period ..nl " draws interest for one period, and so forth. .

20

Economic Evaluation and lnveslment Decision Methods

tt r

tll-0

n-1 A

periods

1

...

2

periods

A n-2

1

period

r

A

A

n-l

n

F=?

Figure 2-3 Time Diagram Development of Equation 2-3. writing the information given in Figure 2-3 into an equation to determine the cumulative future worth of these equal investments for a compound interest rate of "i" per period gives Equation2-3. Note that each 'A" value in this equation corresponds to a present single sum "P" as used in Equation 2-1.

F=A(1)+A(l+i)+A(1+i)2+...

+A(t+i;n-l

Z-3

Multiplying both sides of Equation 2-3by the quantity (1+i) yields: F(1+i) = A(1+i)'+ A(1+i)2 + A(1+i)3 +. .'^. -+ A(1+i)n

2-1

Subtrccting 2-3 from2-4 gives:

F(1+i)-F=A(l+i)n-A or F=A

[(l+i)n-1]/i

2-5

The factor [(1+i)n-l]/i is called the "uniform series compound-amount factor" and is designated b r- F/A:,r. This factor is used to calculate a future single sunt, "F", that is equivaient to a uniform series of equal end of period payments, "A".

EXAMPLE 2-3 Uniform Series Compound-Amount Factor lllustration

calculate the future value forty years from now of a uniform series of $2,000 Roth IRA investments made at the end of each year for the next forty years if interest is 12/" per year compounded annually.

Solution:

A=2,000 A=2,000 A=2,000 767.0914 F=2,000( F/A 1 2/o,40)=1,534, 1 83 2. 40

uhapter 2: Compound lnterest Formulas

21

SINKING'FUND DEposIT FACT.R. To determine the amount of 'A", that must be sunk into a fund at the end or ,,n,,

money,

periods at

"i"

e*h

interest per period to accumurate .,F, doilars, -:--*'"'

l-S;br'A". A=F{il[(t+i)n_i]]

p".iod. for

,irir. "" eqrr,i"^ 2_6

.The factor i/[(]+i)tl-tr is cailed trte "sinking-.fitrtd depositfactor,, and is NF i.r. T'he factor is used to calculate a uniJorm series of equal ::.t-:t,:::, ,U,' paynents, ,,A,,, €u&-oJ-p€t'tttLl that are equivalent to afuture sum, ,,F,,. Ncte that Mi., = l/(F/Ai,n) EXAMPLE 2-4 sinking-Fund Deposit Factor ilrustration calculate the uniform series of equar investments made at

the end of each year for the next forty years that wourd jenerate a

$2,000,000 varue forty years from now. Assume a nominar interest rate of 12o,t" per year compcunded annually.

Solution:

...... A=? F = $2,000.000 3.... -40 years

A=? 0.00'130 A = 2,000,000 (A/F1 2/",40) = 2,600

CAPITAL-RECOVERY FACTOR. To relare a uniform series of end of period payments, 'A", to a present sum, ..p,,, conrbine Equations 2_6 and 2-

I

as flollows:

A = [P(l+i)n]til(l+i)n-ll = p[i(l+i)n]/[(l+i)n_l] Z_7 The factor i(l+i)n/[(r+i)n-t] is calred the "capitar-recoverl,factor,, ancr ,:_1r:::::,,rO,Ul *rr,o This factor is used to catcutctte o unyonn series of end oJ perod pa)'ments, "A", that are equit,alent to a pr"seni single rum if nxone)r, "P".

EXAMPLE 2-5 Capital-Recovery Factor A retiree has a Roth rRA with a current varue of g2,000,000. rf the total account varue was invested today in a 2s-yea,l-nnriiv based on 7.o/" annual interest, what annual annuity revenue will be available over the next 25 years?

-Economic Evaluation and lnvestment Decision Methods

22

Solution: P=

2,000,000

A=?

01

A=? 2

A=?..,....A=? 3.... .*..25 years

0.08581 A = 2,000,000 (NP7"y",25) = 171,620

UNIFORM SERIES PRESENT.WORTH FACTOR. TO dCtETMiNE thE present single sum of money, "P", that is equivalent to a uniform series of equal payments, '4", for "n" periods, at "i" interest per period, solve Equation 2-7 for "P". P=

A[(1+i)n-t]lli(l+l)nl

2-8

The factor [(1+i)n-1]/[i(1+i)n] is called the "uniform series presentvt'orth factor" and is designated by P/A;,rr. This factor is used to calculate the present sum, "P", that is equivalent to a uniform series of equal end of

period payments, "A". Note that

Mi,n = l/(P/Ai,n)

EXAMPLE 2-6 Uniform Series Present-Worth Factor lllustration Calculate the present value of a series of $1,000 payments to be made at the end of each year for six years if interest is 10% per year compounded annually.

Solution: A=91,000 P

A=91,000

=? 4.355

where at time zero, P = $1 ,000(P 1A19"7.,6) = $4,355 Note that the uniform series present worth factor P/A1O%,6 brings

the six equat $1,000 values at periods one through six to time zero (which is the beginning of period one), and not to time one, which is assumed to be the end of Period one.


23

SUMMARY OF COMPOUND INTEREST FORMULAS Name, Formula and Symbol Designation lltustration Single Payment Compound_Amount Factor -- i1.;,n = F/P;,n Single Payrnent present-Worth Factor = 1/(1+i)n = P/F1,n

p-

'

gtve{i

'"l,rlr n F=a(FlP'

U...... ^.

\

---P=F(PiF; trl-) '

"

0...........

n

E I

gtven

Uniform Series Compound_Amount Factor

=

[(1

+i)n-1]/i

- FiA;,,1

F=A(FiA;,6)

Sinking-Fund Deposit Factor = i/[(1+i)n-1] = 1ro,,n

A=F(A/F;,p) A

o'l .........n

tr

' gtven

Capital-Recovery Factor = i(1,i)n/[(1+i)n-1] = tuPi,n

Pgiven A=P(A/P;,p)

Uniform Series present-Worth Factor = [(1+i)n-1]/[i(1+i)n] = p/A;.,1

P=A(P/A;,,1) Agiv,:n A

1...

1....n

ln applying the formuras as shown in the iilustration corumn, the symbol letters

arways arternate in each equation. rno*ing this may assist the reader in remembering the correct factor to use for various applications of the formulas. There are several different timing considerations that impact how an investor should place his or her doilais on a time diagram. up to this point

all dollar values have been treated from a time value Jr

*on"f

viewpoint

as though a check u'as being received or written at the end or each compound_ ing period. Such values are often referred to as cliscrete ertd of period dollur v'alues. However, not ail doilars are actuaily realized at the end of a com_ pounding period. In addrtion to end of period values, investors can also work with beginning of period varues ani nid-period varues. A fairly common €xample of beginning of period values might be lease payments. Usu_ ally, lease payments are made up front to secure the use of the asset for the

following period, whereas morr-sage

pry*.ni, u;" il;^p""oo oor*"rr,


made after the accrual of interest during each compounding period. When invesrors are unqe-rlais as to the exasl-fi$jBg, or'.whg. n,,4ollars, rye l"r.9ly flowing throughout a time period, it is common evaluation procedure to utilize the mid-period approach. As the name implies this approach literally allocates dollars to the middle of each time period on a diagram. Each of these different timing considerations are illustrated in the Example 2-7 .

EXAMPLE 2-7 TimeVatue of Money Factors and Timing Considerations A person is to receive five payments in amounts of $300 at the end of year one, $4OO at the end of each of years two, three and four, and $500 at the end of year five. lf the person considers that places exist to invest money with equivalent risk arl9"/" annual interest, calculate the time zero lump sum settlement "P", and the end of year five lump sum Settlement "F', that would be equivalent to receiving the end of period payments. Next, determine the.five equal end of year payments "A", at years one through five that would be equivalent to the stated payments. Finally, recalculate the present value assuming the same annual payments are treated first, as beginning of period values and second, as mid-period values.

Solutions Based on Discrete End of Period Dollar Values:

$3oo $4oo $4oo $400

p-2

$5oo

tr-2

0.7084 0.7722 0.8417 +4o0(P/Fgy",a\ +400(P/F9"/op) + 400(P/Fg"/",2) P = 300(P/Fgo/o,1)

0.9174

0.6499 + 500(P/Fg "/o,S\= $1,529 or,

0.9174

2.531

P = 300(P lF g"/o,1) + a00( P/A97.,3)

(

0.9174 P/Fg

"/o,1)

0.6499 + 500(P/F97., 5)

= $1,529 I

I

L--

Chapter 2: Compound lnterest

1

For:mulas

.412

F = 300(F/Pgyo,q)

1

2i

.295

+ 400(F/p9 o/o,g)

1

+

i.090 + 400(F/P9"/",1) + 500 =

.188

OO(F/pgo;21

$2,gS3

t

of,

1.412

3.278

1.090

F = 300(F/Pg"t",q) + 400(F/A9 V,g)(F/pg"y".1) + 500 = $2,353 OT,

1.5386

F = 1,529(FlPgy..S) = $2,359 equivalent annuar payments "A" can be carcurated .ingThe by spreadthe present

varue of $t,seg foruvard into five or by spreading the future value of $2,353 back into "qr"ipayments five equal pay-

ments as follows:

0.2571

0.1671

A = 1,529(NPg"6,s) = $Bg3 or, A - z,osg(NFg.y",5) = $3g3 Present value Based on Discrete Beginning of period Varues:

P=?

$3oo $400 $4oo $4oo 012345 0.9174

0.8417

P = 300 + 400(P/F9o/o,1) + aOOp/Fg"7",2) +

$500

0.7722

40}(ilrnr.rl

0.7084 + 500(P/F97",4)

-

91,667

or,

2.531 0.7084 P = 300 + a00(P/A9"/o,g) +SOO(P/F9%,4) = $1,667


PresentValueBasedonDiscreteMid'PeriodValues: g

$3oo $400, $+oo $4oo -')

0.5

MathematicallY, P/Fi,n

0.9578

P=

2.5

1.5

3.5

$soo

;4.5

= Uh 0'8787

0'8062 400(P/F9"/",2.5) 300(P 1Fg"1",g.5) + 400(P/F9o/oj.5) + 0.6785 0.7396 + 400(P/F9%,3.5) + 500(P/F9"/",4.s\ = $1,596

Note that the mid-period present value is about half way between the beginning of period and end of period results'

2.3 Nominal, Period and Effective Interest

Rates Based on

Discrete ComPounding of Interest

of interDiscrete compounding of interest involves using a finite number specified by est compounding perioor p". year. Interest rates are normally a compounded interest with financiai agencies On a nominal annual basis inter5'0% pays a bank specified n.Irmber of times per year' For example, if annual interest rate is e.st compounded daily, tnii means that the nominal by 365 days which is 5.07o ancl the daily p"rioa irrterest rate is 5.07o divided receive interest would 0.0137va per day. with daily compounding a depositor of each daily end on the principat and accrued inteiest in his account at the be 365 period. Relating this to our compound interest formulas, there would to the effect periods p.. y"ui with the period interest rate i = 0.01377o. Due per year, period Lf .o*pounaing with moie than one interest compounding interest nominal the tota^l u*orni of interest paid per year is greater than the rare' commonly rate, "r", times initial principal. Tht t"tn] anrutal percentage with a nominal interchangeable are referred to as'APR" ,-and simple interest Be aware persons. interest rate as the terms are used by banking and finance differvery that engineering economists often use the term "simple interest" interchangeable meaning with "add-on" or "flat" interest ently as having

described in Section 2.6.

i

L-


Period interest rate =

27

i=rlm

\r'here m = number of compounding periods per year, r = nominal interest rate = mi ?

An eft-ective interest rate is the interest rate tilat when applied once per year to a principal sum wili give the same amount of interest equal to a nominal rate of "r" percent per year compounded "r)" times per ye;r. Arrnual Percentttge Yield (.APY) is the standard term used by the banking industry to identify an effective interest rate. The development of the formula for an effective interest rate "8" follows.

Ihe future worth "F1" of "P" dollars invested ati%o per period for,.m', periods is:

F1

=P(F/Pi,*)=P(1+i)m

2. . . . .. . . m periodsiyear

If an effective interest rate, "E", is applied once per year a future worth, "F2", results from investing "P" dollars. Fz=P(F/PE,1)=P(1+E)l period/year Since the initial principal, "P", is the same in each case, it is necessary to set F1 = F2 to make the total annual interest the same for both cases. This gives Equation 2-9 for "E".

Effective Annual Interest, g =

(I+i)m-l

2-9

If an investor knows the effective interest rate but wants to determine the equivalent period i value, compounded m times per year, Equation 2-9 can be rearranged to solve for i as follows:

i=(1+E;l/m-1 So, if an investor wanted an annual effective interest rate of l2.OVa, but the period interest rate.irvas to be compounded monthly, the relevant period interest rare i = l.nll/12) - I =0.009489 orO.94|9vo per monrh.


28

EXAMPLE 2-8 Discrete Nominal, Period and Effective

lnterest

Rates

.*

Aninvestorisscheduledtoreceiveannualpayment$of$1,000at years one, two, and three. For an annual interest rate of 10% compounded semi-annually, calculate the time zero present value and ihe year three future value of these payments.

Solution: Use semi-annual periods and interest in the first solution.

P=? 0 1

$t,ooo

2 3

$1,000

4 5

$1,000 F=?

6semi-annual periods

Semi-annual period interest i = 1Qo/"12 = 5"/".

0.8227

o.gozo

0.7462

P = 1,000(P lF 5.y.,2) + 1,000(P/F io/",q) + 1,000(P/Fs%,6) = $2,47 6

1.2155 F

=

1,000(F lP 5y",4)

1.1025 + 1,000(F/P S"/",2) + 1,000 = $3,31 8

Checking the answers; P - 3,318(P/F5"/o.6\ = $2,476 F =2,476(F/p5y.',6) = $3,318

With semi-annual periods, the uniform series factors, P/Ai,6 and be used, because you do not have a series of equal $1,000 values at the end of each semi-annual period. However, by using annual periods with an annual effective interest rate that is equivalent to 10% annual interest compounded semi-annually, the uniform series factors can be used as follows in the second solution. F/A;,,1 cannot

P=? 0

$1,ooo $1,ooo

1

2

$1,ooo F=?

3annual periods

Effective interest rate per !€dr = (1+.05)2

-

1

= 0.1025 or 1 0.25%


29

2.476

P = 1,000(P/A10.25,3) = 1,000[(1.10es;3 = $2,476

-

1yt(1.1025)3(0,1025)]

ot-,

0.9070

0.8227 o.7462 + 1,000(P/F1 g .Zlo/o,Z't + 1,000(p/F 1 0.25"/o B) + 1,000(1/1 .1025)2+ 1,000(1/1.1025)g ='1,000(1/1.1025)1

P = 1,000(P lF fi

.2g"/", 1 )

= $2,476

3.318

F = 1,000(F/A1 0.25"/"5) = 1,000[(1.1025)3 - 1]i0.1025 = $A,318 o[, 1

F = 1,000(F/P1

.21

55

1

.1025

O.ZSy",z) + 1,000(F/P10.ZS"/o,1) + 1,000 = $3,918

2.4 Nominal, Period and Effective Interest

Rates Based on Continuous Compounding of Interest If the number of compounding periods, "m", per year become very large the period interest rate, which is the nominal interest rate, "r", divided by "m", becomes very small. In the limit as "m" upproaches infinity, period interest "i" approaches zero resulting in a continuous interest situation. with continuous compounding of a nominal interest rale, "r", an infinitesimal amount of period interest, "i", occurs an infinite number of times during the evaluation period. This requires differential calculus derivation of continuous interest time value of money factors equivalent to the discrete compounding of interest factors developed earlier in this chapter. In Appendix B factors are developed and illustrated for continuous interest and discrete dollar values, while Appendix C factors are developed and illustrated for continuous interest and continuously flowing dollar values. Following are the continuous interest compound amount and present worth factors for discrete values from Appendix B.

Continuous Interest Single Discrete Payment Compound Amount Factor (F/Pr,n) F/P,,n = srn

30 .


Continuous Interest Single Discrete payment Present Worth Factor (p/FAn) P/F1,n =

1

etD

Where: r = nominal interest rate compounded continuously n = number of discrete evaluation periods e = base of natural log (ln) = 2.7183 . . . The Equation 2-10 formula for an effective discrete interest rate, .,E,,, that

is equivalent to a nominal interest rate,"r", compounded continuously follows for a discrete initial investment of "p" dollars: calculate the future worth, "F1", of "P" discrete dollars invested at a nominal discrete interest rate, "f" , compounded continuously for one year. P

F1 = P(F/P.,1) = Pler(1)l

1y"*

0

Now calculate the future worth, "F2", of "P" discrete dollars invested at an eftbctive discrete interest rate, "E", compounded discretely for one year. F2 = P1F/PE,1.1 = P(1+E)1 ;r year Setting

Fl

= F2 gives Per = P(1+E) or E = er-1.

E = er-1

2-10

If an investor knows the effective

discrete interest rate, E, but rvants to determine the equivalent continuous interest rate, r, Equation 2-10 can be rearranged to solve for r as follows: r=

ln(l+E)

So, if an investor wanted an annual effective discrete compound interest rate of 12.07o, but the nominal interest rate "r" was actually compounded continuously, the equivalent continuous interest rate r = ln(1.12) = 0.1 133 or 11.337o. In other words, money earning a continuously compounded interest rate of ll.337o is effectively earning a discrete.nominal interest rate of 12.l%o compounded annually.

Chaoter 2: Compound lnterest Formulas

31

lb illustrate, calculating rhe present worth of $l for a yeil using a dis_

creie interest rate of. l2.0vo, compounded annually, the present value is s1(1/(l+.12))1 = $0.8929. The present value of $1 usinq a conrinuous inreresr rare of ll.33vo is $1(r/e0.I133; = $0.g929. A6imilar equality must ai:rr exist when comparing future values. For investors u,orking with continuous interest, results (such as rate of return) may be expressed as a continuously compounded nominal rate, or converted to their effectir,e discrete equivaient annual value. Given that most frnancial markets deal u,ith discrete interest, some investors working with continuous find it easier to choose the later and compare numbers by convefiing their continuous results to discrete values using Equation 2-10. There really is no economic analysis advantage to rvorking with continuous interest rates rather than discrete values for i. In fact, in many ways it simply adds confusion to the analysis procedure. Continuous compounding is legai, just not real cornmon in the t'inancial markets. Therefore, vast majorities of econcmic evaluations are made using discrete compound interest. Finally, Iinancial calculators and functions built into most spreadsheets today are designed to work with discrete interest rates based on discrete dollar values. This is just another advantage for utilizing the discrete methodology. The effective interest rate, "E", described in Equation 2-10 is the discrete interest rate that is economically equivalent to a nominal interest rate, ,.r", compounded continuousl v. To illustrate these tir,,o different types of interest rates, consider the l.5vo monthly period interest charged on some credit card accouuts. This 1.57o monthly period interest rate corresponds to a monthly compounded nominal

rate of 18.07o. using Equation 2-9, the effective annual interest rate that is equivalent to a nominal rate of lBVo compounded rnonthly is E = (1+.0t5;12 - I = 0.1956 or r9.56vo per year on any unpaid balance. If the

nominal rate of 18.07a was compounded continuously instead of monthly, Equation 2-10 would be used to get an effective annual interest rate, E = 19.727c fromE-..18- t. The continuous interest compound amount and present value factors for continuously flowing values from Appendix c are developed as follows:

Continuous Interest, Single Continuous payment Compound Amount Factor (F/p*1,n) (FiP*r n) = [(er- 1)/r]("r(n-

I )1

32


(er-1)/r is the continuous interest, continuous flow of money, single payment compound amount factor for one period. It converts continuously flowing funds to a discrete end-of-period ,urn. 1er(n-l); is the continuou. interest, discrete dollar value single payment cornpound ameunt factor that takes the end.of-period discrete sum forward to the end of period "n." Continuous Interest, Single Continuous Payment Present Worth Factor (P/T*1,n) (P/F*r,n) = [(er-l)/r] [1/ (ern)]

(er-1)/r is the continuous interest, continuous flow of money single payment compound amount factor for one period. It converts continuously flowing funds to a discrete end-of-period sum. 1i(em) is the continuous interest discrete dollar value, single payment present worth factor that takes the period "n" discrete sum back to period zero. Equation 2-10 also results from considering a continuously flowing ,,p,, dollar investment during year one as follows:

calculate the future value, "F" of "P" dollars invested uniformly during year I at a nominal interest rate, "r," compounded continuously for one year.

Ft = P[(er-1)/r] Now calculate the future value, "F2," of the discrete present value equivalent of "P" dollars invested uniformly during year one at a nominal interest rate "r" compounded continuously for one year. consider that the discrete present value equivalent of the "P" dollars invested uniformly during year one earns interest at an effective rate, "8," compounded discretely for one year.

P[(er- 1 )/r] [ 1/er]

F2 = P[(er- I )/r][ l/er]( 1+E) I

Equating F1 and F2 gives E = er-1 which is Equation 2-10


EXAMPLE 2-9 Equivarence of continuous and Discrete

lnterest Calculations

Caiculate the present value of $100 to be regeived in the future assurning a continuous compounci interest rate 5f 1s"/" per year for the foilowing tirning assurnctions: A) The $100 is a discrete sum received once at the end of year six. B1 The $100 is a discrete surn received once at the beginning of year six, which is the end of year five. C) The $100 is received uniformly during year six. D) The $100 is a discrete sum received aiperiod 5.5 cqmpounded with the equivalent effective interest rate of E e.15_1 = = .161g or 16.18%. E) Treat the halves of $j00 flowing during year six as g50 at year five and $50 at year six, using the equivarent discrete rate of '16.18% shown in part D to be equivarent to 1s% compounded continuously. I

i

Sotution:

A) P=?

F = $100

1........4

0

5

.6

0.4066 P = 100(PiF15%,6) = 100(1/e.15(6)) = $40.66

where the 15% annuar interest rate is compounded continuousry.

(see Appendix B for the continuous interest factor deveropment

for discrete values.)

Note from Equation 2-10 an effective annual discrete interest rate is.equivarent to 15% compounded continuousry since :f,19]8% 0.1618 = €'rc - 1. using the discrete interest rate of 16.1g% to calculate present value: 0.4066 P = 100(P/Fro.r8%,6) = 100[1/(1.1618)q] g40.66 =

I

l


i

0.4724 P=

100(PiFtsz,s) = 100(1/e'15(5)) = $47.24

where the 15% interest rate is compounded continuously'

(See Appendix B for the continuous interest factor development for discrete values.) For equivalent discrete compounding as presented in part A: 0.4724 P = 100(P/FtO.t8%,S) = 100[1/(1.1618)5] =$47.24

'F

c) P=?

1......4

0

= 9100

5

0.4386 P = 100(PiF"15%,6) = $43.86

(See Appendix C for continuous interest, continuous flow of money, single payment present worth factors.) F = $100

D) P=?

0

1......4

5

5.5

6

0.4383 P = 100(P/Ft O.t8%,S.5) = $43.83

This discrete mid-period value, discrete interest result is very close to the part C continuous interest, continuous flow of money result which shows the approximate equivalence of these analysis methods.

Chapter 2: Compound lnterestFormulas

E) P=?

35

F=$50

0.l

F=$50

1........4 ?

0.4724 0.4066 P = 50(PiF16.1B%,S) + 50(P/F16.18%,6) $43.95 =

This result is also similar to the part c and D results, which demonstrates the equivalence of discrete vaiues and discrete interest to continuous flow of values and continuous interest if proper tim_ ing of values are used with equivalent discrete effective interest rates. It is irnportant to recognize that the time value of money factors introduced in section 2.4 and developed in Appendices ti and c, usually determine a discrete sum at a future point in time rather than a continuously flowing value. The only exception to this rule pertains to tne capital recovery lactor for conrinuous interesi and continuous flowing funds (A/P.p,p), and the sinking fund deposit factor for continuous interest and cbntinuous flowing funds (A/F"r.n). when making hand calculations with these factors, the resulting'calculated values are continuously flowing values, not discrete sums. lf further discounting or compounding of the calcuiated series rs required, the appropriate factors developed in Appendix c must be utilized. consider a uniform series of continuous flowing funds in years three, four and five that are to be discounted to t[e present. with methodology utilized in this text, the two most common approaches to discount the values would be: (A) Discount each cash flow back individually with a PlF. r.n factor, or, (B) Reduce the number of factors by employing a combihation of p/A.r.n and p/F, n factors for the appropriate periods. Notice the differehbe in these'factors with the (B) approach. The P/A*r,n factor is taken from Appendix c, which is developed for continuoub flowing funds. However, the p/A.r,,.., gives us a discrete sum at the beginning of year three, or the end"ot yea, two. Again the key word here is a discrete sum, not a year two continuous flowing fund. Therefore, we must discount this sum back to time zero with ihe appropriate p/FT,nfactor utilizing the factor for continuous interest for a discrete sum of money as developed in Appendix B of the text. This process will yield a result within roundoff error accuracy provided in the tables.


Throughout this textbook the reader will find constant reminders to )arefully consider all relevant timing considerations in laying out a ime diagram for any investment. This continuous interest application s another timing related issue to be considered in proper[y accountng for time value of money.

1.5 Applications of Compound Interest Formulas

:XAMPLE 2-10 FlPi,n Factor lllustration With lnterpolation What is the future worth six years from now of a present sum of i1,000 if: A) The interest rate is 8% compounded quarterly? B) The interest rate is 7.5o/o cofipounded annually?

iolution:

\) 8% Compounded Quarterly ) = $1,000

ll

1.608

2....24

F=$ 1, 000( Fi

P

2"/",24)=$1, 608

75% Compounded Annually

) * $1,000

1.543

2....6

F=$1,000(F lP7 .5"7",6) = $1,543

ln Part A, the nominal rate of 8% divided by four quarterly comrounding periods gives us a period interest rate of 2.0"/", (available n Appendix A). ln Part B, the7.5% interest rate is not in the tables. [o solve this problem, substitute the values into the FlP7.5"y..6 factor 'ot nathematical definition and solve that factor for an "i" value 7.5.k 0.075) and six compounding periods. This gives (1.07S;6 = 1.543. \n alternative approach is to interpolate, as is illustrated below, tsing the known Appendix A values of 7"/" and 87".

Chanter 2: Compound lnierest Formulas

lnterpolation lor F1P7.5"7o,6

'

37

FlP7.5oy",6:

587

= 1.501 + .5(1.587-1.501) = 1.544

ln general, FiPl.S.t",6= 1.501 +

544

''a"

'_al

SC1

and a'b = c,'d sD, a = b(cid) = (7 .5-7 .Ai(1 .587-1 .s01 )r'(8.0-2.0) = (.5)(.0860) = 0.0430

<.b+ <-C_+

7.0% 7.5"/o g.Oo/o Figure 2-4

lvlathematic ally F/P7.5o/o,6 = (1+0.07S;6 = 1 .543

The difference in the 1.543 mathematical result and the 1.s44 interpolation result is due to interpolation error since the factor varies non-linearly rather than linearly urith changes in the interest rate. EXAMPLE 2-11 F/p|,n Factor lilustrated with tnterpolation years will it take for a present sum of 91,000 to grow to . How many $2,000 if interest is 10% compounded annually? (or, in general, how long does it take to double your money invested at 1c./" fier annum?)

Solution: P = $1,ooo

$2,000/$1,000 =

F = $2,000 = $1 ,000(FlP 10"/o,n) (F lP

1

g"7",n) = 2.O

Go into the 10% tables in the F/p;.p column of the 10% table for F/P1g.7",n= 2.000. No factor listed giili:s exacily 2.000. Choosing the closest values, linear interpolation between Fip19"7",7 1.g4g and = FIP 1g"7.,6 = 2.144 gives: n = 7 years + (1)[(2.000-1 .949)t(2.144-1.949)] 7.26 years = A general rule of thumb that can be used to obtain the approximate period of time required to double your money at a given interest rate per "o-pound period. This rule often is called the Rule of 72 for an obvious reason: Number of Periods to Double Money

i

=

72 Z_11

38


In Example 2-L1 note that 72110 gives 7.2 years needed to double your money which is very close to the 7 -26 year result calculated. To double money at 6Va interest per year takes 72t6 = 12 years, while to double money at l2%o per year takes only six years. Approximation error [ecomes very significant when dealing with three or less compounding periots, or interest rates above 307o.

EXAMPLE 2-12 PlFi,n Factor lllustration

What is the present value of two $1,000 payments to be made three and five years from now if interest is 8% compounded semiannually?

Solution: P

$1,000

=?

0

1 2 3 4

5

6

7

$1,000

I

I

10

ln this example, "n" is expressed in semi-annual periods. Therefore, the appropriate compound interest rate "i" is 4/" per per;od.

0.7903

0.6756 P = $1 ,000(P/F4"7,0) + $t ,OOO(PIF41",10) = 91,465.90

EXAMPLE 2-13 PlAi,n Factor lllustration

What is the present value of a series of $100 payments to be made at the end of each month for the next five years if interest is 1 2/" compounded month ly? Solution: t-)

7)

0

$100 $100 1 2

i = 1"/" per month 44.95 P = $'100(P/A1%,60) = $4,495

$100

.60months

+


EXAMPLE 2-14 Factor lllustration for Multiple uniform series calculate the present, future and equivalent annual end of period vaiues for the series of incomes presented on the follcwing iime line diagram. Assume an interest rate of k

15o./o.

l'-

41=$.100 A1=$100 42=9150 A2=g1S0 A3=g200 43=g296 t

1......5

6......10

11 ..

..

15

tr-2

Solution: Present Value

3.352

3.352 0.4972

3.352

0.2472

5.019 3.352

5.847

5.019

P = 100(P/A15,5) + 150(P/A15,5)(p/F15,S) + ZOAp/A15,5)(p/F15,1 g) = $751

3.352 or

= 1 00(P/A1 S,S) + 1 50(P/A1 = $751

5, 1 g-P/A1 S,5) + 200(p/A1 5, 1 5-p/A1

S,

1

O)

Future Value

6.742 6.742 2.011 6.742 4.046 F = 200(FlA1S,S) + 150(F/A15,sXF/p1S,S) + 100(F/A15,5)(F/p15,1 g) 8.137 = $6,'l 1 0 or, 751 (F/P15%,1S) = $6,1 1 0

Equivalent Annual Value oJ71A2 0.02102 A = $751 (NPlsy",1s) = $128 or, $0,11 o(NF15"7",ts) = $1zg EXAMPLE 2-15 NFi,n Factor lllustration

what uniform annual cost for ten years is equivalent to a single

$10,000 cost ten years from now if interest is g% per year?

Solution: 0.06903 A = $10,000(NFB/",10) = $690.30 A =$690.30 F = $1o,ooo

.

Economic Evaluation arlci lnvestment Decision Methods

:XAMPLE 2-16 Equivalent Annual Payments lllustration Wh,at payments each year are equivalent to $3,000 payments at ne'ehO of years three, six'and nine from now if interest is 10olo Cornpunded annually? Determine the future worth of these pgyments at he end of year nine.

iolution: $3,ooo

$3,ooo

$3,000

0.30211 Payments Annual = $906'33 each $3,000(A/FtOr,g) Equivalent = of year three the end at years. the Spreading $3,000 rear for nine gives equivalent Same the rniformly ovei years one, two and three lnnual cost as spreading the $3,000 at the end of year six uniformly )ver years four, five and six. The argument is similar for ye"ars Seven, >ight and nine. 13.579 Future Worth = $906.33(F/A167",9) = $12,308

1.331

1.772 ev = $3,000(FlP rc%p) + $3,000(F/P1 97",6) + $3,000 = $12,308

$3,000 times a uniform series compound amount factor for the 107" interest rate will not work here because the equal payments do rot occur at the end of every annual compounding period. The calcutation must be made as shown unless you determine an effective rnterest rate per three year period and work the problem with three periods of three years each. The effective interest rate approach to this type of problem is based on using Equation 2-9: E = (1.10;3- 1 = 0.331 or 33.1% ln this case, "E" is the effective interest rate per three year period. 4.1027 $3,000(F/A33. 1 "/o,3) = $1 2,308

r.

I

Chaoter 2: Corpound lnterest Formulas

41

EXAMPLE 2-17 NPi,n Factor lllustration What annual end of year mortgage payments are required to pay off a $10,000 mortgage in five years if interest is 10% per year?

Solution:

A=?

A=?

P = $10,000

A=

0.2638 $1 0,000(A/P1 0%,5) = 92,638/year

Figure 2-5 shows how these payments amortize, or pay off, the

mortgage:

END OF

PRINCIPAL

MORTGAGE

OWED DURING YEAR

PAYMENT

YEAR 1 2 3 4

$10,000 8,362 6,560 4,578

r,39q --5 TOTALS

INTEREST AMOUNTAPPLIED

=.10

TO REDUCE (PRTNCTPAL) PhtNCtPAL

NEW PRINCIPAL

OWED

$2,638 2,638 2,638 2,638

$1,000 $1,638 $8,362 836 1,802 6,560 656 1,982 4,578 458 2,180 2,398 r!g__ __?fg9_ __ _____atgg__:::i_

$13,190 $3,190

$10,000

Figure 2-5 Mortgage Amortization Schedule "Add-On" or "FIat" Interest (Applied to the ..Rule of 78,') while compound interest is applied to unpaid investment principal and accrued interest each interest compounding period, aflat or add-on interest rate is applied to tlrc initial investment principal each interest compounding pariod. This means that the cumulative interest paid or received by a borrower or lender on a flat interest loan is proportional to the length of the loan period and not affected by the payment schedule. If "p" = initial investment principal, "i" = flat interest rate per period, and .,n,' = number of 2.6

interest paying periods;

Cumulative Add-On Interest for "n" ComPounding

&

Periods

= (P)(iXn)

2-12

42

Economic Evaluation and lnvestment Decision Methocls

Be aware that engineering economists often use the term ,.simpre,, interest to refer to "flat" or "add-on" interest as.defined here. The common banking and financial marker use of the teim-'"isiihpte,",inteieil1;,*ty|dif_ ferent, generally being interchangeable with "nominal', or ;Srnuur percent: age rate" interest. Note that the interest paying periods are not necessarily years but may be days, weeks, months, or other periods. when it is necessary to calcurate the interest due for a fraction of a year or period, it is common practice to take ioan principal,"P," times the flat interest rale,,:i,,, times the fraction of the o. applicable period. For example, $1,000 principal at flat interest )'"-* of 9.07o per annum

for 100 days gives flm interest equal to:

(0.09X$ 1,000X1 00/365) $24.66. =

It is best to always state expricitly the type of interest and compounding you are using in financial calculations, otherwise, confusion about the rneaning of the results can occur. It is generalry considered to be compretely lcgitimate and ethical to use either a flat interest or compound interest rate irr financial dealings.

EXAMPLE 2-18 compounding with "Frat" or .,Add-on,, tnterest What annual end olyTr. mortgage payments are required to pay a $10,000 mortgage in five years i-f the interest rate is 10% per off year using "flat" or "add on" compounding of interest? Assume that the loan is paid off with five equar principar pr-us interest payments in years 1-5.

Solution: lnterest Owed Each year: Principal Owed Each year: Total Annual payment

$1 0,000(0.'t 0) = $1,696 $10,000(0.20) = $2,996

$3,ooo

Note that from Exampre 2-17, ror a 10"/" compound interest rate, the_annual payment is g2,68g; a differenc" oi $goi p", y"ur, or, $1,810 over the five year loan rife. The frat or add-on interest rate that would be equivarent to a 10% compound interest rate can be carculated by setting the annuar cost oi each alternative equar to each

other and solving for,,i',:

i I

I

I

I

r-hapter 2: Ccrnpound lnterest Forrr,,llas

(Annual Cost = AC) AC of compound lnterest Rate paym€nt = AC of Flat lnterest Rate Fayrnent $2,63s = $2,000 principar payment + $10,00c(iltfrat inter-est payment $6rE = $10,000(i) i = $638 / 910,000 = 0.0688 oi- 6.3g% pei. j/-ear

so, a nominal interest rate of 1a.o% compounded annually is equivalent to a flat or add-on interest rate of 6.3g% for this five year mortgage example. The "Rule of 78" is a rnethod of arlocating the principal and interest components of a loan with flat interest as illustrated in Example 2- l g. The inter_ est component of each payment is calculated by multiplying a changing tiaction each compounding period by the rotal amount of flat interest to be paid over the entire loan period (use equatiott2-lz to compute the total loan interest). This fraction is determined in the following *un,r".' The numerator is the nurnber of remaining compounding periods in the loan, (therefore changing with each cornpounding period) while the denominator represents the sum of the numbers thai represent the compounding periods over the entire loan lil-e, (this nunrber remains constant.) The number of compounding periods are rrost easilv calcuiated using Equation 2_13: Strm of Conrpounding Per-iods =

[n(n+l)] I 2

Z_13

\\'hich is equivalenr to = I + 2 + -l +... + n_l + n The principal due each compounding period represents the.difference between the total period payment and the accrued interest.

EXAMPLE 2-19 lltustration of the Rute of 78

' I

i

using Example 2-18,1the add-on interest rate based payments of ^ $3,000 per year were

amortized based on the Rule of 7g, what would the principal and interest components of each payment be for the five year loan?

I

I

Total Flat or Add-on lnterest paid = g1O,0OO(.10)(5 years) $5,000 =

I I I!

s

I * 'l

Evaluation and lnvestment Decision Methods .-Economic

Sum of Compounding Periods = [5(5+1)l lZ 15 periods = ..lnterest,,: Year ':- 'Principal 1 $5,000(5/15) = $1,667 $3,000-$1,667=$i,ggs 2 $5,000(4115) = $1,333 $3,000 - $1,333 t,O6Z 3 $5,000(3/15)= $1,000 $3,ooo-$1,000=$2,000 4 $5,000(2/15) = 667 $3,ooo-$ 667=$2,993 5 $5,000(1/15) = 333 $3,000-$ 333=$2,667

..,

i$

$ $

$5,ooo

$10,000

2.7 LoanPoints and Buying Down Interest

Lending organizations such as banks, savings and loans, and insurance companies charge "points" on loans as a way of indirectly increasing the eft'ective interest rate being charged on loans. points are a percentage of kton value and vary widely at dffirent tintes. In a related way, develipeis v'ho want to sell assets sometimes "buy-down" interest rates on loan miney by paying an up front lump-sum fee to lending agencies so that the lending agencies can loan to persons at a lov,ter rate. However, the developer typicaiiy adds the buy-down charge to the price of the property being sold. The buyer needs to understand the economic principles related to ..points', and

"buying down" interest to make valid investment decisions that involve

these considerations.

EXAMPLE 2-20 lllustration of Loan points and Buying Down lnterest Rates lf a lender receives six points (a fee of 6% of loan value) to make a $100,000 loan at a 12/" interest rate per year with uniform and equal annual mortgage payments over twenty years, what actual inteiest rate is the lender receiving?

Solution: The mortgage payments are based on the 9100,000 loan value at year over twenty years.

i

j !

12"/o interest per

0.13388 A = $100,000(A/P 12/o,ZO) = g13,3BB per year payments

I


45

From the lender viewpoint, the actual net loan amount is: $100,000 - $6,000 point fee = $94,000. Net loan =

$94,000

Payments/year =

$13,39i9

$13,399

Annual Worth Equation: $94,000(A/Pi,ZO) = $13,388 where "il' is the actual net loan interest rate for the lender. NP;,2O = $13,388/$94,000

=0.14243

by interpolation:

NPISo/",20= 0.15976 NP12./",20 = 0.13388 i = 12/" + 3%[(0.14243 -0.13388)/(0.15976 -0.13388)] = 12.997"

12.99% is the interest rate actually being realized on the net loan by the lender. lf the borrower is paying points, the borrower is paying 12.99"/" interest on the actual net loan amount. lf someone else is paying the points, such as the seller of a house to be financed by an FHA loan in the United States, then the borrower is really only paying 12/" annual interest. The concept of buying down interest rates involves similar calculations. Consider a realtor who would like to buy down the interest rate on $100,000 of twenty year loan money f rom 1 2h lo 10%. What upfront lump sum payment, similar to points in the previous calculations, will enable the lenderto loan $100,000 at 10% instead of 12"/" with uniform and equal mortgage payments over twenty years? Let X = buy-down payment at time of loan 0.13388 0.11746 ($100,000 - XXfuP1 2"/o,20\ = $100,000(NPrcy",ZO)

($13,388 - $'l 1,746)/0.13388 = X = $12,265 Equivalently, we could have discounted the actual mortgage payments to be received (based on 10% interest) at the desired yield (which, in this example, is 12%\ and subtracted that amount from the loan value to determine the required up-front payment.

Economic Evaluation and Investment Decision Methods

o.11746 Actual Payments to be Received = $100,000(A/Prct",ZO) = $11,746 7.469 Required Buy-Down Payment = $100,000 - $11,746(P/A12/o,2e) = $100,000 - $87,731 = $12,269 $12,269 is within round-off error accuracy of the previous method rf explicitly solving for the payment which was $12,265. The $12,265 rp-front payment makes the lender net loan of $100,000 - $12,265 = F87,735 dl 12"/" interest which enables the lender equivalently to oan $100,000 allO% interest.

1.8 Arithmetic Gradient Series In many economic evaluation situations, revenues and costs increase or Jecrease from period to period in an arithmetic gradient series. For examrle, escalation or de-escalation of projected incomes and operating costs [rom period to period, including the effects of inflation and supply/demand :onsiderations, often may be approximated by an arithmetic gradient series cf values. Figure 2-6 illustrates a general arithmetic gradient series where the first term in the gradient series is "B" and the constant gradient between period terms is "g".

B I

B+g Z

B+2g

3

B+(n-2)g B+(n-1)g

... ........ n-1

n

Figure 2-6 A General Arithmetic Gradient Series

Computing the present worth or future worth of the series of values "n" separate calculation terms using the single payment present worth or future worth techniques. To reduce the labor involved in these calculations, it is desirable to be able to convert this arithmetic gradient series of payments to a uniform series of equal payments which can then be converted to present or future worth single sums with P/Ai,n or F/Ai,n factors. shown in Figure 2-6 requires

Cfiapter 2: Compound lnterest Formulas

The uni.forun serit.s of equol pa.\,rnents which is equivrtlent to an arithoJ payments may be found using Equation 2-r4: .

metic gradient series

,

A=B*g(A/Gi.n) rvhere A/G1,n =

2-14

(1ii)-in/[(1+i)n-1]] is a factor tabulated in Appendix A for

various interesr rates, "i", and project lives, "n". The derivation of this factor is presented in rire Appendix E. Example 2-21 ilrustrates the use of the AlG1,n factor.

Note two things about the gradient series calculations. First, the gradient,

"9", i. plus or minus depending on whether the gradient series is increasing or decreasing. Second. the arithrnetic series factor is calculated for the ,,n" period project iif'e, nor the n-1 periods the gradient is applied. This is due to the wav the gradient fiictor derivation works ollt as developed in Appendix E, fbr readers interested in the matlrematical development of the factor. In other words, always include the base yalue period in determining the appropriate value of "n" for the A/Gi,n factor. The A/G factor helps when making hand calculations, by reducing.the number of factors required in a present worth equation.

EXAMPLE 2-21 Arithmetic Gradient Series

consider an arithrneticaiiy inireasing series of royalty payments, where the first payment (the first term in the gradient series) is B = $1 ,100. The payments increase by $t 00 each year, so g = $100. The interest rate is B"/" per annum and royalty payments are expected to last ten years (n = '10). Calculate the series of equal royalty payments or incomes that is equivalent to the gradient series of values using Equalion 2-14. Solution: $1,100.00 0

Uniform Equal Payments, A = $1

t

10

,1

3.871 g1 ,487.10 00 + $1OO(A/G6 "/",10) =


48

This result indicates that if money is valued al EP/" per period, the uniform series of ten equal end of period payments of $1,487.10 is exactly equivalent to the original gradient series, as the following diagram illustrates: t

0

EXAMPLE 2-22 Arilhmetic Gradient Series Factor lllustration A man expects to invest $1,000 in common stock this year, $1,100

next year and to continue making each year's investment $100 greater than the previous year for twenty years. lf he earns a 10"/" rate of return on his investments, what will the investment value be twenty years from now? Assume end of year investments.

Solution:

-

$1,000 $1,100 $1,200 $1,300 $1,400

6.508

-

grad. series

-

$2,900 F=?

57.27

F = [$t,000+g100(A/G1 O%,201lFlAl1o/o,20) = g94,500

EXAMPLE 2-23 A Uniform Series with an Arithmetic Gradient Series

Find the present worth of the series of payments shown on the diagram if interest is 10% per period. P-'.)

-

$SOO

$500 $SOO $600 $700 -

grad. series

-$1,700

Solution: Conversion of the series of gradient terms to a uniform series of equal payments from years three to fifteen yields;


Years 3

to 1 s: A = $s00+g10o,r"r;;:,s) = $eog.go

This gives the following equivatent time diagrakr: P=?

$s00 $soo $e6e.e0 $959.e0

$969.9C

4........'15

Present Worth:

1.736

7.103

.8264

P = 500(P/A10"/",2) + 969.90(PlAlO"t",l3Xp/F1

O%,2) = $6,561

1.736 7.606 1.736 or = $Qg1plAlA"t",Z) + 969.90(PlAl1o/o,15 - p/A1

eo/o,2)

= g6,561

2.9 Alternative Time Line Diagram and the concept of cash Flow An alternative time line diagram rhat often is used to more graphically

illustrate the c
is allttcating ,ttt-

-f!ov's and inflox's o.f ntottet' ar rite poirtt in time clctsest to yvhen doilar ,trtttruxt.\ crre octuullv erpected Io be incurterl. when we speak of inflows lurd outfiorvs of money, on a befbre-tax basis, we are referring to revenues arrd capital or operating costs associated with a particular inveltment. How_ e'cr, only if your projects are tax-free should you consider making economic evaluations on a before-tax basis. In general, economic evaluations are made on an after-tax basis utilizing cash flow generated from a project fbr the investor's unique tax positron. In Chapters 2 through 6, the dollar value per period generated by adding revenue and capitar oJoperating c,srs in a time period should really be thought of as repiesenting positive and negative cash flow in that period. As mentionea in ihapt". t]tt"r" first six chapters ignore the specifics of tax considerations to ,"dr"" confusion while gaining an important understanding of the basic concepts of time value of money. If we had a specified investment "p" at time zero that was expected to generate equal annual revenues 'A" over years one through four, the modified time line diagram may appear as follows:

&


50

+A +A +A

4^î

+A

ltll llll,-2-3-4

I

I

v

-P Figure 2-7 Alternative Time Line Diagram Approach EXAMPLE 2-24 lllustration of the Alternative Time Line Diagram

lnvestments at time zero of $5,000 and the end of year one of $10,000 are expected to generate revenues of $7,500 each year, for ihree years, beginning at the end of year 2. Draw the modified time line diagram for this proposed investment and then calculate the present value of'future revenues and costs associated with this project for a discount rate of 15%.

Solution:

+++

+7,500 +7,500 +7,500

ttl lll

i -5,000 -10,000 Present Value @ 15h =

2.283

0.8696

0.8696

7,500(P/A1 5"7",g)(PlF1S"/"J) - 10,000(P/F15"/o,1) - 5,000 = +$1,194 The alternative time line approach is not utilized extensively in this textbook because of the author's preference for a straight line time diagram. However, this approach may be preferred by the reader in laying out solutions to problems in the text.

Ch.'pl;r 2: Cc",oOund ln:r.resr :Ormular

2.10 Introduction of Rate of Return Analvsis In the exampres presented to this point, the period interest rate (or rate of rc;uir) iias bee, specifico and w.e have caicuiarcd equilalcnt present worth, fr;1irre r.'.'orih. or aflirr-l.ii clLiairiitics ibr Eile i; r.iit,tl-s ,,_ it",ruur.a'or costs. If ,i.e ciiar!se arrl'oirhe:c erample probiems by.spec;fying both the costs and rerenues u'hiie leai'irrg tne pcriocl inicrest rate, ',i,', unkr,,,ra.n, we generati: a rate oire:';rn caiciilaiion t1.pe probie:n. To irir:itraie ti:is conji:;ji, cc;isider chan:_ inr Exarnple2-ii to read, "r.vhat b,rrowed mone).annual'compound interest rare is being paid on a^$10,000 mortgage ttrat wir be paid offin fiu. y"u., with five equal end of 1'ear *ortgug" payments of $2,63g?,, The annuar morrgage interest rate that the borrower is paying is the annual rate of return that the lender or investor is recei'ing on the unamortized mortgage invest_ ment principal each year. To determine this rate of return we must w,rite an equation compz*ing costs and revenues at the same point in time or on an eqtiivalcnt annual or period basis using an unknown rate of return per period, "i''. For exampie, for an unknorvn "i;, we can equate the annuar rnortgare payments of $2,638 to the equivalent annual investor cost as we did in Exirrnpie 2-i7 to carcurare the mortgage paymeilts for a given iiiteresr raie. S 1t1.000(A/Pi.5) = $2,638 (Ar"1,5) = $2.638i$ 10,0C0 0.2638 = 81' trial anci error procedures we can check the factor .A/p; in the tables fi;r di1'lerent inte.est r'ates until u,e find in the l0% tables 5 rr,'rive1,5 giu., the desired value of 0.2639. Therefore, the rate of retu,r, ,.i',, is rovo per vcar' tisualiv ihe "r" r'alue rve are loot ing lbr oc":urs between irjterest rat,J:i -tiren i, the t.roies and we rnllst use iinear interpolation ro calculate the..i,, ialLre. The dctails of this inierpolation procedure and general rate of return calculations are iilustrated in the next chapter.

EXAMPLE 2-25 lilustration of the Use of All Factors An investor is to make the foilowing payments for a parcer of rand: $.5,090 down payment at time zero, a gradient series of payments starting at $2,000 1t t!" end of year one and increasing by a constant gradient of $500 per year in years two through elght prus a lump sum "balloon" payment of $10,000 at the end of"year eight. For an interest rate of 12h compounded annually, calculate: A) The present wo.rlh of the payments, i.e. the singre time zero payment that wourd be equivarent to the given seri6s of payments.


52

B) The future worth of the payments, i.e. the single end of year eight payment thal would be gCgivalent to the given series,of paymenls.

C)The equivalent annual payments, i.e the uniform annual payments at the end of each year that would be eqqivalent.to the given series of payments.

Solution: $5,000 $2,000

glo,ooo $2,500

2.913

-

gradient series

-

$5,500

0.4039

4.968

A)

P=5,000+[2,000+500(A/G t 2,A)](P/A1 2,g)+ 1 0,000(P/F 1 2,g) = $26,2

B)

F=5,000(F/P

2.476 1

2.913

11

12.300

2,6)+[2,000+500(A/G 1 2,dl(F/A1

2,

g)+ 1 0,000 = $64,895

0.0813 2.913 0.2013 C) A=5,000(A/P 1 2,g)+2,000+500(A/G 1 2,g)+1 0,000(A/F1 2,8) = $5,276 2.11 Summary In this chapter, seven different time value of money factors have been introduced that incorporated a number of variables which are summarized below in order of occurrence through the chapter, rather than alphabetically:

P F A i i+ n m r

= = = = = = =

E = E =

present single sum of.money future single sum of money

uniform series of payments,like mortgage payments period compound interest rate period compound interest minimum rate of return number of compounding periods in an evaluation life number of compounding periods per year

nominal (annual) interest rate which in banking terminology is often referred to as the 'Annual Percentage Rate" or APR, or just a "simple" interest rate. If an investor has annual compounding, the period interest rate, "i" is equal to the nominal interest rate, "r." effective interest, in banking this is analogous to the term Annual Percentage Yield, or APY. It represents the annual interest that compounded once per year, will yield the same interest as a period rate compounded "m" times per year.

(t+i)m-l

Chapter 2: Compound lnteredt Formulas

The seven time value of money factors include:

P/Fi,n

FPi,n

Calculates the present value of a single future sum. Calculates the future value of a single presery sum.

P/A1.n calculates the present value from a uniform series. on a time diagram, this present varue wilr arways be one compounding period to the left of the first varue in the uniform series of payments. FiAi,n calculates the future value from a uniform series. on a time dia_ gram, this value is recognized at the end of the last period _future for which there is

APi,, .i

AFi,, dGi,,

I $

a value.

Calculates the series of uniform values at the end of each compounding period that is equivalent to a given present sum. This factor is most commonly used to calcuiate mortgage payments when "compound interest,, is relevant.

calculates the series of uniform values at the end of each compounding period equivalent to a given future quantity. calculates a uniform series of values equivalent to an increasing or decreasing arithmetic series of cash flows.

The factor symbolism is designed so that the first value always designates what is being calculated. For some readers, the forward slash, .7', is often thought of as representing the word "given" and the second letter represents the known quantity. So, using the prFi,, factor as an example, you calculate p

given F at the period compound intereii'rate for n compouraini periods. Four types of compounding were introduced in chapter 2 incruding: 1) Discrete End of period $, Discrete Compounding App

A

2) Discrete End of period $, Continuous Compounding _ App B 3) C,xtinuously Flowing $, Continuous Compounding _ App C 4) Add-on or Flat Compounding

_ Finally, the timing of cash flows was another relevant issue in this chapter. lt was emphasized that while most cash flow,s are considered to be discrete

end-<.rf-period

dollar yalues may also be described as beginning of 'alues. period. mid-period or conrinuously flou,ing funds. These timing issues only influence where evaluators should place dollars on their time

discounting methodology itself remains relativelv bonstant.

dlagram. The

54


PROBLEMS

I

i

.t

I

2-l Your project'has

an estimated cost for lahtl rbilaniatioir to be iealized at the end of 20 years from today for $70,000,000. If curyenr long term

bond interest rates are 7.}Va compounded annually, wliat would you need to invest in zero coupon (or deep discount) bonds today to cover the estimated cost after 20 years? In other words, calculate the present value "P" at time zero of $70,000,000 20 years from today for an interest rate of 7.}Vo compounded annually. What uniform series of payments at the end of periods I through 20 would also cover the year 20 cost? Use the same annual interest rate of 7.07o. .ta

Calculate the present value, "P" at time zero and the corresponding future value, "F" at the end of year three for a series of $15,000 payments to be made at the end of each of years one, two and three. Assume that no payment is realized at time zero. Use a nominal interest rate

2-3

2-4

of

15.j%o compounded annually.

An investor has a series of three $15,000 payments expected to be realized at the end of each of evaluation years three, four and five. Calculate the present value, "P" at time zero, and the corresponding future value, "F" at the end of year 7. This analysis assumes that no payments are realized in periods zero, one, two, six or seven. Assume a nominal interest rate of 15.07o compounded annually. A loan of $15,000 is incurred today and is to be paid off over the next 4 years. Calculate:

A)

The uniform anrunl mortgage payments '4" at the end of each of years I through 4 that would pay off the loan for an interest rate of I 5 Va

B)

compounded annually.

The urriform monthly mortgage payments at the end of each of months 1 through 48 that would pay off the loan for a nominal interest rate of l5%o compounded monthly.

2-5 Suppose you have a newbom child and want to begin covering the estimated cost of tuition and expenses for four years of college beginning l8 years from now Your estimated cost is $30,(n0 per vear beginning at the end of year l8 and running throu-eh the end of year 2l (4 years). Assume a nominal investment interest rate

of l0.0Vo compOunded annually.

.l

I


2-6

A)

what is th-e-value of the paymenrs ar rhe beginning of year 1g (end of year t7)?

B)

How rnuch wourd you ha'e to invest today (time zero) to cover the estimated cost of four years of college? !

c)

what uniform series of deposits at the end of each years of 1 through 17 would be required to cover the tuition costs in years 1g through 21?

: ,

,

Two alternatives are being considered to finance the acquisition of a new vehicle. The purchase price is assumed to be $20,000 for all sce_ narios (no discount for a "cash" purchase artemative). The investor,s minimum rate of return is a nominal ro.,vo monthry.

"o*pornd"d

Altemative A is to accept the dearer finance package which includes a nominal 6.5vo interest rate based on ,.add-on,, or ,.-flat,, compounding. A down payment equar to 20.ovo of the purchase price is required and the loan payments (principar and interest) are spread out uniformry over months 1_36. Alternative B is to finance the acquisition through a bank at an annual percentage rate of g.jvo compounded monthly (normar compound interest). A down payment of 20.ovo is required and the monthry loan payments are uniform over months 1 through 36.

1)

2)

Based on the monthry payments, which alternative would you select? CalcLrlate the present worth cost @ i*=1070

compounded monthly for A and B and compare with the $20,000 cash alternative.

2-7 what future

amount of money will be accumulated r 0 years from now by investing $1,000 now plus $2,000 5 years from now at6Vointerest

compounded semi_annually

?

2-8 what

is the present value of $500 payments to be made at the end each 6 month period fbr the n.î 10 years if interest g7o

pounded semi-annually

2-9

is

of

com-

?

S/hat equivalent annual

end_ot-_-vear payntents for the next 6 years are equi'alent to paying $5,000 roo,urd-$10,000 years 6 from now if

interest is 67o compounded annuallv?

56


2-10 What monthly car mortgage payments for the next 36 months are requirgd to ,amortize ra p{esent loan of $3,000 if interest is l2Vo compounded monthly?

?

2-11 what amount of money must be deposited in a savingt account at the end of each quarter for the next 5 years to accumulate $10,000 in 5 years if interest is 6Vo compounded quarterly? 2-12 What is the present value of $1,000 payments to be made at the end of each year for the next 10 years if interest is \Vo compounded semiannually?

2-13 A company can lease an asset for the next four years by making lease payments that are equivalent to annual payments of $3,000 at year 0, $6,000 at year 1, $7,000 at year 2, $7,000 atyear 3 and $4,000 at year 4. Use a lZ%o minimum discount rate determining the year 0 present worth lease payments, yer 4 future worth lease payments and year 1 through 4 equivalent annual lease payments.

2-r4 An investor plans to invest $1,000 at the end of every year for 20 years at lOvo interest compounded annually. what is the expected value of this investment 20 years from now? What single sum of money invested now at l0% compounded annually would generate the same expected future worth? I

2-15 A payment of $2,000 will be realized today with additional payments of $1,000 at the end of each of the next 5 years. Assume a nominal interest rate of 20Vo is appropriate and calculate the following:.

A)

Determine the future value of all the payments at the end of five years from now.

B)

What is the future value 5 years from now of the $2,000 plus $1,000 annual payments if the 20Vo nominal interest rate is compounded semi-annually?

C) What semi-annual each year

if

payments are equivalent to the $1,000 payments the 20Vo interest is compounded semi-annually?

2-16 what effective annual interest rate is equivalent to 20vc compounded quarterly? 2-17 what nominal annual interest rate is equivalent to an effective annual interest rate equal to 2O7o if interest is compounded quarterly?

'j

i I i

l D

f I

ii f,

{

T

I

I

I


2-18 What sum of money will be accumulated in 40 years if:

'

A) '$1,00ois irivested at the end'of each'of'the'ii8it?o ye ari at a 15vo rate of return compounded

B)

annually.

r

$1,000 is invested at r.he end of the first yeaq $1,100 is invested at the end of the second year and each succeeding year's investment is $100 higher than the previous year's investment for 40 years at I 5Vo

compounded annually.

C) what

single investment at time zero would generate the part .A,' future worth?

2-19 calculate the present worth cost of service for the following cash flows. The minimum rate of return is a nominal rl.ovo compoundid monthly.

BTCF Years

Months

-8,000 -2,000

012 01224

-3,000

-2,500 a

J

36

A) Use monthly compounding periods. B) use annual compounding periods and the appropriate effective annual interest rate that is equivalent to 12.0vo compounded monthly. 2-20 company A owns a patent with 15 years of remaining life. Company B is paying royalties to Company A for a license to the patent. It is estimated that royalry payments for the next 15 years will be per year for the ' first 5 years, $8,000 per year for the next 4 years and$6,000 $10,0b0 per year for the last 6 years. company B offers to pre-pay the expected royalty payments for $70,000 now. If company A considers 70vo per year to be its minimum acceptable return on investment, should it accept the pre-payment offer for $70,000 now or take the royalty payments year by y"*i 2-21 what uniform annual payments for the next 15 yeers are equivalent to the non-uniform series of royarty payments described in problem 2-20? 2-22 A machine which has a l0 year rife will cost $11,000 now with annual operating costs of $500 the first year and increasing $50 per year each

of the next 9 years. If the salvage value is estimated to be $2,000 at the end of rhe l0th year, what is the equivalent an,ual cost of operating this machine for rhe next l0 years if other opportunities exist to have the investment dollars invested where they would earn an gva annual rate of return?

&

t

s8


2-23 An investor expects to realize I 8 monthly payments of $ 100 starting at the end of rnonth 3.1 .and running through month 48. For a discrete nominal annual interest rate of l\Vo, calculate the present worth of these payments for the following cases: ?

A)

Use monthly discrete periods and assume the l5To rate is compounded monthly.

B)

Use annual periods and assume the 15Vo annual discrete rate is compounded annually. Put the payments into the analysis at the closest year to which they are incurred. Compare this result with the result obtained by handling the time value of money using the effective annual interest rate of 16.07Vo that is equivalent to the annual rate of l5%o compounded monthly.

C)

Work case "B" for continuous compounding of interest by using the annual continuous interest rate equivalent to 16.07Vo discrete.

D)

Use annual periods assuming continuous flow of payments during year 3 and 4 with continuous interest (as in case "C").

2-24

$600 $700 $800 $900 $1,000 $1,100 $1,200. . . . $1,200

7........

6

For the values shown on the time diagram and

a

l0

I27c nominal interest

rate, find:

A) B)

Present value at time 0.

Future value at the end of year 10.

2-25 Determine the sum of mcney that must be invested today at 9To interest compounded annually to give an investor annuity (annual income) payments of $5,000 per year fbr l0 years starting 5 years from now. 2-26 Determine the monthly mortgage paymenrs that will pay off a $16,000 car loan with 36 equal end-of-month payments for:

A) B)

'.-

10Vo per year add-on interest rate.

llVo

annual interest compounded monthly.


59

2-27 rt you borrow $15,000 at an ApR of ll.5vo compounded monthly, calculate the equal monthlymortgage payments thtt will pay offthis loan over four years.

t 2-28 C = Capital Cost, OC = Operating Cost, L Salvage Value =

All

Values are in Thousands.

C=$1,500

OC=$400

OC=$500

OC=$600 L=$300

other opportunities exist to invest money at a nominal ll.ovo interest rate compounded annually. Calculate the present, future and equivalent annual cost (end of years I -3) for proriding service with this asset over the 3 year life. 2-29 A person is to receive five revenue payments in amounts of $300 at year one, $400 at each of years two, three, and four and $500 at year five. If the person considers that places exist to invest money with equivalent risk at 9.ovo annual interest, calculate the year zero lump sum settlemeni and the year five lump sum settlemeni that would be equivalent to receiving the payments. Then determine the five equar yearly payments at years one through five that would be equivaleni to the stated payments for the following assumptions, (this is a re-statement of Example 2-7 for which we w-anr ,o onuryr. tr-,.'roiro*ing continuous interest solution variations):

A)

use 9.07o per year continuous interest and assume that dollar val_ ues flow continuousry during the year in which they are incu*ed.

B) Assume the 9.jva is a discrete compound interest rate and the timing of the discrete dollar values in Example 2-7 are correct. Adjust the interest rate and dollar values for valid continuous interest,

continuous flow of funds analysis. This requires assuming that half of each annual discrete value flows uniformly througi the preceding year, and the other half of each discrete value florrls uniformly through the following year.

c)

Assuming that the timing of the continuorrs florv of lnoney values in part (A) is correct, adjust the interest rate to make a valid discrete dollar discrete interest rate anarysis using mid-period 'alue, discrete present worth factors.

60


2-30 An investor expects to realize annual revenues of $120 uniformly during years L,2 and 3. Calculate the present value of the revenue assuming a 127o nominal interest rate compounded annually. Assume the revenues to

be:

*

A) Discrete end-of-year values. B) Discrete beginning-of-year values. C) Discrete mid-year values. D) Consider that the annual revenues of $120 flow

uniformly during years 1, 2 and3. Place the revenues at the closest discrete annual point in time to which they occur by putting the month 1-6 revenue at year 0, month 7-18 revenue at year 1, month 19-30 revenue at year 2, and month 3l-36 revenue at year 3. This is the desired approach for putting revenues and costs into analyses based on discrete compounding of interest with discrete annual end of period values, such as most financial calculitor and spreadsheet analyses.

2-31 Assume a 10 megawatt power plant is operating at full capacity 6,000 hours per year producing electric power that is sold for $0.04 per kilowatt-hour (kwh). Calculate the time zero present value of revenues to be generated over years one through ten assumin g a llVo nominal interest rate for:

A)

Discrete end-of-year revenues with discrete annual compounding

of interest.

B)

Discrete beginning-of-year revenue with discrete annual compounding of interest.

C)

Discrete mid-year revenue with discrete annual compounding of interest.

D)

Revenues flowing uniformly during each year are treated as discrete revenues at the closest annual period to which they are incurred. In other words, revenues incurred within plus or minus 6 months of an annual period are treated as discrete sums at that period. With this approach, put month l-6 revenue at time 0, month 7-18 revenue at year l, month 19-30 revenue at year 2, and month 31-36 revenue at year 3.

l

_l

f1

l I

+)


61

&

,: :

'

2-32 As a recent new hire in the tlnancial department of your company you have been handed the folrowing schedule of payments and the corresponding variable annual interest rates on an outsranding loan note. The note does not include a stateti current loan valueryoo linra been esked

t.

detcrrnrne rhe roan vaiue today (time zcro), and then to deveiop a for the paynients, brezrking each payrnent

loain arnoriizario;t-schedultr

down into rhe reievant pri*cipal iari rlterest pay::rInts ea:h yelr. Assunie the interest is based on discrete annual compounding and payments iue made at the end of each year. All dollar values are in (000,s) Year 0

I 2 3

4 5

Loan

Payment

Interest Rate

0.00

1,100.00 1,100.00 1,100.00 700.00 700.00

6.5Vo 6.5Va 7.0Vo 7.5Vo 7.5Vo

CHAPTER 3

*

PRESENT, ANNUAL, AND FUTURE VALUE, RATE OF RETURN AND BRBAK.EVEN ANALYSIS

3.1 Introduction The five economic.decision method approaches in the title of this chapter

are the basis of virtuhtty all economic analysis decision methods used today. when applied properly, any of these approaches reads to exactry the same

economic conclusion. proper application of these diffelrent approaches to analyzing the relative economic merit of alternative projects depends on the type of projects being evaluated and the evaruation situation. As was mentioned in chapter 1, there are two basic classifications of investments that

are analyzed I

)

Revenue-producing investment alternatives

2) Service-producing investment alternatives

The application of present worth, future worth, annual worth and rate of return analysis techniques differs for revenue and service-producing projects. In addition, there are a variety of crifferent break-even anarysis approaches that can be applied to analyze both i,come-producing and service-producing investment alternatir"r. In this chapter we will concentrate on the application of present worth, annual worth, future worth and rate of return techniques. variations in the apprication of present worth analysis techniques using either net present varue, present worth cost, present value ratio or benefit cost ratio will be addressecl. Break-even analyses of various types are also discussed in this chapter and subsequent chapiers of the text. These techniques are presented here on a before-tax analysis basis to avoid confusing the reader with tax considerations ht the same time new evalua_

I

ch:.:pter 3: Present, Annual and Fut0re Value, Rate of Beturn and Break-even Analysis

tii)n techniques are introduced. Starting in chapter g, these techniques are appiied after-tax, to varying evaluation situations. Valid analyses must be 'lone after-tax unless taxes are not relevant oi significant. k

.{.2 Break-even and Rate of Return (ROR) Calculations Using Present, Annual, and Future Worth Equations

in chapter 2, several examples and end-of-chapter problems relate to calr'i-ilating the present worth of a future stream of revenue for a given interest r';rte or desired rate of return. This calculation gives the initial cost that can he incurred for the future stream of revenue if you want to receive the speci-

iled rate of return on your investment dollars. This really is a break-even type of calculation involving calculation of the initial investment cost that i.'ill enable you to break-even with a specified rate of return on your ilvested dollars. Example 3-1 illustrates this type of break-even calculation tirr ditrerent conditions, to emphasize the importance and effect of the time r"alue of money. Break-even calculations can be made for any single project parameter such as initial cost, annual revenue, salvage value, or proieit iife, to name a few. one equation can always be solved for one unknown, either c.xplicitly or by trial and error. The evaluator has a choice of letting the uirknown be any desired break-even parameter or evaluation criterion such ii: fiite Of returnEXAMPLE 3-1 Present worth Revenuc Equars Break-even Acquisition Cost Determine the present worth of the revenue streams, ,,1',, given in alternatives "A" and "B" for minimum rates of return of 10./" and 20"/o. This gives the initial cost that can be incurred to break-even with the 10"/" or 20"h rale of return. Note that the cumulative revenues are the sarne for the "A" and "8" alternatives but the timing of the revenues is very different. .

P=? A)

0

p-2

I=$200 l=$300 l=$400 1

2

34

l=$soo

l=9400

l=$300

l=$S00

l=$200

B) 4-

64


Solution:

0.8264

0.9091

i = 1O/o, PR = 200(P lFlo"t",l) + 300(P/F

1Oo/o,2)

0.7513 + 400(P/F10o/o,3)

0.6830 + 500(P/F1 0"h,4) = $1 ,071.76

1.381

3.170 oI, P4 = [200 + 100(A/G1 O%,4)1(PlA1O"/o,4) = $1,071 .78 1.274 2.589 i = 20o/o, P4 = [200 + 100(A/G2 g/",a)](PlA2O/o,4) = $847.64 Note that to get a 1oo/o rate of return you can afford lo pay $224 more than the $847 you would pay to gel a 20o/o rate of return.

1.381

i = 1Oh, P6 = [500

-

3.170 100(A/G1 g"/",a)l(PlA1O/o,4) ='$1 ,147.22

i = 2Oo/o, Pg = [500

-

100(A/G2 g/",a)]PlA2O/o,4) = $964.66

1.274

2.589

Note that in Example 3-1, while you still pa"y more to get a70Vo rate of return than to get a207o rate of return, the cost that you can incur for the "8" stream of income for a given rate of return is greater than the corresponding cost that you can incur for the'A" income stream. Since the cumulative revenue is identical for'A" and "B", the difference is due to the time value of money. Getting the revenue more quickly in "8" enables us to economically

justify paying more for it initially. It is important.in evaluation work to account for the time value of money correctly by treating costs and revenues on the time diagram in the amounts and at the points in time most expected. If you think of the incomes in 'A" and "B" as being analogous to mortgage payments that will pay off an investor's loan, you will note that the bigger payments in early years for "B" amortize the investor's principal more rapidly than in '4". Compound interest rate of retum is directly analogous to the interest rate you receive on money in the bank, or to the interest rate you pay on a loan. In all cases rate of return refers to the interest rate that the investor will receive on unamortized investment principal each interest compounding period. The term "unamortized intestruent principal" refers to the investment principal and accrued interest that have not been r€cov-€r€d through profits, saltage, savings, nxortqage payments, tuithdrawals from a

+'e

chapter 3: Present, Annuar and Future Varue, Rate of Return and Break-even Anarysis

bank account, or other revenue forms, depending on

the investment situation. It is important that you recognize that the rate of return which banrcs pay as interest on a savings accint is exactly analogous to

clLarged on compouttd interes.t mofigage roans whlch in

the rate of return thar.we

the interest rate turyis onologous to

wiil be ,ii,ig ,o ,;;r;;";;;,ff"'i"',ri'a,0",

nrcrLis where costs and revenues are ktlown or projected.

oJ.invest_

In the next example it is shown that results such as those carcurated in Example 3-1 can be.obtained using other types of calcurations such as annual or future worth equations. wlen we talk about writing an equation in economic analysis worl, this usuary means we are going to equate costs and revenue terms on an equivalent basis. However, sometimes we write an equation to equate rhe costs of alternativ" ,"rri""lp;;; projecrs. By revenue terms we mean ail incomes, profits, ,uuingr, ,uirug" or other receipts of revenue. On an after_tax basis, revenu", .Iau""a ;;-;;;; costs and income taxes are represented by positive cash flow. costs refer to all expenditures or outflows_Lf *on"f ,nd a.e represented on an after_tax basis by negative It arso is iossibre to refer to before-tax positive t9y and negative cash"Th flow if income tax coirsiderations *. n"gi"o"d. Foilow_ ing are the three basic equations used in economic evaluation work: Present worth (pw) Equation: present worth Costs = present worth Revenues Annual worth (AW) Equation: Equivarent Annuar costs = Annual Revenues Future worth (FW) Equation: Future wo.tt costs = Future worth Revenue In economic evaruations, whether you are making rate of return anarysis, break-even or net value. carcurations, you compare costs and revenues at any desired point in time. you are not "u, rimited io pr"r"n, *orth, annuar worth and future worth.equations, although they are most commonty used. consider the Example 3-2 to illustratJthe aiprication uJ-"iuirurence of present worth, future worth and annual worth equations. )

EXAI,IPLE 3-2 ilrustration of present worth, Future worth and Annual Worth Equations A rental asset is expected to yield $2,000 per year in income after all expenses, for each next ten years. tt is also to have a "JJlqin ten y""i.. resale vatue of $25,000 How much "*p".t"O for this asset now if a 12/".annuar compound interest rate oiiJtrrn before taxes is desired? Note that the'worJing of this. exampre courd be

";;;;;rid


66

changed to describe a mineral, petroleum, chemical plant, pipeline or other general investment and the solution would be identical.

C = Cost, I = lncome, L = Salvage Value andi = 12h.

C=? 0

l=$2,000 l=$2,000 l=$2,000 ...10 1 2

L=$25,000

Present Worth (PW) Equation:

Equate costs and income at time zero. The present worth of income and salvage at a ROR of 12"/"is the break-even cost that can be paid at time zero; PW Cost = PW lncome and PW Salvage (At the 12.0% minimum rate of return)

0.3220 5.650 + 25,000(P lF C = PW Cost = 2,000(P/A1 z/",10) p"1"1g) = $19,350 lf $19,350 is paid for the property at time zeto, a 12"/" rale of return per year on the unamortized investment will be realized. lt is instructive to note the same result could have been obtained by writing a future worth equation or an annual cost equation.

Future Worth (FW) Equation: FW Cost

= FW lncome and FW Salvage

(At the 12/" minimum rate of return)

17.55 3.106 C(F/P1 2"/",10) = 2,000(F/A12,/o,19) + 25,000 C = [2,000(17.55) + 25,000]/ 3.106 = $19,350

Annual Worth (AW) Equation: AW Cost = AW lncome and AW Salvage (At the 12.0% minimum rate of return)

4.1770 C(fuP1

z/"j1)

0.0570 = 2,000 + 25,000(fuF 12/",19) so C = $19,350


It should be noted in Example 3-2 thata present worth quantity, ,,C,,, was to be calculated, and it was obtained from a future worth and aunual worth equation, as well as a present worth equation. In fact, an written to equate costs and income ut uny point ";r;;;;J#; in time and the same break_ even acquisition cost of $19,350 wouid be obtained Norv consider the use of present, future and annual worth equations to determine project rate of return. This requires writing the equations with the unknown rate of return, "i", in evaluation situations where both the costs and revenues are specified but the rate of return, ,,i',, is unknown. A trial and error solution generally is required to calculate "i" as illustrated in the following examples. EXAMPLE 3-3 Rate of Return (ROR) lllustration lf you pay 920,000 for the asset in Exampre 3-2, what annuar compound interest rate of return on investment dollars will be received?

Solution: C = Costs, I = lncome, L Salvage Value, i _ ? =

C=$20,000 l=$2,000 l=$2,000

t=$2,000

2 ........

10

L=$25,ooo

The onry unknown in this probrem is the rate of return, ,,i,,. A present worth, future worth or annuar worth equation ,r, t" used to obtain "i" by triar and error carcuration. rn fact, ,n may be written setting costs equar to income at any point "qr"tion in time (the beginning or e1d of any period), to determine the project rare of return, ,,i,,. The result is the same regardress of the point in time chosen to write the cosi equals income equation. Present Worth (pW) Equation at year 0 to Determine .,i,, PW Cost = PW lncome + pW Salvage Value 20,000 = 2,000(P/Ai,1 O) + 2S,000(p/F;,1 g) Mathematically the equation is:

20,000=2,000[(1 + i;10_ 1ili(1 + t;101 +25,000[1/(1 +

i)101

68


There is no mathematical way to explicitly solve this equation for "i"' A trig!.-qnd,error sglution is requirgd and inis is more easily done using the equation in factor form, rather than mathematical form. By trial and error we want lo find the "i'1that rnakes tbe.right.side,of the equation equal to the left side, which is 920,000. This is done by picking an "i" value, then looking up the factors from the tables in Appendix A. To approximate an "i" value in the correct rate of return range, the following approximation is good for project tives of g to l0 periods or more. Approx.

i=

Arithmetic Average

lncome

Cumulative lnitial Costs or 1Oo/" for this example.

2,000 20,000

= 0.'10

salvage- value is neglected in this approximation because it is far enough into the future for lives of 8 to 10 periods or more that due to time value of money, it has litile effect on analysis results. That is why this approximation is only valid for relatively long lives of g to 10 periods or more. i

.

=

10"/o= 2000(6.145) + 25,000(.3g55) = g21,9OO

i=? i= 12"/o=2000(5.650)

= $20,000 + 25,000(.3220) = g19,350

Because there are no 1 1% tables in Appendix A, we must interpolate between the 10% and 127o values:

i=10"/" +2%l(21,930

-

20,OOO)/(21,930

- 19,350)l = 11.5o/o

Graphically, the rate of return interpolation for this problem and fale of return analysis in general is presented in Figure B-1 on the following page. Note that although the present worth equation used to calculate the rate of return, "i", has two unknown present worth factors, each factor is a function of only the single variable, "i". There is no limit to the number of unknown factors that can be in an equation used to calculate rate of return as long as all factors are a function of the same variable, "i".


Present Value

$21,930

Rlght Side of Present Worth Equation

c

li li lr li

\

-i-..i_,

lo. I

i

S

l--."---.--.----...-..--.--..---- b

f i:

BOR (i)

Figure 3-1 Graphic rnterporation for Rate of Freturn, ,,i,, Rate of Return, i = 10"/o + a

Tl," smail triangre with sides "a" and ,,c,, is geometricaily ,, simirar to the larger triangre with sides "b" and 'iJ,.in." both triangres have equa!

angles. Therefore, the sides of the two triangre. ur" prof,oitionat. (a/b) = (c/d), therefore a b(c/d) = Substituting these values gives: a=

2%-10"/")(21 ,990-20 ,OOO)/(21,930_19,g50) = Rate of Return, i = 10"/" + 1.5./" 11.5"/o = (1

1 .So/o

This is the same result shown on the previous page.

I

L


Future Worth (FW) Equation to Determine

"i"

20,000(F/Pi,10) = 2,000(F/A;,10) + 25,000 Trial and error solution of this equation gives the same result: i = 11.5o/o. Only interpolation error will cause the resul?to be slightly different from the present worth equation result'

Annual Worth (AW) Equation to Determine'oi" 20,000(A/Pi,1O) = 2,000 + 25,000(A/Fi,1O)

Trial and error solution of this equation gives the same result: i = 11.5o/o.

The "i" value of 11.5% in this example is the rate of return per year that the investor would receive on unamortized investment principal. ln this text, rate of return often is referred to as "ROR". Other authors sometimes use the term internal rate of return, or, "lRR", or, return on investment, or "BOl", to refer to the same quantity. Whichever term is adopted, it is helpful and important to note the exact analogy between rate of return on investments in general, and rate of return on mortgage investments or bank account deposits. ln each case the ROR to the investor is calculated each year (or period) based on unamortized investment, which is the investment value that remains to be recovered. Project ROR is not the return received on initial investment each period, it is the return received on unamortized investment each period lt is very important to look at the unamortized investment values to which rates of return apply in using the rate of return criterion for investment decision-making. A large rate of return that relates to small unamortized investment value can have very different meaning than a smaller rate of return that applies to a larger unamortized investment value. That is why investments with the largest rates of return often are not the best investments from an economic viewpoint as is illustrated in Chapter 4. The timing as well as the magnitude of costs and revenues that go into discounted cash flow analysis criteria calculations is extremely important as the following example illustrates.

:,

I

.t

chapter 3: Present, Annual and Future varue, Rate of Return and Break-even

Anarysis

z1

EXAMPLE 3-4 Rate of Return Sensitivity Calculate the investor rate of return on a g1000 investment that doubles in value: A) ln one year B) ln three years C) ln five years

Solution:

C=Cost, l=lncome

Ar C=$1,000 l=$2,000 ,.tnl PW Equation: 1,000 = 2,0A0(plFi,1) 2,000(1/1 + i) = 1,000(1+i) = 2,999 1,000 + 1,000(i) = 2,000 1,000(i) =2,000-1,OOO i = ROR = 1.0 or 1OO%

This solution was expliciily obtained, without trial and error for this simple case.

o, et

C=$1,000

l=$2,000

PW Equation: '1,000 = 2,000(p/F1,3) at i= 25"h: = 2,000(0.5120) = .1 ,024.0 3t i= 30"/": = 2,000(0.4552) = 919.4 ROR = i = 25o/"+ S%[(1,O24 1,000)/(1,024

-

ROR=i=26.05"/o

-910.4)]


72

1

c)

;

c=91,000 0

1

PW Equation : 1,000 =

2 ., - 9' ,..

l=$2,000

.1,,-,,..,.

2,000(P/F;,5)

.Q., .r,,,,,.

*

?t i= 15"h: = 2,000(0.4972) =994.4 at i = 12/": =2,000(0.5674) = 1,134.8 ROR = i = 12/o+ 3%[(1,134.8

- 1,000)/(1,134.8-

{

994.4)]

ROR=i=14.88'/" The difference in these ROR results shows that the timing of doubling your money as well as the magnitude of the numbers is very important in determining the economic analysis result.

3.3 Rate of Return and Cumulative

'E.--,1

Cash Position

The cumulative cash position diagram is a graphical means of explaining the meaning of rate of return and gives results that are analogous to the tabular results presented in Chapter 2 for Example 2-17 to explain the meaning of the 10Vo interest rate, or rate of return, in that example. The advantage of the cumulative cash position diagram is that a picture is often better than words or tabular figures to clearly explain something. By definition, cumulative cash positii)ru is the investment principal and accrued interest that has not been amortized by revenue such as profits, salvage value, savings, mortgage po.\ments or other inflows of money. Lettirtg revenue terms have a positive sign gives costs a negative sign, so in iwestment situations, the cumulative cctsh position starts in a negative position crnd works back to zero ot tlrc end of a project lfe when the time value of ntonet' is handled at the project rate of return. Each compounding period the cumulative cash position is adjustei in the negative direction for new investments and accrued interest, and in the positive direction for revenues received. Unamortized invest-

ment is another term with a meaning synonymous with cumulative cash position. Witlt cumulative cash diagrams, it is intportant to recognize that v'e are not re.ferring to a plot of indiyidual project cash flotvs, but rathe4 are monitoring the overall cash position of an iru,estment oppcrtuniry. For a sirtgle investment situation, the concept of cunutlatiy'e caslt position is arrctlogoLts to graphing the balance in a bank savings accoutlt over tinte.

J

chapter 3: Present, Annuar and Future varue, Bate of Return and Break-even Anarysis

Think of an investor nnking an initiar deposit, (the investment ,f fun.ds into :: btt'k savings accoutxr wourd be anarogous to negative cash frow) that deposit eccrues interest so the account barance increases witrt time. In rris ('Lise, it tneans the barturce becomes more negative. the

investor $ext, o1 the Jitncrs .froi, trt<, accoL.ltt (trte itti,es.tLr })ortion rrrilrid,' positiv'e cash flow). Ttu redttced aL.(.oitt1[ bcrance accrues irtter,st Li 't;ittg tlte r.ext ce-tinpoundittg pe riod otrcl, evertiuatty the crtttbinatiott ,f ;'it(i{}.\'€s

io

t1.itrrclrolt'_(i

'iitiidroy..als.irom the accrued. interest o,rrt

pri,r,rip;;";;r;';;re

balance b :i:ru ai tirc end of the project rife or sot'i,gs period. The folrowing four i''xarnples illustrate rate of return calculation:i and provide an explanation of iireir meaning using the cumulative cash position concept.

EXAMPLE 3-5 rfiustration of ROR and cumutative cash Diagram For a $10,000 investment now, an investor is to receive $2,6gg income at the end of each of the' nlxt five years ,nJ r"ro sarvage value. carcurate the rate of returh and diagram the investor,s cumurative cash position for the project rite. rvot6 tne iolniiti-bltween this problem and Example 2_1i.

Solution: writing an annuarworth equation as we did in Exampre 2-17 gives: 10,000(A/Pi,5) = 2,63g, so i= 1O% The carcuration of "i" wourd be by triar and err.or if we had not already determined the result in Ex#ple Z_t Z. -10,000

+2,638

+2,638

_2,000

cuM.

-4,000

CASH

POSrr. -6'000 _8,000 _10,000 il

:i

iil

$

{ $ ?l

* ;{

Figure 3-2 Cumulative Cash position Diagram at ROR

=

1xo/o


74

Plotting the cumulative cash position for the 10% project rate of return in Figure 3-2, we start out in a -$10,000 cash position at time zero. During each year we expect to receive d 1O"/o rate of return'onlunamortized

investment at the beginning of that year, so the negative cumulative cash position is increased each year by 10% of the uhamortized investment at the beginning of the year. For example, if the investor wanted to terminate this investment at the end of year one, he would need to receive a total of $1 1 ,000 at year one to realize a 1O"/" ROR on the initial $10,000 investment. lf he wanted to terminate it at the end of year tvvo, the cumulative cash position diagram shows that he would need to receive $9,198 at the end of year two in addition to the $2,638 received at the end of year one. The cumulative cash position diagram shows that the investor'S rate of return of 10"/" makes the cumulative cash position zero althe end of the pro1ect life. Any other interest rate makes the final cash position either positive for "i" less than 10"/" or negative for "i" greater than 10%. ln other words, for "i" less than 10% the $2,638 annual payments are more than enough to pay i% interest on the unamortized investment each year, with money left over at the end of the fifth year. The opposite, of course, is true for "i" greater than 10%' Note in the cumulative cash position diagram that the rate of return of 10% is applied to the unamortized investment each yea(, a declining amount of money each year in this problem. lt is not applied each year to the initial investment and it is not applied to the income each year' On the contrary, the income each year is used to reduce the amount of principal that the rate of return is applied to in the following year.

EXAMPLE 3-6 Rate of Return When Salvage Equals lnitial Cost Work Example 3-5 with a salvage value of $10,000 instead of zero.

Solution: lntuitively you can determine the rate of return in your head when initial cost and final Salvage value are equal with uniformly equal revenues each period. Regardless of the physical investment, you can always think of this analysis situation as being equlvalent to putting the investment dollars in the bank at an interest rate equal to the project rate of return, withdrawing the interest each period, (which is equivalent to the period revenue), and wilhdrawing the investment

:

I

'9.i

chaprer 3: Present, Annuar and Future varue, Rate of Return and Break-even Anarysis

principal from your account at the end of the evaluation life (which is equivalent to salvage value). The project interest:,.rate, or rate of irrturo per period, in this situation, where initial invesiment cost equals salvage, is always equal to the period revenw divided by the r;',riai investmeni (or salvage value since they are equal). using the fr:iio*ing present worth equation you can veriiy that the rate or return rs lr-i.I:3".ro for th!s example.

PW Equation: 10,000 = 2,63g(p/Ai,S) + 10,000(p/F;,5) i = 26.38"/" -'10,000

2,638

-2,000

cut/.

-4,000

cASH POSIT.

-6,ooo -8,000

-10,000 -12,000 -12,638

Figure 3-3 Cumulative Cash position Diagram at ROR =

26.3go/o

l-i'ic cunrillative cash position diagram in Figure 3-3. illustrates that when salrage value equals the initial investment and annual incomes are unifbrm,

the investor's cornpound interest rate of return is applied to the initial i,r'estment, which is also the unamortized investment value each com_ l'.'oundin-q period ($10,000 per year in this case). when this cash frow situatior-r occurs, the compound interest rate is eqrivalent to a flat or add-on interest rate. If the initial cost and terminal .ilrug" are the same, and the in'estment yieids a uniform income each compounding period, the RoR ma1' be explicitly calculated by using the approximation technique described in Example 3-3. This technique (a,erage income / cumulative initial investment) contputes the flat or adcl-on interest rate fbr an investment and uses that value a3 a starting point for the triai and error solution for regular RoR. Using this approach in Example 3-6, the rate of return is 2.638 I 10,000 = 0.2638 or 26.38Vo.


76

EXAMPLE 3-7 Rate of Return for lnitial Cost and a Single Revenue

'

Evaluate the rate of return an investor will receive if $10,000 is invested at time zero and this investment generateB a future lump sum income of $16,105 five years later. Develop the cumulative cash position diagram at the project rate of return.

Solution:

c

= $10,000

I = $16,105

PW Equation: = 16,105(P/Fi,5), so P/Fi,5 = 0.62093 i = 10/" per year, by trial and error

10,OOO

Notice that the unamoriized investment gets larger each year when no annual revenues are received to offset the accrued interest. Also note that Examples 3-5 and 3-7 both involv@ 1Oo/" rate of return projects with a $10,000 initial investment and a five year evaluation life. However, the unamortized investments that the 10% rate of return relates to in Examples 3-5 and 3-7 are very different after the first year for each of these projects. The cumulative cash position diagrams clearly show this. +1 6,1

-10,000

05

-13,310 -14,641 -16,105

Figure 3-4 Cumulative Cash Position Diagram at ROR = 10"/"


The next example illustrates a rate of return problem with costs at two

EXAMPLE 3-8 Rate of Return with More Than

on"Lo"t

Consider the investment of $10,000 at time zero and $S,O0O at the end of year one to generate incomes of $9,000 at the end of year two and $9,500 at the end of each of years three, four and five. what is the annually compounded projeci rate of return? Develop the cumulative cash position diagram at the project rate of return.

Solution: C=$10,000 C=$S,000

l=$9,000

l=$9,500

[=$9,500

l=$9,500

PW Eq: 1O,0OO

+ 5,000(P/Fi,1) = 9,000(P/F;,2) + 9,500( p/Ai,)(plFi,2)

Rearranged Format: 3 = -10,000 DIV Eq: @

-'l 0,000

-

-

i=

5,000(P/Fi,1)

* 9,000(p/F;,2)

+ 9,500(p/Ai,3)(piF;,2)

30%:

5,000(.7692) + 9,000(.59i 7) + 9,500(1 .816)(.5917) +$1 = ,687

oWEq: @i=40"/": -1 0,000

- 5,000(.7143) + 9,000(.5102) + 9,500(1 .SB9)(.51 02) = _g1,278

3y interpolation: = ROR = 30% + (40%

-

30%)t(1,687

-

0)/(1,687+1 ,278)J= 35.69%

triiminating interpolation error by interpolating over smaller ranges of than 30% to 40% gives gs.2s4% as the coirect'RoR resurt.

'-

78


-5,000 +9,000

+9,500

+9,500

+9,500

Cumulative Cash Position

-10,000

Figure 3-5 Cumulative Cash Position Diagram at ROR = 35.254/o EXAMPLE 3-9 Rate of Beturn for Discrete or

Continuous lnterest For the cash flows presented below calculate rate of return for the following assumptions regarding the type of compounding and flow of funds in a project.

CF=j1__9f-13 012

CF=+13 CF=+14 CF=a15

CF=+19

345

A) Assume discrete end of period dollar values and discrete compounding of interest. B) Assume a continuous flow of dollar values with continuous compounding of interest. C) Relate the solutions from (A) & (B) to show that they are equivalent and interchangeable.

Solution: A) Discrete End of Period Cash Flows, With Discrete Compounding of lnterest PW Eq: '19(P/Fi,5) + 15(P/Fi,4) + 1a(P/F;,3) + 1B{p/Fi,2) + 13(P/F;,1)- 21 = 0

chapter 3: Present, Annuar and Future Varue, Rate of Return and Break-even

PW @ i= 507o: 19(.1317) + 1s(.1975] +_14(.2963) + 1B(.G667) - Zl = +3.06

+

Anarysis

zg

1B(.444a)

PW @ i=70o/o: 19(.OZO4) + 1S(.1197) + 14(.2005)110(.3a68) + 13(.5882) - 21 = -2.82 By lnterpolation: i = 50"/" + 20%{(3.00

-

Oy[9.06

- (-2.87)]] = 60.37o

Eliminating interpolation error by interpolating over much .increments smaller of "i", the true ROR is S-g.OSZ.

B) Continuous Cash Flows With Continuous Compounding of lnterest I CF=-2t I CF=+13 I CF=+13 I CF=+14 I CF=a.15

0129456

Factors are calculated from Appendix

;

CF=a19

I

c definition of p/F*r,n

PW Eq: 19(P/F.1,6) + 1S(p/F.r,g) + 14(p/F*r,4) + 1S(p/F.1,3) + 13(P/F.j- 21 (p/F.y,1) = 0 ,2)

PW Eq @

r= 40%: 19(.1115) + 15(.1664) + 1 4(.2482)+ 13(.3703) + 13(.5525) -21(.8242) = +2.28

PWEq @ r=50"h:191:9616]+ 15(.1065) + 14(.17s6) + 13(.2895) + 13(.4773) - 21(.7869) = -1.27 By lnterpolation: i = 40"/" + 1 0%{(2.7 8

-

O)

t[2.2 B

-

(-1 .22)]]

=

46.9"/o

Eliminating interpolation error by interpolating over much .increments smaller of "i", the

true ROR is 46.4"h.

C) Equivalence of Discrete and Continuous

Compounding Results

Given E = ef - 1, From Eq. 2-10. "E" is the effective discrete compound interest per year, which is the Part (A) result. Nominal period interest compoundeb continuously is given by "r", the part (B) result. Natural log base e is given by,,e,,.

-.*-

80


E = s0.464

-

= .5g04 ot sg.o4o/o 'ar:: The equivalence of the discreie'ano ioniinuous interest rate of return results is demonstrated. once you have either result (discrete or continuous), you can calculate the other using Eq. 2-10. 1

|

3.4 Alternative Methods to Obtain Annual Initial Cost, C, and Salvage Value, L

Value From

Determination of the rate of return for problems such as Example 3-6

where initial cost equals salvage, and incomi is uniform each period can be simplified by introduction of another method to obtain unnrui value. until now the normal way to obtain annual value for given initial cost, ,,C,,, and salvage value, "L", has been to use Equation 3_1:

- L(A/F1,,) = Equivalent Annual Cost 3_I for investrrrent, "c", and sarvage, "L", shown on the folrowing time diagram: C(A/Pi,n)

C = Investment

L = Salvage Value

An alternate method to convert an initiar investment, ..c,,, and a salvage value, "L", into equivalent annual value is to use Equation 3_2:

(C

-

LXA/P1,,) + Li = Equivalenr Annual

Cosr

It may not be immediately evident that Equations 3lent so the proof follows. Working only with Equation

3-z

I

and 3-2 are equiva3-2:

(C-LXA/pi,n)+Li = C(fuPi,n) + L(i = C(A/Pi,n) +

- A/pi,p)

L{i - [i(1

+

i)n/(l + i)n _ 1]]

= C(A/P1,n) + L{t(i(1 + i)n = C(A/Pi,n)

-

- i) -

(i(1 + i)nl/(l + i)n_ 1}

L(A/F1,n)

This completes the proof showing that Equation 3-1 is equar to Equation 3-2. Either equation may be used to reduce a first cost, .,i,,, and salvage, "L", to an equivalent annuar value. one situation where the use of Equation

Lirailte!- 3: Present, Annual and Future varue, Rate of Return and Break-even

Anarysis

g1

-3-2 is very useful occurs when salvage value equars the initial investment. writing the annual value equation foiExampre 3-6 in the form of Equation .: 2 I ieiii:; lhe ann,-ial w'orth equation shown:

i.'icmEr

3-6:C=$10,000 I=$2,63g I=52.63g

0 A\1' Eq' (10,000

r

- 10,000)(A/pi,5)

....

5

*

L=$10,000

+ 10,000i = $2.63g

i'ireretbre, i = .2638 or 26.3gvo explicitly, rvithout trial and error. This rat: of return result is the ratio of average annual income ($2,63g) divided b1, irritial investmenr cosr (9tr0,000). Th; is the approximaie ROR calculati,)il iniroduced in Example 3-3. s'hen initial cist and salvage value are euual and income is uniform and equal. the RoR obtained from the Exarn_ i,rc 3'3 RoR approximarion given eariier is always the exact project RoR. t\ riiing an annuar worth equation usine Equation 3-2 i'steaiof a present \\r)rlir equatron makes this intuitively evident. An erample iliustrating that Ecluation 3-1 and 3-z give the same equivalent annual cost foilows. EXAMPT-E

3-10 lllustration of Two Equivatent Annual Cost Calculation Methods Determine the equivarent annual cost for eouipment with an estinratecj ten year life having initial cost of $30,000, estimated salvage vatue in ten years of 910,000 for a minin':um RcR of 1s"/o per year. tJse both Equation_3-1 and Equation 3-2 to show that they give the sanre result. see Example 3-1'l to illustrate another application of Equation 3-2 to rate of return calculations.

Solution: Using Equation S-1: 0.1 9925 A = 30,000(NP1S%,10)

-

0.04925 10,000(A/F1S%.10) = $5,485

Using Equation 3-2: A = (s0,000

L

-

10,000XAPp]3'z:o) + r0,000(.1s) = $5,485

82

3.5


Rate of Return on Boncl Investments

Bond and debenture offerings are very popular ways for companies and governments to raise debt capitai. Implicitly then, the analysis of bond and debenture investments is an important investment anElysis for potential investors. Bonds and debentures are similar debt paper except bonds are backed by the assets of the issuing organization as collateral whereas debentures are only backed by the name and general credit of the issuing organization. These types of investments are really very straightforward to evaluate but people often get confused between new bond interest rates (or rates of return) and old or existing bond rates of return. Another factor that adds to bond analysis confusion is that bond interest usurtlly is paid semi-annually which means you must work with semi-annuul periods and a semi-annual period rate of return to hcndle the time value of money properly in bond yield calculations. Three other important factors to remember in bond evaluations are: (l) at the maturill\ date of a bond, the holder of the bond will receive its face value as a salvnge or terminal value, (2) bond cost or value will vary between the initial offering date and the maturity date as interest rates fluctuate up and down in general money markets, and (3) Bond call privileges are yvritten into most corporate or municipal bond offerings to give bond issuers the right to pay ojf a bond issue early (before the maturity date) at any date after the call date. Early payoff of bonds is desirable for the bond issuer if interest rates have dropped significantly (by several percentage points) so the old bonds can be refinanced with new bonds at a lower rate. Bcnd call privileges often affect the value of old bonds significantly, so it is very important for potential purchasers of old bonds to account correctly for bond call privileges. U.S Treasury Notes and Bonds generally are considered to be the highest quality investments available. They are semi-annual interest bearing obligations of the U.S Treasury, are issued in denominations from $1,000 to $1 million, and are unconditionally guaranteed by the U.S Government. Treasury Notes have maturities fiom two to ten years and are not callable by the U.S. Government prior to maturity when they mature at face value. Treasury Bonds are similar to Treasury Notes except that maturities are eleven to thirty years. Some Treasury Bond issues are redeemable at par five years prior to maturity and are known as term bonds. The U.S Treasury sells new note and bond issues at various times, as money is needed, either to raise new firnds to finance the Federal deficit, or to refinance existing bond or note issues. A tax advantage associated with U.S. Government Securities is

ffi

; ti

-lliapier 3; Present, Annual and Future Value. Rate of Return and Break-even Anarvsrs

tirat stat,: income tax does not hxve to be paid on interesr tiom U.S. Trea_ sury Bonds, Notes or Bills (see section 3.6 ibr Treasury BiII discussion;.

EXAI\,tPLE 3-11 New Bond Rate of Fleturn Using Eq. 3-2

analysiJ

calculate the nelv bond rate of return for a new issue of $1,000 bcncs uriih a maturity date twenty years after the issuing cjate, if the new bond pays interest of 940 every six month period.

Solution: C = Cost, I = lnterest lncome (Semi-Annual), L Maturity Value = j

,t

C=

S1,000

l=$40

l=$40

l=$40

#

* fr

If * T

I

,

{

PW Eq: 0 = -1,000 + 40(p/A;,49) + 1,000(p/Fi,+O)

The "i" value may be determined by triar and error. However, since the initial cost is equal to the sarvage (or maiurity) value and period income is uniform and equal; use the apprcxi;naiicn technique from Exarnple 3-3 to explicitly solve for the raie of return. ROR, i = 40/1,000 = 0.04, or 4.Oo/o par semi-annual period

The nominal rate of return is equal to the 4.0"k x 2, or g.o"/o per year compounded semi-annually. ln bond broker terminology the term "yield to maturity," is used to describe this nominal rate of return and may be listed by the acronym ,,yTM.,, EXAMPLE 3-12 Old Bond Rate of Return Analysis lf the bond described in Example s-11 was initially offered six years ago and now sells for $800, what rate of return would an investor who holds the bond to maturity receive? Note that only the cost and evaluation life are different from the Example 3-11 new bond evaluation. Fourteen years (28 semi-annual periods) of life remain from the original 20 year life.

84


Solution: C = Cost, t = lnterest lncome (Semi-Annual), L = Maturity

C=$800

l=$40

l=940

2....

Value '#

l=$t0 L=$1,000

..28semi-annual

When initial cost does not equal salvage value there is no advantage in writing an annual worth equation over a present worth equation to calculate rate of return. Present Worth Eq: 800 = 4O(PlAi,2g) + t,000(PiF;,26)

ln selecting the initial trial and error "i" value, remember that,,i" is a semi-annual period rate of return. lf we paid 91,000 for the bond we know i = 4'h, so try a higher value: i = 6/"= 40(13.406) + 1000(.1956) = $Zgt.e+ i = 5"/"= 40(14.898) + 1000(.2551) = 9651.62

By interpolation i = 5.43"/o per semi-annual period. The nominal or annual rate of return is 10.86% per year compounded semi-annually, which is the yield to maturity. Another broker letm, current yield, is defined as annual interest divided by bond cosf. For this example old bond current yield is 80/800 equaling 0.10 or 10%, which compares to the previously calculated yield to maturity of 10.86% compounded semi-annually. EXAMPLE 3-13 Old Bond Rate of Return With and Without Catt Privileges

consider that the bond described in Example 3-11 was initially offered 6 years ago and now sells for 91,200 (interest rates have dropped so the bond price has increased). The bond is callable 10 years after the initial offering date (which is 4 years from now) at par value (the original $1,000 value). Calculate the rate of return that an investor paying $1,200 for this bond would receive if, (1) the bond is not called early and is held until normal maturity 14 years from now and (2) the bond is called 4 years from now. '

chapter 3: Present, Annuar and Future varue, Rate of Return and Break-even Anarysis

Solution: (1) No early cail privireges, so use a'.14 yEai (2g.semi-anndar

periods) life.

c = cost, I = Interest rncome (semi-annuar), L Matlrity = C=

$1,200

l=$40

l=940

2.....

tir

I

$

varue

l=$40 ...

..28

L=9.1,000

Present Worth Eq: 1,200 4O(p/Ai,2g) + 1,000(p/F;,2g) = The semi-annuar period rate of return is 2.ggyo, and the annuar rate of return (yierd to maturity) is 5.96% compounded semi-annuaily. (2) lf call priviteges exist and can be expected to be exercised.

(since interest rates.have dropped), use a fouiyear (eight semi-annual periods) life to the call date.

C = Cost, | = lnterest lncome (Semi-annual), L = Maturity Value C=

$1,200

l=g40

1=940

l=$40 L=91,000

Present Worth Eq: .1,200 40(p/A;,6) + 1,000(p/F;,g) = i i

i ir

)

The semi-annuar period rate of return is 1.35%, and the annuar rate.of return (yierd to cail date maturity) is z.zoy" compounded semi-annually. lf you pay 91,200 for this-bond and it is cailed four vears later, your rate of return is considerabry ress than the 5.96% annual rate of return to maturity alyear 14.

lf an investor wants to receive a 6% rate of return compounded semi-annually to the cail date, the present worth of interest and maturity "call" value for g semi-annuar periods at 6.00/o/2, or 3.0"/" rnterest per semi-annual period gives the bond value: Varue or

calabre Bond = ,oei;Z::,a) + r,000(p?/38.|.]u, = *,,oro

86

3.6


Rate of Return Related to T-Bill Discount Rates

' u.s. Treasury Bills are short-term (beginning.in'2001, T:Bills are sold. as either three month or six month instruments but in previous years 1 year instruments were atsohvailable) obligations of the E.i ;;;;."*it" offer investors a high quality return and minimal risk. Treasury bills are sold at a discount so interest is received when the investor receives the maturity value (or par value) back from the governrrrent. Treasury bills are sold in denominations of $10,000, (or $5,000 increments above $10,000). Treasury bill interest rates often exceed most bank certificates of deposit (cD,s). Buying Treasury Bills at a discount effectively pays th; interest up-front instead of at the maturity date of a Treasury Bill. This makes the T-bill investment compound interest rate of return greater than the T-biil discount rate, as the following example illustrates. EXAMPLE 3-14 T-Bill Discount Rates and Rate qf Heturn

A $10,000, six month r-bill with a1o/o discount rate can be purchased. what is the equivalent compound interest rate of return on this investment? Solution: T-bill discount rates are always given on an annual basis, so !0% I 2 = 5.0"/" per 6 months. The 5% discount is realized when the T-bill is purchased by 5% reduction in cost. The face value of the

T-Bill is paid as maturity value at the six month maturity date. Discounted Cost = $9,500

Maturity Value =$1 0,000 6 months

PW Eq: 9,500 = 10,000(p/F;,1) Trial and Error, i = 5.263/o per six months

Doubling this six month period rate gives an annual rate of return

of 10-526"/" compounded semi-annuaily as being equivalent to a 107", six month T-Bill discount rate.

chapter 3: Present, Annuar and Future Varue, Bate of Return and Break-even

3.7 Financial cost of capitar

vs

Anarysis gl

opportunity cost of capital

This section differentiates the two common approaches used by investors determine their minimlrm rate of retum, or discount rate.

to

Irf thii textbook, a ,ri,imum rate of return, or discount rate, is distinguished from a project ."*l pound interest rate of return by placing an asterisk immediately urt"i i, hence, i*. so, how is an investor's minimum rate of retum, i* established? Going back to chapter one, discussion was presented to introduce the reader to two schools of thought concerning this subject. Most managerial finance authors suggest that the appropriate methodology is to determine the investor's financial cost of capital which, for a privatery held company may be the average ccst of financing projects under consideration o. ueing financed. For a pubricly traded company it's a little more "r.r"nily comprex as the reiurn that stockholders are demanding in order to buy the companies stock must now be considered in addition toihe after-tu, of debt. iven thougtr we are focused on before-tax applications at this point, "orithese calculations will in reality, always account for the deductibility of applicable interest. The theory behind the use of the financlai cosior capital is that at a mini_ mum, if investors have nothing else to do with their money, they could always pay down existing debt, or buy back stock. This theory hoids true for an investor with unlimited resources so that any project undeiconsideration could be financed. Howevel in practice most investors are financially constrained and therefore, it's not the financial cost of capital that is the relevant minimum rate of return, but the opportunity that will bi foregone. This concept is illus_ trated in Example 3- 15 and further discussion on this topic wil follow.

EXAMPLE 3-15 Financiat Cost of Capital Versus Opportunity Cost of Capital

A firm has the opportunity to invest g100,000 in a project that is expected to generate a revenue minus operating cost before-tax cash flow of g'r5,000 per year, at years one thiough ten with a $100,000 sale value projected tor year ten. The firm,s weighted average financial cost of capital (from a combination of borroried money debt and equity capital) is 10%. capital budget dollars are limited and other opportunities for investing capital are tnougnt to exist that would give a 20"/" rale of return. UsL rate of return arialysis to deter_ mine whether or not the firm should invest in the project. $100,000

88


Solution:

c

l=$'15,000

= $100,000

l=$15,000

L=9100,000 l=$15,000

0

PW Eq: 0 = -100,000 + 15,000(P/A1,1g) + 100,000(P/F;,19) By trial and error, (or Eq 3-2); i = ROR = 15o/"

Although the project ROR of 15% is satisfactory compared to the the rate of return ol20o/" representing other opportunities for investing capital (opportunity cost of capital). lf investors are attempting to maximize profit from limited investment dollars the decision would be to reject the 15% ROR project and invest elsewhere at2O"h to optimize the use of investment capital from an economic viewpoint. Obviously if borrowed money were unlimited, investors would want to consider all projects with rates of return exceeding the cost associated with raising funds to finance investments (the financial cost of capital). 10ol" financial cost of capital, it is unsatisfactory compared to

Marginal lnvestment Flate of Return Hypothetical Budget Limit (X) With Capital Raiioning

a

Maximum Desirable lnvestment ( Y) With No Capital Rationing

Opportunity Cost of Capilal (OCC)Wlth Budget X

occ

=

Fcc

-TiT:1c"-:L9i Capital (FCC)

Projects'l

234

Cumulative lnvestment Budget

($)

..................n X

n+1

y

Figure 3-6 Financial Cost of Capital vs Opportunity Cost of Capital

chapter 3: present, Annuar and Future varue, Rate of Return and Break-even

Anarysis

gg

The Figure 3-6 curves for marginar investment rate of return and financiar cost,of capitar versus cimurative investment budget show graphicaily why the concrusion of Exampre 3-15 to reiect the 15.0% ROR is correct. This graph is a plot of investment ;;i return versus curnurative brrdget doilars with investments one through ,,n+1,,graphed from ieft to right in the order of decreasing project rate of return. For a given budget this marginar cnJrement"r) 3how9.the the marginar investment in the tasipgect wiil give.'roi"*"rpre, under

*,"'"i;;ilil;;r,

:lp]!?l rationing with g budget of gX, investments with rates

of return of 20-0"/" would have to be rejected to accept and finance the 1s.0olo RoR project described. This creirry wourd make suo.optirat use of investor

capital by not maximizing the future varue that.* o" dnerated from available budget doilars. An investor wourd *oneyl-y ,no"rtut ing the Example 3-15 project, but the inve_stor has the pot"ntiar of making more money by rejecting the Exampre 3-15 project ano inrl"ting budget dollars ersewhere at a20.o/" rate oi ,gtr.rn:. under capitar rationing if the opportunity cost of capitar is rarger than the tinancii cost of capitar as shown in Figure 3-6, then oppoitunity cost oi."pitrri. tn" correct minimum discount rate to use in discounted cash flow.rr"riltioi;';; investment decision-making However, it must b" that the opportunity cost of capitat represents other opportuiiiis for investing 'projea, capital now and in the future over the tife of iiirg anaryzed. when using a tisting of proiect rates of return todiy to determine opportunity cost of capitar, it is assumed that today,s projects are representa_ tive of future and current .investment ipportunitie.s. rf future investment opportunities are projected to be different tro, toorv s- investment opportunities, then it is necessary to change opportunity cost of capitar with time and use technrques other than rate of return, as iiluslnajvlis trated in Section 4.4 of Chapter 4. lf capitar is not rationed it is desirabre to fund ail projects up to the budget linrit where rnarginar investment rate of return equars financiar cost of capitar. As rong as additionar investments earn a rate of return greater than the financiar..cost of capitar, additionar investments are economicafiy desirabre. when an additionar investment wiil give a rate of return less than the financiur .ori of capitar, it is not desirabre. opportunity cost of capitat (tne rate tt return on additionar investment opportunities thought to exist) and financiar cost of capitar are equar at the investment rever where the rasr additionar investment earns a rate of return just equal to the financiat cost of ."iriirl. n, Jescrioed in

,"t"

;;;;;red

90


the last paragraph concerning opportunity cost of capital, when financial cost of capitatrs assumed to equal,,opportunity:cost of capitali because capital is not rationed, it is assumed that financial cost of capital will be the same in the future as it is todty. This is the basis upon which many finance textbooks are written. lt is assumed there that all projects with rate of return potential greater than the cost of raising money (financial cost of capital) will be done. Financing often is not as easy to obtain as it seems in textbooks. ln industry practice capital usually is rationed, making the opportunity cost of capital a bigger rate than financial cost of capital. To determine the opportunity cost of capital requires evaluation of potential future investment opportunities as well as today's investment opportunities. As shown in Figure 3-6 for the $ymaximum desirable investment level, opportunity cost of capital and financial cost of capital are equal. For a budget in excess of gYthe financial cost of capital would exceed the rate of return on the marginal investments so it would.not be economically desirable to fund projects beyond tne $y budget amount. lt is very important to note that although Figure 3-6 presents projects in the order of decreasing rate of return, the reader must not imply that rate of return is a valid project ranking technique. This is emphasized and illustrated by example in chapter 4. project one is not necessarily better economically then project two, just because the rate of return for project one is bigger than the rate of return for project two. All we can conclude from Figure 8-6 is that the cumutative investment of "$X" in projects one through four is better than replacing any of these projects with projects five through "n.,' ln other words, rate of return is an "accept or reject" criterion for each project, relative to investing elsewhere, and not a valid method of ranking projects. Also, when we say investing in projects one through four is preferable to investing in any other projects, we are assuming projects one through four are indicative of future investment opportunities (in addition to today's opportunitiesl. Most investors make this assumption, but sometimes changing projected future investment rate of return from what it is today may make more sense. Just because an investor has many high rate of return investment opportunities today often does not mean that similar opportunities will exist in the future. This requires changing the opportunity cost of capital with time by reducing future opportunity cost of capital from what it is today, as illustrated by Example 4-6 in Chapter 4.

.4.

i"ilapler 3: present, Annuar and Futrre varue. Rate of Return and Break-even

Anarysis gl

Alihough it is opportuniry cosr of capirar rather than financiar cost of cap_ iiai rhat is of prim;*y importance.in discounted cash flow

ii''rirs iificl decision-rraking, inicstors nlust arwal.s cost of capital does not exceed their ooportuniti, 'erif1-

analysis carcura_

that;:;fi#;i

c<.rst of capital. Therefore, is an exp.rar.irion of a tyirical bcfoie-ta>: ,pp.ou.r, thar many r;ri-ior cornpanies use. to arrir.e ar ttreii ,.financicrl :i'eilf i.s trt'ated e.t the tr,'_righttrr orrr*|gn (.ost Jj"ort o1'i,)iral,, r.vhich gen_ equlty rrd debtfirtctrtcirtg (tii it pr0.tp( <:tit'e busis.l-herefore. we must project ih.,.,r.t of equity capiml r:rriidcbr capital anrl prora-re them using the cornpany oJ,r.quiry ratio. Cost ,f .quity capitar represents-the economic return that company 'ititireltolders expec:_r? receive iri trze forrn of clividend paynlents and/or ;t,'t<-k appreciation.

i"i';'rirg

ll

i $

fl

$ s il I i

i

This expected equity return equals that which the share_ irtlider investors perceive could be reairzea on alternative investments of comparable risk. since company costs and revenues are yet to be incurred in riic iurure, this anary'sis is on a prospective basis. using the past as a guide u'herr and if it is corisidered reievant. , The cost of equitl' capital financing generally exceeds the rate of return on long term u.s. Treasury bonds u"cuJ,-rI of the greater risk thai equity share_ hriders assume. If an in'esror can invest in us. Treasury u"ro. which are 'o,sidcred to be rs safc as any in'estment in the woitd today by most people. why u'ould the in'estu in'est in corporate common stock with great'r risk. fbr the same rcturn'J To account for a relatively risk_free u.S. i-reasurl' Lriind rare acr.iLrsteci for risk. ,.carrital a esset pricing Model,,, cAP'\'l) ,pproach ro esrimati,g the c.st of equity capital is used by many !o.nrrilric\ todav. This sinrplisticaily ini,orves three steps: (1) estimating the c''rrirnt f ield on long-term_U.S. Treasury bonds, considered io be the ..riskir;'c'' intercst rate (assumed.to be g.070 per year for this illustration), (2) esti_ nrating an equity risk premiurn reratetl io the spread between common stock rcrrir*s ancr U.S. Treas.ry boncr yields (assumed tobe 6.07c for this analysis), lrrd (-l) estirnating the companies "Beta" factor rvhich measures stock price tolatilit-r'tbr the company rerative to tire o,erall ,rarket (assume an average risk ccrnrpari'. so Beta rs r.0). This gires a cost of equity capital of r4.ovo tt Iiich equals the 8.0c26 risk-free ,ot" I 6.yEo risk rate ; f .O n"iol Tlrc cr'''st of debt is trte interest rare a con:pon), vvourd pay for new rongtcnn -firtatrcirtg b1' bo*ow,i,g riitrctrt'from banks or thr-ougi nex, bond or dtbenture type offerings. Assume uri.oq"debt rate for our illustration. For a company with an assumed 40.ovo debt, 60.0vo financiar structure. the weighted average financial "q;i;ybe 12.0vo, cost of capital would based on the assumed rates as follows:

92


Cost of Cost of

Equity:

14.\Vo x 0.60 =

8.40Vo

Debt: 9.0%o,x0.40 =

3.60Vo

Cupitul-- =

12.007o

Financial Cort of

i Many variations occur in the financiar cost of capitar calcurations done by dffirent cornpanies and investors. For exampre, income generaily tax must be taken into account for valid analysis so you need. to work with afteitax debt rates, risk-free investment rates and risl adjustment rates. Also, the risk adjustment rate can be attained with varying m.agnitude using financiar datafor dffirent historical periods. The riskfree rate-can be based on short_ term u.s. Treasury Bills or intermediate term u.s. Treasury Notes instead of bonds. Debt-to'equity ratios can be based on current market value of common stock and debt rather than historical book value of outstanding debt and equits,^ securities. Ail of these variations generally cause change in the financial cost of capital results making it obvious thai it is an estimate rather than a precise number. Finally, many people have begun to question the validity of beta to measure the relative risk of returns uu-uiluut". In fact, in a June 1, 1992, edition of Fortune Magazine, Eugene Fama, and Kenneth R' French of rhe university of chicago declared, "beta is bogus,,, and therefore, not capable of telling investors much about investment return potential. Beta was designed to measure a correlation in risk return in the stock of one company, rerative to an industry or the market. It was never designed to account for the risk often associated with project evaluations such as political, environmental, engineering, marketingôr geotogical risk, which is the fbcus of this text. As previously mentioned, for many companies the financial cost of capital is the basis for determining the company's minimum rate of return, i*. However, it is equally important to recognize that in many of these cornpanies; even though they may utilize a discount rate of say, L2.Tvo,projects earning a 12-07a return are rarery accepted. Instead the company imposes a ..h*idle rate" (a higher rneasure of performance, say rg.TEoi foi alr projects. Hurdle rates indirectly recog,ize that the company is not willing to accept projects that are just earning the weighteo aveiage financial cost-of capital. instead the company demands a higher return foithe budget dollars rationed ro rhem for a given period of time (a hurdle rate). unfortunately, hurdie rares are a short cut to recognition of the true opportunity cost foi an investor. Hurdle rates may not properly account for all relevant time varue of money considerations. This can only be consistently achieved by using a discount rate that reflects the investor's perceived long-term opportunity cost of capitar.

chapter 3: Present, Annuar and Future Varue, Rate or Return and Break-even Anarysis

The reader can get more detail on rhe many variations in financiar cost capitai calculations from any good managerial finance textbook.

3'8

Rate of Return and the Revenue Reinvestment

of

eutstion

ls reinvestment cI-project revenue or positirc caslr flow related to the physical nreaning of project compound inreiest raie of return? This is a question ihat iias confused many knorviedgeable investment anarysts for many years. fi'-'itrt'esrniertt of revenue.r is rtot impticitry or explicitry tied to the priysical ntcanirtg o.{ ordinorv raie of r€lurn. If reinvestment of investment revenues u as ded to the meaning

of

orclinary compound interest rate of return then the rneaning ot'conventional bond RoR and i"ro bond RoR would be the "orpon sanre. and rhey are not the same, as the follou,ing example shows.

EXAIUPLE

3-I6

Rate of Return With and Without Revenue Reinvestment Meanin g

Consider either investing $1,000 in a new 30 year zero coupon bond yielding a rate of return of g% compounded annuaily or investrng $i,000 in a new 30 year conventionar bond yierdinj annuar dividencis that give tire investor a rate of return of gx iompounoed annrialiy. what assumptions are expricitry or impriciily invorved in cietermining the bond investment that wourd generate the greatest future value at year G0? Assume annuar bond dividends (i'arhei- than semi-annual) of $g0 per year."onr",itionar caicuiate the bond varues if market interest rates drop to 6.0% or rise to 1a.0% or rs.o"/" within weeks of having purcnaseo the g.0% interest rate bonds

Solution Zero Coupon Bond

c

= gl,0cc

10.063

1

2....30

F = $1,ooo(F/p67,39)

= $10,063

ROR = 8% compounded annually Conventional Bond

C=$1,000

0

l=$80 l=$BO l=$g0 1 2.......30

ROR = 8% compounded annually

L=$'1,000


Both bonds have identical 8% compound interest rates of return. However, only. the zero coupon bond has reinvestment of accrued -;a 'Y 13 lil interest (equivalent to dividends) at the 8% compound rate of return ,.. l; tied to its meaning because the zero coupon bond BOR has growth ll ll ROR meaning as introduced in the next Section 3.9. Whenever one or more investments generate a single future revenue, the meaning of the investment ROR is growth ROR and reinvestment of all accrued interest (return on investment) is tied into the growth ROR meaning. When several revenues are generated by an inveStment as with the conventionai bond, reinvestment of revenues is not tied into the ROR meaning. lf reinvestment of conventional bond dividends at a rate of return of 8% per year was explicitly or implicitly tied to the meaning of the conventional bond 8% ROR, then investing in the conventional bond would give the identical future value as investing in the zero coupon bond. While reinvestment of dividend revenues is guaranteed with the zero coupon bond investment so the year 30 future value is known explicitly at the time the zero coupon bond is purchased, assumptions must be made by the investor concerning the conventional bond dividend reinvestment rate that can be realized from year to year. There is no explicit or implicit reinvestment of the conventional bond dividends at any rate tied to the meaning of the 8% conventional bond rate of return. lf an investor assumes that the conventional bond dividends can be reinvested at i 0% interest per year, then the conventional bond future value is: I

I I

I

164.494 F = $80(F/At oz,go) + $1 ,000 = $14,1 59

lf reinvestment of conventional bond dividends at 6"/" per year is assumed, then: 79.058 F = $80(F/AOZ.,3O) + $1 ,000 = $7,324

These dividend reinvestment assumptions enable an investor to project whether the zero coupon bond investment or the conventional bond investment combined with reinvestment of dividends will generate the maximum year 30 future value. The reader can see that the revenue reinvestment rate assumption has a very significant effect on

chapte;' 3: present, Annuar and Future Vaii.ie. Rate of Feturn and Break_even Anaiysis

iloteniial future varue to be accrued, giving a factor of two difference in tire future values for revenue reinvestment rates of 10.0% versus 6.0o/o. Howe*er, reinvestment of bcnd dividends or project ,rrunrr-Jir";;; wa,v physicaily rerated to the mganing of the ion,cntimitiona ar general investment ordinary rate of retui.

Tne sensitivity of 8.0% interest rate bond varues to changes in rr:arkei interest rates is made assuming the g.0% interest bond was purciiased severar weeks ago, so yearJof rife are un.nung.o. Zero Coupon Bond Sensitivity from g1,000 Value: 6.0% rnterest: year 0 vatue 10,063(p = This is a75.2A increase in value.

irllllr=

$.r,7s2

.

0.0573 10.0% lnterest: year 0 Vatue 10;063 = {p/F1g.7.,g0) = $5Oz This is a 40.57," reduction in value. 0.0151 15.0% lnterest: Year 0 Value j0,063(p /FlSj,e,gd= g.152 = This is a84.B9L reduction in value.

Conventional g.0% tnterest Rate Coupon Bond, Sensitivity from g1,000 Value: 6 0% rnterest: year 0 Vatue =

This is a 2Z.Sohincrease ,,

l,ogglrr?l!l,lr.

,otrzltulrlSol

;rlr'J:'u

0.0573

9.427 i0.0% lnterest: Year 0 Value 1,000(p/F10%,30) g0(p/A10%,SO) = +

This is a 18.9% reducrion

',

;,lu"l'

0.0151 6.566 15.0% lnterest: Year 0 Value 1,000(p/F1S%,gO) gO(p/A-15 = ./"SO) + This is

a

46.0"/"reducrion ,,

:

;,lu.1o

96


Notice that both conventional and zero coupon bond market values are very sensitive to interest rate .changes, but zero coupon mar[gt ,.,.,* values are much mors sensitive ihan conventional bonds.'Due'iii '''E interest rate changes, investors in honds and deQentures can make ..,.+ or lose money just as fast due to market interest rate changes as

investorscanmakeorlosemoneyincommonstockorrealestate

of

investments. This is another way of emphasizing that there is risk losing capital investment value with all types of investments as well as the possibility of increasing investment value.

An evaluation situation that involves partial reinvestment of revenue meaning associated with rate of return is in analyses involving effective interest rates. Calculation of effective interest rates for either discrete or continuous compounding of interest explicitly assumes accrued interest (revenue) will be reinvested at the period interest rate during the effective interest rate period. However, this reinvestment assumption only applies during the effective interest rate period which is usually one year and does not apply between periods. If you make a 10 year life project analysis using annual periods with costs and revenues each yearly period and calculate the project RoR to be z0.0vo compounded annually, this 20.0vo RoR is an effective annual interest rate. Reinvestment of annual revenues from year to year is not implicitly or explicitly tied into the 20.0vo RoR meaning, but during any year money is considered to grow (be reinvested internally) at the minimum rate of return.

EXAMPLE 3-17 Rate of Return Without Revenue Reinvestment Meaning

consider the investment of 9100,000 at time zero to generate five uniform and equal revenues of $37,'185 at each of years one through five with zero salvage at year five. calculate the project rate of return and discuss the meaning of the result.

Solution: Cost =

$100,000

Rev =

$37,185

Rev = $37,185

.

c'apter 3: Present, Annuar and Future varue, Rate of Return and Break-even Anarysis

PlV Eq: 0 = -100,000 + g7,185(p/{,5) Trial and Error, i = ROR 2i.0o/o =

'f're ratu'of

on rhis invcstment is 25ao.

L

ni,rl,rr,rutenr of year r,ytnr 25vo rate of

ot:e ti;roughJit'e'erurn re,enues at 2svo tied into trte rneattirtg

reittrit? l-lnt is rrte crassical rate of return r€venue reitn,estntent quesrion. 'l!utt-; rtnolysts ruu'e said yes, butitre ans,*br is ,o. Reinvestment of rev_ ettues i.t rtot tied to the meaning of any ordinary compounrd-i-tlterest rate of rerLtrn and that answer can and wilr be exp'rainei in severar dffirent rrrr,i'.s. First, an intuitive approach is used. Consider the g100,000 inyest_ ment in Example 3-17 to be a 10an that would be paid off by n"" rrorrgage payments of $37,1g5 at years "qr"r one through five, the compound nrL\rrsage interest rare received by the lender una"friJ oy tr," borrower r',.uld be25To.Intuitivery it is knt*,n that the *.uiing #,n. 257o com_ p.unc interest rate, from the r,iewpoint of either tne uoiowe. or iender, is unaffected by rvhat the render does with the mortgage fuy,irr"nr, as they are received. The economic rverfare of the lende. *iit-""riui,rry be affected by uhat the lender does with_the mortgage

payments as they are received, but the physicar nieaning of the 25vo rite of return is not af cted by what is done rvith the revenue each year. A second int,itive expla,a.tion can be made of u'hy rein'estment of revenues is not related ,o',ir. meaning of .rdinary rate of return by.thinking of project investments as-being anaro_ go,s to deposits in a bank u..ornt. Then consider that the bank interest rate equars the project RoR with project revenues being equivalent to withdrawals from

the bank acconnt .r.t year with the last withdrawal reducing the account barance to zero. Any reasonable person knows that u'hat an investor does with money taken from a bank u..orn, each year has no effect on rhe pLrysicar meaning of the bank accouni .o*pouno interest rate of return. Simirar rationareipplies to ,o-pourrJ interest pro_ ject rates of return in general.

Looking at the cumulative cash position diagram illustration of project ROR, this is a more explicit ,uy oi showing that compound interest rates of return in general, .ilut" to remaining unamortized investment each period and have nothing to do with reinvestment of revenue at any rate of return' For this example the cumurative cash position diagran, is shown in

Figure 3-7.

98


-100,000 +37,185 +37,185 +37,185 +37,185 +37,185

CF

'5

CUMULATIVE

-37,185

CASH POSITION

-90,730

-100,000

-109,769 -125,000

Figure 3-7 cumulative cash position Diagram lor z5

/o

RoR project

It can be seen by analyzing Figure 3-7 that the 25vo RoR applies to remaining investment value to be paid off (amortized) each year and that interest from reinvestment of revenues at any rate of return is not related to the diagram in any way. only if an investor is interested in calculating the rate of growth of investment dollars does reinvestment of revenues affect the calculations and meaning of rate of return results. This relates to the concept of grorvth rate of return that is introrjuced and illustrated in the next section. 3.9 Growth

Rate of Return

Growth rate of return is the compound interest rate at which investment dollars grow. All compound interest rates of return, however, are not growth rates of return. To determine the compound interest rate at which iuvestment

dollars will grow requires making an assumption concerning the rate of return that will be received from the reinvestment of project revenues (or positive cash flow) as they are received. The minimum rare of retLrrn (denoted as "i*") reflects other opportunities that existfor the inttestment of capital nov, and in the future, so the minimum rate of return is th.e reinvestnlent rate thctt slrculd be u.sed in growth rate ctf return calculations. Note that you do not assume that you will be able to reinvest initial project revenues at a rate of return equal to initial project RoR. you assume ihut you. reinvestment rate represents other opportunities for the investment of capital thought to exist now and in tlie future over the project life. That is exaCtly what the minimum rate of return represents as it relates to all discounted cash flow analysis calculations and the proper application of discounted cash flow analysis results for economic decision purposes. with some analysis techniques such as NPY the meaning of minimum rate of return just described is

chapter 3: present, Annuar and Future varue, Rate of Return anc Break-even Anarysis

irnplicitly built into the calculations, bLrt ri.ith growth rate of return the mini_ murri rare of return revenue reinvestment meaning is explicitly built into the analvsis calculations. To illustrate specific growth rate of rcturn calculations, Eryrmple 3_17a con_ 'itiers an extension of $100,000 investmenr case study used in Example 3_r7 ar the basis for discussing rate of return and the revenul reinvestment question.

EXAMPLE 3-17a Growth Rate of Return Analysis Consider the investment of $100,000 at time zero to generate five uniform and equal revenues of $7;1gs at each of yeais"one through five' with zero salvage value at yeaitive. The rate of return on this investment was shown. in the pruri*. Exampre 3-17 to be 25/" and was given in severar oitrerent ways to emphasize that the 1:-:T-::r rernvestment of the $37,1g5 revenue each yeir nu. n,iining to do with the meaning of the 25% RoR. Ho*.r"r,lnJllz'?rrriio". no, repre_ sent the rate of growth of investment doilars (Growth RoR). rc carcu_ late .Growth RoR, reinvestment or i*u"nr". as they are received must be tied into the anarysis carcurations.-Assume that the investor minirnum ROR is 120/" over the five year project rife which means other opportunities for investing capital 'ar a 12yo RoB are thougnt to exist both now and in the futurd over the next five years. carcurate the investment GroMh RoR and deverop the cumurative cash position diagram showing its meaning.

Solution: C = Cost, R = Revenue, F Future Terminal Value =

lnitial Project C = $100,000 with ROR = ZSo/o llolo

|

1_

R = $37,185

........5

C=$37,185..C=$37,1g5

Revenue Reinvest Qt i* =

R = $37,.1gS.

m

6.353 where Future Terminal Value, F = $37, 1gS(F/\2%,5)

F=$236,236

100


Total C = $100,000 lnitial+Reinvest

0'

't

........5

F=$236,236,.

PW Eq: -i00,000 + 236,236(p/F1,5) =0 Irial and Error, i= GroMh ROR 1g.26% =

:i,:,::::J:,:3,i:l1r

36vo is the compound interesr RoR on rhe com_

:iil:)"ii,,i,"#_ :,Ti*"ili:,lH,illlr"::".""t".;;";",,;;;fi;#,-il;:J';:: j,1; ;T:JX':,::,:"i:ll.:1'"::::::-:1""::yll;ffi

t# 33 :::- ::":Y "'; ;'";;; ;, ;ffifi;' :':','iil ff#; li H:Tl-i::."""1":i,gr:1y*"!,",;;;Hi.#;r:IJ[HI t or i1:::

the meaning of

projeclRoR

costs,

i C;;;;6i.

-100,000

+236,236 E

CUMULATIVE CASH POSITION

-100,000 -1 18,760 -141 ,039

-167,498 -198,921 _236,236

Figure 3-g cumurative cash position ror 1g.76"hGrowth RoR The Growth RoR of 1g.76vo rerates to larger

and rarger unamortized investment values each year whereas ,rr" rni,iur project RoR of 25vo rerates to smaller

and smaller unamortized investment varues each year as was illustrared in Figure 3-i rn n*ampre :-i;;. oR of z5vo and the Growth RoR of 6vo not onty airtt. i, ,ugniiri. tri ,t .1g.7 ,.tu," to completely different investment varues each year after the first"y year. A question that often oc:urs to peopre at this time is, ,.why isn,t the Growth RoR bigger than the initiar piojec, non ,in." future varue from reinvest_ ment of revenues is being aoo"a to the iniiiar project?,, The answer rerates to the fact that reinvestment costs as welr as the revenue reinvestment future value are added to the initiar project ,o g.i c.orth RoR, and the reinvest_

il;i;t,,"ir.":"ii',i

chapier 3: Present. /rnnual and Future Varue, Rate rf Return anc Break-even

Anarysis

10.1

meri costs and future value correspond to a 12(/c project RoR. Adding the I2Vo project to the initial prolect which has a 25Vo :.I:"i:^t:inv.eTm:nr t:tJK. lhe combinarion of the two pro.iects will hli,",e a weighteO average rate ut rerurn berween 12vo and z|vo.rn ihir.ur. ri...,Jgrri,i ri.".ug" RoR ,

i.7i,'i.

tite

projr;i

is

Gro,.r.ril R.OR.

A tn;rjtlrity of investiirs do not utiiize Growth RoR for evaluating the ecoit.)iriu potential of iiivestntettt projccts. Horvever, it r,,,i11 be shown in Chapi:-r 4 iiret there are fu'c evaluation situations where if you \4,ant to make rate i::'rctuin anarysis of r'iojects, u.sing Growth RoR is either necessary cr a i;'sirable alternatire tc ordinary RbR analysis. Remember, however, that ri:ere iire no evaluation situations where it ii necessary to usl rate of return rnaly'sis' lbu aiu'avs have the alternative choice of using pr"r"nt, annual or li.iture v'alue, or brcak-even analysis to analvze th" potential of ciiirer incorle ""Jntmic or serl-ice_producing alternatives.

A scc',d crowth hoR exarnple w,l nou, be presented to re-emphasize c:-owth RoR conceprs and actditional related considerations.

EXAMPLE 3-1g Growth Rate of Return with Murtiple costs Consider the time zero initial investment of $SS,0O0 and a year one investrnent of $.ri5,000 in project ,,A,, gener.ate incomes of $30'000 per year foi'years two thiough ten with zero salvage. Deteri-nrne project Rofi. Then assume th;t each year incornes are rein_ vested in other investment opportunities yierding a 12k non. Refer to revenue reinvestment at a 12% RoR ai proleit ,,8,i Determine the overall growth rate on the time zero and'yei, one investme,ts by combining the initiar and revenue reinvestrnent projects ,,A,,and ,,8,,.

Solution: All Values Are tn Dollars A)

C=55,000

PW Eq: 0 = -55,000 i = 25"/o: -55,000

C=45,000

I = 30,000

2

-

I = 30,000

.........10

4S,000(p/F;,1) + 30,000(p/A;,9)(p/F1,1)

- 45,000(.g000) + 30,000(3.463X.8000) = _7,800 i = 20"/": -55,000 4S,OO0(.8333) + 30,000(4.031)(.8938) = +8,272 i = 0.20 + 0.05[(8,2 7 2 O) / (8,27 2 + t,BO))J = 0.226.o r 22.6"/" -

102


Without interpolation error, the ROR is 22.82% No reinvestment assumption or requirement is associated with the meaning of the initial project rate of return, Reinvestment of revenues only relatds t6 growth rate of return calculations. Reinvestment otrevenues al lzyo is project "B".

C=30,000 C=30,000 B)

F

2 .........10

0

= 443,280

14.776 where Future Reinvestment Value, F = 30,000(F/A12y",g) = 443,280 The overall groMh rate over ten years on the initial project "A" time zero and year one investments of $55,000 and 945,000 respectively can be found by combining the cost and revenue numbers on the project "A" and "B'time diagrams by adding them together. Combining project "A" with a 22.37"/o ROR and project "B" with a 12/o ROR must give an ROR between 12"/" and22.37% on the combination of the projects. A+B)

C=55,000

C=45,000

2..

..

10

F

= 443,280

PW Equation: 0 = -55,OOO -45,000(P/Fi,1)+ 44g,2BO(plFi,1O) By trial and error:

i=

=-55,000 -45,000(.8696) + 443,280(.2472) =+15,442 i=20"/" = -55,000 -45,000(.8333) + 44g,2$O(.1ti1S) =-20,909 therefore, i = 15"/" + 5%[(1 5,447 - 0)/(1 5,442 + 20,909)] = 12.1"/o The rate of return on the combined projects "A" and "8" is 17.1"/". This 17.1% ROR is a compound int-.rest rate of return, but it is a special compound interest RoR that represents rate of growth of 15"/"

investment dollars. whenever a single revenue (such as $44s,29g at year 10) is generated by costs at an earlier point in time, the project ROR is Growth ROR.

A variation of GroMh ROR is to bring any net costs or negative cash flows incurred in future periods back to time zero at the minimum ROR, instead of at the unknown project ROR, "i". This gives

chapter 3: Present, Annual and Future Value, Rate of Return and Break-even Analysis

103

Growth RoR on the period zero equivalent present worth investment cost instead of Growth RoR on the costs from the points in time where they are actually incurred. This modified present worth cost Growth ROR calculation for Example 3-19

follows:

t

0.8929 Modified PW Eq: 0 = -55,000

-

45,000(p/Fi*=12/o,t) + 44Z,2gO(p/F;,16)

or 0 = -95,1 80 + 443,280(p/F;,1 g) By trial and error, i = Growth ROR

=

16.2"/"

This Growth RoR is slighfly different (lower) in magnitude from the 171% result because it is based on different (bigger) initial investment value due to the modification in the handling of the year one cost. This result is equivalent to the non-modified 17.17o result and therefore is equally valid. Although both types of results are useful, in some analysis situations, it will be introduced later in chapter -4 that

in using Growth RCR to rank independent income-producing alternaiives, it is necessary to use Growth RoR results based on the maxinrum caprial exposure in a project. This is not always the present worth of all negative cash flows but is described in more detail begin_ ning in Example 3-21 and Section 3.11 of this chapter. Finally, this modification of Growth RoR is similar to the procedures employed in some spreadsheet models. The Excel function known as Modified lnternal Rate of Return (MIRR) is always determined from a single investment at time zero growing to a single future value at the end of a project. 3.10 Net Present Value, Net Annual Value and Net Future Value Methods of Analysis

It will have occurred to many readers before now that to determine if a given project is satisfactory by usin,E rate of return analysis, it is not neces_ sary to go through the trial and error calculations required to obtain the pro_ ject RoR. Depending on the type of equation you have written, comparison of present, annual or future revenues and costs calculated at the minimum rate of return, "i*", tells you if there is lnore or less than enough revenue to cover costs at the minimum rate of return.

't04


Net Present Value (NPV)

=

Present Worth Revenues gr Savings @ - Present Worth Costs @ i*

or = Net Present Worth Positive Net Annual Value

=

(NAV)

i*

€

and Negative Cash Flow @

i*

,

Equivalent Annual Revenues or Savings @ i* - Equivalent Annual Costs @ i*

or = Net Equivalent Annual Positive and Negative

Cash Flow @

i*

Net Future Value (NFV)

=

Future Worth Revenues or Savings @ - Future Worth Costs @ i*

or = Net Future Worth Positive

i*

and Negative Cash

Flow @ i*

A positive net value indicates a satisfactory investment relative to investing clsewhere at the minimum RoR, i*. To illustrate the application of Npv, NAV and NFV to evaluate projects consider the following example.

EXAMPLE 3-19 Net Present, Annuat and Future Value Analysis The time zero initial investment of $55,000 and a year one investment of $45,000 in project "A" generate incomes of g30,000 per year for years two through ten with zero salvage. For a 12.0"k minimum discount rate, determine investment net value on a present, annual and future basis. Then analyze the effect on net value from including reinvestment of revenues at lhe 12.0"/o minimum discount rate in the net value analyses. Refer to revenue reinvestment at 12.0% as project "8."

Solution: All Values in Dollars A)

C = 55,000

C = 45,000

I = 30,000

I = 30,000

.2 ..........10

Chapter 3: Present, Annual and Future Value, Rate of Return and Break-eiren

0.8929

NPV4 = -55,000- 4S,000(p/F 12,i + = +$47,541 > 0, ok

0.17698

Analvsis

5.328 OO,OO0(P/A

0.17698

10S

0.8929

p,9)pF.rr,-;,, ?

NAVA = NPVR(&P12,19) = 47,i41(A/pi2,1g) +$g,414 > 0, ok =

3.106

3.106

NFV4 = NPVA(FIP12,1g) = 42,541(Flp12,19) +$1 42,662 > 0, ok =

or

3.106 = -55,000(F/P1 ZjO)

= +$147,665

2.773 -

14.776

4S,OOO(F/P12,9) + 30,000(F/A12,9)

B) Net Value based on Reinvestment of Bevenues at 12.0%

0

1

F=443,280 14.776

where F = 30,000

(F I

A12,$ = 44B,2gO

4.32197

5.3283 0.89286 30,000(piA12,9)(p iF1Z,1) = $O This indicates a break-even with investing elsewhere at i- 1z.o"h = NPVg = 443,280(P/F12,19)

0.1 7698

NAVB = NPVg(tuP 12,10) = (0)(tup12,10) = gC

Tlris indicates a break-even with investing ersewhere at i* 12.0yo. = . 14.776 NFV3 = 443,280- 3O,O0O(F/A12,9) $0 = This indicates a break-even with investing elsewhere at i* 12.0o/o.

=

A+B)

C=55,000 C=45,000 F=443,280

2_10

NPV4*3 = NPVA + NPV3 = +47,541 + 0 = +$47,541

0.32197

ot = 443,280(P/F1 = +$47,544 >

O

2. 1 O)

-

0.89286

4S,OOO(P/F12,1)

_55,000

106


The $3 difference in this NPV result and the initial project NpV is due to factor round-off error: :

NAV4*g = NAVA + NAV3 = +8,414+ 0 = +$8,414

or

'i 0.17698 0.17698 = NPVA+S(A/P12,1O) = +47,544(NP12,19) = +$8,414

NFV4ag = NFVA + NFVB = +147,665+ 0 = +$1 47,665

ot = Qy',J,280 = +$1

_-,45,000

47,665

(F lP

p,g) -

55,000(

F lP

p

1 g)

Reinvestment of project revenues at the minimum discount rate is implicitly built into net value analysis of projects, but since it has no effect on the final net value results (because investments earning at the minimum discount rate have zero net value), there is no need to explicitly go through the revenue reinvestment calculations as you must do for Growth BOR analysis. The absence of the need to make trial and error calculations, as is required

for rate of retum analysis, is a major advantage of net value analysis. The time value of money is accounted for properly and analogously by both rate of return and net value analysis. The meaning of the minimum ROR, "i*", is identical for both rate of return and net value analysis as well as all other valid discounted cash flow methods of analysis. Net value analysis provides a quick, easy and therefore very useful way of screening mutually exclusive aiternatives to determine which is best as discussed in detail in Chapter 4. Finally, remem.ber that a project earning a rate of return equal to the mininrunr ROR, "i'", has a zero net value, so projects with a positive net value are better than investing money elsewhere at "i*". An imporfant consideration to be aware of when applying NAV or NFV to analyze unequal life income-producing altematives is that a corrrmon evaluation life must be used for all altematives. Normally the li{'e of the longest life alternative is selected but any corrrmon life may be used. If unequal lives are used for different alternatives, the time verlue of money considerations are different in annual value and future value calculations and the wrong altemative may appear as being best. This means that NFV must be calculated at the same future point in time for all alternatives, or NAV must be calculated by spreading costs and revenues over the sarne number of years for all altematives. Analysis related to comparison of unequal life income-producing altematives is presented in Chapter 4.

chapi'-'r 3: Present, Annuai and Future Vaiue, Rate of Return and Break-even Anarvsis

107

EXAMPLE g-20 NpV Apptied to project Vatuation Determine the maximum cost that an investor can incur to acquire lanc for an industriar deveropment righis'tor-[etroteum or foiminerai mining deveropment) and rearize a 12.0"/" rate 6f hturn'on invested ooiiii:-s' rf the ranc i: deveropmer':r is exper-1ed to occur two "g!yirjo" ;ears iater for a cost of $1,700,000. Developnient is projected to gen_ erare riet revenues after operating costs (beicre-tax $Jii;; cash frow) of $2t0,000 at year three, g225,0"00 year at fcur, $250,000 at year five, with a $2,000,000 sare varue arso expected to be rearized at year five.

Solution, All Values in Thousands of Doltars:

AcqCost=? C=1,700

l=200

l=225

| = 2,250 5

on a before-tax anarysis basis, positive NpV represents additionar ccsi tiiat can be ir:curred at the point in time y;here NpV has been calculated and give

the investor the minimum rate of reiurn on invest_ ment capitar. lt wifi be iilustrated rater in the text that on ,n after-tax basis, I'JPV represents additionar after-tax cost (which is negative cash flow) that can be incurred and give the investoi the after_tax minimum rate of return on invested cap,tat. ceneratty,'the tax savings from tire tax deductions reraied tc adJitionar costs the break_ even acquisition cost to be bigger than "rur", after-tax NpV. NPV at the 12.0"/" minimunr rate of return equals:

0.7972

-

1,

700 ( P/F

0.7118 0.6355 0.5674 e%,2) + 2OO(p / F p/o S) + zzs(p / F e:;,1 + Z,2SO{p /F p1,s)

= +$206.758

$206,758 is the break-even time zero acquisition cost that courd be incurred and sti, project glnerate a 12.0"/"rate of return, that is, to have NpV @ 12.0%equalfi r"ro as follows:

Trlr!: 0.7972

0.7118

0.6355 1,700(P/F12/",2) + 200(p/F1 + 225(p/F12/",4) 2o/o1) 0.5674 +2,250(P/F12%,5) $O =

-206.7

-

108


Net Present value (NPV) may also be expressed in graphical form by plotting the.sum,gf.the project's discounted cash flow for each co*pornding period, (usiially years). In this diagram. the vertical y axis ,.p..r.n,, the project cumulative NPV over time, the horizontal $ axis reprisenting project life. The final data point in this diagram is the overall project Npv. Generally referred to as cumulative Npv, this graphical illustration can be used to illustrate not only NPY but other evaluation criteria including discounted payback and maximum capital exposure which is useful in ratios. Discounted payback or payout is defined as the point in time where the cumulttti,-e discounted cash flows for the project equal zero, or the point on tlte cumulative NPV diagram where NpV equats zero. The shorter the payback, the less capital exposure to the investor in terms of the.time that is required to recover his or her investment. However, discounted payback is

not always an accurate tool for assessing the economic profitability of investments, as is discussed in Chapter 9.

The final consideration relevant to the cumulative Npv dibgram involves the development of ratios. Much confusion exists in practice in determina-

tion of the appropriate denominator to use in ratios. This subject is addressed in detail in the next section but is easily illustrated using the lorvest point on the cumulative NPV diagram. This lowest point is sometimes defirred as project maxirnum capital exposure or maximum capital at risk.

Using the cumulative NPV diagram makes it easier to determine the cash flows that need to be included in the denominator in order to develop ratios that rvill yield economicaily consistent results.

The following example uses the cumulative Npv diagram to make a comparison of RoR and NPV. Note that both criteria are developed from the same present worth equation but offer different approaches to assessing the profitability of an investment.

EXAMPLE 3-21 ROR, NpV and Their Graphicat Meaning

A five-year project requires investments of 9.120,000 at time zero and $70,000 at the end of year one to generate revenues of $100,000 at the end of each of years two through five. The investor,s minimum rate of return is 15.0%. calculate the project RoR and develop the cumulative cash position Diagram to illustrate the meaning of the RoR result. Also, calculate the Npv and develop a

chapter 3: Present, Annuar and Future Varue, Rate ci Return and Break-even Anarvsis

109

cumurative NpV diagram to show the net present varue of the project uver trme..Using the cumurative Npv diagram indicate the project disccunted payback and maximum capitat or"* un ivPv Profile to show how the varue of the "-6;il;;.;nit,y, proleci iiirpr.t"o by the seie:ied discouni r-al*.

$olution:

-$i20,000 -970,000 $100,000 $100,000 $100,000 $100,000 Flate

of Fteturn (ROB)

PW Eq: 0 = -120,000 70,000(p/Fi,1) + 100,000(p/A i,flelF1,l @ 25% = 12,928 @ 30"h.= -7,212 i = 2byo + S%(1Z,g2g/ZO,14A) 2g.2y.

=

By calculator or spreadsheet, i 2g.1 = "h. The rate of return exceeds the minimum of 15.0% and suggests that the project is economicaily satisfactory. The "curnutative cash position riiagrrm"-iilustrates the meaning of the project rate of return. The slope alsociateo with each compounding period is a function of the projeci rate of return at 2g.1%. Cash Flow

100,000 ; : -2,., t;;;, P

100,000

;,,,:,..::llQl,il!t:

rojec-t,Lif e, (Ye.a'is)

l. .

-100,000 :1531718

-178,06s -239,006 -286,580

Figure 3-9 Cumulative Cash position Diagram; ROR, i =

.l

-a--

2g.1o/o

110


Net Present Value (NPV) @ i*

-

lio/o

0.8696 -

-120,000

70,000(P lF 1So/",1) + 100,000

2.8550 0.8696

(P/\SV,yXPIF1

:

So/o,1)

= $67,389 To generate the necessary data points for the Cumulative NpV Diagram, an alternative approach is presented that yields the same NpV.

NPV After Time

0

O

- 120,000(P/F1 Sy",O)= -120,000

1 -120,000 - 70,000(P/F 15"/",1) = -180,870 NPV After Year 2 -180,870 + 100,000(P1F15"1,,2) = -105,255 NPV After Year 3 -105,255 + 100,000(P/F1g"yo,g\ = -39,504 NPV After Year

4 NPV After Year 5 NPV After Year

-39,504 + 100,000(PlF 15"7o,4) 17,672 + 100,000(P/F15"7o,g)

= =

12,622 O7,gBg

This calculation demonstrates the impact on NpV of one additional year of project cash flow. Note that 967,989 is the time zero project NPV at 15.jYo which is greater than 90, indicating acceptable economics compared to investing elsewhere. The yearly measure of Npv represents the data points for the cumulative NPV diagram as shown. 100,000 6)

6

50,000

Discounted Payback, 3.7:Years

c

o a (,

0

(L I

o

-50,000

o (U

3

-100,000

E

o

-150,000 ,l'l

-200,000

l,.f

Project Year

Figure 3-10 Cumulative NPV diagram

Ll;rspter 3: Present. Annuar

ai^rd

Future Varue, Hate of Serurn and Break-even Anarysis

111

Using the cumurati,re r.Jpv Diagram, Discounted paybackoccurs when the cumulative discounted cish frows sum to zero, vi"r"rrv, ni, can be seen as occurring during year four. specificaily,-in. o"Li'"rry be identified as

follows:

r

0.57''8 3 Years of Cash Flow + (99,504/100,000 (piF15"L,4)) = 3"2 years lf t.e cash frows were considered discrete end-of-period varues, the period r"ouid be rounded up and descrioed u, i.o-yuar payback. Discounted payback is a measure of financiar risk " cimmonty com_ puted with other economic measures such as rate of return and net Fresen! value. ln this case, it measures the time required r", tni ii'_ counted cash frows to sum to zero. see section g.2 in chapter o tor more detairs on payback and the different ways it might be determined.

j

Unfortiinately payback is an inconsisteri *errlr€ of ,,sssnsrnjs prcfitability" becaLtse it teils us nothing about a project,s benefits or costs (cash flow) once the payback is achieved. lvlaximum capitat Expo2ure is the present worth measure an investor has "exposed', in the project. lt is the capital of capitar an investor wouid need to have in the bank today, avairable to invest direcily irlr" project, or in other opportunities at i* to be able to cover immediate and downstream negative cash frows in a project not offset by positive cash flor,v in earlier years. Maximum capital-exposure is always the lovyest point, or most negative ,raiue on the Curnulative NpV Diagram. This is always the appropriate monetary measlr'e when considering tiic denominator for any ratio, discusseciin the next section. Finally, by carcuiating NpV for a range of discount rates the tionship between Npv and-RoR may be graphicaily iilustrated rera_ as for_ lows. substituting values of i from 06/. to qoin inrc the present worth equation, the NpV is computed in 5% increments and graphed as shown. Neglecting tim: value of money (i"=0yo), the value oi this prolect is $210,000, but as investors associate more time varue with their money, the project value is diminished. Note that this retationsrrip-is not linear as is assumed when interpolating for the rate of return. The compound interest rate, i, that makes the project NpV equal zero is the project rate of return. Deveroping an NpV profire is a commofl graphical sensitivity to address unceriainty concerning the true magnitude of the opportunity cost of capitar and the impaci it may have in an investor's economic decisions.

112


it :

:.:tB

;:

o

3

roo,ooo

o

a o

5o,ooo

2

. .',."I

25?:tF

r.. : .1t:it,t r'

'. .:J;r,r".f

a:.

,,

-,

; e' .: ., . :. i;l .lft-
Discount Rate,

I

I'

Figure 3-11 NpV profile

3.ll

Benefit-Cost Ratio and present Value Ratio

Instead of looking at the difference in present worth positive and negative cash flow to analyze projects with net present value calculation reiults, some,investors prefer to look at the ratio of present worth net positive cash flows to the present worth of net negative cash flows. This ratio, commonry called Benefit Cost Ratio (B/C Ratio) is defined as follows:

Benefit Cost Ratio =

Present Worth Positive Cash Flow @

lPresentwo@

i*

A B/c Ratio greater than 1.0 iiidicates satisfactory project economics, a B/c Ratio equal to 1.0 indicates break-even economics with investing elsewhere at an RoR equal to the minimunr RoR, ..i*,,, and a B/c Ratio less than 1.0 indicates unsatisfactory project economics compared to invest_

ing elsewhere a[ "i*". B/C Ratio is also sometimes ref'erred tô profitabilas iq' Index, or; "P1," or Discounted profitctbility Index, ,,Dp\.,, The numerator, "Present Worth positive Cash Flow @ ix,,, is the present worth of positive cash flow not required to pay off future negative cash flows. The denominator, "present wortr\ Negaitve cash Frow @ i*,,, represents the absolute or positive value of present worth negative cash

chapter 3: present, Annuar and Future Vaiue, Rate of Feturn and Break-even

Anarysis

113

flou' tlrut /r,'c not be,en paict rbr b1, prsitive .ash J.row generated in earrier prolect yeors' The denominoi'to, ii ,rltrctive of thor,, ,h ii*s that cause Ttroject Ctntulative NpV to ,"r,rn rti'irirrest point n, ,,moii,nunt cctpital at risk" in the project. r\{aximum cr;it;i l-11' "fiiis is tire r:roney rrn inlestor ar Risk u,as intr6.trriced in Eranrpre wrii:ld ha'e to set arile today, (rime ':li-o) at the minirnurrr rate crr return in orcer to cover att or tiie-negati'e cash Il.w rcllizcd in the project that ir no, p-i.cted to be cor,.ereci b1, project positirc c,sh trows in earrier 1'cars. Do*,nstream cosis that are coverrrd ()r I-';lid oJ'f by positir'e cash flow in eariier years do not impact the ratio rlcnominator because from an analysis viervpoint, no additionar investrncnt needs to be made ".ono.ra uv,rr"-i"".stor. For example, down stream e'xpansion, abandonment or reclamation costs that are otrset, or can be paid ofi either by revenues in rhe .vear ,n;;;; incuned or by preceding positi'e ia-sh-tlorv do not irnpact ratio denomiruiorr. often times. management wants to knorv the value of this ratio basecl on ['udgetary capirar evpendijur.er i* , p;;j;;r. such a rrtio may be con.sidered uselul in assessing frnanciit.bu,rg",*y"i..rorrnance but not necessarilv economic prorrtabiriry. This is irusiatei in-Crrapter 4 when rarios are ut,iz,.d to rank a'ailable non-nrutuaily excrusivc in'estment opportunities. From ir, econonric ,ier'r'point. a ilollar ipe,t is an oufflor-,. of available funcls, whetiicr tirose dolrars are representati\c of operating expenses or capitar outrays. The positi'c or negative cash flou'g"n..ri.,r 'e. in each e'aruation period rcpre_ ror arl ,,,ary,", Lrsing eirher ratios. ner presenr Ii:::hTl::l::li::,0",,,rs

A r"ari.rio, of B/c Ratio ofter: useci in inclustry practice is pr.esent varue Raiio (P!'R) r',,hich is definecl as: Presenl Value Ratio =

Net Present Value @

i*

lP."sert

A PVR greater than zero inclicates satisfactor-y pro.ject economics, pVR a equal to zero indicates break-e'en ..onu,rri", *,ith in'esting elsewhere at an RoR of "i'r"', a.cr a pvR le.ss than r".o'i,r,ri.ures unsatistactory project eco_ ttomics co.'pared to investing ..ix,,. Sotne companies "rr.*rr".. ui the rninimum rate of return refer ,o

,ii, .rrio o, irtort*"rt

Efficiertct,;accuratery reflecting the meaning of rhe calcur"ri;; ;; pvR is u *Luru.. of th" pr"r"rt worth profit (Npv) being generur"a p., p..r"r, worth investment doilar as describecl earrier. It is arsolwonr, ,otilg ["." tt,ut the denominators of both B/C Ratio and pVR are the same.

L

114


Illustrating the mathematical relationship between pvR and B/c Ratio: NPV @ i8 PVR -

lPresentwo@

or. Ietting PW = present Worth and CF = Cash FIow,

PVR

IPW Negative CF @ i*

|

or,

PVR =

PW+CF@i* lFw - cF @l{

lpw-cF @ ixl - lP-wlcF @ t.T

or,

P\/R=B/C Ratio- 1 or, pVR+ 1 =B/C Ratio why the break-even criteria of 1.0 for Bic Rario and zero for ^.}':.:lo,ains PVR differ by unity. Since most investors carculate Npv for projects being analyzed economicany, it is easy to get pvR by dividing Npv by the abso_ lute value (positive yury"l of thl prJsent worth net negative cash flow, not offset by positive cash flow in earlier years. If the investor wants to carculate B/C Ratio, adding one to PVR quickly gives the desired B/C Ratio result. The utility of ratios will be described in detail in chaprer 4. However, the principal application is to assist investors in ranking alternatives when more opportunities exist than- capital budget dollars. Since ratios are used to allo_ cate this limited capital,.many p"opt" want to measure the Npv generated per capital budget dollar invested. Financially, it may be of interest, but eco_ nonrically, all costs represent outflows of fu.ds and there is no relevance to distinguishing capitar and. operating costs in a before-tax evaluation. Froln an economic viewpoint. the focus should be on the project,s negative cash flow and resulting max]11m capital .^poru." as ilrustrated in the four cases

that make up Example 3_22.

L:r,apter 3: Present, Annuar and Future Varue, Rate cf Return anc Break-even Anarysis

115

EXAMPLE 3-22 carcuration of correct pvR and B/c Ratios To illustrate correct ratio denominator calculations for both pVR and B1c Ratio, consider the following four cases wl-ich .or"irri t-n', rrifereni situations that occur in determining correct ratio denomina_ tcrs. calculate the proper denominator, pVR and B/c Ratio for each case, given a minimum rate of return of 15"/". Case

1

C=100 Rev=59

Rev=50

2

50 Rev - 50 3.......10

Rev =

Correct Ratio Denominator = 100 5.019 BiC Ratio = SO(p/At 5%,10)fi00 = 2.5045 > 1, so satisfactory 5.019 PVR = [50(P/A1S%,10) - 100y100 = 1 .5045 > 0, so satisfactory Note, PVR + 1 = B,,C Ratio for allthe cases.

EL tt'v nn

z

o)

!50 ) o=0 -50 -

100

project Life (years) Maximum Capital at Risk

Figure 3-12 Case 1, Cumulative NpV Diagram @ 1S% Figure 3-12 shows that the maximum capitar exposure is rearized at time zero with the initiat investment in this prolebi.

"8i-

116


Case 2

C=100

C=90 Rev=50

Rev =

50

Rev

='50

Rev =

50

correct Ratio Denominator = 100 + (g0-50)(plF15"/o,1)

4.772 .8696 .8696 B/C Ratio = 50(P/At g"1",g)(PlF1S%,i11100 + 40(p/FtSZ,t)l = 1.5394 > 1, so satisfactory 4.772 .8696 PVR = [50(P/A157",9)(P/F1 So/",i

-

.8696 a}Q,Flyo/o,1)

.8696

- 100il100 + 40(P/FrsZ,r)l = 0.5394 > 0, so satisfactory 100

zo. ou (E

E E o=

-so

-1

00

-1

50

Project Life (Years) Maximum Capital at Risk

Figure 3-13 Case 2, Cumulative NpV Diagram @ Note that the denominator is:

l-t 00 -

15"/o

0.8696 4O(p/F 115/",1) | = 1 35

It doesn't make any difference whether the year one net negative cash flow (-40) is derived from a cost (-g0) and income (s0), or if its just an investment of (-a0). The same cumulative NpV position of -135 is realized either way.

,.:$

chapter 3: Present, Annuar and Future varue. Rate of Return and Break_even

_ C=100 Rev=59 Case 3

C=190 - 50

'

Rev

2

0

Rev --

Anarysis

S0

+.

112

Fiev = 56

....10

Cori'ect Ratio Denominator = .100 + [(190-50)(p/F15.y",1_ 50Xp/F1 S"/o,1)

f;,'C Raiio:

4.487 .7s61

.8695 .8696 _ SO](P/F1 + Ap/FtS,/",1) S%,2)/t100 [1 S%J)] = 1.444 > 1, so marginally satisfactory = SO(PiAtS%,8)(Pirt

PVR:

5.019 = [50(P/A1

5,

1

0)

-

.7561 i 90(p tF t 5,2)- -r 0oy{1 00 +

.8696 fi a}p/i1sJ

= 0.044 > 0, so marginally sati.sfactorv

.8696

_ ) s0](p/F1 s j )}

20 7

0

10

-20 -40

zoo

-60

=

-'100

-80 E f -120

o

-140 -1

60

-1

80

Project Life (years)

\

Maximum Capitai at Risk

Figure 3-14 Case 3, Cumulative NpV Diagram at Note that the denominator is:

0.8696

l-roo - (140(p/F

1so/o,1) + so)(

0.8696

p/;;;;,1)l

= 162

1S"k

118


Case 4

C=100

01

C=90 Rev = 59

Rev = 59

Correct Ratio Denominator = 100

Rev

-

50

Rev

- 50

3.......10 ;

4.487 .8696 B/C Ratio = S0(PiAt S,gXp/Ft S,e) - 40(p/Ft S,e; + SOle/15,1 ;

.7561

.7561

100 1.829 > 1, so satisfactory =

4.487

.7561

PVR = 50(P/At S,eXP/Ft S,Z)

-

40

.7561 .8696 (p/F 1g,2) + S0(p/F15 1)

- 100

100

= 0.829 > 0, So satisfactory

Since the yeai 2 cost of $90 is entirely offset by the year 1 and 2 revenues of $SO per year, only the initial year 0 cost of $100 is included in the PVR or B/c Ratio denominatbr. The initial g100 cost is the only cost "not covered by project revenue in the year the cost is incurred or earlier years.,, 100

80 60

iqo z o2o

(Ev

7tr

-zo

3

-qo -60 -80

-1

00

-Maximum

Project Life (Years) Capital at Risk

Figure 3-15 Case 4, Cumulative NpV Diagram at

15"/"

Note that the year two net cost of $40 is offset by positive cash flow in year one and does not cause more capital to be at risk to the investor.

,.,gi

chapier 3: Present, Annual and Future varue, Rate of Beturn and Break-even Analysis

119

EXAMPLE 3-23 Cash Flow, Rate of Return, NpV and Ratios An investment project will involve spending $200,000 at time zero arrd $35J,000 at the end of year one. These invesiments wili generate qioss revenues of $333,000 at the end of year one and $5S6,000 at tile end of each of years two through eight. A roy,alty cost of $33,000 in year one and $56,000 in years two through eight will be incurred along u"'iih operating costs of $200,000 in year one and $320,000 at the erd or each oi years two through eight. caiculate the pr.oject before-tax cash fiow and then, for a minimum rate of return of 15.0%, calculaie tne rate of return, net present value, present value ratio and benefit cost ratio to determine if the project is economically satisfactory.

Solution: All Dollar Values in Thousands Year

2-8

Gross Revenue - Royalty Costs - Operating Costs - Capital Costs Before-Tax Cash Flow

-200 -200

333

s56

-33

-56

-200 -350 -250

-320 180

0.8696 4.160 0.8696 NPV = -200 - 250(PlF15,1) + 180(P/A1 5,7)etF15,1) =.+$233.8 $233.8 > 0, so project economics are satisfactory. P;-cjeci FOR is the "i" vaiue that makes NpV O = 0 = -200 - 250(P/F;,1 ) + 1 80(PiA;,7)(p/F;,1 ) By trial and error with interpolation: i = ROR = 29.6"/" > i* = 157o, so the project is satisfactory

PVR

=

NPV @ i-

tr-

233.8

=

-* , '1 zoo.7ffi "r"n = o'56

0.56 > 0, so satisfactory B/C Ratio =

PW +CF @ i.

lrw -cr o

i.

'1.56 > 1.0, so satisfactory

_ I

s"t".ilp lF rcx,i = 1.56 200 + 250(PlF67,,l

1Bo(P I Al

120


Note that PVR + 1 = B/C Ratio. Also note that all four evaluation criteria give the same economic conclusi9n oI very satisfactory economics compared with investing etsewnere at a 15;/";;i;;f return. PVR results always give correct economic

.on.rrion, if proper ,.net cost

not offset by project revenue" is used in the denominatolr of'each ratio instead of the present worth of costs without any consideration of revenue or savings available to offset part or all of costs. This applies to reclamation or abandonment costs at the end of projects as well as to intermediate pro_ ject period costs. The following variation of Example 3-23 illustrates the correct handling of downstream costs in ratio denominator calculations. EXAMPLE 3-23a Downstream cost variation of Exampte 3-23 suppose the operating cost in year eight of Example 3-23 could be delayed one year till the end year nine without effecting the revenue gen_ erated in year eight. For the revised cash flow diagrarn-shown below,-calculate the project rate_of return, net present value, present value ratio, and benefit cost ratio. The minimum rate of return is siitt ts.oz.

Solution: All Dollar Values in Thousands Year

Ftow

2-7

_250 180 500 .rr"

Before-Tax Cash _€20 -2OO The cumulative cash frow is ih" as for Exampre B-23 except the final $320 negative net cash flow is delayed one year to year nine. 0.8696

NPV

3.784

0.8696

0.3269 - -200 - 250(P/F.15,1) + i80(piA15,6)(p/F1S, j) + 500(p/F15,6)

0.2843 - 320(P/F1S,9) = +247.4

This NPV resurt is rarger than the Exampre B-2s resurt of $2g3.g as expected intuitively, since delaying the year g cost one year, with other values held the same, must give improved economics. ln the denominator of pvR ano guc Raiio, we onry use the absorute value of the present worth of negative cash frow noi offset by positive cash flow in earlier years. since the year nine negative cash flow obviously..is offset by year eight ,"r"nu", onry the year zero and year one negative cash flow are relevant for the pVR'and grc Ratio denominators.

L'napter

i:

present, Annuar and Future varue, Rate of Return and Break-even Anarysis

Correct PVR =

121

247.4 = +0.59 + [200 250(P/F15,1)]

This Pr/R resurt is gi'eaterthan the 0.56 pvR ro,, J*"rpre o-2s as knoiv it shourd be. The Bic Raiio resurt is arso nigGil, foilows: 've Cor:"ect B/C Ratio equals pVR +

=

I = j.Sg or:

180(PArs,oXP/Frs,lF [500 - 320(p/F1g r)l(P/F15,6) [200 + 250(P/F15,1)]

= +1.59

Note thai the numerator of B/c Ratio contains the present worth of revenues not offset by costs in the year revenues are incurred or rater

yea's anc the denominator is the same as for pvR. rhe mistake c'ften made in ratia carcurations aoni in industry practice is to incrude in ratio denominators the present vattie of do*iitreaiieiative cash flavvs that are offset by positive rb* in earrier years. For this

example that means incruding the"uripresent varue of the year nine negative cash frow in the ratio deiominatti. 1",i. gives tne tottowing pvn:

lncorrect PVR

=--

247.4

t20t;2s0(p/Fj

I

i

I

. szqell 5pf

= +0.49

This is a lower pVR than the 0.56 resurt for Exampre 3-23. This indicates that deraying the year eight cost gives ress desirabre economics than incurring the cg,sl alyear 6ignt. io, intuitivery see that something is wrong vvith this pVn cat6utation, "rn since deraying the cost must and does give better economics as verified by the rvev"anarysis. lncorrect B/C Ratio 180(P/At s,O)(p/Fr s, t ) + 500(p/F15,6) = 200 + 2\O(P/F 15,1) + 020(p/F15p) = +1.49 This B/c Ratio resurt is ress than the .r.56 resurt for Exampre 3-23. The same comments mentioned tor incorrect pv* are appricabre.

122


3.12 Effect of Income-producing Project Life on project Economics Because of the time value of monei, costs aiid revenues that occirr more than ten years from now are not nearly as important tgproject economics as the costs and revenues that will occur within the first t6n years of project life. However, it is important to recognize that the investment evaluation situation significantly affects the sensitivity that project life has on evaluation results such as rate of return. In general, the rate of return for a project with relatively good prohtability will be helped much less by lengthening project life beyond 10 years compared to a project with marginal profitability. The effects of investment profitability and project life on project rate of return are illustrated in the following example.

EXAMPLE 3-24 Sensitivity of project ROR and NpV to Changes in lnvestment Size and project Life

consider rate of return and NPV analysis of a project that may have an initial cost of either $20 million or $36 million depending on final engineering considerations. The project is expected to generate profits of $6 million per year for either s, 10 or 20 years with zero salvage value. Use a 10% minimum discount rate. Solution, All Values Are in Millions of Doltars: 5.Year Evaluation Life

C=20

0

l=6

l=6

1..........5

PWEq: 0=*20+6(P/A;,5) So, i=ROR= 1S.Z% 3.79'l NPV = -20 + 6(P/A1 O%,5) = +$2.75

C=36

l=6

0

'1

PW Eq: 0

l=6

..........5

=-36 + 6(P/A;,5) so, i = ROR = -S.B%

3.791 NPV = -36 + 6(P/A1 O/",5) = -$13.25

cirapter 3: Present. Annual and Future Value, Rate of Return and Break-even

Analysis

1zs

10 Year Evaluation Life

c=$20

0

l=$6

l=$6

1....

...10

?

FW Eq: 0 =-20 + 6(P/A;,10) so, i = ROR 27.3yo = |'JPV =

G.1t 4 + 6(PiA10%,10) -20 = +$i6.86

lv=Jb

0 FVt' Eq: 0 =

l=6 ...10

l=b

1....

-36 + G(P/A;,10) so, i= ROR = .10.6%

6.144 NPV = -36 + 6(P/A10%,10) = +$0.84 The effect on RoR and Npv results from doubling the evaluation life rom 5 to 10 years has been very significant for both cases. when the evaluation life is less than i0 years, changing the life used has a very significant effect on evaluation results for both economically good and rnarginal projects. Now double the evaiuation life from 10 to 2o years: f

20 Year Evaluation Life

C=2A

0

l=6

1....

l=6

...20

PW Eq: 0 =-20 + 6(P/A;,2g), so, i = ROR =2g.5"h 8 514 NPV = -2O + 6(P/A1 O%,zd = +$31.08

C=36

0

l=6

1....

l=6 ...20

PWEq: 0=-36+6(P/A;,2g), so, i=ROR =1i.go/o 8.514 NPV = -36 + 6(P/Aj O%.ZO) = +$15.08

124


Percentage change in ROR and NPV for 10 year to 20 year life change:

t

$20 Million Cost Project

RoR Percent chanpe rv

--

NPV Percent Change =

(29'5

- Z't3) x 100 +8o/o = 27.9

(31 .08

- 16.86) x 100 16.86

=+84o/"

$36 Million Cost Project ROR Percent Change =

Npv perceint chr lllge

(15.9-10.6)x100 10.6

(15'08

=+507"

0'86) x 100

=ffi=+1,6537o

The purpose of showing these percent change calculation results is to emphasize how much more sensitive positive NPV (and PVR) results are to increases in evaluation life beyond ten years in comparison to ROR results. This greater sensitivity of NPV is due to the fact that the 10% discount rate used in the NPV calculations is much smaller than ROR rates of 29.5% and 27.3/" in the $20 million cost analysis or 15.9"/" and 10.6% in the $36 million cost analysis. The present worth of future values beyond year 10 is relatively insignificant for the larger BOR discount rates, but the present worth of future values beyond year 10 is much more significant for the 10% discount rate used in the NPV calculations. 3.13 Mining and Petroleum Project Analysis Mining and petroleum projects are evaluated in the same manner using the same evaluation techniques applicable for evaluating non-mining/ petroleum projects. Only income tax considerations differentiate the analysis of mining, petroleum, real estate, chemical plant or other non-mining/ petroleum type of projects. On a before-tax analysis basis, in all indusl"ries, all types of costs are outflows of money and revenues for all sources are inflows of money. In

u^hapier 3: present, Annuar and Future Varue, Rate of Return and Break-even

Anarysis

125

iayirr.l out a time diagram \&,ith co.sts, revenues and salvase at different rimes' except tbr tax considerarions, economi. ;nrff;r 'arue a proiect is not .ri-ft';rcd b1' the industry .\Jurce of the costs anil rei,cnues. Anaryses shourd be drne;rlter-ta.x in iiil situations where income tax consicrera*:ns are relevant, ,,:- ijctrils of pripc' .ifiei-rar anall,sis of miriing. p",,ri.u., ancl general i.::: i;r,.'e:riillerlti project:. are prcscnted in Chapters 7 tllrough I 1. .\ithr-rugh evaruation consirrerarions rre the same for anaryzing projects in all iricil5irir's on a before-tax ba-sis, there are siq,if.icant differences in common i"r-rriinologl' used to rerer to costs, rel",eilues and ownership interests

in

differ_ ind-ustries. The peroreum industry especiarly tends to use unique terminol_ r'sr''

c,r

l\'fineral and petroleum rights a.qri.rtion

.orr,

*"

unrtogou,

to rand and i ,tient acquisition cosis in general industr--y anai1,.ses. Research t; ies are cquivalent to exproratioir costs ilrcurred in searching for petroreum and r"incrllis' Project developnient costs in aii non-rnining/ petroleum inclustries are :;irir llenl to deveroprnent costs in petrole*m and minirg pro.jects.

irt,,n;ll'i;d;:

Arr ob'.ious differcnce betu.een most niining and petroreum projects is the capiial inrensive narure of the mining industry. Siartup .ori, ono timir_e L-;lccrs arc olten much.greaier in minin! compmed to onshore petroleum pro_ 'ii';iion projects, but the tirne a,cl invJrtment in core drirling, metalrurgicar Ic-rr;ng and mine design ofien reduce exiraction and processirig risks rerati'e t'r oil iurd gas in'erstmenrs. oftihore peti-olcum project a.".ioi*",rt is more .,t:.113r,,us to niinirrs in terrns of capirai inten,i,,.enc.,s itnd dclaled proclire tiorr t;;ri,!. I{i,i.g re'enues .fren are siow comirg io ," ;;;0r.,,, of a mine ' \ r;iL- tlLrss are uortcci.otir of the,iining, rniiling, and transportation systems. This is difterent fio*i oil and gas ra,hereiroduction often is greatest

up front. tr'lining can be segmented into se'eral different categories including hard rrck' coar, and aggregates. For each of these categories, mine designs can L'e based on open pit, undergrorrncr. or a combination of both depending on thc ore Lrody. site rc'rcation, enl,iron,,ental. and other intangibre issues. Hard rocii nrinin-q includes gold. sil'er, read. ,i,.,.. .opp"r, molybrJenum, bauxite anrl a variety of otherprecious. strategic and non-strategic minerals. Due to the vertical nature of these industries,-mine economics may go substantially bc-r'ond the mine itserf include miiling and smerting considerations .to as ri'ell as transportation issues that impaci the ability of a mine to turn a profit. In addition to those paramerers ihat affect alr project economics such as product price, other key parameters in hard rock eJonomic evaluations include ore grades, metallurgical recoverv rates, etc. In.most hard rock min_

.

126


ing evaluations, the product is purchased and sold on a variety of markets but two o{ the primary markets include the New.york Mercantire Exchange (NYMEX) and rhe London Metals Exchange (LME).

t

coal mining is vastly different in that in some situations, the product

extracted from the pit may be ready for sale. However, coal often requires treatment to improve the quality of the product by reducinj *u,". removing non-coal materiars, etc. Coar is sold "on,rr,, diiectry to a"consumer such as a utility, typically on a contract basis, although ,poi-u.t"i, do exist for short term demand and may represent the bulk of sales for a mine or con_ sumer over time. There are many other investment anarysis terms unique to minerar project evaluations but the terms described give the reader sufhcient background to

understand many mineral project evaruation statements and sorutions,

presented in this text.

as

EXAMPLE 3-25 A Mining project Anatysis An investor has requested that you evaruate the economic potential of purchasing a gord property now (at time ieiojior a gr miilion ,.1::rr1 rights acquisition'cost. Mining equipment costs of $3 miilion will be incurred at year one. Mineral development costs of $2 million will be incurred at time zero with an additional $.1.5 ,ifftn spent in year one. Production is projected to start in year one of 150,000 tons of gord ore, witir uniform production witn tne mining of 250,000 tons of gold ore per year in each of two, three and four. Gord ore reserves are estimatê^d_to be v"ur. depreted at the eno oi year four. Reclamation costs of $0.5 mirion *ri-'" incurred at ine end of year four when $1.0 miilion is projecteo to be rearized from equipment salvage varue- Ail gord ore is estimated to have ,r"r"g" grade of 0"1 ounces of gord per ton of ore *ltn "n *"trrtrg;;dovery esti_ mated to be 90%. The price of gotd is estimat;d to 6L $300 per ounce in year one and to escarale 157" in year two, zo"t" in year three and- 10"/"in year four. operating costs are estimated to be g20 per ton of ore produced in year one (or $222_22 produced and sord), and to escarate 'crt"rirte of gotd sz p", y"rr. the project RoR, Npv and pvR for a minimum iate of return of 15%.

r;;;;."

;

chapter 3: present, Annuar and Future varue, Rate of Return ano,Break_even

Anarysis

127

Solution, All Values in Millions of Doltars: c=costs, l=lncome, !!=onerating costs, L=salvage vatue Net Cash Ftow (Net CF) = in.oru-* sur;;g.

-;il[r"r,"'

l=4.05 L=1.0 OC=3.00 cRecl=o'5 l=V.763 l=g 315 -CD"r,, =2.0 CDev=1.50 i=1A.246 ci,i'n.Rts=i.o 0 CEo CC=5.400 =3.00 lr',irn.Rlg=i OC=S.832 OC=6.298

i

_--<. ear

BTCF

0

-3.0

_3.45

23 +2.363

4

+3.483

+4.448

Production/yi,x OunceVfon x Ounces Recovered x price Revenue = Y,ear 1 Revenue, (Revenues in Millions of Dollars); M Thousand

.

= r'50,000 tons/yr x 0.1ozr,ton x 0.g recovery x g300/oz g4.65p11y = Year 2 Revenue at the year 1 Selling price: 250,000 tons/yr x 0.1o2lton x 0.g recovery x g30O/oz 66.751,414 =

Accounting for serting price escaration year 3, and 10"/" in year 4:

of

1so/o

in year z,2ao/o in

Year 2 Revenue $6.750 x 1.15 = = $7.763MM Year 3 Revenue 97.768 x 1.20=

=

Year 4 Revenue $9.315 =

$9.31SMM

x 1.10 = $10.246MM

Operating Costs. Year 1 j50,000 tons/yr x $2Olton = $g.00MM Year 2 Operating Cost at the year 1 Cost Rate 250,000 tons/yr x $2olton = $S.000MM Year 2 Operating Cost = $S.O0O x 1.0g = $5.400MM Year 3 Operating Cost = $5.400 x 1.08 = $5.g32MM Year 4 Operating Cost = $5.g32 x 1.0g = $6.29gMM

128


NPV @ 15% using the beforetax net cash flows on the time diagram. -3.9 -,3,45{P1F15,1 ) + 2.8ffi(P/F1 5;d, + 3.4$(P/F15,d + a.M\(PlFg.d = +$0.6e0

t

Rate of Return (ROR) = 1g.4ro is the ,,i,,value that makes NpV 0. = Present Value Ratio (PVH) = 0.620/(3.0 + 3.45(plF1S,1)) +0.103 = NPV and PVR greater than zero and RoR greater than l* lsyo = consistently indicate economic acceptability of this project compared to investing elsewhere al a 15/" rate of return. In the petroleum industry, exploration or development well costs are often broken into two components and referred to as intangible drilling costs (the cost of drilling oil and gas wells to the point of completion) and tangible well

costs (the costs for tubing, producing equipment, tank batteries, separators and gathering pipelines necessary to "complete" bringing a well into production. The acronym "IDC" often is used to refer to "intangible drilling costs,' in petroleum drilling Fojects. Thx deduction considerations are the primary differences in tangible and intangible well costs as discussed in detail for different types of investors in Chapters 7 and 8.

when considering oil and gas investment opportunities, most fall into one of four general categories. First, you can evaluate the project from the viewpoint of the party holding all rights to the properry, which might be considered as "development economics," or drilring a property "heads up.', Both terms imply that the intent is to look at all rer,'enue and costs as pertinent to the investor.s economics. Second, you, the investor might consider passing the rights to develop on to a second party who wourd put up all or part or thl drithng costs and well completion costs. Typically the developer gets all or most of the revenues until the costs are recovered and then you back-in at that future point in time for a working interest in the property. This is sornetimes referred to as "famring out" a property. Third. you could also look at the opposite viewpoint of a "larm-out" and consider developing a property that someone else owns for an interest in the property. This often is referred to as "farming in,, a property. The "farm-in" party is on the opposite sicre of a petroleum development agreement as the "fann-out" party. Finally, you could consider selling a property and keeping a royalty, with no liability towards development of the p.op"rty. This sometimes is referred to as retaining a "carried interest,' or an ..overriding royalty'' in a properly

..i ialrier 3:

:iesent, Annual and Futtrre Value, Rate of Return and Break-even Analysis

129

The rbi.rowing

discussion off-ers sorne of the basic terminology that is fairry ('om*on to manl' oil and gas dears. Fio:i an.egonomic evaruation

viewpoint, of capitar and i';:craiiitg expendiiures paid by the investor inl'ulved in .,f-arm-out,, '"{;in-in" evaruiition.i. or 'a{ous These ;green-,.rr, .rn bc r,-ery aynrrii.'or,"r a propeny iitc. ill,w'ing for i,terests in costs an,.l revenues to change at vanous points in iire p;'uject rif'e as introduceci in the cjerirition of some otit,,e iottor"ing terms: these terms herp to estabrish royarties rc ue paia and the portion

Lr

states. il is possible to se'er the interest betrveen the rand (suiface and the nrineral interest rtsetilminerar right.s). Rights to the minerar inte*st itserf are transfered uy u *in..rr deed

. riehis)'-'io,t

:J,:lr;lilJr$:r|i*'

or fee simple title. Before a mineral rights to a lease may be..tuin"o by paying

r,.rared,op.ou"iffi

"#il;i:,:;.,ff

,HH:ffi I#J]

J;:-,"fl i,f :rs l/Sth (125%) of grr;ss..u.r.r., or rhe weilhead varue of the I{'r'aliies are usuailr' the rrrst e.\pense

oil and orgas. u" o.au"ted from ,t gro* yalue of ;''roduction to determine the net revenues a'ailabie " to the producer to cover

i"

costs. il'crenue afterroyahies often is referred [o as the,,net revenue interest,, a propert\,.

il

A "rvorkirg interest" is the i,vestor,s percentage of responsibility for costs related to an oil ancr i:as lease. when there is jusr one investor, that person or ilrin i-; responsibre ror a, c'srs assticiated u itrr creveropirg ;i;;r. and are sr,id tc ic:t3ip 3 10076 1i6p1,1rg ir,.r".lir rh. l"ur". l, inq"*lrf
ties' o'erriding royalties or carried lrt"."rtr, reversionary

interests, etc. As mentioned earlier, those net revenues that are available often are referred to as a "ne[ revenue interest." ceneraily. *ortJ,g inrerests ,turo ira.-p"ndent of the re\enue ir)terest in a lease. Again, , na,'a*nue interest can be defined as rev_ ettue lcrss royalties, overriding royalties, carried interests, etc. This is a frac_ lional intcrest based on the gts ,"r.rue potential from a lease. Net revenue rLrpresents the dolrars available to the invJstor to recover dollars invested to cover the working interest costs to extract the lease resources (driling costs, lelrse costs, operating costs, etc.).

on the other side of development economics-is the royarty owner. A royalty interest is a minerar or rani o*r..t ,t *e of production free and crear of all operating expenses and capital costs. Royarty owners usuary are respon_

L

130


sible for their share of applicable severance and excise taxes. These taxes usuaily are paid directly b-y the operator. The operator then recovers the appropriate prorated portion of these taxes by reducing royalties paid by the prorated tax amount.

Overriding royalties or interests are predominately baseci on the net revenue interest but could be based on anything from the gross value ofthe oil and gas to the net revenue adjusted for operating expenses. This interest is free and clear of all production and capital expenditures. It is a royalty in addition to the landowner's royalty. Overriding royalties may also be described as being carved out of the working interest, so in a standard 1/8 of 8/8 royalty situation, a 1/8 overriding royalty may come out of the 718 net revenue interest. A carried interest is similar to an overriding royalty; this is a fractional interest in a lease that gives the holder no obligation for operating or development capital costs. It is a royalty and may be caiculated on gross revenue ot gross revenue adjusted for various out of pocket expenditures incurred by ihe investor.

In many deals, the interest in the property may change as the property is developed. This relates to reversion(ary) interests which represent the portion of the working interest that reverts or is allocated to another party after a specified occurence, such as payout of investments by specified criteria.

In reversion calculations, a reversion point is that point where working and or revenue interests in a property are transferred or revised, based on a previous agreement in the lease. This is usually determined by some type of payout calculation. In many cases, payout is based on the before-tax capital expenditures (intangible and tangible) and rnay involve revenues that reflect anything from gross revenues to gross revenues adjusted for all out of pocket expenditures. After-tax cash flow (discussed beginning in Chapter 7) is rarely used in these calculations due to the complexity and variation of tax positions for each investor in the property.

The following example illustrates a basic drilling evaluation neglecting complex issues of multiple participation, reversionary interests, etc. Note the similarity of these calculations to the other examples throughout the textbook.

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131

EXAMPLE 3-26 A Petroleum project Analysis Analyze the economics of the following petroleum project in which an in"'e stoi' has a 100% working interest. costs are in thousands of dollars, prorluction is'in thousands of barrels and crude price is in doila:s per barrel. Production Crude Oil Price/Bbl lntangibles (lDC's) Tangible Compl Cost Lease Bonus Operating Costs

80 20

4520105 21 22 23

50

55

900 300 200

Year012345

60

65

24

70

Royalty costs are 16% of revenues. Net salvage and abandonment l'alue at the end of year 5 is estimated to be zero. Determine the investment RoB, NPV and pvFt for a 12/o mlnimum rate of return.

Solution, AII Values in Thousands of Dottars:

12345

Year

Revenue

-Royalty Costs -Operating Costs

-lntangibles(lDC's) -900

1,600 945 440 230 -256 -151 -70 47

-50 -55 -60 _65

-Tangible Compl Cost -300 -l-ease -200 BTCF -1,400 1,294

120 _19 _70

Bonus

739 310 128

31

NPV @ 12/o = -1 ,400 + 1 ,294(PlF 12,1) + 739(p/F e2) + 310(P/F12,3)+ 1ZB(P/F12,4) + 31(piF12,S)

= +$664.0 ROR = 43.5% is the "i" value that makes NpV = 0. PVR = 664.0/1,400 = +0.474 NPV and PVR greater than zero and ROR greater than i* = 12,/o all consistently indicate economic acceptability of this project compared to investing elsewhere al a 12./" ROR.

132

Economic Evaluation and lnveslment Decision Methods

3.14 ROR, NPV and PVR Analysis of Service-producing InvestmentsWith Equal Lives .,

: :.

In comparing alternative ways of providing a service,l$ is common to have onlv costs and maybe some salvage value given for eich alternative being considered. This means that rate of return for individual alterative *uy, of providing a service is usually negative, and often minus infinity if no net revenue is involved in the analysis. Therefore, rate of return analysis of the individual investments to provide a service does not give useful information for economic decision making. Similar considerations relate to Npv and ratio analysis of service alternatives. In providing a service, the most common situation is that you know the service is needed and from an economic viewpoint you want to provide it as cheaply as possible. For rate of retum, net value or ratio analysis of alternatives that provide a service, investors must meke an incremental analysis between the alternatiy,es. Incremental implies the difference in two alternatives. Incremental analyses are made to deternine if the additional up front investment(s) in the more capital-intensive alternative generates sfficient reductions in' downstream operating costs (increntental savings) to justify the investment. rf the rate of return on the incremental (or additional) investment is greater than the minimum rate of return, or if incremental net value is greater than zero, or if incremental pVR is greater than zero, then the incremental investment is satisfactory from an economic viewpoint and selecting the larger investment alternative is econonrically justihed. The following example illustrates RoR, Npv and Ratio analysis of this type of service-producing alternative evaluation problem for equal life alternatives. The analysis of unequal life service-producing altematives is addressed in Section 3.16 and Examples 3-29 and3-30.

t $

chapter 3: Present, Annuar and Future varue, Rate of Beturn and Break-even Anarysis

I

:..

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fl

*

i

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133

EXAMPLE g-27 ROR, NpV and pVR Anatysis of Serviceproducing Alternatives with Equal Lives considering the instariation of autongated ,^ ri 1a 1o*p"ny.is processing operation to reciuce rabor opurrrng equipment costs from Sj00,000 io $220,000 in year one, from $SS0,OOO to $240,0C0 in iear two, from 9360,000 to $260,000 in year three and from s+00,000 tc $2g0,000 in year four. The automated equipment wiil c-.st $200,000 now with an expected salvage value oi $SO,OOO in four j,ears. The minirnum rate of return, "i*", is2oy". use RoFr, Npv and P /R analysis to detei'mine if the equipment should be installed from af economic viewpoint. Then consider an increase in the minimum =cF to 40"h trom 2a"/o and re-evaruate the arternatives. Solution, All Values in Thousands of Dollars:

In setting up a sorution for service arternatives, investors have a cncice in sign conveniion. one approach is to raber ail doirars as capi_ :ai and operating costs or sarvage income. with this approach, costs aie treated as positive varues ind sarvage (misceiia,i"or" income) r''iii eventuaily take. on a minus sign so negative incrementar operating :osts become savings when the present worth equation is modified. Tris approach is iriustrated in the first solution. The second approach s tc address the costs in the context of cash flow where costs are an -uifiow and treated as negative varues. This approach is utirized in r;e second solution. Both approaches are used in various exampres -^rd probrems throughout the textbook as they are in industry. lncrementar setup Approach #1 - Laber Doilar vatues C=Capital Cost, OC-Operating Cost, L=Salvage A)

B)

C=200 OC=220 AC=240 OC=260 0

C=0

e

OC = 300 OC = 336 OC = 360 2

A-B)

C

='200

3

OC = -80- OC = -99 OC = -1gg

OC=290 4

OC = 400

L=50

L=0

OC=-110

L=50

134


lncremental present Worth Equation: 200

- 80(PiF1,1) -90(PtFi,2) -

1OO(p/Fi,g)

- 1 1O(p/Fi,g= 50(p/F1,4)

ln this first equation the present worth of a[ {ncrementar capitar and negative incrementar operating costs appear on the reft-hand

side of the equation. These varues a-re set equar to the present worth of revenues (sarvage) on the right side. However, since'the operating costs are negative, they represent the operating cost savings that wiil pay for the investment. To illustrate how negative incremental operat_ ing costs are identicar. to savings, rearrange the equation by adding the present worth of the operaing costs and subtracting the capitar investment to each side of ine eqritity, this resurts in theioltowing: 0 = -200 + 80(p/Fi,t) + 90(p/Fi ,2)

+

1OO(p/Fi,3) + 160(p/Fi,4)

lncrementar setup Approach #2 cash Frow sign convention The alternative method that yierds an identicar present worth equation involves using cash frow sign convention where costs are negative values, and revenues (such as 6arvage) are positive values as foilows: A)

-200

-220

-240

-260

-290 4

-300

B)

A-B)

-330

0

-200

-360

50

-400 4

80

90

100

110

50

This approach indicates immediatery the savings to be derived on the time diagram and reads to the deveropment o"f the present worth equation:

."r"

0 = -200 + 80(p/F;,1) + 9O(p/F;

,2)

+ 100(p/Fi,g) + 160(p/F;,4)

either approach, sorving for the incrementar rate of return using trial and error provides the iollowing:.

Itilq

..,:aprer 3: Presert, Annual and Futuie varue, Rate of Rerurn anc Break_even Analysis

135

., 30% = 16

.,4

4A"h =

-19

i= g0% + 10%(16/(16+19)) = 3a.6% An incremental rate of return oi g4.6o/L is gi-eaterlthan the zo.o% n"iirrirnum Interpolating:

rate of return and therefore, is satrifactory; so accept tlre irrvestment in the automated equipment. lncrernental Net present Vatue @ 2A7o

0.8333 0.6944

0.5787

-200 + 80(P/F2g,1 ) + 90(P/F20,2) + 100(p/F2g,3) + = +64.2 > 0, so accept automated equipment.

1

0.4823 6O(p/F2g,4)

lncremental present Value Ratio

I

;

64.2i2O0 = 0.32 > 0, so accept automated equipment. RoR, NPV and Ratio analyses arways give the same economic conciusions if the techniques are useo properry. chapter four wiri emphasize proper use of these techniques in oitterent anarysis situations.

Changing the minimum rate of return, i* changing the minimum RoR to 40"k trom Zo.h changes the economic conclusion to rejection of the auiomated equiprient with all

evalLraiion criteria.

_comparing the incrementar RoR of 04.6y" to other opportunities at indicates reject the automated equipment. lnvestors do not want to purchase equipment that provides an inferior return com_ pared to other opportunities thought to exist for the investment of ycur capital. This is one illustration of why it is important to know what your true returns from other opportunities reaily are because the discount rate can and will influence your decision in many investment situations. 4092"

lncremental NpV @ 40%

0.7143 0.5102 0.9644 -200 + 80(P/F49,1)+ 90(P/FaO,2) + 100(p/F4g,3) =

-18.8 < 0, reject the investment

0.2603 + 160(PlF49,4)

the automated equipment.

136


lncremental PVR -18.8/2OO = -0.094 < 0, so rgect automated equipment.

consistently with all analysis techniques, the eocnomic choice has shifted to rejecting the automated equipment investment. For an increase in the discount rate to 40"/", all methods indicated an economic preference for the labor-intensive approach. Bigger discount rates will always make it more economically difficult to justify capital intensive alternatives versus less capital intensive alternatives. 3.15 cost Analysis of service Producing Alternatives That provide the Same Service Over the Same period of Time. Instead of making an incremental analysis of sen ice producing alternatives using rate of return, net value or ratios, it is equally valid to analyze lhe present, annual or future cost of providing a service for a common evaltration life.witlt any of these methods, economic selection is based on the alternative that provides the service for the minimum cost. consistent with discussion in section 3.14, these evaluations can be handled with different sign conventions. The minimum cost analysis is based on using the cost and revenue sign convention vvhere costs are positit,e and revenues are negatit;e. Note this is the opposite of the cash flow sign convention v,here revetute.s are positive and costs negative. Consistency in application and proper interpretotion of results is realh, the key issue. As mentionecl, both ctpprooclrcs are equally valid. Exctmple 3-28 will focus on the cash _flow s igrt corn,ention a pproach.

t. n

s

r !

EXAMPLE 3-28 Present, Annual and Future Cost Analysis of Service Producing Alternative With Equal Lives Evaluate alternatives "A" and "B" from Example 3-27 shown on the following time diagrams using present worth cost analysis. support those conclusions with an annual and future cost comparison to illustrate the economic equivalence of the results. The minimum rate of return is 20.0%. Finally, re-evaluate the alternatives using present worth cost only tor a 4O.Oo/" minimum rate of return. :,

;i! ::,

,).

.:

t

}, :1

'I

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chapter 3: Present, Annual and Future Value, Rate of Return and Break-even Analysis

137

Cash Flow Sign Convention Time Diagrams

-200

A)

-220

0

-240

n

-300

B)

-260

2

1

-290

3

-330

*

50

--400

-360

Solution: All Values in Thousands Present Worth Cost (PWC Al @ 2A"/o 0.8333 =

-200

-

0.6944

220(P /F 20, 1 )

-

24A(P

F

2g,2) -

4.5787 26Ap /F20, 3)

0.4823

-

24Q (P tF 2g,4)

-$B'16.2, least negative value is "A" (provides minimum cost)

Present Worth Cost (PWCg) @ 20% 0.8333 = -300(P/F20,1)

0.6944

-

0.5787 330(P/F26,2)- 3G0(PlF29,g)

-

0.4823 4OA(?F26,4)

= -$880.4

The incremental analysis presented in Example 3-27 r,rras based on looking at the difference in the A-B costs or cash {lows. lf you consider the difference in the PWCA - PWCg or, -816.2

- -880.4 = +64.2

This was the incremental NpV for A-B.

Because the calculations are very similar, (both involve discounting at iJ some investors fail to recognize the difference in an incremental NPV analysis and an individual cost analysis of the available alternatives. with a cost analysis the investor always wants to minimize cos.t, wnereas in NPV analyses, the objective is to maximize value generated from savings or income from initial investments.

Annual and future cost calculations are alternative methods that will always arrive at the same economic conclusions from a present wcrth co:;t analysis as follows:

Annual Cost (ACll

@ 2oo/o

0.38629 -816.2(A/P 20"h,4) = -$315.3, Least cost alternative is "A"

138


Annual Cost (AC6l @ 2}yo 0.38629 -880.4(A/P26,/o,4) = -$340

Future Cost (FC4) @

1

*

2lo/o

:

2.0736 -816.2(F/P 20"/",4) = -$1 ,692.5, Least cost alternative is Future Cost (FCg) @ 2To/o

,,A,,

2.0736 -880.4(F/P ZO"/",4) = -$1 ,825.6 changing the minimum RoR to 4oy" trom 20"/o changes the economic choice to rejecting the automated equipment i,itn ail cost analysis criteria, just as it changed the economic choice with RoR, NPV and PVR anarysis in the pievious exampre. present worth cost analysis is presented here.

Present Worth Cost (pWC p,) @

0.7143 = -200 -220(P1F40,1)

-

40o/o

0.5102 0.3644

0.2603

240(p/F4g,Z) _260(ptF4o,g) _240(ptF4g,4)

= -$636.8

Present Worth Cost (pWC g) @

0.7143 = -300(P/F40,1)

-

40o/o

0.5102

330(P/F40,2)

a.9644 0.2603 - 360(p lF4s,s) - 4oo(ptF4g,.4)

= -$6'18.0, least negative value is

,,8,,

other opportunities exist to invest your money and . ^Y!"n earn a 40.0% i-ate of return, the economics favor the more labor intensive alternative "8." The decision is to take the cash ,no gei the better return potential from other opportunities. The increm-ental NpV is presented to support this cost analysis and illustrate the equivalence of the result. Again, if you consider the difference in the pWC4 incremental NPV 3t i.={e7 is. -636.8

- -618.0 = -19.g < 0, reject A, select B.

_ pWCg, the

,j

I

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chapter 3: Present, Annuar and Future value, Rate of Freturn anc Break-even Analvsis

139

tir li ti

i {

i

This example and the previous exarnple illustrate

that the minimum rate oi return is a significant. evaluation parameter that has a definite effect on economic conclusions with a, eu-aluition criteria. It must be serectecr care-

fu.lly to represent the attainable rate of return thought tG€xist from investing iii orher alter,atives. The rninimum rate of returnlnusr nor be an arbitrarily determined "hopeci fbr', number.

3'16 Comparison of unequar Life Arternatives that Provide the Same Service Li this section methods that may be used to compare unequal life altema_ tives that provide the same service are introducea ano iliustrated. In general, lhe r:rerhods pr:esenred here are not valid for comparirg;;-";;

fife income_ producing altematives, or.for comparing service-producing projects thar do not re sult in the same service. AssumptiJns for the evatuatiin of unequal life income-producing projecr, -" p."r."ted and illustrated in chapter 4, whire 'ural1'sis of service arrernatives ihat provi,Je diff"** ;;;;e il ao.tresseo rn Se ction 3. i 7. To get a nreaningfur comparison of unequal life artematives that provide Ihe sanre service, assriinptio,s or estimates must be made to permit compari'':oo of ihe alternatives on an equ:.rr rife or equar study periocr basis. you are contpar.ing diffc-rent tiitrrl sen,ice il -you compare project costs tor unequal tii'L'pci'iocis. Three wi, be prese.ted rhat corer the possible *,,avs of c.rnfarine u,equarir:.:l"dt lir-e projects. Combinarions of these *"ril;;';;r;i; be used. The methods are not listed i, their order of importance or varidity. It is irnpossible to say that one method is best for ail situaiions. project circum_ stances usually dictate the method that is best for given evaluation condi_ lions' However, method 1 is seldom a desirable or valid approach to use. you should be familiar with it because many textbook and literature authors ha',e aili'ocated its use. Methocrs 2 ancr 3 are trrc approae.hes that in generar sltould be used Io olttcrin a con.tmon studt, period fbr unequat life projects tlrttt prot'ide thet same sen'ite. The three mlthods io. ot,trinrng a common stud) r-'criod for urequai life service-producing alternatives are described and illustrated usin-q the fo,owing tlvo unequal life alternatives .A,, and ..B,,. C = Cost, OC = Operating Cost, L = Salvage Value

o,

r

C,q

0

OCAI

1

OCA2

z

,

,A

140


Cg

u,'0

OCgl |

OCBZ 2

OCg3,

3

rB

once a common study period is estabrished usinf one of the folrowing methods, then the resulting equal life alternatives c-an ue compared using the fir'e basic approaches presented earlier in this chapter which are present, annual, and future worth, RoR or break-even analysis. However, for the results from any of these calculation procedures to be valid for economic decision-making, an investor must be iooking at the costs for each arternative to provide the same service per day, *"Lk o, yearry period as welr as for a common evaluation life. This means the service alternatives must be compared on the assumed basis of generating identical revenue streams.

Lethod 1: Replacement In-Kind

. This method of getting a common study period for unequar life arternatives is based on the that pro;ects ,A,, and ,,B,,^can O" .as.sylntion in-kind for the same initial costs, opeiating costs and salvage values until a common study period equal to the lowest common denominator between the pro.iect lives is attained. For alternatives ,A,, and ,,B,, replacement in_ kind gives a six year study period as follows:

r"r*;

A)

B)

Ca oCar

01 Cg

OCe2 2

OCnt ocgz

CA OCet OCRZ C4 OCat OCn" ,LA-r--L4 t-A A 3 4'6 oce: r ^ Ce oCgl oCnz oCg: J

LBm-LB

The obvious shortcoming of this method is that escalation of values is neglected from year to year. Experience of recent decades emphasizes the fact that costs can, and generalry do escarate as time goes uy. To assume replacement in-kind involves closing our eyes to the true situation that costs change. Economic anarysis is only valid wiren you use actual expected costs and revenues and actuar timing of costs and revenues. A method such as replacement in-kind that does not project actual expected costs should not be used. As mentioned earlier in this section, replacement in-kind is presented here to familiarize the reader with the disadvintages of the methods since it is mentioned and advocated widery in other texts.and journal literature.

I.

ci

'rpter 3: i'resent, nnnuar

ani

Future Varue, Rate of Return and Break-even

Anarysis

141

Anorher

approach recommended b1, some aLlthors is to consider each .lt".ialir, c based soleiy on theii equivaic,t annuai cost. with this method no adjrrstmenls are made to .o*p.nrate f,-rr critferences in ser'ice rives, each as:ci iS rneasured based on its own useiur lif-e ancr urro{"r"J costs. This :r:.:i iroil irn;,ii-'itl-r' os:iullcS rhe replaccnre,r iu kind-philosoohy. I3y evaruat_ irte rrirc.l,-iai l,ie s.'r', ice aricrrriri'"'es r.i'ith crl,,ivareiit a,rnu.rl yo, u." ,laiiy assuming iiiat assets calr be replaccii nirw ,fld irr "urr, for the future the srrn:e cupital and operating cost structure stai:cl ror the initial use periotl.

\'fcthod 2: Neglect Extra Life of Longer Life Alternatives

of getting a common study period for unequal life artemaon the extra rife of ronger life alternatives over .based 'egrecting sliorter iife altematives and letting sarvage varues at the end of the shorter lite ref'lect rentaining estimated value of-the assets from sale or use of the .ls.sei's elsewhere. This otien is a assumpri.n if the seru.ice provided is r'nly expectcd to be needed for the'alid shori arternaiirre rife. Appll,ing this methoils to alternatil'es 'A'' and "8" gives a two year study period as follows:

.

Thi.s method

tii'es is

A)

CA

OCat ocaz 2

,,,

La

oc'lll octsr .^ , 0 | I .B

9L

Notc that the sarvage r,alue of "B" is designated L,g at the end of year t\\'o to designate it differentry from Lg Normally "would we expect L,g to bc greater that Lg since the asset wilihave bee, used for a shorter time at lear [\\'o con-ipared to Vear three. N4ethod 2 is a valid method of obtaining a common study period in many

pl:1 sicrrl ei i.rluution situations.

llethod 3: Estimate Actual

Costs to Extend Shorter Life Alternatives to a Longer Life Common Study period

This method of getting

a common study period for unequal rife artematives is based on extending the life of the shorter life alternatives by estimat_

142


ing actual replacement or major repair and operating costs needed to extend the service life of an alternative. This method may result in a study period equal to the longest life alternative or it may ..rult in a study period unre_ lated to the alternative lives. If we apply this method alternatives .A,,.and

"8" for a common

{p

study period of three years we get the following:

CA oCAt o9A2, C'4 oC43 -. L's o r 2 tA 2

o,, .;r, ")

B)

Cg

3

OCer OCSZ OCs3

Lg

Note that at the end.of vear two the replacement cost c,4 is not assumed to be the same as the initiar cost, C4. c;4 may be greatei tr less than c4 depending on whether it is new o. os"d .ft tn. end of year three note that the operating cost, OC3, and "quip.*nt. the ialvage value, U4, are desig_ nated differently than earlier valuJs to emphasize that they pro6iuty will not be the same as the corresponding values for earlier years. Engineering and business judgement is required on the part of the investment analyst to use correct assumptions in different analysis situations. In many cases the correct assumptions tc use are not clear cut and it may be best to look at a problem from several analysis assumption points of vierv. It is very important to get into the habit of clearly stating all assumptions used in making an analysis so that the person with the reslonsibility of evaluat_ ing anaiysis calculations, assumptions and economii conclusions has the necessary information to make his or her evaluation properly. Economic analyses involve intangible considerations (such as the besi assumptions to use to convert unequal life projects to equal life projects) and the decision_ making manager must know the basis on which an analysis has been made before analysis results can be utilized to reach valid economic decisions.

once again, before illustrating these three methods, it is important to reemphasize that these methods or combinations of these methods are valid only for altematives that provide the same service and not for income-producing projects. Methods 1, 2 and 3 are almost never valid for the of income "o*pu.iron or service-producing projects that provide different services oi benefits.

(

"apter

3: Present, Annuar and Future Varue, Rate of Return and Break-even Anarysis

143

EXAMPLE 3-29 Comparison of Unequal Life Service

Alternatives

Cd000 Ar

OC1=1,599 OC2=2,996

L=1,000 ;

C=10,000 OC1=1,egg OC2=t,+00 OC3=1,800 OC4=2,2gg

t]\

L=2,000

c; -'

c=L 0

,}

1

Z

soo

3

4

S

L=5,000

Three alternative methods "A", ug' and ,,c,, with costs, sarvage varues and lives given on the time diagrams are being considered t6 carry out a processing operation for the nex three yeals. rt is expected that the process will not be needed after three years. A majoi repair costing $3,000 at the end of year two wourd extend the rife of arternative ,,A,, through year three with a third year operating cost of $2,s00 and the salvage value equar to 91,000 ai the end of yiar three instead of year two. The salvage value of arternative "8" is estimated t; $3,000 at the end of year three and the sarvage varue of arternative ,,c,, is estimated to be 97,000 at the end of year three. For a minimrn ,.ur"-ot reiurn of 159'., which arternative is economicaily best? Use equivalent annuai cost analysis for a three year study period.

;;

solution, The three year study period time diagrams fortow with all values in dollars:

C=6,000 OC=1,500

B)

C=3,000 OC=2,000

OC=2,500

2

3

L=1,000

C=10,000 OC=1,000 OC=1,400 OC=1,800 0

C=14,000

1

2

3

OC=500

OC=600

OC=700

L=3,000

L=7,000

144


The annual cost calculations follow:

0.43798 0.907 0.28798 AC4 = 6000(A/P1 S"/",g) + 1500 + S00(A/G1 5/"p> 1000(A/F1 sy"S) 0.7561 0.43798 + 3000(P/F 1 S%,2)(Np1 S%,g) = $5,287 ttvr: 0.43798 0.902 0.907 O.ZgZgg ACg = 10,000(A/P 15"/"p) + 1000 + 400(fuG1 _ 3OOO(A/F S"/",g) 15"/",g)

= $4,897

0.43798 0.907 0.28798 AC6 = '14,000(A/P 15"/",g) + SOO + 100(A/G1 7O0O(A/F1S%S) So/oB)= $4,706

select alternative "c" with the smallest equivalent annual cost. This problem is presented again on an after-tax evaluation basis in chapter 10 and we then find alternative "A" is worst, ,,B,, best and *c,, the next best choice which emphasizes the impori"nce of tax con_ siderations and that analyses must be done after-tax to be valid. once again, taxes are being omitted in the early chapters of this text to avoid confusing the reader with tax considerations at the same

time evaluation concepts are being developed.

3.17 comparison of Service-producing Alternatives that provide Different Service

If different service-producing

arternatives are projected to give different ser'ice per operating period, and if the extra r".ri." produceiby the more productive alternative can be utilized, then it is necesiary to get the alterna_ tives on a con,non service-producing basis per operating p"iioa as welr as for a common evaluation life. This simply means that if two old assets are required to do the job of one new asset and sen,ice provided by two old assets is needed, then you must either compare the co.st of service for two old assets to one new asset, or one old asset to one-harf of a new asset, or some other 2to lratio of old to new assets. Differences in annual operating hours of service as well as volume of work differences per year between alternatives must be taken into account to get all alternatives on , .o*rnon service basis before making economic compaiisons.

{ { !

I I {

i

chapter 3: Present, Annuar anci Future Vair-re, Rate of Return and Break-even Anarysis

14s

EXAMPLE 3-30 comparison of Arternatives that provide Ditferent Service Three used machines can be purchased for g45,000-each to provide

a needed service for the next three years. lt is estin:dfed that irie salvage value of these machlnes will be zerc in three ,.ears and that they will need to be replaced at year three wrin two n:,..r r-irachines that each qive 1 50"h of the productivity per machine being r+alized with each old rrachine. The nelv machines r,vould cost 9145,0ii each at year three. The service of the old and new machines is needed for the next five vears. so use a five year evaluation life assuming the salvage value of the new machines will be $50,000 per machine it year five. Operating ccsts per machine for the old machines are estimaiec to be $30,000 i; ycai one, $35,000 in year two, and 940,000 in year three. Operating costs per new machine are estirnated to be g33.000 in year four and $36,000 in year five. Another alternative is to buy ihe t*o n"u, ,'nachines now for a cost of 9125,000 per machine to provide the neecjed service for the next five years with annual cperating costs per machine of $25,000 in year one, g30,000 in year two, g3s,000 in year three, $40,000 in year four, and $45,000 in year five. sarvage varue at year five would be $15,000 per machine. For a nrinimum Bon of ts.r,., Lise present worth cost analysis to determine the most econornical ailernative way of providing the needed service. Then cjetermine the

uniform and equal revenues for each alternative that would be required at each of years one through five to cover the cost of service at the 15o." before-tax minimum ROR.

Solution, All Values in Thousands of Dollars: compare the economics of three used machines with two new machines for a five year life to get the alternatives on a common service basis for a five year common study period. C=290 A) "3 Used"

B) "2 New"

IT

C=135_

!9=99_99=105 OC=120 OC=66

OC=72 L=1 00

0

C=250

OC=50 OC=60 oc=70 oc=80 oc=90

L=30


146

0.7561 0.6575 0.8696 PW Cost4 = 135 + 90(P/F15o/oJ) + 105(P/F157" ,Z) + 410(P/F157" 0.5718 0.4972 + 56(P/F1 S"/.,4) + Q2 - 100)(P/F1S7",S) = $586

$,

0.7561 0.6575 0.8696 PW Costg = 250 + 50(P/F1 S"/o,1) + 60(PiF1 S"/",2) +7O(P/F15"/oS)

0.5718 0.4972 + 80(P/F1 5y",4) + (90-30)(PlF6"1o,g\ = $460, select min. cost

Alternative "8" with the minimum present worth cost is the economic choice.

On a before-tax anatysis basis, equivalent annual cost represents the equivalent annual revenues required to cover cost of service at a specified minimum ROR as the following calculations illustrate. Annual Revenues = Annual Cost = (PW Cost)(A/Pi*,n) 0.29832 Annual Revenues4 = ACA = S86(A/P1 S"/",5) = 174.8 0.29832 Annual Revenuesg = ACB = 460(A/Pt S"/o,S) = 13'7.2, select minimum The alternative with the minimum revenue requirement is the economic choice. This is alternative "8" which in this case and in general, always agrees with the cost analysis results.

A class of sen'ice investments requiring additional economic evaluation consideration is the decision as to which type of environmental remediation is best for a given site, with remediation work having po impact on income generation. In many situations there are several alternatives that might be considered, all of which provide the same service of cleaning the site, but the time required may vary by ten years or more. For example, in cleaning contaminated ground water the alternatives may include the use of activated carbon absorption versus ultra violet radiation, or maybe bio-remediation. In these situations the alternatives may be treated as all providing the same service over different time periods with no impact on income generation. If

rr

chapter 3, Present, Annual and Future value, Rate of Return and Break-even

Analysis

142

rncome is afiected

in any way by the ser'ice provided, income must be of iir,: aiicrnaiives. If income is rrot affectecl Lry the remediation or service rrork. then service aliernatives can be fairlv compar"ed us*rg clifferent ser_ 'reL'i,ics ior pioviding the remeiliation or generai service in alternative r".L1s. Tiie following e.xantple illustrates sen'ice analysis fbr clitferent serincludecl in ttre analysis of each alternative lbr valid economic evaluation

r rce lii'e alternatives that have no impact on inconre generation.

EXAMPLE 3-31 Environmentat Hemediation Alternatives Two soil vapor extraction alternatives are being considered in ilre cieanup of a former plant site. For a minimum rate of return of 10.0% per year, which option would you consider to be economically best? Use present worth cost and net present value analyses in supporting youi- cecision assuming the same remediation service is realized r,sing either alternative 1 or 2 with no impact on revenue generation. Alternative 1 woutd involve drilling a total of twelve wells on a 50 foot radius. The well costs are incurred at time zero and are estimated to be $10,000 per well. End of year operating and mainte_ nance costs are estimated to total $1,000 per well per year. The total iiie of this operation is estimated to be five years. Alternative 2 would involve drilling a total of six wells on a i00 fcot radius. The well costs are incurred at time zero at a cost of s10,0u0 per weil. End of year operating and maintenance costs are estimated to total 91,000 per well per year. The total life of this operation is estimated to be fifteen years.

Solution:

-.. .-120,000 -12,000

o

1....

...._-l2,0oo

.....5 3.7908

PW Cost: -120,000

-. 2^-60,000 Att 0

PW Cost: -60,000

t

-

12,000(P1A1g"7",5) = -$165,490

-6,000. 1.... -

.._6,000

...15

7.6061 6,000(PiA rc%,15) = -$105,637, Select A2.

148


Att 1_2 -60,000

-6,000 -6,000 6,000 1........,5

3.7908

NPV = -60,000 =

_959,953

-

.6,000

. 6.........15 6.14# 0.6209

6,000(P/A10%,5) + 6;000(P/A 1Oo/o,1gXP/F1g7",5)

The NPV is less than zero, so reject the incremental investments in A1, select 42.

3.18 Summary several decision criteria were introduced in this chapter to help investors evaluate a variety of different investments. Those methodologies and related issues are summarized here along with the related acronyms. 1. Rate of Return, i

(RoR or IRR in Many calculators and spreadsheets)

Rate of return, "i" is the compound interest rate received on the unpaid portion of the dollars invested over the project life. It is also defined as the compound interest rate that makes NPV equal zero. Solving for RoR involves a trial and elror process and interpolation between the two interest rates that bracket a present worth equation equal to zero. To illustrate the meaning of "i" the cumulative cash Position Diagram was developed to show how this measure of profitability is identical to the compound interest rate received in a savings account. The RoR is compared with other returns from other perceived opportunities to determine if a project is economically acceptable.

2. Financial Cost of Capital (FCC)

Two schools of thought exist in terms of what should be considered in establishing an investor's minimum rate of return. The theory in utilizing the financial cost of capital is based on the assumption that financing is unlimited and that as a minimum, a company can always use excess cash to pay off existing loans or buy back existing stock. Therefore, a weighted average cost of capital is determined based on the investor's average cost of debt (normally after-tax recognizing the deductibility of most interest) and the cost of equitlz (stockholders desired return determined using the capital Asset Pricing Model, CAPM). The company's debt/equity ratio is then applied in proportion to each. This approach should not rely on the current

(inapter ll. Present, Annual and Fulure Value, Rate of Return and Break-even Analysis

149

uost of capital but rather, thc- expected cost oi capital over the period of time Ior which cash flows will be discounted. Although not advocated, FCC is a (lo;Irnoit t:irsis for determining a conlp.rn), mi;-,imurn rate of returlr. i.,.

t3. Opportunity Cost of Capir:rl (OCC.

i*)

l'hg r-rpportunity cost of capital is the preferred approach to esteblishing lir'i invest()r's minimum rate of return. The oCC is based on the "*p..t"d rciur-irs tr,r be generated in future years (mavire the nert l_15 vears) from ii:',,cstmeni in new projects. Although this approach ,Joes not adhere to any specific coctrine. in general it is the a\,erage retum measured in terms of corxPolrild interest, in-!'estors are looking for and beiieve to be sustainabie ovcr thc long mn. Opportunity cost of capital may also tre referred to as a dis.:ount rate anC is identified in the text by ,.ir.." .1.

Hurdle Rates

Hur,Jle rate is a common term interchangeable- wiih opportunit.v- cost used invcstors who discount at their perceiveti financial cost of capital, FCC. Investor's realize thc FCC does not represerlt the return tney are wilting to accept iiorn the investtnent of their funds. In this situaiirrn, investors fbrce p'iojccts to equal or exceecl an irnposed "econonric hurdje,'' such as a larger di-ic()Lult ra.re (rate of rcturn) or a set clollar value lor Npv, or a pvR of iay 0..'5 rnsiead of zero. The hurdie is a u,ay,of recognizing that projects earning

bf

the fuiancial cost of capital ininimum rate of return do noi add value to a portfolio but are a breakeven economically with those other perceived opportunities. This approach is very common, but may lead to inconsistent economic conclusions if compared to proper use of a discount rate based on opportunity cost of capital. 5. Reinvestment l\Ieaning Related to ROR N{any' analvsts and authors have been confused on the subject of the irnplicd reinvestrnent nreaning associated r.vith RoR. There simply is no ;nrplied rc-investment meaning associated with intlividual project compound interest rates of return. However, if an investor or company wants an investment portfolio to grow over time at a given rate, then all projects now and in the future must achieve that level of return to sustain the overall "rate of grorvth."

150


6. Growth Rate of Return (GROR or

MIRR in some spreadsheets)

Growth Rate of Return does consider the reinvestment of funds.but not necessarily at the project rate of return. Instead, project positive cash flow is assumed to be reinvested at the project minimurf,iate of return, i+, which should reflect other perceivecl investment opportunities both now and in the future. 7. Net Present Value (NpV)

NPV is defined as the sum of all cash flows discounted to a specific point in time at the investor's minimum rate of retum, or discount rate, ix. NFv is the measure of value created by investing in a project and not investing money elsewhere at the minimum rate of return. Npv greater than zero is acceptabll compared to investing elsewhere at i*. An Npv equal to zero is a breakeven with investing elsewhere while a negative Npv is unacceptable. The cumulative NPV Diagram illustrated graphicalry the value of one additional year of cash flow and its impact on the overall project NpV. This diagram also introduced the concept of discounted payback and maximum capital exposure, which identified graphically the correct measure for denominators in ratios such as Present value Ratio and Benefit cost Ratio summarized next. Related "value" approaches include Net Annual value (sometimes referred to as a Net uniform Series) and Net Future value. Like Npv, these measures look a revenue minus costs but as described, on either an equivalent uniform series basis, or at some future point in time. 8. Present Value Ratio (pVR, Also known as

Investment Efficiency, or IE) PVR is defined

as;

NPV/ lMaximum Capital Exposurel

It is a measure of present worth profit being generated per present worth dollar invested. This profit is measured in terms of dollars ubou" what is required to earn the desired minimum rate of return. pvR greater than zero indicates a satisfactory project, while a pvR equal to zero is a breakeven with investing elservhere. From the viewpoint of an economic assessment of individual projects, PVR is somewhat redundant in that you already know the project is acceptable by the magnitude of its numerato;, Npv. The utiliry of ratios is described in Chapter 4.

ci

apter 3: Present, Annuar anc Futur€ varue, Rate of Return and Break-even Anarysis

151

9. Benefit Cost Ratio (BlC Ratio, also knorvn as

Protitability rndex, pr, or Discounted profitahility rndex, DpI)

B/C i{aric is dcflned as:

P\\' i:'csitive Cash Florv/ lMaxinrum Capit:ri Exposrr-e

! I

ii/c

I'l'rtio is ir measure of present woi.th revc.ue beir-,g generated per pre_ s;tli l't-ti'th ilollar invested. B/C Ratio results that are g,,i""t". tlian o1e incii*ri': srtisrircrory projects *,hile a B/c Ratio equal to o,ie is a breakeven

with

i,"rstirrg cisei'here, (present w{,r'th re,v,erlues are equal to the present worth ci::ts). if tire Eic ratio is less than zero the result is unsatistactory. I0. Evaluating service Arternatives using rncrementar Anarysis In evaiuating replacement alternatives that involl.e costs, incremental iirilhsis is required to Lrtilize the various income criteria introduced in this i'ilrlpier' Iucremental means difference. So iricremental

calculations involve capital-intensive airernative from an alternative requiring a greater initial inr"estment, or simply, biggc-r initial investment minus sr,allcr. This difference should leave ilincrementai diagram with an initial cosi that generates c'per.ting co,;t sa\ings in future uJo., to prcrvide the tiesired rcturl on the up-f.ant in,,.estl,e*t. To properlr. interpret .ir the increniental cush florv strealn nru.st Lregin ,i,itl.,'u cost. For .v-our analy_ instance. if ,,io!l;1rq are not investerj up front. there is no \\ay you can talk about rate of r(riirrl r.ireariirs in your i'csults. This toprc will be amplificd in chapter.l. sirbtracting a

I

Lr.ss

l. Evaluating Service Alternatives rvith Cost Analysis

Replacement alrernari'es may arso be evaruated individualry by rooking at tire present, future or equivalent annual cost of operating an asset over a s.pccified servicc period. present u,orth cost is by iar the irost common of tlir- three rrerh.ils described, but each is consistent when properry applied. Proper interpretation of sign con'ention is the key consicreiati.n arong with recognition that the e'aluation is a cost analysis, not an income evaluation,

:o rninimizing cost is the key. Fi,ally, with all service evaluations; arternati'es musr be compared based on obraining the same service over the same period of tirne. If an investment alrernative offers additional productivity thit can be ulilized, then adjust_ ments must be made to account for the additional use in comparison with a

Iess productive approach

I


1s2

PROBLEMS 3-1

A foreman in a processing plant wants to evaluate whether to rebuild and repair five existing assets or replace them with togr new assets that are more productive and capable of providing the same service as the current five machines. Four new assets can be acquired at time zero for a total cost of $240,000. The total maintenance, insurance and operating costs for the new equipment is $20,000 at time zero, $40,000 dt yeu one, $50,000 at year 2 and $30,000 at year 3. The anticipated salvage for these assets after three years is $i00,000. The alternative is to repair the existing machines for total cost of $50,000 at time zero. However this approach will realize much higher operating costs over the next three years. In addition to the repair cost, the total operating costs for the repaired assets is estiinated at $20,000 at time zero but that escalates to $140,000 in each of years one and two and $70,000 in year three. The salvage for the existing assets after three years of service is anticipated to be zero. The used machines have no salvage value today in the marketplace due to their current condition. The

desired minimum rate of return on invested capital is a nominal l5.0%o. Use present worth cost analysis and support the cost findings with an incremental analysis using rate of return and net present value to determine which altemative is economically preferred. 3-2

The owner of a patent is negotiating a contract with a corporation that will give the colporation the right to use the patent. The corporation will pay the patent owner $3,000 ayer at the end ofeach ofthe next 5 years, $5,000 at the end of each year for the next 8 years and $6,000 at the end

of each year for the final 3 years of the 16 year life of the patent. If the owner of this patent wants a lump sum settlement 3 years from now in lieu of all 16 payments, at what price would he receive equivalent value if his minimum rate of return is 87o before income taxes? J-J

'A"

has an initial cost of $50,000, an estimated service period of 10 years and an estimated salvage value of $10,000 at the end of the 10 years. Estimated end-of-year annual disbursements for operation and maintenance are $5,000. A major overhaul costing $10,000 will be required at the end of 5 years. An altemate Machine "B" has an initial cost of $40,000 and an estimated zero salvage value at the end of the l0year service period with estimated end-of-year disbursements for operation and maintenance of $8,000 for the first year, $8,500 for the second

Machine

.ir

cirapter 3: Present, Annual and Future value, Rate of Return and Break-ev,r

Analvsis

1s3

year and increasins $500 each year thereafrer. using a minirnum of lOvo, compare the present worth costs of tO--year service RoR from N,iach.ines

*:''l

'4,'

and

(.B,'.

!

A project has an initial cost of si20,000 and an estimarttl sar'ags

y21xe

arler 15 vears of s70,00c. Estimateti average arinuar re.eipts ar-e s25 000. Fstirnuted averd:ie annual clisburserrrrnts iire gl-5,00a). Assuming that ar,rual receipts a,d disbursements r.vilr be unifomi, coilipute the prospec_ tile rate of return beforc taxes.

l-5

A couple plans to purchase a home for $250,000. property raxes are crpected to be s r,900 per yea.r- wrrile insurance piemiu,is are estimated to be $700 per year. Anr:ual repair and maintenance is estimated at $1.'100. An arternative is to..ri o house of about the sarre size for per nronttr (approxiniare usi*g $1g,000 per year) lll00 falments. If a 8.0ozi return before-taxes is the coupie's mi,iinunr rate'of return, what rrr"rst the resare'alue be 10 years from today fbr the cost of o*n".ririf .o equal the equi','urent cost of renting? Finalry, given the brcirrieven year 10 sale value just askecJ for, what is the corresponding annual price escalatir)n oi'de-escalation in the house over the ten y.*ri

3,-6

old 30 l,ear tif'c $1000 bond marures in 20 years and pavs semi_ di'ide*ds ol'$40. what rate of rc-rurn co,rpounded semi_anlrually does rhc bond yicr.i ir r ou pai $g00 tbr it a,d holcl it until maturity

A,

anr-,u^l

assu,ring the analy'sis is rnade on the first day of a new semi-annual divi_ denci period. If the bond is calrable g years fiom now at face value, what could an investor pay for the bond and be assured of recei'ing annual returns of 6vo compounded semi-annuaily? Negrect the possibility of bankruptcy by the bond issuer.

3'7

what ca, be paid for the bond in probrern 3-6 if a 107o return compourded semi-annuaily is desired ancr a bond call is considered a neg_ ligiLrle probabilitv?

3-8

An investor has 950,000 in a bank account at |vo interest compounded annually. She can use this sum to pay for the purchase of a prot of land. She expects that in l0 years she wiir be able to sell the land for $r30,000. During that period she will have ro pay $2,000 a year in property taxes and insurance. Shourd she make the purchase? Base your d"cision on a rate

of return analysis and verify your conclusion with:future

L

'alue

anarysis.

154


3-9 A new poject will require development costs of $60 million

,

at time zero aod $ 100 million ar the end of 2 years, from time zero with incomes of $40 million per year at the end of years 1,2 and3 and incomes of $70 milrion per year at the end of years 4 through l0 with zerotalvage value predicted at the end of year 10. calculate the rate of return for thisproject.

3-10 Equipment is being leased from a dealer for $500,000 per year with beginning of year payments and the lease expires three years from now.

It is estimated that a new lease for the succeeding four years on similar new equipment will provide the same service and will cost $750,000 per year with beginning of year payments. The frst payment under the new lease occurs at the beginning of year four. The equlpment manufacturer is offering to terminate the present lease today and to sell the lessee new equipment for $2 million now which together with a major repair cost of $600,000 at the end of year four shoulc provide the needed equipment service for a total of seven years, after which, the salvage value is estimated to be zero. use present worth cost analysis for a minimum rate of return of 20vo to determine if leasing or purchasing is economically. the best approach to provirJe the equipnrent sen.ice l'or tlie next se\.en years. Verify your conclusion rvith ROR and NpV analysis.

3-l1A

person is considering an investment situation that requires the

investment of $100,000 at time zero and $200,000 at year one to gen-

erate profits of $90,000 per year starting at year two and .urnirg through year 10 (a 9 year profit period) with projected sarvage value of

$150,000 at the end ofyear 10.

A)

Determine the compound interest rate of return for these discrete

end of period funds. Draw the cumulative cash position diagram for time zero through the end of year three at the project rate of return.

B)

Determine the continuous interest rate of return for these discrete investments assuming all dollar values represent discrete end of period funds.

c)

Determine the continuous interest rate of retum for this investment assuming all dollar values represent continuously flowing funds. Treat the time zero investment as a continuous flowing sum (year 0l) and the year I investment as flowing during the folrowing year 12. Profits flow continuously from years 2-3 to years 10-r 1. Salvage flows continuously during year ll_12. .

cnapter 3: Present, Annuar and Future Value, Rate of Beturn and Break-even Anarysis

. D)

Determine the continuous interest rate of retum for this investment assuming the year zero investment is a discrete value and all dollar values after time zero represent conrinuously flowi,g funds. The year one investment is assumed to fl.w from ve# 0-1 and profits

i'iorv r-rssinni,g from year r-2, to )'ear 9-10. Sarvage florvs from 1'ear

E)

ILi-li.

calculate the project rate of return assuming all dollar values after ti*re zero are discrete values realized in the middle of each compounding period. Assume the time zero investment is a discrete sum at that point.

3-12 A firm is evaluating whether to lease or purchase four trucks. The lbur trucks can be purchased for a total cosr of $240,000 and operated tbr inaintenance, insurance and general operating costs of g2O000 at year 0, $40,000 at year l, $50,000 at ye* 2 and $30,000 at year 3 iputting operating costs at tlie closest points in time to r,vhere they are incurred) with an expecte,C salvage value of $100,000 at the end of year 3. The four trucks courd be ieased for $120,000 per year, for the 3 years, with monthly payments, so consider $60,000]"are ."rt at year 0' $i20'000 per year ar vea.s i and 2 and $60,0c0 at year 3, putiing lease costs at the closest point in time to u,here they are incurred. The lease costs include maintenance ct'lsts but cio not include insurance and general operating costs of $1f),000 at year 0, $20,000 per year ar years I a.d 2 a*tr $10.000 ar year 3. If the minimum rate orreturn is r5vo belore tax considerations. use present worth ce-rst analysis to determine if economic a,alysis dictates leasing or purchasing. verify your conclusion rvith ROR, NpV and pVR analysis. 3-13 500.000 lbs per year of raw material are needed in a production operatic;, (treat as end-ol-year require,rents). This material can be purchased for $0.24 per por-rnd or produced internaily for operating costs of $0.20 per pound if a $40,000 machine is purchased. A major repair of $10,000 will be required on the rnachine ifter 3 years. For a 5 year project life, zero salvage value and a 30vo minimum rate of return before tax considerations, determine whether the company should purchase the equipment to make the raw material.

A) B)

A

Use Incremental ROR, NpV and pVR analysis.

verify your result from (A) with present worth cost analysis.

156


3-14 Earnings per share of common stock of XyZ corporation have grown at L2% per year over. the pastren.years and the price of XyZ common stock has increased-proportionary at r2vo p". yi* from $30 per sha.e ten years ago to $93 per share,today. The comtton stock paid annual dividends of $2 per share in years I through 5 and $4 per share in years 6 through 10. Assume the dividenJ, *"r. deposited when received in a money market account yielding g7, interest per year. Determine the growth rate of return over the past decade on money invested in Xyz common stock ten years ago today. Hint: use an analysis basis of one share of common stock bJught ten years ago. 3- 15

A corporation has invested $250,000 in a project that is expected to gener_ ate $100,000 profit per year for years 1 ttlrough 5 plus u $iso,oo0 sarvage

value at the end ofyear 5- It is proposed that the profits and salvage varue

from this investment will be reinvested immediately each year in real

estare that is projected to have a $2,000,000 value aithe end of 6 years from now. calcurate the project rate of retum and growth rate of retum.

3-16 A heavy equipment manufacturer plans to lease a $100,000 machine for 30 months with an option to buy at the end of that time. The manufacturer wants to get. lVo per month compound interest on the unamortized value of the machine each month and wants the_unamortized value of the machine to be 925,000 at the end of the 30th *or,t,. wtat uniform montl'ily beginning-of-month lease payments must the company charge for each of the 30 months so that these payments plus a $25,000 option to buy payment at the end of 30 months will recover the initial $ 100,000 value of the equipment prus interest? what interest is paid by the lessee during the third month of the lease? what is the unamortized investment principal after the fourth payment is made?

3-17

A 20

year loan is being negotiated with a savings and roan company. $50,000 is to be borrowed at gvo interest compounded quarterry with mortgage payments to be made at the end of eacir quarter ou", u 20 yei, period. To obtain the 10an the borrower must pay ..2 points,, at the time he takes out the loan. This means the borrower must pay 2Ta of $50,000 or a $1,000 fee at time zero to obtain the g50,000 loan. Iithe borrower acceprs the loan under these conditions determine the actual interest rate the investor is paying on the loan. This nominar interest rate, compounded quarterly, is often referred to by lenders.as an ..effective interest rate.,,

,i

$

chapter 3: Present, Annual and Future value, Rate of Feturn and Break-even Analvsis

157

l-lint: The eft-ecrive interest rate on a loan is based on the actual doilars available to apply to the purchase of a home, etc. While effective rates Iiom a savingr account viewpoint (see development of Equation 2-9) measured the annuai rate that if compounded once per _v-.Ear, would accrue lirc same interest as a nominal rate conipoitnded "m" times per yezr. 3-18 T\r'o developnrent alternatives exist to bring a new project into produciion. The first de..,elopment approach u'ould involr.e equipment and de'elopment expenditures of $ I million ar year 0 and $2 million at year I to generate incomes of $1.8 million per year and operating expenses of S0.7 million per year starting in year I for each of years I through 10 r,"'hen the project is expected to terminate with zero salvage value. The second development approach would involve equipment and developmenr expenditurcs of $l million at year 0 and experrses of $0.9 million at year I to generate incomes of $2 miliion per year and operating expenses of $0.9 million per year sta-rtirig in vear 2 for each of years 2 through l0 when the project is expected to terminate with zero salvage value. For a minimum rate of reium of l5vo, evaluare which of the alternatives is econornically better using rate of return, net present value and present value ratro analysis techniques.

3-

l9

An investor is interested in purchasing

a company and wants you to deter-

mine the maxirnunr value of the company today considering the opportu-

nity cost of capital is 2A7o. The company assets are in two areas. First, an existing production operation was developed by the company over the past two years for equipment and development costs of $100,000 two years ago, $200,000 one year ago and costs and revenues that cancelled each other during the past year. It is projected that revenue minus operating expense net profits will be $ 120,000 per year at the end of each of the next 12 years when production is expected to terminate with a zero salvage value. The second companv asset is a mineral property that is projected to be developed for a $350,000 future cost one year from now with expected future profits of $150,000 per year srarting two years from now and terminating l0 years from now with a zero salvage value.

158


3-20 Based on the data for a new process investment, calculate the before-tax .. annual cash flow, ROR, NPV and PVR for a l5Vo minimum ROR. Then calculate the break-even product price that, if received uniformly from years 1 through 5, would give the project a157o invtutment ROR. Cost dollars and production units are in thousands: Year

Production, units

012345 62 53 35 24

t7

26.0 26.0 26.0 27.3

Price, $ per unit

28.7

Royalty Costs on the patents are based on l4%o of Gross Revenue Research

& Develop. Cost

Equipment Cost Patent Rights Cost Operating Costs

750

250 670

100

175 193 Zl2 233

256

Liquidation value at year 5 is zero.

3-21 Based on the following data for a petroleum project, calculate the betbre tax annual cash flow, ROR, NPV and PVR for a l5Vo minimum RoR. Then calculate the break-even crude oil price that, if received uniformly from years I through 5, would give rhe project a l57o ROR. Cost dollars and production units in thousands. Year

Production, bbls Price, $ per bbl

012345 62 53 35 24

t7

26.0 26.0 26.0 27.3

28.7

Royalty Costs are based on I47o of Gross Revenue Intangible (IDC) Thngible Equipment Mineral Rights Acq. Operating Costs

750

250 610

100

Liquidation value at year 5 is zero.

175 193 Zt2 233

256

chapter'3: Present, Annual and Futurs. Value. Rate cf Returrr and pfeai(-ever Analysis

't

59

3-22 The following clata relates to a mining project with increasing wasre rock or overburden to ore or coal ratio as the mine life progresses, givirrg declining productirn per vear. calcuiate the before iax annuai cash flow, RoR. NPV and pvR for a 157o minimum I?oR. Then carculate tlie break-eve n price per ton of ore or coal that. ia received uniformly Irom years 1 through 5, rvould gir.e the pro,:ect a i5Zc ROR.. Cust 6s1iu.t and production are in thousancls. Y'e;rr

62 53 35 24

Prcduction, lons Selling Price, $/ton

17

26.A 26.0 26.0 27.3

28.7

Royalty Cosrs are based on t4Tc of Gross Revenue Ir,Iine

Develcpment Equipment

I,liling

750

250

lvlineral Rights Acquisition 100 Operating Cosrs

57A

175 tg3 212 ?33

256

Liquidation value at year 5 is zero. 3-23 A machiire in use now has a zero net salva_ge valLre and is expected to irave an additional trvo years of uset'ul iif'e but its service is needed for another 6 years. The operating costs v,rith this machine are estimatecl to be 54,5()0 tor the next year oi use at y,car I and $5,500 at year 2. The salvage value will be 0 in two years. A replacem"nt -u.hin. is estimated to cost $25,000 at year 2 with annual operating costs of $2,500 in its first year of use at year 3, increased by an aritnmetic gradient series of $500 per year in following years. The salvage value is estimared to be $7,000 after 4 years of use (at year 6). An alternative is to repl,ce the exisring machine now rvitir a ne\\, machine cosdng $21,000 arid annual operating costs of $2,000 ar year i, increasing by an arith_ nretic gradient of 9-500 each following year. Tlie sah'age value is estimated to be $4,000 at the end of year 6. Compare the economics of these alternatives fbr a minimum RoR of 207o using a 6 year life by:

A) Present worth cost analysis. B) Equivalent annual cost analysis C) Incremental NpV analysis

TT


160

3-24 Discount rates on U.S. treasury bills (T-Bills) are different from normal compound interest discount rates because T-Bills interest effectively'is paid at the time T-Bills are purcha.sed'Iather than.at the maturity date of the T:-Bills. Calculate the nominal annual rate of return compounded semi-annually to be earned by invesfing in a 6-month $10,000 T-Bill with a 5Vo annualdiscount rate. 3-25

l, I

I

company wants you to use rate of return analysis to evaluate the economics of buying the mineral rights to a mineral reserve for a cost of $1,500,000 at year 0 with the expectation that mineral development costs of $5,000,000 and tangible equipment costs of $4,000,000 will be spent at year 1. The mineral reserves are estimated to be produced uniformly over an 8 year production life (evaluation years 2 through 9). Since escalation of operating costs each year is estimated to be offset by escalation of revenues, it is projected that profit will be constant at $4,000,000 per year in each of evaluation years 2 through 9 with a s6,000,000 salvage value at the end of year 9. calculate the project rate of return, then assume a 157o minimum rate of return and calculate the project growth rate of retum, NPV and PVR.

A

3-26 For a desired rate of return of 15Vo on invested capital, determine the maximum cost that can be incurred today by an investor to acquire the development rights to a new process that is expected to be developed over the next two years for a $1.5 million cost at year 1 and a $2.0 million cost at year 2. Production profits are expected to start in year 3 and to be $1.0 million per year at each of years 3 through 10. A $3 million sale value is estimated to be realized at the end of year 10. If the current owner of the development rights projects the same magnitude and timing of project costs and revenues but considers l07o to be the appropriate minimum discount rate, how does changing the discount rate to l0o/o from 15Vo affect the valuation?

I

l

I

Lraptei'3: Present, Annual and Future Value, Rate of Reiurn and Break-even Analysis

161

-q-17 To achieve a 2avc rate

of return on invested capital. determine the maximum cost that can be incurred today to acquire oil and gas minerai rights to a Dropertv that r.vill be developed 3 years from nor,v for estimated escalared doliar (actualy drilling and well eompletion costs or Si,,i00,0t"r0 r.',,iih an errpected net profit of $i,50r_i,000 r,rojected to L're generateci at year 4 ar;d decrining by'an arithmetic gradient of s:00.u00 per vear in each vear afrer year 4 foi- eight y,ears of production (evaluation vears 4 through li). sal'age'alue is

zero.

Il

estimated to be the oil and gas mineral rights are already owned b1. an investor

whose minimum RoR is 20vo, what range of sale'alues for these mineral rights rvould make the owner's economics of selling better than holding the property for development 3 ye.ars from now for the steted profits'?

3-28 Your integrated oil and gas company ov/ns a l\CIvo working interest in a lease. The iease cost is considered sunk an
Drilling

Intangible $250,000 Thngible Completion $100,000 ProdLrction

(bbls/yr) (g/bbl) (g/bbl)

Selling Price operaring cost Royalty (12.5Vo Gross, or in

17,500 9,000 6,500 $20.00 $20.00 $21.00

$4.00 $4.00 $4.00

$/bbl) $2.50 $2.50 $2.625

3.000

$22.05

$4.00

$2.756

162


3-29 Evaluate the economics of problem 3-2s it you were to farm out the property described in that statement based on the following terms. You would re ceive a 5 .OVo overriding royalty (5 .TVo of gross revenues) untii the project pays out, Payout is based on un$scounted ,net revenues minus operating costs' to recover before-ta-x capital costs estimated at $350,000. Because of the additional overriding royalty, the producer will realize a 82.5Vo net revenue interest to payout. For this example, neglect any potential sale value or abandonment costs in year 4. Upon payout, your company would back in for a 25.07o working interest and a27.875Vo net revenue interest (25.0Vo of 87.5Vo). After payout the 5.lvo overriding royalty terminates.

3-30 A gas pipeline manager has determined that he will need a compressor to satisfy gas cornpression requirements over the next five years or 60

months. The unit can be acquired for a year zero investment of $1,000,000. The cilst of installation at time zero is $75,000 and a major repair would be required at the end of year 3 for an estimated $225,000. The compressor would be sold at the end of year five for $300,000. The alternative is to lease the machine for a year 0 payment of $75,000 to cover installation costs and beginning of month lease payments of $24,000 per month for 60 months. Lease payments include all major repair and maintenance charges over the term of the lease. The nominal discount rate is l27o compounded monthly. Use present wofth cost analysis to determine whether leasing or purchasing offers the least cost method of providing service. verify this conclusion with equivalent monthly cost analysis.

3-31 A new $10,000, ten year life U.S. Treasury Bond pays annual dividends of $800 based on the 8.07o annual yield to maturity. (A) After purchasing the bond, if interest rates instantly increased to !O.OVo, or decreased to 6.0Vo, how would the bond value be affected? (B) If the bond has a 30 year life instead of a ten year life, how do the interest rate changes to l0.0%o or 6.0Vo affect the value? (C) If the 30 year 8.0Vo bond is a new $10,000 face value (yr 30 maturity value) zero coupon bond instead of a conventional bond, how is value affected by the interest rate changes?

c"apter 3: Present, Annuar and

Futr,rre Varue, Rate of Hetuin and Bi.eak-even

Anaivsis

163

3-i2 An existing remediation system will require end of year l though 7

operating and maintenance bosts of $50,000 each year to complete the An upgrade to this system is estimated to cost $75,000 today 1at time zero). This investment is expected to reduce {he current annual opcrating and mlintenirrce costs b1 $15,000 each year. Further, the upgrade modifir:ations u'ould accelerate the remediation process with cieanup compler,-'d alier just 5 years irorn today. pi"ocess.

For a minimum rate of retum of 9.0% per year, should the current system be retained or modified? use present worth cost analysis and verify your results u,ith net present value analysis.

3-33 Two remediation alternatives are currently under consideration. you have been asked to evaluate the economics of each alternative. Alternative I involves the installation of an active vapor extraction system with off gas trearmeirt. This alternative has a time zero cost of $170,000 and an operating life of 5 years. operating and maintenance costs arc estimated at $35,000 at the end of each of years one through five. Estirnated salvage at the end of yezr s is $50,000. Alternative 2 in'olves a bio-remetiiation process that will require l0 years of treatment rvith q,arierr\/ sarnpling estimated to cost $12,500 per quarter through the end of year rcn. For a minimun'I rare of return ol 9.ovo compounded quarteriv, should Alternative I or Airernative 2 be selected'l Use p..r"nt wr-rrth cost anaiysis and verif\ your resurts with net present value anarysis. 3-34 Evaluation of tu'o off gas rrearment options for a project with an estimareci life of five years is being considered. For a minimum rate of return of 10.07c compounded monthry, use present worth cost and net present value analyses to determine rvhich alternative is best. option 1 use a catalvtic converter with a time zero capitar cost of - end of month operating s70,000 and and maintenun." .ori, of $1,000 per mouth.

option 2

-

Install two r,000 pound carbon canisters at a cost of

$7,500 each at time zero. Replacemelrt of the carbon is estimated to be required six times in each of years one and two, fcur times in year three, two times in year four and one time in at the end of month six in year five (middle of the year). The cost of carbon replacement is estimated to be $4.00 per pound.

E

CHAPTER 4

,

MUTUALLY EXCLUSIVE AND NON.MUTUALLY EXCLUSIVE PROJECT ANALYSIS

For economic anarysis purposes, income-producing and service-produc-

ing alternatives must be broken into two sub-classifications which are: (l ) c-ontpariso, of mutuaily excrusive alrcrnatives, which means making an analysis oJ several alternatives from which only one can be selected, such

as selectittg the best way to provide service or to improt,e an existing operation or the best way to develop a new process, prodict, mining

op"iorio, o,

ctil/gas reserve; (2) comparison of non-mutuaily excrusivz ahernatives v'hich nleens analyzing severar arternatives which more than otle can from be selected depending on or budget restrictions, such as rankirtg _capital researclt' development exrloratio, projects to determine the best pro-a11 jects to fund with available dollars. Analysis of mutually exclusive alterna_ tives will be presented first in this chaiter with non-mutually excrusive alternative analysis discussed in the ratter pages of the ;il;;. It will be shown that valid discounted cash flow criteria iuch as rate of return, ner present value and benefilcost ratio are applied in very different ways in proper analysis of mutually exclusive und nor-rnutuaily exclusive arternative investments.

4'l

Analysis of Mutuaily Excrusive rncome-producing Alternatives Using Rate of Return, Net Value and Ratios

In any industry, classic illustration of mutually exclusive alternative anal_ ysis often involves evaluation of whether it is economically desirable to improve, expand or deverop investment projects. whenever ylu must make an economic choice between several alternative investment choices, and selecting one of the choices excludes in the foreseeable future being able to 164

chapter 4: Mutuaily Excrubive and Ncn-Mutualy Excrusive project Anarysis

155

in'est in the other choices, ),ou are invorved with mutually excrusive alter_ native anaiysis. In Chapter 3 we have already illustrated iow incremental analysis of differences betwecn alternati.,,e w,a's of providing tlie same ser_ vice was the key to rate of return, net present r.true or raiio anarysis of :iervi.e prr:.iucing alternati'es. It u,ill be illustrated in the ibllowing exam_

pies that increntental runhsis is flrc kcl. to correct analysis of mutuaily

e-tclusive inconte -producitrg arternatit.es v,ith ail dirrru,r.ira cash fr,.tw ctnalysis techniEtes. The rvorJs incremental, difference and marginal are u,sed interchangeably by persons involved in evaluaiion work when ref-er_ ring to changes in costs or revenues that are incurred in going from one alternative to another or from one level of operation to another.

EXAMPLE 4-1 Mutuaily Exclusive lncome-producing Alternatives consider the anarysis of two different ways, "A'r and ,,8,,, of improvi.lq :n existing process. As shown on ths foilowing time diagrams, "A" involves a small costing $50,000 with iavings and sal_ vage as iliustrated. 9.!ange "8" involves a much larger cnange costing $500,000 which includes the "A" chanEes with savings and salvage as shown. Assume $S00,000 is availible to invest "and that other opportunities exist to invest any or all of it at a 1s% RoR. which, if either, of the mutuaily excrusive arternatives ,,A,, and ,,8,, shourd we select as our economic choice? use RoR anarysis, then verify your economic concrusions with Npv, NAV NFV and irVB anarysis.

solution; HoR Anarysis, vatues in Thousands of Doilars C = CoSt, | = Savings, L = Salvage Value A)

C=50

0

l=50

l=50

1...........5

L=50

PW Eq. 0 = -50 + S0(p/A;,5) + 50(p/F1,5) By trial and error, i = RORA 1AA"/o > i"=157o, So satisfactory = B)

c =l9Q 0

l=250

1...........s

L=500

PW Eq. 0 = -500 + 250(p/A;,5) + 500(p/F;,5) By trial and error, i = RORB = 5Oo/o > i"=1S7o, so satisfactory

166


The RoR resurts can arso be obtained by dividing annuar savings by initial cost and multiplying by 100, since initial cost-equals satvagJ. Many people think in terms of the project with the raigest rate of return on total investment as aiways being economicatlyiest, but in fact, the largest rate of return prolect is not atways tf;e best economic choice. ln this case, although ,,A,, has a totai investment iaie of return of 100%, which is twice is large as the,,B,,rate of return,

the investments differ in magnitude by a factor of ten. A smaller RoR on a bigger investment often is economicaily better than ger ROR on a smalrer investment. rncremental analysis a bigmust be made to determine if the extra, (or incrementar) ga50,000 that will be invested in "B" over the required "A" investment, wili be generat-

ing more or less profit (or savings), than the $4S0,000 would earn if invested elsewhere at the minimum rate of return ol 1s%. The incremental analysis is made for the bigger project ,,8', minus the smaller

project "A" so that incremental ibst'is ioitowed by incremental income giving:

B_A) C=450

0

l=200

l=200

1...........5

L=450

PW Eq. 0 = -450 + 200(p/A;,5) + 450(p/F;,5) ;

= RORB-

A=

44.4"/"

It should be clear from an economic viewpoint that if $500,000 is available to invest, we would be better off with all of it invested pro_ ject "8". our incrementar anarysis has broken project ,,B,, in into two components, one of which is like project ,,A," and the other is like the incremental project. Selecting proleit ,,8', effectively is equivalent to having $50,000 of the capital invested in project ,,A; earning a 1oo"/" RoR and $450,000 incrementar investment earning a 44.4./" RoR. s-u1ely_selecting "8" is better than selecting ,,A', which wourd give a 100% ROR on the 950,000 capital invested in ,,A,, and require invest_ ing the remaining $450,000 elsewhere at the 15% minimum ROR. lncreme.ntal analysis is required to come to this correct conclusion and notice that it requires rejecting atternative ,,A,, with the largest ROR of 100% on total investment.

chapter 4:

r"4utuarry Excrusive

and Ncn-r\4ri,:aily Excrusrve pr-o.iect Anary"ris

Evaluation of mutually exciusive multiple investment alternatives (the situation where only one alternative may be selected from more than one investment choice) oy rate of return analysrs requires both tctal investment and incremenial investment r-atu of return analysis. T'he rate of reiurn analysis cancept for muttaity exclusive alternafi'ves is baseci on testrng to sce that each satisia-ctori,, level of invest_ ment meets two requirements as foliows: lr) The iate of return an total individual project investment must be greater than or equat to the minimum raie of return, "i"; (2) The rate of return on incremental investment compared to the last satisfactory level of investment must be greater than or equal to the minimum ion, ,'i"". The largest tevel of investment that satisfies both criteria is the economic choice.Anal_ ysis of total investment rate of return alone will not always lead to the correct economic choice because the project with the largest total investment rate of return is not always best. lt is assumed fl-rat money not invested in a particular project can be invested elsewhere at the minimum rate of return, "i*". Therefore, it is often preferable lo invest a large amount of money at a moderate rate of ieturn rather than a small amount at a large return with the remainder having to be invested elsewhere at a specified minimum rare of return. These evaluation rules and concepts appty to grow,th rate of return anatysis as well as regular rate of return analysis, since growth rate of return is just a special type of regular return.

NetValue Analysis (present, Annual, Future) of Mutually Exclusive Alternatives ,.A,, and ,iB,' To illustrate the application of NpV NAV and NFV to evatuate mutually exclusive investment alternatives these techniques will now be applied to evaruate alternatives "A" and "B" for the previously stated 15orl, minimum RCR.

atC=$50 0 B) C = $500

l=$50

l=$50

I = $250

| = $250

1...........s

1...........5

L=$50

L = $500

't

68

B


-

A) c-=

$$q

I = $2oo

| = $2oo

1...........5

L = $450

.4922

L

3.352

NPV4 = 50(P/At S./",5) + 50(P/F15 "/",5) NPV3 = 250(P/At S"/",5) + S00(P/F1

-

S"/",5)

.14832

NAV4 = 50 + 50(A/F1

So/o,S)

-

S0 = +g1+ZISO

-

500 = +$5g6.60

.29832 50(A/p1S%,S) = +$42.50

NAVg = 250 + 500(A/F1 S"/",5) - SOO(A/PI5%,S) = +$175.00 6.742 2.011 NFV4 = 50(F/At5%,S) + 50 - S0(F/p15%,S) +$286.50 = NFV3 = 250(F/At S./",5) + 500 - 500(F/p1 S/",5) = +$1,1g0.00 we see that all the net value results are positive which consistenily indicates that both aiternatives "A" and ,,B,,are satisfactory since they generate sufficient revenle to more than pay off the invLstments at the minimum RoR of 15"/". To determine which alternative is best we must make incremental net value analysis just as we did for RoR analysis. we can get the incremental net value results either by looking at the differences between the total investment net values for the bigger investment minus the smailer, which is "B-A,, in this case, or by working with the incremental costs, savings, and salvage.'Exactly the same incremental net values are obtained either way.

NPV6-4 = NPVB

-

NPVA = 586.6

-

3.352 .4972 or = 2Qg1p I Al S"t",S) + 450(P lF 1 S"/",5) directly from the incremental data:

142.5= +$444.10

-

4S0 = +$444. 1 0

NAVB-4 = NAVB - NAV4 = 175.0 42.5= +$132.50 NFVB-4 = NFVB - NFV4 = 1,180.0 - 286.5 +$893.50 = ln each case the incremental net value results are positive, indicating a satisfactory incremental investment. The reason it is satisfactory can be shown by looking at the net value that would be received the $450,000 incremental capital elsewhere at ilo1_ilvesting i = 15"/".

cnapier 4: Mutually Excrusive and Nor,'-Mutuaily Excrusive project

c = $450 at i.= 15%'

.F

1.........5

Anarysis

169

,E^t-tn \ a^^ . ^= 450(F/P I St",S) = +$904.95

0.4972 IJPV = 9C4.95(P/F15%.5)- 450 = $0 Similarll,, NAV

-

$0 anij NFV = $0.

fu1oney invested at the minimum RoR, "i*", arways has a zero net value. obviously the positive incremental net vatue results for,,B-A,, are better than the zero net value that would be obtained by iinvesting the money elsewhere at "i*"

ln summary, the net value analysis concept for evaluating mutuatty exclusive alternatives is based on two testsi 1t1the net vatie on to,tit individual project investment must be positive; (2) the incremental net value obtained in comparing the totat investment net value to the net vaiue of the last smaller satisfactory investment level must be positive. The largest level of investment that satisfies both criteria is the economic choice. This is always the alternative with the targest positive net value. This means, if you have a dozen mutually exclu-sive alternatives and calculate NpV, or NAV or NFV for each, the economic choice wili always be the alternative with the lar.gest net value. when you select the mutually exclusive investment alternative with the largest net value as the economic choice, ycu are not omitting incremental anaiysrs. Experience shows that incremenial analysis-always leads to selection of the project with the biggest net value on total investment as the economic choice. you can mathematically convert between NPV, NAV and NFV and therefore you must get the same economic conclusion using any of these techniques NPV = NAV(P/A;-,n) = NFV(p/Fi",n)

Ratio Analysis of Mutually Exclusive Atternatives ,,A,,and ,,8,,

n) c

=l!Q 0

B) C = $500

I

t

l=$50

l=$50

1...........5 I = $250

I = $250

1...........5

L=$50 L = $500

170


PVR4 = NPVA / PW cost = 142.srso 2.g5 > 0, so satisfactory = PVRB = NPVB / PW Cost = 5g6.6/500 1.17 > 0, so satisfactory = Project "A" has the bigger total investmenl ratio byt the smaller project "8" ratio relates to ten times larger investment iatue. Getting smaller dollars of Npv per present worth cost dollar invested on larger investment often is a better mutually exclusive investment

choice. Incremental analysis is the optimization analysis that

answers the question concerning which of mutually exclusive alter_ natives "A" and "8" is the better investment. This ii true with ratios the same as was illustrated earlier for RoR and net value analysis.

B_A) C = $450

o

I = $200

I = $200

1...........5

L = $450

PVRB-4 = NPVB-A / PW lnvestment = 444/450 = 0.99 > 0 satisfactory pro. Accepting the incrementa! "B-A" investment indicates accepting ject "8" over'4", even though the total investment ratio on ,,8,, is less than "A". As with RoR anatysis, the mutualty exclusive alternative with bigger RoR, PVR or Benefit cost Ratio on totar individuar project investment often is not the better mutualty exclusive investment. lncremental analysis along with totat individual project investment analysis

is the key to correct anarysis of mutuaily excrusive choices. since benefit cost ratio equals pVFi plus one, it should be evident to the reader that either pVR or Benefii cost Ratio analysis give the same conclusions, as long as the correct break-even ,itio. of zero for PVR and one for Benefit Cost Ratio are used.

4.2 unequal Life Mutually

Exclusive Income-producing Alternatives and The Handling of Opportunity Costs in Evaluatlons As discussed in Chapter 3, it is important to recognize that when using RoR' NAV or NFV techniques to anaryze unequar life service-producing alternatives that generate revenue, you must use a common evaluation life for all alternatives, normally the life of the longest life alternative. The only exception to this rule involves the evaluation of alternatives that do not have the opportunity to have revenue allocated to them, such as remediation work' Analysis of unequal life income-producing ahernatives is not a prob-

chapter 4: Mutually Excrusive and Non-Mutuaily Excrusive project Anarysis

171

leni rvith NPV or ratio analvsis because time zero is a common point in time for calculating Npv or ratios ot either equal or unequal life altematives. If you have unequal lives for different alremati,res, the tirne value of molle)' consideratiofis are difi-erent in rate oi return. annual value and future vaiLie calcuiirtiolts and you may chocrse the wrong aile5nslir. as being best if lt-ru do not get a coir.lulon evaluation lii'e. This merely meaus that 1-ou inu51 calculate NFV at tlie same future point in tirne for aij alrematil,es. or you inust calculate NAV by s;.readilig costs anci revenues over the same nurnber of years for all alternadves. hr RoR, tret value or ratio anctbsis of Lm.:quol life income-producirtg altenntit,rs, treat all projects as hat,irig equ"al lives whk:h are equal to the longest life project with net reyenues and cosrs of zero in the later rears of shorter life projects. Note that this is not the same technique presented in chapter 3 to convert unequal life sen,iceproducing alternatives that have revenues associated with thim, to equal life aiternatives using either Method 1,2 or 3. when projects have different stalting dates, net present value must be calculated at the same point in time tbr all proiects tbr the results to be comparable.

opportunity cost is the current market cash value assigned to assets alreadl' owned which will be used in a project instead of being sold. passing up the opportunit-v to sell the assets in order to keep and use thenr creates an opportunity cost equal to the foregone market cash salc value. If an asser is not saleatrle, the oppor-tunity cost effectively is zero. Actions taken b1 ma,agement to deiay expenditures mav create a nega-

tire opporiunity cost, or actually add value to the property. An exampie ir.ri'.i'es invcsti*g adtlitio*al capital in a negative profit general business unit (or an off.shore petroleum platform or mining operation) in order to delay abandonment or reclamation costs. As long as the net present value of

the altematives calling for additional investment to defer abandonment has greater value than the net present viilue of abandonment no$,, deferring abandonment would be preferred from an economic vie.,vpoint.

Finally. in analyzing either equar lif'e or unequal life income-producin_: or ser'ice-producing alternatives, c.hangirtg, the mirtimtun discount rate Lnay chartge tlw ec'ottontic choir:e. You cannot use net y,alue or ratio results t:ulculaled at a giv,en discount rate such as l2ch to reach valid economic decisions for a dffirent minimum discount rate such as 25To. you m.ust use net value and ratio results cctlculated using a discount rate representative of the opportunitl, cost of capital for consistent economic decision making. The follorving examples illustrate these considerations.

172


EXAMPLE 4-2 Anatysis of Unequal Life Mutuafly Exclusive lncome Producing Alternatives and Opportunity Costs Analyze whether it is economically desirable to sefthe development rights to a new process or property for a $150,000 cash offer at time zero, or, to keep the rights and develop them using one of two development scenarios. The project net before-tax cash flows for each alternative are presented on the following time diagrams. Use NPV, RoR and PVR analysis techniques to make this economic decision for a minimum rate of return of 15.0%. Then, consider the sensitivity of changing the discount rate to 2o.o%. All dollar values are in thousands. (A) Develop

(B) Develop

(C) Sell

-200:gl_100j00

150. . . . . . 150

23

10

-300 -400 200 2ao

200..

...200

150 10

Solution for i* =

15.07o:

since much confusion exists regarding the applicability of different criteria to different investment situations, this solution looks at each of the decision criteria independent of another to show the overall equivalence of each. (A) Develop

-z!q:35qjll too 150...... 150 3

NPV4 = -200

-

4........8

350(P/F1S,1) + 100(P/A1S,2XplF15,1)

+'t 50(P/A1 5,5)(p/F1 5,3) = -32.37 < 0, so, reject A.

I

10

chapter 4: Mutuaily Excrusive and Non-Mutuaily Excrusive project

ROR4

Anarysis

17g

= 13.1g%

is the compound interest rate that makes NPV = 0. 18.ig"/o

<

15.O"A, so, reject A.

t

PVR4 = -32.87 t[2AO + 350(prF1S,1)] = -0.064 < 0, so reject

A.

All criteria indicate that Alternative ,,A,, is not economically ccnpeti_ ive with rnvesting money elsewher.e at i* 1S.0%. This leaves only = "8" and "C" for further consideration.

(B)

Develop

NPVB = -300

{9100 J00_!00 01 -

2oo.

.

. . . 200

400(P/F15,1) + 200(p/A15,9)(p/F15,1)

= +182.0 > 0, so, B is acceptable.

RORB

=

21.62o/" is the compound interest rate that makes NPV = 0.

21.62"k > 15.0"/", so, B is acceptable.

PVRg

= 182.0/

[3OO +

400(piF15.r)] = +0.2809 > 0,

so, B is acceptable

All criteria indicate that Alternative ,'8,, is economically acceptable compared to investing money elsewhere with equivalent risk at i" = 15.07". (C)

Sell

150 0

-

NPV6 = +150.0 > 0, so, C is acceptable. HOR6 = *o/o > 15.O/", So, C iS aCCeptable. PVH6 = - since your denominator is zero > 0, so c is acceptabre. All criteria indicate that Alternative ,,c,, is also economically acceptable.

174


Proper economic analysis of mutually exclusive alternatives requires incremental analysis to determine the optimum choice. However, as previously illustrated in Example 4-1, a proper incremental analysis will always lead to selecting the alterqgtive with the largest individual NPV. Applying this concept here, Aiternative ,,8,, with a maximum NPV of $182.0 is the economic choice. However, note that "B" does not have the largest individual ROR or pVR. With any type of compound interest rate of return or ratio analysis, you must make a proper incremental analysis when evaluating mutually exclusive alternatives. As just shown, of the three alternatives only "8" and "C" are preferable to investing elsewhere at i* = 1s.o/", so incremental analysis is provided between "B" and "C".

-450

(B-C) Dev - Sell

0

The concept of opportunity cost is formally introduced in chapter g, section 9.3, on an after-tax basis, but notice here that the Alternative "B-c" incremental analysis automatically converts the sale cash flow of +$150 to an incremental opportunity cost of -$150. lf an investor passes up the opportunity to sell for +9150 in order to keep and develop, an opportunity cost equal to the forgone sale cash flow must be built into the economic analysis of the alternatives. lncremental analysis of mutually exclusive alternatives will always properly account for opportunity cost considerations as in this Develop minus Sell analysis where the time zero incremental cash flow of -$450 results from a -$150 opportunity cost and a -$300 development cost. To prove that selecting the largest individual NpV is best consider first the incremental NPV analysis which can be solved for with two different approaches:

NPVg-g = -450

-

400(P/F15,1) + 200(P/A15,9)(P/F15,1)

= +32.0 > 0, the incremental investment

is satisfactory,

accept B. or,

=

NPVB

-

NPVC = 182.0

-

150.0 = +32.0 > 0, accept B.

chapter 4: Mutua,y Excrusive and Ncn-Mutuary Excrusive project Anarysis

175

To support the Npv conciusion that Arternative ,,8,, generates the most economic varue, an incrernentat anarysis is reqiired for both RoR and PVR anarysis. RoRg-6 is the compcurrd interesi rate that makes NPV3_6 = 0. t

HORg-g

=

15.g8% by calculator > i*

=

15.A"/o, so,

accept B.

Note that Alternative "c" (selling), with an infinite incividual RoR is not the economic choice. By serecting "8", the additionar capitar

invested i! "8" is generating a oigger rate of return than if that money were earning the minimum rate of return. i" 1S.07o. That translates = into more value \,vith "8", as was refrected in the Npv anaiysis.

PVRB-6 = +32.0 /{4SO

+ a00(p/F15,t)J= +0.04 > 0, so, accept B.

Analysis of changing the Minimum Discount Rate to Za.ao/o: Frorn the individual economic analyses jrst compreted, cr"iy A[ter.natives B and c have rates of return competitive wiih investing else_ where at i* = 20.011. since the rargest NpV is arways the eccnonrjc choice when evaruating mutuariy exilusive alternatives, onry NFV will be iliustrated here for i* = 20%. NPV4 @ 20"/o = -104.8 < O, so, reject A. NPV3 @ 20"/" = + 3g.5 > 0, so, B is acceptable compared to investing @ i*= ZO.O"/". NPVg @ 20% = + 150.0 > 0, so, C is the maximum NpV select

C.

.. Tl" maximum project NpV is the Arternative c, which indicates the investor shourd sell today and invest the g150 in other opportunities where it could earn a 2o.o% RoR and maximize the investor,s economic value. This same sensitivity to discount rates can be expanded to a graphical format, for a range of i* values. This is often referred to as an Npv profire and is iilustratbd in Figure 4-1.

176


$1,200

br,ooo o 5

G

co o o

$8oo

\.- .'.

$6oo

. :.,.\]

$400

o-

zo)

$200 $o

10%;

($eool

Discount Rate, i* lntersection points between the projects indicate the "i." values that make the prgjects a break-even. They also indicate the incremental rates of return between the respective alternatives. For example, RoRa-c is equal to 1s.gg% as previously calcula.ted- Since selling is a time zero value, its NpV is unatfected by the discount rate, with NPV remaining constant at $150 for all disccunr rates.

Figure 4-1 NPV Profile for Example 4-2, Alternatives A, B and C.

A variation of the analysis in Example 4-2 is presented to show when mutually exclusive projects have different starting dates, you must calculate NPV for different alternatives at the same point in time for results to be comparable.

EXAMPLE 4-3 Analysis of Mutually Excrusive Arternatives with Different Starting Dates With NpV, ROR and pVR.

Re-analyze mutually exclusive alternatives ,,8', and ,,C,, as described in Example 4-2 using Npv, RoR and pVR analyses for i. = 15.0"/", assuming the "8" development project starts at the end of year two, instead of time zero. All values are still in thousands.

-300-400

(B) Develop 0

200..

...200

crrapter 4: Mutually Exciusive and Non-ly'utuallv trxclucive project Analysis

177

150

{C) Sell

1 2 3 Solution for i* = 15.0o/o: NPVg = -300(P/Fr

S.e)

-

4...

...12 +

400(p/F15,3) + ZAOptA15,9)(p/F15,3)

= +137.6 > 0, so, ,,8', is acceptable. NPVg = +150 > 0, so, "C" is acceptable and largest

NpV.

selecting the maximum Npv project "c" (or selling) is the economic choice. This is a different economic decision than was reached for the project timing described in Example 4-2 where ,,8,, Deveiop started at time zero with NpV of +192.0.

lncremental Hate of Heturn Analysis

(B-C)Dev-Sell -150 - -300-400 200

o 1 2T.-:z

PW EQ: 0 = -'150

-

.....200

300(PtFi,z) -a00(ptF;,3) + 200(p/A;,9)(p/F;,3)

By calculator, FOR, i = 14.56"io < 15.0% so, reject Develop, accept Sell

lncremental PVR Analysis

PVRg-g = lncremental

= =

NPV

/ lncremental PW Costs

(137.6 - 150 ) / {1S0 + 300(p/F1 5,2) + 400(p/F15,3)} -12.4 / 639"83 = -0.0194 < 0

so, reject Develop, accept Sell

When properly applied, NpV, ROR, GROR, pVR and B/C Ratio criteria will lead to the same economic conclusion in the evaluation of mutually exclusive alternatives. lf projects have different starting dates, for valid NPV analysis of mutuaily exclusive alternatives, project NPV's must be calculated at the same point in time before proper interpretation of the results can be ma'de.

178


EXAMPLE 4-4 Mutually Exclusive Proiect Analysis Case Study

An existing production facility must-be shut,down unless an environmental capital cost of $1so mittion is incurred now at year 0. This improvement will enable production to continue ano @herhte estimated profits of $60 million per year for each of the next g years when salvage value of the facility is projected to be zero. An alternative under consideration would combine process improvement and expansion with an environmental cost change for a cost of $200 million now at year 0, plus $150 million cost at year 1 to generate estimated project profits of $00 million in year 1 and $120 million profit per year in each of years 2 through 8. The minimum ROR is 12ol". Evaluate which of these alternatives is better using ROR, NpV and PVR analysis. Then, change the minimum ROR lo 2S/o and analyze the alternatives using the same techniques.

Solution, allvalues in millions of dollars:

61C=150 l=60

01

B)

I=60 l=60 l=60 l=60 2 4 ......8

l=60 C=200 C-150 l=120

l=120 l=120

l=120 .8

ROR Analysis

A) PW Eq: 150 = 60(P/Ai,g), i = ROR4 = 36.7"/o > i* =

12o/o

B) PW Eq: 200 = -90(P/Fi,r) + t 20(PlAi,)(P/F;,1) i = RORB = 28.60/o > i* =

12o/o

Both projects have acceptable economics, but incremental rate of return analysis is required to determine if the extra incremental investment in "B" generates sufficient incremental revenues to justify the additional $50 million cost at time zero and the additional g150 million cost at year one.

chapter 4: Mutua,y Excrusive and Non-Mutualry Excrusive project Anarysis

B-A) C=50 C=150 l=60

l=60

B

l=60

3

0

-A ) PW Eq: 50 = -'150(p/F;,1) i = RORB-

179

l=60

4......8

+ 60(p/A

A= 2jo/o > i* =

i,Jyprci1 12o/o

So select "8".

Note that once again, the project with biggest RoR on totar investment is not the economic choice. NPV Analysis 4.968

NPV4 = 60(P/At 2y",8)

- 150 = +$148.1 4.564 '0.8929

0.8929

NPVg = 12a(PtAt2.7",)(ptF1zoh,1) _ 90(plf;;. _ZCO ,ry = +$208.7 lncrementar anarysis verifies the serection of the project with the largest total investment NpV which is ,,8,,. NPVB-4

-

NPVB

-

NPV4 =ZOg.7

-

14g.1= 60.6 > 0, so select,,B,,.

lncrementar anarysis.of mutuaily excrusive arternatives arways leads to selection of the investment project with rargesi rvpv on totar investment. often this.is not the project with tre tarjest Rbn or pvR on total investment. However, incrementar anarysil gives tne same economic conclusion with all techniques " of analysis. PVR Analysis

PVRn=r+h=#=o.ee>o PVR.L'=

NPVB

PW Costg

208.7 200 + 90(P/F1 2/o,i)

=0.74>0

180


Which alternative is better, "A" or "8"? "A't is not necessarily preferred just because it has the largest ratio on total investment. As with ROR and NPV incremental analysis must be made and Present Worth Cost "B-A" is taken from incremental "B-A" time $iagram and does not equal Present Worth Cost B minus Present Worth Cost A because of the effect of year 1 income on the year 1 total and incremental investment net costs.

NPVg-4 208.7 - 148.1 PVR3-4 = PW Costg-4 50 + 150(P/F12/"J) = o'33 >

t'

;L'*''ff1'ffi;",.::fitent

with the

See the incremental "B-A' time diagram for verification that the incremental costs are 50 in year 0 and 150 in year 1 . Benefit cost ratio analysis gives the same conclusions following the PVR analysis procedure. Remember that benefit cost ratio equals PVR plus one, and one is the break-even ratio with benefit cost ratio analysis while zero is the break-even ratio with PVR analysis. Change i* lo zso/o: ROR Analysis

A) RORA = 36.7"h > i* = 25"/o B) RORg =28.6/o>i* =25"/o Project ROR for each alternative is greater than the new minimum ROR, indicating acceptable economics for both. lncremental analysis is the optimization analysis that tells whether "A" or "B" is the better economic choice.

B-A) RORB -A=21o/o
chapter 4: Mutuarry Excrusive an.d Non-Mutuaily Excrusive project Anarysis

181

NPV Analysis.for i* =21o/o 3.329

NPV4 = 6O(P/AZSy",A)

-

150 =

3.161

+$49.24 ,

0.8000

NPVg = 120(PlAZ57.,)(P/F2So/o,1)

-

0.8000 90(p/F25

"7",1)

-

2OO

= +$31.46

At i* = 25"/o, alternative "A" gives the largest NpV and is therefore the correct economic choice. lncremental evaluation confirms this:

NPVg-4 = 31.46

- 49.74=

-18.28, so reject,,B,,

. Note that you cannot use the Npv results calculated for a 12o/o discount rate to achieve valid economic conclusicns for a 2s% discount rate. PVR Analysis for i* = 27r/o

PVR1 =

PVRg =

NPV4 PW

Cost4

NPVg

49.74 1S0

=.33>0 31.46

PWC"ttB=2@='25>o

PVRe-4=

pwc"ffi=ffiNPVg-4

31.46

49.74

,,8,' = -.'1 0 < 0, so reject

NPV and incremental RoR and pVR indicate that alternative ,,A,, is the correct investment choice when the minimum rate of return is2s"/". Note that NPV and PVR must be recalculated at the appropriate minimum RoR in order to make correct economic decisions. ln this example, NPV and PVR at i* 12/o cannot be used to evaluate = the alternatives when the minimum rate of return has changedro 25%.

&

182


4.3 Mutually

Exclusive rnvestment Anal.ysis using Growth Rate of Return and Future Worth profit Methods

'

It

was mentioned in the summary of the Example 4.1 Res. analysis that Grorvth RoR analysis is applied to evaluate mutually excluiive alternatives in the same way regular RoR analysis is applied. This means calculating

Growth RoR on both total investments and incremental investments to verify that both are greater than the minimum ROR, ,,i*,,.

Looking at future worth profit for decision purposes is just a variation of the Growth RoR or Net Future value evaluation techniqu"i. Th, objective of all investments from an economic viewpoint is to maximize the profit that can be accumrilated at any specified future point in time. from a giien amount of startirtg capital. Instead of using analysis methods such as RoR, Growth RoR, NPV NAV NFV or PVR to achieve that investment objective, another ralid evaluation approach is to directly calculate the futuie worth profit (future value) that can be generated by investing a given amount of capital in difl'erent*'uvays and assuming the prohts can be reinvested at the minimum RoR, "i""', when the profits are received. The investment choice that gives the maximum future worth profit is the same choice we would get using RoR, Growth RoR, NPV NAV NFV or pvR analyses. The following ple illustrates the Growth RoR and future worth profit techniques. "ruo,-

EXAMPLE 4-5 Growth RoR and Future worth profit Analysis Use Growth RoR analysis for a nrinimum rate of return of 15% to determine which of the following mutually exclusive alternatives, ,,A,, or "B", is best. Verify the result with future worth profit analysis and NPV analysis. All Values in T'housands of Dollars, A)

C=200

c

= cost, I = lncome, L = salvage

l=80

l=80

0

B)

C=300

l=150

L=140

Declining Gradient

L=200

L=0

chapter 4: Mutuary Excrusive and Non-N4utuary Excrusive project Anarvsis

183

Solution: Both Growth

FroR and Future worth profit anarysis, as weil as Fii'}v, NAV I\JFV and reguiar RoFi anarysi. ur.qEn" ttiaiiesiouar cap_

itai nct invested in one.of the projects and iircomes as they are leceiygd can be invested ersewhere at the minimum RoR, which is i =15"/" for this anaiysis. This gives the foilowing orowtn RoR and future worth income (profit)

.r,.ilrtion.,

Growth Bate of Return Analysis

Alternative "A,,i

A) C =-?e9

l=80

0.1

Reinvest "l" and ,,L,at i"

=

l=80

L=200

1S"h

C=200 C=80 .8

C=80

F = 1,298

where F = B0(F/A 1b"/",g) + 200 = +1,29g A + Reinvestment of lncome

C=200 F = 1,298

Growth ROR4 pW Eq: 200 1,29g(p/F1,6) = Growth ROR4 i = 26.Syo > i" .157o, so satisfactory =

Alternative "8,':

B) C=-300 l=150

0

l=140

declininggradient,'L=0

- 15"/o _____9= 1s0 C = 140 declining gradient O l----,.......8

Reinvest lncome ali*

where F =

['t S0

-

1

O(A/G1 S1l^,g)]F/ nlSy",e) = 1,627

F=1,677

184


B + Reinvestment of lncome C

=_t99_

F = 1,677

0

Growth RORg PW Eq: 300 = 1,627(plFi,g) Growth RORg, i = 24.1o/o > i* = 1]5/o, so satisfactory

lncremental Growth ROR Analysis (.,8-A',) (B_A) C = 100

F=379

Growth ROR3_4 PW Eq: 100 = 379(p/F;,g) Growth ROR3-4 = i = 1 8.4/o >i* = 157o, indicating Select,,B,, Project "B" does not have the largest GroMh RoR on total investment but incremental analysis indicates that "B" is the economic choice.

Future Worth Profit Analysis Future worth profit analysis uses the future worth profit calculations from the Growth ROR analyses. lf $300 thousand is available to invest, putting.$200 thousand of it into alternative "A" and reinvesting the profits at = 15% will generate a future value of g.l,2gg thousand in I years. lnvesting elsewhere the $100 thousand of our g300 thousand that is not needed if we choose ,,A,,gives:

i

(B-n; c =

3.059

100

1...........8

F

= 100(F/P1S%.8) = $305.90

Therefore, the total future value g years from now of $200 thou_ sand invested in project "A" and 9100 thousand elsewhere is g'1,2gg thousand + $30S.9 thousand or g,|,604 thousand. This is not as great as the future worth of "8" calculated to be g1,677 thousand, so select "8". This is the same conclusion reached with Growth RoR analysis. NPV analysis indicates select,,B,, as'follows:

chapter 4: Mutuarry Excrusive and Non-rJutualry Excrusive project

Npv4

=

-200

Anarysis

1g5

. sop,/ll:;,g + eoo(#i33rlr, = +$zz4

NPVg = -800 + [1S0

-

2.781

4.4t7

10(A/c1s%,g)Xp/ArSZ.e) = +$248

ordinary compound interest RoR anarysis cannot be used on this problem and many similar problems because of difiiculties eflclr.;r_ tered with the incrementar RoR anarysis. Evaruation of ordinary com_ pound interest RoR on totar investment shows that the rates of return on projects ,'A,'and ,,8,, are satisfactory. tncre*entrt gives the following time diagram:

,;;;;

(B.A)

C=

100 l-70 1

l=60.....1=0 2........8

L=

-200

This time diagram has inbrementar cost foilowed by incrementar income followed by negative incrementar sarvage vatue which is the same as incremental cost. This is the type of inalysis situation that generates the duar RCR probrem discussed rater in this chapter. Regular RoR anarysis cannot be used in this situation for reasons that will be given iater. This is why it is important to be familiar witn other techniques of anarysis src-h as Growth RoR, Fr-,ture worth Profit, NPV NAV l',jFV or pVR.

.

4.{

Changing the }Iinirnum Rate of Return with Time The minimum rate of return (opportunity cost of capital or hurdle rate) represents the rate of return that we think we could get by investing our rTl(,r'Iey elservhere, both now ancl in the future f,rr the p"iioa of tlln. coverecl by the project evaluation life. There is little reason to expect that our other ,pportunities rvill remain uniformly the same over a long period of tirne. while other opportunities for the irwesrmenr of capital ,o* -oy i" *"I'= lAcic, we may expect a major project with a proiectid 207c RoR to be deveroped starting three years from norv which could absorb all of our available capital and raise "i*" t-o-zoEo. For anaryses with minimum rates oJ return that change with time, Npv NFV pvR a-nd Future worth profi:.t anarysis are recontmended as the best and reail1' the onry usa.ble analysis methods. Regu_

lar RoR and Growth RoR are not always consistent

o"i.i",

"riteria

if

you

186


do not have a single minimum RoR to which you can compare them. Similarly, you cannot calculate NAV,with different,minimum rate of return values at different points in time. For analysis simplification reasorls, most investors including major companies assume theii minimum Rofr is uniform and equal over project evaluation lives. However, changing the minimum rate of return is illustrated in the following example to demonstrate proper economic analysis techniques for this situation.

EXAMPLE 4-6 Effect of changing the Minimum Rate of Return With Time

compare the economic potential of mutually exclusive investments "A" and "8" using NpV and rate of return anaiysis. Assume the minimum rate of return is 30.0% in years one and two, changing to 12.a"/o in years three through ten. Then re-evaluate the investments using a 12.0% minimum rate of return over the entire ten year project life and then again using a 30.0% minimum rate of return over the entire ten year project life. comparison of these results emphasizes the importance of the minimum rate of return in investment decisionmaking from an economic viewpoint. A)

B)

C=20 l=10 l=10 C=30 l=12 l=12

l=10

l=10

3.

.10

l=12

l=12

L=20 L=30

Solution, All Values in Thousands of Dollars: Net Present Value, Changing i* From 30olo in years 1 &2 to 12o/" Over Years 3 to 10 NPV4 = -20 + 10(P/A3g,2) + 10(p/A1 2,e)(p/fr1,r1 + 2O(P/F 12,6)(P/F3g = +22.ZB ,2)

N

PVg = -30 + 1 2(P l Agg,2) + 1 2(p t A1 2,g) (p /F gg,2) + 30(PiF12,8)(P/FA0,2) = +28.78

These results indicate virtual break-even economics between ,,A,, and "8" with a very slight one thousand dollar present value advan_

chapter 4: Mutuaily Excrusive and Non-Mutuaily Exclusive project

Anarysis

1g7

tage to "8." However, if we consider the minimum rate of return to be uniform and equar over time at either 12.0% or 30.070, as most investors usually do, we get different results. Net Present Value, i*

=

1i2o/oover

the entireirolect Life

NPV4 = -20 + 10(p/A1 2,10) + 2O(P/F12JA) = +42.94 l{PVg = -30 + 1Z(plA1Z,td +30(p/F12.tO) = +42.46 Select B v;ith Largest NpV Net Present Value, i* 30% Over the Entire project = Life NPV4 = -20 + 10(piA3g,1g) + 20(piF3O,1 0) = +12.17 Select A with Largest NpV l.JPVg = -30

+ 12(p/Agg,1g) + 30(piF3O,10) = +9.27.

Assuming uniformry equar discount rates of 12.0% or 30.0% has . given different economic conclusions than the break-even economic conclusion reached by changing the discount rate with time in the ini_ tial analysis. Most companies oL not want to get involved in the addiiional level of confusion invorved with changin-g ilre oiscount rate with respect to time. Therefore, even thougn a Joripanv Lno* they have other opportuniti". investing .rpit"t at a reraiiv"ry nigr, rare of J?f return such as 30.0% for the next'severar years, tottoweo by an assumed rower rate-such as 12-o"/o, tl"y simprify the anarysis by using a 12.0% rate of 'return over the entire evaluation life. This exam_ ple shows that such a simprifying assumption, with respect to the evaluation discount rate, can nivJan effect on economic investment decision-making. Rate of Return Analysis

I,v_Eq.A 0 = -20 + 1l(lfi,r6) + 20(p/F;,16) By Trial and Error. i = RORA lbO.OV; > i*'="C0.07o

or

12.0o/o,

so satisfactory

lW_Eq B 0 = -30 + 12(pifi,ro) + 30(p/F;,1s) By Trial and Error, i = RORfi'J 1O.OU > i*'-50.0% or 12.Ooh, ' so satisfactory


lncremental rate of return analysis is needed now to determine the optimum qhqice .since both "A' and "8" have satisfactory.total invest ment rates of return compared with investing money elsewhere at either i" = 30.07" oI i* = +

12.O"/".

B-A)c19

l=2

0

l=2

!=2.-.'.-,-.-,-.-.'

E

3...........10

t=lo

PW Eq B-A 0 = -10 + Z(PlAi,1O) + t0(P/F;,19) By Trial and Error,

i=

RORB-A =20'0"/o < i* = 30'07o but > 12'0%

The investor cannot tell with rate of return analysis whether the incremental "B-A" investment is satisfactory or not relative to investing elsewhere at 30.0% over the next two years and at 12.O% over the following eight years. The fact that rate of return analysis of projects often breaks down and cannot be used when discount rates that vary with tirne possibly is one of the main reasons that changing the discount rate with time is not more commonly applied in industry practice. A large majority o{ companies emphasize rate of return analysis over the other techniques of analysis and that cannot be done if discount rates are changed with time. Notice that for a uniform minimum discount rate of either Q.A"/o or 30.0% over all ten years, incremental rate of return analysis gives economic conclusions consistent with NPV analysis conclusions. Select "B" if i* = 12.0o/", select "A" if i* = 30.0%. Finally, if firms are utilizing a discount rate based on financial cost of capital, that number is very likely to be changing over project lives the same as opportunity cost of capital changes. once again however, rather than trying to forecast those future changes, most companies use a financial cost of capital calculated today, aS reflecting the average financial cost of capital over the project life. We do not advocate the use of financial cost of capital unless unlimited financing is assumed to exist so that financial cost of capital equals opportunity cost of capital. Remember that it is always opportunity cost of capital that is relevant for valid discounted cash flow analysis of investments.

Chapter 4: Mutually Exclusive and Non-Mutr;ally fy11r.;u" prcjeci Analysis

139

4.5 Differences Between Net varue Anarysis and cost Anarysis Thcre is a tcndency for pe,lpre to get confused c.ncernin-e the .iifference between Present worth (pw), Annual worth (Aw), or Euture *onr, (FW) cost aral-vsis of sei'rice-producrng alterniilives an.i Ner pre-sent Value (Npv), Net A-rmual \alue (NAv) and Net Furure Value iNFV) an*l1,sis of income-producing

aitei-;;atives or dirferences betrveen senice-p;,;d,rcing allernatives. 11ey are simi_

lai'b.rt very difi.'rent because of sigri uoniinirr;n dir.r.rences. co.sr nm!.ysis is

u'seti ro evaius.le ser,-ice-producittg inve:rmerti.t. lvhen L.cst., can1. a positive sign and any revenlies or s,lvag,e a:'e negatiu'e, the net c.ost lpicsen.t, annuar or l,arues discounted

at

.futurc,

"i"')

is a positi,e nurnber wirh this ,ign rorrr"ntionfor minimunt cost option is

Lrnau-Zing altemcfiy,e wa\'s of prcv,idbry a serv,ice, the selected- Net vcilue analysis, horvet,e4 i.s used to asses.i

i*orte-protrucing

"hhu projects or incrcmental differences betw,een seruice-producing pt-ojects usirtg corn'entional cctslt flow' anarysis sigti ctstweniitrt t'herc ,rrt, i,i negadte and t?t'i.turcs are positive, so the altenntit.e yielding nu:xilnrun net v-altts is selecud. To uiliize cost e*ralysis in rhe e'aluation of sirvicetroarcing altematives, you work with the given or estimated costs for each individual alteriative way of pro_ viding a service. To utilize net value anaiysis in service evaluations, you must work with incremental savings that incremental costs will generate. Net v'iue analysis is just a short-cut form of rate of rerurn anelysis. Foinet or rate of tetLlm attalyses, )'ou must look at the incremental 'alue differences betwecn altemative rt ays of providing a se^,ice. The foliowing exarnple illustrates these techniques. EXAMPLE 4-Z A Comparison of present Worth Cost and Net present Value Analysis Criteria

Economic anarysis of the optimum thickness of insuration for a steam line needs to be made for an investor with a minimum rate of return, i. = 12.0"/". Engineering has arrived at the following estimates for installed insulation costs and the annual heat loss resulting for tlre different amounts of insulation. This data is summarized "o.t, in the following table: lnsulation Thickness

0" 1"

2" 3"

Cost of Annual Heat Loss (Per Year

$o

$ 60,000 $ 85,000 $1 18,000

$40,000 $20,000 $10,000 $ 6,000

190


Assume the insulation and project life are eight years with zero salvage, valuq,,and.bgse your analysis resutts on.both present wsrth cost analysis and net present value analysis. The 0"'option represents the current situation, so in calculating net presnt value, compare the current situation with the other amounts of insulation.

Solution: Present Worth Cost Analysis

0" lnsulation

40,000(P/A12,g)

= g1gg,704

1" lnsulation 20,000(PlA12,g) + 60,000 = g1Sg,g52 2" lnsulation 10,000(PlA12,g) + g5,000 = g134,676* Minimum Cost 3" lnsulation 6,000(PlA12,g) +11g,000 = g147,g0s " Selecting two inches of insulation will minimize the present worth cost. This is illustrated in Figure 4-2.

200,000

o 150,000

oo o

100,000

= o a o

o-

50,000

0

0 lnches

1

lnch

2 lnches

lnches of lnsulation Figure 4-2 Present Worth Coit Analysis

3 lnches

Chaprer 4: Mutually Exclusive and Non-MutLrally Erclusive project AnalVSis

191

Net Present Value (lncremental Analysis) For NPV analysis, carcurate the incrementar savings for one, two and three inches of insulation compared to the currenisituation of no insulation. ln other r/,;ords, if $00,000 is spent Today for one inch of ir:suiation, the investor can s.i.,,/e $20,000 in annuai heat loss costs -20,C00 - (-40,C00) = +20,000 savings

1"--0"

2"-0" 3"-0"

20,000(P/A 12,8) 30,000(P/A12,S) 34,000(PiA12,B)

=$c - 60,000 = $3g,352 - 85,000 = $64,028 * Maximum NpV - 118,000 = $S0,ggg

* seleciing two inches of insulation now maximizes the net present vaiue obtainable from these investment alternatives. This is illustraied in Figure 4-3. 70,000 60,000

o

: ;o

(E

o

50.000 40,000

E

30,0c0

i

zo,aoo

o.

10,0c0 0

0lnches

1 lnch

2 lnches

lncremental lnches of lnsulation

3 lnches

Figure 4-3 lncremental Net present Value Analysis

lncremental Analysis (lnch by Inch) lnstead of comparing each alternative with the current scenario of zero inches, each additional one-inch of insulation investment could be

thought of as representing mutuaily excrusivery income producing (savings) alternatives. These alternaiives rangs from doing nothing (0,, option) to spending the money tor 3" of inJulation. whei the alterna_

192

€conomic Evaluation and lnvestment Decision Methods

tives are compared with the current "do nothing,,scenario, the individual economics were determined. Next, the inih-by-inch incremental analysis for each of these mutually excniriv" Oras;;;:'

1o4'

20,000(P/A1e;A)

-

"it"iriiiiv.r'is ,, ---

60,000'E $39;B5Z

b

As discussed in previous examples in chapter 4, the incremental NPV of $39,352 teils us that 1" is better than doing nothing. Therefore, the next level of initial capital investment is coripared to the last satisfactory level, as follows:

2'-1'

10,000(P/A12,g)

-

25,OOO =

g24,676

5a,2" add value over the 1,, option. Note aiso that the NpV2,

- NpV 1,, gives the same result:

$gg,os2 = g24,676 comparing the next level of investment to the last satisfactory level gives: $64,028

-

3'-2'

4,000(P/A1 2,g)

- 93,000 = -$1g,1 29

select the largest level of investment for which incremental economics are satisfactory. This is two inches of insulation as deter-

mined by the earlier incremental NpV analysis. The inch by inch incremental process would be more essential had rate of return analysis been asked for in this problem as investors can't expect individual total investment rates of return to consistenfly determine which alternative is best. ln this example, the results would be the same by relying on individual RoR criieria, but this is not always the case.

Alternative

Rate of Return

1',-0"

29"/"

2"-0' 3',-j',

31"/"

2',-1" 3',-2',

23% 37% -17o

2" is preferred to 1" 2" is pretened to S"

9o, 2" is the largest level of investment satisfying both the individual and incremental economic criteria and is the-ec6nomic choice.

criapter 4: Mutuarly Excrusive and Non-Mutuaily Excrusive project Anarysis

4'6 Effect of Evaruation Life It

193

on Economic Anarysis Resu.rts

n'as illustrared in chapter 3 Example 3-24 thatproject Iife has little

eti'ect on analysrs results when you get biyond 10 orrl5 years, depending on

tite p.oiitabiiity ,ri the projects being evaluated. However, for shorter life pr.;jects ri irh er.'aluation lives under 10 years, the evaluation life used can atiict tiie economic choice significantry. For example, sometimes the life o\cr \\lrich u'e choose to evaluate a process improvement is very *uit*riiy chosen due ro the uncertainty assocLted with irojecting savings in certain prodess analyses. The following illustration shows how evaluation life can affect economic results in this relatively short evaluation life situation.

EXAIIPLE 4-B Effect of Evaruation Life on comparison of rwo Alternatives Evaluate two different levels of improvernent being considerec for an existing process. The new equipment costs anc f,rojected annuar savings in labor and materials are as fcllcurs:

Equipment

projected Annual Savings

.--qg$_ Level 1 $200,000 Level 2 $350,000

$125,000 $18O,0OO

For a minimum RoR of 2oo/o evaruate Levers 1 and.2 using NpV analysis assuming zero sarvage varue for (A) a 3 year evaruati6n rife, and (B) a 5 year evaruation life. (c) For what evaluation rife would there be no economic differences beiween the arternatives?

Sclution, All Vatues in Thousands of Dollars: A) 3 Year Life 2.106

NPVI = 125(P1AZO"/"5)

-

ZOO

= +$63.25 Select Maximum NpV

2.1 06 NPV2 = 1 9O(P/ AZA%,3)- 3S0 +$29.08. =


194

B) 5 Year Life

lncreasing the evaluation life enhances the economics of both alternatives. However, the economics of bigger initial cost alternatives are always enhanced relatively mo!'e rapidly thantmaller initial cost alternatives by lengthening evaluation life (or lowering the minimum discount rate). ln this case the economic choice switches to selecting Level 2 for a 5 year life whereets Level 1 was preferred for a 3 year life. 2.991

NPVI = 125(P|AZO"/",5) -200 = +$173.88 2.991 NPV2 = 180(P/Az O/",s)

-

350 = +$188.38 Setect Maximum NPV

C) Break-even Life "n" When there are no economic differences between the alternatives, NPVl will equal NPV2. lf we write an equation setting NPVl=NPV2 for an unknown life "n", we can solve for the break-even life "n". 125(P / A21o/o,n)

-

200 = 1 80(P/A2g7",n)

-

350

or, 150 = 55(p/A2g7",n) (PlA2g"1",n) = 150/55

=2.727

By interpolation in the P/A1,n factor column of the 20% tables we get o = 4.34 years. Select Level 2 for an evaluation life greater than 4.34 years. Select Level 1 for an evaluation life less than 4.34 years.

4.7 Investment Analysis When Income or Savings Precedes

Costs

\\''hen income or savings precedes cost, ROR analysis leads to the calculation of "i" values that have rate of reinvestment requirement meaning instead of rate of return meaning. These results must be used very differently than ROR results since "rate of reinvestment requirement" results greater than the minimum ROR are unsatisfactory (instead of satisfactory with regular ROR). This is illustrated in Examples 4-9 and 4-10.

Chapter 4: Mutually Exclusive and Non-Mutually Exclusive Proiect Analysis

195

EXAMPLE 4-9 Analysis of Mutually Exclusive Alternatives When lncome Precedes Cost Consider the foilowing problem. Evaluate the. follorving two mutuGrov;thhOR, Future \l/orth Profit, NPV and PVR. The minimum rate of return i* = 107o.

all;v exclusive aliernatives using ROR,

A)

c = $100,000

B)

c

= $100,000

1............5 I = 941,060

L = $305,200

I = $41.060 --='

L = $0

Rate of Return Analysis A) PW Eq: 0 = -100,000 + 305,200(P/F;,5), i = ROF1A =25"/" B) PW Eq: 0 = -100,000 + 41,060(P/A1,5), i = RORB = 30"/o Since the initial costs of projects "A" and "8" are equal, many people conclude there are no incremental differences in the orojects, so, "8" is the choice, since "8" has the larger ROR on total investment. This is incorrect! Looking at "A-8" so incremental cost is followed by incremental revenue we get the following: (remember negaiive incremental income is equivalent to cost) A_B)

C=$0

0

c

= 941,060

c

= $41,060

1............5

L = $305,200

A-B) PW Eq: 0 = -41,060(P1A1,5) + 305,200(P/F;,5) ROR4-3, i = ZO.O"k > 10.0% so, accept "A" Even though project "8" has the largest ROR on total investment, project "A" is the economic choice from incremental analysis. Differences in the distribuiion of revenues to be realized cause incremental differences in the projects that must be analyzed. The year one through five incremental costs of $41,060 per year are referred to as "opportunity costs" by many people since they result from the following rationale. Selecting project "A" causes the investor to forgo realizing the project "B" revenues each year. Revenues or savings foregone are lost opportunities or "opportunity costs", so selecting "A" causes opportunity costs of $41,060 in each of years one through five.

I'

196


lf you look at "B-A", you get incremental income followed by incremental cost so the following rationale-applies:,.,. ;i. ;;

B-A)

C=0

0

l-$41,060 l=$41,060 + c = 9305,200 1............5

B-A) PW Eq: 0 = 41,060(P/A1,S) i = 20.A/"

-

305,200(P/F;,5)

> 10.0% so, reject

,'B,'

(This B-A "i" value does not have rate or return meaning. lnstead, it represents the rate at which funds must be reinvested to cover the year five future cost of 9005,200. See the foltowing discussion)

The incremental numbers and trial and error "i" value obtained, are the same for "A-B". However, note that on the "B-A" time diagram incremental income is followed by cost. tt is physicaily impossible to calculate rate of return when income is fottowed by cost. You must have money invested (cost) foilowed by revenue or savings to calculate rate of return. when income is foilowed by cost you calculate an "i" value that has "rate of reinvestment requiretnent,' meaning. The "B-A" incremental "i" value of 20"/" means the investor would be required to reinvest the year one through five incremental incomes al20"h to accrue enough money to cover the year five cost of $305,200. lf the minimum ROR of 1Oo/o @presents investment and reinvestment opportunities thought to exist over the project life, as it should, then a reinvestment requirement of 2a% is unsatisfactory compared to reinvestment opportunities of 107", so reject ,,8,, and select "A". This is the same conclusion that the "A-8,, ROR analysis gave. Summarizing several important considerations about the RoR analysis for this problem, for the incremental RoR analysis of alternatives "A" and "8" we discussed the need to subtract alternative ,,B,,from alternative "A" so that we had incremental costs followed by incremental revenues. Then we discussed what happens if you incorrectly subtract alternative "A" from "B" as follows: B_A)

C=$0

| = 941,060

1=941 060

c

= g3o5,2oo

chapter 4: Mutuaily Excrusive and Non-Mutuaily Excrusive project Anarysis

1S7

lncrementar "B-A" incomes of $41,060 each year precede the $305,200 incrementar "B-A" cost at the end ot'yeaitive. when income precedes cost, the "i" that we carcurate is the interest rate tnat must be obtained through the reinvestmeet of the income each period, for the finar value ot ine cumuiative incomes and compound irrterest to equal the cost at that time. A required reinvestment rate greater than the minimum RoR is unsatisfactory, whereas an RoB gi-eater than the minimum RoR is sairsfactory. rigure 4-i shows lhe cumulative cash position diagram for this situaion. Note that the cumulative cash position in this example is positive durin! the entire project life. whenever the cumularive cash position is jositive, no investment is invorved and the interest rate ,,i,, means the rate at rvhich money must be reinvested and not the rate of return on investment.

sbo,ooo

"B-A"

264,492

CUMULATIVE

CASH

200,000

POSITION tor i = 20"k

100,000

345 Figure 4-4 cumurative cash position for rncome preceding cost

Given that

the minimum RoR is 10%, do we accept arternative "A" or "B" for the exampre just discussed? As previously mentioned,

if investment preceded income, we wourd accept ariernative ,,A,, because an increm entar 20"/" RoR is better than investing else-

198


where at a 10% ROR. However, if "B-A, incremental income precedes cost, we would be rejecting projec! ,,B',, becausE the "B-A" rate of reinvestment required al2o"/" exceeds the other opportunities we have to invest capital.which is assumed to be @o/o. , Future Worth Protit Anatysis from 9100,000 tnitial lnvestment A) FW Profit = $305,200 6.1 05

B) FW Profit = $41,060(F/A10%,5) = 9250,671 Select Project "A" to maximize future profit. Since the $100,000 initial investment is the same for both ,,A,,and "8", maximum future profit (value) on total investment is desired.

GroMh Rate of Return Analysis A) Growth ROR4 is equal to the regular ROR4, = 2ilyo B) Growth ROR3, PW Eq: 0 = -100,000 + 2SO,6Z1(p/F;,5) GROR3 =i=20.2/o

since the same $100,000 inilial investment is involved with both "A" and "8", we want the alternative with the largest Growth ROR, "A". lncremental analysis gives the same conclusion. A-B) GROR PW Eq: O = 54,529(P/F;.5) GROR4-g = i = o/o > 10.0% SO,' SeleCt "A" OVer "8" Net Present Value (NPV) Analysis 0.6209 NPV4 = 305,200(P/F:o"7",$

-

100,000 = +$89,500 <- Select,,A,, With Max. NPV

3.791

NPVg = 41,060(P/A16./.,5)

-

1

00,000 = +$SS,70O

NPV4-g = 89,500 55,700 = +$33,800 Therefore, Select "A", consistent with selecting the project with the larlest NpV.

-

C,rap1s|.4: Mutually Exciusive and Non-lvlutuaiiy Exclusive prolect Analysis

'199

Present Value Ratio (PVR) Anatysis PVR4 = 89,500 / 100,000 = .895 > 0, acceptable PVFtg = 55,700 / 100,000 = .557 > 0, acceppble 3.1 699 55,700)14i,060(PlA1 A"h,4) = 33,800i130,155 = C.26 > 0, so select "A".

PVR4-3 = (89,500

*

When each of these evaluation methods is properly applied you consistently come to the same economic conclusion. you only need to utilize one method to make a proper evaluation. Here, as in other examples throughout this text, multiple criteria solutions are presented to illustrate the consistent results obtained with any of the evaluation methods. Proper incremental analysis is the key to the evaluation of mutuaily exclusive alternatives where only one alternative may be selected.

EXAMPLE 4-10 Rate of Return, Net present Vatue and Rate of Reinvestrnent in Analyses When lncome Precedes Cost Your cornpany has been asked to consider a proposal to accept a payment of $12 million today and $25 million at the end of one year from now in order to handie cirsposar of a waste product from a facility for each of the next 10 years, (enci of years one through ten). your esiimated costs for disposal of this material include a time zero capital investment of $5 million with end of year one through ten operating costs of $4 million per year. The minimum rate of return is '15.0%. Use ROR and NPV analysis to determine if the company should accept this opportunity?

12.0 25.0 Costs: -5.0 -4.O -4.A. Year 0 Net CF: 7.0 21.0 lncome:

...-4.O

1

NPV @ 15.0o/o: 7 o - 4.0(P/A15,9)(P/F15,.1) ! ?](PiFrs,r) > 0, acbeptable $8.66 =

200


This analysis has income preceding costs, so the meaning of the calculated "i" value that makes Npv eq-ual zero is rate of reinvestment req ui rement, not rate'of i.eturn. Therefoie, a req'uired,rnua-strl.i: r"i" that is less than our minimum rate of ret nu"ri* nt o p'p oiii, n ti e s) s acce ptanr Ji'x' ;[Et] " ment greater than i* = 15.0% wourd be an unacceptabre project. PW EQ: 0 = 7.0 + 21(ptFi,1) - 4.0(p/Ai,9)(p/F1,1) i

i

#?;?y:i ili?'l*f

i

Rate of Reinvestment = i 5.06% < 15.0"/o, acceptable. =

q,

= 810

oq o

1o

zo

'llk 15k

2Ao/o

ZS/, ZO/o g1o/o

(s)

A}a/a 43% S07o

(10)

Discount Rate

Figure 4-5 Npv'vs i. with Rate of Reinvestment Meaning

As illustrated in Figure 4-s, when NpV increases with a corre_ spondingly higher discount rate it is generally the result of income preceding costs and the presence of rate of reinvestment meaning associated with each i* varue. This situation courd be thought of in two different ways; First, rarger discount rates arways diminish the present vafue of future cash flows more rapidty than imailer discount

rates. second, as other opportunities for ine Lse or iipitut increase, the money received up front at time zero and year one can generate more future value creating more profit relative to the estimated down stream cosfs. once again note that the "i" carcurated when investment precedes income has compretery different meaning than when income preoedes investment. Difficurty arises if these two types of projects are mixed in incremental analysis because the interest ,,i,,has two differ-

chapter 4: Mutualry Excrusive and Non-filutuaily Excrusive project Anarysis

ent meanings in the same equation. Techniques for analyzing this type of investment project situation are presented in the foll6wing

section. lt will be shown that the cumulative cash position diagram is a useful tool in the evaluation of this type of prublem.

4.8 Alternating Investment, Income, Investrnent: The Dual Rate of Return Situation In the last section, examples 4-9 and 4-10 illustrated situations where income preceded costs. Discttssion then focused on how the meaning of the calculated "i" r'a]ue was not rate of return, but the rate at which the rev"enues (or positive cash l1orv.t tlust be reinvested to ilssure sufficient revenues exist to cover future costs. When this occurs, thc resulting rates of reinvestntt:nt that are less than the investor's minimum rate of retum are considered economically satisfactory. Extending this concept, when a time dia,qram contairs cash flows with an

initial investment, foilowed by income and then more investment(s), the related present w'orth equation will yield muitipie ..i', values. Thcse ,.i,, will airval's contain a combination of meariing related to both rare of 'alues retJill on the iriitill irtveslntcnt as weli as rilte of reinvestment requirement relarod to ectrnomicallv corering rl:e cost(s) in the tuture years. These,,i,, vaiucs are olren rei'erred to as "Duai Rates of Return,,and generally speak_ ir.ls. iire nor signilicant in assessing the econornic potential o1a project. This is due to the fact that despite the label, "Dual Rate of Return,,, neither solu_ tio, has pure "rate of return" meaning. Instead, as mentioned previously, the results contain a combination of rate of return and rate of reinvestment meaning, which may be good part of the time, but unsatisfactory part of the time. This forces investors to consider other criteria such as Npv or a modified forr, of rate of return ilrustrated in the foilowine examples. Cash flows containing in'estment-income-investment timing may occur in a variety of investment situations. One such case involves the incremental analysis of mutualll, exclusive alternatives that have different project lives. This is often defined as an acceleration probrem and is to oil and gas as well as mining investments. In depleting a finite, "o,rr*on resource, the decision to accelerate the production rate through immediate capital expenditure(s) will shorten the life of the project. The incremental analysis of these alternatives creates the crassic in'estment-incorire-investment siiuation.

202


Alternating investment, income, investment analysis situations occur in a variety of situations. The most cornmon, which is illustrated in the next example, occurs from looking at incremental differences between uneqdal life alternatives where the bigger investment alternative has bigger period;revenues.and sltorter project life. This is the classical acceleration problem nientioned previously cofirmon to mineral and petroleum development type projects where a given mineral or petroleum reserve can be depleted more rapidly by making a bigger initial investment. This evaluation situation also commonly occurs with acceleration type investments in many different types of general industry situations. Other examples of cost, income, cost include (1) An investment in a building or project that generates income for several years after which the building or project must be razed or restored to different condition; (2) Mining projects that generate income followed by significant reclamation costs; (3) Forest planting investments followed by clear-cutting which generates income but must be followed by forest replanting costs where environmental laws or company policy require it; (4) Offshore platform development fbr petroleum production that must be followed by significant platform ree lltmation costs.

EXAMPLE 4-11 Analysis of Mutually Exclusive Unequal Life Acceleration Type Projects lnvestments "A" and "B" are mutually exclusive ways of developing a project. Which is best if the investor desires a minimum rate of return of 20%? Make a valid ROR analysis using either Growth ROR or one of two present worth cost modifications known as the "Escrow Approach" or "Year by Year Approach." Verify those conclusions with NPV.

Solution, All Values in Thousands of Dollars: I = Revenue, L

A) C = 182

-

Salvage Value, C = Cost

l=

0

B)

C=250

100

l=

184

l=

100

2

1

l=

100

l=

184

Get equal life alternatives by assuming the lile of "B' is 3 years with net revenue and cost of zero at year 3.

chapter 4: Mutually Exclusive and Non-Murually Exclusive prolect Anaiysis

203

Rate of Return Analysis, (ROR) By trial and error the RORA = 3Oo/o and RORB 3Oo/o both of which = exceed the 20% minimum rate of return requirel for the inr,,estment of capiial. The investments and project live.s are ufiequal so it is difficult to teii intuitively if "A" or "B'' is best for i' = ZO"/o. projects with equal

toial investment rates of return are not necessariiy economicaily equivalent. lncrernental analysis gives: B_A)

C=68

l=84

l=84

ROR PW Eq: 68 + 100(P/Fi,3) = B4(p/A;,2) or, in NPV format: B4(P/Ai,2) - 100(p/F;,3)

C = 100

-

6g = 0

i = 0"/" : 84(2.000) i = 10"/": 84(1.736)

- 100(1.0000) -68 = 0 - 100(0.7513) - 08 = +2.69 i = 15"/": 84(1 .626) - 100(0.6575) 08 i.2.89 - = i = 20"/": 84(1.528) - 100(0.5787) OS +2.48 - = i = 307" : 84(1 .361) - 100(0.4552) 68 = +0.80 i = 40"/": 84(1 .224) - 100(0.3044) 6S - = -1 .62

i=0% and i=33"3% are duar rates of return by triar and error.

+

--t--F+-_]%

50%

100%

rL

++ --f-

-r

rt + -T Figure 4-6 NPV vs Discount Rate For Cost, lncome, Cost

204


100

Cumulative

Cash

50

Position in

Thousands for

i=0% i=

33.3%

0.0

-50 -68 -1 00

Figurc 4-7 Cumulative Cash position Diagram and The Meaning of Dual "i" Values A graph of NPV versus the discount rate "i" as illustrated in Figure 4-6 emphasizes the parabolic variation in NpV with the discount rate changes for this cost, income, cost situation. This is very different from the declining exponential variation for NpV versus "i" when cost is followed by income as illustrated earlier in chapter 3, Example 3-21. However, the term "dual rates of return" is really a misnomer because neither "i" value means rate of return. Both "i" values have a combination rate of return, rate of reinvestment meaning as the Figure 4-7 cumulative cash position diagram shows. Note that a oo/o rate of return is bad compared to = 20/o percent whereas a reinvestment rate o'f 0/o is good compared lo 20/" reinvestment opportunities. Similarly, a 3o"/o rate of return is good but a 33% rate of reinvestment requirement is bad. Both dual ROR values are good part of tne time and bad part of the time.

i

Looking at Figure 4-7, whenever you are in a negative cumulative cash position, the meaning of "i" is rate of return. On the other hand, whenever you are in a positive cumulative cash position, the meaning of "i" is rate of reinvestment requirement. Using the cumulative cash position diagram, it becomes evident that the dual ,,i,, values have different meaning at different points in time. Therefore, an analysis method other than regular rate of return should be utilized. lf

ihapter 4: Mutuaily Excrusive and Non-Mutuaily Excrusive project Anarysis

the investment decision must be based on a compound interest rate of return measure, severar modifications qan oe'rnaJe to eriminate the cost-income-cost sequence in the tro*.. wili be introduced and incrude Growth RoR, upe iscrow Approach and the Year-by-year Approach. The rast two ippror"nus are some_ times referred to as "present worth cost Modiiications.,, r.rpv .*uv aiso be utilized if an alternative to RoR analysis is acclptable. The net varue techniques are varid arternative anarysis tecrrniques inai avoid the "dual ROR,'problem.

i;;." ilffi;;

.*n

Net Present Vatue Anatysis, (NpV) For i* = 20"/o,lime zero is a common time for all projects. 2.1 06 NPV4 = 100(P/AZO%,3)

-

182= +$28.6

1.528 NPVB = 184(?lAZA"k,Z) -.ZSO +$31..1 +_ Select ,,8,,, = Largest NpV

Growth Rate of Return Anatysis There is no need to carcurate Growth RoR for the ,,A,, dnd ,,g, total investments. we arready know that ,,A,, and ,,B,, are satisfac_ tory from totar investment RoR anarysis, so we onty neeo to Growth RoR anarysis to the incrementar investments. "il[ the Growtn RoR reinvestment step eriminates tne Note that art"rnliin"g* investment, income, investment siiuation and gets us back to incrementar investment foilowed by incrementar revenue, the RoR analysis situation. B-ur.;c = $68

01

l=$84

l=$84

C = $1oo

23

Reinvesting incremental year 1 and 2 incomes ati" =20"k:

C=$84

C=$84

F = +$221.8

2.200

1.200 where F = 84(F / A2g"/",2)ff /p 2O%,1) = +$221

.B

206


B-A + Reinvesting incremental income: B:A) C=$68

F l-== +$121.8 3 +,:

PW Eq: 68 = 121.8(PlFi,g), i = Growth RORB-4=21.4/o > 2O/" Select "B" ROR Using the Escrow Approach

Another modification for ROR analysis that many individuals and companies use to eliminate the alternating investment, income, investment situation is a present worth modification of the final cost. By discounting the final cost at the minimum ROR, you convert the problem to a regular cost followed by income type of evaluation. Working with the incremental "B-A" diagram, discount the final year 3 cost of 9100 thousand at i* = 20%, giviig the following modified iime diagram: 0.5787-

l=$84

01 Modified PW Eq: 0 = -125.87 + 84(PlA1,2) i = 21.6/o > 2Oo/o, Select "B" NPV

Figure 4-8 NPV vs. Discount Rate For Modified PW Cost Analysis

chapter 4: Mutually Excrusive and Non-Mutuailv Excrusive project Anarysis

Explanation of the Escrow Approach This present worth cost modified RoR analysis really involves adding

an outside investment earning at the minimum discouni rate to the initial cost, income, cosi project. By seiecting the nra$nitude of the outside investment so it wiil generate income in the later years equai to costs foliowing income in the initiai prolect, cost following income is eliminated.

C=$68

B-A)

l=$84

l=$84

6 = 9100 3

+

An Outside lnvestment C = $57.g7 at i = 29"1o

= Total

c

| = 9100

23

0.5787 where "C" at time 0 = 57.87= 100(P/F;.= 20"/o3) l=$84 l=$84 = $125.87

_

Modifiect Present Worth Equation: i = 21.6"/" PW Cost Modified ROR ROR Using the Year-by-year Modification

0=-125.BZ +84(p/Ai,2)

it is not necessary to present worth costs following income all the way to time 0 to eliminate costs following income. lt iJonly necessary

to bring costs following revenue back one year at a time until they are offset by income, as the following illustrates. B_A)

C=$68

l=$84

+

An Outside lnvestment at i* = 20"/o

C=$68

0 Modified PW Eq.: 0 =

i

c

= 9100

2

3

C = $83.3

I = 9100

23

whereCatyear2= = Total

l=$84

0.8333 1 OO(P lF 2g"/", t ) = $83.3e

l=$84

I = $0.7

-68 + 84(PlFi.1) +.0.7(PlFi.2) = 24.4/o PW Cost Modified HOR

3

208


Adding an outside investment of 83.9 at year 2 to the uB-A" project does not .weight,the PW',Modified RoR,as. lswâs.adding,the investment of 57.87 at time zero. However, note that the zq.4y" modified RoR result relates to very different unamortized invesment values each year than the time zero 21.6% pw Modified RoR relates to. Both results are economically equivalent even though different in magnitude. The modification that eliminates cost following revenue and modifies the analysis as little as possible is felt by many people to be the most desirable modification to use, so the latter year 2 modification often finds use in industry practice.

All of the analysis methods utilized for this example, other than regular ROR, have selected alternative "B" consistenfly. Any of these techniques can and should be used in place of regular rate or return analysis when the investment, income, investment type of analysis is encountered. The combination rate of return, rate of reinvestment meaning associated with cost, income, cost dual rates of return is what makes the dual RoR results useless for valid economic decisir:ns. The existence of dual rates is algebraically caused by the sign changes in cost, income, cost equations. This can be illustrated for incremental "B-A" analysis in this example.

B-n; c

:igq

l=$84

l=$84

C = $100

PW Eq: 0 = -68 + 84(PlFi,1) + 84(p/F i,2) - 100(p/F;,3) Mathematically: 0 = -68 +

Bailt(+i)) + 8a(1/(1+i))2

-

100(1/(1+i))3

Substirute X = (1/(1+i)): 0 = -68 + 84X +84X2

-

100X3

This is a "third order polynomial equation" as a function of X. Alge-

braic rules indicate that polynomial equations may have as many


positive roots as sign changes, two in this case. sorving for X: X = 1, and_X = 314 gives i = O7o and i = g3.A%. These are tf,e same dual ROR results obtained earlier by trial and erroi..

t

EXAMPLE 4-12 Recramation costs can cause the Duar RoR problem

. An investment project requires the initiar investment of g70,000 to generate a projected stream of positive g40,oo0 per year cash flows inêach of years one througi'r tive. However, a reclamation cost of $140,000 is expected to be required at year 6 (the year 6 reclamation cost could relate to restoration of srrface land to origi_ nal contours for an open pit mining operation, recramation of an offshore platform for an offshore pitroreum production project, or reciamation costs for land cleanup from chemical contanrination, to name several possibilities). The minimum RoR is 2c7". Analyze iire econornic potential of this project using both Npv and RoR analysis.

Solution, All Values in Thousands of Doltars: C =7O

l=40

l=40

l=40

l=40

l=40

C=140

when cost foilows revenue, correct RoR anarysis requires use of one of the modified noR analysis techniques intioduced in Example 4-11 (Escrow Approach, year-by-year Approach, or Growth Rate of Return). ln this exampre, the sum of ine posiiive cash frows is $200 while the negative cash frows totar $21d. crea;ry,;ome of the early cash flows will be utilized to pay off the initiii investment, providing a return on that investment, while the remaining positive cash flows will have to be reinvesied at an interest rate if enough cash is to be generated to cover the future obrigation. This iilustrates the rate of return and rate of reinvestment-meaning associ_ ated with all dual "i" values.

210


ROR Analysis Using an NPV Type of Equation

PW Eq: 40(P/A;,5)- 140(P/Fi,6) i = O"/" i = 5"/" i = 8"/"

-

70 = 0

40(5.000) - 140(1 .0000) 40(4.329) -140Q.7462) 40(3.993) - 140(0.6302) i = 15/" 40(3.352) -140(0.4323) i = 20o/" 40(2.991) -140(0.3349)i = 25"/" 40(2.689) -140(0.2621) i=30% 40(2.436) - 140(0.2072) -

+.

70 = -10.0 70 -1 .3 70 +1.5 70 +3.0 70= +2.7 70 +0.9 70 -1 .6

= = =

= =

Note that due to the "parabolic variation" of the NpV type equation results versus "i", by interpolation, Npv = 0 fgr the dual rates of return of 6.40oh and 26.780/, Each of these rates makes the cumulative cash position zero at the end of project-life. Both rates involve a combination meaning of rate of return on investment in early project life and rate of reinvestment rate in the later project years. These dual rates cannot be used direcily for decision making pur poses as ROR results. However, the dual rates do provide some useful information because they bracket the range of minimum rate of return values for.which project net present value is positive. This tells the range of 'ri " for which the project is satisfactory. whenever dual rates exist, it is easiest to rely on NpV analysis for decision purposes. However, going to Growth RoR analysis or present worth cost modified RoR analy.sis is equally valid, but generally more work. NPV calculated at "i^" always leads to correct economic decision in this situation, whereas the dual rates problem makes RoR analysis more confusing. 2.991 N

PV @ i* =2O/" =

qA(P I AZOo/o,S)

-

0.3349 ltr;O(P lF 2g/",d

-

70 = +$2.754 > 0

The NPV analysis is quick and simple to make and the positive NPV result tells us the project investment is satisfactory, although the NPV of +2.754 is only slightly greater than zero compared to the magnitude of costs that generated it, so project economics effectively are a break-even with investing elsewhere al2O"/o.


211

-t

-60 Figure 4-g NpV vs Discount Rate for Cost, lncome, Cost |

Now before getting into the detairs of the modified RoR anarysis, note that Npv at i = 0."/o is negative for this exampre, whln has duar positive rate values of 6.4% ind 26.7g%. whenever the NpV at i = 0% is negative for an investment, income, investment situation, due to the paraboric variation of NpV with changes in ,,i,,, ouat positive ,,i,,varues exist if any real interest rate solutions exist for the Npv equation. lf NPV at i = 97 is positive, duar rate varues exist with one being negative and the other positive. This test of NpV at i = 0% tells an investor where to rook for the dual rates if it is deemed desirable to determine them. Beforg presenting the modified RoR anarysis carcurations, note that in calculating the duar rates at the beginning of this sorution that as ,,i,, increased from 0% to 15% that NpV i-n"r"rr". rather than decreases. This is a unique resurt of cost foilowing revenue in the cost, income, cost analysis situation. whenever you firid Npv increa.ing-*itn increas_ ing interest rates (rather than de'creasing as it always does for cost, income analyses) it is the author's expeiience tnat ihis is caused by cost following revenue in the analysis.

.

--r&..

212


ROR Using the Escrow Approach To eliminate this investment, income, investment situation, present worth the finai cost or costs at the minimum RoR to an equivalent t present value, giving the following diagram:

0.3349

6 = $70 * $1a0(P/FZO*,O)

0

|

1............5

6

PW Cost Modified ROR PW Eq: 116.88 = 40(P/Ai,5) Modified ROR = i = 21.1"/" > 2oo/o, so satisfactory. since this modified RoR result is based solely on income following cost, it is valid for economic decision-making purposes as a rate of return result. Dis: counting the year 6 cost modifies the magnitude of our RoR result but not its validity for comparison to i* = 20"/o for the economic decision.

Growth ROR

combine reinvestment of revenue at "i*" with the initial project to eliminate cost following revenue as follows:

lnitial Project

C=$70 0

l=940 l=940 1..........5

C=9140 6

Reinvest Flevenues @ i*=2Oo/o

_ 0

c=940 c=940 1..........5 7.442 1.200

6

F=$357.20

where, F = 40(FiA 2O%,5)(F /P 21o/o,1) = $3SZ.Z

Combine lnitial + Reinvestment Revenues

C=$70 o

-

C=9140"-

Growth ROR PW Eq: 70 = ZlZ.2(PlFi,A) Growth ROR = i = 20.9"/" > 2oo/o, so satisfactory.

F=$357'2

chapter 4: Mutuaily Excrusive and Non-Mutuaily Excrusive project Ana,ysis

213

Both the present worth cost modified RoR analysis and Growth RoR condulion, which is in this case and in general always consistent with Npv analysis eccnomic conclusions. The key, to correct RoR analysis of cost, income, cost anaiysis situations is to modity tne analysis to eiinrinate cost following revenue before making ihe HoR anaiysis. The present worth cost ani Growth RoR nrcciifications ai.e the turc basic approaches used to elimi_ nate cost following revenues. Hcwever, there are several variatiorrs in the v'ray different people apply these modifications. one in particular is worth noting. - ln applying the escrow approach it is not necessary to bring costs following revenue all the way to year'0. lt realry is only n*""r.rry to pre_ sent worth costs following revenue year by yea, until those costs are ar-ialysis have given the same economic

offset by project revenue. This gives the foilowing modification referred to as the "year by year Modification.,,

HOR Using the Year-by-year Modification

.

Discount the year 6 cost year by year until offset by project

income.

C=19.90 I=40.00

Net l=20.1 0

3'

\

Dttr

C=63.88\

C-110.66\

\ l= 4o.oo t"ro,\"=ro9 20,1. - \Net +4!!A e)rro.., \Net c= C=23 88 76.66

56

4

New Modified Diagram

c=70 l=40 l=40

I

= 20.1

PW Eq: 0 = -70 + 4O(p/Ai,2) + 20.1(p/F;,3) i = l,{odified ROR

= 22.75% > i* =

ZO"k so satisfactory

This modified RoR result is several percent bigger than the initial present worth cost modified result. This is oecause the two modified RoR results relate to very different initial year 0 investments. The results are really equivalent and give the same economic conclusion when compared to the 20% minimum RoR. Remember that you cannot and should not look at the magnitude of RoR results and think

214


big is best' A projggt with a big RoR that rerates to a given invesr ment an d stream ingome m.ay. not .be,a.s..desi rable qs project.with 9f , a smarrer noR that'r"ritei dirrerent stream of b

income.

EXAMPLE

i t, biffiffi;Ji;"r",;;

"

4-rs A petroteum tnfifi Driiling Accereration probrem lnvolving Dual Rates of Return

A producing oil field has wells drilled on 160 acre centers. lt is proposed to infill drill wells on g0 acre centers to accelerate petroleum production and give more efficient drainage of the petroleum reservoir. Present and proposed costs and net revenues are shown on the following time diagrams with varues in thousands of doilars:

Present C

0129456

o

Accelerate C=

0129456

For a minimum RoR of 12yo, use rate of return anarysis to determine if the acceleration drilling program investment is satisfactory from an economic viewpoint.

Solution: The-"present" producing project is crearry satisfactory since costs for deveropment.have arready been incurred (so they are sunk). For no additionar costs to be lncurred, the ,,present,, project revenues are projected to be generated. This relates to an infinite percent return on zero dollars invested in the present project, an economically satisfactory project. The accererated plolect total investment RoR is 2oo%. However, this RoR does not need to be calculated because knowing the present project is satisfactory, we can go to incremental analysis to determine if incremental invest_ ment dollars spent on the accererated production infiil driiling pro_ gram are justified economically by incremental revenues. The incremental diagram involves cost, income, cost as follows because the negative incrementar incomes in years 4, 5 and 6 are

effectively costs as follows:

chapter 4: Mutuary Excrusive and Non-t\,.lutuary Excrusive project

Anarysis

215

lncremental Rate of Return Analysis Accelerated C= *Present o

1

2

s

,4

5

O

6

lf you i,vrite a conventionar present worth eqtration for the incre_

nrental diagranr values, you get dual rates as follows: PIV Eq: 0 = -735 + 850(p/F;,1) + a50{F iFi,2} +50(ptF;,3)

-

310(p/F;,4)_ 280(p/Fi,5) _ 1S0(p/F;,6) . rrial and error, duar "i" varues of 12"/, and 2so/o resurt. An investor that treats either of these resurts as rate of return co*pareo with the minirnuin RoR of 12/, wourd concrude that the incrementar project economics are satisfactory. This turns out to be an incorrect conclusion. Both of the duar rates of 17% and 25"/" have rate of reinvest_ rnent meaning as weil as rate of return rneaning at different pc:nts in time..Required rates of revenue reinvestment of 1 Zo/" and 25"k compared with 120,L reinvestment opporiuniiles inoratei oy trre mini_ mum ROR indicate a very unsatisfactory incrementar investment whereas rates of return of izx and 25"/. compared to the 120n minimum RoR rook satisfactory. The unsatisfactory rate of reinvestment meaning is stronger than the rate of return melning as t!:e foilowing NPV and Escrow ROR analysis results show. Net Present Value Anatysis

NPV= -7OS + 850(p/F1 2"/",1) + 450(p/F1 2y.,2) + S0(p/F1 2%,g)

-

310(P/F1 2/o,4)

-

12y",5) - 150(p/F1 2/.,6) = -13.6 < 0, so slighfly unsatisfactory. 2BO(P/F

Escrow Approach ROR Analysis tulod. Yr 0 Cost

=735 + 3i0(p/F1 2"k,4) + ZgO(p/F12o/o,S) + 150(P/F12%,6) - $1,166.90

Mod. PW Eq: 0 = -'t

,1

66.9 + 850(p/Fi,1) + 4S0(p /Fi,2) + 50(p/F1,3)

Trial and error: i = Modified ROR

=

11"/o

216


Note that any investor who treats either of the positive dual rates of ond 257" as rate of r€turn comes to,the wrong economic conclusion in this analysis. when cost follows revenue, you must modify the analysis to eliminate cost foilowing revenue oefoi5you can make ROR analysis. 17o/o

EXAMPLE 4-14 cost, lncome, cost from lncremental Service

producing Analysis

It is necessary to evaluate whether an asset should be replaced today for a $20,000 cost or two years from today for a 925,000 cost with operating costs and salvage values as shown on the diagrams for a six year evaluation life. Use incremental RoR analysiJfor a 15% minimum RoR to reach the economic decision. Allvalues are in thousands of dollars on the diagrams. Replace Now (A)

C:?9__OC:I_-9C=Z OC=3 'OC=4 OC=S OC=6 0

1234

L=4

C=25

Replace Later (B)

C=0

oc=6 oc=g oc=1

oc=Z OC=S OC=4

L=9

Solution: Analyze "A-B" to get incremental cost followed by incremental savings. However, it is impossible to avoid having incremental costs and negative incremental salvage (effectively cost) in years 3 through 6, giving the dual ROR problem. R = incremental savings which are equivalent to revenue.

A_B)

C=2O R=5 R=31 C=2 C=2 C=2 C=2

L=-5

A modified RoR analysis is needed to eliminate cost following revenue or savings. There is litfle value obtained from calculating the dual rates of return except to satisfy curiosity, but the dual rates for this analysis are +22.32/" and -13.03%.

Chapter 4: Mutually Exclusive and Non-Mutually Exclusive project Analysis

Mod. Cost = 20 + ZtPiA157",4)(p/F15"/o,Z) + 5(p/F1 S"/".6) = 26.4g PW Eq: C = -i6.48 -,.5(p/F1,1) + 3.t(p,,F;,2) E i = incremental investment mocjified ROR

= 1g.05% > i. _ 1syo

Accept the incremental investment in "A" an,c reject,,B', (repiace later). iircremental i{PV anarysis verifies this modified Ron anatysis conciusion.

0.8696

0.7561

2.855

0.7561

NPV4-B = S(P/F1 S"/o,i + 31(P/F15"/",2) -2(P/A15o7o,4)p/F115r",Z)

-

0.4323 5(P/F15%,6) -20 = +1.31 >

0,

so accept,,A,,, reject "B"

To compiete oui'discussion of the investment, income, investment siiuations, it is inrportant to point out and emphasize that if income fciiows the second investment, a duat RoR situation may not exist. This means that investment, income, investment, income analysis situations may not give the duat RoR prablem but investment, income, investment always does. Look'at the project cumulative cash position diagram for the positive ,'i,, value calculated to test whether combination rate of return, rate of reinvestment meaning is associated with the "i" value at different points in time. Remember, if the cumuiative cash position does not go positive at any time, rate of reinvestment meaning does not exist aid ihe meaning of ,,i,, is rate of return for the entire project evaluation life. The foll6wing example illustrates this important analysis consideration.

EXAMPLE 4-15 An lnvestment, lncome, lnvestment, lncome Situation Where Conventional ROR Anatysis is Vatid Diagram values in thousands of dollars.

C=50

l=30

C=100

l=30

l=60

l=60

l=60

A development project will require investments of g50,000 at time zero and $100,000 the end of year 2 as shown on the time diagram, 9f with incomes of 930,000 at the end of years 1 and 2, and $60,0-00 ai

Eoonomic Evaluation and lnvestment Decision Methods

tggnd of years 3, 4 and 5. For a minimum

rate of return of 2070, use RoR. analysis to, evatuate. the economi'.desirabirity..i tni. oroject. rs the "i" value that you calculate meaningful for economic decision mak_ ing as RoR? Does NpV anarysis verify'your conctusioni--

Solution, All Values in Thousands of Do[ars: NPV Eq: 30(P/A;,2) + 60(p/A;,3)@/Fi,2) _ 100(p/F;,2) _ 50

-

0

Since the NPV at i = O"/o is positive, only one positive ,,i,, exists that will make NPV = 0. By trial and error, i 22.46"/" makes NpV = g. 1s = this "i" value of 27.46o/o a rate of return result or is ii onl of a pair of "dual rates of return" that have combination rate of-rrirrn, iate of reinvestment requirement meaning at difierent points in time? lf a companion dual rate. of return exisG, it would be'negative. However, it is possible to avoid the search for the possinte-iompanion ouat Rpn-lv testing to see if rate of reinvestment meaning is associated wilh 27.46"/" "i" value at any point in rime. Dual rates if return arways have combined rate of return, rate of reinvestment meaning at different point in time. lf rate of reinvestment meaning does not exist at anv time, then dual rates of return do not exi$ f6r this probtem and the 27.46% "i" varue can be treated as RoR for decision'pirpo,i'"1'"

Project years

Figure 4-10 cumurative cash position Diagram of a cost, lncome, Cost, lncome Rate'of Return Analysis

chapter 4: Mutua,y Excrusive anri Non-Mutuariy Exclusive project Anarysis

219

Evaluation of the cumrrrative cash position diagram for this project. at the project "i" varue of 22,46/" does show thatih; cumurative cash position never goes above zero at any ti*" Jriirgiie prolect tife. Therefore, the "i" of 27.460k means*rate of Eturn,,over rhe entire project life and never means rate of reinvestment. onry ,/.,ien ::e cumulative cash position goes positive does ,,i,, nave tne rate of re_ investment meaning.

{.9

Alternating Income, Investment, Income Situations For the situation when you have alternating income, in'estment, income on the time diagram, dual rates of return occur for the opposite conditions rlescribed in the previous section for the inr.,estment, income, investment situittion. For income, in'esiment, income situations, duAl ..i,, varues that are hr;th positive wiii exist if Npv ar i = O,ra is p<-rsitive, and positive duar ..i,, .'alues *'ill not exist if Npv at i = avc is negaiive. Analysis'rules rre similar to tlrose given in the last sectior. Hon,ever. note that fbr income, investmerit, income siruations, a groject is acceptable for an ;.r;.;;;;;;ii.,un tt. rrgesr project posrtive dual "i" value because this is the iegion of positive net present when
r

trates those considerations.

EXAMPLE 4-16 lncome, Cost, tncome and Rate of Return

Analysis

As a variation to.Exampre 4-10, a company has been asked to consider a proposal to accept two payments of $12 million, one at time zero and the second at the end of year one, arong with a payment for $23 miilion at the end of ten years from now ,,.Ir,en the pro_ ject is to be compreted. These payments are to provide for the cost to your company in disposing of waste product from a processing facirity for each of the next 10 years, (end of years on" inlorli ten). your estimated costs for disposar of this materiar incrude a tinie zero capi_ tal investment of $7 miilion with end of year on" inrortnien operating costs of $4 million per year. The minimum rate of return is 15.0%. Use ROR and NpV analysis to determine if the should accept this opportunity? "ornprny

220


Solution: All Values in Millions of Doltars lncoine: 12.0 Costs: -7.0

01 5.0

Year:

Net CF:

.-12.O

23.0

-4.O

-4.0

8.0

NPV @ 15.0o/oi 5.0 +€(P/F1S,1)- a.OplA15,gXp/F15,1) + 19(P/F15,1g) = $1.04 > 0, acceptable, but close to break-even

This analysis has inconi.E preceding costs, followed by more income, which is another dual "i" value situation with 0% and g.6% being the dual "i" values. ln this analysis, the meaning of the calculated "i" value is rate of reinvestment requirement in the early years as inilial positive cash flows pay off the negative cash flows in some of the early years of the project. tn the later years, the ineaning of "i,, switches to rate of return as the remaining cosfs are paid off-by the positive cash flow in year /0. This analysiJrequires a cash flow modification to eliminate the multiple sign changes in the present worth equation. The same procedures used in cost-income-cost situations must be employed here, giving an that an initial investment generating positive cash flow. such a modification is presented below in the form of Growth RoR. ln this anarysis, all positive cash flows are taken forward to the end of the project, while negative cash flows are discounted to time zero. Net

CF:

5.0

8.0

-4.O.

-4.0

Year:

rime Zero Modiried cost:

19.0 10

-4 ofiill#x8;Piil

End or year 10 Future Varue: 561p0i;:0) PW Eq:0 = -15.61 + 67.37(P/Fi,1g), GROR = i = 1 5.75% > 15.0"/o, acceptable

.

) = -15.61

8(3l3ll,nn,

. , e = 62.32

Chapter 4: Mutualiy Exclusive and Non-i\,4u+ri::Ii,, f {clusive i.oject Analysis

The following diagram illustrates that when income-cost-income exists, the analysis relationship between NpV and i* is inverted when compared to

cost-income-cost.

*

2.0c

o

1.50

E c

1.00

o o

E 0.50

o-

zo

0.0c (0.50)

'47o -27o Figure

Oo/o

+I1

2% 4% 6% g% 1oo/o 12% 14/o

16%

Discount Rate

NpV vs i* for lncome_Cost_lncome

As illustrated in Figure 4-11, the parabolic relationship between

NjPV and the discount rate in this income-cost-income analysis tends

to vary in an inverse fashion to that in a cost-income-ccst situation. once again, to get varid rate of return resurts, you must modify the analysis to have dollars invested initially followed by income in later

years to provide a return on the invested capital.

4.10 Evaluation of Non-Mutually Exclusive Investments Non-mutually exclusiYe invesiments :Ire investment zrlternatives from which more than one choice can be selected deoending on available capital or budget resrictions, such as selecting research, developrnent or exploration projects from many altematives. The ranking of drilling prospects in the petroleum industry is a classic example of non-mutually exclusive alternative analysis. The objective in analyzing non-mutually exclusive projects is to maximize the cumulative profitabiliry that can be generated from the available investment dollars. To maximize the cumulative profitability that can be generated by investing available investment capitar in several non-ntutually exclusive alternativ'es, select the combination of pr
222


value or cumulative future worth profit. use of any of these methods requires looking at all the different possible combinations.of projects:to determine the group of projects that is best for a given budget. To rank non-mutuallv exclusive projects in the o:rder thafiou will want to select them to maximize curnulative net value or future profit for a given bttdget, ttt'o ranking techniques may be used. They are growth rate of return and ratio analysis using either PVR or B/C Ratio as calculated earlier The following examples will show that often this does not involve selecting the project with the largest individual project net value, and that ranking the projects in the order of decreasing regular RoR on project investments does not properly rank non-mutually exclusive alternatives. of the basic evaluation techniques discussed earlier in this text, only Ratio analysis and Growth RoR consistently rank non-mutually exclusive alternatives in the order that you want to select them to maximize cumulative profitability. If a true opportunit.tt cost o1 capital is used as the minimum discount rate, all proiects with positive NPV will be accepted for investment and there is no need for ranking mdependent non-mutually exclusive alternatives. Hoyvev'er, inv'estors usually do not have an exactly correct opportttnity cost of cupital, so under conditions of rationed capital (limited budget dollars), the ranking of independent or non-mutually exclusive projects is necessary.

EXAMPLE 4-17 Evaluation of Non-Mutually Exclusive Alttirnatives Compared to Mutually Exclusive Alternatives With Net present Value consider the following four investment alternatives which all have a 5 year life and zero salvage value. Assume that i. = 2oo/o before taxes. Alternative

1 2 3 4

lnvestment, $

Operating Cost Savings Per Year, $

10,000

6,000

25,000

10,000

35,000

15,000

50,000

17,000

A) lf $50,000 is available to invest and alternatives 1 , 2,3 and 4 are mutually exclusive (only one alternative can be chosen), which alternative should be selected?

chapter 4: Mutualry Excrusive and Nori-rvi.rtuary Excrusive project Anarysis

B) lf 93s,000 is avairabre to be invested, and the arternatives are non-mutuaily excrusive, shouid we choose aiternaiivli or arternative 1 plus 2? we do not have enough money to finance arternative 4 so for financiai reasons (rather thai economic regsons) it must be reft cut of this analysis. Solution: A) Mutually Exctusive Alternatives NPV anarysis wiil be presented here because it is generary the simplest method to use to evaruate n.,rtrrrrv *.i*iu"'p)oi".t..

NFVI = 6,000(p/A133r1,r,- 10,000 +$7,e46 = NPV2 = 10,000(plAZO"7 ,S)_ 25,000 = +$4,910 NPV3 = 15,0C0(p /AZO1",S) _ 35,000 = +$9,g65 ,, NPV4 = 17,000(p /AZO"7 _ ,d SO,0C0 = +$g47 Alternative 3 has il-re largest NpV at i* = 20o/o, sofor mutually exclusive arternatives, arternitive 3 is ,re triJ.".

".onon.,i.

B) Non-Mutualty Exclusive Alternatives lf we were to pro.ceed we did in part (A) and pick the rargest 3s NpV, we wourd serect arternative 3 for this anarysis arso. However, it shourd be very obvious that if these arternatives are not mutuaily excrusive, we can make better use of our g35,000 by selecting rft"r*tives 1 plus 2, rather than 3, since this gives us $iO,OOO in savings each yearfor the $35,000 investment, ra-ther tfran tfre $15,000 .iring. obtained form atternative 3. cumurative r.rpv ro,, J,"rn"rr"i"""r;; is $12,8s6 compared to NpV3 = $9,965. Therefore, note that tne prolect.witi biggest NPV is notrYnvorveo in tre economic choices. onry with mutu_ ally exclusive arternatives is the biggest NpV projeciur*rvi

t

0".t.

. Y.h:' a smail group of arternatives is being considered, cumurative NPV is usuary the easiest approacrr utirized to determine optimar value for a rimited bydget, However, *n"n the number of arternatives increases, the use of ranking tecnn(ues such as ratios can herp evaruators rank arternatives in the economic order of serection to maximize

cumulative NpV.

L

224


EXAMPLE 4'18 Ranking Non-Mutua[y Exctusive projects Using Determine whether-a.manager should spend $5O,OOO on project

1

or projects 2 and 3 if the projects are non-mutually txclusive for a minimum ROR 10%. 1) C = $5o,ooo

2)

C=$30,000

ROR = 40% I = 920,000

I = $20,000

ROR = 337.

l=$10,000

.l=$10,000

0

3)

L = $so,ooo

C = $20,000

|

L=$30,000

=$5,000. : .. .. . . . . I =$S,OOO

L=$20,000

Solution: Maximize Gumulative NpV

1.736

.8264

NPV1 = 20,000(PlAtO.t",Z) + 50,000(p lFrcy",Z)

-

S0,OO0 = +$26,033

3.791 .6209 NPV2 = 10,000(PlAlO.t",S) + 30,000(p lFfin,S)- OO,O0O = +$26,535 4.868 .5132 NPV3 = 5,000(P/A 10"/",i + 20,000(plFlyo/o,7) _ 20,000 = +$14,605

Maximum Cumulative NpV = NpV2 + NpV3 +$41,140 = Note that ranking by regurar RoR or Npv does not give the correct answer. However, ranking the alternatives by Growth ndR or pVR does rank the projects correcfly as is illustrated in ihe following calculations.

Growth Rate of Return

Use a common evaruation rife of 7 years for each arternative assuming net revenues and costs of zero in the later years of shorter life alternatives "1" and,,2,,.

cnapter 4: Mutuaily Exclusive and Non-Mutually Exclusive project Analysis

1) .t C=$50,000 I=$20,000 l=$20,000

225

L=9S0,000

C = $50,000

Rein-

-01

VESt

C=$20,000 C=$20,000 2. .. .. .7

F = +$148,210

1.772 1.611 where, F = 20,000(FlP13"7",6) + 70,000(Flp19"7",5 ) = +$148,210 t+ Rein- v^$50,000 VESt F = +$148,210 1 ....

.......7

Growth ROR PW Eq: 0 = -50,000 + 148,210(p/Fi,7),

Gi'owthHCR=t=17"t'o

Z\ C=$30,000 l=$'t0,000 l=910,000

0

1...........s

L=$30,000

c=930,000

Rein-

-0

VESt

C=$10,000 C=$1C,000

1

.....5

6.1 05

where, F =

[1

7

P=+$110,170

1.21

0,000(F/A1 0%,5) + 30,000](F/p {0"/",2) = +$1 10,1 70

2+

ReinVESt

^

v-

$30,000

1

.... .......7

F=+$1 1 0,1 70

Growth ROR Eq: C = -30,000 + 110,170(plFi,7),

GrowthROR-i=20.4/"

J) ^\ C=$20,000 l=$5,000 l=$5,000.L=920,000 ^

ReinVest

C=$20,000

C=$5,000 C=$5,000 1...........7

9.487 where, F = 5,000(F/A 1O%,) + 20,000 = $67,430

F=$67,430

226


3+ ReinVest

1

F=$67,430

...........7

v

Growth ROR Eq: 0 = -20,000 + 67,430(plFi,7), Growth ROR = i= 19.1% Alternatives 2 and 3 with the largest and next largest Growth RoR values are the economic choices for the available investment budget of $50,000. Ratio analysis verifies these choices as follows: PVR = NPV @ i. / PW Cost @ i* PVRI = 26,033/50,000 = 0.52 PVR2 = 26,535/30,000 = 0.88 PVR3 = 14,605120,000 = 0.73

B/C Ratio = PVR + B/C Ratiol = 1.52 B/C Ratio2 = 1.88 B/C Ratiq = 1.73

1

Both PVR and B/c Flatio results indicate projects z and 3 are the economic choices consistent with Growth RoR and cumulative Npv results. Properly calculated ratios and Growth RoR results will always rank non-mutually exclusive alternatives in exacily the same correct order. It is instructive to note that a higher minimum RoR such as i* = 257o causes the choice to switch to alternative 1. when you have good reinvestment opportunities, the short life high RoR project 1 is economically more desirable than when reinvestment opportunities are poor.

NPV for i* =

25o1o

1.440 N PV 1

= 20,000(

P I AZS"k,Z) + 50,000(

= +$10,800

.6400 P lF ZS"1",Z)

N

2.689 .3277 PV2 = 1 0,000(P / A25"y.,5) + 30,000( P /F 25"/",5) = +$6,700

N

PV3 = 5,000(P/ AZS"t",i + 20,000(P I F 2gy",)

3.1 61

-

S0,

-

gO,0O0

000

.2097

-

20,000 = +$0

Maximum Cumulative NPV = NPVI = +$10,800, so select prolect 1.

chapter 4: Mutually Excrusive and Non-rr4uluaily Excrusive prciect Anarysis

Growth ROB for i* = 25"/o GroMh ROR calculations for i* =

ZSo/o verity that project 1 is best based on the following results: Growth RORI =bl.gt", Growth ROR2 = 28.S1L, Gro'.*h BOR3 = 25oh. Aithough the GrowihhOR results for p?c_ jecis 1 and 2 are equal, with a $50,000 budEet we can either Cc project 1 vrith a28.6% GroMh RoR cr the combination projects 2 anc 3 which nave Growth RoR results at 29.6% and Zs"/o respectively. lntuitively you know tiiat ihe Growth HoR of 28.6% on $30,000 inveited in projects 2 and the Growth RoR of 2s"k on 920,000 invested in project 3 has to be less desirable than the Growth RoR of 2g.6% on all'gsb,ooo in project 1, so select project 1 as the economic choice.

PVFI

for i* =

25o1o

PVRl = 10,800/50,000 =A.22 PVRz = 6,700/30,000 =0.22 PVR3 =

0120,000 = 0.00 since the Growth RoR results for pro.iects 1 and 2 were equal, we expect the ratios to be equal. once again, as rvith groivth RCil, h,ro mutually exclusive choices exist for spending $sc,obo on ncn-mutually exclusive projects 1, 2 and 3. we can either invest in project 1, or in the combination of projects 2 and 3. PVRl

=

0.22

PVR2a3

=

(6,700 +

O)

/ (30,000 + 20,000) = 0.13

Since the same $50,000 would be invested either way, select the maximum PVR which is project 1. This is consistent with Growth RoR results and conclusions. with both pvR and Growth RoR results, the projects were put in the desired selection order but the budget constraint caused us to analyze several mutually exclusive choices before making the final investment decision. The next example has a primary objective of emphasizing the necessiry of netting togerher all inflows and outflows of money ana wJrking with thl resultant net cash flow each compounding period in either Growth RoR or Ratio calculations.

228


EXAMPLE 4-19 Ranking Non-Mutually Exclusive Projects Using . PVR ald Growth i

, ,

.

ROR

Rank the following non-mutually exclusive alternatives ''*' using PVR for i* =

" l=$110 l=$110 l=$110 C=$100 C=$2OO OC=$0 OC=$0 A) 0 1 2 ....... . . .8 l=$150 l=$150 6=$90 OC=$40 OC=940 B) c = 9100 01 2 ..........I 157o.

Solution: All Values in Dollars Net Present Value (NPV) 4.487 NPV4 = 1 10(P/A1s%,8)

-

4.160 NPVg = [110(P/A15./",2)

-

.8696 200(P/F1 s"/",i- 100 = +$219.60 .8696 90](P/F1

S"/",i

-

100 = +$219.60

These two projects are identical economically and financially from an out-of-pocket cost viewpoint. To rank them equally with PVR (or Growth ROR) you must first net costs and incomes together that are at the same points in time and work with resultant net costs in the cienominator of PVR.

Present Value Ratio (PVR) PVR4 = 219.6 / [100 + (200

-

1

.8696 10)(PiF 15"/o,1)] = +1 .23

.8696 PVR3 = 219.6 / [100 + 90(P/F15%,1)] = +1 .23 lf you incorrectly do not net the project "A" year 1 cost of $200 against the revenue of $1 10 before calculating the ratio denominator, you calculate .8696 PVR4 = 219.6 / [100 + 200(P/F1 S%,i] = 0.80 = incorrect PVR4 This ranks project "A" as economically inferior to project "8" which is not the case.

chapter 4: Mutually Excrusive arrd Non-Mr-rtuaiiy Exciusive projecl Analysis

GroMh Rate of Fleturn To use Growth RoR as a ranking technique, project net costs must he discounted at the minimum RoR to a common time zero before making the Growth RoR calculations for a c*:;non evaluation life. This makes the cost basis for Growth RoB calculations identical to the PVR denominator. A)

C=$100 NetC=$90

0 Reinvest lncome @ i" =15"/o

l=$110

l=9110

2..........8

C=$110 C=$110

1

2........8

F=$1,217.40

1"i.067

where, F = 1 1O(F/A1 S./",7) = $1,217.40 A+ Reinyest lncome @ i*=15"/o

c=$100

C=$90

2

.......8

F=$1,217.40

Tb properly rank projects using Growth RoR, bring the net costs beyond time zero back to time zero by discounting at the minimum RoR, which is 1So,a for this anarysis. This enables us io calculaie rate of growth of a single year 0 sum of money which wilt always be identical to the denominator of a properly calculated pvfr or B/c Ratio. By caiculating Growth RoR on year 0 single sums of money, it is clear that projects with the largest Growth RoR are our choices. Those projects must generate the greatest future value possible from available investment dollars.

Growth RoR

pw Eq: 100 .

e0(p/P1T"7.: 1) = 1,212.4(p/Fi,s)

GrowthROR=i=27.14"h since the net costs and incomes on the "8" diagram are identical to the "A" diagram, Growth RORB = Growth ROR4 =27.14/o As with PVR analysis, it should be evident that if the year 1 costs and revenues for project "A" are not netted together, a Growth ROR is calculated based on year 0 cost of $100 and year 1 cost of $200 and the 23.79% Growth RoR4 result would rank "A" inferior to "8". This is not a valid result.

230

Economlc Evaluation and lnvestment Decision Methods

Finally, if costs such as major repair; reclamntion or expansion costs occur in project years after poiitive net income has been realigd fur one or ntore years, cnlv the resultant present wofth net cost that is not covered by the present worth of net income from the early project years goes into the denrytinator of pvR calculatiorts or as part of the cost basis for Growth RoR carculations.

when investments have different starting dates, whether they are mutually exclusive or non-mutually exclusive, it is necessary to use a common evaluation time for NPV or PVR results that lead to valid economic decisions as the following variation of Example 4-18 illustrates. EXAMPLE 4-19a Analysis of Mutually Exctusive or Non-M.utually Exclusive Alternatives With Different Starting Dates Re-evaluate alternatives A and B in Example 4-19, first as mutually exclusive alternatives and second as non-mutually exclusive alternatives, considering that investment alternative B starts closer to year 1 than year 0 for i. equal to 't5%.

l=$110 l=$110 l=9110 C-$100 C=$200 OC=$0 OC=$0 A) 1 2........8 I

B)

C = $100

$150

I |

9150

= = 6=$90 gg=940 66 = 940

2

3........9

Solution:

lf you incorrectly calculate Npv and pVR results at the start of

each project, you get the results shown in Example 4-19 which lead

to break-even economic conclusions whether the alternatives are

mutually exclusive alternatives with equal NpV results of g21g.60 or non-mutually exclusive alternatives with equal pVR results of +1.23. when projects have different start dates, for valid analysis, you must calculate NPV and PVR results at a common evaluation date, so we arbitrarily select year zerc for this analysis.

chapter 4: Mutually Exclusive and Non-lvlutualry Exclusive project Analysis

231

Net Present Value (NPV) 4.487 NPVA = 1 10(P/AtS%,8)

-

0.8696 200(PlF tS%,1)

- 100 = +$219.60

4.160 0 7561 0.7561 NPV6 = i 10(PiAt 5"r ,)(PlF11orc,Z) - 9O(P/F1 S"/",2) 0.8696 gC.gB 1 C0(P/F1 Sozt,,1 ) = +91 lf A and B are mutually exclusive, the maximum Npv (alternative A) is the economic choice. This is a different economic result than the break-even economic conclusion reached in Example 4-ig where the same projects were assumed to have the same starting dates. Therefore, project start date affects mutually exclusive alternative analysis conclusions. lf the alternatives are non-mutually exclusiie, rank them with PVR as follows. Present Value Flatio (pVR) 0.8696 Year 0 PVR4 = +219.6/[100+(200- 110)(p/F15,1)] +1.23 =

0.8696

0.756'1

Year 0 PVRg = +190.98/[100(P/F1S,t)+ 90(p/F1S,2)] +1.23 =

l'he alternatives are economically break-even non-mutually exclusive investments as was concluded for the equal starting date investment alternatives in Example 4-18. Notice rhat since pvR is rhe rario of Npv divided by present worth net cost not offset by revenue, pvR results calculated at any year are the same, because discounting or compounding both the numerator and denominator of PVR (or benefit cost ratio) the same number of years has

no impact on the PVR result. However, the numerator and denominator of PVR must both be at the same evaluation tirne. The following example concerns ratio analvsis ranking of non-mutually exclusive alternatives. It has been shown in numerous earlier examples that changing the minimum rate of return can and often will change economic choices, which emphasizes the need to use a minimum rate of retum in all analysis situations that represents other opportunities thought to exist for the investment of capital. Straying from this requirement causes inconsistent

232


and often incorrect economic decision-making. Similarly, with ratio analysis it is necessary to use break-even cutoff rates of zero for PVR and one for

B/C Ratio. You will not get economic decisions that are consistent with results from other valid techniques if you raise the ratio cutqff rate to say, 0.25 for PVR or 1.25 for B/C Ratio as an alternative to raising the minimum ROR and leaving the cutoff at zero for PVR and one for B/C Ratio. The following example illustrates considerations related to these points.

EXAMPLE 4-20 Ratio Analysis Considerations Related to Net Present Value and Rate Of Return Alternatirres "A" and "B" may be either mutually exclusive or nonmutually exclusive investments. Discuss the analysis of these alternatives for either situation using NPV, PVR, B/C Ratio, ROR and Growth ROR for minimum ROR values of 't2o/o, 15% and 20/". Analyze some of the effects of changing the PVR cutoff value. as an alternative to changing minimum ROR. A) C = $100 0

I = $34.67

I = $34.67

B) C = $100

I = $16.67

t= $16.67

Solution: By Trial and Error, ROR4 = 21.7"/", RORB = 16.0%

Decision

Criteria

i* =

12o/o

i* =

15o/o

i* =

20"/o

+

3.70

+

NPVA NPVg

.*+25.00

+16.20 + 4.80

PVR4 PVRg

+ +

.25 .25

+ +

B/C Ratio4 B/C Ratiog

+ +

1.25 1.25

+ 1.16 + 1.05

+ 1.04

13.1% 13.1%

15.6%

20.1% 18.4"/"

Growth ROR4 Growth RORg

.16 .05

15.2P/"

-18.50

+

.04 .18 .82

chapter 4: Mutuarry Excrusive and Non-tvrutuary Exclusive project Anarysis

23t

i

For = 12/". alrernatives "A" and "B" are economicaily equivarenr whether lhey are mutually exclusive or non-mutually exetusive alternatives. Raising "i-" to 1s"h or 20% causes the economi: cho;ce to shift to .rrr. -b" wrtn r.(, favoring r.1vurrilg"A" A over ,,8,, with all tech*iques. Not+ NotS lJpvp. I is greater than NPVB fcr both i* = 15% and i' rru "A" knovr kro,nr,,A,,is is =bay.: better than ''8" if we conside'rhe arternatives tc be 6rrifr;alry excir.Is;ve. Explicit incremental analysis calcuiations nrust be done tc vei.ify tilat conclusion with PVR, B/c Ratic, RoR and Growth RoF, but ait techniques give the same economic conclusion.

lf "A" and "8" are considered to be non-mutually exclusive alternatives, PVR, B/c Ratio, anci Growth RoR, are the valid ranking techniques and all indicate "A" is better than ,,8,, for i* 157o and 2O%. = Now iet's analyze the effect of using i* = 1}o:o tc handte tirne value of money in ratio calcurations, but effictivery in.r"rul ii;,, by raising the cutoff for acceplable projects from zero to 0.25 with pvR and from 1.0 to 1.25 with B/c Ratio. This is an evaluation aporoach sonretimes used in industry practice. Notice ii leads to the conclusion in this case that "A" and "B" are equivalent,',r,h€ti,lr the-,.are r:-ir_i._;aliy exclusive or non-mutually exclusive. Actually raising ,,i*', to 15% or 20"/" shows clear economic advantage to "A" for the higher ,,i*,' values with all standard techniques of analysis. Aiso notice that for i* i2'lo. all merl-rods show the aliernatives are equivalent. This means= incr"emental RoR analysis ("8-A" to get incremental costs followed by incremental revenues) will give the incremental ,,B-A" RoR to be 12'/". Treating alternatives "A" and "B" as eco*nomically equivalent is the conclusion given by ratio calculations at i* = 12/o with increased cutoff values of 0.25 for pVR or 1.2s for B/c Ratio. This expliciily means that a 12/" incremental RoR on "B-A" is satisfactorv. This is a misleading conclusion if other opportunities really, exist to invest dclkrs-at an RoR greater than 12%, which raising the cutoff rates on PVR to 0.25 and B/c Ratio to 1.25 implies. Since this approach involves inconsistencies in economic conclusions, it seems undesirable to use it. However, it can be argued that the differences in results from increasing the ratio cutoff instead of increasing ,,i*,, are small and not significant. Regardless, it is important to be aware that this approach may give economic results and conclusions that are

inconsistent with other standard evaluaticn techniques.

234

4.ll


Summary of Mutually Exclusive and Non-Mutually Exclusive Alternative Analysis

MutuallyExcIusiveAlternativeA:ralysis...:.:.b. Rate of Return or Growth Rate of Return. With either regular ROR or Growth ROR analysis of mutually exclusive alternatives you must evaluate both total investment ROR and incremental investment ROR, selecting the largest investment for which both are satisfactory. Use a common evaluation life for Growth ROR analysis of unequal life alternatives, normally the life of the longest life alternative, assuming net revenues and costs are zero in the later years of shorter life alternatives.

Net Value Analysis. With NPY NAV or NFV analysis you want the mutually exclusive alternative with the largest net value because this is the alternative with the largest investment that has both a positive total investment net value and a positive incremental net value compared to the last satisfactory smaller investment. When using NAV or NFV to evaluate uncqual lit-e altematives vou must Llse a common evaluation life, normally tht life of the longest lit'e alternative, assuming nel revenues and costs are zero in the later years of shofier lit-e alternatives as with ROR analysis. Ratio Analysis. With ratio analysis of mutually exclusive alternatives using either PVR or BiC Ratio, it is necessary to evaluate both total investment ratios and incremental investment ratios. Analogous to ROR analysis, the mutually exclusive alternative with the biggest total investment i'atio often is not the best economic choice. A bigger investment with a somewhat smaller ratio often is a better mutually exclusive alternative choice. Future Worth Profit Analysis. Calculate the FW Profit that can be generated by each alternative*project if profits and salvage are reinvested at the minimum rate of return, "i'''". Select the project that maximizes FW profit for the investment money available to invest, With this method ),ou must compare the FW profit for the same investment dollars in all cases. by assuming the budget money not required in one project will be invested elsewhere at "i*".

Dual-i-Values Dual-i-values relate to the multiple solutions that occur when cash flows result in investments generating revenue that are followed by more investments. All dual-i-values contain a combination of rate of reinvestment and

chapter 4: Mutuallv Exclusive and Non-Mutually Exclusive project Analvsis

235

rate of rerurn meaning. rvhich makes them useless for RoR analysis in

terrns of being used to reliably assess economic profitability. If a compound interest RoR measure of economic value is desired, three modifications to

the cash flows were described incruding Grc,urth RoR. the Escrow Approach anij thc Ytar-by'-!'ear Approach. In euc:h present north m,;dificrti0n. cash flou's are adjusted at the minirnum rare of return tit eliminate r\e exisience of cost-income-cost. Each of the present u'cri'th rnodificatiops resulis in the same NPV. This n'reans that cven rhough the i:pproaches genrrate nrodifii-:d ROR results of varying rna-tnitudes, the eccr.riirnic conclusions r.vill always be consistent with NpV.

Non-Mutuall_r, Exclusive Alternatives

Rate of I{eturn or Growth Rate of Return. Regular RCR analysis cannot be used to consistently rank nou-mutuallv exclusive alternatiyes. Use cror,',rh RoR and rank the alternatives in ti-re order of decreasing Growth RoR. This will maximize protrt from avaiiable investmr:nt cirpital. use a common evaination lif'e r'or Growth ROR anall,sis of unequal life alternir_ iives. normally the life of the longest alternative. Net value Analysis. with NPV NAV or NFV analvsis of non-murua:11, exciusire pi'ojects, seiect rhe gloup of projects that will maxirnize cumulative net r-aluc for tlie dollars available to invest. This does not necessarily involve seiecting rhe project with lar-sest net valnc on inclividual project investrnent. when using NAV or NFV to evaluate unequal life alternatives you must use a common evaluation life, normally the life of the longest life alternative. Ratio Analysis. Ranking projects in the order of decreasing present value ratio (PVR) or B/C Ratio is the easiest u,ay of selecting non-mutually exclusive projects to maximize cumulative NpV for available investment dollars.

l'uture.lYorth Profit Anall,sis. carcurate the FW profit that can be genif profits and sall,age are reinvested at the minirnum rate of return, "i'". use a comnon evaluation life equal to the longest life alternative if unequal life projects are involved. Select the group of investment projects that will maximize the cumulative FW profit for the money available to invest, analogous to the way Npv and NFV are applied. Assume that any odd dollar amounts can be invested elsewhere at '.i*;', the erated by each alternative project

same as with other techniques of analysis.

236


Remember, with all of these evaluation methods, the minimum rate of return, "i*", 1s the rate of return that represents the other opportunities which are felt to exist for the use of available investment capital. This term is sometimes called the hurdle rate;..discount rate, opportunityrost of capital or just cost of capital.

If you use a valid minimum ROR and apply the discounted cash flow analysis techniques as described in this chapter and summary, you will reach correct economic decisions for your cost, revenue, and timing of cost and revenue project data and assumptions. It is always important to recognize that economic analysis calculation results are just a direct reflection of the input data and analysis assumptions. With any techniques of analysis, economic evaluation conclusions are only as good and valid as the data and assumptions on which they are based.

chapter 4: Mutualry Exclusiveand Non-Mutually Excrusive project Analysis

237

PROBLEMS

4-1 If a time zero cost of $300,000

is incurred for equipment replacement, an existing project '4" is expected to maintqin the generation of $450,000 before-tax profits each year for year one through year 10. Two alternatives are being considered for improvement (project ,,8") and improvement combined with expansion (project,,C;,) with pro_ jected costs and revenues as shown on the time diagrams. All dollar values are expressed in thousands of dollars.

A) C=300 0

B) C=900

J=450

I=450

I=450

I=550

I=550

I=850

I=850

1

I=550

I=750 c) C=1,200 C=800

0

I

2.........10

For a minimum rate of return of r5vo and considering the alternatives to be mutually exclusive, determine whether project ,A,,, ..8,, or ,,C,, is economically best using RoR, Npv and pvR. Then increase the minimum RoR to 257o from l5vc over the entire l0 year evaluation life and re- evaluate the alternatives using any valid analysis.

4-2 A company is analyzing the economics of leasing

a parcel of land over the next 6 years for time zero lease payment of $g0,000. The land would be used as a product marketing center requiring construction of a $200,000 building on the land in time zero and it is estimated that it would generate annual revenues of $290,000 and annual operating costs of $160,000 at the end of years one through five. The lease contract stipulates that at the end of year six the building must be torn down and the property restored to its initial conditions and this is estimated to cost $360,000 at the end of year six. Use rate of return analy_ sis to determine if this project is economically viable for a 20To minimum ROR. Verify your results with NpV.


4-3 A double

pipe heat exchanger with steam in the shell is to be insulated to reduce heat loss to surroundings. The thickness of insulation, initial cost and projected annual cost of heat loss are given in the following table. If the minimum RoR is 2ovo before taxes, detdrmine the optimum thickness of insulation for an insulation life of six years with a zero salvage value.

insulation Initial (Inches) Cost, g

Annual Heat Loss Cost, g

Thickness

00 1 2 3 4

r400 1200 1800

800 600 500 400

2500 3500

Base your results on Net Present value Analysis, then verify them using Present Worth Cost Analysis.

4-4

You have been asked to make rate of return analysis to support or reject the economic viability of project development that has the estimated potential of generating $450,000 revenue per year and operating costs of $310,000 per year for each of the next l0 years (assume end_ of-year one through 10 values). A company has agreed to construct and finance the project for deferred payments of $50,000 per year at the end-of-year two, three and four with final lump sum purchase cost of $ 1.450,000 to be made at the end of year five. The year ten salvage yalue is estimated to be $300,000. Make rate of return analysis to eval_ uate project economics for rninimum rates of return of (a) 5ro, (b) 15Vo and (c) 507o. Do NPV results support your conclusions?

4-5

To achieve labor cost savings on a manufacturing process, a company rvill install one of two possible equipment automation changes. New capital equipment costs and projected savings are as follows: Equipment Cost

Projected Annual Savings

Change I

$150,000

$ 80,000

Change 2

$230,000

$115,000'

chapter 4: Mutuary Excrusive and Non-Muluary Excrusive project Anarysis

231

(A) For a six year evaluation life, which change, if a,y, should be selected if ix - 40va beforc tax? use Net piesent vaiue Analysis aitd assume zero salvags lrlues. (B) Is rirere a,y evaiuariori lifc orher rhan ,ii y"r^ rhar *,ould swirc:h the eco.omic evaiuai;on rcsuii f,un..l in (A)? Ii yes, what is the b,eak-even rife for r'hich there are no econonlic ciifferences hc:tweefl the alternatives?

4-6

R'ank non-mutuaily exclusive alternatives Anall'sis for i* = l5Vo. C

A)

$200

= g=U00 I=$110 I=$110 2

C = $200

'A"

C = $230 I = $110 I = $110

and ..B,, using pvR

i = $110

4... . ..8

J

.

C = S180

B) C=$100 I=$i10 I = $20 I = $110 I=$110 I = $1to 4. ...8

4-7

Two muiualry excrusive unequai lir-e investrnent alternatives, .A,, and "8".,rust be evaruated to deierminc rhe best econonric choice for i* = 20vc.T..e in'estrnenis, "c". a,cj incorres,'.I". and salva-ee values,.,L,,, are sirou,n on the trrne clial,rams in thc.usands of dollars.

A) C=100 0

B) C=150

J=40

I=40

I=40

I=40

I

2

3

4

I=60

I=60

I=60

I=40

L=100

L=150

Determine the dual rates of return that resurt from a direct incremental ROR Analvsis of ,,B_A',. ii) Evaluate the projects using NpV Analysis.

iii) Evaluate the projects using incremental Growth ROR Analysis. iv) Evaluate the projects using pW Cost Modified ROR Analysis. v) Is the economic choice affected by reducing the rninimum RoR to l07o from207c'!

240

.


A friend offers to give you 10 payments of $1000 ar annuar time periods zero through lo.:excepr year three if you give,him $9000 at year

4-8

three as shown on the time diagram.

All

values are in dollars.

I=1000 I=1000 I=1000 C=9000

I I=1000 4..

I = i000 10

Evaluate this income, investment, income opportunity shown on the time diagram for before-tax minimum rates of retu oi loEo and 20vo.

^

A) Use Net Present Value Analysis. B) Analyze the project using present worth cost Modified 4-9 A new process

can be developed and operated at Levers

RoR.

,d,, 6. ,,g,,

with capital costs, sales and operating costs as shown. All values are in

thousands of dollars.

Sales = 75

100 Op. Costs = 45 Sales

B)

C=150

0.

=

1....

Sales = 75

Sales

=

100

Op. Costs = 45

.....5

If i* = 207o, which level of investment should be selected? use Npv Analysis and verify your resulrs with RoR and pvR Analysis. If the minimum RoR is reduced to 12vo, what is the economic choice? If the minimum RoR is r2vo for years one and two and 2ovo for years three, four and five, what is the economic choice? 4- 10 Improvement and expansion of a production facility are under consideration. Ai the present time profits are $450,000 per year, and in the future

escalation of operating costs each year is expected to exactly offset escalation of revenue, so profit margins .re projected to remain constant at $450,000 per year for each of the next years one through 10. Two alternatives for improvement and improvement combined with expansion are being considered with projected costs and revenues as shown on the time diagrams following this problem statement, with all dollar values expressed in thousands of dollars.

Chapler 4: Mutually Exclusive and Non-Mutually Exclusive prolect Analysis

I=450

A) 0

1

I=450

241

I=450

2.,......10 t

B)

C=

80{-)

I=700

I=7C0

I=700

2........10 I=850

c) C=1,300 C=900 I= 1,050 I= 0

1,050

2........10

For a mirrimum rate of return of l5%o. and considering the altematives to be rnutualiy e.rclusive, determine whether the present'A,', improvement

"8", or ir,rprrcrvement plus expiinsion projeci "c" is economically

best

using RoR. i\iPV and pvR. Then incrcase rhe minimum RoR to 25% from 15vc over the entire l0 year evaluation lit-e and re-evaluate the alternatives using both RoR and Npv analysis. Then assume that the nrinimurn I?"oR is increased from 15% to 25vo tbr evaluation vears orie and two ancl then. stz*ting in year three, the minimum RoR reverrs to 157c again through vear 10.

4-1

I

An ethylene plant manager is considering the investment of $32,000,000 today (time zero) to integrate heavy duty industrial single shaft gas turbines to reduce energy costs for a 750,000 metric ton per year liquids cracker. The time zero investment is expected to generate energy operating cost savin-qs of $12,000,000 at the end of each of years one thiough

five. An alternative is to consider using aero-derivative turbines (based on jet engines) that cost $34,000,000 but require an adclitional $4,000,000 time zero invesrment in design modifications. These rurbines are slightly more efficient and generate annual energy savings of $14,000,000 but also require additional maintenance or $soo,oOO per year at the end of each of years one through five. If the desired minimum rate of return is 15vo, use rate of return analysis to determine which of these two alternatives should be the economic choice. verify your conclusion with NpV analysis.


242

'A"

involving the investment of $240,000 to generate a,series of equat end-of-year revenues of $50,000 per year for five years plus a salvage value of $240,000 at the end of year five and project "B" involving the investment of $240,000 to generate a series of equal end-of-year revenues of $98,500 per year for 5 years with zero salvage value. The projects are mutually exclusive and the minimum ROR is l}%o.Yeify your results with NPV Analysis.

4-12 IJse ROR analysis to compare project

4-13 Dctermine the best economic way for a research manager to allocate $500,000 in the following non- mutually exclusive projects if 1* = 207o'

A)

c

B)

C =-$1Q0,999

,Offi

L=0

0

C = $300,000

Ll

L=0

,000

L = $100,000

0

Use NPV Analysis and verify the results with Growth ROR Analysis and PVR Analysis

4-14lf alternatives 'A" and "B"

are mutually exclusive projects, use ROR Analysis to determine which is economically best rf i* = l5Vc.

A)

C=$20,000 I=$12,000 .....I=$12,000

B)

C=$28,000 I=$14,000 ...'.I=Sl4-000

0 0

l I

.......r2 .... .i

L=$20'000

L=$28'000

Verify the ROR Analysis results with NPV and PVR Analysis.

cnapter 4: Mutua'y Excrusive and Non-Mutua,y Excrusive pr.ojeci Anarvsis

?43

4-15 Three unequai lit'e investment alternati'es'u,ith costs, profits and vage values as shown on the time diagrams are being considered.

Al

C = Sl60.0o(i

tr

I=Sl-50,000

=

0

C = $,120,0(1.1

ts)

I = $27.':.i;00.

C = 5436,000

I = $500,000

"$

I50,000

L = $50.000

I = $275,000 L = $70.000 .4

0

c)

sar_

I = $500,000

L = $100,000

Assume $490,{100 is avairabre ro invest and other opportuniries exist to irr'est all avaiial',re d,ilars at a 2avc RoR-

use Nl-v'anary.si.

,o a"r"._ n'rine how the 9490,000 shourcr be spent from an economic viewpoint if the altematii'es are non-rnutuairy exclusive. Then use Npv Analysis

to determine r.ihiih of the alic.n;itives. .A,,. .!ll,i or ,.(J,,, is best if the iliernarives are inutuary excrusivc-. \'erify these results r.vith pvR

Analysis.

4-16 For an in'estor with a minimum rate of return of 20.070: (1) Rank the fo'owing non-mutually exclusive alternatives. (2) For a time zero budget of $500, which of these projects wourd you setect? con_ sider that year one or later year budgets wilr cover costs in year o,e or later years n't co-r'ered by project revenues in earlier years. (.3) Deve lop the "cunlulative NI,v,,biog.o.n for alternative ..D,, and indicate the poini ott this dia-uranr that represenfs tle investor,s m..r:inrum capital at risk.

Yr A) B)

c) D)

-200 -300 -300 -300

60

-90 2s0 100

100 100

250 r50

-200 200 250

-350

300 300

400

250

-600

400

300

400

244


4-17 lmprovement to a coal mine haul road is being considered to shorten the overall.haul distance and reduce ffuck cycle times thereby increasing the total number of cycles per day. The cost of the improvements is estimated to be $6,0U),000 at time zero. Currently, the trucks make 16 cyclgs per day from the mine pit to a port load out facility which translates into 2,000,000 tons of coal being hauled each year. If the road improvement is made, the number of cycles is anticipated to increase by 257o to 20 cycles per day which translates into an additional 500,000 tons of annual production from the mine. Total mine reserves are estimated to be 14,000,000 tons and the coal is sold at an average selling price of $32.00 per ton with operating costs of $12.00 per ton. If the haul road improvement is made, the life of the mine will be shortened from seven to six years due to the accelerated production schedule. With the modified haul road, total production in years one through five would be 2,500,000 tons per year with year six production of 1,500,000 tons. For this acceleration problem, use rate of return analysis to determine if the investment in the haul road improvement should be made today. The minimum acceptable rate of retumis l5%o.

4-18 Improvement to a mine haul road is being considered to shorten the overall haul distance and reduce truck cycle times, thereby increasing the total number of cycles per day. The cost of the improvements is estimated to be $6,000,000 at time zero. Currently, the trucks make 16 cycles per day from the mine pit to a crushing facility rvhich, given the truck perfornrance, capacities and haul distance, translates into 2,000,000 tons of ore - being hauled each year. If the road improvement is made. the number of cycles is anticipated to increase by 257o to 20 cycles per day which translates into an additional 500,000 tons of annual production tiom the mine. Total rnine reserves are estimated to be 14,000,000 tons. If the haul road inrprovement is made, the life of the mine will be shortened from seven to six yeilrs due to the accelerated production schedule. With the modified haul road, total production in years one through five would be 2,500,000 tons per year with yezr six production totaling 1,500,000 tons at which tirre the resenres wor-rld be depleted. For each alternative, the average ore is 0.09 ounces per ton with a 0.85 recovery rate. The average selling -erade price is $350.00 per ounce while the operating costs are S 190.00 per ounce beginning in year 1. The minimum acceptable rate of return is 15.0Vo. For this acceleration problem, use rate of retum analysis to determine if the haul road improvement can be economically justifred. Then verify your findings with a net present value analysis.

Chapter 4: Mutually Exclusive and Non-l,4uri.rally Exclusive project Analysis

4-19 Two alternatives exist for you to invest $100,000 as shown on the fol_ lowing time diagrams. consider risk to be identical for each investnrent. Assuming your rninimum discounr rate is L5.\va, determine the economically better choice using the rate of retu?n, net present value, present value ratio, grorvth rate of return and future worth profit.

4-20

A)

-250

B)

-250

11532

.....115.32

1,206.7

A proposed investment

has the following projected net cash flow

stream from development ccst and-product sale profits, ancl abandonment costs. use rate of return analysis to determine if this investment

is satisiactory tbr a minirnum discount rate of lz.ovo. All cash flows

are in thousands.

BTCF

4-21

-1!70 +1,705 +897 +103 _617 _-559

_2g4

A natural gas distribution company is evaluating the economic desirability of replacing or repairing existing gas mains in a small town. At the present time, with the existing gas mains, it is estimated that 100,000 Mcf of gas is being lost per year and that this gas could be sold to colporate customers at $2.00 per Mcf. The cost of replacing

the gas mains is estimated to be $1,000,000 at time zero. Replacement

of the mains ri'ould effectivery eriminate all gas loss for the next 10 vears. The cosr of repairing the gas mains is estimated to be $400,000 rvhich would reduce annual gas loss to 25,000 Mcf at year one (allocate to the end of the year), with gas loss increasing by a constant gradient of 6,000 Mcf per year in years following year one. Use present rvorth cost analysis for a l0 year evaluation life and i* = l5.0vo to determine from an economic viewpoint if the gas mains should be replaced, repaired, or left in the present condition. verify your cost analysis results with incremental NpV.

CHAPTER 5

ESCALATED AND CONSTANT DOLLARS

5.1 Inflation

and Escalation in Economic Analysis

Many people use ihe terms inflation and escalation interchangeably but generally, as usec in this textbook they have different meaning. The term inflation is often associated with increases in prices for goods and services that is the result of "too many dollars chasing too few goods," and is commonly measured by various indexes, the most common of which is the consumer Price Index or cPI. A more refined definition of inflation might be, "a persistent rise in the prices associated with a basket of goods and services that is not offset by increased productivity." Deflation could be defined as an overall decline in prices for a similar basket of goods and services . Escalation ort the other hand refers to a persistent rise in the price of specific

comntodities, goods or serv,icis due to ct combination of inflation, .sttpply/demand and other effects such as environmental and engineering changes. Note that escalation deals with specific goocls and services, while

inflation is concerned with a basket of goods and services, so inflation irrvolves et'eroge change. Inflation is just one of several factors that contributes to cost or price escalation and it should be evident that escalation is what affects the actual costs and revenues that will be realized for a project. Therefore, escalation of costs and revenues is the consideration that must be accounted for in economic evaluation of investments. Figure 5-l enumerates these considerations.

246

Chapter 5: Escalated and Constant Dollars

247

lnflation Supply/Demand Technological Changed Market Changes Environmental Effects Political Effects Miscellaneous Effects

:

r.:

Items Affectin g Escalation

,

Figure 5,1 items Related to,,Escalation" T'he importance of cost and price escalation and the pyramiding effect it can have on the escalation of investment costs and revenues has been driven

honic forcefully to investment decision makers around the world in the i970's and 1980's. Examples are innumerable. To cite one, in 1969 tire Alaskan pipeline was esrimated to have a cost of $900 million while in 1977 the final cost estimate w,as close to $g biliion, or about agoavo escalation iiorn the initial estimate. Certainly not all of this cost escalation rvas due to inilatron. sLrpplyidernand cffects on rabor and materials and other factors such as environnrental and engineering changes caused very significant cscalarion of the project costs. Fortunateil,, crude oil price escalirtion prior to the 1986 price decline rvas sufficient to cover the escalated costs. This exatnple helps emphasize the vcry signilicant differences berween inflaticn raies and escalation ri.tes and that in econonric elaluarion u,ork rve are con_ cerned 'uvith the effects of escalation of costs and revenues rather than inflation effects alone. Hou,ever. int-ration often iias such a siguificant effect on escalation rates for specific goods and services that we should digress and discuss some of the causes and effects of inf'lation before later proceeding to present two explicit approaches for handling inflation and escalation properly in economic analyses. Literally millions of words have been written on the subject of inflation in tlre past decade. we discussed in cliapter 2 that 6zo compound interest u'ill double capital in twelve years but have you thought about the fact that 6%' inflation will cut the purchasing po*er of currency in half in tu,elve years'l The sarne compound interesl factors that work fbr us with savings apply inversely when accounting for the effect of inflation on purchasing power. who benefits and rvho gets hurt mosr by inflation? There is no firm answer to this question but generally individuals and governments with large amounts of borrowed money benefit to some aegiee from inflation because they are able to pay off their debt with inflated future dollars that have lower purchasin-e power than the dolrars originally borrowed. on the

248


other hand people on fixed incomes with little or no debt clearly are hurt by inflation because the purchasing power of their in9.n$9 gI capital decreases each year approximately proportional to the annual inflation rate. It is necessary to discuss.how inflation rates are derived to anrplifythe meaning of the last stiltement.

As previously mentioned, in most countries the quoted inflation rate is derived from the change in an index, made up of the weighted average of prices for a "basket of goods and services." In the United States, the broadest measure of this index is known as the Consumer Price Index or CPI. As currently defined the CPI can be broken down into 8 major categories including food and beverages, housing, apparel, transportation, medical care, recreation, education and other items including the cost of hair cuts, cosmetics and bank fees. Approximately 80,000 items in total may be surveyed each month. The. CPI may be calculated regionally or by other socio-economic parameters. The later categories include an index for all Urban Consumers known as the CPI-U and a similar index for all Wage Eamers known as the CPI-W. A survey of the buying habits of more than 29,A00 families in these different categories forms the basis for the items selected each month. For more information check out the Bureau of Labor Statistics website at http://stats.bls.gov/cpihome.htm. It is important to note that an increase in the level of one price, or group of prices does not constitute inflation. If the prices of crude oil, iron ore, and uranium rise while the prices of automobiles, wheat and beef fall, we can not be certain whether there has been inflation. In other words, it may not be certain whether the average level of prices has risen or fallen. Lower prices in one set of goods may have offset higher prices of other. An increase in the price of one individual commodity is just that, an individual price increase. It may be due to inflation, excess demand or short supply or a combination of both. This further emphasizes why in economic evaluation work that escalation of costs and revenues for specific goods and services is relevant to the analysis rather than a given inflation rate. Even if inflation is dirninished tc l7a to 17a levels over an extended period of time, different financial instruments such as indexed bonds and the more traditional non-indexed variety do exist in the financial marketplace today and require unique consideration to make a proper economic comparison. Further, although escalated dollar evaluations are the most common approach we find utilized in the United States, some U.S. companies and many foreign companies prefer to utilize constant dollars. Therefore, it is important to understand how inflation can influence project evaluations and the various assumptions that investors make on this subject.


249

There are two basic evaluation tecrmiqttes that can be used with equal ualidity to handle inJlation and escalation properly in economic analyses. Tht.t are escalated doLlar analy'sis and cansrant rJollar analysis. The econornic conclusions that are reached are always identical with either appi.)ach. Escalateci tiollar values refer to actual tlollars of reventte or cost tiicti will be realizetl or incLtrred at specificfuture poittts irt tinte. Elsewhere in tita literarure yott find escalated dollars ;"eferred to by the tenns crffrent aL.,iitit's, ir{lcted dollars, nominar doilars, cmd dollars i7 tlr" day. corrstant dollur v-alues refer to hypothetical constant purchaiing power dollars obtained by discounting escarated doilar values at the inflii)n rate to some arbitrary point in time, which often is the time that corresponds to the beginnilry of a project but may be any poini in time. Constait doilars qre referred to as real dollars or deflated dollars in many places in the literature. Figure 5-2 relates the definitions of escalated dollars and constant dol_ lars to a variation of Figure 5-1 used earlier to illustrate how inflation and

escalation are related.

lnflation

Constant Dollar Items

SupplyiDemand Technological Changes Market Changes Environmental Effects

Escalated Dollar Items

Political Effects lu4iscellaneous Effects

Figure 5-2 "Constant Dollar,, ltems That Relate to 'Escalated Dollar" ltems constant dollar analysis advocates take the point of view that since escalated dollars have different purchasing power at different points in time it is necessary to convert all dollar values to some hypothetical constant purchas_ ing power value before making an economic anaiysis. constant dollir analysis is a valid analysis approach but it is no bettei than making the analyiis directly in terms of escaiated dollars and the constant dollai calculations require extra work and chance for error. It must be remembered that the purpose of economic analysis is to compare alternative opportunities for the investment of capital to select the investment alternatives that will maximize the future profit that can be accumulated at some future point in time. If you stop to think about it, the alternatives that will maximize your future profit in escalated dollars must be exactly the same alternatives that will maximize your future profit expressed in terms of constant purchasing power dollars

250


referenced to some earlier date. The key to proper analysis and consistent, corect economic evaluation eonclusions lies in recognizing,that you cannot nrix escalated dollar analyses and constant dollar analyses. To do so is analoqous to comparing apples and oranges. You must eithercornpare'all alternatives using escalated dollars or you must compare all altematives using con-

stant dollars; and you must remember that the minimum ROR must be in relation to other escalated dollar investment opportunities for escalated dollar analyses while the minimum rate of return must be expressed in relation to other constant dollar investment opportunities for constant dollar analysis. A very common constant dollar analysis mistake is for an evaluator to calculate constant dollar rate of return for a project and then compare it to other escalated dollar investment opportunities such as a btrnk interest rate, attainable bond interest rates or another escalated dollar ROR opportunity. Consider how escalated and constant dollar estimates for costs and revenues are obtained. The evaluatirrn of projects starts by estimating project costs and revenues in today's prices or values. This establishes what the project costs and revenues would be if the project occurred today. Next, since projects do not occur instantaneously, the today's dollar values are adjusted to project the actual or escalated values that will be realized. If the increase can be described on a percentage basis, single payment compound arllount factors, (F/P1,n) can be used to determine the actual value anticiprted at the future point in time. The escalation rate is used as a substitute value for i. This approach is utilized in many text examples, but may not be appropriate in actual evaluations depending on the forecasting techniques utilized. Also note, if you think values may decline, negative escalation rates might also be used. The actual values anticipated to be realized over the prolect life form the basis for the escalated dollar evaluation. N4any investors choose to utilize the anticipated inflation over future vears as an approximation for escalation. This is one forecast of the future, but there is no reason evaluations cannot be based on as many separate escalation or de-escalation rates as there are parameters that make up the model. Curnmodity prices, the price for construction equipment, steel, concrete, labor and energy to name a f'ew, may not move in direct correlation with the rilte of inflation. The use of inflation as a proxy for escalation is addressed in more detail later in the chapter concerning the use of today's dollar values in evaluations. If a constant dollar analysis is desired, the escalated dollar cash flows for a pro.iect must be adjusted to have the anticipated intlation removed from each expressed

Chapter 5: Escaleted and Conslant Dollars

value

ac1

o'er the proiect life.

Remember. that escalation includes inflation. Inflation can be removed from escalated dolrar.cash,flows by murtiprying each escalated ritirrar varue by trre singre payrnent present worth factor (P/F1.n), at the assumed inf'lation rate .,fl roi tn" nu*b., otferiods needed to t,r'ing each value to its cqui'elent constant purch:rsing po*".. Note that escalation rates and inflation rates are trearecr jusr rike compound interest rates. As prel'iously indicated, this means t"hat ttre factor, (F7l'e.n) r'vhere e represents the arnnual rate of escalation, and (p/F1,n) where f represents the annual rate of inflation can be utilized in the same manner but for different objectives than accounting for the time value of money. . so, if the selling price of a product *u. $zoo per unit this year (today,s dollen value is $260) and the varue was expected to escarate 5.|va nextyear, the actual selling price one year from today would be: Escalated $

Yr I Price = $260(Flp5%,) = $273

[1 inllation is3.07o during the same period, the rea] or constant purchasing power price of the product would be:

Time 0 Consranr $ price = $273(p/Fg qo,fl $265 = This is still a year one price, but now expressed in terms of time zero condollars. Many in'est'rs w,ill approximate this varue by taking a shorrcut. This approach n'ould argue that if prices escalate 5vo and

st.,t

inflation is 3vc, then approximately, there should be a 2o/o real or constant dollar increase in the price of the product. Constant $ = Today,s g x (Escalation _ Inflation) Constant $ $260 x (1.02) = $265

=

when escalation and inflation are row, the approximation is very close for those wanting to work with constant dollars. However, this approximation is not utilized in the text exampres and problems due to confusion that can develop when working with after-tax cash flows. Afler-tax cash flows reflect the impact of various tax deductions, wrrich often (but not alrvays) are escalated deductions. Improper mixing of dollar values only dilutes the quality of the economics so the .no." p.""i.e, longer winded approaches are utilized throughout the text. As a final introduction, gord has often been thought of as good hedge against the long-term i*pact of inflation. Looking bacl to 1990, the price of

252


gold was $420 per ounce. In mid 2000 the price had fallen to $275 per ounce. That translates into an annual price decline of. 4.LVo per year (negative escalation) over 10 years. During that same period, U.S. inflattn (as measured by the CPI) averaged approximately 3.OVo pelyear. Had gold increased in value at the rate of inflation, the value in 2000 would have been. t.3439 $420(FlP3Eo,10) = $564 per ounce

Instead, the actual price dropped to $275 per ounce and the corresponding constant dollar equivalent price of gold (in terms of 1990 base year dollars) dropped as well to: 0.7441 $275(P83o7o,10) = $205 per ounce

It is worth noting that the term "constant purchasing power" has nothing to do with the price remaining constant each year. Instead, it has to do with the relative purchasing power of that money over time, If the price of gold had actually escalated 3.0Vo per year so it just covered inflation, the real purchasing power of the ounce of gold would still be $420. Obviously more than inflation is influencing the price of gold and other parameters that make up our economic evaluation models. Finally, it is aiso important to note that inflation can have the same impact on interest rates as that just described in terms of cash flow or the price of a product. Further, discount rates are influenced irr the same manner as is introduced in Example 5-l: Consider the following investment analysis example to illustrate how today's dollar values are the starting basis to get both escalated dollar values and constant dollar values and the corresponding rates of return. EXAMPLE 5-1 Escalated and Constant Dollar ROR Analysis Today's Dollar Values, C=Cost, l=lncome

co 0

$aoo

1

2 years

Convert today's dollar values to escalated dollar values assuming cost escalation of 2O"h per year and income escalation of 10% per year. Calculate the escalated dollar rate of return. Then assume inflation is 15% per year and calculate the constarlt dollar project ROR.


Solution:

., Use singre payment compound amount factors (F/p;.p factors).at the appropriate escaration iates to convert gday,s ooillo to esca_ lated dollar"s as shoir",n on the followirrg oragram.

Escalated Dollar Value Delermination C0 =

$to0

1.200 C1

t5!q/P20%,1)=$1

1.210 B0

le = $+OO(VP rco/",2) = $484

Escalated Dollar ROR Analysis when escarated costs and revenues have been obtained, they are ihe basis for the time varue of money carcuration to determine the ciesired evaiuation criterion, which in this anarysis i. non. PW equation: 0 = -100 180(ptFi,1) + 4g4(p/Fi,) i = Escalated Dollar ROR 47.7"/" by trial and error = This escalated doilar RoR is compared to the escarated doilar minimum RoR to determine if the proleci is economicaily satisfactory.

Constant Dollar Value Determination Now convert the escarated doilar varues to constant doilar varues assuming an annuar infration rate of 15%. This is achieved by dis_ counting the escarated dollars from the previous analysis at the rate of inflation to express ail doilar varues in terms of time zero purchas_ ing power. Use the singre payment present worth factor (p/F;,n factor) for the assumed infration rate as iilustrated on the tottowing diagram: Co =

$ru

Ct

=

$1eo(p/Frcx.; = g156

lZ = $4Aq(P lF

I Sk,Z) = $3G6

. The,meaning of the constant doilar costs and revenues is as forlows: $156 at year 0 wourd purchase the goods and services that $180 would purchase at year 1, if inflation is tSZ p"ry*, and g366 at year 0 wourd pulghase the goods and services tnat $+84 wourd purchase at year 2, if infration is15% per year. The meaning of these constant purchasing power doilars is onry varid when ,"t"t"o to purchasing the average goods and servicei tnat make ,p in" infration index that was used to project 1S% inflation per year.

254


Constant Dollar ROR Analysis PW equatiooi-O = -100

-

tS6(P/Fy,1) +

366(piFy,2), :?

i'= Constant Dollar ROR = 2B.S% by trial and error

:

r

This constant dollar RoR is compared to the constant dollar minimum RoR to determine if the project is economically satisfactory. Either escalated or constant dollar analysis of this project, or any other, will give the same economic decision. To reach a decision concerning the economic viability of this project using either the escalated or constant dollar analysis results we must know the other opportunities that exist for the use of our money and whether these other opportunities are expressed in escalated or constant dollars. For illustration purposes consider that other opportunities exist to have our money invested at an escalated dollar RoR of 3s%. comparing the project's escalated dollar RoR of 47.7"/" to the minimum rate of return of 35% we conclude that the project is economically satisfactory. lf we compared the project's constant dollar RoR of zg.s% to the other escalated dollar opportunities we would be making an invalid comparison. we cannot compare constant dollar results with other escalated dollar project results. All results must be calculated on the same consistent basis before comparison is made. To make a constant dollar analysis result comparison, we must convert the escalated dollar minimum RoR of 35% to the corresponding constant dollar minimum rate of return, which we then compare to the project constant dollar RoR of 28.s%. Mathematically, for escalated dollar RoR analysis, the factor P/Fi.n does the job that the product of factors P/F1,6 x P/F;,,n does for constant dollar analysis of the same starting escalated dollar values where "i" equals escalated dollar RoR, "i"' equals constant dollar ROR and "f" equals the inflation rate. P/F;,n = P/F1,p xPlFi,,n, gives: 1l{1 + i)n = 1/(1 + f)n x 1/(1 + i',)n, or:

1+i=(1 +f)(1 giving

i'=

+i,)

[(1 + i)/(1 +

5_1

f)]- i

Equation 5-1 and its rearranged form are equally valid for the calculation of project rates of return and minimum rates of return. This means the value of the escalated dollar project.rate of return, "i," can


be replaced by "i*," the escalated dollar minimum rate of return, and the constant dollar project rate of return ,'i,,1, may be replaced by the constant dollar equivalent minimum rate of return ,,i",.,, Foi this example, the escalated dollar minimum ROR, i" 857o, the infiaiion rate f = = 15"/", so this gives a value for i*, as fcllows:

i

'

=[(1 +.35)/(1 +.1S)] -1=.1Z4ori7.4h

comparing the project's constant doliar RoR of zg.s%to other constant dollar RoR opportunities of 17.4"/,leads to the economic con_ clusion that the project is satisfactory, consistent with the escalated dollar analysis conclusion based on comparing the 47.2"/o project escalated dollar RoR to the 35% escalated ooilar minimum RoR. Note that if our other opportunities were represented by an dollar bond investment RoR of 10"n, tor 15% annual inflation,"r.ur"t"J the corresponding constant dollar minimum RoR would be negative. = [(1 + .10)/(1 + .15)]- 1 = -.043 or *4.O/"

i

. -Mgney earning a 1o"/" escalated dolrar RoR in a 15?k per year inflation climate is losing average purchasing power al a 4.3"/" rate per year. Yet, if that is the best other opportuniiy that exists, it is betier than doing nothing with the money and rosing the average purchasing power at a 1 S"/" rale per year. EXAMPLE 5-1a Today's Doilars Equar Escarated Doilars Assumption: Escalation _ 0% when escalation is predicted to be oo/o pa( year on all costs and revenues over the life of a project, today's dollar values equal esca_ lated dollar values. To illustrate, consider 0% escalation per year of the Example 5-1 today's dollar costs and revenues and calculate escalated dollar project ROR.

solution: Escatated Dollar vatues tor o"h Escatation per year

C0 012

'1.000

1.000 F/POZ,2) = $400

That the F/P factors al o"/o equal 1 .0 is clearly seen in the mathematicalformat, where

F/PO%,1=(1 +i)n=(1 +0)1 =1.0and F/Pg"y",2=(1 +0)2=1.0.

256


Escalated Dollar Analysis PW Eq: 0 = -100 - 150(p/Fi,1) + 400(p /Fi,2) i = Escalated Dollar ROR gg.6% by trial and =

errort

This 38.6% escalated dollar RoR is different in magnitude from lhe 47.7% escarated dollar RoR result from exampte E-1 because the cost and revenue escaration assumptions are J;;r"i"ry;;ff.; ent. Assuming that all costs and revenues escalate at oi/" per year in this Example 5-1a is very different than assuming that ail costs escalate at 20"/" per year and revenues escalate at loy. f"iy"", as was assumed in Example 5-1. EXAMPLE 5-1b Today,s Doilars Equal Constant Dollars Assumption: Escalation lnflation = when escalation is predicted to equal the inflation rate per year on all costs and revenues over the rife of the project, toi"v,J doilar varues equal constant doilar varues. To iilustrate, consider isz p"iy"",, escalation of the Exampre 5-1 today's doilar costs ,."u"nr.. (15% is the assumed annual infration rate for Example"nd s-t) ano catculate escalated doilar RoR and constant doilar RoR. Then use Equation 5-1 to verify the interchangeable character of the results. solution: Escarated Dortar varues for Escaration Equarto 15% lnflation per year

c

=

sz,t) = $tzz's

t = $+oo

(F/P1s%d= $528'8 2

Escalated Dollar Anatysis: PW Eq: 0 = -100 - 1 72.5(plFi,1) + 528.8(p/F;,2) i = Escalated Dollar ROR Sg.4% by trial and = error. This result is different from either the Exampre 5-1 or Exampre 5-1a results, because the cost and revenue escalation assumptions are different.

Constant Dollar Values:

C=$100

C = $172.5(P/FtSZ,t) =

o----.--.------1=-2

g150 t= gSZe.a(p/f.,

5.h,2)= $400

Escalating a[ varues at the rate of infration (1s% in this exampre) and then discounting all values at the same rate of inflation for the


same number of periods makes project constant dollar values equal plF1,'n = 1/(1+f)n is the recipi,ocal ot Flp1,n= (1+f)n.

to the original today's doilar values. This occurs because Constant Doilar Analysis: PW Eq: 0 = -100

-

1S0(p/F;,,1 )

+

4OO(plFy,2)

i'= Constant Dollar ROR = 9g.6% by trial and

error.

The constant dollar RoR is numericall! the same as the Example 5-1a escalated dollar RoR result, but the 3g.6% constant dollar FloR result must be compared to a minimum RoR expressed in constant dollars whereas the Example 5-1a escalated dollar RoR result must be compared to a minimum RoR expressed in escalated dollars, analogous to the decision-making discussion near the end of Example 5_1. once the escalated dollar RoR has been carculated, when infiation is assumed to be uniformly the same rate each year, Equation 5-1 can be used to calculate constant dollar HoR instead of discounting escalated dollars at the inflation each year and then calculating constant dollar RoR by trial and error from prop", constant dollar present worth equation. " To prove thts approach for this example: i = escalated dollar ROR Sg.4%

=

f = inflation rate = 15.0"/" From Equation 5-1, 1 + i = (1 +f)(1 + i,) so i'= [(1 + i)/(1 + f)] - 1, therefore;

i'=

[(1

.594)l(1.15)]-

1 = 0.386 or 38.6%

A simplification of Equation s-1 often used by bankers, brokers, economists and evaluation people alike is as follows: 1 + i = (1 + f)(1 + i') = 1

+f + i, +fi,

Neglecting the fi'interaction term and rearranging gives the following approximation:

.

i-f=i'

5-2

For Example 5-1b, escalated RoR of sg.4/" minus the inflation rate of 15.0% is approximately a 44.4/" constant dollar RoR. The

258


difference in the approximale 44.4% constant dollar ROR and the actual 38.6% result is 5.8ol" approximation error from dropping the fi' interaction term which equals (.15)(.386) or 0.058 or 5.8%. A diagram explanation of the relationship between efcalated dollar present worth calculations and constant dollar present worth calculations is shown in Figure 5-3 which follows.

Escalate Using F/P",n

Today's $

Escalated Dollars

Discount Using P/F;.,6

Net Present Value

t-

I e=escalationrate I f=inflationrate I i' = escalated $ discount rate I i.'= constant $ discount rate I i = escalated $ rate of return i i'= constant $ rate of return

Discount

Using P/F1,n

Discount

Using P/F;*,,n

.

1,,, I

l'/ '#,ii?i'

Figure 5-3 Equivalent Escalated Dollar and Constant Dollar Present Value Calculations EXAMPLE 5-2 Escalated and Constant Dollar ROR and NPV

Analysis

A proposed investment has the following projected today's dollar costs and revenues: C0 =

$20,000 Cr = $40,00Q

01

Rev = $70,000 Rev = $70,000 OC = $20,000 OC = $20,000

L=0

Use ROR and NPV analysis to evaluate the economic potential of this investment project for the following evaluation assumptions. For escalated dollar analysis assume the escalated dollar minimum ROR is 15%. Use Equation 5-1 to calculate the equivalent constant dollar minimum ROR for constant dollar analysis.

Chapter 5: Escalate,l and Constant Dcllars

A) Assume year l development cost will escalate 7"/" per year. year 2 and 3 revenues wiil escarare gr" per year. operating';;i. ili ' escarate go/o per year. Mako escatateo c"rrrr "nlv.].. B) the part (A)_escaration assumptions an*given escarated dor.For lar minimum RoR of 1s"h, make constant doilar anarysis of the

c)

project for assumed infration or 6"r. per year over the project rife. use the given today's dollar cost and revenue estimates to evaruate the pi'oject and discuss the impricit escaration and infration rate assumptions involved.

Solution, All Values in Thousands of Doltars: A) Escalated Dollar Anatysis 1.166 1.260 Rev=70(F/P 8y",2)=$.az zo(F lp gy",g)=88.2 '1.188 OC=Zg1p/pn

co=20

Cl=40(F/Pl"n

i=42.8

1.295

"/o,2)=EJ9

2O(F/pg"/",3)_el

Net Rev =57.86

=62.3

The today's doilar costs and revenues are converted to escarated dollar values using singre payment compound amount factors carculated at the escalation rates. Note that the uniform series compound amount factor, F/Ai,p, has not been utirized to escarate the uniform revand operating costs. The uniform series factor would give cumu_ :l.r"s lative revenue or operating cost at year 3, rather than year by year carculation of the desired escalated oottar costs and revenues. Therefore, the uniform series factors are not usefulfor escalation calculations. Escarated $ Npv

=

-20 -

4r Bpi:l::/",1)

+ 57.86 pilu,!!r",r1

.657s + 62.3(P/F1 s"/"5) = +27.49 > 0, accept the project

Escalated $ ROR = 42.5/oi this is the ,,i,,value that makes escalated dollar NpV = 0. 42.5% > i" = 157o, accept the project

260


B) Constant Dollar Analysis The escalated dollar costs and net revenues from part (A) are the basis for the constant dollar calculations.

t

Constant Dollar Vatues

.9434 .8900 .8396 Ct =42.8(P /F 6y",1) RZ=57.86(P/F Rg=G2.3(p IF 6y",2) A11^.g)

^ CO-20___:j9fB

=51.49 '

-

=S2.AO"'"'"'

0

constant dollar values are obtained by discounting escalated dollars at the rate of inflation of 6% per year. For constant dollar NpV analysis we must convert the 15% escalated dollar minimum RoR to the equivalent constant dollar minimum RoFl for the assumed 6% per year inflation rate. use Equation s-1 for this calculation as follows: 1l

i

= [(1 + i)/(1

+f)]- 1 = (1.1S/1.06)-

Constant $ NPV = -20

-

1

=0.C849or8.49yo

.9217 .8496 40.98(P/Fg.4go/o,1) + 51 .4g(plFg.49o/o,Z) .7831

+ 52.30(P/Fg.qgy.p) = +27.48 > 0, so accept Constant $ NPV =-20-4O.gB(1t1.08+S11 + 51.49(111.0A+S;2 + 52.30(1/1 .08+S;3 = +27.48 > 0, so accept

Note that the constant dollar NpV is identical to the escalated dollar NPV within round-off error calculation accuracy. constant dollar NPV equations are mathematically equivalent to escalated dollar NPV equations, so of course give the same results. The mathematical equivalency follows: Escalated $ Calculatiohs = Constant $ Calculations (Escalated $ Val ue) ( P/F;*, 6) = ( Escalated $ Value) ( p/F1, since P/F;.,n = (P/F1,n)(p/F;.,,n)

p)

(p/F;.,, n)

as discussed earlier in the development of Fquation 5-1. For this analysis in year Z:

Chapter 5: Escalated and Constant Dcilars

P /F

1

S"/",2 = 0.7561 = (p lF 6"7o,2)

261

p

/F

a.qg6,Z)

= (0.8900X0.8496) = 0.7561

constant $ RoR = 34.41a; this is the "i" var0e that makes constant cjoliar net present value equal to zero. g4.4"'o > i* 8.49o/o, aqgepi the prOject =

C) Today's Dollar Anatysis

Using today's doilars as the basis for evaruation carcurations .involves one of two

different assumptions. Eithei yo, that today's doilars equar escarated doriar vaiues "rrrre that o,, you assume today's dollars equar constant dofiar varues (see Ex. 5-1a and s-1b), Today's Dollars Equal Escatated Doltars Assuming that'today's coilars equar escarated doilars expricitry assumes that costs and revenues in the future will be the same ai tiiey wouid be ioday. This implicifly involves tne assumption that cc_cts and revenues wiil escaiate at oi/" per year over the evaruation rife. This is an escalated doiiar assumption simiiar to, but different fr-orn the pai't "A" assumptions, so an escaraterj doila: minimum RoR must i:e used in NPV carcurations and for RoR anarysis decisions. The

"today's doliar varues equar escarated doilar varues,, ,nrrvai, foilows: I'JPV

= -20

-

4O(,PlF1S"/",1) +

= +15.9 > 0 accept

'

SO(p/FlS,t",2)+ S0(p/F1 S"/"5)

ROR = 32.2% > i" = 157o, accept Note these escalated dollar NpV and RoR results are significanily different frorn the part "A" escalated dollar analysis results. There are an unlimited number of different ways that esiarateci costs and rev_ enues can be projected. Assuming today's doilars are equar to escaiated dollars is just one specific eslalation assumption. lt is to understand specific escaration assumptions used in important anaryses because they v,,ill affect evaluation results. A variation of the today's doilar equar escarated doilars assumption is to escalate ail capitar costs (such as acquisition and deveropment costs) at specified rates and to assume that the escalated dollar net revenues or profits in the income generating years wiil equar today,s dollar net revenues or profits. rnis is often cailed ,,the

washout

262


assumption" since it assumes that any escalation of operating costs each year will be offset (washed out) by the saqe dolfar escalition of revenue. Note there are no revenues in the development years, so costs must be escalated in pre-revenue years. The warhout assumption only applies to revenues and operating costs in the revenue generating years.

Today's Dollars Equal Constant Doilars since constant dollar values are obtained by discounting escalated dollars at the rate ol_ inflation, the only way today's dollars can equal constant dollars is if all today's dollar values escalate each yeai at the rate of inflation. Escalating each cost and revenue at the rate of inflation and then discounting the resulting escalated dollars at the rate of inflation to get constant dollars brings the dollar values back to the starting today's doilar values. This is constant dollar analysis assumption, so a constant dollar minimum RoFl (g.4g"/" for this analysis) must be used in the constant dollar Npv calculations and for the economic decision with constant dollar RoR results. The today,s dollars equal constant dollars Npv and RoR results follow:

i

NPV = -20

- 40(PlFg.ag"/",1)

+ S0(P/F8.49 o/",2) + 50(p/Fg.497.,3)

= +24.8.> 0, so accept ROR = g2.2%, i*'= 8.4g"/o,So accept

Note the today's dollars equal escalated dollars NpV result is significantly different from the today's dollars equal constant dollars NPV result. Although the RoR results are 32.2/o for both cases, different minimum rates of return are used for the economic decision with RoR in the two cases. These two today's dollar analysis cases are very differeht and can lead to different investment decisions. obviously, understanding the escalated dollar or constant dollar inflation and escalation assumptions being made is very important for correct economic decision making.


263

EXAMPLE 5-3 Escalated and Constant Dollar Cost Analysis of Service Producing Alternatives Compare tvuo alternatives that provide a service using escalated and constant dollar analysis. Consider alternati'.re "A" to be capital intensive, requiring the expenditure of $100,000 at time zero and no operating costs in years one or tr.ro to provi,Ce a -service for two ye.l!"s while alternative "8" is labor intensive, requiring end-of-yeai"one ard two escalated dollar operating costs of $60,000 and $72,000 respectively. Salvage is zero in both cases. Make present worth cost analysis for an escalated dollar minimum rate of return of 30%, then for i* = 20"h, then for i* = 107o. Assume inflation at 20% per year for all ccnstant dollar calculations. Verify the present rvorth cost results with incremental NPV analysis.

Solution, All Values in Thousands of Dollars: Escalated Dollar Diagranrs

(A)c--too 012

CC =

(B)

Soiution:

60

aC =72

-

Escalated Dollar PW Cost Analysis 1) for i* = 307o, remembering P/F1.,n = (1/1+i.)n PW4 = 'lQg PWB = 60(1/1 .3) + 72(111.3)2 = 88.75, Select "8"

2) tor i" = 20"h

Ptll4 =

1gg PWB = 60(1/1 .2) +72(111.2)2

=

1OO

Results indicate break-even economics.

3) for i* = 10"h PW4 = 100, select smaller Present Worth Cost, "A" PWB = 60(1/1 .1) + 72(111.1)2 = 114


Constant Dollar Diagrams for lnflation Rate, t =

(A) c

2}o/o

=.

012t OC

(B) -

= AOP|IZOW,|

=50

OC

= Z2(PlFZot =50

,Z')

Constant Dollar PW Cost Analysis: 1) for an escalated dollar minimum RO*,R, i* = 307o, the corresponding constant dollar minimum ROR, i*' is 8.93% PS/4 = 'lQQ PWg = 5O(1/1.0833) + 50(1/1.08eS12 = 88.75, Select Smatter, "B" 2) for f - zC,/o, constant dollar i*'is 0 PW4 = 100 Break-even PWg = 50(1/1.0) + 50(111.0)2 = 100 Results indicate break-even economics

3) for i* =

10o/o,

constant dollar i"'is -g.33%

PW4 = 100, Select Smaller, "A" PWg = 50(1/.9166) + 50(1/.9166)2 = 114 Note that not only do escalated and constant dollar analysis give the same conclusion, but those conclusions are based upon identical present worth costs. If we look at the incremental difference between "A" and "8", we can show that incremental NpV analysis gives the same result for escalated or constant dollar analysis.

lncremental Escalated Dollar Diagram

(A-B)

100 012

C=

Savings

60

72

Escalated Dollar lncremental NPV Analysis i*

=3}o/o, NPV4-B = 60(1/1 .3) + t2(1 11.g)2 -1.00 = -1'1 .2S < 0,

select "B"

Ji

Cnaprer 5: Escalated and Consiant Doiiars

265

i*-ZO%, NPV4-g = 60(1 11.2) + 72(111.2)2

- 100 = 0, break-even i* =1}o/o, NPV44 = 60(1 /1 .1) + Z2(1 11 .\2 - 00 = +1 4.0 > 0, 1

select "A"

lncremental Constant Dollar Diagram

A-B) c =

,00

Savings

50

50

Constant Dollar lncremental NpV Analysls For i* = 30% and t=20o/o, equivalent

i*'=

NPV44 = 50(1/1.0833) + 50(1/1.08SS;2 = -11.25, select "B', For i* = 2O/o and f=2Oo/o, equivalent i"'

NPV4-B = 50(1/1.0) + 50(1/1.0)2

-

=

8.387o

- 100 Oo/o

100 = 0, break-even

For i" = 10% and t=20"/o, equivalent i*' = -8.337o NPV44 = 50(11.9166) + 50(1/.9166)2 - 100 = 14.0, select "A"

Note the identical incremental Npv results for both escalated and constant doliar analysis, so of course the same economic decision resulis either way. Now that we have established that either escalated or constant dollar analysis properly handled gives the same economic analysis conclusions, are there reasons for preferring one method over the other? In general it takes two present worth calculations in constant dollar analysis to achieve the same result that can be obtained with one present worth calculation in escalated dollar analysis. r.;ewer calcrrlations means fewer chances for math elrors. a point in favor of escalated dollar analysis. To make proper after-tax anaiysis, tax calcuiations must always be made in escalated dollars as must borrorved money principal and interest payments. For constant dollar analysis. this requires careful diligence to avoid improper mixing of escalated and constant dollars. with escalated dollar analysis all values are in escalated dollars so this is not a problem, another point in favor of escalated dollar analysis. From a practical and ease of calcuiation viewpoint, there is little to be said for constant dollar analysis that cannot be said more favorably

266


for escalated dollar analysis. However, for those evaluation people who .want to make constant dollar analysis rather than escalated dollar analysis, .i the following steps should be fbllo*ed:

1) Determine the escalated dollar values for all project costs &rd revenues. 2) Convert all escalated dollar values to the corresponding constant dollar

3)

values for the assumed inflation rates each year. Convert the escalated dollar minimum ROR, "i*", to the corresponding constant dollar value, "i*"', if the minimurnROR is initially expressed in terms of escalated dollars.

4) Calculate constant dollar NfV using i*', or calculate constant dollar ROR, "i"', and compare to i* for the economic decision. One situation that can give constant dollar analysis a potential intangible advantage over escalated dollar analysis is in evaluation of a project to determine and negotiate the break-even selling price that a purchaser may be u'illing to pay for a product. Since in inflationary times a given constant dollar minimum rate of return is always less than the equivalent escalated dollar minimum rate of return. it may be easier to convince a buyer to accept paying the price needed for you to ger a l57o constant dollar ROR than a higher but equivalent escalated dollar ROR for a given rate of inflatiorr. This is a potential marketing or negotiation advantage rather than an economic analysis advantage. The following example shows that breakeven analysis economic calculations (such as break-even selling price) will be exactly the same with either rscalated dollar analysis or constant dollar analysis.

EXAMPLE 5-4 Break-even Selling Price Analysis in Escalated and Constant Dollars. The investment of $3C,000 today is estimated to produce 100 prod-

uct units each year for the next two years when the product is expected to become obsolete. Year 2 salvage value is expected to be zero. Today's dollar operating costs for years 1 and 2 are estimated to be $8,000 per year. lnflation is expected lo be 7.0"/" per year for each of the next two years. lf product selling price is projected to escalate 8o/" per year and operating costs escalate 10% per year, calculate the year 1 and 2 escalated dollar seliing price that will give the investor a desired 157o constant dollar ROR on invested dollars.


257

Solution, All Values in Dollars: Let X = Today's Dollar Selling price per Unit Year Escalated $ Escalated $

---

Year 2

Saies Xl199lt.1tA"/o,t) = 1OB.0X

oC

EscalatedSFrofit Constant $

1

Profit

X(100XF/ps

"/",2) = 116.6X

a99o(rie1oz,r) = 8,800 B0J0(F/P101;,2) = 9,6a0

108.0X-8800

(108.0X

116.6X_g,OS0

- S,800XP/F7%,1) (1i5.6X

-g,6g1)(ptF7%,2)

As shown, today's dollar revenues and operating costs are converted to escalated dollar values using the selling price and operating cost escalation rates of B% and 162 respecti-vely. lf you want to work in escalated dollars you use the resultant escalated dollar profrts shown, which are a function of the unknown today's dollar selling price per unit, X. lf you prefer to work in constant dollars you present rvcrth escalated dollar profit at the 7"k per year rate of inflation to convert escalated dollars to constant dollars. Escalated Dollar Present Worth Equation Catculations To write an escalated dollar present worth equation we mu*qt use escalated dollar values and handle the time value of mcney at the escalated dollar minimum RoR that is equivalent to the desired 15% cor,'stant doliar minimum RoR for assumed 7./" per year inflation. Usirg Equaticn 5-1:

i'= (1 +f)(1 +i"') -

1 = (1.07X1.1S)

-

1=.2305 or23.O5"h

.81268 30,000 = (108.0X

- 8,800XP/FZ3.OS,t)

.66045 + (t 1G.6X

-

9,68OXp/FZS.AS,Z)

Solving for X = g164.26lunits = Today,s Dollar Selling price 264.26(FlPgg/o,1) = $285.40/unit = Year 1 Escalated $ Selling price

264.26(F/P8"/o,Z) = $808.24lunit = year 2 Escalated $ Selling price

Constant Dollar Present Worth Equation Using Constant Dollar Profits Handle the time value of money using the 15% constant dollar minimum ROR:

Economic Evaluati6n and lnvestment Decision Methods

30,000 =

(1

08.0X

+

(1

1

-

6.6X

8,800XP lFT"/o,1)(P lF

-

6y",1

9,680XP 1F7"7",21(P lF 1 S"/o,Z)

solving for X = $264.26lunits = Today's Dollar Selling Price 264.26(FlPg%,1)= $285.40/unit = Year 1 Escalated g Selling Price 264.26(FlPg,/o,Z) = $308.24luoit = Year 2 Escalated g Selling Price

The same break-even selling prices result from either escalated or constant dollar analysis. To conclude the inflation and escalation discussion, a few words should be said about forecasting escalation rates for different commodities or segments of industry. The past often is a good indicator of the future, so analysis of past cost trends for a particular commodity or asset is one way to get an indication of what the future might hold. This approach was very poor in the 1973-1980 period due.to the significant increase in energy costs around the world, but no method of analysis is going to predict consistently the effects of that kind of upset to the world economy. The U.S. Bureau of Labor Statistics publishes 70 or more price indices for commodities and materials, 5 or more indices or wage rates and an index of engineering costs which can be obtained to give past trends. "Chemical Engineering" consolidates these

Bureau of Labor Statistics into the CE Plant Cost Index published on a bimonthly basis as a measure of the cost of typical chemical plants. "Chemical Engineering" also presents the Marshall and Swift equipment cost index bimonthly. No one has a crystal ball to forecast the future accurately, but the use of available indices can be helpful in determining cost and price trends and rates of change of these trends needed to forecast meaningful escalation rates for costs and revenues needed for investment analyses.

In summary, proper evaluation of investments in various escalated dollar or constant dollar analyses requires understanding how to apply the following three different kinds of rates: 1. Escalation Rates: used to convert today's dollar values to escalated dollar values. 2. Inflation Rates: used to convert escalated dollar values to constant dollar values. 3. Time Value of Money Rates: used to account for the time valug of money using "i" oa "i*i' in escalated dollar analyses, and "i"' oa "i*"' in constant dollar analyses.

Chapter 5: Escalated and Constant Dclars

These rates are

all usetl in the foilowing example illustrating general

escalated and constant dollar analysis calculations involving different escaiation rates a:,d infla: ioil ratcs each y:i:r. +

EXAMPLE 5-5 RoR and Npv Anatysis with changing Escatation and lnflation Rates Each year

A cost of $100,000 today is projected to generate today,s dollar incomes of $75,000 per year at the end of eaoh of years 1, e and 3 with today's dollar operating costs of $25,000 per year at years 1, 2 and 3. saivage value is zero at year 3. lncomes and operating costs are projected to escalate 10% in year 1 , 1z/o in year 2, and 1s"1" in year 3, so net income minus operating cost escalates at the same given rate each year. calculate the prolect escalated dollar RoR and NPV assuming the minimum RoR for each of the 3 years is 15% in escalated doilars. Then assume inflation rates will be 10% in year 1, 8'A in year 2 and 6% in year 3, and calculate constant dollar RoR and NPV. Solution: Today's Doliar Values (ln Thousands of Doltars)

C=100

Netll=gQ

Net

12

=

$Q

Net

13

= $Q

Escalated Dollar Values

C=100 Netll=gg 01

Net

12

= 61.6

Net

13

= 70.84

where, 1 .100 Net l1 = 50(F/Pt O"/"J) = 55.0

1

Net

12

Net

13

.100

= 50(F/Pt

1

.120

g"/",iF|P12./",i

.100

.120

= 61.60

1 1 1 .150 = 50(F/Pt g"7"1)f /P12.7"1)fflP11"/b,i =70.84

270

Ecbnomic Evaluation and lnvestment Decision Methods

Escalated Dollar Present Worth Equation for ROR Analysis 100 = 55(P/Fi,1)+ 61.6(P/F1,2) + 70.84(P/F;,3) i = Escalated Dollar ROR = 37.4o/" by trial and error

i

.

37.4% > 15"/o, so satisfactory

Escalated Dollar Net Present Value

.8696

.7561

.6575 NPV = -100 + 55(P/F1 S/o1) + 61 .6(P/F 15"/o,2) + 7A.84(PlFt SZ,g) = +$41.0 > O, so satisfactory

Constant Dollar Vatues C = 100 Net 11 =

50

Net

12

= 51

85

Net

13

= 56.25

where escalated dollars discounted at the annual inflation rates give the following constant dollar net incomes: .9091

Net l1 = 55(P/Ft O"/"J) = 50.0 .9259 Net l2 = 61 .6(P/F 10"/o,1)(P/Fg/.,1) = 51.85

.9091

.9091

Net

13

.9259 .9434 70.84(PlFlOo/",1)(PlF67",1l!lF6"y",1) = = 56.2S

Constant Dollar Present Worth Equation for ROR Analysis 100 = 50(P/F;,,1)+ 51 .85(P/Fy,2) + 56.25(P/F;,,3)

i'= Constant Dcllar ROR = 26.3%

> i*'values shown in next section, so satisfactory.

Constant Dollar Net Present Value Analysis For constant dollar NPV analysis we must calculate the constant dollar minimum ROR each year that is equivalent to the 15% escalated dollar minimum FiOR for the assumed inflation rates of 1O"h in year 1, 87o in year 2 and 6% in year 3 using a.rearranged version of Equation 5-1.


271

i" ={(1 +i.)/(1 +f)}-1 Year 1, i*' = (1 .15/1.10) - 1 = 4.54o/o Year 2, i*' = (1 .15/1 .08) - 1 = 6.48o/o 1, 'r€3r 3, i*'= (1.15/1.06) - 1 = 8.4g,/o .9566 .9s66 .9391 NPV = -100 + 50(P/F4.54 + 51 .BS(P/F4 .S4o/o,1)p/F6.agh,i "/o,1)

+ 56.25

.9566 .9391 4"/", 1) (P 6. a6"7", i (P

(P / F 4.S

/F

.9217 /F

g.qg"t,i

= +$41.0 > 0, so satisfactory

Note the equivalence of constant dollar NpV and escalated dollar NPV results. Even with different escalation and inflation rates each year, correct analysis gives the same escalated and constant dollar NPV results, so of course the same economic conclusions from eithei' analysis. similarly, with RoR analysis results in either escalated or constant dollars, as long as you compare project RoR to the minimum RoR expressed in the same kind of dollais (escalated or constant), you get the same economic conclusions from either escalated or constant dollar ROR analy'sis. 5.2 Exchange Rate Effects on Escalation and cash Florv Analysis

lr'henever project costs and revenues involve more than one currency, e

\ciran-qe rates must be projected fbr each evaltration period to perrnit analysis

of the project in terms of one currency. Economic analysis reiults tend to be

very sensitive to exchange rate projections. In general, exchange rate changes often reflect current changes (or perceived tuture changes) in relative inflation rates between countries. However, this is not always the case. Differences in country interest rates and balance of pal,nrent deficits can be major factors that affect exchanse rares. In 1995 the U.S. clollar declined l57o to z\vo against btrth the Jnp:rnese yen anrJ the Cennun mark alrhough U.S. inflation rvas relatively lorv at 3Lh per year or less. usnally however, devaluation of a country cuffency a!,ainst the U.S. dollar, Japenese yen, or European currencies is caused by current or projected future inflation rate differences between the countries. Therelbre, exchange rate projections often implicitly account for future inllation effects on escalated dollar analysis. The following example illustrates the mechanics of handling exchange rates in cash flow analysis as rvell as the sensitivity of evaluation results to exchange rate effects.

272


EXAMPLE 5-6 Exchange Rate Analysis Variations

A company is considering investing,$1,000 in Country'12" whose time zero (now) currency exchange rate is lOOZ Units per$1.00 U.S. The following time diagram. giveis the relevant cost, ptoduction, selling price and exchange rate data, with valdes in today's U.S. dollars. Production = 10,000 units in years one and two Selling Price = $0.06/unit in years one and two Operating Costs = $0.01/unit in years one and two Cost

=

$1,000

$600 Flevenue $600 Revenue $100 Op. Costs $100 Op. Costs Salvags = $400

2

Analyze the following evaluation cases: Case

A)

Evaluate the project ROR and NPV @ i* = 10% using a U.S. dollar analysis assuming zero percent escalation olall values per year.

CaSeB) Evaluate the project ROR and NPV @ i* = 10% using

Country Z currency analysis assuming zero percent escalation of all values per year. Case C) Change the Case B analysis to reflect projected 50% per year devaluation of Country Z currency against the U.S. dollar. Assume all project costs, revenues, and salvage values are incurred or realized in U.S. dollars, but do the analysis in terms of Z units. (This could be a Country Z mining or petroleum project where product is sold internationally in U.S. dollars.) Case D) Re-work Case C assuming sales revenue is realized by selling product (such as electric power) in country Z at the uniform selling price of 6.0 Z units {(1002/$1.00) x ($0.00/Unit)) per product unit each year. ln other words, product is being sold to the country Z general population and the economy of the country will not permit passing inflation effects through currency devaluation on to the consumer. For simplicity, also assume salvage value is unaffected by exchange rate changes. However, operating costs are assumed to be affected by exchahge rate changes.

Chapter 5: Escalated and Constant Doilars

case

E)

Re-anaryze case B using u.s. doilar financiar borrowed assuming $g00 U.S. (g,OO0 2 units) are borrowed at year 0 at 107. annuar interesi to be repaid with year 1 Z mortgage payments of $200 U.S. ban principar prus accrued interest prus a barioon (rump sum) roan principar payment of $400 ar the end of year 2tl pay off the loan when the project is sold.. Sales in 'Corntr, Z currency units are converted tc U.S. dclars to pay oft tf,e toan. Re-anaryze,case D using the reveraged borrowed money conditions described in Case E.

, money leverage

*l

case

F)

Solutions: Case

A) U.S. Doltars Analysis

$600 revenue/yr 19,990 uniisiyr ($0.06iunit) -10,000 unitsiyr ($0.0ilunit) = _$.t00 operatlng cosVyr Before-Tax Cash Flow/yr $500 + = $400 Salvage atyear

=

-$1,000

$500

2

$500+$400 Satv.

ROR = 23.11"/" NPV @ 10"/" = $198.05 U.S.

Case

B) Country Z Currency Analysis

-100,0002

50,0002

ROR = 23.11"/" NPV@ 10%- lg,Bg5ZUnits

Case

C)

-Country Z Currency Analysis With Exchange Rate Changes

Year 0 100 Z Units = $1.0 U.S. Year 1 150 Z Units = $1.0 u.s. Year 2 Z2S Z Units $1.0 U.S. = Z Currency Unit Cash Flow:

(100 Z Units $1.0 U.S.) = 50,0002 + 40,0002 Salv.

274


012

Year

Revenue Operating Costs Capital Costs

-

Cash Flow

.

-100,000 -100,000

90,000* +75,00O

225,0a0

+202,500

$600 U.S.(1502 Units / $1.00 U.S.) = 90,0002 Units **-$100 U.S.(1502 Units / $1.00 U.S.) = -15,0002 Units ROR = 84.667" NPV @ 1Oo/o = 135,537 Z Units

The devaluation effects have worked for the investor and given better economic results for the assumption that revenue escalates proportional to devaluation. This would relate to an export project such as mining or oil and gas production project where product is sold internationally in U.S. dollar prices. Often it is very difficult or impossible to pass on to dornestic consumers price escalation due to currency devaluation effects. The domestic consumer may not have the

financial means to pay higher prices. Case D shows that this assumption gives very different economic results. Case

D) Z Gurrency Unit Cash Flow Analysis as in Case

C, No

Revenue Escalation Year

0

Revenue - Operating Costs - Capital

-100,000

-

-100,000

Costs

Cash Flow

12 60,000

-ru,ooo 45,000

100,000

-22,500 77,500

ROR = 13.36% NPV @ look =+4,958 Z Units

These results are economically less desirable than the "no devaluation" Case B results, because devaluation is assumed to negatively affect operating costs but to have no off-setting positive effect on revenue. When leveraged money is involved, currency devaluation can have much greater effects as the following two cases show.These cases relate to borrowed money analysis considerations introduced


in chapter 11, but applied here to illustrate the significant impact of exchange rate changes on leveraged evaluations. Refer to chapter 1 1 , Example 11-1 if the handling of borrowed money (leverage) in ',lris analysis is not clear.

E)

Case

Z Currency Analysis With Borrowed U.S. Doilars, No Exchange Rate Changes

012

Y'ear

Revenue - Operating Costs - lnterest - Loan Principal - Balloon Principal + Borrowed Dollars - Capitai Costs Le,reraged CF

60,000 -10,000 -9,000

-i33;333

:

-20,000 +22,000

100,000 _10,000

-6,000

-4oooo +24,000

Le'reraged BOR = 77.58% Leveraged NPV @ 10o/" = 1g,Bg5 Z Units

This leveraged RoR result is much betier than the 29.11% RoR tor cash investment case B. However, note the NpV for case B is ti:re same as for case E because the cost of borrowed money which is the 10% interest rate is the same as the 10% opportunity cost of capital minimum discount rate. Case

F)

Z Currency Analysis With Borrowed U.S. Dollars, Exchange Rate Changes Affect Operating Costs, Loan Principal and lnterest, Not Revenue

'/ear

0

Revenue - Operating Costs - lnterest - Loan Principal + Borrowed Dollars +80,000 - Capital -100,000

12 60,000 -15,000 -12,000 -u.:..

100,000

-22,500 -13,500 -135,000

Costs

Leveraged CF

-20,000

+3,000

-71,000


276

Since the cumulative negative cash of 91,000 exceeds the 3,000 positive cash flow, it is evident that the leveraged FIOR for this Case is negative. Leveraged ROR = negative infinity Leveraged NPV @ 1Q"/o = -75,950 Z Units The effects of devaluation of currency can be devastating economically on a leveraged project using a U.S. dollar loan (or another hard currency loan) to be repaid with the devalued currency. Unless revenue in terms of the devalued currency escalates at a rate proportional to the currency devaluation rate, the economics of leveraged investments deteriorate. 5.3 Summary Escalated values are also defined as actual, current, then current or nominal dollars. They are always inclusive of the effects of inflation and other parameters including technological, environmental, market and related issues. Constant values are escalated values that have had the effects of inflation discounted from them to a base period in time which typically is time zero, but it could be any point. Constant doliars are also referred to as real or deflated dollars. The only difference between escalated and constant values is the inflation rate each year related to the host currency. Previous variables introduced in Chapters 2 and 3 are now defined more explicitly to reflect the proper handling of inflation, they include: e = escalation rate

f = inflation rate

i* = escalated $ discount rate i*' = constant $ discount rate i = escalated $ rate of return i'= constant $ rate of return Escalated dollar and constant dollar project rates of return and minimum rates of return can be explicitly related for any uniform annual inflation rate using Equation 5-1, which was developed in Example 5-1. Eq 5-1:

lai

Rearranged'

=

(l+l)(1+i')

i'=

{(1+i)/(1+0}

-

1

A commonly used approximation to Equation 5-l is the following:

7'=i-f

Chapaer 5: Escalateci and Constant Doilars

277

PROBLEMS

5-1

Consider the following ,,today,s dollar,, cash flows

C=I00

C=200

For cases A, B, rate of return is

A)

c

Rer,-600 OC=100

and D below, assume the escalated dollar minimum

LS.OVo and:

câlculate the project escalated dolrar Npv if revenue escarates at -10.lVo per year, and all costs escalate at 6.TLoper year.

B) using

the escarated dolrar results from case A, calculate the project constant dolrar Npv if inflation is 5.\vo per year and escalation of costs and revenues is the same as i, caie A. For evaruation con-

:rrslency' adjust the escalated dolrar minimum discount rate of 15,,vo using text Equation 5-r to calculate the equivaleri.onrtunt dollar minimum discount rate for use in this constant

analysis.

c)

D)

xpv

Calcuiate the projecr escarated do,ar Npv assuming today,s dolrar values equal escalated dollar values. State the explicit cost and

enue escalation assumption built into this analysis.

I

aotta,

rev_

Calculate the project constant dolrar Npv assuming today,s clolrar values equal constant dolrar values. State the explicit cosiand

enue escalation assumption built into this analysis.

reu-

5-2 In I988,

the U.S...gross domesric producr (GDp) increased to $4.90 tril_ lion at year end. from the l9g7 yiar end level of $4.54 trillion in acrual

escalated dolrar varues.-In

the same year, the consumer price index rose approximatery 4vo. what was the escalated lcurrentj dolrar percent increase in GDp? what was the constant (real) dolrar p".""niincrease in GDP?


5-3

An investment related to developing a new product is estimated to have the following costs and revenues in "today's" or "time zero" dollars.

I=$200,000 Co=$5o,ooo

o

I=$200,000

C1=$150,000 OC=$100,000 OC=$100,000

1

2............5

A)

Evaluate the project escalated dollar RoR if both capital cosrs and operating costs are estimated to escalate at l|vo per year from time zero with income escalating at l\Vo per year.

B)

Make constant dollar RoR Analysis of case 'A" assuming the rate of inflation for the next 5 years will be t\Vo per year.

c)

use escalated dollar RoR Analysis to analyze the investment assuming a washout of escalation of income and operating costs with a l\Vo escalation of capital costs in year one.

5-4 A product that sells today for $100 per unit is expected to escalate in price by 6vo in year one, 8vo in year two and,lovoln y"a. three. calcuiate the escalated dollar year three product selling price. If inflation is expected to be 5vo in year one, gvo in year two and 12vo in year three, determine the year three constant dollar product selling price. 5-5

An investor has an opportunity to buy a parcel of land for $100,000. He plans to sell it in two years. what will the sale price have to be for the investor to get a 25vo constant dollar before-tax RoR with inflation averaging 1\vo annually? what escalated dollar annual rate of increase in land value will give the needed sale price?

5-6 Determine the break-even escarated dolrar selling price per unit required in each of years one and two to achieve a 15% constant dollar project RoR, assumtng a 12vo per year inflation rate. AI clor;"r varues are today's dollar values.

C=$10081

Sales=$1116661 Sales=$X(1000)

___99=$50,000

OC=$50,000

Selling price escalation is l\va per year from time zero when selling price is $X per unit. operating cost (oc) escalation is r5vo per lear from time 0. 1,000 units are to be produced artd sold each year.

It


279

5-7 what can be paid now (today)

ro acquire a property that will be developed 2 years from now and which engineeri .rii-ut" wilr have today,s dollar costs and revenues shown on th. forlowing time diagram. All values are in thousands of today,s dollars. C

C=$200

0inow)

Rer,=5159

Rev=$ 1 59

Rev=$ I 50

OC=$50

OC=$50

OC=$50

2

Starting from now, it is projected that inflation will beTvo per yea*. The escalated dollar minimum RoR is l5vo. Evaluate the acquisiiion cost that can be paid now to acquire this property for the foilowin! five cases:

case 1. Make the analysis using today's dolrar cc-rsts and revenues assuming they are a reasonable projection of escalated dollar capital costs, operating costs and revenues to be incurred (this assumption effectively assumes zero percent escalation of all costs and revenues each year or lhat escalation of all capital costs and operating costs will be offset by escalation of rev_ enues so that escalation of costs and revenues will have zero eft'ect on economic analysis results).

Case

2.

Make the anarysis using the today's dollar costs and revenues assuming they represent consrant dollar values. (This assump_

tion is valid if you assume that all capitai costs, operating costs and revenues will escalate at the same rate of inflation each year. Discounting these escalated dollar values at the same rate of inflation to get constant dollar values gives the original today's dollar values for this assumption.) case

3. use escalared

doilar anarysis assuming capital cost (c) escaration will be rzvo per year, operating cost (oc) escararion wi, be 10Vo per year and revenue (Rev) escalation

Case

will be lOTcper

year.

4. Make constant dolar analysis for trre Case 3 escaiation assumption assuming 7Vo inflation as given.

case

5.

use the escarated dolrar analysis assuming capital costs escalate at l2%o per year and escalation oioperating costs is exactly offset by a like-dollar escalation of reveiues each year which gives uniform profit margins each year. This com_ monly is called.,the washout assumption,,.

Economic Evaluaticin and lnvestment Decision Methods

5-8

An investor has paid $100,000 for a machine that is estimated to produce 5000 product units pe1 year for each of the next three years when the machine is estimated to be obsolete with a zero salvage value. The product price is the 'unknown' to be calculated; so it is estimated to be $x per unit in year one escalated dollars and to increase l}Vo per year in year two and 67o inyear three. Total operating costs are estimated to be $8000 in year one escalated dollars and to increase l5%o in year two andS%o rn year three. The annual inflation rate is estimated tobe77o. What must be the year one, two and three escalated dollar product selling price if the investor is to receive a L2Vo annually compounded constant dollar ROR on invested dollars?

5-9

Reclamation costs on a project are expected to be incurred over a 30 year period from 27 to 56 years in the future from now. Reclamation. costs are estimated to escalate 5.07o per year in the future. Using a 5.l%o anntal discount rate as representative of annual reclamation Cost escalation, the year 0 piesent worth of the 27 to 56 year future reclamation costs is estimated to be $1.0 million. It is assumed that money today and in the future over the 56 year life of this analysis can be invested in U.S. Treasury Bonds paying 9.lVo interest per year. Determine the magnitude of year 0 investment that an investor needs to make in U.S. Treasury Bonds paying 9.07o anuJal interest to cover the year 27 through 56 future reclamation costs.

5-10 The following time diagram before-tax cash flows are today's dollar values. The investor has a constant dollar minimum rate of return of 1.0.0Vo and annual inflation is forecasted to be 3.0Vo over the project life beginning in year 1. All values are in millions.

-r00

-200

r50

150

150

150

A) Assuming the rate of escalation for all cash flows is equal to the inflation rate, calculate the escalated dollar project NPV and ROR. B) Based on your calculations in part A, determine the constant dollar equivalent cash flows and resulting constant dollar NPV and ROR.

Cirapter 5: Escalated and Constant Dollars

c)

Neglecting A and B, if the rate of escalation was forecasted to be ,Vo per yeir over the project life, calculate the conespoJi"i lated dollar NpV and RbR. tnRution "r"u_ is still forecasr to be 3.0Va

per year.

D) Again

neglecting A anij B, if the rate of escaration was forecasred to be }vo per year over the project life, bur inflation is stiil forecast at 3',vo per year, carculate the corresponding

and ROR,

:

I

I I

t

)

I

I ,

t

constant doilar

Npv

CHAPTER 6

UNCERTAINTY AND RISK ANALYSIS

6.1 Introduction In this age of advancing technology, successful managers must make informed investment decisions that determine the future success of their companies by drawing systematically on the specialized knowledge, accumulated information, experience and skills of many people. In evaluating projects and making choices between investment alternatives, every manager is painfully aware that he cannot and will not always be right. Management pressure is increased by the knowledge that a company's future depends on the ability to choose with a high degree of consistency those investment and market opportunities that have a high probability of success even though the characteristics of future events are seldom precisely known. In the previous chapters in the text investment analyses all were considered to be made under "no-risk" conditions. That is, the probability of success was considered to be 1.0 for each investment evaluated. This means that by expressing risk and uncertainty quantitatively in terms of numerical probabilities or likelihood of occurrence, where probabilities are decimal fractions in the range of zero to 1.0, we implicitly considered that the probability of achieving projected profits or savings was 1.0 for investment situations evaluated. We are all aware that due to risk and uncertainty from innumerable sources, the probability of success for many investments is significantly different than 1.0. When faced with decision choices under uncertain conditions, a manager can use informal analysis of the risk and uncertainty associated with the investment or he can analyze the elements of risk and uncertainty in a quantitative manner. Informal analysis relies on the decision makers experience, intuition, judgement, hunches and luck to determine whether or not a particular investment should be made. The quan282

Chapter 6: Uncertainty and Risk Analysis

.

283

titative analysis approach is based on analyzing the effects that risk and uncertainty can haveon an investment situation by,using a logical and con_ sistent decision strategy that incorporates the effects or iist< and uncertainty

into the analysis results. use of the quantitative analysis approach should aot be considered to imply that the iriformal analyiis considerations of experience, intuition and judgement are not needed. on the contrary, the purpose of quantitative analysis of risk and uncertainty is to provide the decision maker with as much quantitative information as possible concerning the risks and uncertainties associated with a particular investment situation, so that the decision maker has the best possible information on which to apply experience, intuition and good judgment in reaching the final deci_ sion. The objective of investment decision making from an Jconomic view_ point under conditions of uncertainty is to invest available capital where we have the highest probability of generating the maximum possible future profit. The use of quantitative approaches to incorporate risk and uncertainty into analysis results may help us be more successful in achieving this

objective over the long run. No matter how comprehensive or sophisticated an investment evaruation may be, uncertainty still remains a factor in the evaluation. F,ven though rate of return or some other economic evaluation criterion may be calculated for a project with several significant figures of accuracy usinj the best available cost and income data, the decision maker may still feei uneasy about the economic decision indicated because he or she knows the assumptions on which the calculations are based are uncertain. If the economic evaluation method used does not reflect this uncertainty then every assumption built into an economic analysis is a "best guess', and the final economic result is a consolidation of these values. Making decisions on the basis of such ,,best guess" calculations alone can be hazardous. Consider a manager who may select investment alternative 'A" with a 20vo RoR over inv-estment ..B,,

which has a 157o RoR based on the "best guess" RoR calculation approach. would this decision be justifiable if the probability of success of alternative 'A" was 50va (or one chance in two) compared with probabila ity of success of alternative "B" of 90va? It is evident that the manager needs some measure of the "risk" involved in each alternative in addition to

the "best guess" or most likely rate of return results.

Tlrcre ore seve ral different approaches that can be used to quantitatively ittcorporate risk and uncertainty into analyses. These inctuie sensitivity analysis or probabilis_tic sensitivity analysis to account for uncertainty associated with possible variation in project paramercr;, and expected

284


value or expected net present value or rate of return analysis to account for risk associated with finite prubabiility of failure., The,use of, sensitivity anatysis is advocated for most economic analyses and the use of expected value analysis is advisable if finite probability of projectfailure exists. Sensitivity

analysis is described in the first half of this chapter and expected value analysis is described in the second half. Sensitivity analysis is a means of evaluating the fficts of uncertairul,^ on investment by determining how investment profitability varies as the parameters are varied that affect economic evaluation results. Sensitivity analysis is a means of identifying those critical variables that, if changed, could considerably affect the profrtability measure. In carrying out a sensitivity analysis, individual variables are changed and the effect of such a change on the expected rate of return (or some other decision method) is computed. Once all of the strategic variables have been identified, they can be given special attention by the decision maker. Some of the typical investment parameters that often are allowed to vary for sensitivity analysis include initial investment, selling price, operating cost, project life, and salvage value. If probabilities of occurrence are associated with the various levels of each investment parameter, sensitivity analysis becomes probabilistic sensitivity analysis. It may now be evident to the reader that the term "uncertainty" as used in

this text refers to possible variation in parameters that affect investment evaluation. "Risk" refers to the evaluation of an investment using a known ntechanism that incorporates the probabilities cf occurrence for success and failure and/or of dffirent values of each investment parameter. Botll uncertainty and risk influence almost all types of investment decision, but especially investment involving research and development for any industry and exploration for minerals and oil or gas.

6.2 Sensitivity Analysis to Analyze Effects of Uncertainty

As described in the previous section, sensitivity analysis refers to analysis

of how investment profitability is affected by variation in the parameters that affect overall profitability. For a case where rate of return is the economic criterion used to measure profitability, sensitivity analysis involves evaluation of how rate of return varies with parameters such as initial investment, profit per year, project life, and salvage value. It is frequently


2As

used to determine how much change in a variable would be necessary to reverse the decision based on average-value or best-guess estimates. It usually does not taiie into consideration the likelihood of variation. The rate of

change in the total outcome relative to the rate of change in the variahle being considered u'ill indicate the significance of this variable in the overall evaluation.

Example 6-1 will introduce a single variable sensitivity analvsis. tri is important for the reader to keep in mind that in this analysis, no dor.r'nstream effects are considered relevant to the evaluation. In other words, each parameter is independent of the other so changing the magnitude of the capital investment will have no impact on any other operations or the magnitude of project operating costs, etc. To further illustrate, when the years of income are reduced, there is no adjustment in the residual value of the assets. Such numbers may increase or decrease but again, are neglected here to simplify the introduction of this evaluation procedure.

EXAMPLE 6-1 Single Variable Sensitivity Anatysis Annual profits of $67,000 are shown on the time diagram for ihis

$240,000 investment case with an expected salvage value of $70,000 after five years. Evaluate the sensitivity of project RoR to plus or minus 2o'h and 4oo/" varialions in initial investment, annual profit, project life and salvage value.

Profits

C=$2a0,000 $67,000 $67,00Q $67,000 $67,000

$O7,OOO

L=$70,000

Solution: Using the most expected cost and revenue parameters gives:

PW Eq: 240,000 = 67,000(P/A;,5) + 70,000(p/F;,5) The "most expected" project ROR is 18%. How will this ,,most expected" 18% ROR vary as parameters are changed?

286


A) lnitial lnvestment Sensitivity Analysis lnitial 'Change in Percent Change ih 18.0% investment Prediction ROR ROR Prediction 144,000 192,000 240,000 288,000 336,000

-40 -20 0

+20 +40

42.0 27.5 18.0 11.2 5.8

133.3 52.9 0

-37.9 -67.7

The percent variations in the ROR from changes in initial investment costs are very significant. ln general, changes in parameters close to time zero (such as initial investment and annual profit) have a much more significant effect on investment ROR than changes in parameters many Vears in the future from time Zero (such as salvage value). B) Project Life Sensitivity Analysis Project Life

Change in Prediction

3

-40 -20

4

ROR 5.6

5

0

13.4 18.0

6

+20 +40

20.9 22.9

7

Percent Change in 18.0% ROR Prediction

-68.8 -25.5 0 16.3 27.1

Note that this sensitivity analysis really involves changes in total cash flow as well as project life. lf project life was longer (say 10 years or more), changes in life would have a less sensitive effect on ROR. C) Annual Profit Sensitivity Analysis Annual Profit

Change in Prediction

40,200 53,600 67,000 80,400 93,900

-40 -20 0 +20 +40

ROR

3.6 11.0 18.0 24.8 31.5


-80.2 -39.0 0 37.9 74.8

Chaorer 6: Unce(ainty and Risk Analysis

287

Percent variations in ROR due to changes in annual profit are very significant beciuse the changes start occurring close to time zero. lndividual parameters such as selling price. production rates and ope;ating costs affect profit.

D) Salvage Value Sensitivity Anaiysis Salvage Value

42,004 56,000 70,000 84,000 98,000

Change in

Prediction -40 -20 0

+20 +40

ROR


15.9 16.9 18.0 19.0

20.0

-11.9

-

6.0 0 5.4 10.8

Sensitivity analysis shows that accuracy of salvage value is the Ieast important of all the parameters that go into this ROR analysis because salvage value occurs far in the future from ti,-r:e zero. Also, in this case, cumulative salvage doliar value is small compared to cumulative profit. ln some evaiuatjons this is not the case and sali,age value has a much more sensitive effect.

6.3 The Range Approach to Sensitivity Analysis The range approach involves estimating the most optimistic and most pessimistic vzrlues (or the best and worst) tor each thctor in addition to estimating most expected values. This approach will make investmcnt decision ruraking easier fbr the cases where (1) a project appeats desirable even when pessimistic values are used and therefore obviously should be adopted from an economic viewpoint, (2) when a project appears to be undesirable even when optimistic values are used and therefore rejection is dictated on economic grounds. When a project looks good with optirnistic values but bad with pessimistic values further study of the project and the risk and uncertainty surounding the project should be made. Application of this method is shown in Example 6-2.

288


EXAMPLE 6-2 Range Approach Sensitivity Analysis Use the range approach to evaluate the investment described in Example 6-1 for best and worst case sensitivity analysis using the plrrs or minus 2Oo/" parameter variations with a five year life and a minimum rate of return of 15"h.

Solution: Best Case

Expected Case

Worst Case

240,000 67,000 70,000

288,000 53,600 56,000

5

5 3.7

lnvestment 192,000 Annual Profit 80,400 Salvage 84,000 Project Life in yrs. 5 ROR, % 36.4

18.0

The results indicate that the project is satisfactory for the best and most expected conditions but unsatisfactory for the worst conditions. More information is needed on the expected probability of occurrence of the worst case conditions to reach a valid and meaningful decision. The best, worst and most expe-cted sensitivity analysis results give very useful information that bracket the range of project ROR results that can reasonably be expected. This is the type of information that managers need to reach investment decisions. It is very important to recognize that although a project ROR greater than the minimum rate of return is predicted for the most expected parameters, the result is based on parameters which are subject to variation. This variation should be analyzed over the full range of possible results utilizing the best engineering and management judgments of people involved with a project.

6.4 Probabilistic Sensitivity Analysis The application of probability distributions to relate sales volume and prices, operating costs and other parameters to probability of occurrence

Chaoter 6: Uncertainty and Bisk Analysis

289

permits "probabilistic analysis" by the Monte Carlo simulation technique. A brief description of the method follows, then the method will be illustrated.

A weakness of traditional tecirniques for the evaluation of projects is an inabilitv to combine ilfcrmation fron-, a number of sources into a straightforivard and reliable profitability indicator. The major factor in this problern is the large number of variables that must be consi.dered. Ai: additic,nal facior is that it is not possible to obLain accurate single-valued estirnates of n:anr oi the r,ariable:. Probability theory is the study of the uncertainty of events. A basic tool of probability theory is the use of a range of values to describe variables that cannot adequately be quantified by single value estimates. For example, the determination of the least, greatest and most likely values of a variable will more accurately quantify the variable than will the average value. The distribution and relative possibilities of values assigned to a given variable will remain characteristic of that varjable if factors affecting the variable remain constant. Figure 6-l illustratejs three possible distributions of values. The values are plotted on the horizontal axis and the respective probabilities of their occurrence on the vertical axis. Analysis of the three examples indicates that the uncertainty of the parameter described in Example 3 is much greater than that in Example 2. The uncertainty of Example 3 is indicated by the w'ide range of vllues. A m:{ority of the parameters in ieal evaluations otien give an intermeciiate ranse of parameter values as illustrated by Example l. Ideally we would like to have a very smirll range tor all piuameters as illustrated by Example 2. In practice we get a combination of small, intermediate and large ranges of parameter value variation for different parameters in actual evaluation situations.

The drstributions in Figure 6-1 all have their values symmetrically distributed around the most likely value (the familiar bell-shaped curve is an example of this distribution). A distribution of this type is called the normal distribuiion. If the most likely value is shifted to either side of the center of the distribution, then it is refered to as a skewed distribution. The nonnal distributiotr and the skewed distribution are botlt special types of a general cLass of cttrves known as densitl, functions. In rhe remainder of this discr,tssion normal distributions will be referred to as such and the terru density function will be used to describe probability distributions that are not normal. The shape of the distributions will be determined by the nature of the variables they describe.


290

trl

uJ

o zu

O

zul

(f cE f

- MEAN

,r'

tr (r :)

o o o IL o

() u-

tJ

F

co

co

)d]

=

o E

o cc

(L

INTERMEDIATE RANGE EXAMPLE 1

tl UL

o-

LL

SMALL RANGE EXAMPLE 2

uJ

o

zul 0c

c(

f

o o IL o F

-)

6

c0

o (r (L

PARAMETER VALUES, LARGE RANGE EXAMPLE 3

Figure 6-1 Frequency Distribution Graphs lllustrated (LL = Lower Limits; UL = Upper Limits) In principle, a relative frequency distribution graph is converted to the

equivilent cumulative frequency distribution graph by moving from the left .nO of the distribution to the right end and computing the total area that is less than or equal to corresponding values of the parameter within the range evaluated. The cumulative area to the left of a given parameter value divided by the total area under the curve is the cumulative probability that a random parameter value ivill be less than or equal to the given parameter value. Figure 6-2 illustrates typical cumulative frequency distribution graphs that would result from converting the relative frequency graphs shown in Figure 6-1 to the corresponding plots. Note that Example 3 with a large range of parameter values has a much flatter cumulative probability curve than Example 2 which has a small range of parameter values. Example 2

Chapter 6: Uncertainty and F{isk Analysis

has a much higher percentage of the parameter values in a given range of curnulative probabitity of occurrence compared to Example 3.1.0

1.0

no

i ,!

0.9

,_a

0.8

0.7

0.7

0.6

0.6

0.5

0.5

,:1

:C

o-! oi >x

=* E

f O

Pa:a;'le'er Values lnt€rinei;ate fiange Example

o.4

0.4

0.3

0.3

4.2

0.2

0.

0.1

1

0.0

i-:

1

Para;rete:- ', a jues tJL Small Range Example 2

0.0

'1.0

0.9

io' 4.7

=

.c.6

oI .>;

0.5

6LJ

E= tr

0.4

= O

0.3 0.2 0.

LL

Values

parameter Large Range Example 3

1

0.0 UL

Figure 6-2 Cumulative Frequency Distribution (LL = Lower Limit, UL = gppsr Limit) Now to describe the probabilistic sensitivity analysis approach, consider that we are about to evaluate a new development project. we migtrt be interested in the effect on our economic evaluation criterion of changes in prolect

parameters such as initial investment, product selling price per unit, priaucdon rate or

292


sales projections, operating costs, project life and salvage value. For each of these variables and/or other variables to be investigated, a frequeney distribu-

tion plot of probability of occurrence versus para=meter vatue similar ,o tr," 6-l is prepared by the person or persons most familiar with and capable of projecting future values of the parameter involved. These frequency distribution data are then converted to cumulative probability of occurrence versus parameter value graphs similar to the examples in Figure 6-2. when these graphs are available for each parameter that is considered to vary, the use of Monte carlo simulation is applied. This generally involves curve-fitting a mathematical expression to describe each cumulative probability of occrurence versus parameter value so that picking a random number between 0.0 and 1.0 analogous to the cumulative probability of occurrence will automatically fix the parameter at the corresponding value. Different random numbers are selected to fix each of the different parameters being varied. Then using the randomly selected parameter values, the economic analysis criterion such as rate of return is calculated. Then you iterate and do the same thing over again picking a new set of random numbers to determine a new set of parameters used to calculate rate of retum. This is done over and over again for somewhere between 100 and 1000 times and then a histogram (frequency distribution plot) of these rate of return results versus probability of occurrence is prepared. The number of iterations (RoR results) that are required is the number that will give the same shape of frnal RoR results frequency distribution graph that would be obtained if many more iterations were made. To illustrate. examples in Figure

consider hypothetical cases

I and 2 shown

s

\o

o

o

() 25

:20 o o

o15

15

o

't0

Ero

a6 -oo5l (L1

o 5 o-

0

30

8zs

20

= o 6

ih Figure 6-3.

5

't0 15 20

Rate of Return, 7o 100 Simulations (Note multimodal maxima)

Case

1

25 Flate of Return,

9,o

500 or 1000 Simulations (Constant shape, unimodal) Case 2

Figure 6-3 Frequency Distribution of RoR for Varying simulations

Cnapter 6: Uncertainty and Risk Analysis

293

'rhe

histograrn fbr case t has murti-modar (more than one maxima) peaks indicating that statisticalry we have not mad-e enough simurations to get ,a rate of return frequency distribution with constant shape. If the input data frequency clisuibution is unimodal (having one peak as'iitusirated in Figure 5-l; the ourtrrut rare of return freirue,cy ciistribuiion plot str,;urd be uni_ mcdal. case 2 iiemonstrates for this hyporheticar situation thar if we to to 5il0 or 1000 simulations we g"t a unimtdar graph *.ith the same shape indi_ ;;iting that 500 simulations are sufiicient fbr anaryzing this hypotheticar iuresrment using probabilistic sensitivity analysis. An iniportant thing to.note about probabilistic sensitivity anarysis resurts is that y'ou i-ro not gei a single resrrt,iut instead you get a iarg" over which the results vary as a function or prouaurtity of occunJr""aJ*so you get a m'st expected result. In many iuvestmenisituations tt.,tup" ot this curve is more important rt, varue. For exampri a project with a most expecred RCR ofln:-ryl,rr.*p..i.o 25vc with ara,ge of po-rsibre ,.Ji; from negative RoR to positi'e 4c7o might be consideri ress cssirabt. tt un a r_,rcrject wirh a i,lr)it expecied ROR oi lgvo and a range u.f pos.;ible resurts from rQvo to 25vc because the certainiy associateJ ,rirtr, tn. tsz, mosi-expected RoR inr"estment is greater.tho,t th"..rtointy r.socrieted with the other investment. Prohabilistic sensitivity anarysis .*.ot.i-tr," decision get a firmer fecirng concerning rhoeff-ecis or.irirra "r"t.r^," uncertainty on economic anarysis resuirs than an-v otlrer analysis approxsh gi1.sE.

The weak pr:int of trre prooauriistic ariarysis method ries in the subjectire assi,uning of probabiritie. of occurrence to the rer,.els of pararneters that go int. the arrarysis' it is gene'aily considered to be best to specificallv staie the probabililies 'o::.u:r, of o.-.r.."n.. based on the best judgement of peopic invoh'ed rvith a project and then to ba.se the anaiysis on these esti_ maies even though they are subjective in nature. In the finar analysis any evaluatio, technique is only as gooo as the estimates of the input parameters and rnust be used in conjunction with good engineering logic and managerial judgement' Assigning probabilities to"pararneter estimates is just one more step in quantifying the assumptions thai are made. These rechniques provicle managcment with addirionai toors to aiir in the decision-.uung'p.o."rr. Folloiving is a simprified exampre that ilrustrares the principles of apprying probabilistic analysis to project ROR sensitivity analysis.

EXAMPLE 6-9 A Simple probabilistic Sensitivity Analysis A new product to be produced by one of two different processes. rt is felt that there is a 60% prooaoitity that the process serected wiil

294


have initial cost of $50,000, a life of five years and zero salvage value. There is a 40"/" probability that an improved process will be selected with iriitial cost of $4O,OOO, a life of five years and zero salvage value. With either process there is a 50ol" probability that annual profit will be $20,000 for the five year project life and a 25"/o probability that the annual profits will be $15,000 or $25,000 per year. Plot project ROR versus probability of occurrence assuming the parameter values are independent of each other.

Solution: The following table presents the possible combinations of different investment costs, annual profits, probabilities of occurrence and ROR.

lnvestment

50,000 50,000 50,000 40,000 40,000 40,000

Annual

Profit

15,000 20,000 25,000 15,000 20,000 25,000

Probability of Occurrence (.6X.25) (.6X.50) (.6X.25) (.4X.2s) (.4X.50) (.4X.25)

=.15 =.so =.15

=.to =.zo =.10

P/Ai.s

3.334 2.500 2.000

2.667 2.000 1.600

RoR (%) s.2 28.7 41.1 25.4 41.1

56.0

1.00

The most probable project ROR is 41.1"/" with a 35% probability of occurrence. The cumulative probability diagram shows a cumulative probability o175"/" that the ROR will be 28.7% or above in Figure 6-5. Instead of mathematically determining the probability of occurrence of the various ROR results for this problem we could have used the general Monte Carlo simulation technique to get the same results. The general idea of this method as described earlier is to first develop curves for cumulative probability of occurrence versus the economic ltarameter similar to Figure 6-4. Then random numbers between zero and one are related to cumulative probability of occurrence so that selection of a random number fixes the initial investment for a calculation. Selection of another random number selects the cash flow for that calculation. ROR is then calculated using these values. This procedure is then repeated several hundred or maybe a thousand times until the shape of the ROR versus probability of occurrence curve does not change with additional calculations. For a large number of Monte Carlo simulations the results using this technique for Example 6-3 will be identical to those

Chapter 6: Uncertainty and Risk Analvsis

295

given in Figure 6-5. The Monte carlo simulation data are evaluated by forming a histogram from the RoR results. For instance, if one thousand runs are made tlren approximately tluee hundreci of the RoR rcsults ,rrorr.i bc 2g.i7o, since we know the mathematical probability of occurrence of this resylt is .30. In general, the variations ir.r input crata such as seiling price, oirL-raiing cost. production rale, borrowed money interesr rete and so forth that are evaluated will be continuous functions of cumulative probability of occurrence rather than the step functions illusrrated in Figure 6-4. Coniinuous input data give a continuous RoR versus probabiliiy of occurrence histograrn graph for lrlonte Carlo simulation in general.

Cumulative Probability

of Occurrence

1.00

1.00

.80

.80

.60

.60

.44

.40

.20

.24

0

0

15,000

Initial lnvestment

Profit

(a)

(b)

25,000

Figure 6-4 Cumulative Probability Diagram

Probability of Occurrence

.30 .20 .10 0

1.00 .80 Cumulative Probability of

Occurrence

.60 .40 .20

0L 0

10 20 30 40 50

60

ROR, %

Figure 6-5 Cumulative probability of Occunence

296

6.5


Expected Value Analysis (Economic Risk Analysis)

Expected value is defined as the dffirence between expected profits and expected costs. Expected profit is the probability of receiving a certain profit times the profit and expected cost is the probabitity that a certain cost will be

irtcurred times the cosr. If you define cost as negative profit and keep the signs straight, you can do as some text authors do and define expected value as the algebraic sum of the expected valuq of each possible outcome that could occur if the alternative is accepted. Either definition leads to the same expected value result, which sometimes is called a "risk adjusted" result. Several examples of expected value analysis when time value of money considerations are not relevant or significant will be presented first. Then time value of money related examples will be illustra.ted.

EXAMPLE 6-4 Expected Value Analysis of a Gambling Game

A wheel of fortune in a gambling casino has s4 different slots in which the wheel pointer can stop. Four of the s4 slots contain the number 9. For $1 bet on hitting a g, the gambler wins $10 plus return of the $1 bet if he or she succeeds. what is the expected value of this gambling game? what is the meaning of the expected value result?

Solution: Probability of Success = 4/54 Probability of Failure = SOIS4 ExPected

Var

ue

:,?ffi,'*,T

:lffi ;;: * -"-::l

The meaning of the -$0.195 expected value result is that it is the rverage monetary loss per bet of this type that would be realized if he gambler made this bet over and over again for many repeated trirls. lt is important to recognize that the gambler is not going to lose 80.185 on any given bet. over a large number of bets, however, the oss per bet would average $.18s. This result should make it evident hen that a positive expected value is a necessary condition for a satsfactory investment, but not a sufficient condition as will be dis:ussed later.

Chapter 6: Uncertainty ar':C Flisk Analysis

297

EXAMPLE 6-5 Expected Value Analysis of a Simplistic

. ,:,DrillingVenture

I

lf you spend $500,000 drilling a wildcat oil well, geologists estimate the probability of a dry hole is 0.6 with a probabiiity of 0.3 that

I i

the well will be a producer that can be sold immediately for

l

$2,000,000 and a probabiiity of 0.1 that the weil will produce at a rate that will generate a $1,000,000 immediate sale value. What is the oroject expected value?

!

I { {

, f

Solution, All Values in Thousands of Dollars: Expec'ied Va ue

I

il:ffi

I I

:

I

= +$200

:;ffi ; :lT:::::H,

_ o 6(500)

or rearrangtng:

I

Expected value = 0.3(2,000) + 0.1(1 ,000)

- 1.0(500) = +$200

I I I i

1

t

i I I

tf

It

lI I *

t

Over the long run, investments of this type will prove rewarding, but remember that the +$200,000 expected value is a statistical longterm average profit that will be realized over many repeated investments of this type. The expected value of an investment alternative is the average profit or loss that would be realized if many investments of this type were repeated. ln terms of Example 6-5 this means if we dritied 100 wells of the type descrrbed, we expect statistics to begin to work out and assuming our probabilities of occurrence are correct, we would expect about 60 dry holes out of 100 wells with about 30 wells producing a $2,000,000 income and about 10 wells producing a $1,000,000 income. This makes total income of $70,000,000 from '100 wells drilled costing a total of $50,000,000 leaving totai profit of $20,000,000 after the costs, or profit per well of +$200,000, which is the expected value result for Example 6-5. Certainly if you have enough investments of this type in which to invest and enough capital to invest, statistics are very favorable to you, and you would expect to come out ahead over the long run if you made many investments of this type. However, if the loss of the $500,000 drilling investment on a dry hole would break you, you would be foolish to invest in this type

298


of project because there is only a 4avo chance of success on any given try. only if you can stick with this type of investment for many times can you expeci statistics to work in your fuvor. This is one important reason why most large companies and individuals carry insurance oi various types even though in nearly all cases the expected profit from self-insuring ii positive and therefore favorable to the company or individual. If a disaster from fire can break a company cr individual financially, that company cannot afford to self-insure. This is why insurance companies spread large policies over many insurance companies. If disaster does strike a large poti.y holder, the loss will be distributed over several companies, lessening the iikelihood of financial disaster for any one company. It is also the r"u.on most individuals carry fire insurance, homeowners or tenant insurance, and car insurance. The direct financial loss or lawsuit loss potential is so great that most of us cannot afford to carry that risk alone even though exp-cted value is favorable to us if we self insure. The conclusion is thit a pisitive expected value is a necessary br.tt not a sfficient condition for a satisfactory iivestment. It should now be evident that although expected value has deterministic meaning only if many trials are performed, if we consistently follow a decision-making strategy based on selecting projects with positive expected values, over the long run statistics will work for us and income should be more than sufficient to cover costs. on the other hand, if you consistently take the garnbler's ruin approach and invest or bet on investments or gambling games with negative expected values, you can rest assured that over the long run, income will not cover your costs and if you stick with negative expected value investments long enough, you will of course, lose all your capital. This is exactly the situation that exists with all of the gambling garnes in places such as Las vegas, Reno and Atlantic city. The odds are alrvays favorable to rhe house, meaning the gambling housl has a positive

expected value and therefore the gambler has negative expected value. The gambler has absolutely no realistic hope for success over the long run under

these negative expected value conditions. He will lose all the money set aside to gamble with if he sticks with the games long enough.. The reader should notice that in Example 6-5, two different, but equivalent equations were usec to calculate expected value as follows:

Let P = probability of success, I Expected Value = (P)(Income

or

-

-p

= probability of failure

Cost)

-

(1 _ pXCost)

=(P)(Income)-(i.O)(Cost)

6-2 6-2a

C;,apter 6: Uncertainty and Risk Analysis

6.6 Expected

NPV, Expected pVR, and Expected ROR Analysis

when time value of money considerations are significant, expected Npv, PVR and RoR anarysis are merhods of incrudrrg ir" p."ir"uiii,;"s of suc_ cess and failure in analyses when costs and revenues occur ar different ptrints in time. If we use appropriate time ralue of monel. present rvorth fac_ ttlrS ro convert costs and profits at different points in time to lurnp sunr r.al_ ucs at time zero or some other chosen tinre, the expected value anarysis approach can be applied to determine if rhis type crt.- investment woulcl be snitable over the long run for many repeated investments of the same type. \{'ith e-rpected Npv and pvR *r u.., of course, looking for alternatives with a positive expected with expected RoR anarlsis we calcurate the expected RoR value,'arue. "i", that will rnake the expected Npv equation cquai to zero. An acceptabre expected RoR must be greater than the minimum ROR.

EXAMPLE 6-6 Expected Vatue Apptied to ROR, NpV and pVR

Analysis

A research and deveropment project is being considered. The project is expected to have an initial ihvestment cost of $90,000 and a

probability of 0.4 that annual profits of $50,000 will be reatizeO during the 5 year life of the project with a probabitity of 0.6 of faiting. Sal_ vage value is expected to be zero for success or failure. Assume the minimum discount rate is lOoh on a risk_free basis.

should the project be done? compare expected varue, nel present value, expected rate of return and expectedexpected present

value ratio analysis with corresponding non-risk adjusted evaluation results.

Solution, Values in Dollars: C = $90,000

P=0.4

|

=$50,000 . . . . I =$50,000

1............s

P=0.6

L=0

i

300


A) Expected vatue Anarysis rncruding Time varue of Money at i' = lgolo 3.7908 EV = 6.4150,000(P/A1 Oo/o,S)

-

9O,O0O)

- .6(90,000)

= _$14,1g4

The negative expected varue indicates rejeci investing in the project.

The expected varue approach is most usefur when statisticaily more complex models are utilized. ln many of these cases, the various outcomes that are possible in a model are defined, associated chance factor is derived for each branch. rn this "no ", t*o outcomes exist which can be rabered ,,success', "r."rpr" and ,fairure." By carculating the NPV for each separatery, the chance factors (in .outcome this modeljust the probabirity of success and fairure) can to.the appropriate outcomes to determine the identicalbe appiied expected value (EV) as shown: Expected value per outcome (Npv of outcome)(chance = Factor) summing ail of the project's expected varues per outcome gives the overall project expected value. 3.7908 EV Success = {50,000(p/Aj _ 39,g16 Oy",S) 90,000X0.4) EV Failure = (_go,ooo)(0. = -54,000 Project Expected Value = -14,184 The same calculation was introduced in the expected value solution to this problem.

=

B) Expected Net present Value This analysis is identicar to the (A) anarysis, except we generaily use Equation 6-2a to determine Expected Npv. Thi; just is a rear_ ranged form of the Expected Value equation from part,,A,,. 3.7908 ENPV = 0.4(50,00O)(ptA1O%,5)

-

1.0(90,000) = _$1 4,184

Since the ENpv is negative, we shourd not invest in the project from an economic viewpoint. Note the expected varue anarysis in oart (A) accounting for the time varue of money ro7" p"iyear gave e result identical with the EV result from Case "t A.

Chapter 6: Unce(alnty and Risk Analysis

For these statistically simplistic models, another approach for calculating the ENPV is to first, risk adjust the cash trows, and then, carculate ENPV using the same present worth equation format previously developed. This approach is helpful in making expected value calcuiations on a hand calculator but again, it shor.rli be emphasized that the resuiting cash flow siream does not represent the expected ca-sh flows for any single investment. lnstead, ihe calculated values reflect the average cash fiows that mignt be expected after many repeated investments of this type with success occurring 40% of the time and failure occurring 60% oi the time.

Cash Flow.

-$90,0i,09.0)__!9900!04).. .. s0,000(0.4)

0 Risk Adjusted Cash Ftows -$90,000

ENpv = -e0,000 + 20,000(priliijr, *14,184 = This approach can be more difficurt to appry to more comprex statistical models than summing the expected vaiues for all outcornes. lf you look at non-risk adjusted NpV (risk free NpV) which implicitly assumes 1007o probability of success, the project economics look very acceptable Risk Free NpV = (50,000)(pr"^133^,r,- (e0,000) +$ee,540 > 0 = obviously adjusting for risk of failure or not adjusting for risk of fail_ ure has a very significant impact on economic results aid conclusions.

C) Expected Rate of Return Expected RoR is the "i" value that makes Expected NpV equal 0. Expected Present worth lncome @ "i" present worth cost ,,i,,= @

-

50,000(P/Ai,S)(.4)

-

0

90,000 = 0

by trial and error, "i" = Expected RoR g..ay" < i* 10% so reject = = Non-risk adjusted or risk free rate of return analysis gives a different conclusion:

302

50,000(P/A1,5)


-

90,000 = 0

by trial and error, f i" = ROR

=

47.60/o> i*

= 10% so accept

This result is much greater than the 10% minimum RoR indicating very acceptable economics. As a variation of expected RoR analysis, some people account for risk by increasing the minimum ROR by what they consider to be an appropriate amount. The difficulty with this approach is that there is no consistent, rational way to adjust the minimum RoR appropriately to account for risk in different projects. For this example, the risk free RoR is about 42.6% (based on probability of success of 1.0 instead of 0.4) compared to the expected RoR of 3.6%. Expected RoR of 3.6% compared to risk free minimum RoR of 1oh indicates the project is economically unsatisfactory. To get the same conclusions based on comparison of risk free RoR and risk adjusted minimum RoR results requires increasing the minimum ROR to about 132% (where l1yo/3.6/" = Xl4Z.6/q therefore X = 13:2%). A majority of people who attempt to compensate for risk by adjusting the minimum RoR end up significantly under-compensating for risk. For this example many peopre wilr propose increasing the minimum RoR inversely proportional to probability of success 6to.q) from 10"/o to 25% when as previously discussed the increase should be from 10% to about 1go%. However, there is no way to know this without making an expected RoR type of analysis. Expected value" analysis using RoR, NPV or. PVR is the preferred approach to incorporate risk into economic analysis calculations.

D) Expected Present Value Flatio Analysis EPVR = ENPV / Expected PW Cost = -14,184/90,000 = -0.16

The negative expected PVR indicates the project economics are ;nsatisfactory for the project parameters built into this analysis. Risk Free PVR = +99,540/90,000 = +1..l1

consistent with the other criteria, the risk free pVR indicales very rcceptable economics which is the opposite conclusion reached with lxpected PVR.


Expected value anarysis in general involves constructing a diagram showing investmenr cosrs and all subsequ"r, riilues that are anticipated. Standard si;mbolism"t*."^;;;;;, and dolrar uses circles to designiite chance nodes tiom which different delrees of success and fairure may be -rh,"rn ttr r-rccLlr. l''c sum of the proh.b'ilities of occurrence on the dittbrent branches ernanariirg from a .hu,.,"" node must adil up a r.o. Trtese e-rpec'ted r'.;lua L!trigrtuns" are someriutes called ,,rlecisiori rree d.iagrants,, . be-cause decisior: options concerning whether to proceed in one or more di ferent wa)'s or to terminate the project aiways .îrt prio, to each chance node where different degrees oi ,r.."r, or failure may occur. These diagrams often ha'e murtiple branches and rook u.ry -u.h like a drawing of a tree' which has led to the name "decision ree analysis,, being used in industry pracrice to refer to this type of analysis. In typicai decision tree anaiyses, at dift-erent stages of projects, probabilities of success and failure cha3ge, As you progress from the i"r.ur.t or exproration stage of a project to.c.eveiopmenr and production, risk oi' failure ,rgniii.unrry. This is illustrated in the follorving two examples. "hung",

,:

EXAMPLE 6-7 Expected varue Anarysis of a petroreum proiect Use Expected NpV anarysis for a minimum RoR of 2o"hto evaruate the economic potentiar of buying and driiling an oir lease with the following estimated costs, ,."rlnr". and suc-cess probabirities. The lease would cost g.100,000 at time 0 and it is considered 100% certain that a wefi wourd be driiled to the point of .ornpr"tion on" year later for a cost of $500,000. There is a boz pronaoitity that welr lggs will took good enough.to comprere the 1 for $400,000 compretion .o.i. tf tre wirr rogs are ,n.rtirtr.tory a an abandonment cost of $40,000 wi, be incurred at year 1. rf the we, is completed it is estimated there will be a 50"/" prooroitity of generat_ ing production thar will give $450,000 per year net income for years 2 through 10 and a 3S^y" probability of generating $000,000 per year net income for years 2 through io, witn a 1so/o piobanitityof the weil comple-tion being unsuccessiur, due to water or unforeseen compretion. difficulties, giving a year 2 salvage value of $2SO,OO0 for pro_ ducing equipment.

*!riril"ar

304


Solution, in Thousands of Doltars: l=450

l=450

(c)

(D) C=40

Times 1 and 1+ are effectively the same point in time, the end of year 1. Normally drilling and completion time are separated by weeks or months which puts them at the same point in time wittr annual periods. Times 1 and 1+ are separated on the diagram to make room for the probabilities of occurrence associated with different events. Expected Value

. *Determine the possibte different outcomes and the subsequent NPV for each. Apply the probability of each outcome to the calculated NPV's giving ENpv per outcome. summing each gives the project's overall ENPV. There are four (4) possible outcomes:

A) Successful Development leading to incomes of $450 per year. B) Successful Development leading to incomes of $300 per year. C) Failure with a salvage of $250 at the end of year two. D) Failure with an abandonment cost of $40 at year one. For Case A, the chance factor is; (0.6X0.5)r For Case B, the chance factor is; (O 6X0.35)\ For Case C, the chance factor is; (O.OiiO.f S)

\

I-100 ?l t-100 9) t-100 D) t-100 -

1)

eoo(p /F 20,1) + a50(p/A 20,eXp/F zo e00(p tF 20"1) + 300(p/A

2['!11er

rltlsol

;;:j,]

e00(ptF 20"1)-1-250(p/F i6"ill(0.00i"" 540(P/F16"1)l (o 4) Project Expected Net present Value lftrln4

l---

= +198.48 io.zr i = +33.13 '

=

-60.87

= -219.99

=

-49.26

Chapter 6: Uncertainty and Risk

Analysis

g0

Expected Net present Value (ENPV) at

4.031

2}o/o

4.031

.8333 {[450(PiA20,eX.5) + 300(piA20,ex.ss) + zs0(n{1X. j5) _ 400](.6 .8333

-

40(.4)

- 500xP/Fz},i _ 100 = _49.26

This

resurt is onry- srighfly ress than zero compared tc the totar pro. ject costs of $1 miilion, therefore, srighily unsatisfact"iy o,, break. even

economics are indicated.

Alternate Form of (ENpV) Equation

4.031

.8333

4s0(P t Azo, (. 5) (P/F20, e)

.6944 +zso (p'/ F 20,z) (. 1sX. 6) .8333 -500(P/F20,1X1 .0)

-

1

_

X.

4.0g1 .8933

4oopi

100 =

.ffi33

6) + 000(p/A2s e) (. 35) (p/Fio, r X o)

; ;;.

1 X. 6)

_ +o

.8339

pir"ll1; i: +1

-49.26

Risk Adjusting the Cash Flows

-500(1.0) 450(0.6X0.5) -jlBIS

cF(prob Year

) _i00(1 0)

RiskAdj

cF -1oo -zs6

O

1

?i IBBIB\Y'Y/\v' Bi[B ?Ei tet

450(0.6X0.s) 3oo(0 6iio 3b) g_10

ffi

ENPV = -100 -756(p/F20,1)+ 220.S(p/F29,2) +'t 98(p/A2 0,A) (p ff ,0,r1 ENPV = -49.26 EXAMPLE

6-8

Expected value Economics of a New process _ use Expected NpV and pVR anarysis for a minimum rate of return of 2a.0"/" to evaluate the economic potentiar of buying ano Jevetoping the rights to a new process with the foilowing estimated costs, revenues and success probabilities. The process rights would cost g100,000 at

J

306


time zero, and, it is considered 100% certain.that experimental devel_ pilot plant work wiil be done one year raier for 9ql"ll a cost of $500,000. There is a 60.0% probabirity that the experimentar development results will look good enough to iake the project to production flr a $400,000 capital cost at year 6ne. (This capital cost is estimated to be incurred in the first six months of year one which is closer to year one than year z.) lf the experimental development results are unsatisfactory, a pilot prant abandonment cost of $+o,ooo wiil bL-incurred at year one- If the project is taken to production, it is estimateci there will be a 50.0% probabirity of generating production that wirr tive g4s0,000 per year net positive cash flow for years two through ten, i 35.0% prob_ ability of generating $300,000 per year net positive cash frow for years two through ten with a 15.0% probabirity of the project deveropment being unsuccessfur due to unforeseen iechnicai oiiticutties giving a year two salvage value of $250,000 for production equipment.

Solution, in Thousands of Dollars:

l=450

!=1OO lru

P=1.0 C-

P=0.6

l=450

(A)

P=0.35 l=300 l=300

01

P=0.4

(D) C=40

P=0.1 5

(c) Salv=250

Expected Value

..letermine the possibre different outcomes and the subsequent

App]y the probabirity of each outcome to the carcul\lv. Igl "uch. lated NPV's giving ENpv per outcome. Summing gives the project's overall ENpV. "r.n There are four (4) possible outcomes:

A) successfur Deveropment reading to incomes of $450 per year. B) Successful Development leadin! to incomes of per year. c) Faiture with a sarvage of $2s0 ai tne end of y"ri$gOO t*o. D) Failure with an abandonment cost of $40 at'year

one.

Chapter 6: Unce.tainty and Risk Ahalysis

For Case A, the chance factor is; (0.6)(0.S)r

\

\ - e00(PlF zo,t) + 450(P/A 20 eXp/F 20 il fo.eoy= +1e8.48 e00(P/F zo',t) + 300(p/A 2['eylerr zo. jlj tt'.z1)= +33.13 !) i-t00 - e00(Pilzo',1) .250(piF i6',ill (c.obi" 9) f-100 = -60.87 D) [-'100 - 540(P tF zo',t)] (0 4) = -219.99 =l'roject Expected Net Present Value (ENPV) = _49.26 1)

t-100

Risk Adjusting the Cash Flows

-500(1.0) 450(0.6x0.5) 450(0.6X0.s) -400(0.6) 300(0.6x0.35) 300(0.6)(0.35) CF(Prob.) -100(1.0) -40(0.4) 250(0.6X0.1s)

Year0lzg_10 Risk Adj CF -100

zzo.s

-7s6

198

ENPV = -100 -756(PlF20,1)+ 220.5(plF29,2) + 1 98(p/A2 g,g) (p /F 2g,2) ENPV = 49.26 Expected Net Present Value (ENPV) at20o/o

4.031

4.03.1

{1450(P I A2O, 9X. 5) + 300( P/A20, g) (. 35)

.8333

+

ZSo (p I F

20,

(. 1 1)

5)

- 4001 (. 6)

.8333

-

40(.4)

- s00)(p/F2},i

-

100 = _4s.26

This result is only slighfly less than zero compared to the total pro-

. costs of million, ject therefore, slightly unsatisfactory or break_ $1 even economics are indicated.

Expected Present Value Ratio (EPVR) EPVR = 49.26 / 100+[500 + a00(.6) + 0(.a)](p /F20,1) = -0.07 The small negative EPVR result indicates the same slighily unsatisfactory or break-even economics shown earlier with ENpV analysis.

308


EXAMPLE 6-9 Expected NPV Anatysis Calculate the,,Expected NPV of a project which will cost $270,000 at time zero. This investment has a 60.0% probability of generating downstream development and equipment costs-at tne=end-of yeai one estimated to total $500,000. lf the secOnd expenditure at the end of year one is successful, there is a g0.0% probability it would lead to the generation of net cash flows totaling $400,000 per year at the end of each of years two through ten. Failure from the time zero investment would result in no additional cost or benefit to the investor. However, should the project fail after the year one cost, a net cost of $250,000 would be realized at the end of year two from dismantlement costs and salvage of equipment. Should this project be accepted? Use a minimum rate of return of 1Z.Ooh.

Solution: All Values in Thousands.

.-$270

P=(0.60)

-$500

P=(0.4)

(c)

P=(o.eo)

$400.

..$4oo

,......-*

(n)

P=(0.10)

$o

(B)

-$2s0

Expected Value There are three (3) possible outcomes in this solution: A) Successful Development, Chance Factor,

P=(0.6)(0.e) =(0.54) p = (0.6X0.1) = (0.06) c) Failure at Time Zero, Chance Factor, p = (0.40) B) Failure at Year 2, Chance Factor,

EVA) [-270-500(P/F1 2,1) + 400(P/\2.9XP/F p,i)(0.s4) = +640.71 EVs) [-270 - 500(P/F1 2,i - 250(PlF p,2)] (0.00) = -54.94 EV6) [-270] (0.40) = -108.00 Project Expected Net Present Value (ENPV) = +477.77

Chaoter 6: Uncertainty and Risk Analysis

Risk Adjusted Cash Flows

Year

0. -

CF(Prob.) Risk Adj

-270fi.q

CF -270

ENPV = -270

-

309

.,,

1

3-10

2

400(0.6x0.e) -500(0.6) -2s0(0.6x0.1) 400(0.6x0.e) 201 216 -300

300(P/F1 2,1) +

201(PlFp,2)

+ 216(PlA12,g(PlFe,Z)

= +477.77 > 0, accept Expected Net Present Value

ENpv @

12.0o/o

= +0o17;ffiX0.e)3/8Fe122:?,,0 0.79719

-

250(P/F1 2,2)(0.1X0.6)

-

u,

0.89286 500(P/F12,1X0.6)

- 270 = +477.77 > 0, ok Example 6-94 Utilizing the data from Example 6-9, what additional cost could be incurred at time zero tor either research or geological/geophysical data and give the investor the same risk adjusted NPV of $477.77? Assume the additional time zero cost will increase the probability of success in year '1 from 0.6 to 0.8. Therefore, the probability of failure in the same period will be reduced from 0.4 to 0.2.

Solution: Let X equal the additional cost to be incurred at time zero.

-x ^-270

P=(0.8)

400

-500

400 (A)

0 P=(0.1)

P=(0.2) 0 (c)

,!

-250

(B)

310


ENPV Approach:

Ecohomicary equivarent, mutuary_excrusive

arternatives wi, always have equar net preseirt r"ir".. Therefore, this sorution rooks at equating the current project Npr;ith the Npv br tnlr"uired prob_ abilities based on the new invest*.ni, x at time zero as forows: 477 .77 = [400(p/A eX,g)(0.9) _ 500J(p/ F ex,[email protected]) - 250(p/F12/",2)(0.1)(0.S) _ z7O _ X 477.77 = [400(s.s2825X0.9) _ so0](0.89286)(o.B) _ 2s0(0.7e71exo.1x0.8) _270

_X

477.77 = 1012.98_ 15.94

X

_270_X

= 249.27

Risk Adjusted Cash Flow Approach, ENpV:

-x -270 P=(0.8) _5W P=(0.9) 4OO P=(0.2

400.

.400

;; . -,

(n)

P=(0.1)

0 (c)

-250

(B)

Risk Adjusted Cash Flow Catcutations: -270(1.0)

-500(o.s)

01

Risk Adjusted Cash Ftows: _4oo

-.270

400(0.72) -250(0.08)

4oo((0.72)

0.......... 268

288

10

. . .288

3..........10 ENpv = -

27

o-

400(p?F81Yji"?, + z6s@

4.9676 + 2Eg(p / A1

ENPV = 727.02

o.7g71g

2"7",g) (p / F 1 2y",2)

/{1\ir,


,11

The difference between the New ENpv of 227.oz and the originar ENPV ot 472.27 is 249.25 which represents the additional cost that could be incurred at time zero and ailow the invest,oi to obtain the desired ENPV of 477.27.

It was emphasized earlier in this section that expected value represents the average gain or loss per investment that an investor would realize over many repeated investments of the type being analyzed.

whether we work with expected vaiue, expected Npv, expected pvR or expected RoR, the average meaning of results is similar. A common misconception that some people have about expected value analysis is that it oft;n is not valid because investors serdom repeat the same type investments over and over. These people have missed the basic .*p".t.d value analysis premise that even though each specific investment
investment, if we consistently select investment altematives having positive expected value, ENPV or FPVR, (or having expected ROR greater than the minimum R.oR), over the'long run our average rate of return on invested capital will be greater than the minimum RoR. Similar to material pre_ sented earlier in chapter 4 for risk free analysis,

to rank non-mutually exciusive alternatives using risk adjusted reiults, you must use either expected PVR or expected Growth ROR resultr. fo evaluate mut,ally exclusi'e alternatives with risk adjusted results based on any valid anarysis technique, irlcremental analysis is the key to correct economic decision making. These ruies hold true even though each investment decision rerates to a different investment prospect with different probabilities of success and failure at different stages ofeich project.

6.7 Protrability of Survival (Financial Risk Analysis) Probability of survival refers to the probability that you will not go bankrupt

with a given amount of capitar to invest in-projecis with estimated probabilities of succes.s. This concept is a financial .irt unutyris considera_

tion rather than specificaily relating to risk analysis. A project with a large positive expected value and ".ono*ic a smail probability of success may economically look better than ail other investment opportunities under con_ sideration. However, if.failure of the project would lead to bankruptcy, the project most likely will be considered financially unacceptable due to the

financial risks. The itaricized statement preceding Example 6-6 says ..a positive expected value is a necessary but not a sufficient condition for a satisfactory investment." This meanr ihut u positive expected value indicates an

312


economically satisfactory investment but not necessarily a financially satisfactory investment..small investors in the oil and gas drilling, business,usu. ally attempt to reduce rheir financiar risk of total failure liankruptcy; by -inteitaking a small interest in a large number of projects rather than large ests in a few projects. This diversified investment portfolio upp.oo.h giu.. the probability of success a better chance to be realized over time as the fol-

lowing example illustrates.

EXAMPLE 6-10 Probability of survival Applied to Exploration

consider an exploration manager who is faced with the task of determining whether to invest 91,000,000 of a small company's money in 10 independent exploration projects which will each cost $100,000 with a 10% probabitity of achieving a gs,000,000 profit above cost; lf the company will go bankrupt if not successful on at l"3pt ] exploration project (or if the exploration manager will lose his job), discuss their probability of survival. Solution: Expected Value = 5,000,000(.10)

- .tOO,OOO(.9) = +$+t0,000

The positive expected value is economically satisfactory and is a necessary but not a sufficient condition for this investment to be satisfactory. The question to be answered now is what is the probability of survival, or stated another way, what is the probability of getting at least 1 success out of 10 tries. This relates to the financial acceptability of the project. Probability of at Least one SucceSS = 'l

-

probability of Zero Successes

Probability of Zero Successes = (.g)10 = .34g5

Therefore, probability of survival = 1-.84g5 = .6515 or 65.15% which is certainly considerably less than a sure thing (100% probability of success). Note that ten projects each having a 10% chance of success do not give a 1oo% probability of overall success. companies often get together in joint ventures on large exploration projects to have more capital available so that enough exploration projects can be made to make the probability of survival higher than it would be if they operated alone. If in Example 6-r0 a sum of $2,000,000 is available to invest in 20 projects, the probability of survival increases to l-.1214 .g7g6 =

C,rapter 6: Uncertainty and Risk Analysis

and

313

if $3,000,000 is available the probability of survival is .95i7. Joint

ven-

tui'es can increase survival probabilities to very tolerable levels when exploration work is being done in areas where enough geological and geophysical work has been done to predict reasonable probabilities of success on any gir cn tr).

The same type of reasoning can be applied to research and development projects in all types of industries, but it is very difficult to come up with truly meaningful probabilities of success. However, if these probabilities are not estimated explicitly, managers will base a decision impiicitly on "gut feel" for data, and these authors feel it is better to explicitly state the bases upon which decisions are made. The statistical basis of the probability. of survival calculations just presented is the binomial distribution for mutually exclusive altematives. The general binomial distribution equation is: Probability of Exactly "r" successes from "n" tries = glpr11-pyn-r where

C? = Combination of "n" things taken "r"

nl

n factorial r factorial times (n-r) factorial

r!(n-r)l nl = n(n

-

at a time.

lXn

-

2)

----

(3) (.2) (t)

0! =lbydefinition P = Probability I

of success on a given try.

-P = Probability of failure on a given try.

Note that for Example 6-10, the probability of zero success from

c1o0r.rlor.o)'0

6.8 Risk

=

.ffi

l0 tries

is:

(1x.9)10 = (.p110 = 0.3485

Due to Natural Disaster

Another type of risk to be considered is that due to natural disaster. The following example illustrates the evaluation of data using probabilistic expected value concepts for this type of problem.

314


EXAMPLE 6-11 optimum lnvestment to Minimize Flood Damage Costs A manufacturing company plans to build a plant on low land near a river that floods occasionally. lt is considered necessary to build a levee around its facilities to reduce potential flood damage. Four different sizes of levees that give different levels of protection are being considered and the plant manager wants to know which size levee will minimize total expected annual cost to the company from the sum of (1) amortization of the levee cost over 20 years for a before tax "i " of 157": plus (2) maintenance costs, plus (3) expected damage to the plant and levee if the levee is not high enough or strong enough to hold back flood water. Analyze the following oita to determine the optimum levee size.

Levee Probability of a Expected Damage Annual size cost Flood Exceeding Levee if Flood Exceeds Maintenance Size During year Levee Size

Levee

1 2 3 4

$120,000

.20 .10

140,000 160,000

.05

200,000

.025

$ 8o,ooo 110,000 125,000 150,000

$4,000 5,000 6,000 7,000

Solution: Expected Damage

Levee

Size 1 2 3 4

Annual Levee

Cost

+

0.1598 120,O00(A/P15,20)=19,180

Total

Annual

Expected

PerYear + Maint. = Annual Cost 16,000

4,000

$39,'180

= 22,370 160,000(0.1598) = 25,570

11,000

5,000

38,370

6,250

200,000(0.1598) = 31,960

3,750

6,000 7,000

37,820 42,710

'140,000(0.1598)

Levee size number 3 is the optimum selection to minimize

expected equivalent annual cost.

Cilapter 6: Uncertainty and Risk Analysis J,3

{i.9

k.

Option pricing Theory Related to ENpV Patent rights to new technorogy, Iand and minerar rights to minerar prop_ erries ofren se, at prices signifi-c'#rrv rrigrr", ;il;;il'*"ff erty' Those caicurations ur. uur.J-Jn tf," *ort .-r..,.0 of the said prop_ production rar.es, p.q..t or the ."u,on 'lr'' pri ci n g erre* rnay, u" r"r u t" J io ;;;,i:,

i* ;#"

il:ff'J#n:::l;:f*T":nf.r:' -

il ?:,H#ffi::"ffi;, T:#Hl

me)' occur if actual production rates or product prices than the mosr expecred productio;;;". and prices turn out to be higher costs tum out to be lowei ro. ir," a*ual operating than trr. rnort expected ,utu"r), may also rerate ro the decision ^p."ar.ii* grt option pricing ,o J.ruy Ly ura caprure a greater procuct nrice in the future, o, ,.urir" g."rr..'.*#es than what is rhe ten t i ai ;;;;. u. io XJ,x,JI'J:H i or dec re a s e

ii

[', because j

i

"r

ll:t {',:.Xl' }'i:' ,;;i #;:rt"#.:Hption

i,r Ji'"., rerationshif

" pricine

"n""i

options are referred to as_either puts. or cars. A generar review of each ,lffi u-ni ; *" d in ch ap,er 1* ", -io,

:tffi:rH:: i, opti on p.i g, "i,

th

;.:;

x;3;*ti*

.r; ;#:;,[HJ ::,"]X#t :tii,f oir,"i Derivatiu;,

ar, reader uno'

;:lli:f ;i&lliT |i,:lll'

iilf

;; ;;k-schores

and

a carl option gives rhe buver rhe right to buy

'd:l'Jii*^:'Ji:rt::l's' ouncesorgora,RounilTf ._Tff as a srike price) for a specifrec p"rioo or tir. rn9'.u,i,i;;;":r* an opdon rs generallv limited to a specifiLa p".iro-."r"rred ro by;; ;;;., expirarion dare when a ca, option d,k";;;;;',11, asset, rhe calt is said price or the to be .,in_the ,"rr.l"; The differen;;;;;;;, value of rhe asser the marker r,ro. i,n* is often referred io ..inrrinsic value'" when a call"i1.1T as option strike price ir'g..ur". than the current of an asser. the varue.of ,rr. ..our_of_the_rnoney.,market value ,il il be no intrinsic varue. Ther"j".", since it has "pir",i'i. ,t"^";d;, based solely on specuration future price movement. of a The ou..rrirur* of any option is iefened to as a componenrs known as inrrinsic fl:tfr,T*JlJ,J:*"ms mav

a

:lT#:Iff X,!?:S;L:i::,?#*,::iil*; g" il ;#;;#;t

r:;;

"*''i'l*"

Typically, investor

caroptioni";;;;;'J:1JxiTil:ffJ;Jr:",TTfi

,?:,",ffi

::Tfl ffi I

316


price and make a profit, if the stock price moves above the strike price. Based on either today's prices or the most expected future prices, a patent or mineral property may have much smaller Npv value compared to the value based on higher potential future prices and or lower future production costs. This potential additional value can be captured with either of two approaches. First, a mathematical formula calted the Black scholes equation usually is used to value common stock options. Modification of the Black-Scholes model can make it applicable to valuing patents and mineral rights. Most people consider this approach to be very mathematically sophisticated, and therefore, often difficult to explain and sell to management. The second approach to capturing the call option value of a patentor mineral right type of asset is to use expected net present value (ENpv) analysis. we have seen in previous examples that ENpv can measure both uncertainty concerning the probability of failure and uncertainty concerning the magnitude of a parameter and the timing when that parameter might bi incurred.

Advocates of the option pricing analysis (real-options analysis) technique often state several perceived disadvantages associated with discounted cash flow (DCF) valuation of investments using Npv. Four of these commonly perceived disadvantages fol low:

l.

DCF analysis tends to under value investments because use of average production and product prices and costs may not capture value associated with possible higher future production, prices and productivity or lower future costs.

2. Use of today's expectation of investment future cash flow is not indicative of the true value of an investment and prohibits accounting for the embedded investment opportunities in long-lived assets.

3. Traditional DCF analysis fixes the knowledge base today for determining future cash flow over the life of a project, which prohibits accounting for large investments occurring in stages.

4. DCF analysis constrains investment timing by assuming that investments cannot be deferred, so investment decisions must be based on today's information. The authors of this text disagree with these perceived DCF analysis disadvantages and feel they represent misperceptions of the assumptions and calculations upon which proper DCF analysis should be based. If you based


317

DCF analysis on the assumptions given in the four DCF disadvantage sratements, then DCF analysis would be improper and disadvantageous. Following are our rebuttals to the perceired disudvanrages: 1. Economic analyses of all types require projecting the future. Assuming today's production rates, prices and costs will stay the same over a project life typicaliy is a very weak economic analysis assumption, You must project the actual future production rates, product prices, and costs by projecting escalation rates for the parameters realized over the project life. If different future cash flow scenarios are felt to be possible, probabilities of occurence must be projected for the different scenarios so that the cumulative probabilities equal 1.0

2. Investors dc not need to use today's dollars and probably should not use today's dollars as the basis for most DCF analyses. Nerv project and development or expansion options can be built into any year of an evaluation. There is no reason to feel that DCF anaiysis precludes you from accounting for possible future changes in production, product prices, costs, productivity and other evaluation parameters. You must project the future possibilities at the time you do the analysis for either' DCF analysis or option pricing analysis. And the future assumptions that you make will have a very significant impact on your analysis results with either DCF analysis or the option pricing methodology. 3. DCF analysis does not have to assume that project development will start today. With ENPV analysis you can build development and expansion into analyses over unlimited years, taking advantage of new technology and product pricing information available at different stages.

4. Investments can be deferred to future times vrith DCF analysis the same as with options-pricing analysis. Evaluation results with any method of analysis reflect the input data and the assumptions applied to the input data such as timing, escalation rates and tax effects in after-tax analyses. ENPV analysis can be done today for a project expected to be developed several years in the future. In Example 6-12,

the development occurs today at time zero, but development could easily be deferred several years with similar analysis procedures.

ur8


Example 6-12 ENPV Analysis to Capture Call Option Value Consider that you are interested in acquiring the rights to develop a project. To minimize calculations needed, assume the project has a four-year life with expected production, prices per unit, operating costs per unit and capital costs shown in the following table. Calculate the project NPV for minimum discount rates of 10"/" and20%.

Atl values in 000's except selling price Assuming a1O0o/" Probability of Occurrence, Most Expected!

Year

0

Production, Units Selling Price, $/Unit Gross Revenue, $ Operating Costs, $ Capital Costs,

$

BTCF

1234 100 100 100 20 22 24

20oo 2200 2400

100

26 2600

-1200 -1400 -1600 -1800 -2400

-2400 800 800 800

800

NPV @ 10% =+136 NPV @ 20/"=-329 Based on these valuations, an investor satisfied with a 10% rate of return on invested dollars could pay $136 at time zero to acquire the project rights and an investor desiring a 2O'/" rate of return on invested dollars would need to be paid $329 at year zerc to take the project rights and just gel a 20"/" ROR. Next, instead of using average or most expected values, assume there is a 6O"k probability of the scenario just described actually occurring. Further, there is a 10% probability of a worse case scenario and a3O"/" probability of a better scenario as summarized below:

Assuming a6O/o Probability of Most Expected Values

Year

0

Production, Units Selling Price, $/Unit Gross Revenue, $ Operating Costs, $ Capital Costs,

$

BTCF

NPV NPV

@ @

10% = +136 20/" = -329

1234 100 100 100 20 22 24

2000 2200 2400

100

26 2600

-1200 -1400 -1600 -1800 -2400 -2400

800

800

800

800

Chapter 6: Uncertainty and Risk Anatysis

3't9

Assuming alao/o probabirity of worst case occurrence

Year

0

Production, Units Selling Price, $/Unit Gi-oss Revenue, $ Operating Costs, g Capital Costs, -Z4OO _2400 BTCF

g

1234 100 90 20 20

90 20 1800 1800 1800 -1S00 -1800 -210O

2000 -12A0 800

90 20

300

-300

NPV @ 10o/o=-1750 NPV @ 2O/"=-1770

Assuming a3oo/o probability of Best Case Occurrence

1234 100 130 .30 16C 22 25

Year

Production, Units 190 Seliing Price, $/Unit 35 Gross Revenue, $ 220A 3250 4800 6650 Operating Costs, g -1200 -1300 -1400 _1500 Capital Costs, -Z3OO BTCF -2300 1000 1950 3400 5150 NPV @ 10"/"= +6293 NPV @ 2Oo/"= +433g

$

Combining these results with the probability of occurrence yields the project ENpV as follows: Expected Value al1O"/.

Expected

Worst Best

136(0.6)

=

Expected Value @ 20"/o 82

-1730(0.1)=-178

6293(0.3) = 1969 Expected Value = 1797

Expected -3rr(0^6)-1r?

Worst Best

Expected

-1770(0.1) = -1Zl 4309(0.3) = 1302

Value

=

g2g

The ENPV ar 1o"k indicates an investor courd pay thirteen times as much ($1797 vs $136), for the project as was det,ermined by the initial expected value analysis based on average expected values. Ar a 2o"k discount rate, the ENPV is $geg versus _$SZg with the ur"r"g" or most expected value analysis. These differences reflect the idea in *option pricing effects" which were discussed earlier. The ENpv captures this concept, but the average or most expected value analysis alone did not.


320

The '100% probability of occurrence NPV results based on average or most expected values may be best for making a "develop" versus "do nothing" decision. However,'the ENPV results capture the "option pricing value" by accounting for the 30% probability of realizing higher production and prices and lower costs. An investor may not want to base a develop decision based on 30% probability of occurrence data, but the same investor may be willing to pay a higher price to acquire and hoid the project to realize the opportunity to profit from the best case scenario if it occurs. 6.10 Summary

Sensitivity analyses are a means of identifying those critical variables that if changed, could considerably impact the profitability measure such as rate of return or net present value. Three types of sensitivity were addressed including: 1. Single Variable 2. Multi-Variable or Best Case - Woist Case 3. Probabilistic Sensitivity (Monte Carlo)

Two software models that address the probabilistic sensitivity (Monte Carlo) are known as @Risk and Crystal Ball. These template programs interact with most spreadsheets providing users with a low cost methodology to these approaches. Risk analyses identify the likelihood of project failure and the subsequent cost to the investor. In this text the subject of project risk was addressed utilizing the decision theory approach based on expected value, (EV) which is defined as:

EV = Expected Profit

-

Expected Cost

Examples then illustrated several approaches that could be used with equal validity for the level of complexity presented in the textbook examples and problems. These approaches included:

1. Chance Factor, (Expected Value)

2. Expected NPV, (Incorporated the probabilities of success and failure directly into present worth equation) 3. Risk Adjusted Cash Flows, (By applying probabilities of occurrence directly to the expected values each compounding period, one risk adjusted cash flow results which can then be manipulated in a manner consistent with examples and problems prior to Chapter 6.

L


of lr

321

these methods, the Chance Factor, Expected value methodorogy broadest range of acceptabilrty for thi more statisrically comprlx

lT[1j*

An alternative method sometimes used to measure financial or political risk involves ac1-iusting the ,,1i.scouut rite, i'i lor greater chances of farlure. \vhile common in inclustry practice, it is not a prefen-ed approach. This is due to the ditficurties that a'se whe;; comparing prdects ti:at might

be competing for limited capital. Discussion in tt. expectei rate of return soru_ tion to Exarnple 6-6 provided further amplificatio..on tt is topi, For more information on these and rerated topics, see; ,.Decision Anarysis.for?etroleum Exploration," by paul D. Newendorp, or, ,,Mineral Exploration Decisions," by Deverle p. Harris.


PROBLBMS

6-l A roulette wheel

has 38 different stopping slots numbered from I to plus 36 0 and 00. Eighteen numbers are re4 18 are black. with 0 and 00 green. calculate the expected value for the following situations where the term "payofF' means in addition to return of the bet or

"profit above all costs."

A)

The payoff is $35 for a $1 bet on any number.

B) The payoff is $17 for a $1 bet on any line between two numbers (meaning the bettor wins if either number hits).

c)

The payoff is $8 for a $1 bet on any corner between four numbers.

D) The payoff is $1 for a $1 bet on red or black or odd or even. 6-2

Due to uncertainty in development costs, the cost of a new manufacturing process is considered to have the following possibilities:

($) 5,000 8,000 10,000 14,000

Cost

Probability of Occurrence 0.10 0.30 0.40 0.20

What is the expected cost of the new process? 6-3 An electronics manufacturer is considering entering into a research and development venture. The research and development investment requires $100,000 at time 0 and, if successful, generates $g0,000 in profit each year for 5 years. Salvage value is estimated to be zero. Experience suggests that such projects have a 40vo probability of success. If the before-tax minimum rate of retunl is 2ovo, should the manufacturer undertake the project? Use expected NpV and expected ROR Analysis. 6-4

l.

If you bet $5 on the outcome of 3 football games,

you can win $25 total (not above the bet cost) if you correctly pick all 3 winners. If the odds are considered even in each game or if point spreads are specified that are considered to make each game an even odds bet, what is the expected value of your bet if you neglect tie game situations?

Chapter 6: Uncertainty and Flisk Analysis

6-5 An initial

of

investment

S50,00i1

323

is projected to generate cash flows

over a three year project life as follows: Year

Cash

Flow

Probabilit-v of Occurrence

$25,000 18,000 30,000 20,000 35,000 25,000

1

J

.40 .60 .50

.s0 .70 .30

Evaluate the expected NPV of this investment for a minimum rate

of

return of l5Va.

6-6 A research

and development manager associates a probability of suc9.6) with a research investment at time zero being successful and generating the need for an additional $300,000 developcess

of

60Vo

ment investment at the end of year one which is estimated to have a probability of 907o (0.9) of successfully generaring profits of $200,000 per year for year two through 10, assuming a washout of escalation of operating costs and sales revenue. If failure occurs after the time zero research investment a reclamation cost of $100.000 rvill be realized at the end of year one. If failure occurs after the year one investment, the salvage value will be $250,000 at the end of year two for equipment salvage. To achieve a before-tax expected ROR of 25Vo in this investment, use expected NPV Analysis to determine how much money can be spent on research at time zero assuming the year l0 salvage value is zero? What is the risk free project NPV valuation?

6-7

Calculate the expecte,j net present value of a project which will cost $70,000 at time zero, considering there is a 5OVo chance that the investment at time zero will be successful, which will require an additional investment of $120,000 at year one. There is a70Vo chance of success of the year one investment yielding profits over a six year period (years 2 to 7) equal to $125,000 per year. Failure at time zero will result in an abandonment cost of $10,000 atyeat one and failure at year one will result in a salvage value of $50,000 at year two. Should this project be considered from an economic viewpoint if the minimum rate of return

is 20Vo? Compare your expected NPV result with risk-free NpV.

324


What additional cost could be incurred at time zero and still give the investor an ENPV of 2.0 (or $2,000) at time zero? Assume the additional investment in R&D or geological or geophysical work would improve the initial probability of success.from 507o to 8-07o..reducing the initial risk of failure at year 1 to 20Vo.

6-8 A project

that would have a time zero cost of $170,000 is estimated to have a 407a probability of generating :ret income of $60,000 per year for each of years one through 10 with a zero salvage value, a 307o probability of generating net income of $50,000 per year for each of years one through 10 with a zero salvage value, a 207o probability of generating net incomes of $40,000 per year for each of years one through 10 with a zero salvage value and a lU%o probability of failing and generating a $20,000 salvage value at year one. For a minimum rate of return of 20Vo calculate the project expected NPV. What is the project expected ROR?

6-9

'

Two years ago, a petroleum company acquired the mineral rights to a property for which an offer of $1 million cash has been received now at year 0. Development of the property is projected to generate escalated dollar cash flow in millions of dollars of -1.5 in time zero, and +1.0, +1.8, +1.2, +0.8 and +0.4 in years one through five respectively. If the minimum DCFROR is 207o, should the company keep and develop the property or sell if there is considered to be a 607o probability of development generating the year one through five positive cash flow and 407a probability of failure generating zero cash flow in years one through five?

6-10 Two alternatives are being considered for the development of an investment project. Alternative 'A' would start development now with estimated development and equipment costs of $10 million at time zero and $20 million at yea-r one to generate net revenues of $6 million in year one and $12 million per year uniformly at years two through l0 with a zero salvage value. Alternative 'B' would start development ttvo years from now for estimated development and equipment costs of $15 million at year two and $30 million at year three to generate net revenues of $9 million in year three and $ 18 million per year uniformly at years four through 12 with a zero salvage value. Use NPV Analysis for a minimum ROR of 207o to determine the economically better alternative and verify the NPV results with PVR Analysis assuming:


A) rclEc probahility of

325

success is associated with all investments.

B) Alternative 'A' has a 60Vo probability of success associated with year 0 cost and 1007o probabiiity of success associated with the year 1 cost. with zero net salvage r.alue to be realized if failur-e occurs at year 0. Alternative'B' has an 807o probability of success associated rvith the year 2 cost and a l00Vo probability of success associated u,,ith the year 3 cost, with a zeto net salvage value to be realized if failure occurs at vear 3. 6-t 1 An investor has paid $100,000 for a machine that is estimated to have a 70Vo probability of successfully producing 5000 product units per year for each of the next 3 years when the machine is estimated to be obsolete with a zero salvage value. The product price is the unknown to be calculated, so it is estimated to be $X per unit in year I escalated dollars and to increase lla/o in year 2 and 6Vo in year 3. Total operating costs are estimated to be $8,000 in year I escalated dollars and to increase l5%o in year 2 and 87a in year 3. The annual inflation rate is estimated to be 7Vo. What must be the year 1, 2 and 3 escalated dollar product selling price if the investor is to receive a 127a annually compounded constant dollar expected DCFROR on invested dollars? Consider zero cash flow to be realized the 307a of the time the project fails. This assumes that equipment dismantlement costs will exactly offset any salvage value benefits. 6-12 A plant operation is scheduled to be developed for a time zero capital cost of $400 million with year I through l0 revenues of $200 million per year less operating costs of $100 million per year with a zero salvage value. Assume a washout of escalation of operating costs and revenue each year. "Washout" means any operating cost escalation is offset by the same dollar escalation of revenue (not the same percent escalation) so profit rernains uniform at the today's dollar value profit.

A)

Evaluate the project ROR and analyze the sensitivity of the result to changing project life to 5 years or l5 years.

B)

Evaluate the sensitivity of project ROR to increasing the time zero capital cost to $600 million and $800 million for the l0 year project life.

CHAPTER 7

DEPRECIATION, DEPLBTION, AMORTIZATION, AND CASH FLOW

7.1 Introduction Depreciation, depletion, and amortization are means of recovering in before-tax dollars your investment in certain types of property used in your trade or business, or held for the production of income. The basis upon which depreciation, depletion, and amortization are calculated is normilly the cost of the property, although property acquired as a gift, or in other manners, may have a basis other than cost. The cost of buildings, machinery, equipment, and trucks are examples of business property costs that may be recovered by depreciation over the useful life of the asset. Acquisition costs and lease bonus costs paid for mineral rights for natural resouries such as oil and gas, minerals, and standing timber are examples of investment property costs that may be recovered by depletion. Numerous other business costs such as the cost of acquiring a business lease, research and development expenses, trademark expenses, and pollution control equipment costs may be recovered by amortization. Depreciation, depletion, and amortization all achieve essentially the same thing, that is, recovery of the cost or other basis of investments in before-tax dollars through allowable tax deductions over a specified period of time or over the useful life of the investment. If depreciable property is sold, all or a portion of any extra depreciation claimed in prior years may have to be recaptured as taxable income. This concept is addressed in Chapter 8. Figure 7-1 shows that the only difference between before-tax cash flow and after-tax cash flow is state and federal income tax. To calculate income tax, you must properly calculate taxable income. This requires reducing rev326

Chapter 7: Depreciation, Depletion, Amoi'iization, and Cash Flow

BEFORE and AtrTER-TAX CASH FLOW CALCULATION:

327

EXPANDED AFTER.TAX CASH FLOW CALCULATION:.

Revenue

Revenue

-

- Operating Costs - Dc'nreciation :-' :-:'-""" III uc_nietton

-

Operating Costs Cal,.ita_l Costs

ijefore-Tax Cash Fiow (BTCF)

-

Income Tax

After-'fax Cash Flow (ATCF)

-

;

_

ffi:"fi?:'"" J

_:

- -i" .

)

non-cash

capital cosr aeducrions

Taxable Income

-

Income Tax (Fed. & State)

Net Income + Depreciation + Depletion + Amortization + Write-offs - Capital Expenditures

After-Tax Cash Flow (ATCF)

Figure 7-1 Before-Tax and After-Tax Cash Flow enue by both operating expenses and "non-cash deductions"

for depreciation, depletion, amortization and write-offs, as appropriate. "Non-cash deductions" get their name because they do not represent physical cash costs. These depreciation. depletion. amortization and write-offs are permitted by tax law to enable investors to recover their out-of-pocket cash "capital costs" in before-tax dollars over the deduction period of tirne prescribed in the tax law. The term "capital cost" means the cost is deducted for tax purposes over more than one year by depreciation, depletion, amortization (D, D & A) or write-off deductions rather than being "expensed" or deducted from revenue in the year the cost is incurred. Tax laws of countries dictate costs that can be expensed v.rsus those that must be capitalized and d:ducted over a specified number of years by D, D & A or write-off deciuctions. Chapter 7 covers the details of utusiness costs that may be "expensed" for tax deduction purposes (Section 7.3,7 .6, and 7 .9) and costs that must be capitalized and depreciated (Section 7.4 and 7.5). depleted (Section 7.7) or amortized (Section 7.8). Write-off of costs not previously deducted is addressed in Example 7-3b and Chapter 8, Sections 8.5, 8.9 and 8.10. Write-offs generally relate to deductions for the remaining undepreciated, unamortized or undepleted value of capital costs or to the original costs for land or working capital costs which generally are not deductible

328


by depreciation, depletion or amortization under the tax laws of any western

world country. Tax informaiion in the following sections of this chapter and Chapter 8 is based on U.S. Internal Revenue Service (IRS) Publication 34, the "Tax Guide for Small Business" and IRS Publication 17, "Tax Guide for Individuals." These are free publications available by calling or writing any local U.S. Internal Revenue Service office. 7.2 After-Thx Cash Flow in Equation Form

Figure 7-1 illustrates two alternative ways of calculating after-tax cash flow. In equation form, these after-tax cash flow calculations may tle expressed as follows in accordance with the right column of Figure 7-1:

After-Tax Cash Flow = Net Income + Non-Cash Deductions - Capital Costs = Net Income + Depreciation + Depletion + Amortization + Write-offs - Capital Costs

7-r

Alternatively, in accordance with the left column of Figure 7-1, the same after-tax cash flow results may be obtained from calculating revenue minus lll out-of-pocket expenditures. After-Tax Cash Flow = Sales Revenue - Operating Costs - Income Taxes - Capital Costs

1a l-L

flow represents revenue or savings minus all cosfs, no matter w,hat trocedrn'e you ase to calcttlate it. Therefore, for any operating period, after.ax cash flow is the after-tax lrloney that is available to an investor (individral or corpol'ate) to invest in new projects, pay out in dividends, pay off old lebt, or retain for future investment purposes. It is imperative that after-tax :ash flow calculated as described here be the basis for the correct after-tax Cash

rnalysis of all types of investments.

7-1 Calculating Cash Flow =XAMPLE An ongoing project is expected to generate revenues of $100,000 text year with operating costs of $30,000 and income taxes totaling

Chapter 7: Depreciation, Depletion, Amortrzation, and Cash Flow

329

$24,000 for the same period. Further, capital costs totaling $29,000 for equipment are also expected to be incurred during the year. Cal. ct.:late the anticipated project cash flow for next year.

Solution: (Using Equation 7-2) cash Flow = $100,000 - $30,000 - $24,000 - 920,000 = +$26,000 Of these four values (revenues, operating costs, capital costs and income taxes) the only value that is different from determining before-tax cash flow is the income tax. The remainder of this chapter and Chapter 8 will examine deductions from revenue, both *out-ofpocket" and "non-cash" deductions including depreciation, depletion, amortization, loss carry forwards and write-offs of book values that

allow us to determine our taxable income and the appropriate income tax to be paid or tax savings to be realized. obviousry, fairure

to recognize the tax component in Example 7-1 would have a very significant impact on the value of the anticipated cash flow and resulting economic analysis calculations.

EXAMPLE 7-2 Betore-Tax and After-Tax Rate of Return Analysis of a Bank Account

An investor deposits $10,000 in a bank account certificate of deposit and will receive 12"/" inlerest at the end of each year for the next three years and will receive maturity value of 910,000 at the end of year three to recover the initial investment. Assume the investor is in the 40% income tax bracket and calculate before-tax ROR and after-tax ROR on the investment. Solution: Before-Tax Time Diagram, C = Cost, I = lncome, L = Salvage

C=$10,000

l=$1

,200

l=91 ,200

l=$1,200

L=910,000

Before-tax ROR is 12.0% There are no allowable tax deductions on bank account, bond or debenture type investments. Bank account deposit investments are neither depreciable, amortizable nor depletable. The deposit cost is

330


deductible against the maturity or salvage value which results in zero gain or loss on the salvage value in this analysis: lnterest income each year is fully taxable whether withdrawn from the account:or not. 40% ol the interest income of $1,200 per year goes to taxes, so $720 = ($1 ,200X"1-.40 tax rate) is the after-tax cash flow from interest each year. The year three maturity value of $10,000 just recovers the initial cost so it is cash flow, that is, none of it is paid out in income tax.

CF=-$10,000

6P=$720

CF=$720

CF=$720 CF= $10'000 J^

Therefore, using Equation 3-2, since initial cost (or negative cash flow) equals {inal salvage (or positive cash flow) the aftertax ROR is 7.24/o cotTrpared to the before-tax ROR o' 12h; a significant difference. lt is important to account for income tax considerations.

If you want to compare the economics of leaving your money in a bank investing in real estate, minerals or petroleum, it makes a difference 'o whether you comparc before-tax or after-tax results. Investments in general rnvolve costs that are either depreciable, amortizable, depletable, expensed, rr not deductible at all except against liquidation or salvage value, as in the :ase of land or bank account deposits. This makes the relationship between

refore-tax and after-tax economic analysis results uniquely different for :ach investment situation analyzed. This necessitates comparing the ecoromics of investments on an after-tax basis to properly and fairly evaluate he economic potential of different investments from the viewpoint of taxraying individuals or organizations.

/.3

Business Costs That May be Expensed

For tax purposes, the fastest method of deducting costs is to "expense" or leduct them in full in the year incurred. Investors would prefer to treat all :osts in this manner because the faster you get tax deductions, the faster you ;et the tax benefits from your deductions, and this improves project ecotomics. However, tax law does not permit "expensing" all costs so we need

o enurnerate some of the important costs that can be "expensed"

as

rpposed to those costs that must be "capitalized" and deducted for tax pur)oses over a period of time greater than a year.

Chapter 7: Depreciation, Depletion, Amortizatjon. and Cash Flow

331

operating cosrs that lnay be expensed include costs for direct labor, i,tiirect labor, materials, parts and supplies used for product produced

and cosrs associated rvith product actually .old nroy be deducted. This introduces the reader to the subject of working capital .rri.n involves drrllus tied up in prociuct inventory, spare parts inventory, accounts receivable, required cash on hancr, etc., that are not deductible for tax purposes until such items are actually used up or sold, as in the case of produci and spare parts inventory. A more complete discussion of working

sc'ld.

only

capital and the accounting methods that affect its value is presented in chapter g. Some other common costs in the operating expense category include utilities, freight and containers, borrowed money interest paiJ, royatties, sever-

ance taxes, sales taxes, ad valorem taxes and certain ixcise taxes. The cal_ culation and determination of some of these later items is expandea upon in Section 7.9,

Research and Experimental Cosls including labor, supplies, etc., are considered to be the equivalent of operating costs and *ryb. expensed in the year incurred. Alternatively, an investor may elect to amortize these costs straight line over five years, but this election is seldom made because slower tax deducrions give slorver tax savings and less desirable project

economics.

Mining Exploratio, coits are expenditures required to delineate the extent and quality of an ore body and may incrud" io." drilling, assaying, engineering fees, geological fees, exploratory shafts, pits, drifts,

etc. Expro-

ration costs ma!- be either capitalizecl into the cost iepletion basi.s (as dis_ cr,tssed later in this chapter') or expensed in the amouttt in the year fuli incurred by individual taxpayers. However, with the exception of subchapter s corporatiotts, corpor,tions may expense onry 70vo of ntining rction costs irt the y-ear incurred with the remaining 30vo cleducted "rpiou'sing straight line amortization ot,er a 60 montlt period, with the first year deduction proportional to the month such costs are paid or incurred. Ifexploration expenditures associated with successful ventures have been deducted by the expense option, these deductions are subject to recapture as follows: The taxpayer may elect to forego taking depletion deductions until the cost basis of exploration charges is fuily recovered, or the taxpayer may restore a dollar amount equal to the previously expensed exproration charges as income. If the rater option is elected, ti," tu*puy"r may add the

332


additional income alnount to the cost depletion basis for the property. Recapture may be avoided by not expensing exploration charges but instead, adding such charges to the cost depletion basis ofthe property.

hlining Developmen costs are defined as expenditures incurred after the determination has been made that an ore body is economically viable and the decision has been made to develop the property. Development costs mav include exploration type costs after the decision has been made to develop a

mine. Mining development costs typically include costs for overburden stripping, underground shafts, drifts, tunnels, raises, adits, etc. Mining development costs are not subject to recapture at any point in time and may be expensed in the full amount in the year incurred by individual taxpayers and, analogous to mining exploration costs, corporations (except Subchapter s corporations) may expense only 70vo of mining development cosls in the year incurred with the remaining 30vo deducted straight line over a five )'ear, 60 month period, with the first year deduction prorated from the montlt such costs are paid or incurred. As an alternative to expensing all or 707o of mining development cost, tax law gives investors the option of capitalizing these costs and deducting them by units of production depreciation over the producing life of the mine. Development expenditures end when a mine reaches a level of full production. Then, costs that previously were mine development costs are treated as operating expenses from that time forward. Petroleum Intangible Drilling Costs (IDC's) are defined as the cost of drilling oil and gas wells to the point of completion and may include: Costs of agreements with operators and drilling contractors. Survey and seismic work related to location of a well. Road cost to well location to be used during drilling. Dirt work on location for pits, etc. Rig transportation and set-up costs. Drilling costs including fuel, water, drilling mud, etc.

Cost of technical services including engineers, geologists, logging and

drill

stem test services.

Cost of swabbing, fracturing and acidizing. Cementing of surface casing and main casing (not the cost of casing).

Reclamation of well site.

Chapter 7: Depreciation, Depletion, Amortization. and Cash Flow

333

sinrilar to mining development costs, intangible drilling costs may either be capitalized into the cost depletion basis (as discussed later in this chapter) or expensed infutt amount in the year incurred by individuals or corp-orations that are not "integrated" producers. (An ,,integrated,,

petroleum

pn-tdtrcer refnes nlore than 50,000 barrels of crude oil per day

for

average

dcily production over a year, or has retail sales of oil and gas products exceeding $5,000,000 per year) "Integrated" petroleum producers may onlv- expense 70vc of intangible dritling costs in the year incurred and are reqttired to amortize the remaining 30vo of their intangible drilling costs straight line over a five year or 60 month period beginning in the month the costs are paid or started to be incurred. This provision does not affect the option to expense dry hole intangible drilling costs in the year the dry hole is incurred by both integrated and non-integrated producers. Non-integrated producers often are rcferred to as independent producers.

7.4 Depreciation The term depreciation is used in a number of different contexts. some the most common are:

of

1. Tax deduction or allowance 2. Cost of an operation 3. A method of funding financing for a plant replacement 4. Measure of falling value

In the first context, annual taxable income is reduced by an annual depreciation deduction or allowance that reduces the annual amount of income tax payable. The annual depreciation charge is merely a paper or ,.book,' transaction and does not involve any expenditure of cash. In the second context, depreciation is considered to be a manufacturing cost in the same way as labor or raw materials are out of pocket cash costs. This is a common application of depreciation for internal company cost accounting purposes. In the third context, depreciation is considered as a means or providing for plant replacement. However, in rapidly changing modern industries, it is doubtful that many plants will ever be replaced because the processes are likely to have become obsolete during operation, so this approach is outdated. In the fourth context, a plant or a piece of equipment miy have a limited useful life, so deducting cumulative depreciation from initial value gives a measure of the asset's falling value, usually called book value or adjusted basis.

334

Eeonomic Evaluation and lnvestment Decision Methocls

ln this text, the term clepreciation is usual\t used in the context of a tax allowance. Depreciation is a tax deduction comprising, ,,a reasonable allowance for the exhaustion, wear and tear and obsores[nce of property used in a trade or business, or ofproperty held by a tax payer for the production of income." Land, however, may not be depreciatei. ttre basis or value that would be used to find investment gain oi loss if property were disposed of is the basis that shourd be used fir determini,g'oepr"ciation. The first step in computing depreciation is to determine the estimated useful life of the asset or its allowabre depreciable life. No average useful life is applicable to ar depreciable assets in different types of businesses and the allowable depreciabre rife is often different irom the expected economic life. The question of

whether certain expenditures are repair expenses (deductible in the year incurred as an operating expense) or a capital expen_ diture deductible through annual depreciation deductions often is important in relation to the tax deduction tranating of equipment and major.facility repair/rebuild type costs. The distinction between repairs and capital expen_ ditures is not always crear but here is a usefur guide: A repair is an expendi_ ture for the purpose of-keeping property in an ordinary efficient operating condition. It does not add to the varue o. ur" of the proplfty. It merery keeps the property in an operating condition over its probable useful life for the uses for which it was acquired. Repair costs may be expensed as operating costs for tax deduction purposes. on the other hand, capital expenditures are alterations, additions or improvements that increase the asset useful life or vaiue or make it adaptabre for a different use. Capital expenditures must rarher rhan expensed using the depreciation life of an equiva_

i^"-*f:.:::ed rent new asset.

Depreciable property generalry is tangible while amortizabre property generally is intangibre. Tangible property genera,y is any physical properry that can be seen or touched. tntarglUtl property generally is paper_type assets such as a copyright, patent or franchise. o"prJ"iuut" p.op".ty may be personal 0r real' Personal property is property such as machinery or equipment that is not real estate. Rear property is land and generally anything that is erected on, growing on, or attached to land such as buildings. However, land itself is never depreciabre, so rand is non-depreciable ieal property whereas buildings are depreciable real properry.

Chapter 7: Depreciation, Depletion, Amortization, and Cash Flow

Property is depreciable

335

if it meets these requirements:

1) It must be used in business or herd for the production of income.

2)

It must hare a determinable life Year.

and that life must be longer rhan one

3.)

Ir must be something that wears out, decays, gets used up, becomes

J)

It is placed in service or is in a condition or state of readiness

obsolete, or loses value from natural causes.

available to be piaced in service.

In general, depreciable.

if property

does not meet all four of these conditions,

and

it is not

Finally, the method of financing the purchase of assets has no effect on depreciation deductions. whether you puy cash or borrow all the money to acquire an asset, you get the same depreciation deductions. Interest on bor_ rowed money is the only tax deduction that is difi'erent with project analyses involving borrowed money instead of cash investments as is discussed in Chapter 11.

7.5 Depreciationl{ethods Depreciation of tangible properry placed in service after l9g6 is based on using Modified Accelerared cost Recovery System (MACRS) depreciation for: (1) the applicable depreciation methoa, (z) ttre appticable recovery period (depreciarion life), and (3) the applicable first y.ui^d.pr".iation con_

vention. MACRS depreciation calculations relate to the following three

depreciation methods:

1) Straight Line 2) Declining Balance 3) Declining Balance Switching to Straight Line

A fourth method used to a lesser extent for tax deduction purposes but to

a greater extent

for pubric company sharehorder reporting purposes is: 4) Units of Production

The following secrions 7.5a through 7.5e ilrustrate the application of

these different merhods. The MACRS depreciation method is addressed in section 7.5e. Prior to introducing depreciation methods, several timing conventions that affect the first year depreciation deduction need to be

addressed.

336


Under the curent u.S. tax law, there are three applicable conventions that have an effect on the ailowable depreciation deducti_on in the first ye3r. These conventions apply io itre MACRS merhod, the'straight iinu'lrrrA'C(S method (which is straight line depreciation for the recovery period allowed by N{ACRS depreciation) and the arrernative ACRS method iwhich applies longer depreciation lives for straight line depreciation of special categories of assets such as foreign tangible property or tax exempt use or bond financed property). The recovery period begins when an asset is placed in

service under the applicable first year deduction convention. The terms

recovery period and recovery property are tax terms meaning depreciation period and depreciable property. Personal Property First Year Depreciation Conyentions 1. Half-Year Convention

in

First year

Under the half-year convention, applicable to personal property which is recovery property other than residential rental and non-residential real prop_ erty, all property is deemed to be placed in service in the middle of the year. Therefore, one half of the first year normal depreciation is allowed in the year that the property is placed in service, regardless of when the property is placed in service during the year. In industry practice, the half-year convention usually is assumed to be applicable in evaluations involving depreciable personal property so this convention is emphasized in text examples and

problems.

2. Mid-Quarter Convention

in

First year

If more than 40vo of annual depreciable property costs in a given depreciation life (recovery period) category are incurred in the last 3 months of a tax year, the taxpayer must use the mid-quarter depreciation convention. under the mid-quarter convention, ail property placed in service during any quarter of a tax year is treated as placed in service at the quarter's midpoini. This has the effect of increasing your depreciation deduition for property placed in service in the first two quarters of the tax year and deireasing your deduction for property placed in service in the last two quarters. Instead of deducting 50vo of a full year's depreciation for all p.op".ty placed in service during the year, the mid-quarter convention gives yo, un 87 -5vo deduction for property placed in service in the first qu-arter (which equals 10.5 mo-ll2mo. x 100), 62.5vo deduction for second quarrer property, 37 -5vo for third quarter property and l2.5vo for fourth quarter property.

Chapter 7: Depreciation, Depletion, Amortization, and Cash Flcw

Real Property First Year Depreciation Convention

Mid-Month Convention For residential and non-residentiar real property the mid-month first y,ear convention ap-plies. Under this convention, qualifying property is deemed to be placed in service during the middle of the monrh. irre attowaule deduc_ tion is based on the number of months the property was in service. Therefore, for a calendar tax year, assets placed in service in January of that year would receive a first year deduction equal ro the fraction it-sttztimes the calculated year one depreciation. Thus, one half month's cost recovery (depreciation) is allowed for the month the property is placed in service with full month deductions in subsequent months.

7.5a Straight Line Depreciation Straight line depreciation is the simplest method of computing depreciation in all countries but, unfortunately, it also is the slowest method of

depreciation. The faster you get depreciation tax deductions, the faster you get the tax benefits from the deductions if income exists against which to use the deductions, and the better the economics of any project will be. Therefore, straight line depreciation is generally not desirable for tax deduc-

tion purposes if I'aster methods are allowable.

with the straight line merhod, depreciation per year is determined by multiplying the cost basis of a property times a straight line depreciation rate which is one divided by the allowable depreciation life, .,n,, years. In

equation form:

Straight Line Depreciation per year =

(Cost)(l/n)

l_3

EXAMPLE 7-3 lrlustration of straight Line Depreciation Assume you purchase a new machine for g1o,0o0 in January of a tax year that corresponds to a calendar year. Assume the asset is placed into service in August of the same tax year. The estimated life of the machine is eight years when salvage value is estimated to be $3,000. Determine the annual allowable depreciation deductions by the straight line method assuming the machine is in the five year depreciation life category and that the half-year convention is applicable in the first year.

338

Economic Evaluation and lnvestment Decision Methocts

Solution, All Vatues in Dollars: The actual estimated asset use rife and salvage value have no effect on depreciation carcurations under current tax taw

Depreciation

Yearsl Years 2 to Year

6

:($10,000)(1 t1)(1t2)

5 : (910,000X1/5) : (910,000)(1/s)(1t2)

Cumulative Depreciation

=

= :

$1,ooo 92,000 $1,ooo

=

910,000

Note that because of the harf-year one convention, it takes six years to fully depreciate the asset. The six annual depreciation deductions are related to cash frow carculations in the next example.

EXAMPLE 7-3a straigh.t Line Depreciation Rerated to cash Frow Assume the 910,000 cost asset in Exampre 7-3 is incurred in evaruation time zero and generates annual incomes ano operating of $8,000 and $5,000_respectivery in years one tnrough costs eight, assuming a wash-out of annual operating cost and income escalation. Year eight sarvage varue is projected to be zero. since the asset goes into service closer to evaluation year one than zero, start depreciation in year one. Use an effective income tax rate of 40./" and calculate the after-tax investment DCFROR.

Solution, All Values in Doilars: Year

Revenue -Operating Costs -Depreciation -

Taxable -Tax @40"/"

Net lncome +Depreciation -Capital

Cost

Cash Flow

-

2-5

7-8

g,OO0 g,0OO g,000 B,O0O _5,000 _5,000 _5,000 _5,OOO

-1,000

_2,OOO _f

,OOO

2,000 1,000 2,000

3,000

-800 _400 _800 _1,200 1,200 600 1,200 1,g00

+'1,000 +2,000 +1,000 -10,000

-10,000 +2,200 +2,600 +2,200 +1,g00

Chapter 7: Depreciation, Depletion; Amortization, and Cash Flow

339

PW Eq:0 = -10,000 +2,200(PlF;,1)+ 2,600(P/A1,a)(P/F;,1) + 2,200(P lF i,6) + t,800(P/A;,2)(P/F;,6) By trial and error, i = DCFROR = 16.82/"

Comparing this after-tax DCFROR result to the corresponding beiore-tax ROR gives: Before-Tax PW Eq: 0 = -10,000 + 3,000(P/A;,g) By trial and error, before-tax ROR = 24.95%.

The before-tax ROR is about 50% bigger than the after-tax DCFROR indicating the significance of tax considerations in a proper analysis. Note that after-tax cash flow each year is the $3,000 annual beforetax cash flow of $8,000 revenue minus $5,000 operating cost reduced by income tax. This always gives the same cash flow as net income plus depreciation minus capital costs. Capital costs such as land and working capital are not depreciable under the tax law cf the United States and virtually all other countries. Working capital investments typically represent money tied up in raw material and product sale accounts receil'able. Chapter 8, Section 8.10 gives a more detailed explanation of working capital. Land and working capital investments are deducted for tax pulposes as lump sum deductioiis equal to the original investment cost in the year of sale or disposal of the assets. These deductions commonly are called "write-offs." Sale value, if any, from liquidation of land or working capital assets is treated as income in the year that a write-off is taken. Example 7-3b illustrates these considerations as a variation of Example 7-3a.

EXAMPLE 7-3b Non-Depreciable Cost Variation of ExampleT-3a Re-work Example 7-3a with an additional time zero non-depreciable cost of $2,000 for surface land (or working capital to cover inventory and accounts receivable costs). Land and working capital costs are not depreciable but are deducted by lump sum deductions called "write-offs" against final sale (liquidation) value which is assumed to be $3,000 at the end of year 8. Assume the final sale value is treated as ordinary income for income tax purposes.

340


Solution: All Values in Dollars Revenue - Op. Costs - Depreciation - Write-off Taxable - Tax @ 40/"

",7

2-5

0

Year

8,000 9,000

9,000

8,000

-5,000 -5,000 -5,000

-u,30

-,,10 -r,30 -,,30 2,000 1,000

-800 -400 1,200 600

11,000*

-5,000

-2,000** 4,000 -800 -1,200 -1,600 1,200 1,900 2,400

2,000

3,000

Net lncome + Depreciation +2,000 +1,000 + Write-off +2,000 Cap. Costs -12, 000 Cash -12,000 +2,200 +2,600 +2,200 +1,800 +4,400 "lncludes sale value of 3,000 and revenue of 8,000 **Land working or capital time zero cost DCFROR = 13.62"/" after-tax Before-Tax ROR = 20.54%

Flow

Note that money tied up in land and working capital has a negative effect on both the before-tax and after-tax economics of this project when compared to Example 7-3a with no land or working capital investment.

7.5b Declining Balance Depreciation

Declining balance depreciation applies a depreciation rate from the straight line rate of 1/n to 2/t't to a declining balance each year. Many gor,ernments specify applicable declining balance rates for different assets such as .1, .2 or.3, to be multiplied by the aJjusted cost basis. Others specify the depreciation lives and type of declining balance depreciation, such as 1507o or 2007c declining balance rates now applicable under U.S. tax law, and require the investors to calculate their own rates. For example with 150% declining balance, 1.5/n, (where n is the depreciation life), is the depreciation rate. 20A7a declining balance is often referred to as double declining

balance since the rate is twice the straighrline rate or, 2ln. The declining balance rate is applied to a "declining balance" or "adjusted basis" each year as shown in Equation 7-4.


341

Declining Balance Depreciation per year

-

(Declining Balance RateXAdjusted Basis)

7-4

u,here. for any depreciation method:

Adjusted Basis = Cost or Other Basis - Cumulative Depreciation previously

Taken

7_5

The term "book value" or "tax book value" often is used interchangeably with "adjusted basis." The terms "diminishing balance,,and ,,written down value" have interchangeable meaning with "adjusted basis,, in canada and Australia. All these terms represent the remaining undepreciated value of a depreciable asset.

EXAMPLE 7-4 lllustration of Decrining Balance Depreciation Assume the $10,000 cost in Example 7-3 is to be depreciated using 200"/" declining balance for a five year depreciation life and that the half-year convention is applicable in the first year. Determine the annual depreciation deductions.

Solution: All Values in Dollars The 20a% declining balance five year rife rate is 2/s, or 0.40.

Year 200% D.B. Rate, (2/5) x Adjusted Basis =

1 2 3 4 5

6

.40 x 1/2 .40 .40 .40 .40

10,000 8,000 4,800 2,880 1,729 1,037

Cumulative Depreciation After 5 years

200"/o D.B.

Depreciation

2,000 3,200 1,920 1,152 691

$8,963 Note that the adjusted basis is not fully depreciated in five years. _ This is a problem with declining balance d'epreciation. lt literally takes infinite years to fully depreciate a given adjusted basis with declining balance depreciation. Switching from declining balance to straight line depreciation enables an investor to fully depreciate an asset in the depreciation life plus one year as shown in the next section.

342


7.5c srvitching from Declining

Balance to straight Line Depreciation

In the u.s., all depreciation rates,for the MACRS depreciation of personal property are based on either l1avo or 200% declining balance switching to straight line. 1r is desirable to srl)itch to straight line from declining balance in the year when you will get an equal or bigger deduction by switching than if you do not switch. This occurs when the straight line rate equals or exceeds the declining balance rate because when you switch the remaining adiusted basis is depreciated straight line over the remaining years of depreciation life. EXAMPLE 7-5 Declining Balance Switching to Straight Line Depreciation

Assume the $10,000 asset described in Example Z-4 is to be depreciated using 200% declining balance switching to straight line for a 5 year depreciation life, use the"same harf-year convention in the first year of depreciation. calcuraie the annual depreciation to depreciate the asset as rapidly as possible

Solution, All Values in Dollars: ln switching to straight line from declining balance depreciation, when you switch methods, the remaining adjusted basis is depreciated straight line over the remaining years of depreciation life. ln this analysis it is desirable to switch in year four when two and a half

depreciation years remain. The straight line rate of 112.5 (or 40%) for switching in year four equals the declining balance rate of 4o/o, so switching in year four is economically desirable (switching in year three, the straight line rate would be 1/3.5 which is less than the 20O% DB rate ot 4O%).

Year

1 2 3 4 5 6

Method

Rate

2OO% DB to St Line

x Adjusted Basis

2OO%D.8..4000(1t2) 200"/" 200% St. St. St.

D.B. D.B.

.4000 .4000 .4000 .4000

Line Line Line .4OOO(1/2)

$10,000 8,000 4,900 2,890 2,990 2,990

Cumulative Depreciation After 5 years

=

Depreciation $2,000 3,200 1,920 1,152 1,152 576

$10,000


343

The straight line rate of 0.4000 equals 112.5.

Note that the straight line depreciation rate is applied in each of years four, five and,six to the adjusted basis at the time of switching in year four and not to the new acjusted basis each year. since a half-year deduction is taken in year one, tne final haliyear deduction occurs in year six. ln generai, with 200% declining balance switching to straight line, you always switch methods in the year after the mid-year of the depreciation life.

7.5d Units of Production Depreciation Units of production depreciation deducts the asset cost over the estimated producing life of the asset (instead of over a given depreciation life) by taking annual depreciation deductions equar. to the prodict of the ,,asset cost,,, or other basis, times the ratio of the "units produced" in a depreciation yea4 divided by " expected asset lifetime units of production,,, such as initial mineral reserves. This method permits the asset to be depreciated for tax purposes in direct proportion to asset use. Units of production depreciation is allowed as an alternative to expensing mine development costs under u.S. tax law. Also, units of production depreciation ii used by mining and petroieum public companies for calculating financial net income and cash florv for shareholder reporting pulposes.

EXAMPLE 7-6 Units of production Depreciation Assume that the 910,0oo cost to be depreciated in Example 7-3 is estimated to produce 50,000 product units over its useful life. lf 14,000 product units are expected to be produced in year one, and 12,000 in year two, calculate the year one and two units of production depreciation.

Solution: Units of Production Year

, 2

$r0,000 x (14,000/50,000) = $10,000 x (12,000/5O,O0O) =

$2,800 $2,400

344

7.5e lVlodified Accelerated


Cost Recovery System (MACRS) Depreciation

The cost of most tangible depreciable properfy is recovered for tax purposes using the modified accelerated cost recovery system methods. Cost and recovery methods are treated the same whether property is new or useci. Salvage value is neglected in computing the appropriate depreciation deduction.

MACRS depreciation methods for personal property include 200Vo declining balance switching to straight line, and 150Va declining balance switching to straight line. These rates are sometimes called "accelerated" depreciation rates because they give deductions faster than with straight line depreciation. Alternatively, you may irrevocably elect to use straight line depreciation over the regular depreciable life. The straight line method of depreciation is required for residential rental real property or non-residential real property purchased after 1986. Straight line depreciation often is called "straight line MACRS depreciation." MACRS depreciable property is often called recovery property. All recovery property is depreciatbd over one ofthe following lives: 3 years,5 years, 7 years, 10 years, 15 years, 20 years, 27.5 years or 39 years. Depreciation lives or recovery periods are determined based on Asset Depreciation Range (ADR) mid-point class lives that were in effect prior to the introduction of ACRS depreciation rates in 1981 (see Table 7-1). Recovery periods follow for some typical depreciable assets: 3 year property includes property with an ADR class life of four years or less. Under the ADR system, property with a midpoint life of four years or Iess includes: special tools and handling devices for manufacture of products such as food, beverages, rubber products, finished plastic products and labricated metal products.

5 year property inclLrdes property with an ADR class life greater than four years and less than 10 years. Cars and light general vehicles are in this class which also includes chemical plant assets, research and experimentation equipment, qualified technological equipment, bio-mass properties that are small power production facilities. semi-conductor manufacturing equipment, and heavy, general purpose trucks including ore haulage trucks for use over the road.

year property includes property with an ADR class life of l0 years but less than l6 years. This class includes office furniture, fixtures, equipment such as mining machinery and oil and gas producing equipment and gather7


345

ing pipeiines, railroad track and equipment used to manufacture certain rubber products, metar products, steel mill and automotive products, food a.d beverage products, etc. Typical oir and gas equipment thar wourd qualify for this depreciation class include: Surface and vrell casing, tubing (including transportation) \\'cllhead (christmas tree), pumping system Do.;rnhole equipment including guicle shoes, centralizers, etc.

Salt water disposal equipment. Tank battery and separators including site preparation, and operation roads. Installation of flow lines and other equipment.

.

Typical mining equipment that wourd qualify for this depreciation class

includes:

Loaders. dozers, scrapers, drils, large compressors, haul trucks, conveyors and other equipment for quarrying metallic and non-metallic minerals (including sand, gravel, stone and clay) and the milling, beneficiation and other primary preparation for such material. 10 year propertj' includes properry r.vith an ADR class life of 16 years but less than 20 years. This class includes equipment assets related to petroreum

refining, nranufacture

of tobacco produits, grain mill prociucts, sugar and vegetable oil products, and synthetic ,aturar gas from coar gasification. 15 year property incrudes properry with an ADR class rif-e of 20 years but less than 25 years. This includes waste-water treatment plants, telephone distribution plants and comparable equipment, liquefied natural gas prants, gas utility trunk pipelines and related storage facifities, commercial contract petroleum and gas pipelines and conveying systems,

20 year property includes property with and ADR crass life of 25 years or more, excluding real property with an ADR midpoint of 27.5 years or more. Municipal sewers, cable and transmission lines, electric utility steam power plants and water utility plants are included in this class. 27.5 year residential rental real property includes buildings or structures with respect to which 80 percent or more of the gross rentar iicome is rental income from residential dwelling units. If any portion of the building or structure is occupied b_y the taxpayer, the gross rental income from tne pioperty includes the rental value of the unit oicupied by the taxpayer.

346


propefi is real property that is not (l) residential rental.property,.or.(2) property with a class.life of less ,than 27 .5 years. This class includes property that either has no ADR class life or whose class is 27.5 years or more. 39 year non-residential real

Table 7-1 identifies the depreciation method, ADR class lives and recovery period (depreciation life) for each class ofrecovery property.

Thble 7-1 Personal Property Depreciation Lives and Methods Recovery

I

Mid-Point ADR

Period Dqpqgglq{ion Method* Class Lives*x 3 Yr 200Vo DB Depreciation Switching to St. Line 4 yrs or less 5 Yr 200Vo DB Depreciation Switching to St. Line 4.5 to 9.5 yrs 7 Yr 200Vo DB Depreciation Switching ro St. Line 10 to 15.5 yrs 10 Yr 200Vo DB Depreciation Switching to Sr. Line 16 to 19.5 yrs 15 Yr l50Vo DB Depreciation Switching to St. Line 20 to 24.5 yrs 20 Yr l50Vo DB Depreciation Switching to St. Line 25 or more yrs

Depreciation deductions can also be computed using the straight line method over the applicable recovery period. ' ' Depreciation life (recovery period) is determined by the mid-point life of Asset Depreciation Range (ADR) lives for assets. The ADR lives applicable are for the ADR Class Life Depreciation System in effect before 1981. An investor must use mid-point ADR class lives for Alternative Ivlinimum Tax depreciation. Those who elect to use the'Alternative Depreciation System" (ADS) which is also called 'Alternative MACRS Depreciation" must use straight line depreciation over the mid-point ADR lives. Thble 7-2 Real Property Depreciation Lives and Methods Recovery

Period 27.5 Yr 39 Yr

Depreciation Method

ADR Class Life

Straight Line Depreciation Straight Line Depreciation

As an alternative to computirrg the depreciation deductions each year for each class of depreciable personal property, the following Thble 7-3 rates taken times depreciable cost give annual depreciation for three, five, seven and ten year life qualifying depreciable property that would qualify for the

347


half-year convention in the first year. The rates ir] Table 7-3 are not vtlid for personal property subject to the micl-quarter convention or lbl rcal property. Under those cirbumstances, the reader will be required to make his or her own depreciation calculations or refer to yeal one mid-quarter convention rate tables published by the IRS and various accountiirg house companies. Table 7-3 Modified ACRS Depreciation Rates* Recovery Period is: 3

Year

Year I 2 J

4 5

6 7 8

9 10

l1

t2 13

t4 15

16

5

Year

7

Yeat

10

Year

15

The MACRS Depreciation Rate

33.33 44.45 14.81 7.41

Year

20Year

7o is:

20.00 t4.29 10.00 5.00 32.00 24.49 18.00 9.50 19.20 t7 .49 14.40 8.55 rr.52 12.19 11.52 7.70 6.93 9 .22 .52 8.9 3 7.37 6.23 5.76 8.92 6.55 5.90 8.93 5.90 4.46 6.-s5 5.91 6.s6 5.90 6.55 5.91 3.28 s.90 5.91 5.90 5.91 2.95 11

3.750 7.219 6.677 6-177 5 .7

13

5.285 4.888 4.522 4.462

4.46t 4.462

4.46r 4.462

4.461 4.462

4.461

t7

4.462

18

4.461

t9

4.462

20

4.461

2.23t 2l * Thrr" rates times initial depreciable cost give the annual depreciation cleductions for personal property. The 3, 5, 7 and 10 year life rates are

based on 200Vo declining balance switching to straight line with the halfconvetiiott applicable in )'ear l' The 15 and 20 year life rates are

)-ear

based on t 50Va declining balance switching to straight line with the half-

year convention applicable in year l. Do not use these rates quarter first-)'ear depreciation convention is applicable'

if

the mid-

Economic Evaluation and lnvestment Decision Methods i

EXAMPLE 7-7 lllustration of Modified ACRS Depreciation Machinery and equipment have been purchased and plaged.int.oService prior to the final quarter of the tax year for a cost of $100,000. The modified ACRS depreciation life is seven years which means the asset is depreciable by 2OO% declining balance switching to straight line when appropriate for a seven year life. Determine the annuallllowable depreciation deductions assuming this is the only depreciable acquisition of the year so the half-year depreciation convention applies in year one. Then look at the effect on depreciation deductions of not getting the asset into service until the final quarter of the year, assuming this is the only depreciable acquisition of the year so the mid-quarter convention applies in year one. Solution:

when depreciable assets in a given depreciation life category whose cosf ts 60% or more of total depreciable cosfs for a tax year are ptaced in service during the first three quarters of a tax year, the hati-year convention applies in the first depreciation year to all depieciable cosfs in that depreciation life category that are placed in service in that year. 2OO"/o Declining Balance Switching to St. Line Depreciation With The Half Year Convention in Year 1. 7 year depreciation life rate = 217 = 0.2857 2OO% DB to St Line Basis x Adjusted Rate = Depreciation

2OO% declining balance

Year

1 2 3 4 5 6 7 I

Method

$14,285 2t7(.5) 200% D.B. $100,000 24,489 85,715 2/7 200% D.B. 17,493 61,226 217 200% D.B. 12,495 43,733 2/7 200% D.B. 8,925 31,238 1/3.5. St. Line 8,925 31,238 1i3.5 St. Line 8,925 31,238 1/3.5 St. Line 4,463 31,238 1/3.5(.5) St. Line $100,000 Cumulative Depreciation *The straight line rate of 1/3.5 equals the declining balance rate of 217 or 0.2857, so switching to straight line is economically desirable.

i

II

Chaoler 7: Depreciation. Depletion; Amo!'trzation, and Cash Flow

Table 7-3, Modified ACRS Rates for 7 Year Life Depreciation 7 Yr Life Rate 1 0.1;i29 2 0.2449 3 0.1749 4 0.1249 5 0.0893 0.0892 6 7 0.0893 I 0.0446 Cumulative Depreciation Year

x

2a0/" DB to St Line

= $100,000 100,000 100,000 100,000 100,000 100,000 100,000 100,000 lnitial

Basis

Depreciation

$14,290 24,490 17,490 12,490 8,930 9,920 9,930 4,460 $100,000 Within round-off error accuracy, the Table 7-3 rates give the same depreciation deductions as the declining balance switching to sti'aight line depreciation calculations. The mid-quarter convention applies in the first depreciation year when assefs in a given depreciation life category whose cost exceeds 4C% of total depreciable cosls for the year in that depreciation life category are placed in service during the final quarter of the tax year. Cosfs for each quarter, not just the final quarter, are affected by this convention. A half-quarter deduction is allowed in the quarter assels are placed in service. 200% Declining Balance Switching to St. Line Depreciation With the Mid-Quarter Convention in Year 1 and 8. 2OO% DB to Stline Year Method Rate x Adjusted Basis = Depreciation

1 200%D.8.2t7(1.5t12) $100,C00 $ 3,571 2 20C% D.B. 2/7 27,550 96,429 3 200% D.e. 2/7 68,879 19,679 '14,056 4 200% D.B. 217 49,200 5 200%D.8.2t7 35,144 10,041 6 St. Line 112.875- 25,103 8,731 7 St. Line 112.875 25,103 8,731 8 St. Line 112.875(10.5112) 25,103 7,640 Cumulative Depreciation (Within Round off) $100,000 .The straight line rate of 112.875 equals 0.3478.

350


7.6 Election to Expense Limited Depreciable Costs .U.S. tax law permits limited expensing

of depreciable,property ,that would otherwise be treated as a depreciable capital cost. Under current law, up to $24,0m of depreciable personal proprry costs can be expensed in the'year'the property is placed into service in 2001 and 2002. This amount increases to $25,000 in 2003. The IRS refers to this as a Section 179 deduction. Should the cost of qualified depreciable property placed into service during the year exceed $200,000, the expense amount is reduced dollar for dollar by the excess above the threshold. Further, the total cost of the property that nray be expensed is limited to the taxable income of the taxpayer as derived from any active trade or business. For more information on this deduction, check IRS Publication 946.

7.7 Depletion Methods The owner of an economic interest in mineral deposits, oil and gas wells, or standing timber may recover his or her cost through federal tex deductions for depletion over the economic life of the property. You have an economic interest if through investment you have: (1) acquired any interest in minerals in place or standing timber, (2) received income by any form of legal relationship from extraction of the minerals or severance of the timber, to which you must iook for a return of capital. Oil, gas, and mineral depletion is computed by two methods: (1) cost depletion, and (2) percentage depletion. Only cost depletion applies to timber. For petroleum and mining both cost and percentage depletion must be computed each year. The result that gives the largest allowable tax deduction, accounting for the 507o or 1007o percentage depletion limits applicable to mining and qualifying petroleum producers, is used as described later. You can switch methods from year to year with the exception that integrated oil and gas producers may only take cost depletion on oil and gas properties. For tax law definition purposes, an "integrated" oil and gas producer, as defined earlier in Chapter 7, Section 7-3, is an oil and gas producer that refines more than 50,000 barrels of crude oil per day (average daily production over a year) or has retail sales of oil and gas products exceeding $5,000,000 per year. An "independent" producer does not refine crude oil or sell oil and gas products in excess of the "integrated" producer limits. For percentage depletion calculation purposes, independent producers and royalty owners are subject to a "small producer limitation" which limits the production on which percentage depletion is applicable to 1.000 barrels per day of crude oil production or 6 million cubic feet of natural gas production,

Chapter 7: Depreciation, Depleticn; Amortizaticn, and Cash Flcw

35't

or the combined equivalent of both oil and gas production using 6.000 cubic feet of gas as equivalent to one barrel of crude oil. If independent producer prr..duction exceeds these limits. percentage depletion may be taken on the equivalent of 1,000 barrels per day of crude oil or 6,000,000 cubic feet of naturai gas production prorated over total annual production. 7.7a Cost Depletion

The capitalized costs that generally go into the cost depletion basis for petroleum and mining projects are for mineral rights acquisition and or lease bonuses or their equivalent ascertained costs. Mining exploration costs and petroleum intangible drilling costs may be capitalized into the cost deptretion basis but seldom are. Cost depletion is computed by dividing the total number of recoverable units in the deposit at the beginning of a yer (tons, barrels, etc., determined in accordance with prevailing industry methods) into the adjusted basis of the mineral property for that year, and multiplying the resuliing rate per unit either by the number of units for v,hich payment is received during the tax year, if you use the cash receipts and disbursements method, or by the number of units sold if you use an accrual method of accounting. The adjusted basis is the original mineral rights acquisition or lease cost plus any additional costs capitalized into the cost depletion basis (such as mining exploration or petroleum drilling costs) for the mineral property, less the total depletion previously allowed or allowable over the life of the property. The depletion allowed or allowable each year is either percentage depletion or cost depletion as discussed in the following pages. The adjusted basis for cost depletion calculations can never be less than zero. In equationfornt cost depletion equals: (Adjusted BasisX

Mineral Units Removed & Sold During the Year Mineral Units Recoverable at Beginning of the Year

where Adjusted Basis = Cost Basis + Adjustments

-

Cumulative Depletion

Mineral rights acquisitiott, lease bonuses, and otlter equivalent ascertained costs, including geological and geophysical survey costs, and recording, legal and assessment costs, nornlally are the primary costs that go into the cost depletiort adjusted bosis. The IRS permits most other capital costs to be depreciated or amortized and companies generally do not put any costs in the cost depletion basis that are not required to be put there because percent depletion can be taken even if the cost depletion basis is zero. This means that you normally want to deduct every capital cost possi-

352


ble by a means other than cost depletion because you still get percentage depletion which is usually larger than cost depretion anyway. Integrated oil and gas producers have lost this benefit since they are no longer eligible for percentage depletion on revenue from oil and gas production. However, all types of mining operations get the larger of allowed percentage or cost depletion on mining mineral revenue.

EXAMPLE 7-8 Cost Depletion Calculations tilustrated You own an oil property for which you paid g1SO,00O in mineral rights acquisition costs last year. Recoverable oil reserves are estimated at 1,000,000 barrels. 50,000 barrels of oil are produced this year and are sold for $29.00 per barrel. your operating and overhead expenses are $180,000 this year and allowable depreciation is $120,000. You also expect the same production rate, operating costs, and selling price next year. calculate the cost depletion for this year and next year assuming we do not use percent depletion this year (percent depletion is calculated later for this exampre). This is an integrated petroleum company analysis, since integrated petroleum companies are only eligible for cost depletion on oil and gas production.

Solution: Year 1:Cost Depletion = ($1S0,000X

50,000 barrels 1,000,000 barrels

= 97,500 Year 2: Cost Depletion = (9150,000

-

$7,500X

50,000 barrels 950,000 barrels

= $7,500

lf percentage depletion had been taken in year one, rather than

cost depletion, the cost depletion basis for year two would be the year one basis minus year one percentage depletion. This is illustrated in Example 7-9.

7.7b PercentageDepletion Percentage depletion is a specified percentage of gross income after royalties front the sale of ruinerals removed from the mineral property during tlle lax ),ear. Hovveve4 the deduction for depletion under this method catxnot

Chapter 7: Depreciation, Depietion, Amortization. and Cash Flow

353

€-\t:€ed 507o of mining ta-rabre inconre or r00To of oil and gas taxabre iitcorue from the property after alr deductions excipt depretion and ross cctrrv fonv,rd tledur:tions. If percentage depletion exceed

i soqo @r l}ovo) of taxable income, you are allowed to take 507a (or lCf,vc) of the taxable ;n.ome before depre tion as your percenrage depletion deduction. (There is *.r lirnitation on the maximum annual cost depLdon that can be taken in a given year, except tirat the accrued cost deplition cannot exceed the cost basis of the prope(y.) However, unrike depreciation and cost depletion, per-

centage depletion accruals are not limited to the cost basis of the property; percentage depletion may be taken after the cost basis in the property has been recovered. The amount of percentage depletion that an investor can take in a given year is affected only by tie 50% or r00vo limit on percentage depletion and possibly by a 65vo of gross income limitation for oil and gas producers. In applying the 50vo or r\ovo rimit on percentage depletion

rvhen operating loss carry forwards from previous y"u., *" available, the depletion calcularions and the 50vo or l\oEo limiti.rt ur" applied before loss carry forwards are deducted. This prevents loss carry forward deductions from affecting the 50vo or r00vo limit on percentage depletion. In addition to the percentage depletion l00vo limii on taxabre income before depletion, oil and gus protrucers erigibre for percentage depretion also are subject to a 65vo gross income litnit v,hich aiserts thrit percentage depletiort may not exceed 65Ta of the taxpayer's taxabre income front all sources, not just tlte propertl, being et,ctluatecl. Since taxable income from all sources often is not known at the time of project economic analysis, the 657o gross income limitation often is neglected. Finally, since october 11, 1990, eligible producers may take p".""riug" depletion on oil and gas properties that were transferred from integrated producers. An effect of this new tax law is to allow integJated petroleum producers to transfer oil and gas properties to an independent producer eligible for percentage depletion and thereby increase the value of the property. Thble 7-4 Procedure to Determine Allowabre Depletion

Percent

Depletion

\ \

507c Mining Limit or lO0Vo Oil & Gas

.,.

The Smaller is rhe

,--)

'Allowed Percent Depletion"

'Allowed Depletion" on the Investor's Tax Return

Limit on Percent Depletion

The Larger is the

Cost Depletion

354


Gross income afrer royalties from a mining or petroleum property eligible for percentage depletion generally must be based on.the amount for which the taxpayer sells the oil and gas or mineral product in the immediate vicinity of the well or mine.-If the oil and gas or mineral product is processed or transported or both, the average market or field price before processing or transportation is the basis for percentage depletion calculations. percenLge depletion rates for oil and gas and various minerals are shown in Table 7-5.

Thble 7-5 Applicable Percentage Depletion Rates Percentage Depletion,To

Mineral Deposits

Oil and Gas*

15

Sulphur and uranium; and if from deposits in the U.S., asbestos, mica, lead, zinc, nickel, molybdenum, tin, tungsten, mercury vanadium, and certain other ores

andmineralsincludingbauxite

........22

If from deposits in the U.S., gold, silver, copper, iron

ore,oilshale,andgeothermailiquidsorvapor ...... 15 Coal**, lignite and sodium chloride . . . . l0 Clay and shale used in making sewer pipe, bricks, or used as sintered or burned lightweight aggregates.

. .7 yz

Gravel, sand, stone Most other minerals and non-metallic

ores.

. . . . 14

*

l1 Int"g.uted petroleum producers are not eligible for percentage depletion. 2) The fixed contract pre 2fir75 natural gur p"r""ni depletion rate is 227o.3) stripper well (less than 15 barrels per day ur'"rug" production) depletion is increased l%o for each $1.00 per barrel that the average annual crude oil price is less than $20.00 per barrel, subject to a maxi_ mtrm lAVo rate increase from l5%o to 25Va.

*] rn" percentage

depletion rates for coal and iron ore drop to gvo and, 72vo respectively after the cost depletion adjusted basis has been recovered by allowed percent or cost depletion deductions.

Chapter 7: Depreciation, Depretion, Amortization, and cash Frow

.

i)5

T'he percentage

crepletion rates from Tabre 7-5 areappried to the,.gross income from the pr?per-ty after royarties," which is the gross income from

mining or well head price for oil ,ra gur. In addition ,i ,fr. ;;;;#; # nrinerals from the ground, "mining" inciudes tr-eatment p.;;rr", appiied by tlrc miue owner or operator to the minerals or rhe oro unJ,ronsportation that is not over 50 miles from the poinr or extrecrion ,o',rr. pi*, or mill in rvhich allowable treatment pro..rrl, are apptied. r."utrn"*'p*cesses con_ sidered as mining depend upon the ore or minerar mined, Lo generariy include those processes necessary to bring the mineral or ore io the stage at which it first becomes commercially marketabt"; tfris-usuali meuns to u shipping grade and form. Howerer, in certain.ur"r, uaaiiiorui'p.o."rses are specified in the Internar Revenue service regulations and are considered "mining." Net smerter r."tT*, or its equivalent, is the gross income on which mining percenrage depretion.co--onryl. uur"o. niy"rrv r*r"., g", p".centage depletion on royalty incorne so cor onsrossincomeaf terroyaliies.sma,i,;;f,iH;"T::::::?:;*:r"[T: eligible to take percent depletion wt ite inilgrated oil and gas producers are ' only eligible ro take cost depletion.

EXAM'LE 7-g percentage Depretion and cash Frow Anatysis For the case study described in Example 7-g with 50,000 bbl of oil g29.00/b;i6r.",i*"" to be after royalties), operating costs of gtgO,0O0 and depreciation of $tZ0,OOO, compare percentage and cost depretion tor this next year (years one and two) if"-rnarysis is ftr an'inoependent .assuming producer erigibre for either perclntage or cost depretion. The 1 s% oir and gas percentage depleiion rate is appricable. Assume severance taxes are 930,000 this year. Use income tax rate and carculate cash flow for this year. îo"t" prod,uction this year, a seiling price of

6;;uni

Solution: owners get percentage depretion on royarty revenues and - -Royalty pay severance tax on royarty revenues so we can base the anarysis on net revenues after royalties.

356


Year

Net Revenue, (50,000 bbl @ $29.00/bbD - Operating costs - Severance Tax

'

: !qpl99!e!9!

Taxable lncome Before Depletion - 100% Limit for % Depletion (1 .0X1,120,000) - Percentage Depletion (.15)(1 ,450,000) - Cost Depletion (From Example 7-8)

$1,450,000 :190,0Q0 -30,000 -120,000 1,120,000 1,120,000*

-217,500 7,500

902,500

Taxable lncome - Tax @ 40%

-361,000 541,500

Net lncome (Profit) + Depreciation

120,000 217,500

t_e9p!9!9!_rakel:

*

1

$ 879,000

Cash Flow From Sales This Year

Since the $217,500 percentage depletion is less than the 1OO% limit for percentage depletion, $2'17,500 is the allowable percentage depletion. This is greater than the $7,500 cost depletion so $217,500 is the largest allowable depletion deduction. ln year two, if the revenues and deductions are assumed to be the same as year one, percentage depletion is the same. However, the cost depletion deduction differs in the second year because the cost basis must be adjusted for the actual depletion deduction taken. ln this example the year one depletion deduction was for percentage depletion. Remembering from Example 7-8 that the property was acquired for a cost of $150,000, the year two cost depletion is calculated as follows: Year 2 Cost Depletion = (150,000

-

217 ,500)(50,000/950,000) < 0

No cost depletion is allowable when the cost basis is negative. There is no cumulative limit on percentage depletion so it again would be selected in year two.


357

EXAMPLE 7-10 lllustration of Depletion Calculations for Co-Product Ores A mining operation yields annual sales revenue of g j,500,000 from an ore containing the co-products lead, zinc and silver. g'1,000,000 of the revenue is from lead and zinc, and 9500,000 from silver. operating costs are $700,000 and allowable depreciation is g100,000. Determine the taxable income assuming the cost depretion basis is zero and that operating costs and depreciation are proportional to revenue from different ores.

Solution: Since silver is a 15% depletion rate minerar whire read and zinc are

22'k depletion rate minerals, sales revenues must be prorated for percent depletion calculation purposes. Sales Revenue - Operating Costs - Depreciation

$1,500,000 -700,000 -100,000

Taxabie Before Depletion 50% Limit on % Depletion Lead, Zinc "/" Depletion (.22)(1,000,000) Silver % Depletion (.15)(S00,000)

-220,000 | select -75,000 J ZgS,OOO

Taxable lncome

$

700,000 350,000

-..p*

7.8 Amortization It is permissible for a business to deduct each year as amortization a proportionate part of certain capital expenditures. Amortization permits the recovery of these expenditures in a manner similar to straight line depreciation over five years or a different specified life. As a general rule, amortization relates to intangible asset ccsts while depreciation relates to tangible asset costs. Howevcr, only certain specified expenditures may be amortized for federal income tax purposes.

Thirty percent of the cost associated with either mining deveropment costs incurred by a corporation or intangible drilling costs incurred by an integrated oil and gas producer must be amortized over a 60 month period.

A corporation's organization

expenses under certain conditions may be

358


amortized over 60 months. The cost of acquiring a lease for business pur-

poses (other than a mineral lease) must be recovered by amortization deduc-

tions over the term of the lease. Research and experimental .;;;;;; be amortized over a 60 month period or longer, oi deducted currentry as a business expense ifconnected with your trade or business. The cost ofcerti_ fied pollution contror facilities may be amortized over a period of 60 months for installations in plants that were in existence prior to 1976. start_ up expenses incurred by a taxpayer in connection with ietting up an active trade or business, or for investigating the possibility ofcreatini or acquiring an active trade or business, may be amortized over 60 monthi. The cost of acquiring an exclusive right to a patent is amortized over the remaining patent life, 17 years for new patents. The cost ofnon-exclusive patent rights may be expensed when incurred. The annual allowable amortization for qualifying expenditures is carcu_ lated in the same way that straight line depreciation is carculated for depre_ ciable assets, with first year amortization proportioned to months of service.

EXAMPLE 7-11 ilrustration of Amortization carcutations A mineral investor is assumed to be involved in either mining or oil and gas development. A one million dollar mine develojment cost (or petroleum intangible drilling cost, IDC) is incurred in July of a calendar tax year (seventh month of year 0) for either mine deveropment or oir and gas well drilling. consider.the investor is a corporation from a mining viewpoint ,n.integrated producer from an oir and gas viewpoint. calculate"lg the allowable tax deductions from this one million dollar year 0 cost.

Solution:

only 70% of corporate mine deveropment or integratcd producer intangible drilling costs can be expensed in the yea"r incurred, with the other 307" amortized over 60 months, beginning in tre month the cost is incurred or praced on the tax books.-Therifore, 9700,000 of the year 0 cost can be expensed in year 0. thirty percent'of the cost, $300,000, is deducted using straight rine amortization over 60 months, with the first year amortization deduction (year 0 in this anal_ ysis) prorated from the month the cost was incurred. From the seventh to twelfth month of year 0 is 6 months so a 6/60 amortization deduction is taken in year 0. Arternativery, the year 0 deduction courd

Chapter 7: Depreciation, Depletion, Amort,zation, and Cash Flow

359

be described as taking 6112 of the straight lins s..ral deduction of 1/5. This and subsequent years are summarized below

Year

AmortizationDeductions

0 1-4 5

(6/60x$300,000) = (6/12x1l5x$300,000) = 930,000 (12l60X$300,000) = (12t12X1l5)($300,000) = $60,000 (6/60X$300,000) = (6/12X1l5X$300,000) = 930,000

7.9 Royalties, Production Payments, Severance and Property Taxes

Earlier in this chapter, discussion in section 7.3 addressed the subject of costs that could be expensed. Under operating costs, it was pointed out that all royalties, severance taxes, ad valorem (property) taxes and some excise taxes are an allowed expense deduction in computing taxable income. As was also previously mentioned in Section 7.7b, royalty holders are generaily entitled to the allowed percentage depletion deductions on royarty revenue or cost depletion on their mineral rights cost basis for a property. This is the same as for any producer, because they are considered to retain an "economic interest" in the prope(y being depleted. An investor retains an economic interest in a property if the investor has acquired any interest in minerals in place or standing timber or receive income by any form of legal relationship from extraction of the minerals to which the investor must look for a return of capital. In the mining and petroleum industry, the word "royalty" as applied to an existing mineral lease means compensation provided in the lease to the owner of the property for the privilege of developing and producing mining minerals or oil and gas from the leasehold, and consists of a share in the value of the minerals extracted. In the oil and gas industries, 'overriding royalties' and 'carried interests' are interests carved out of the lessee's share

of the mineral value, also called the "working interest," or "net

revenue

interest" as distinguished from the owner's interest. Royalties, whether overriding, carried or basic, are payments that extend to future production. In contrast, production payments generally terminate when they reach a total cumulative sum. A mineral production payment is treated fbr tax purposes as a loan by the owner of the production payment to the owner of the mineral property. The loan is to be repaid by an amount or fraction of annual production, analogous to the way mortgage payments pay off an initial loan amount. Therefore, as with a loan, the initial payment is

360

Economic Evaluation and lnvestment Decislon Methods

llol tax dcductibie by ttre lender nor is it taxable income to the borrorver. All income from the propefiy including the value of production payments, is taxable income to the owner of the property working interest and that owner is entitled to the applicable mineral depletion allowance..,The..owner.of the production payment is not entitled to depletion on that amount. In the process of calculating mineral project cash flow, the first items deducted from applicable mineral sale revenues are the royalty costs because in most instances, royalty owners are allowed to take depletion on royalty revenues. Therefore, the investors in mining or petroleum properties only get to take percentage depletion (when applicable for oil and gas) on the net revenue after royalties which in the oil industry is often referred to as the 'net revenue interest.' Landowner royalties vary with different mineral industries. In oil and gas, thb standard royalty is 1/8 or l2.5va of the gross value of the product extracted from the property and may include an up fiont fee on a dollar per acre basis, this fee is sometimes referred to as a lease bonus or rental and generally goes into the cost depletion basis for a lease. An overriding royalty or caried interest is similar to a landowner's royalty and is subject to the same percentage depletion considerations. participating interests also previously defined as net revenue interests and working interests may change over a project life, but they too represent an economic interest in the property and are subject to applicable percentage depletion considerations. on the other hand, percentage depletion is not allorved for lease bonuses, advanced royalty payments, or any other amount payable without regard to actual production of minerals. only cost depletion is applicable in rhis situation.

A

'severance tax' is a state tax imposed on the severing of natural

resources from the land and is based on the value or quantity of production. Severance taxes are also defined as mining taxes, excise tax or a net proceeds tax dependent upon each states unique wording. However, the tax is alu'ays based on actual production or output value of either oil and gas or rnining minerals and tirnber. Severance taxes usually are calculated on a flat

dollar per unit produced basis, or as a percentage of the value of the

resource extracted.

A property tax (or ad valorem tax) is levied by rhe appropriate taxing authority on the value of a property. property taxes can be levied by different cities, counties, states or school districts. property taxes can be levied against either real or personai property. Real property is taxed in all states, but many states do not tax personal property. Royalty recipients are also subject to their portion of severance and ad r.,alorem taxes. Such taxes are

Cnapter 7: Depreciation, Deoletion, Amortization. anC Cash Flow

ofren paid in

fuli by the producer u,ho then allocates

351

rhe taxes to the appro_

p'ate parties such as royalty recipients, thereby reducing the royalty income actually distributed. Properly accountin-e for the detailed tax implications

of all the resource property taxes, rol'alties, and production payments goes 'arious well beyond the objectives of this textbook. In all evaluation situations, the investor's tax position rnust be carefully considered with appropriate tax personnel or material to determine the rlsulting tax benefits and liabilities or obligations of each investor. 7.10 Four Investor Financiar situations That Affect cash Frow All investors whether individual, corporate or government, fall into one of four financial situations for cash flow calculation purposes. The four

financial situations are described as follows:

Expense An investor has other taxable income from existing salary, business or investment sources against which to use negative taiable income from new project operating expenses and deduction, i-, any year. In this sit-

uation the new project is credited with tax savings from the operating expenses and deductions in the year incurred. This assumes the new project deductions are "deducted" against other income in the year deductions

incurred, so they' are "expensed', against other income.

are

stand Alone An investor is assumed to not have other taxable income against which to use negative taxable income generated by new project deductions in any year. Therefore, in analysis of inew projeci negari'e taxable income must be carried forward to be used against pio.y"ct revenues when they are realized. This approach is often refened to ui *uting project economics "stand alone."

carry Forward An investor with sufficient corporate loss carry forward deductions from negative taxable income on existing projects would anticipate paying no regular federal income tax for a numbei years of into the foreseeable future. Such companies have significant exposure to Alternative Minimum Thx, (AMT), which is inrroduced in chaptei g of this textbook. This category of investor will utilize a different straiegy of focusing on rhe minimization of Alternative Minimum Taxable Income and the resulting minimum tax since regular federal income tax liabilities have already been eliminated as previously mentioned.

362


BeJbre-I-ax An investor is a tax-free entity such as a government or charitable organization that is not required to pay any federal or state jncome taxes on revenues associated with investments. Therefore, before-tax analysis of project investment is appropriate.

If other income exists against which to use deductions from new projects, the investor realizes the tax benefits more quickly than when other income does not exist and negative taxable income must be carried forward. The economics of projects will always look better for investors who have other income against which to use deductions when incurred, compared to an investor that does not have other income. Tilx savings are inflows of money, and the faster they are rcalized, the better project economics look. Therefore, it is very important to analyze the economic potential of projects from the viewpoint oi the financial situation of the investor. This is true in all industries and countries. It is incorrect to carry negative taxable income forward in an analysis if other taxable income exists, and it also is incorrect to credit a project with tax savings from negative taxable income in any year if the invesior is in a financial situation that requires carrying negative iaxable income forward. In this text, examples and problems in chapter 7 are presented primarily -from the viewpoint of investors who must carry negative taxable income forward using "stand alone" economics described earlier in this section. Chapter 8 illustrates analyses from both the viewpoints of ,,stand alone,, and "expense" economics. chapters 9 and l0 emphasize analyses from the viewpoint of investors that have other income against which to use deductions, so the "expense" economic analysis financial assumption is emphasized. The next example illustrates carrying negative taxaLle income forward using the "stand alone" financial assumption.

:XAMPLE 7-12 lllustration of the Stand Alone Financial Situation for Mineral Cash Flow Analysis A corporate investor is considering acquiring and developing a min:ral property believed to contain 1,000,000 units of mineral reserves mineral units could be ounces of silver, barrels of oil, etc.). The mineral ights acquisition cost for the property would be g2,000,600 at time 0. Iineral development (or petroleum intangible drilling) cost of $g00,000

h

Chapter 7: Depreciation, Deplelion, Amortization, and Cash Flow

and.tangible equipme.nt costs (mining equipment or oir/gas producing equipment and piperines) of $t,oob,ooO arso pr61".ted to be incurred at time 0. Assume amortization of 3c% "r" of the minerat develop_

ment or intangible drilling cost starts in time 0 with a 6-month deduction. tulodified ACRS depreciation of the g1,000,000 equipment cost will be based on 7 year depreciation rife assets, starting in year 1, assuming the half year 1 convention is applicabre. prcducion is proyected to be uniform for each of years 1 and 2 ar 2oo,o00 units p"i yu". product selling price is estimated to be $30.00 per unit in year 1 unO $S+.OO p., unit in y,ear 2. operating expenses are estimateo to oe g7o0,0oo in year 1, and $800,000 in year 2. Royalty owners will receive 2or" of revenues each year. Assume the mineral is a 15% depletion rate mineral and that the investor is an "integrated" producer if the mineral is oil. Use an effective income tax rate of 4o/". Determine pCIect.u.n ito* for years 0, 1, and 2 assuming the investor does not have otner incom" ,si"i.rt which to use negative taxable income in time 0, so it o" carried fonvaid to make poect economics "stand arone.,,Then carcurate the cash flows using Equalion 7-2.

,r.t

(A) A corporate mining company, and (B) An integrated petroleum company.

Solution: (A) Corporate Mining Anatysis

Depreciation (Zyr life Modified ACRS): Year 1 ($1,000,000)(.1 429) = $1 42,900 Year 2 ($1 ,000,000)(.2449) $244,900 = Year 1 Cost Depletion:

200,000 units produced

1,000,000;;its

in;;;rve

x $2,000,000 Acq. Cost = $400,000

Year 2 Cost Depletion:

200,000 units produced 800,000 units in reserve

x ($2,000,000

-

9720,000) = $320,000

ln year two cost depretion, the g720,000 adjustment is based on the projected year one percentage depletion ictually taken.

364


Year 0

Year

Sales Revenue - Royalties @ 20% Net Revenue - Operating Costs - Development - Depreciation - Amortization

Year 2

1

6,000,000 -1,200,000

6,900,000 -1,360,000

4,900,000 -700,000

5,440,000 -900,000

-142,900 -49,000

-244,900 *49,000

3,909,100 1,954,550

4,347,100 2,173,55A

-720,000"

-916,000.

-560,000

-24,000

Taxable Before Depletion -594,000 - 50% Limit on % Depletion - Percentage Depletion - Cost Depletion - Loss Fonvard Taxable lncome - Tax @ 4A"/"

-594,000

Net lncome + Depreciation + Depletion + Amortization + Loss Forward - Capital Costs

-594,000

400,000

320,000

__qry,ogq _2,605,100

3,531 ,100

-1,042,040

-J,412440

1,563,060 142,900 720,000 49,000 594,000

2,'1'19,660

24,000

244,900 916,000 49,000

-:,elgpgq".

Cash Flow

-$3,800,000

$3,057,960

$3,227,560

Largest Allowable Depletion Deduction Using TableZ_4.

'*capital cost includes g2,000,000 mineral rights acquisition,

$1,000,000 depreciable, and g240,000 amortizable which is 30% of rnining development.

)alculating Cash Flows Using Equation iame Results: 'r

O

CF = -560,000

-

7_2

3,Z4O,OO0 = -$3,g00,000

'r1CF = 6,000,000- 1,200,000-700,000'

r 2 CF = 6,800,000

Gives the

1

,O4Z,O4O= $3,057,960

- 1,3O0,OOO - 800,00 O - 1,41 2,440 = $3,227,560

Chapter 7: Depreciation, Depletion, Amorti:.ttion, and Cash Flow

365

(B) lntegrated Petroleum Company Analysis

Depreciation (7yr tife Modified ACRS): Year 1 ($1.000,000X.1429) = $1 42.900 Year 2 ($1 ,000,000)(.2449) = $244,900 Year 1 Cost Depletion:

200,000 units produced 1,000,000 units ir, ,eserve x $2'000'000 Acq' cost = $400,000 Year 2 Cost Depletion:

200,000 units produced 800,000 units in reserve

x ($2,000,000

-

Year 0

Year

Sales Revenue - Royalties @ 20% Net Revenue - Operating Costs

- IDC -

-

-

Depreciation Amortization Cost Depletion Loss Forward

1

Year 2

6,000,000 -_l_,?qqpgq

:1,39qp9q

4,900,000 -700,000

5,440,000 -900,000

-142,900 -49,000 -400,000 -594,000

-244,900 -49,000 -400,000

2',925,100

3,947]00

-1,tlgp+o

-1,579,940 2,368,260 244,900 400,000 49,000

6,800,000

-560,000

-ro,ooo

Taxable Income -Tax @ 40%

-584,000

Net lncome + Depreciation + Depletion + Amortization + Loss Forward - Capital Costs

-594,000

Cash Flow

$400,000) = 9400,000

24,000

1,755,060 142,900 400,000 49,000 594,000

-:;rq,ogg-. -$3,900,000

$2,929,960 $3,061,160

Capital cost includes 92,000,000 mineral rights acquisition,

$1,000,000 depreciable, and g240,000 amortizable which is 30% of intangible drilling cost (development).

366


Calculating Cash Flows Using Equation 7-2 Gives the Same Results: Yr 0 CF = -560,000

- 3,Z4O,OO0 = -$3,8001000.. Yr 1 CF = 6,000,000 - 1,200,000 - 700,000 - 1,1 7O,O4O $2,g29,g60 = Yr 2 CF = 6,800,000 - 1,360,000 - 800,000 1,SZB,B40 $3,061,160 = 7.ll

Summary

The following is a summary of the various expenditures addressed in this Chapter and the most likely treatment for tax purposes in economic evaluations. Description

Method of Deduction

Operating Costs

All investors All investors All investors

Royalties

Taxes State Income Thxes Severance

Research

Develop Mine Development Experimental

may expense 1007o may expense l00Vo may expen se 1007o

-All investors may expense l007o

All All

investors may expense 100Vo investors may expen se IO0To

Individuals may expense lNVo Corporations may expen se 70Vo and Amortize 30Vo over 60 months

lntangible Drilling

costs

Non-Integrated producers may expense 1002o Integrated Producers may expens e 7 \Vo and Amortize 30Vo over 60 months

3uildings (Real

Prop.)

For all invesrors, if the property is: Commercial Property (Non-Residential) Straight-line depreciation, 39 yrs, Mid-Month Residential Rental Property Straight-line depreciation,

{uipment

(Personal

Prop.)

27 .5

y

rs,

Mid-Month

For all investors. depreciate over the applicable tax life (3 to 20 years) by the MACRS rares in Thble 7-3, or by straight line, or by units of production using a measure such as proven units. Half year or mid-quarter convention may apply. Half-year convention is built into rates

in Thble 7-3.


Description

367

Method of Deduction

intangible Assets (goodwill, For all investors, amoftize over the appropriate patents, copyrights. etc.) life, which varies in accordance with the asset and tax code. For example, the acquisition of exclusive rights to a patent would be amortized over the remaining life of the patent. Goodrvill is amortized over l5 years. Working Capital

It is common evaluation practice to treat working capital like land and simply charge the cost as a capital expenditure when incurred and write-off the cost basis when the property is sold. Section 8.10 in Chaprer 8 offers more details concerning the "Cost of Goods Sold" calculation and the relationship to working capital.

Land (surface rights)

For all investors, write-off the original cost basis in the land when the property is sold. For individuals, gain or loss may be subject to long-term capital gain, see Chapter 8 for more details.

Common Stock

For all investors, write off the original cost basis against the sale of stock to determine the gain or loss. For individuals, gain or loss may be subject to long-term capital gain, see Chapter 8 for more details.

Mineral rights acquisition, lease bonus and recaptured exploration costs are the adjusted basis for Cost

Depletion

Percentage Depletion

For all mineral investors, such expenditures form the basis for cost depletion deductions. Cost Depletion =

(Adj Basis)(

Basis

Units Produced & Sold Proven Reserves at Beginning of

Yr

There is no cost basis for percentage depletion. When applicable, percentage depletion is based on a percentage of net revenue (Gross Revenue - Royalties) and may be subject to a 50Vo limit

in mining or a

100Vo

limit for oil and

gas.

368


other terms introduced in chapter 7 include the folrowing:

-

Amortization is much like shaight-line depreciation and is used to deduct the cost of intangible assets such as goodwill, copyrights, patent acquisitions, etc.

Depreciation - is related to the cost of tangible assets including both personal and real property. The most common methods include straight line, declining balance and units of production. In the united states, the fastest

form of depreciation is declining balance switching to straight line when it is advantageous to do so.

Straight Line: (Cost BasisXl/n)

Declining Balance: (Adjusred Basis)(1/n) up to (Adjusted Basis)(2/n) 'Adjusted Basis" is cost basis less ail depreciation previouiily

taken.

Declining Balance Switching to straight Line: Based on declining balance methodology, this approach switches back to the straighrrine method when a taxpayer can achieve an equal or rarger deductiJn by switching. This occurs when the fraction of (i/Remaining Life) is equar to or greater than the declining balance rate because with straight line, the rate is applied to the remaining cost basis. This is the approach utilized in the MACRS depreciation percentages in Table 7_3. Units of Production:

(cost Basis)(units produced)/(units Available to be produced) In the resource industries, the units produced are based on proven units in reserves or hours of service. MACRS

-

Modified Acceleratecl Cost Recovery system is that area of rhe the details relared to depreciation of personal and real property. This system repraced the Accelerated cost Recovery system (ACRS), which replaced the older Asser Depreciation Range (ADR) method.

u.s. tax code that identifies

write-offs may also be referred to as a book value, tax book value or

written down value - it is a measure of the original cost of an asset that has not yet been deducted by the various methods available such as depreciation,

amortization or depletion. In the case of land surface rights and common stock' the cost basis is oniy deductible when the property ii sord.

Chapter 7: Depreciation, Deptetion, Amortization and Cash Flow

369

PROBLEMS

7'l

Equipment (such as oil and gas producing equipment, mining machinery- or certain general industry r'quipment) has been purchased and put into service for a $2,000,000 cost. calculare the N{ACRS dep:eciarion (20avo declining balance swirching to straight line), and straight line depreciation per year, for a7 year depreciation life, u,ith the half year I convention applicable for both methods, assuming the equipment is put in service in year 1.

7-2 An owner may calculate depreciation

'

using the "units of production,' methods of depreciation. consider a depreciable asset costing $100,000 that is expected to have a useful life of r20,000 units of production. Salvage value is estimated to be negrigible. Estimated annual production for the next 7 years is 30,000, 30,000, 20,000 and 10,000 per year over the remaining 4 years. Use the "units of production"

depreciation method and determine the depreciation each year.

7-3 A company

has acquired a vehicle and right rrucks costing $100,000 in the first quarter of the tax year anc research equipment costing s500,000 in the fourth quarter of the tax year, so the mid-quarter convention is appropriate for depreciation calculations (more than 4ovc of depreciable cost is in the last quarter). The assets qualify as 5 year life depreciable property. compute the MACRS depreciation and straight line depreciation for each year.

7-4

Depreciable residential rental real property has been purchased for $800,000 and put into service during the third month of the taxpayer's tax year. For the applicable 27.5 year depreciation life, determine the annual allowable straight line depreciation deductions. Remember that for real property, the first year depreciation is proportional to months of service and subject to the mid- month convention in the month real property is placed in service.

7-5

Mineral rights to a petroleum property have been acquired by a company for a $500,000 lease bonus fee at time zero. To develop the propefty, estimated time zero capital expenditures include intangible drilling

370


costs of $2'000,000 and 91,000,000 for.tangible welr compretion costs.

It is estimated that production will start in year one with 200,000 bar_ rels of oil produced from initiar reserves estimated to be 1,000,000 bar_

rels' The selling price is estimated to be $rg per barrer oioil. Royarties are l6vo of gross income. operating costs in year one are estimated to be $200,000. Assume 7 year life MACRS depreciation starts in year one using the half-year convention. use a +Tqo effective income tax rate and "stand alone" economic analysis financial situation.

A)

Determine the time zero and year one cash flow assuming the property is acquired and developed at time zeroby a smafl

inaepenoent petroleum company with less than 1,000 bbviay crude production.

B)

Determine the time zero and year one cash flow assuming the property is acquired and developed at time zerc by an integiated petroleum company. Expense Tavo of the IDC at time zero. Deduct the remaining 30vo by amortization with a six month (6/60) deduction at time zero.

7-6 Mineral rights to a gord/silver minerar property can be acquired

by

a company for a $500,000 bonus cost at time zero. A mineral develop_ ment cost of $2,000,000 and mine equipment cost of $1,000,000 will be incurred at time zero. year one production wil be 200,000 tons of ore from initial reserves estimated to be 1,000,000 tons of ore, with net smelter return value of the ore estimated to be $ 1g dollars per ton. Royalties are r6,a of gross income. operating

costs in v"* r are esti7 year rife ulcns a"p.".iution .tu.t, using the half'-year convention. Expense loEo of the mine

mared to be $200,000. Assume

in year I de'elopment cost at time zero. Deduct the remainin g 3ovo by amorti_ zation with a six month (6/60) deduction at time zero. Assume a40vo effective income tax rate and determine the estimated cash flow during time zero and year one fbr a corporation, assuming ,.stand arone,, ross carry forward economic analysis. 7-"7 A minerar reserve containing 100,000 tons of gold ore is to be devel_ oped and produced uniformly (20,000 tons fer year) over years I through 5 by a corporate investor. The estimated net smelter return value of the ore is $120 per ton in year l, and the net smelter return is expected to increase $ 15 per ton in each following year. Royarties are 57o of gross income. A one_time mineral development cost of $900,000

s

Chapter 7: Depreciation, Depletion. Amortization, and Cash Flow

371

is to be incurred in time zero. Expen se 70vo of the mine development cost at time zero and capitalize the remainin g 3ovo to be amortized over

sixty months with a six month (6/60) amortization deciuction at time zero. A mineral rights acquisition cost of s900.000 also will be incurred i, time zero (rhe basis for cost clepletion calculations), along with equipment cost of $1,000,000 to be depreciated using MACRS depreciation beginning atyear one, for a7 year life with the half year one convention. write off the remaining book value from the depreciable asset at yefi 5. $1,000,000 wil be invested in working capital at year 0 for in-process inventories, product inventories and accounts receivable. Assume all assets, including inventories, will be liquidated at the end of year 5 for a sale value of $1,000,000 and write off all remaining book values to determine the gain or loss. Treat the gain or loss as ordinary income. Operating costs are $900,000 in year one, escalating $100,000 per year in the following years. other income and tax obli-eations do not exist against which to use negative taxable income, so losses must be canied forward to make the project economics stand alone. Determine the project cash flows foi years 0 through 5, assuming a 40vo effective inco,e tax rate, and calculate the projeci after-tax DCFROR and NpV for i* l2Vo. =

7-8

You are considering investing in a new processing facility. The process patent rights will cost $2,000,000 in time zero, and must be amortized over 60 months (take 6 months of amortization in time zero). Research and development costs of $1,500,000 will be incurred during rime zero and fully (l00Ea) expensed in that year. No other income exists against which to use deductions, so all negative taxable income will be carried

forward until used against project income (stand alone analysis). Processing equipment costing $3,000,000 at rime zero will go into service in year one and be depreciated using the 7 year life MACRS depreciation with the half-year convention. production is estimated to be 350,000 gallons of product per year, starting in year one. product selling price is estimared to be $22.00 per gallon in year one, $23.00 per gallon in year two, and $24.00 per gallon in year three. patent roy_ alties are 10.07o of gross revenues each year. operating costs are expected to be $3,000,000 in year one, $3,300,000 in yeir rwo, ancl $3,600,000 in year three. The effective income tax rate is-40.07o. Determine the cash flows for years 0, r, z and 3, without taking write-offs on remaining tax book values at year 3.

372


i-9 A mining project involves production from a 5,000,000

ton reserve.

The time ze.ro mineral rights acquisition cost of $2,000,000 is the basis for cost depletion. Development expenses of $1,500,000 will be

incurred at time zero. No other incorne exists against which.to use deductions, so all negative taxable income will be carried forward until used against project income (stand alone analysis). protlucing equipment costing $3,000,000 at time zero, will go into service in year one and be depreciated using the 7 year life MACRS depreciation with the half-year convention (the half-year convention is included in Table 7-3 rates), beginning in year one. production is estimated to be 350,000 tons per year, starting in year one. product selling price is estimated to be $22.00 per ton in year one, $23.00 per ton in year two, and $24.00 per ton in year three. Royalties are lo.\vo of gross revenues each year. operating costs are expected to be $3,000,000 in year one, $3,300,000 in year two, and $3,600,000 in year three. The allowable percentage depletion rate is 15.\vo. The effective income tax rate is 40.0vo. Determine the cash flows for years 0, 1,2 and 3, without' taking write-offs on remaining tax book values at year 3, assuming:

7

A)

The investor is a corporation. Expense 70va of mineral development in time zero. capitalize the other 3ovo and deduct by amortization over 60 months assuming a six month deduction (6/60) in time zero.

Bj

The investor is an individual. Expense ro\vo of mineral development cost in time zero.

-10 A petroleum project involves producrion of crude oil tiom a 5,000,000 barrel reserve. Tirne zero minerar rights acquisition cost (lease bonus)

of $2,000,000 is the basis for cost depletion. Intangible drilling expenses of $1,500,000 will be incurred in time zero. Tangible producing equipment costing $3,000,000 ar time zero will go into service

in vear one and be depreciated using the 7 year life MACRS depreciation with the half-year convention (the half-year convenfion is included in the Table 7-3 rates), beginning in year one. production is estimated to be 350,000 barrels per year, starting in year one. wellhead crude oil value before transportation costs is estimated to be $22.00 per barrel in year one, $23.00 per barrel in year two, and $24.00 per barrel in year three. Royalties are ro.ovo of revenues (well-

Chapter 7: Depreciation. Deol'eticn, Amo.tization. and Cash Flow

373

head value) each year. operating cosrs are expected to be s3,000,00c in.year one, $3,300,000 in year twci, and $3,600,000 in year rhree. The allowable percentage depletion rate is 15.07o. The efI'ective income tax rate is 40.07o. No other income exists against which to use ceductions, so al1 negative taxable income u,ill be carriecl forward until used

against project income (stand alone analysis). Determine the cash flows for years 0, 1,2 and 3, without taking write-offs on remaining tax book values at year 3, assuming:

A)

The investor is an integrated producer. Expense 707o ofintangible drilling costs in rime zero. capitalize and deduct the other 30vo by straight line amortization over 60 months assuming a six month deduction (6160) in time zero. only cost depletion *uy be taken by integrated producers

B)

The invesror is an independent producer with ress than 1,000 barrels per- day average procluction. Expense r00vo of intangible drilling costs in time zero; the larger of percentage (subject to'the 1007c limit) and cost depletion is allowed on production up to 1,000 barrels per day. To simplify the calculations, assume all production would qualify fbr percentage depletion.

c)

The investor is an independent producer with more than 1,000 barrels per day. Expense 1007o of intangible drilling costs in time zero; only cost depletion is allowed on incremental production above 1.000 barrels per day.

7-11 Petroleum mineral rights have been acquired at time zero for a lease bonus of $200.000. A well is projected to be drilled and complered ar time zero for an intangible drilling cost (IDC) of $500,000 and tangible producing equipment cosring $400,000. The tangibre producing equipment is depreciable over 7 years using modified ACRS depreciation starting in year one, rvith the half year convention. Gross oil income is expected to be $700,000 in year one, $600,000 in year two, $500,000 in year three. $400.000 in year four and $300,000 in year five. Royalties total 15.\vo of gross revenues each year (for a net revenue interest of 85.07o). operating costs are estimated to be $50,000 per year in each of years one through five. Depletable oil reserves are estimated to be 200,000 barrels with 30.000 barrels produced in year one, 27,000 barrels in year trvo, 24,000 in year three,21,000 barrels in

374

_Economic Evaluation and lnvestment Decision Methods

year four and 18,000 in year five. Assume no other income exists so all negative taxable income will be carried forward and used against project taxable income (stand alone economics). use a 4o.0vo t'ax rate and calculate the cash flows for each of years 0 through 5. Deduct write-off's on remaining tax book values at year 5, when a sale value of $250,000 is received. Determine the project DCFROR for: .

A) An integrated petroleum

producer eligible only for cost depletion and required to deduct 30vo of IDC's using straight line amortization over 60 months. Assume the time zeto amoftization deduction is 6/60 of 307o of the IDC.

B) A small independent producer

(non-integrated) with less than

1,000 barrels per day of crude oil production, ,o eligible for either percentage or cost depletion and who may expense t\ovo of IDC's

in time zero.

. c) An independent producer (non-integrated) with more than 1,000

barrels per day of crude oil production, so eligible only for cost depletion, but may expense l00Vo ofIDC's in time zero.

7-rz A new processing facility will be developed if a $600,000 patent acquisition cost is incurred at time zero. you are to analyze the aftertax project cash flow and DCFRoR based on the following cost and revenue projections. In addition to the time zero patent acquisition cost which should be amortized over five years starting in time zero rvith a six month deduction, a time zero research and experimentation cost of $500,000 will be incurred and deducted for tai purposes in time zero. A $1,000,000 depreciabre equipment cost also will be incurred in time zero but the equipment is not expected to go into service until year one. Therefore, 5 year life MACRS depreciation will start in year one using the half-year convention. projeci revenues are expected to be $2,000,000 in year one and operating expenses are expected to be $1.000,000 in year one, rvith both expected to escalate 10vo per year through year five when the project is expected to be abandoned for zero salvage value due to product obsolescence. Use a 407o eftective income tax rate and make project economics .,stand alone". Take a write off on any remaining tax book values at the end of

year five.


7-13 A new 5 megalvatt hydroelectric plant would cost $6 million dollars at rime zero. It is expected that the plant would operate at its rated full capacity 6.000 hours per year. power is expected to be sold for $0.04 per kilowatt-hour (kwh) and operating costs arc expected to be $0.005 per kwh. Salvage is projected ro be zero at the end of prq""t rife. Escalation of annuar operating costs is projected to be orisei by escaration of annual revenue, giving uniform profit of $0.035 per kwh each year. 1) Calculate before-tax project ROR and NpV for i* = LAVo and i* = l5Vo for evaluation life of A) l0 years, B) 20 years, C) 30 years, and D) 40 years. Assume end-of-year timing for all before-tax cash flows.

2)

For a 30vo effective income tax rate and assuming straight line 10 year life depreciation of the $6 milion prant cosi startfig in year one with a full year deduction, carculate the after-tax project RoR and NPV for i* = lAVo for evaluation life of, A) 10 y"u.., B) ZO years, C) 30 years, and, D) 40 years. Assume .n6_ef_year timing for all after-tax cash flows.

7-14 Assume an investor is considering the investment(s) at time zero of $5,000,000 ro acquire a parcel of rand. $r0,000,000 would then be spent ofl construction of a building for adrninistration and manufactur_ ing for the business. The building is commercial or non-residential real property, (39 year depreciabre rife), and is praced into service in the 10th month of the year l tax year (relates to ttre mid-month convention for real property). In addition to the above-mentioned costs, $15,000'000 would be spent on 5-year MACRS depreciable equipment at time zero. The equipmenr and the building would both goinio service in year l of the project. These three time iero investments are expected to generate year I through 5 incomes of $25,000,000 with corresponding annual operating costs of $r0,000,000 per year. Sales and operating costs are based on 1,000,000 units being produced and sold each year. The business will be sold at the end or tt e nrtn year for $30,000,000. Assume the effective tax rate is 407o and no other income exists against which to use deductions (negative taxable income) in the year incurred so carry losses forward *d ur" against project taxable income (stand alone scenario). when the business is sold, write off arl remaining book values against the sale varue and assume any gain from the sale of the property would be treated as

r

376


ordinary income. Calculate the annual after-tax project cash flows and determine the DCFROR and NPV for an investor desiring an escalated dollar after-tax minimum rate of return of l2%o. 7-15 Using the data in problem 7-14

A)

as the base case, consider the

following:

Suppose this was a mineral project that required an additional time

zero expenditure of $5,000,000 for mine development subject to a 70/30 split with a 12 month amortization deduction taken beginning in year 1. Further, assume the land acquisition cost was really a mineral acquisition cost subject to the cost depletion basis and therefore, deductible by cost depletion and a write-off on the remaining tax book value in the final year. Previously identified production is in tons and total proven reserves are estimated at 10,000,000 tons. The mineral produced is a l}Vo mineral for percentage depletion purposes upon which, the 50Vo limit on taxable income before the deduction would apply. Assume the given annual revenues are net of any applicable royalties. Calculate the annual after-tax project cash flows and determine the DCFROR and NPV for an investor desiring an escalated dollar after-tax minimum rate of return of 127o.

B)

Suppose this v,,as an integrated oil and gas company considering an oil & gas project that required an additional time zero expenditure

of $5,000,000 in petroleum intangible drilling costs, subject to a 70/30 split with a 12 month amortization deduction taken beginning in year 1. Frrrther, assume the land acquisition cost was really a lease bonus cost subject to the cost depletion basis and therefore, deductible by cost depletion and a write-off on the remaining tax book value in the final year. Assume given annual revenues are net of all applicable royalties. Previously identified production units are in barrels and total proven reserves are estimated at 10,000,000 bbls. Calculate the annual after-tax project cash flows and determine the DCFROR and NPV for an investor desiring an escalated dollar after-tax minimum rate of return of 12Vo.

C)

oil and gas company with existing production in excess of i,000 bpd. The company is considering this to be an oil & gas project that required an additional time zero expenditure of $5,000,000 in petroleum intangible drilling Suppose this was a non-integrated


377

costs. FLrrther, assurne the land acquisition cost was really a lease

bonus cosr subject to the cost depletion basis and therefore, deductible by cost depletion and a tax book varue write-off in the final year. Previously given revenue is assumed to be net of all applicable royalties. Previously identifrecl production is in barrels and total proven reserves are estimated at 10,000,000 bbls. calculate the annual after-tax project cash flows and determine the DCFROR and NPV for an investor desiring an escalated dollar after-tax minimum rate of return of l2Vo. 7-16

A man invests

$220,000 at time zero in a repair business including $20,000 for working capital spare parts and $200,000 for depreciable equipment. The equipment is installed in a shed on the back of his property so no cost is considered for the land or the building given that they cannot be separated and sold from the rest of the property, so no opportunity cost is realized. The equipment rvill be depreciated using a 5-year depreciation life, slarting depreciation in year 1. Expected annual income is $100,000 and operating costs are expected to be $40,000 per year assuming a washout in the profit margin for each of the next three years. Further, it is estimated that he can sell the equipment and spare parts in three years for $140,000. Assume the sale value would be taxed as ordinary income and write off the remaining book value of all zrssets against the sale revenue to determine the taxable gain. All taxable income from the business will be subject to a 40.07o effective income tax rate. For an ix = r2To, calculate the aftertax cash flows for the next 3 years and the DCFROR and Npv for this investment opportunity assuming:

A) Straight line depreciation of equipment with the harf year convention. B) MACRS accelerared depreciarion of equipment using Table 7-3.

CHAPTER 8

INCOME TAX, WORKING CAPITAL, AND DISCOUNTED CASH FLOW ANALYSIS

8.1

Introduction

The objective of this chapter is to relate income tax effects, working capital and the concep't of cash flow to discounted cash flow analysis as it relates to rate of return, net value analysis and ratio decision criteria applied after tax considerations. After-tax rate of return analysis commonly is referred to as Discounted Cash Flow Rate of Return or DCFROR analysis. Other names given to this method in the literature and textbooks are Profitability Index or P.I., Investor's Rate of Return, True Rate of Return and Internal Rate of Return. All of these names refer to the same method which easily is the most widely used economic analysis decision method in practice, even though Net Present Value and Ratios have advantages over DCFROR in certain analysis situations described in earlier chapters. The effects of income tax considerations often vary widely from one investment alternative to another, so it generally is imperative to compare the relative economics of investment altematives on an after-tax basis to have a valid economic analysis. Income taxes, both federal and state if applicable, are prcject costs, just as labor, materials, utilities, property taxes, borrowed money interest, insurance and so fbrth are project operating costs. It does not make sense to leave income taxes out of an analysis any more than it would make sense to omit labor, materials or any other operating cost from an analysis. As described in Chapter 7, cash flow is what is left from sales revenue each year after paying all the out-of-pocket expenses for operating costs, capital costs and income taxes. All tax deductible expenses, except income taxes, are put into the operating cost category and capital costs, such as building and equipment costs, are deductible over a period of time greater

chapter B: lncome Tax. working capitar, and Discounted cash Frow Anarysis

379

than a year. In making proper atter-tax anal,vses of investment alternatives, you must be careful to recognize that for economic analysis work the tax considerations thar are reler.ant are what a company or individual does for tax purposes and not what a company does for "book" purposes, u,hich mearis for tinanciz,l accounting or sharehoider reporting purposes. There sornetimes is a tendency for evaluation personnel and accountanB to get confusecl on this point. what a company does for annual report book pr.por", has no relevance to economic analysis of projects. what a company actually does in the way of taking tax deductions and credits on varioui capital and operating costs is what affects the amount of cash flow that new investments will generate at different points in time. To determine valid cash flow each evaluation period for economic analysis requires accounting for the proper costs, revenues and tax deductions at the points in time they will be incurred. It should be evident that this requires using actual tax considerations rather than "book' or "financial report" considerations which are different from actual tax considerations for literally all publicly owned companies.

8.2 Forms of Business organizations and rax considerations This section briefly addresses, from the federal income tax viewpoint, five important types of business organizatio,s in the u.s.: sole proprietorship, partnership, limited liability corporation, sub-chapter S corporation and a regular c corporation. since tax considerations often are extremely irnportaiit in choosing a particular form of busine-ss organization, persons planning to go into business should become familiar with the tax consequences of the different types of business organizations. Different income tax rates apply to corporations and to individuals. Individual income tax rates apply to sole proprietors, individual members of parlnerships and limited liability corporations, and subchapter S corporations. An individual engaging in business alone is a sole proprietor. A partnership must file a partnership income return, which is merely an informaiion return. Each partner is taxed on his oi her shzue of the partnership earnings, reduced by partnership tax deductions, whether or not those earnings are distributed. Limited liability corporations are similar in that income is distributed and taxed like partnership income. They are a hybrid corporation and partnership.

Regular corporate income is taxed to the corporation at corporation rates. corporation earnings, when distributed to the shareholders as dividends, are taxed as ordinary income at the appropriate individual tax rate. However, earnings of a corporation structured under Subchapter S of the Internal Revenue Service code are taxed only once at individual income tax rates, so

380


double taxation of taxable income is avoided with Subchapter S corporations. You may be liable for several types of federal taxes in addition to income tax, depending upon the type of business in which you are engaged. For example, federal excise and employment taxes apply equally whether you conduct your business as a sole proprietorship, partnership, or corporation. Regardless of the form of your business organization, you must keep records to determine your correct tax liability. These records must be permanent, accurate, complete and must clearly establish income, deductions, credits, employee infbrmation, etc. The law does not specify any pafiicular kind of records.

Every taxpayer must compute his or her taxable income and file an income tax return on the basis of a period of time called a tax year. Beginning in 1987, for all taxpayers except regular "c" corporations, a tax year is the 12 consecutive months in the calendar year ending December 31. A fiscal year for "c" corporations is 12 consecutive months ending on the last day of any month, as specified by corporation by-laws. .

8.3 Corporate and Individual Federal Income Thx Rates Both corporate and individual federal income tax rates vary with the incremental level of taxable income after all allowable deductions have been taken. In the year 2000, there are five incremental tax rates for individuals: l5.ovo, 28.47o,31.0va,36.07o and 39.6vo as shown in Thble 8-1. Many high income taxpayers will pay a top rate of more tl'tan39.6vo in 2000 due to current tax

larv phase out of certain itemized tax deductions and personal and dependency exemptions. For example, a marieci couple filing a joint retum for 2000 must reduce their itemized deductions by an amount equal to 3.\va of the difference between their adjusted gross income and $129,950. This in effect increases the marginal tax rate by 1.197o, making the top rate 40.79vc or higher. If a married couple files a joint return and has adjusred gross income above $193,400 their personal and dependency exemptions of$2,550 per person in 2000 zue phased out at a rate of 2.07a of exemptions lost for every $2,500 income rises. This is equivalent to an effective increase in the marginal tax rate of 0.glra for each exemption. currently for a married ccuple with two children the phase out of iternized deduction and exemption deductions makes the upper limit tax rate 44.037o, and with more exemptions it would be higher.

A positive consideration related to individual income taxation is that the levels of taxable income taxed at different rates are indexed (increased) for inflation each year. Indexing taxable income for inflation each year has the effect of eliminating income tax increases due to percent increases in salary or other taxable income equal to the inflation rate in a given year.

chapter 8: lncome Tax, working capitar. anc Discounted cash

Frolry Anarysis

381

1\[arried Taxpayers: If ?00n raxable income is:

0

$

43,8-50

$

11,850

of Taxirble Incorne $6.578 + 287o ofErcess over $43,850 $21,966 + 3lVo ofExcess over g105,950 $41,171 + 36Vo of Excess over $161,450 $86,855 + 39.67a of Excess over g288,350 157a

$ 105,9-50

$i0-i,9s0 $161,450 $i6i,.450 $288,350

$28S,350

and up

Single Taxpayer:

If 2000 taxable income is: Over

$ $

- o

26,250 63,550

$132,600 $288,350

But not over $ 26,250

$

63,550 $132,600 $2gg,350 and up

-

The tax is: 15Vo of Thxable Income $3,938 +28Vo ofExcess over $26,250 $14,382 + 3l7o ofExcess over $53,550 $35,788 + 36?it of Excess over g132,600 $91,857 + 39.6Vo of Excess over $288,350

All of the above income brackets are applicable for the year 2000 and will be adjusted each subsequent year for inflation as meaiured by the consumer Price Index (CPI) for the 12 month period ending August 3l of the previous year. Other relevant information as of 2000: Standard deduction fbr marieds is $7,3-50 Standard deduction for singles is $4,400

If listed as a dependent on another tax return, maximum tax free investment incorne is $700. Itemizcd deductions are reduced by 3va of Adjusted Gross Income (AGI) in excess of $ I 28,950, whether you are married or single. Table 8-1 Individual Income Thx Information In the year 2000 there are six incremental corporate tax rates: 15.07c,25.0go, 34.jvo,35.07o,38.jvo, and 39.\vo.The 39.ovo upper limit rax rare is due to a 5.07o surcharge that congress built into corporate tax law in 19g6 to phase out the tax benefits of the lower 15.0vo and 25.ovo tax rates for relatively high

taxable income corporations. A corporation with taxable income over

382


If corporate t&xable income is:* Over

$ $ $ $ $

tax is: - The 15Vo of taxable income $ 50,000 50,000 $ 75,000 $7,500 +257o ofexcessover$50,000 75,000 $ 100,000 $13,750 +34Zoofexcessover$75,000 100,000 $ 335,000 $22,250 + 39Vo of excess over $100,000

-0

but not over

335,000 $10,000,000 $113,900 +34Vo of excess over $335,000 10,000,000 $ 1 5,000,000 $3,400,000 + 35% of excess over g 10,000,000 $15,000,000 $18,333,000 $5,150,000 + 387a of excess over 915,@0,000 $18,333,000 and up $6,416,540 + 35Vo of excess over $18,333,000 $

*

Thxable income is not adjusted for inflation each year in the manner that individual taxable income brackets are adjusted.

Table 8-2 Corporate Income Tax Rates

$100,000 must pay an additional tax equal to 5vo of the amount in excess of $100,000, up to a maximum additional tax of $11,750. This extra tax operates to phase out the benefits of graduated rates for corporations with taxable incomes between $100,000 and $335,000 and causes the effective federal corporate tax rate to be 397o (34vo + 5vo) on the increment of taxable income

between $100,000 and $335,000. A corporarion having taxable income of $335,000 or more gets no benefit from the graduated rates and pays, in effect, a flat tax at a 34 percent rate on all levels of corporate taxable income up to $10,000,000. Since 1993, corporate taxable income in excess of $10,000,000 has been taxed at 357o instead of34vo. The old corporate sunax on taxable income above $100,000 up to $335,000 has been retained. In addition, a new 3Va surtax was added to corporate taxable income above $15,000,000, up to a maximum of $100,000 in additional tax. The result of this surtax is to assure a flat 357o corporate tax rate on all levels of corporate taxable income for corporations with more than $ 18,333,000 in taxable income. Since both corporate and individual income tax rates vary at different levels of taxable income, it is necessary to use incremental analysis concepts to deterrnine the proper tax rate for new project taxable income. However, for many large corporation evaluations, any new project income will be incremental income above the first $10,000,000 of corporate income from other sources. In this situation, the 357o federal corporate tax rate is relevant for all new project income and therefore is the rate that should be used in evaluating new investment projects. Remember that corporations and individuals also pay state tax

Chapter 8: lncome Tax, Working Capital, and Discounted Cash Flow Analysis

383

in most states, so the overall effective federal plus state tax rate will be greater thin 35Vo for corporations or the appropnate incremental federal rate fbr individLrals. Usuaily the effective rate will be in the range of 35Vc to 407o for corporations, or 30 to 457o for individuals, as will be developed in Section 8.7.

8.4 Corporate and Individual Capital Gains Tax Treatment Cu.rrent tax law continues to make a distinction between capital and ordinary gains and losses. However, in the year 2000, all corporate capital gain is treated as ordinary income subject to the appropriate corporate income tax rates. The maximum long-term capital gain tax rate for individuals is 20.07o. Further, the tax law requires taxpayers to compute capital gain separate from ordinary income as discussed in the following paragraphs. If a gain or loss is from the disposition of a capital asset, individuals or corporations have a capital gain or loss. If a capital asset is held for a period of less than one year and sold, the gain or loss

1t jt.

i

i

from the sale is considered a shortterm gain or loss. Short-term gains are treated like ordinary income. If the same asset is held longer than one year, the gain or loss is considered longterm gain and for individuals, usually taxed at the lower capital gains rate of 20.0Vo. However, persons in the l5.0%o ordinary income tax bracket have capital gains taxed at l0.0Vo. Beginning January 1,2001, taxpayers in the 15.}Vc tax bracket will also see their capital gains rate fall from l0.0Vo to 8.0Vc for assets held for five years or more including assets bought before January 1, 2001. For individuals in the 28.0Vo or higher ordinary income tax brackets, the long-term capital gain tax rate drops from 20.0Vo to l8.0%o for investments made after January 1, 2001 and held for a period of five years. For all individuals, capital gain due to depreciation taken on real property (residential rental or commercial property) is taxed at a 25.UVa rate. Short-term capital gains and losses are merged by adding the gains and losses separately and subtracting one total from the other to determine the net short-term capital gain or loss. Sirrrilarly, long-term capital gains and losses are merged to determine the net long-term capital gain or loss. The total net short or long-term capital gain or loss is then determined by merging the net short-term capital gain or loss with the net long-term capital gain or losses in accordance with tax return procedures. Corporations may only deduct capital losses to the extent of capital gains. Individuals may deduct capital losses from ordinary income up to a maximum of $3,000 per year with the balance carried forward for future years. This includes both long-term and short-term losses.

384


EXAMPLE 8'1 lllustration of lndividual Capital Gains Tax Treatment

An individual investor with $200,000 of annual taxable income purchased a parcel of land 3 months ago for $5O,OO0 and now has an offer to sell it for $60,000. Neglecting the time value of money, what selling price after owning the land 12 months plus one day will give the investor the same after-tax profit as the $60,000 selling price now? Evaluate this problem from the standpoint of an individual whose ordinary federal income tax rate on the incremental income would be 36% and state tax rate zero (.as in Texas and Alaska).

Solution: lf the property is sold now, short term capital gain tax is:

10,000(.36) = $3,600, Therefore a $6,400 profit is realized if the land is sold now. Holding the asset tor 12 months will give greater after tax profit since long term gains are taxed at the 2O/" tax rale.

- lncome Tax $6,400 = X-50,000 - (X-50,000)(0.20) Profit = Gain

where "X" is the before-tax break-even selling price. 0.80X = 6,400 + 40,000, therefore, X = 58,000 Because of the lower 2O.O% long-term capital gain tax rate, selling for $58,000 after one year gives the same profit as selling now for $60,000 and having the gain taxed as ordinary income.

8.5

Thx Treatment of Investment Terminal (Salvage) Value

Whenever an asset such as land, common stock, buildings

or equipment is sold by indi,-iduals or corporations, the sale value (terminal value) is cornpared to original cost, or remaining tax book value of depreciable, depletable, amortizable or non-deductible asset costs to determine gain or loss. If the sale results in a gain, tax must be paid on the gain. If the sale results in a loss, the loss is deductible under the tax rules governing the handling of ordinary deductions and capital loss deductions. As we showed


eai-lier, all long rerm cilpital sains are taxed at the ordinary income tax rates tbr colporat.ions and at a2}.ovo capital gains tax rate for individuals, so it is strll necessary to compute whether ordinary gain or loss, or long term capital sain or loss is realized. If a loss results from the sale, the investor is eligihie for either an ordinary income cleduction 'uvrite-t-ifr or a long term capitai krss deduction write-off, again depending on the type of asset involved

and the tax position of the investor. Although corporations have cctpital gain incorne taxed as ordinary income, corporate capital losses can only be userl against capital gains or carried.forward. when either depreciable or non-depreciable property has been herd for more than one year and sale value exceeds original cost, the increase in the value of property (difference in sale value and original cost) usually is treated as a long term capital gain. For individuals, any gain due to depreciation of personal property is treated as ordinary income. Accountants often refer to the tax on the gain due to depreciation as "recapturing deprebiation." Since the deductions sheltered ordinary income, and were .taken in an amount in excess of the current value of the asset, that component of a sale between the original purchase price and final book value is taxed at the same orriinary income tax rate. one exception is real property, where as mentioned in Section 8.4; for individuals, gain due to depreciation of real property (such as rental property) is currently taxed at a25.07o capital gain tax rate when straight line depreciation is utilized.

EXAMPLE 8-2 Illustration of Sale Value Tax Treatment Assume an asset will be purchased in July of a calendar tax year for a cost of $100,000 and sold 2 years from that date for g1g0,000. The asset is being acquired by 1) a corporation, or 2) an individual. Assume a 35"/o federal and 0% state ordinary income tax rate for both the corporate and individual analyses. calculate the tax and investment DCFROR on the sale for the following scenarios: A) The asset is land, common stock or other non-depreciable property. B) The asset is equipment depreciable over 5 years by MACRS depreciation.

c) The asset is a commercial rental (real) property and depreciated using the straight line method. Assume the building was placed into service in July of year one and the mid-month convention applies.

386


Solution:

Al)

Corporate Analysis - write off the land or common stock investment against the year 2 sale valqq. All corporatg gain is treated as ordinary income.

Year

Revenue - Book Value

180,000

-100,000

Thxable lncome - Tax @ 35%

-28,000

Net lncome + Book Value - Capital Cost

52,000 100,000 0

ATCF

80,000

-100,000

100,000

0

152,000

PW Eq: 0 = -100,000 + 1 52,000(PtFi,2) DCFROR, i = 23.3/"

A2) lndividual Analysis

- write off the land or common stock investment against the year 2 sale value. Assume individual long-term gain is taxed at2O.O%.

YearOlZ Revenue - Book Value

180,000

-100,000


-16,000

Net lncome + Book Value - Capital Cost

64,000 100,000 0

ATCF

80,000

-100,000 -100,000

PW Eq: 0 = -100,000 + 164,000(P/F;,2)

DCFROR, i = 28.1"/"

0

164,000

*

I I $

{

i

$

I t :

t

Chapter B: lncome Tax, Working Capital, and Discounted Cash Flow Analysis

Bl)

corporate anatysis assuming the asset is personar propefi depreciated over 5 years using MACRS applicabre rates. other income exists igainst wniln to u"" rrr-i"o-rltions.

Yeilr 1 Depreciation: $100,000(0.20) g20,000 = Cumulative Depreciation Taken $20,000

\ear 2 Write-off (or book value deduction): $1

00,000

-

$20,000 = $g0,000

Year

Revenue - Depreciation - Book Value Write-off Taxable lncome - Tax @ 35/"

Net lncome + Depreciation + Book Value - Capital Cost ATCF

190,000

-20,000 -80,000 -20,000 7,000 -13,000 20,000

100,000

-35,000 65,000 80,000

0 -100,000 0 7,000 145,000 -100,000 PW Eq: 0 =-100,000 +7,000(p/F;,1) + 14S,OOO(p/Fi,2)

DCFROR, i = 24.0"/"

Since the corporate tax rate is the same for both long-term capital gain and ordinary income, the difference in the sale valu6 and remaining tax book value represents the total taxable income as shown in y.ear 2- This simplified approach may not be applicable for individuals due to the reduced tax rate associated with long-term capital gain resulting from the increase in the asset value comp=ared to the orig-inal purchase price. This is illustrated in Case 82.

L

388


82) lndividual analysis assuming the asset is personal property

depreciated over 5 years,using MAcRs applicable rates. other incon:e exists against which to use all deductions. Long-term capital gain rate applies on gain due to increase in asset value. Gain due to depreciation is taxed as ordinary income. This later component is also known as "recapture" oi depreciation. Year 1 Depreciation: 9100,000(0.20) = g26,999 Year 2 Write-off (or book value deduction): $1 00,000 - $20,000 = $g0,000 Long-Term Ordinary Gain Due to Gain Due to lncrease in Depreciation Asset Value. Year 2 Revenue 100,000 190,000 Depreciation -20,000 - Book Value Write-off -80,000 _100,000 - lnitial Cost Basis Taxable lncome 20,000 80,000 -20,000 7,000 - Tax @ 35% -7,000 - Tax @ 2O/o (long-term gain) -16,000 _13,000 Net lncome 13,000 64,000 + Depreciation 20,000 + Book Value 80,000 0 - Capital Cost 0 0 0 ATCF 7,000 93,000 64,000 -100,000

-100,000

157,000 Note: ln year 2, $100,000 of additional revenue equal to the initial cost has been added in the next to last column. This same amount is then subtracted as the initial cost basis in the final columh. These amounts cancel leaving $180,000 of sale revenue to be considered. The year 2 sum of the two after-tax cash flows is g157,000 which can be reconciled by taking the revenue minus cumulative tax, or; $180,000 - $7,000 - g16,000 = $157,000 PW Eq: 0 = -100,000 + 7,000(plFi,1) + 157,000(plFi,2) DCFROR, i = 28.9oh

Chapter 8: lncome Tax, Working Capital, and Di.scounted Cash Flow Analysis

c1) corporate analysis of a commercial building (real property)

that is depreciable straight line over 3g years assuming the property is placed into service in July of a calendar tax year and the mid-month convention applies with the first year of depreciation in year 1.

Year 1

Depreciation:

9100,000(1/39)(5.Si 12) = $1 ,1 75 Cumulative Depreciation Taken - $1 ,175 Year 2 Write-off (or book value deduction): $100,000 - $1,175 = $98,825 Year

Revenue - Depreciation - Book Value Write-off

-1,175


-1,175

Net lncome + Depreciation + Book Value - Capital Cost ATCF

190,000

411

-764

-98,925 81,175 -28,411 52,764

1,175

98,825

-100,000

0

-100,000

411

0 1

51 ,5gg

PW Eq: 0 = -100,000 + 411(p/F;,1) + 151 ,5g9(p/F;,2) DCFROR, i = 23.3/o

Unlike this corporate analysis, for individuals, gain due to straight line depreciation of real property in earlier years is taxed at a rate of 25%, instead of being taxed as ordinary income as it is for corporations. Note lhe 25"k rate is different than the 20% long-term capital gains tax i'ate and the ordinary income tax rate.


390

C2) lndividual analysis of commercial building (real property) that is dggreglable straight line over 39 years assuming the property ib piaced into service in July of a calenddr tax year and the mid-month convention applies with the first year of depreciation in year 1.

Yearl Depreciation: $100,000(1/39X5.5/12) = $1,175 Year 2 Write-off (or book value deduction): $100,000 - $1,175 = $98,825 Long-Term Gain Due to Ordinary lncrease in Gain Due Depreciation Asset Value.

to

Year

0

100,000

Revenue

- Depreciation - Book Value Write-off

-1,175

Taxable lncome -1,175 - Tax @ 35% (ordinary gain) :" - Tax @ 25% (deprec. gain) - Thx @ 20% (long-term gain)

Net lncome + Depreciation + Book Value - Capital Cost

-gg,g25 1,175

180,000

-100,000 80,000

-294 -16,000

-764

88'1

64,000

0

98,825 0

0

1,175

-100,000

ATCF

-100,000 411 99,706

0

64,000

163,706

Note: Once again, in year 2, the sum of the two cash flows is $163,706 which can be reconciled by taking the revenue minus cumulative tax paid that year, or; $180,000

-

$29+

-

$1

6,000 = $163,706

PW Eq: 0 = -100,000 + 411 (P/Fi,1) + 163,706{PlFi,2) DCFROR, i = 28.2/"

Chapter 8: lncome Tax, Working Capltal, and Discounted Cash Flow Analysis

8.6 Alternative

391

Minimum Tax (ANIT)

The current alternative minimum tax (AMT) was firmly established by the Revenue Reconciliation Act of 1989 to prevent corporations and individuals from usine deductions. exemptions, and cred;ts to completely avcid their regular federal income tax liability. AMT ls computed by determining the taxpayer's alternative minimum taxable income (AIv{TI) and the tentative minimum tax (TMT) related to that income. If TMT is greater than the regular federal tax liability, investors must pay the difference as their altefnative minimum tax. The difference occurs because AMTI is generally based on slower methods of deducting costs than what is allowed for regular Federal income tax purposes. Therefore, more income is initially exposed to tax. As a result, many corporate and individual taxpayers must pay AMT in addition to their regular Federal income tax liability.

A simpiistic breakdown of this tax follows: Regular Tax

Calculation

Revenue

- Operating Costs - Depreciation - Depletion

- Amortization - State Income

Tax

Federal Taxable Income

of Federal Taxable Income = Regular Federal Income Tax on Corporate Taxable Income Above $10 Million. 35Vo

Alternative Minimum Thx Calculation Federal Taxable Income + Net Operating Loss (Loss Forward) + Adjustments + Tax Preference Items (TPIs)

+ Adjusted Current Earnings Adjustment + Other AMT Items - AMT Net Operating Loss Alternative Minimum Taxable Income 20Vo

of AMTI (Corporate Rate)

= Tentative Minimum Tax

Alternative Minimum Tax Based On: Tentative Minimum Tax (TMT) - Regular Federal Income Tax

= Alternative Minimum Tax (AMT)

The starting basis for AMT is regular federal taxable income from all A number of "adjustments" and "preferences" are taken to regular federal taxable income to determine alternative minimum taxable income (AMTI), which is then taxed at 20.0Vo for corporations. The AMT rates for individuals are 26.0Vo on AMTI up to $175,000 and 28.0Vo on sources.

E

392


AMTI in excess of $175,000. AMTI taken times the appropriate AMT tax rate gives tentative minimum tax (TMT). Finally, the difference betwee4 the TMT and'the regular federal tax iiability is thb taxpiyer AMr for a given year, which if positive, is paid in addition to the regular tax, if any. , ,, A partial list of corporate 'AMT Adjustments?' includes; depreciation on assets place into service after 1986 and prior to 1999, mine exploration and development costs, long-term contracts (percentage of completion accountins inetirodology must be used), amortization of pollution control facilities, certain installment sales, and the alcohol tuels credits. An 'AMT adjustment" is defined as the yearly difference between the regular tax deduction for a given expenditure and its allowed deduction under AMT rules. As an example, mining development costs are 70vo expensed and 3avc amortized over 60 months for regular tax purposes, but these costs must be deducted straight line over 10 years for AMT purposes. Therefore, in the first year of such expenditures, AMTI wilr increase, but in later years,

the adjustrnent will reduce the AMTI exposure for that expenditure. The ANIT adjustment on mining development costs can be avoided by using straight-line, 10 year life amortization deductions for regular tax purposes. Unlike adjustments, 'AMT Preferences" (also known as Thx preference Items (TPIs) always increase the amount of AMTI. corporate Tax prefer ence Items include but are not limited to; mining percentage depletion that exceeds the cost depletion adjusted basis for a property, integrated pctroleum producer intangible drilling costs (IDCs), tax exempt interest on certain bonds and reserves for losses on bad debts of financial institutions. A inte-trated oil and gas company can avoid the Tax preference Item for IDCs by amortizing all IDCs over 60 months for regular federal income tax purposes

The 'Adjusted current Earnings Adjustment" (also known as the ACE Ad.iustment) attenrpts to align publicly traded companies taxable income as reported to the shareholders (measured by the ACE) with the company AN''ITI. A corporation's AMTI must be increased by 75vo of the amount by *'hich its ACE exceeds its AIurI, computed without the adjustments for c-ither the ACE preference or AMT Loss Forward deduction. Enough saidl For married persons filing a joint return, the AIVIT exemption is $45,000. This amount is reduced by 257a of the excess above $150,000 and is theretbre, completely phased out if income exceeds $330,000. For corporations, the exemption is $40,000. The AMT Loss forward deduction is known in the tax code as an Alternative Net operating Loss (or, Alternative NoL) and is calculated in a man-

i

j

I

I*


f

393

*

I It *

Ii

l

i $

I I I

fi

I

1

t.

ner similar to other loss forw'ard deductions, except that the Alternative

NoL is reduced by alt tax preference items and tax items that must adjusted for

be

AMTI. This deduction may not offset more than govo of AMTI

computed without the Alternative NOL. A taxpayer may recover any paid AMT in subsequent years that AMT is not paid, through the "Minimum Tax credit" (Ivfrc). A corporation is entitled to a credit against regular tax that equals its full AMT H;bility from previous years, but not exceeding its regular tax payment. Individuals however, must exclude certain adjustments and preferences when calculating their MTC. The MTC is a benefit to the taxpayer if it can be used quickly (due to time value of money). Depending on a taxpayers financial position, if year to year the company is moving in and out of an AMT liability, generally the effects of the tax are neglected due to the limited time value of money issues and the difficulty of trying to estimate the tax. However, for corporations with a longstanding tradition of exposure to AMT, minimization of this tax becomes the objective within the context of economic modeling. oil and gas producers may not use certain tax credits against regular federal income tax when they are under AMT. The alternative fuels credits related to coalbed methane and tight gas sands and the enhanced oil recovery credit cannot reduce regular federal income tax below TMT. This means that in any year a firm owes AM! it may not use the above tax credits. For eligible producers that expect they wilr pay AMT for several years, the loss of the tax credits can be a serious economic detriment. Another onerous aSpect of AMT is its regressive nature. when product prices are low, profits are squeezed which generates losses regarding regular federal taxable income. However, this may create more exposure to AMTI, as the generated deductions may not be fully deductible for AMTI. So, investors are hit with higher AMT when they can least afford to pay higher taxes.

AMT is expensive not only in terms of the additional tax burden it presents, bnt also because of the effort required to calculate the tax each year. corporations must norv maintain several sets of books to account for the regular federal income tax, shareholder financial reporting, adjusted current earnings, AMT and other business transactions. For some investors, the cost of monitoring the AMT calculations may exceed the resulting tentative minimum tax payment. subchapter S corporations, partnerships, real estate mortgage investment conduits (REMICs) and foreign corporations (excluding their income from u.S. business) are exempt from AMT. However, income from these entities

t.

394

i.s


transferred to individuals or shareholders who must then calculate regular

federal income tax and AMT. All other corporations and individuals with income must also compute AMT yearly. Thlrefore, it is necessary to consider the impact of AMT on investments and project.evaluations but given the complexity, usually this tax is negrected unless the investor is assured of a long-term exposure to AMT. As previously mentioned, in such cases the

tax rates reflect applicable AMT rates and deductions are modified to on a long run basis, due to the Minimum Tax credit, AMT should not result in additional tax being paid, it simply is a timing issue that each investor must address individually as to its signifireduce the exposure to AMT.

cance in the projects being evaluated.

8.7 Effective

Thx Rates for combined state and Federal rncome Tax

often it is convenient for evaluation purposes to combine the tax rates of two agencies such as a state and federal government to determine the effective tax rate that accounts for both with one calculation. For the typical indi_ vidual or corporate analysis where, under current tax law, state corporate or individual tax is deductible for pufposes of calculating federal tax, but federal tax usually is not an allowable deduction when calculating state tax. The following effective tax rate results: Effective Thx Rate = s +

f(l-s)

g_l

where: "sl.' is the incremental state tax rate in decimal form "f is the incremental federal tax rate in decimal form

EXAMPLE 8-4 calculation of corporate Effective Tax Rates Consider a $120,000 increment of corporate income subject to the incremental federal corporate tax rate of o4/o and a 6% state corporate tax rate. calculate the effective federal plus state tax rate and the tax to.be paid on the income for this corporation.

Solution: Using Equation B-1: Effective Corporate Tax Rate = 0.06 + 0.34(t_.06) = .3796 or 57.96"/" Total Tax Due = $120,000(0.3796) = $45,552

chapter 8: lncome Tax, vJorking capitar, and Discounted cash Frow Anai.ysis

395

The same resurt courd be obtained without the use of the effective tax rate as follows:

StateTax Total

6.8

=$120,000(0.06)

Tax

=$7,200 = $Sg,OSe

$45SS,

Tax Credits

certain business investments are eligible for an energy or special invest_ ment tax credit. The solar and geothermal energy credits and the rehabilita_ tion credit are referred to as .,Investment Credit.,' solar Energy Property. euarifying solar energy sysrems recei.,,e tax credit. This credit is permanent.

a

ra.Tvo

Geothermal Eiergy property. eualifying geothermal energy properry receives a l0.}Vo credit. This credit is pcrmanent. Enhanced oil Recovery @oR). A 15.Tvc credit applies ro a taxpayer,s cost for tangible properry, intangible ctrilling ana deietopmerlt cosis, and qiialified tertiary injectant exper)ses paid or incurred with an EoR prr-rject locatecl in the U.S. a,d invorving one or more tertiary recovery methods. costs applicable to this credit must be reduced by the credit before being deductcd fbr tar purposes. Research and Experimentation Credit. The research credit calculation is complex with several tests required to be satisfied. Simplistically, incremen_ tal research expenditures that exceed the base weighied uu"rrg" research expenditures for preceding years of investor operation may qualify for a ?o.aTo research tax credit. Research and experimentation i, the exper"ori, inrental or laboratory sense are anou'ed research costs, ar)d generally donot include product development type expenses. The research tax deduction must be reduced by the amount of the research credit. See a tax manual for details ofresearch costs that qualify and the tax credit calculation details.

Rehabilitation Expenditure credits. Certain expenditures related to the rehabilitation of qualified old buildings entitle tuipuy"r, to the following rehabilitation cost tax credits.

396


1) certified Historic Structures - z\vo of qualified rehabilitation costs. Historic structures are either residential or non-residential buildings Iisted in the National Register or locatsd in a registered historic district and certified as being of historic significance to the district.

2) Qualified Rehabilitated Buildings - r\vo of qualified rehabilitation costs for buildings placed in service before 1936.

The depreciation basis of a rehabilitation property must be reduced by the

full rehabilitation credit. Refer to a tax manual for specific tax rules that describe qualified rehabilitation costs.

Alternative Fuels credit. A tax credit is allowed for the domestic production bf oil, gas, and synthetic fuels derived from non-conventional sources (such as shale, tar sands, coal seams, and geopressured brine) that are sold tq non-related persons. The credit is claimed by attaching a separaie schedule to the tax return showing how the credit was computed. The credit is generally $3 per 5.8 million BTUs (energy equivalent of one barrel of oil) produced and sold from facilities placed in service after 1979 and before 1993, or from wells drilled after 1979 and before 1993. Such fuels must be sold before 2003. The time period is extended for certain facilities subject to a written binding contract in effect before 1996. The credit is phased out as the average wellhead price of uncontrolled domestic oil rises fr.m $23.50 to $29.50 per barrel. The credit and phase-out range are adjusted tor inflation. Tax credits from all sources are credits against your regular tax bill and not cleductions fi'om tarable income, sttch as depreciation. A dollar of to.r credit saves a dollar o.f tax cost, vvhile a dollar of depreciation saves onl-t tlrc tax rate tintes a dollar in tax cost. Howeve4 tax credits carutot be used to reduce AMT.

Tax credits

of all types are limited in any tax year to the lesser cf

the

income tax liability or $25,000 plusT5vo of tax liabiliry in excess of $25,000.

EXAMPLE 8-5 Tax Credit Etfects on Cash Flow Compared to

Depreciation Effects

consider a project evaluation year in which annual revenue is 8500,000 and operating expenses are 91s0,000. Assume an investor

chapter B: lncome Tax, woi'king capitar, and Drscounred cash Frow Anarysis

40% effective income tax rate and calculate the annual revenue cash ilow for the year assuming: case 1) no depreciation deduction or tax credits are applicable; case 2) annual depreciation is 9100,000 with no tax credits applicable; Case 3) annual depreciation is zero with a $100,000 tax crecjit appircable. Also caicuiate the maximum tax credit that could be utiiized in the case 3 project evaluation year.

Solution: All Values in Thousands of Doilars Case 1 No Deprec. or Tax Credit Revenue -Cp. Expenses

-Depreciation Thxable lncome -Tax @40"/" +Tax Credit

j 1i

1i

{ ,, 'l

2

Case Case O Deprec. $100 $100 Tax Credit No Tax Credit and No Deprec.

500

-Y

500 -1 50 -1 00

_Y

350

_Y

250

350 -1 40 +1 00

-11

Net lncome +Depreciation

,:

Cash Flow

210

150 +1

00

250

500

'l 310

Note that in case 2 the $100,000 depreciation deduction reduced taxable income by $t 00,000 compared to case 1 with no depreciation or tax credit, which reduced the income tax by g40,000 and increased the cash flow by g40,000 to g250,000 from $eto,oo0. tto*ever, in case 3 the $100,000 tax credit with no depreciation saved $100,000 in tax compared to case 1 with no depreciation or tax credir, which increased cash flow $100,000 to $g10,000 from $2'10,000. ln this analysis, based on a 40"/" effective income tax rate, one dollar of tax credit saves the investor one dollar while one dollar of depreciation saves the investor $0.40. Obviously, a dollar of tax credit is more important to an investor than a dollar oi tax deductions. The maximum tax credit that can be utilized in a year is the lessor of the income tax liability or 925,000 plus 75% of federal income tax liability in excess of 925,000. lf the entire $140,000 income tax liabil-

398

Ecbnomic Evaluation and lnvestment Decision Methods

ity for case 3 is assumed to be federal tax liability, g25,000

+

0.75($1401000'- $2s,oo0) = gl :11,2s0 would be the,maximum usabte tax credit in the year. Any unused credits in a year can be carried forward to subsequent years. lf part of the 40o/o ettective income tax rate is due to state tax, the maximum tax credit calculation would be based on the federal tax obligation.

8.9 Discounted cash Flow Rate of Return (DCFROR), Net present Value (NPV) and Ratio Analysis

DCFROR is defined commonly as the rate of return that makes the

present worth of positive and negative after-tax cash flow for an investment

equal to zero. This is another way of saying DCFROR is the rate of return that makes after-tax NPV equal to zero. Discounting means to ,.present worth" and the name "discounted cash flow rate of return,, comes from the fact that classically, present worth or discounting equations most often have been used to obtain the DCFROR. It already has been shown many times earlier in this text that the same rate of return results from writing an

annual or future worth equation or in general by comparing costs and

incomes (or negative and positive cash flow) at any poini in time, as long as all cash flows are brought to the same point in time. It was mentioned earlier in this chapter that other names commonly used for DCFROR are Profitability Index (P.I.), Investor's RoR, True RoR, and Internal RoR or IRR. Polls of industry presently indicate that DCFROR is the number one economic evaluation decision method used by about two thirds of industrial companies that use a formal economic evaluation procedure to evaluate the economic potential of investments. In this regard it is relevant to note that most major industrial companies use formal discounted cash flow investment evaluation procedures of the type described, discussed, and illustrared in this text.

Net present value (NPV) on an after-tax basis equals the present worth of pcsitive and negative cash flow calcurated at the after-tax minimum rate of return. NPV greater than zero indicates a satisfactory investment. proiect DCFRoR is compared to the after-tax minimum rate of return, .,r*,,, ,-o determine if a particular project is satisfactory compared to other alternative uses that exist for the investment capital. For after-tax analysis using any valid analysis technique, the minimum rate of return, ,,i*,,, must ie expressed on an after-tax basis. comparing an after-tax DCFR)R for a

Cnapter 8: lncome Tax, Working Capital, and Discounted Cash Flow Analysis

prrject w'ith other i-rutestment opportunities expressed on a before_tax basis wou.ld be meaningless. With proper handling of inflation and escalation in analyses it was emphasized earrier that all arternerti,,es must be coinpared on the same basis, either using escalated d'lrar an:ri.sis or consiant doilar an'rlysis. similarly. with after-tax anaiysis, air alterriatives must be on an alier-tax basis including other opportuirities repre,senteo uy ttre minin:um ratc of return, "i*',. After-tar prcsenr value ratio, (pvR). or benefit/cost ratio, (B/C Ratio), is^c_alculated using after-tax positive and negative cash flow similar to the NPV and DCFROR calculations. correct ratio,Jenominators are the present x-orth of project negative cash flow not covered by positive cash flow in the year negative cash flow is incurred or in earlie. yiu.r. pvR greater than zero is satisfactory u'h,e B/c Ratio greater than oni is satisfactory, To calculate DCFR'R, Npv or rdtios, the first thing that must be done is tii calculate after-tax cash flow for each year of the evaruation life. This includes converting a[ capitar costs and salvage values to after-tax values by accounting properry for all appropriate tax considerations. it is desirable to place the after-tax negative and positive cash florvs on a time diagram to insure that the sutisequent evaruation handres them at the correct point in time. After-tax evaluation calculation procedures are ilrustrated in Exanr_ ples 8-6, 8-7, 8-8 and 8-9 for relativell, straightforward evaluation cases. In chapters 9, l0 and rt, DCFROR, N'pv and rarios are appried ro a wide of evaluation situations that will familiarize the reader with the han_ 'irriety dling of these ancl other evaluation methods in a diversity of investment sit_ uations' The DCFR.R, Npv, and ratio calculations u." u"ry straightforward once the after-tax cash flows are determined. The key to correct, successful economic evaluation work rests heavily on experience with proper methods for the following evaruation points: (l) correct handling of all allowable

ii

1i

; I t

1,

i {I

income tax deductions, especiaily depreciation, amortization, depletion and deferred deductions to deiermin" .oi.".t taxabre income; (2) addi,g beck to net income the proper non-cash deductions to obtain cash flow (deferred deductions, amoitized research and deveropment expenses, and special tax write-offs are add-back items frequentry mishandred); (3) proper handling of tax considerations for costs thai ,nuy u. written off for tax purposes in the year in which they are incurred agarnst other income rather than being capitarized and deducted over a perlod of time greater than one yeat; (4) correct accounting for tax effects on salvage value, either tax to be paid or tax write-offs to be taken (in some investment situations i, $r l:i u

400

Ecohomic Evaluation and lnvestment Decision Methods

such as land investment, salvage value may be the major project income so that salvage tax considerations oe tranotea ctrrectly.to get

it is imperative

analysis results); (5) proper handling of incremental tax credit con'alid siderations when looking at differences between alternatives, such as in rnutually exclusive income alternative analysis of projects for which tax credits are applicable. The next example illustrates the sensitivity of DCFRoR analysis for dif_ ferent methods of handling the tax deductions for a given cost. some costs such as equipment and buildings are depreciable by the MACRS

_

rates

or by straight line depreciation. other costs such "ith", as operating expenses, research, development, intangible drilling costs and mineral explorition or

development may, at leairt in part, be expensed in the year incurred. Expens. ing is the fasrest method of deducting costs for tax purposes and depending

on the in'estor financial .situation, may require (r) carrying Iosses "itrr.r, which forward, if other taxable income does not eiist against to use negative taxable income tl year, (making the p-3""t ,.stand alone,,); (21. ,ury crediting the project with tax savings in the year negative taxable income is iucurred by assurning other taxable income will exist against which to use deductions. Economic results are affected significantly'uy trre method that costs may be deducted for tax purposes and by the hnancial assumption concerning r.vhether income will or will not exisi against which to use nega_ tive taxable income in the year incurred.

EXAMPLE 8-6 The Effect of rax Deduction Timing and the rnvestor Financiar situation on DCFRoR Resurts

A $100,000 investment cost has been incurred by a corporation to Jenerate a project with a 5 year rife and estimaied zeio sarvage ralue. Project escalated dollar income is estimated to be $g0,0001n 1_, $44,000 in y_ear 2, $88,000 in year 3, $92,000 in year 1911 -be 4, and 896,000 in year s. operating are estimated to $30,000 n year 1, 932,000 in year 2, "*p"n.er $94,000 in year g, $36,000 in year 4, ind $38,000 in year 5. The effective income tax rate is 40.0%. write>ff remaining book values at the end of project year 5. Determine lroject DCFROR if the initial $100,000 investmenl goes into service n year 1 and is handled in four different ways for lfter-tax analysis )urposes. Consider the 9100,000 investment is:

Cnapter 8: lncome Tax, Working Capital, and Discounted Cash Flow Analysis

A) Depreciable straight line for a 5 year life assuming the half-year

1

convention.

B; Depreciable by MACRS for a 5 year life assuming the half-year

1

conver,ticn.

C) Expensed as a research cost in year 0 with negative taxable income carried forward to be used against project income (stand alone econornics). D; Expensed as research cost in year 0 assuming other taxable income exists against which to use the deduction.

Solution: All Values in Thousands of Dollars lJcte that the cumulative amount of tax deductions is $100,000 by ali four methods of analysis. Only the timing of the tax deductions differs between methods. Since a!r methods of analysis involve the same cumulative revenues and tax deductions, they also involve the same cumulative tax and cash flow. Hou;ever, as you go frcm A) straight line depreciation, to, B) MACRS depreciation, to, C) expensing and carrying forward, to, D) expensing against other income, the timing of tax deductions, and therefore tax savings, is affected significantly causing notable differences to occur in the DCFROR results. ln reviewing the four solutions, note that in each case, the cumulative net income will equal the cumulative cash flow generated by the project. ln all cases, the cumulative value is $102 but to obtain this value for the Case C) net income, combine the cumulative income with the loss forward deductions.

This equality between cumulative net income and cumulative cash flow is not by chance. ln calculating the after-tax cash floy,r, capital costs are charged to the project while depreciation is deducted to determine taxable income and then added back to net income. Therefore, the depreciation deduction is cancelled out, except for the tax effects. ln focusing on net income, depreciation and write-offs represent an allocation of the capital costs, rather than a tax deduction. The timing difference in the two series is what's critical to the economic evaluation process.

402


A) Straight Line Depreciation The half-year 1 deduction causes book value amounting to a harf year depreciation deduction.to exist at the end of year.s.,This book value is deducted (written-off) at year 5. Note that cumulative depreciatlon over the 5 year project rife prus the write-ot"qruis the $100 capital cost. Note that the cumurative income tax paid over the five-year rife is $68 and the cumulative cash flow is $loz. you wili observe that these cumulative numbers are the same for cases B, c and D. This is not a coincidence! only the timing of the deductions varies which affects the timing of the income tax alnd cash flow from cases A to D. Year

5

Revenue -Oper. Costs

-Depreciation -Write-off Taxable lncome -Tax @ 4O/o

lncome +Depreciation Net

+Write-off -Capital Costs -100.0

80.0 84.0

88.0 _34.0

92.0

Cumulative

96.0

44A.O

-30.0 -32.0 -36.0 -38.0 -10.0 -20.0 -20.0 -20.0 -20.0

40.0 32.0 34.0 -16.0 -12.8 -13.6 24.0 10.0

36.0 _14.4

19.2 20.4 20.0 20.0

21.6 20.0

-10.0

-170.0 -90.0 -10.0

28.0

170.0

-11.2

-68.0

+ 41.6(P/F;,4) + 46.8(p/F;,5)

DCFROR = 27.45o/o

r? I .l

16.8 20.0 10.0

-100.0 34.0 39.2 40.4 41.6 46.8 PWEq:0 = -r99 + se.2(p/Fi,2) + 40.4(p/Fi,s) i.:o_o(P/F;,1) Cash Flow

i:

102.0 90.0 10.0 _100.0 102.0

I

I

Chapter 8: lncome Tax, Workrng Capital, and Discounted Cash Flow Analysis

B) Modified ACRS Depreciarion

I I q

i i

Note that cumulative depreciation over the 5 year project life plus the write-off equals the $100 capital cost as with straighi line depre_ ciaiion. However, the timing is different and the taster deductions (bigger deductions in early years) with MACRS depreciation retative to straight line give faster tax benefits and better economics as shown by the 29.0% DCFROR with MACFIs depreciation compared lo 27 .45"/" with straight line depreciation.

Year012g4 80.0 84.0

Revenue -Oper Costs

88.0

5 Cumulative 92.0 96.0 440.0

-30.0 -32.0 -34.0 -36.0 -38.0 -170.0 -20.0 -32.0 -19.2 -11.5 -11.5 -94.2

-Depreciation *Write-off

-5.8

Thxable lncome -Tax @ 40"/"

Net lncome +Depreciation +'vVrite-off

30.0 20.0 34.8' 44.5 40.7 170.00 -12.0 -8.0 -13.9 -17.8 _16.3 _68.0 18.0 12.0 20.9 26.7 24.4 102.0 20.0 32.0 19.2 1 1.5 1 1.5 94.2 5.8

-Capital Costs *100.0

PWEq:0

i-

=

-100 + 38.0(P/F;,1 ) + aa.OftFi,2) * + 38.2(P/Fi,4) + 41.7(P/Fi,5) DCFROR = 29.02/"

5.8

-100.0

-100.0 38.0 44.0 40.1 38.2

Cash Flow

-5.8

41.7

40.1 (p/Fi,3)

102.0


404

C) Expense Research and Carry Loss Forward (Stand Alone) Remember, a "stand alone" financial scenario assumes the project

being evaluated is the only source of income available for the investor. Therefore, any negative taxable income (a loss in that year) must be carried forward and used against positive taxable income in later years. Note again, the cumulative deduction realized equals the $100 cost but the deduction and tax benefits from the deduction are real-

ized more quickly than for either depreciation case. Therefore, expensing the $100 cost gives a better economic result, as the 32.65% DCFROR compared to the smaller depreciation case results illustrates.

4

Year

Revenue -Oper Costs

-Research -Loss

-100.0

Fonruard

lnc.

Taxable -Tax @ 40"h

-100.0

-50.0 -0.8

Flow -100.0 50.0 0 = -100

-1s0.0

2.0 54.0

-100.0 -50.0

-50.0 .2 Forward 100.0 50.0

PW Eq:

Cumulative

80.0 84.0 88.0 92.0 96.0 440.0 -30.0 -32.0 -34.0 -36.0 -38.0 _ll3:3

Net lncome -100.0 +Loss -Capital Costs Cash

5

1

51

-21

58.0 -23.2

-68.0

33.6 34.8

-48.0

56.0

.6 -22.4

32.4

.2 32.4 33.6 34.8

i i

20.0

150.0 102.0

+ 50(P/F1,1)+ 51.2(P/Fi,2) + 32.4(P/F;,3)

+ 33.6(P/F;,4) + 34.8(P/F1,5)

i = DCFROR=32.65%

n

Chapter 8: lncome Tax, Working Capital, 6nd Drscounted Cash Fiow Analysis

D) Expense Research Against.Other lncome Again, the cumulative deduction equals the $100 cost, but having other income on the investor tax return in year 0 against which to use the negative taxabie income (-$tOO; gives the investor the tax benefit of inflow of money immediately in year 0. This gives a better economic result than any of the other cases. ln the U.S., wh:re quar-ter-ly estimated tax payments must be made by all business investors, tax savings from tax deductions generally are realized within three months of incurring tax deductible costs.

4

Year

80.0

Revenue -Oper Costs

84.0

-30.0 -32.0

-Research

-100.0

5

Cumulative

88.0 92.0 96.0 44A.A -34.0 -36.0 -38.0 -170.0 -100

-68.0

Flow -60.0 30.0 91.2 32.4 33.6 g4.B

1.02.0

'170.0

*Capital Costs Cash 'i

0

' Taxable lnc. -100.0 50.0 52.0 84.0 5e .0 58.0 -Tax @ 40% 40.00 -20.A -20.8 -21.6 -22.4 -23_2 Net lncome *60.0 90.0 g1.Z gZ.4 93.6 34.8

PW Eq:

0 = -60.0 + 30.0(P/F;,1)

102.0

+ 31 .2(plFi,2) + Sz.ap/F,,g)

+ 33.6(P/F;,4) + 34.8(p/Fi,5)

i

= DCFROR=44.16%

The reader should observe from these anaiysis results that the faster an investor realizes tax deductions and the tax benefits from the tax deductions, the better the economics of projects become. Expensing the $100,000 cost against other income gives a 60"/o increase in the DCFROR that is obtained by straight line depreciation of the cost. lf you have to carry negative taxable income forward to make the project economics "stand alone', as in part ,,C,,'the project economics do not look as good as when other income is assumed to exist against which to use deductions in the year incurred, as in part "D."


part "D" as Finally, as an alternative to treating the research cost in of 1'0 in rate a wlth I tax ex[ense_ item, consider it to be depreciablepraciice, so in induStry rcat A. fnis iS an approach'sometimes'used :he reader should be'aware of the equivalence of either apprgach aS "D" illustrate' :he following year 0 cash flow calculations for part An Alternative to

"D"

year

o

an

Research as ExPensed Cost Revenue -Oper Costs

-Besearch -Research as DePreciation

-100.0

Taxable lncome -Tax @40"h

-100.0

year 0 Research as a DePreciable Cost

-100.0 -100.0

40.0

40.0

Net lncome +Depreciation -Capital Costs

-60.0

-60.0

Cash Flow

-60.0

100.0

-100.0 -60.0

Example 8-6 involved a single investment cost that was either depreciated invenor expenied for tax deduction purposes. Most businesses also require ,ory .ortr, commonly called working capital' which are not depreciable as discussed in the following section.

8.10 lVorking CaPital worliing capital is the money necessary to operate a business on-a day+o-day inventory basis. It normally is comprised of mouey requireJ for raw material and receivable' in-process ,r,ut"iiul, inventory, product inventory, accounts considered to ready cash. For evaluation purposes, working capital generally is and to be Ue put into a project at the itart of a business or production operation are liquidated' inventories full-v recovered at the end of the project life when Working capital is not allowable as a tax deduction in the year it is incurred capital so it often has a very negative effect on project economics. working inventory until depleted or amortized cost may not be expensed, depreciated, working capital assets are actually used or put into service. One way to explain

E

Chapter 8: lncome Tax, Working Capital, and Discounted Cash Ftow Analysis

407

and w'hy it is not deductible for tax pulposes in the yezr it is incurred, is to con_ sider the determination of the "cost of Goods sold" as it is handled on corpor.rte or individual business tax returns. Table 8-3 illilstrards the sreps necessarv to calculate the annual cost ofgoocls solii for a business operated either by iii; iilCii idual, partiership or corporation. in this cost o1'go64siold calculation, i.riue oi inventories rrt the;,ear end is working capital and is not rax deductible.

Begir:ning of Year Inventory + R.tu lvlaterial Costs from purchases During the Year + Liii:t;r Costs to Convert Rarv Material or parts Into products + lUirterials, Parts & Supply Costs Incured During the year + Other Costs Related to production of products = Cosl of Materials, Supplies ,nd G,,odl*uilubl" f* *- Inventory value at year End Based on Lesser of cost or value = Cost of Goods Sold (Deductible as Annual Operating Cost) Table 8'3 calculation of cost of Goods Sold Related to working capital \\brking capital represents the capital cost required to generate raw material inventories, in-process inventories. product inventories and parts and supplics inventories. As inventorics are used and prociuct sold, working capilal .:c;st items beconre allou'able tax deductions as operating costs through the cost of grlods sold calcr-rlation. However, as inventory items are used they tl pically are replaced so inventories are maintained at a similar level over the project life. If significant increases or decreases in working capital are projected to occur from year to year, positive or negative working capital costs can be accounted for from year to year in pro;eit analyses. Now it should occur to you that raw material and parts in inventory often are acquired at different costs durin_q a year. How do you determine the value of iterns left in inventory at the end of a tax year when items u,ere acquired fbr difi'erent costs during the year with some used and some left in inventory? FIFO, LIFO and average inventory accounting are the three basic inventory accounting systems that detennine the cosr.s of items used during a tax year to be deducted as operating expenses and the costs of items left in inventory and treated as working capital. FIFo is the acronym that stands for,,First-In-Firstout." using FIFo inventory accounting, the first items to go into inventory are considered to be the first items to come out and be used and deducted as operating expenses. Therefore, under FIFo inventory accounting, the last

408


items purchased during a tax year are the cost basis for inventory assets tied up in working capital.

LIFO is the acronyrn for "Last-In-First-Out." Using LIFO inventory accounting, the last items to go into inventory are,considered to be the fust items to come out and be used and deducted as operating expenses. Therefore, under LIFO inventory accounting, the lirst items purchased during a tax year or in inventory from earlier years are the cost basis for inventory assets tied up in working capital. In an inflationary climate the cost of items purchased during a year generally increases and LIFO gives tax deduction for the bigger cost items purchased later in the tax year than are realized with FIFO. In a deflationary climate the opposite is true. A majority of U.S. companies use LIFO inventory accounting and LIFO is the implicit inventory accounting assumption built into the handling of working capital in this text. Under LIFO, assuming uniform quantities of rnaterial in inventory each year, money tied up in working capital is constant each year with changes in item costs accounted for as operating cost changcs. Average inventory accounting is based on using annual weighted average prices or values of assets in inventory for the cost of goods sold and working capital calculations. In most western world countries other than the U.S., LIFO inventory accounting is not permitted. Therefore, FIFO and average inventory accounting are emphasized in countries other than the U.S. In accounting terminology working capital is defined as the difference between current assets and current liabilities. This definition is consistent with our "Cost of Goods Sold" calculation explanation of working capital. When a new business or production facility is started-up, it often generates and produces product for three or four months before product is sold and income is received for product sold. Assuming the business pays cash lor raw materials, parts and supplies during the start-up period, the cost of these items increases the "current assets" of the business. lf all items have been paid for with cash (no time payments), no current liabilities are accrued, so current assets minus current liabilities equals the working capital which is the value of items in inventory including product inventory.

EXAMPLE 8-7 Project Analysis with Annual Changes in Working Capital: FIFO, LIFO, and Average lnventory Accountin g A project is being analyzed that has product selling price, production costs, annual production and annual sales summarized as follows:

L

Chapter 8: lncome Tax, Work;ng Capital, ail._l Discounted Cash Flow Analysis

409

Product Price and Production Cost lnformation Year

Selling Price, $/Unit Production Cost, $/Unit Prod uction/l

$21

$zt

$21

Ci 2

$zt

$10

$15

$tz

1,000,000 1,000,000

1,200,000

nventory lnformation

Year

Units Produced Units Sold

Change in lnventory Cumulative lnventory

500,000

1,000,000

0

800,000

5O0,0OO

500,000

200,000

700,000

s00,000

0

-700,000

700,000

0

Cost of Units Produced $5,000,000 913,000,000 $15,000,000 $8,500,000

To generate this prociuction, ayear 0 capital cost of $10,000,000 is projected to be incurrecJ with I year life MACRS depreciation applicable starting in year 1. The project will terminate at the end of year 3, with zero salvage value and a write-off on remaining depreciable book value. The effective income tax rate is 40"n. calculaie prcject DCFROR assurning the cost of goods soid (coGS) operating cost and working capitai valuations ai.e based on:

A) First-ln-First-Out (FIFO) inventory accounting B) Last-ln-First-Out (LtFO) inventory accounting C) Average inventory accounting

Solution: In comparing the FlFo, LlFo, and average inventory accounting economic results for the analysis, note that LlFo givel the largest cost of goods sold operating cost deductions and the smallest working capital costs in early project years relative to FlFo and average inventory accounting results. Therefore, the best economic results occur with LlFo. with escalating production costs from year to year, LlFo always gives faster tax deductions relative to FlFo and aveiage inventory accounting, which is why virtually all large U.s. companies use LlFo inventory accounting for all major raw miterial and product inventory items. For this analysis, and in general for all analysis situations, average inventory accounting givel project results that are in between FIFO and LIFO results.

410

A)


FIFO lnventory Accounting

Cash plowslijsing Cost Of Goods Sotd and Working Capital . :. Based on FIFO

23 Revenue - Cost of Goods Sold" - Depreciation - Deprec. Write-off

0 0 0

Taxable lncome

0 0

-fax

@ 4Oo/o

16,800,000 21 ,000,000 25,200,000 -8,900,000 -1 3,600,000 -1 9,000,000 -1,429,000 -2,449,000 -1,749,000 -4,373,000

6,471,000 -2,599,400

4,951,000

-1,990,400

79,000

-31,200

Net lncome 0 3,892,600 2,970,600 46,900 + Depreciation 0 1,429,000 2,449,000 6,122,000 - Capital Equipment -10,000,000 - Working Capital*. -5,000,000 -4,1oO,OOO -1,4oO,OOO 1o,50o,OOO. Cash Flow

DCFROR = 15.150/"

-15,000,000

1

,21

1

,600 4,019,600

16,668,900

NPV @ 12o/o= $1,150,709

*Cost of Goods Sold = COGS COGS = Value of Units From tnventory + production Yr 1 COGS: (500,000 * $10) + (300,000 * $13) = g8,900,000 Yr 2 COGS: (700,000 x $13) + (300,000 * $1S) = g18,600,000 Yr 3 COGS: (700,000 x $15) + (500,000 x $17) = g19,000,000 Note: The year 3 COGS represents the gg,500,000 cost of product produced in year 3, along with the $10,500,000 write-off value of inventories drawn down to meet product sales in that year. **Working Capital = Cost of Units produced - COGS Yr 0 Work Cap: 500,000 * $10 - 0 = $5,000,000 Yr 1 Work Cap: 1 ,000,000 * $13 - $8,900,000 = $4,100,000 Yr 2 Work Cap: 1,000,000 + 915 - 913,600,000 = $1,400,000 Yr 3 Work Cap: 500,000 * $17 - $19,000,000 = (g10,500,000). 'Yr 3 Working Capital is a credit for reducing inventory.

-tl

i

j ,! 1

t

n il :i

Cha,:te!'B: Income Tax, Working Capitai. and Dlscounted Cash Flow Analysis

41'l

B) LIFO lnventory Accounting cash Flows using cost of Goods Sold and working capital Based on LIFO Y,aar

0

Revenue - Cost of Goods Sold. - Depreciation - Deprec. Write-off

6,800,000 21,000,000 25,200,000 0 -1 0,400,000 -15,000,000 -16,100,000 0 -1,429,000 -2,449,000 -1,749,000 -4,373,000

Taxable lncome

0 0

-Tax @ 40"/"

'r

4,97't,000 3,551,000

Net lncome 0 2,992,600 + Depreciation 0 1,429,000 - Capital Equipment -10,000,000 - V/orking Capital." -S,000,000 -2,600,000 Cash Flow

DCFROR

2,130,600 1 ,786,800 2,449,000 6,'122,000

0

-15,000,000 1,811,600 4,579,600

-

15.7A%

2,978,000

-1,989,400 -1,420,400 -1,191,200

7,600,000. 15,508,800

NPV @ 12Y"= $1,307,187

*Cost of Goods Sold = COGS coGS = Value of Last units produced + Necessary lnventory Yr 1 COGS: (800,000 * $13) = 910,400,000 Yr 2 COGS: (1,000,000 * $15) = $1S,OO0,0OO Yr 3 COGS: (500,000 * $17) + (200,000 * $13) + (500,000 yr 0 production) x g1O = $16,100,000 Note: The year 3 COGS represents the gg,500,000 cost of product produced in year 3, along with the 97,600,000 write-off value of inventories drawn down to meet product sales in that year. ** Working Capital = Cost of lnventory produced - COGS Yr 0 Work Cap: 500,000 * $10 - $O = $5,000,000 Yr 1 Work Cap: 1,000,000 * giS - 910,400,000 = 92,600,000 Yr 2 Work Cap: 1,000,000 * 915 - $15,000,000 $0 = Yr 3 Work Cap: 500,000 * $17 - $t 6,1 o0,0oo = (g7,600,000). "Yr 3 Working Capital is a credit for reducing inventory.

412


C) Average lnventory Accounting

Cash Flows Uslng Cost Of Gooits Sold hnd Working Capitat BaseC on Average lnventory Year

Revenue - Cost of Goods Sold* - Depreciation - Deprec. Write-off

0 0 0

16,900,000 2'1,000,000 25,200,000 -9,600,000 -13,764,706 -18,135,294 -1,429,000 -2,449,000 -1,749,000 -4,373,000

Taxable lncome

0 0

5,77'.1,O00 4,796,294 -2,309,400 -1,914,519

-Tax

@ 4O%

0 0

Net lncome + Depreciation - Capital Equipment -10,000,000 - Working Capital** -5,000,000

942,706 .-377,O92

3,462,600 2,871,776 565,624 1,429,000 2,44q,OOO 6,122,000 -3,400,000 -1,235,294 9,635,294"

Flow -15,000,000 1,4g1,600 4,OgS,4g2 16,g22,91g DCFROR = 15.35% NPV @ 12o/o = g1,207,03g

Cash

*Cost of Goods Sold - COGS coGS - Average Varue of Units produced and sotd Each yr Yr 0 COGS: (500,000 produced, none sold) g0 = Yr 1 COGS: (($tO * 500,000 + $13 * 1,000,000)/1,500,000) * 800,000 = $9,600,000 Yr 2 COGS: (($tZ * 700,000 + 915 * .1,000,000)/1,700,000) * 1,000,000 = $1 9,764,706 Yr 3 COGS: ($13.70 x 700,000 + $17 * 500,000) g1B,.t gS,Zg4 = Note: The year 3 COGS represents the $g,500,000 cost of product produced in year 3, along with the $9,635,294 write_off value of inventories drawn down to meet product sales in that year. ** Working Capital = Cost of Goods produced - COGS Yr 0 Work Cap: $10 x 500,000 - $O = $5,000,000 Yr 1 Work Cap: g'13 ,r 1,000,000 - 99,600,000 $3,400,000 = Yr 2 Work Cap: g1S x 1,000,000 g19,764,206 = $1,23i,2g4 Yr 3 Work Cap: 917 * 500,000 $t g,1g1,Zg4 $9,635,294" = 'Yr 3 Working Capital is a credit for reducing inventory.

ll

?

i

chapter 8: lncome Tax, working capitar, and Discounted cash Frow Anarysis

413

EXAMPLE 8-8 Rerative sensitivity of DcFBoR and Npv to Project Life and Start of production A new project is being considere,J that would require the investment of S200,000 for processing equipment and $60,000 fcr workir..:g capiiai at y'ear 0. The equipment would be depreciated using year 5 lrfe sti-aight line depreciation starting with the half-year conveni;cn in year 1 when the equipment is expected to go into service. Annual r,uu"nr", are projected to be $300,000 and annuar operating costs are projected to be $200,000 with a wash-out of escaiation of o-per"ating costs ind revenue eacf salvage value and working capital return are expected to -year. total $100,000 at the end of the prolect. The effective income tax rate is 4ao/" and rninimum DcFRoR is 12%. calculate DCFROR and NpV for: Case A) A 10 year project evaluation life Case B) A20 year prcject evaluation life. case c) Assume technical difficulties or envii-onmental permitting delays cause the case A cash flows in years 1 through 10 to be realized in years 2 thi'ough 11 with zeio cash frow in year 1. Solution: All Values in Thousands of Dollars Case A) 10 Year Evatuation Llfe

'

Year

0

Revenue

300

-Operating Costs -Depreciation -Write-off Taxable lncome -Tax @ 4A"h Net lncome r-Depreciation +Write-off -Capital Costs Cash Flow

1 -200

-oo

: 80

60

-32

-24

48

_;

Z_s 300 -2AO

v

7-9

10

300

300

400

-2A0

-200

-200

100

-60 140

-40

-56

-:o

80 -JZ 48

.:

60

+68 +76 +68 +60 . Revenue includes-260 $100 salvage and working

84 +60

+144

capital return. working capital invesiment tax deduction write-off. NPV Eq: 0 = -260 + 68(piF;,1) + 76(p/A; ,4)(ptFi,1) + 68(p/F;,6) + 60(P/A;,3)(p/F;,6) + 1 44(p /Fij g) DCFROR = 25.24o/o, NPV @ 12/o +$160.6 =

.. original

/L

**

414


Case B) 20 Year Evaluation Life

Year

O

Revenue

300

-Operating Costs -Depreciation -Write-off

-200

-!

Net lncome +Depreciation +Write-off -Capital Costs Cash

Flow

7-19

2-5

300 -200

i'

300

-200

-!

80 60 -32 -24

Taxable lncome -Tax @ 40"/"

*

1

48

v

36

:,

80

-32 48

.:

20

300 400. -200 -200 -60 "*

100 -40

140

-56

60

84

;

-260 -260

+68 +76 +68 +60

+144

Revenue includes $100 salvage and working capital return. *" Original working capital investment tax deduction write-off.

NPV Eq: 0

+ 68(P/F;,1) + 76(P/A; ,4)(PlFi,1) + 68(P/F;,6) + 60(P/A;, 1 3XP/F;,0) + 1 44(P /F i,2g)

= -260

DCFROR =26.77"/", NPV @ 12oh=+$251.5

Although beforetax profit literally doubles for the 20 year life project as compared to the 10 year life project, DCFROR only increases 1.5% (trom 25.2/" lo 26.7"h, a 6% change). The change in NPV due to doubling the project li{e, however, is much more significant. The 20 year life NPV of +$251.5 is $90.4 greater than the 10 year life prolect NPV of $160,6 (a 56% increase). Discount rate magnitude is the reason NPV results, in this case and in general, are more sensitive than DCFROR results to changes in cash flow beyond 10 years in the future. 'l-he NPV discount rate of 12/"is smaller than the DCFROR results of 25.2/" and 26.7"/". As a result of the mathematical definition of the single payment present worth factor, (1/(1+i)n), larger values of "i" cause the present worth factors to be much smaller than for smaller discount rates when the evaluation period, n, is 10 or greater. ln this analysis, since the 12% NPV discount rate is much smaller than the DCFROR rates, this causes the years 11 through 20 cash flows to have a much greater impact on the NPV result than on DCFROR result.

Chapter 8: lncome Tax, Worring Capital. and D:scounted Cash Flow Analysis

415

Case C) Rework Case A for 1 year Startup Delay Year

Cash

Flow -260 O

3-6 7 8-10 11 +69 +76 +6g +60 +144

i\PV Eq: 0 = -200 + 68(p/F;,2) + 76(piAi ,4\plFi,Z) + 68(p/F;,7) + 60(PiAi,3)@Fi.) + 144{p/Fi,1l DCFROR = 19.78o/o, NPV @ 12.0% +$1 15.6 =

This is a 21.6% reduction in DCFROR to 1g.7g% from the base case A result o'f 2s.24/o and a zg.a% reduction in Npv to +g115.6 from the base Case A result of +g160.6. These discounted cash flow c:'iteria changes are sign.ificant enough to emphasize the importance of the timing as weil as the magnitud! of cosis ano revenues that go inic anaiyses. 8.11 International project Evaluation Considerations Companies, individuals and sovemments invest and sell products outside their own countries for many difrerent reasons. Reduction of operating costs, penetration of irrternational markets. hedging against curency exchange rate variations and improvement of technical servic.: to intemational customers ;ire so;ne of the reasons- Whether y()u iue evaluating intemational inyestments fiorn the viewpoint of a domestic U.S. cornpany considering investing or seiling product in anorher counriJ. or from the viewpoint of a foieign company investirrg or selling product in another country, tw'o intemational evaluation consideratictrrs tlnt ure not present in clomestic ev,aluations must be given serious atten-

tiort. First, pt'ojecting c,urrenc)- exchange rates for each"year of ttrc rtfe of o- major uncertaintl. in intemational investment or sales agreement evaluatiorzs. whether you are investing or selling in a foreign market' you must project curcncy .*.hurg" rates over ihe life of projects so tirat )'ou can express all project costs and i"u"nu", in terms of the same currency to make a valid economic analysis of these in'estmenrs. projecting projects to be evaluated is

exchange rates invoives significant uncertainty and probably is at least as diffrcult as projecting product selring prices. second, joreign investment projects rnust be evahnted ayer-tax using the tax law of the couiry in which the investnrcnt is located. wcstern world country tax treaties permit companies with a base of operation in one country to use taxes paid on projects in another foreign country as tax credits against domestic income tax. If foreign tax is higher than domestic tax, as is often the case for U.S. companies, then the for-

416

Economic Evaluation and lnvestment Decision Methocls

eign tax cash flow usually is the worst case analysis that the investor should use to evaluate the ecolomic potential of the foreign project. However, differ_ ences in tax deduction methods (for example, depletion usually is not applica_ ble in foreign countries) and asset tax lives, as well as tax iate diffeiences, affect foreign project evaluations. Investors must be careful to analyzeproject economics from both a foreign tax and domestic tax viewpoint to determine the worst case project cash flow stream that is relevant ftr evaluation pur_ poses. In doing the u.S. tax analysis of a foreign U.S. project, ailowable tax deductions generally are different (usually smaller; ttu, ro, a oomestic u.s. project. For example, u.S. mine development or intangible drilling costs can be 70vo expensed, 30vo amortized over five years by specified u.s. domestic producers, but these costs must be deducted straightlin" or"r 10 years for u.s. tax on foreign projects. Allowed depreciation on foreign projects seems analogous to allowed domestic U.s. depreciation for alternative minimum tax. contact your international tax department for specific project tax details.

These tax deductions differences can significantly

tte of a "hung" foreign project in comparison to an equivarent domesiic "conomics U.S. project. The following example illustrates how exchange rate projections are used in an economic analysis involving two currencies, and the sensitivity of DCFRoR and NPV results to exchange rate projection variations. EXAMPLE 8-Ba Exchange Rate variation for DcFRoR and Npv

,.

Sensitivity To illustrate how exchange rate projections rerate to an international. project anarysis, consider the Exampre g-g, case A project to be a u.s- project so arr revenues, sarvage varue, capitar costs, operating costs, tax deductions and cash flows are in u.s. oottars. How_ ever, assume the revenues have been generated by seiling 1,ooo grodugJ units per year in Austraria at a fr-xed contraci price of 9400 Australian per unit for a base case exchange rate of per U.S. $b.ZS $1.00 Australian. This gives $400,000 Australian revenue per year

which converts to $300,600 U.S. per year for the $0.75 U.S. per Aus_ tralian dollar base exchange rate- This yierds the Exampre B-g, case case analysis results of DCFROR 25.24h and NpV @ l^b^q." = 12.0% = $160,600. Anaryze the sensitivity of these base case economic results to changing the $0.75 base case exchange rate to:

1)

$0:65 U.S. per $1.00 Australian due to a strengthened

dollar.

Chapter B: lncome Tax. Working Capital, and Discounted Cash Flow Analysis

2)

417

$0.85 U.s. per $1.00 Australian due to a weakeneci u.s. dollar. (Note that requiring $0.9S U.S. to buy $t.00 Austratian versus $0'75 means the domestic currencv lu.s. doilarf has weakened relative to the Australian dollar.

Solution: All Values in Thousands o{ U.S. Dollars Case 1) Exchange Rate $0.65 U.S. / $1.00 A = Annual Revenue = $400 A ($0.0S U,S./$1.00 A) = $260 U.S. Year 2-5 7-9

260 260 260 -2A0 -200 -200 -20 40 _20

Revenue - Operating Costs - Depreciation * Write-off Taxable lncome

* Tax @ 40"/o

l.let lncome + Depreciation + Write-off - Capital Costs

260 _200 0

10

*360 -2AO 0

*"-50

0

40

2A

40

60

1'30

0

-16

-8

-16

-24

-+U

02412243660 02A402000 0000060 -260

Cash Flow -260 44 52 44 120 * Revenue includes $100 U.S. salvage and working capital return. *" original working capitar investmenl tax deduction wiite-ott.

36

N

PV Eq: 0 = -260 + 44(p lF i,1) + S2(p / Ai, )(p /F i,1) + 36(P/A;,3)(p/F;,6) + 1 20(p/F;, g)

+

44(p lF i,6)

1

DCFROR = 14.19%, NpV @ 1Z.Oyo +$2S.0 = Projecting exchange rates is difficult and involves great uncertainty. lt is necessary, however, in multiple currency analysls due to the sensitive impact that exchange rates have on analyiis results. strengthening of the U.S. dollar to $0.65 U.S. per .OO A from $f $0.7S U.S. per $1.00 A has weakened the economics of this hypothetical project significantly as measured by either DCFROR or6pping to 14.19"h trom 25.24% or, NpV dropping to $2S,OOO from g1Ob,OO6. Anywhere in the world, strengthening oi a domestic currency makes the eco-


418

nomics of export projects less desirable. On the other hand, as Case of a domestic currency makes export project

2 shows,

better.

l

case 2) Exchange Rate = $0.85 U.S. / $1.00 A Annual Revenue = $400 A ($0.es u.s. / $r.00 A) = $340 U.s. (Note the U.S. dollar has weakened relatively to the $0.75 U.S. per $1.00 A exchange rate since more U.S. dollars are required to purchase an Australian dollar) 7-9

10

340

340

-200 -20

-200

"440 -200

2-5

Year

340 340 -200 -200 -20 40

Revenue - Operating Costs - Depreciation - Write-off Taxable lncome - Tax @ 40% Net lncome + Depreciation + Write-off - Capital Costs

Flow

0 0

120

-48

100

40

0 72 60 020402000 0000060

0

140 -56 72 84

0

**-60

120

180

-48

-72 108

-260

-260 92 100

92

84

Cash 168 * Revenue includes $100 U.S, salvage and working capital return. .. Original working capital investment tax deduction write-off. NPV Eq: 0

= -260 + 92(P/F;,1)+ 100(P/Ai,4)(P/Fi,t) + 92(P/F;,6) + B4(P/A;,3XP/Fi,6) + 1 68(P/F;,1 g)

DCFROR =35.48/", NPV @ 12.0% = +$296.2 This weakened U.S. domestic currency analysis shows that export project economics are improved by weakening the value of a domestic exchange rate. The price paid for this benefit is higher prices for imports. lmport project economics are hurt by a weaker domestic exchange rate and import project economics are enhanced by strengthening domestic exchange rates. Governments try to establish and execute exchange rate policies that keep exchange rates in a range that is acceptable to both import and export businesses.

Chapter 8: lncome Tax, Working Capital, and Discounteo Cash Flow Analysis

419

obviously, that is a difficult objective to achieve consistently in this age of interactive international finance. .l

8.12 Nlining and Petroleum Project At'ter-Thx Analysis i\lining and perroieirn': project discounted cash florv analyses are sirnilar to non-*rineral anahses in terms of general procedure. In any discounted cash florv type analvsis you iirst convert all project relenues and costs to positive and ncgatile cash flows. lbu do this by accounting for the tax deductibiliry of costs and the tax to be paid, or tax savings to be redized, on resulting positive or nr:gative taxable income each year.

The unique feuture about discounted cash flow analysis of mining or petroJeum projects. compared to non-mineral projects, is the handling of certain tax cleiluctions. Chapter 7 addressed the tix handling of mining exploration and deve;upment costs und petroleum inrangihle drilling costs and pointed out that lfi)%' of these costs rnal' be expensed bl, "indit,iduar" minerel project opet.Lt-

or "independtnf" pefiolenmptoducers. "corporcte mining operators," or "hiegrated petrolewn producers," nmr- only expense 70vo of these cosrs. The other 307o of the intangible drilling or mining development costs are anwrti
depietion deductions also are unique to minilg and petroleum pro-iects. All nining ltroducers and independent petrolettm producers get to take the greater of allctvt,ed percentage depletiott and cost depletion, while integrated petroleutn

producers are only allowed to take cost depletion on oil and gas productiort. There are some unique severance and excise tax considerations related to mineral and petroleum projects. Most non-mineral, petroleum and mining projects alike involve depreciable costs, so there is nothing unique to mining and petroleum project evaluatjirns in this regard. A typical mining/petroleum project discountcd cash flow analysis is illustrated in the foilowing example.

EXAMPLE 8-9 A Mining or Petroteum project Evaluation Using DCFROR, NPV and pVR

An investor is considering acquiring and developing a mineral property believed to contain 500,000 units of mineral reserves (mineral units could be barrels of oil, tons of coal, ounces of gold, etc.). The mineral rights acquisition cost for the property would be $g00,000 at

420


time zero. A mineral development cost (or intangible drilling cost) of $1,200,000 is anticip-ateq a1 time zero along with tangible equipme.rlt costs (mining equipment or oil and gas producing equipment, pipelines, and tangible well completion costs) of 91,000,000 and working capital costs of $300,000, all projected to be incurred at year 0. Equipment depreciation will be based on modified ACRS 7 year lite depreciation, starting with the half-year convention in year 1 when assets are placed into service. Write off the remaining undepreciated book value at year 5. Mineral production is projected to be 100,000 mineral units per year over the 5 year project life with mineral reserves depleted at the end of year 5. Product selling price is estimated to be $30.00 per unit in year 1, escalating 10% per year in succeeding years. Operating expenses are estimated to be $1,000,000 in year 1, escalating 8% per year in succeeding years. Royalties are 156/o ot revenues each year. The property and equipment are expected to have no net salvage value although recovery of the $300,000 working capital investment is expected from inventory liquidation at the end of year 5. When applicable, assume the mineral produced is a 15% depletion rate mineral (for all mineral or independent petroleum producer evaluations only). Use an effective income tax rate of 32% for individuals and 38/" tor corporations. Calculate the project DCFROR, NPV and PVR for a minimum after{ax rate of return of 20"/", assuming: A) The project is being evaluated by an individual mining producer or small independent peiroleum producer for the following financial positions: 1) Assume other income exists against which to use all deductions in the year deductrons are incurred. 2) Make the project "stand alone"; this requires carrying negative taxable income forward until it can be utilized against project income. B) The project is a mining venture being evaluatec! by a corporation. Expense all deductions against other income as in 41, and begin amortizing 30% of mine development costs with a full 12 month deduction in year 0. C) The project is a petroleum venture being evaluated by an integrated producer. Expense all deductions against other income as in A1, and begin amortizing the 30% of development costs with a full 12 month deduction in year 0. D) A government is considering investing in the project so taxes are irrelevant.

Chapter B: lncome Tax, Working Capital, and Discounted Cash Ftow Analysis

Solution: All Values in Thousands of Dollars A1) lndividual Mining or lndependent petroleum producer, Expense Against Other lncome

1-rt 3,000 3,300 3,630 3,993 4,392

Revenue

I d

* t *

-Royaiiies

-450 -495 -545 -599

Net Revenue -Oper Costs -Deveiopment -Depreciation

2,550

ir

Net lncome rDepreciation +Depletion -rWork Cap Ret.

-143

704 74A -g}z"** 421 180 .129

384 -328

-816

-245

1,407 1,490

-1,2A0 1,025

-Capital Costs -2,20A* Cash Flow -3,016

*

3,086 3,394

-1,000 -1,090 -1,166 -1,200

Before Deplt -1,200 -50% Limit -Percent Deplt -Cost Deplt Taxable -Tax @ 32/"

2,805

-659

3,733 -1 ,260 -1 ,360

-175 -125 -312 1,744 872

2,009 2.061 1,005 1,030

-463

-560

*509

32

1,059

1,281 1,500

-339

-410 871 175 463

-480

1,50i -480

1,020

1,020 312 560 300

1,222 1,986 1,509 1,654

2,199

697 143 382

720 245 421

125 509

The capital cost is $300 working capital, gg00 mineral rights acquisition cost and 91,000 depreciable equipment. *" The write-off of remaining book values is combined with year 5 depreciation. *** The allowable depletion deduction has a minus (-) sign in front of it since it is the largest allowable deduction.

422


Depreciation Calculations Period ACRS Rate

X

Cost

Yearl 0j429 X 1,000 = Year2 0.2449 X 1,000 = Year3 0.1749 X 1,000 = Year4 0.1249 X 1,000 = Year5 0.0893 X 1,000 = Year 5 Remaining Book Value =

Depreciation 142.9 244.9 124.9 124.9

Cost Depletion

(100/s00xe00) = 16s (1 00/400X900-382) = 129 (100/300x518-421) = 32

(100/200x 97-463) < 0

gg.3 229.1

DCFROR Calcutation 0 = -3,0 1 6 + 1,222(P/F;,

) + 1,3g6(p/F;, 2) + 1,50g(piFi,3) + 1,654(P/F;,4) + 2,193(plFi,5) 1

i=DCFROR=39.15% NPV Calculation tor i* 20yo = NPV = 1,222(PlF 20,1 ) + 1,386(p/F29,3)

+ 1 ,654(P/FZO,4) + 2,193(plFZO,5) _ 3,016 =

+$.1

,517

PVR Calculation 1

,517 / 3,016 = 0.S031

The only difference in this case 41 analysis and the following cgse A2 analysis is the financial situation oi tre investor. ln case 42, il is assumed that the investor does not have other income against which to use the year 0 negative taxable income, so it must be carried forward to be used againit project positive taxable income in years 1 and 2. This delays realization of the tax benefits from the year 0 negative taxable income, which gives less desirable eco_

nomics in the following Case A2 relative to Case A.l.

Chapter 8: lncome Tax, Working Capital, and Discounled Cash Flow Analysis

A2) lndividual Mining or lndependent petroleum producer, Carry Losses Forward (Stand Alone) Year

3,000 3,300 3,630 3,993

Fevenue

Revenue -Oper Costs Net

2,SS0

-Development -1,200

2,805 3,C96

3,334 3,733

-1,000 -'1 ,080 -1 ,166 -1,260 -1,360

*143

-245

*175

1,407 704

1,490 740

-382 180

4,20A

-175

reciation

Before Deplt *50% Limit -Percent Deplt -Cost Depli -Loss Forward

*1,200

-1,200 -175 00 -Tax @ 32/" Net lncome -1,200 -175 +Depreciation 143 +Depletion 382 Taxable

+Loss forward

-125

-312

1,744 872

2,009 1,005

2,061 1,030

42',|

-463

129

-509

-560

32 1,29"1

1,500

1,501

-410

-480

-480

884 -283

1,200

601

245 421

871 1,020 1,020

175 125 463 509

-Capital Costs -2,200* Cash Flow -3,400

312 560

175

+\A/ork Cap Ret.

* The capital cost is

4,392

-450 -495 -545 -599 -€59

-Royaities

300

1,5S0 1,442 1,509 1,654 2j99

$300 working capital, gg00 mineral rights acquisition cost, and $1,000 depreciable equipment. ** The write-off of remaining book value is combined with year 5 depreciation. *** The allowable depletion deduction has a minus (-) sign in front of it since it is the largest allowable deduction. PW Equation 0 = -3,400 + 1,550(P/Fi,1)

+ 1,654(P/Fi,4)

* 1,442(PlFi,2) + 1,509(P/F;,3)

* 2,193(P/F;,5)

DCFROR = 37.17%, NpV @ 20% = 1 ,445, pVR = 0.4251

I

- Economic Evaluation and lnvestment Decision Methods

424

B) Gorporate Mining Evaluation,Expense Against other lncome

Year

.

Revenue -Royalties

0

-Development -840

-Amortization -72

Taxable -Tax @ 38%

Sj

3,ggg-- 4,gg2_.

2,550 2,905 3,096 3,394

-Depreciation

Deplt

4--t

-4S0 -49S _545 _S99 _659

Net Revenue -Oper Costs

Before -50% Limit -Percent Deplt -Cost Deplt

g 1 Z 3,000 9,900 3,690

3,733

-1 ,000 -1 ,090 -1,166 -1 ,260 _1,360

-143 -72

-245 -72

-175 -72

-125

***" -312

-72

1,937

704

1,672 836

-382

421

180

-463

129

32

-509

-560

-912

953

987

1,209

1,429

1,501

347

-362

-375

-460 750

-543

-570

-912

1,335 668

1,409

969

2,061

1,030

Net lncome 591 612 -565 886 930 +Depreciation 143 245 175 125 312 +Depletion 382 421 463 509 560 +Amortization 72 72 72 72 72 -Work Cap Ret. 300 -Capital Costs -2,560* Cash 1,199 .l ,3SO 1 ,460 1,5g2 2,10g -3,053 * The capital cost is $300 working capital, $g00 mineral rights acquisition cost, $1,000 depreciable equipment, and 307o of t-ne g1,200 development cost. ** The $72 amortization deduction in years 0 through 4 is ( 1 /5X0. 3X$ 1,200 Minerat Devetopment). *** The allowable depletion deduction has a minus (-) sign in front of it since it is the largest allowable deduction. **** The write-off of remaining book value combined with year S depreciation.

Flow

DCFROR Calculation 0 = -3,053 + 1,188(P/Fi,1) * 1,350(p/Fi, 2) + 1,460(p/F;,3) + 1,592(PlFi,4) * 2,1 03(piF1,5)

i=DCFROR=36.79%

i

Y'

j

i *

I *

Chapter 8: lncome Tax, Working Capital, and Discounted Cash Flow Analvsis

f;

*

I Il

NPV Calculation tar i* = 2}o/o NPV = 1,'188(P/F20,1)+ 1,3S0(P/F20,2) + 1,460(p/F2g,3)

{

+ 1,592(P/FZO,4) + 2,103(P1FA0,S) - 3,053 = + $1,331 FYR Calculation 1,331

i 3,053=0.4360

C) lntegrated Petroleum Producer Evaluation, Expense Against Other lncome Year

Revenue

3,000

3,300

3,630

3,993

4,392

-Rovalties

-450

-495

-545

-599

=659

Net Revenue

-Oper Costs -lniangible -Depreciation -Amortizat!on -Cost Deplt Taxable -Tax @ 38% i\'let lncome

+Depreciation +Depletion +Amortization +Work Cap Ret.

2,550 2,905 3,086 3,394 3,733 -1,000 -1,090 - I, rbO -1,260 -1,360

-14s

-912 347 -565 72

-125

-245 -72

-175 -72

80

-1 80

-1 80

-180

-180

1,155

1,228

1,757

1,981

-439

-467

1.492 -567

-668

-715

716 143

761

925

1,090

1,166

245

180 72

180

175 180 72

-zz**_1-lz

-Capitql Costs -2,560* Cash Flow -3,053 '1,111

* The capital cost is

72

-312

-72

125 180

312 180

72

300

1,258 1,gSZ 1,467 1,g5g

$300 working capital, $900 mineral rights acquisition cost, $1,000 tangible depreciable equipment and 30% of the $1,200 intangible driiling cost (lDC).

** The $72 amortization (1/sxo.3x$1,200 IDC).

deduction in years 0 through 4 is

*"* The write-off of remaining book value is combined with year depreciation.

5

426


DCFROR Calculation 0 = -3,053 + tri1 1 1 (F/F;,1) + 1,2S9(p/Fi,Z).* 1,352(p/F;,3)

+ 1,467(PlF;,4) + 1,958(p/F;,5)

i

= DCFROR=33.12/"

NPV Catculation tor|*

-

ZC;/o

NPV = 1,111(P/F20,1)+ 1,258(plF2A,2) + 1,352(p/F2g,g)

+ 1,467(P|FZO,4) + 1,958(p/F20,5) - g,OSg + $1,023 = PVR Calculation

1,023/3,053=0.3350

)) Before Tax Analysis (ear

levenue

{et

3,000 3,300 3,630

-450 -495 -545

Revenue

2,550 2,gOS 3,0g6

-599

4,392 -659

g,3g4

3,733

3,993

,166 -r 'let income -Work Cap Ret. -Capital Costs )ash

,eoo -1,360

1,550 1,725 1,919 2,134 2,973 300

-3,400"

Flow -3,400 '1,550 1,725 1,g1g 2,1g4 2,67g

The capital cost is $399 working capital, $900 minerat rights acquiition cost, 91,000 tanglb]e equifment, and 91,200 deveiopment or rtangible dritting cost (lDC). ,CFROH Calcutation 0 = -3,400 + 1,550(P/Fi,1) + 2,134(PlFi,4)

i

*

1,225(plFi,2) + 1,g19(p/F;,3)

* 2,670(p/Fi,5)

= DCFROR=45.32/"

427

ChapterS:lncomeTax,WorkinECapital,andDiscountedCashFlowAnalysis

NpV Calculation 1g,1i* =

2}o/o

NPV = 1,550(P/F20,1)+ 1,725(.P1F20,2) + 1'919(P/F2g,3) + 2,1 34(P lF zO,+) + 2,679(P lFzO,S) - 3,400 = + $2'303 PVR Calculation

Z.1iiglg,400=O.67Zs It is irnportant to proiect inflows and outflows of money from either revenues, costs or iax savings and tax costs to fit the investor Situation and physical proiect timing as closely as possible. Only if this is done will evaluation results be valid for economic decision making'

PROBLEIVTS

8-1 A corporation wants you to evaluate the economic

potential of

a

project

with time zero cost of $1,000,000 for research, a $400,000 cost for land, $1.400,000 for a bullding, $300.000 inventory cost for raw materials and spare parts and s2,000,000 for equipment. For income taxes, q.ill be rhe research cost will be expensed at time zero; the equipment depreciate,3 over five years using the MACRS rates from Table 7-3 (star1 depreciation in year 1). The building u'ill be depreciated stiaight line ovei 39 years with a full year deduction beginning in year one. The land, inventory costs, equipment and building book value will all be written off against the escalated dollar sale value of $4,000,000 at the endofy"u.3'Theprojectisestimatedtogenerateendofyearlrevenue of $3,000,000 and operating costs of $1,000,000. Both will escalate 8.07c per year in years 2 and 3. The effective corporate ordinary income tax rate is 40.07c with any gain on the year 3 sale treated as ordinaq.\. income. Determine the DCFROR, NPV and PVR for an aftertrix escalated dollar minimum rate of return of 15'A7a

A)

Assume a "stand alone" financial situation and carry losses forward to be used against project income'

B)

Assume an "expense" financial situation assuming other corporate income exists against which to use losses (negative taxable income) in the year incurred. This means there is no loss forward deduction.

428

'Economic Evaluation and lnvestmenl Decision Methods

8-2 Working Capital

Cost=$50,Q09,. Res & Experiment Cost = $100,000 _ ^ 0

,

Working Capital Return=.$20,000

Sales=$500,000. . . . . Sales=$500,000 rt.vv_vTvvrvvv

As shown on the time diagram, time 0 working capital cost of $50,000 and research and experimentation expens", of $100,000 on a project are expected to generate increased sales of $500,000 per year from :iTlilg process equipment for increased annual operatir! cosrs of $400'000. A washout of annual escaration of operating costl and revenues is assumed. working capitar return (inveniory H{uidation varue) of $70,000 is expected to be iealized at year 5. Tire #"",iu" in.o." tax rate is 40vo. Determine the DCFRoR on investment capitar if:

A)

Research and experimentar costs are deducted for tax purposes as operating expenses at time 0. other income is assumed'to exist against which to use the time 0 negative taxable income.

B)

Research and experimental costs are expensed at time zero with negative taxable income carried forward io be used against project revenues (stand alone analysis).

c)

Research and experimental costs are capitalized and deducted for tax purposes by amortization over years I to 5. ,

D)

Assume this is a u.S. project with all costs incurred in U.s. dollars. Further, assume alr costs rerate to u.S. rabor and materiars and are not affected by exchange rate variations. Assume annual revenue of $500,000 US is realized by selring r,000 units of product per year to

a German company at a fixed contract price of g33.33 German (DM) per for a base exchange rate of $0.60 US per 1.00 flarks 11t DM (note: 933,330 DM x $0.60 us p.bla $500,000 US revenue = per year). This yields the base case (Case A) DCROR of 5l.3Vo. Analyze the Case A sensitivity of changing the exchange rate:

1) 2)

ro $0.70 US per 1.00 DM (a weakening of the US dollar) to $0.50 US per 1.00 DM (a strengthening of the US doliar)

Chapter B: lncome Tax, Working Capilal, and Discounied Cash Flow Analysis

8-3

429

Development of a coal property which a corporation may purchase tbr a mineral rights acquisition cost of $10 miliion is being considered. Mrneral development capital of $10 miliion will be nei'.Ced in evalultion time 0 fol overburden stripping with the cost considercd to be incurred in the tirst month of time 0. Mine equipment costs ot $15 milliun also rviil be incurred in time 0 along with 52 million cost for working capital. 'fhe mine Il'e is estimated to be 5 years. Miiie equiprnent rvill be depreciated over' 7 years using modified ACRS rzrtes, starting in time 0 with the half-year convention. Salvage value and working capital return will be $5 million at the end of yezLr 5 with any taxable gain taxed as ordinary income. The effective tax rate is 4A7o. Coal resen/es are estimated to be 5 million tons and production for years 1 through 5 is projected to be 1 million tons per year. Coal selling price is estimated to be $30 per ton in year 1, escalating l07o per year in years 2 through 5. Royalties are 8Vc of revenue. Nlining operating costs are estimated to be $i2 per ton in year I, also escal.ating by 10Vc per year i:r following years. Calcuiate the project DCFROR and NPV for a minimum DCFROR of 207, to determine if the mine development economics are satisfactory for: A) No other income cxists against which to use tax deductions, so carry negative taxable income forward and use against project income and tax. This makes the project "stand alone."

B)

Other taxable income does erist so realize tax benefits from negative taxable income in the year incurred.

8-1. Annual Cash

cash

Flow

flow from a new investment is projected to be: -$150,000 $60,000 $70,000 $80,000 $90'000

Year

As the year 1 through 4 positive cash flows are realized it is anticipated that they will be invested in treasury bonds paying l2Vc annual interest and maturing at year 4. Calculate the DCFROR and the Grora'th DCFROR on the investment, assumiug a 40c/o income tax rate

is relevant and that after-tax treasury bond interest will be reinvested each year in identical bonds.

rrq

Eeonomic Evaluation and lnvestment Decision Methods

430

8-5 A manufacturing

plant can be purchased for $180,000. An additional $20,000 must be invested in working capital for raw material and spare parts iriventory and accounts receivable money tiedup in.operating costs. The $180,000 plant cost will !e depreciaged gtraight line over 7 years using the half-year depreciation convention in year 1. Actual salvage value is estimated to be $170,000 for used machinery and equipment and working,capital return at year 15. Annual sales revenue is estimated to be $100,000 in year I with operating costs of $40,000. In years 2 to 15, escalation of operating costs and sales revenue are projected to be a washout. The effective income tax rate is 40Vo. The minimum ROR is l5%o after taxes. Calculate the project DCFROR and NPV.

3-6 A mining investor operating as a corporation is considering buying the mineral rights to a small mineral properry. The mineral rights acquisition cost will be $1,000,000 at time 0 and.depreciable mining equipment costs will be $1,000,000 at year I in escalated dollars. lvlodified ACRS depreciation will be used for a 7 year depreciation Iife starting in year 2 with the half-year convention. Mineral development cost of $500,000 will also be incurred at year 1. Production rates will be 100,000 units per year in yearc 2 and 3 and 150,000 units per year in years 4 and 5 which will deplete the reserves. Salvage value of all assets at year 5 is considered to be nil. The mineral product will be sold for $20 per unit while production operating costs are estimated to be $8 per unit. Assume a washout of escalation of operating costs and sales revenue each year and neglect the effect of escalating sales revenue on percent depletion. The mineral produced is in the 227o percentage depletion category. Determine the project cash flow for each year and calculate DCFROR and NPV for a minimum DCFROR of l5Vo.

l-7

XYZ Corp. is evaluating the purchase and development of a petroleum property that can be acquired for $100,000 now (time 0). The purchase

would be followed immediately by intangible drilling costs of $750,000 at month 7 of year 0 and $250,000 at month 7 of year l. Tangible well completion and producing equipment costs of

i. Anticipated production is 200 (BPD) per barrels of oil in year 2, decreasing each year by 40 day $1,000,000 would be required at year


431

BPD in years 3 through 6. Assume oil will sell for $22 per barrel (before royaities of L4%o of sales) in the first producing year, with oil price escalating by 87o annually thereafter. Assume 350 operating days per year. operating costs are estimated to be $175,000 in the firsi producing year and escalating 107c annuaiiy until production is suspended at the beginning of year 7. XyZ corp. has an effective tax rare of 38%o, The company has other oil income against which to expense pre-tax losses, however all oil production by the company is less than 1000 BPD now and in the foreseeable future. The welicompletion and producing equipment costs will be depreciated by modified ACRS depreciation for a7 year depreciation life starting in evaluationyear Z with the half-year convention. use DCFRoR and Npv (for i =L|vo) to evaluate this investment for the 6 year project life if the company is an

,l

*

independent and:

$

fl

{

A)

$

,* E

B)

il $ r

C) D)

I

8-8

Risk of failure is neglected, i.e. the probability of successful well compietion and production through ev'aluation year 6 is l00vo. The probability of success between year b and 1,ear I is 10vo and after 1'ear I is 1007o. tf drilling at year 0 results in faiiure, the company will take a i.r'rite-ofi of the acquisition cost at the end of year 1 when the property mineral rights will be abandoned for a $50,000 cost. The part 'A" and "B" analyses are for an integrated producer. The part 'A" analysis is for "stand alone" loss carry forward economic analysis.

Consideration is being given to rhe investment of $420,000 at time zero for machinery and equipment to be depreciated using 7 year straight line depreciation starting in year I with the half-year convention. Annual sales are projected to be $400,000 less annual operating costs of $200,000. Escalation of operating costs and sales revc;nue is expected to be a washout from year to year. $100,000 for working capital investment is also needed at time zero and working capital ,"tuin is expected to equal the initial working capital investment at the end of the project. Salvage value of the machinery and equipment is expected to be zero. The minimum DCFRoR is r5va and the effective income tax rate is 35Vo. Calculate DCFROR and NpV for:

A) e 9 year evaluation life. B) An 18 year evaluation life.

Economic Evaluation and lnvestment Decision lr/ethods

8-9

The following data relates to a processing facility-

costs and Production are in thousands of dollars. Production units are in thousands of galions and declining because this product is expected to be replaced over the next five years. Yetu

62 53 35 24

Production, (gal)

Research

750

Equipment Patent

Rights

100

Operating Costs Price, ($/gal)

17

250 670

t75 26.0

r93 212 233

26.0 26.0 27.3

256 28.7

Royalty costs on the patent are 14.0Vo of Gross Revenue. Effective Federal/State Income Tax Rate is 40.0Vo Liquidation value in year 5 is zero. Equipment is depreciated using 7 year modified ACRS rates starting in year I with the half-year ionvention . The patent cost is amortized straight line over 5 years starting in year 0 with a full year deduction.

1)

Calculate the after-tax escalated dollar DCFROR, NpV and pVR for a minimum DCFROR of 157o assuming: A) Other income exists against which to use deductions in the year incurred.

B)

Other income does not exist against which to use deductions in the year incurred so project economics must "stand alone". 2) Risk adjust the Part 1A analysis assuming 4OVo probability of success with the year 0 costs and 60Vo probability of failure with failure resulting in a year 1 net abandonment cost of $70,000 to be expensed in that year against other income. If failure occurs a write-off of remaining book value on the patent cost witl be taken at year l. 5) Analyze the break-even product price per gallon that received uniformly over years 1 through 5 would yield the investor a l5.0Vo DCFROR for the Part 1A tax assumptions. 4)

What additional year 0 patent acquisition cost (above the $ 100,000 cost built into this analysis) could be incurred and still give the project a l5%o DCFROR for the data and assumptions for Part lA?

.t

433

Chapter 8: lncorne Tax, Working Capital, and Discounted Cash Flow Analysis

8-10 The following data relates to a petroleum project' Costs and Production are in thousands of dollars. Year

Production, (Bbls) Intangibles, (IDC's) Tangible (Completion) Mineral Rights. Acq. 100 Operating Costs

750

Price, ($/Bbl)

62 53 35

24

t7

250 670

r75 193 212 233

26.0 26.0 26.0 27.3

256 28.7

Royalty costs eaeh year are l4.0%o of Gross Revenue. Effective FederaVState Income Tax Rate is 4O.OVo

Liquidation value in year 5 is zero. Tangibles depreciated using 7 year modified ACRS rates, start depreciaticn in year 1 with the hah'-year convention. For integrated petroleum producers, assume 307o of intangibles are amortizeci using a full year deduction (30'ib of IDC tirores 12160)' beginning in the year the cost is incurred. Initial reserves for cost depletion equal cumulative production. The crude oil percentage depletion rate\s l5%o.

l)

Calculate the after-tax escalated dollar DCFROR, NPV and PVR for a minimum DCFROR of 157o from the viewpoint of: A) Integrated petroleum producer with other income against which to use deductions in the year incurred.

B)

Integrated petroleum producer that does not have other income against which to use deductions in the year incurred so project economics must "stand alone."

2)

Risk adjust Part 1A analysis assuming 407a ptobability of success with the year 0 costs and 60% probability of failure with failure resulting in a year 1 net abandonment cost of $70,000 to be expensed in that year against other income. If failure occurs, a write-off of remaining book values on mineral rights acquisition and year 0 drilling costs will be taken at year 1.

3)

Make the analyses in Parts I and 2 for an independent producer with daily production in excess of the 1,000 bbl/day small producer limitation.

434

Economic.Evaluation and lnvestment Decision Methocls

Make the analyses in Parts t and 2 for an independent producer with daily production less than the 1,000 bbuday small producer limitation.

4)

s) Analyze the break-even crude oil price per barrel that received uniformly from years 1 through 5, wourd make this project yield a 157o DCFROR for the investor and tax scenario in part ta.

what additional year 0 mineral rights acquisition cost (above the $100,000 cost built into this analysis ) could be incurred and still give the project a r5vo DCFRoR for the data and assumptions for Part lA?

6)

8-11The following data relate to a mining project with increasing waste rock or overburden to ore or coal ratio as mine life progresses, giving declining production per year.

Costs are in thousands of dollars.

Production in thousands. Year

Production in Tons Mineral Development Mining Equipment Mineral Rights. Acq. 100 Operating Costs

750

Price, $ Per Ton

62

53

35

24

t7

250 670

175 193 212 233 26.0 26.0 26.0 27.3

Royalty costs are l4.0%o of Gross Revenue. Effective Federal/State Income Thx Rate is Liquidation Value in year 5 is Zerc.

256 28.7

40.AVo

Mining Equipment depreciated using 7 year modified ACRS

rates,

start depreciation in year 1 with the half_year convention. For corporate analyses, assume 3ovo of mine development costs are amortized using a fulr year deduction (30vo of mine deveropment cost times 12160), begi*ning in the year the cost is incurred. Initial reserves for cost depretion equal cumulative production. The mineral percentage depletion rate is l5Vo.

1)

calculate the after-tax escalated dollar DCFROR, Npv and pvR for a minimum DCFROR of l57o from the viewpoint of:

;

Chairter 8: lncome Tax, Working Capitai, and Discounted Cash Flow Analvsis

A) B)

435

Corporate mineral producer with other income against which to use deductions in the year incurred. Corporate mineral producer rhat ooes not have other income against which to use deductions in the year incurred so project economics must "stand alone."

2)

Risk adjust the Part 1A analysis assuming 407o probability of success with the year 0 costs and 60% prci-rability of failure with failure resulting in a year I net abandonnent cost of $70,000 to he expensed in that year against other income. If failure occurs, a write-off of remaining book values on mineral rights acquisition and mine development costs will be taken at year 1.

3)

Make the analyses in Parts

I

and,Zfor an

iraiuiaua mineral pro-

ducer.

4)

Analyze the break-even net smelter return per ton of ore that, re-:eived uniformly from years I through 5, would make this project yield a 15olo DCFROR for the investor and rax scenario in Part 1A.

5)

What additional year 0 mineral rights acquisition cost (above the $100,000 cost built into this analysis ) couid be incumed and still give the project a l5%o DCFROR for the data and assumptions for Part lA?

CHAPTER 9''

AFTER.TAX DECISION METHODS AND APPLICATIONS

9.1 Introduction DCFROR is used more widely as an after-tax investment economic decision method than all other economic decision methods. Npv is the second most used economic decision method and ratios ar.e the third most used evaluation technique. These discounted cash flow investment analysis techniques are the best approaches known today for evaluating the economic potential

of alternative investments. It is important to remember that all of the discounted cash flow techniques are systematic, quantitative approaches to

.

i i :

evaluating investments based on given sets of assumptions and input data. If you put garbage in, you will get garbage out of any analysis carcuration using any technique of analysis. There is nothing magical aLout discounted cash flow results. They are based on the evaluation asiumptions concerning: (l) tax considerations, (2) handring inflation and escalation, (3) risk adjusi_ or not risk adjusting results when flnite probability of 'failure Ing (4) the financial situarion of the individual oi organizaiion "îrtr, for which the analysis is being made, (5) significance of terminaivalue magnitude, timing and tax considerations, (6) cash investment analysis versus leviraged analysii with borrowed money, (7) correct handring of the discounted cash flow analysis calculations whether it involves DCFROR, Npv pvR or another technique, and finally, (8) correct apprication of the discounted cash flow analysis results to analyze mutually exclusive or non-mutually exclusive income or service producing alternatives. Proper use of the discounted cash flow analysis techniques gives investors a better chance of correctly analyzing the potential of alternatiie investments than can be achieved using any other evaruation technique. The key to successful application of the discounted cash flow techniques is consistency. If :

'i 43Q

* *

*t t

Chaprer g: Afler-Tax lnvestment Decision Methods and Apptications

437

\'ou Lrr)alvze a project in escaiated dollars you must compare it with results of orher projects analyzed in escalated dollars, As discussed in chapter 11, if \,(iu le\eiâge a proiect investmeni \.vith bDrroued money, 1,ou should com_ pare it to other projects analvzed x,ith simiiar borro-..ved mcne)r ler.ei:age. Srirriarly, ri'hen considerins an investor's mininrum raie of return, it is ir:rpt.rr".int to recognize that i{ the financial cost of capital approach is to be uLiiizc'J. the cost of Cebt ard the risk free bond rate of retum lrsed in the capital rs:r:t 111eigl rnust be e.rpressed on an aller-tax lrasis. This requires recognition thill tile cost ot iinancing is generally deductible (except for construction loans - :'ce Chapter 1i fbr more details). Further, interest received from an investmenr in bonds would be treated as ordinary income upon rvhich taxes must be paio. These adjustments are easily achieved by multiplving each parameter tinres the quantity (1{ax rate). The resulting product(s) represent the after-tax cost ,.>f borrowed mone)' interest and the after-tax rate of return from a bond

If an invesLor is utilizing a true opportunity cost of capital base-d percgived returns to be realized rr: the f uture, such r;reasures of econoniic rcirirn should be based on after-tax performance using afie.r{ax cash flow in thr: methodology presented in Chapters 7 r.nd g of this text. inve-{tment. u1,rr.ii)

Broad acceptance and utilization of discou:rted cash flcw analysis

occurred in most industries around rhe world in the 1960's and 1970,s. prior to that time, techniques of analizsis such as payback period, which is described in the next section, and several different average and/or accounting rate o1'retttrn calculations which are d*fined and illustrated in Section

9.1()

oi this

chapter were utilized by investors to evaluare the economic Irotentiai of investments. ft will be shown that the older techniques do not properly account for the timing and tax effects related to project costs ancl revenues. Therefore, the older techniques are inconsistent in their usefulness tor evaluating economic differences in project investments. This is the prirnarv reason investors have shifted in recent decades from using these older analvsis techniques to using the discounted cash flow analysis criteria.

9.2

I'a.v-back Period Analysis

Patback period (or payout period) is the time required for positi,e project flow to recover negative project cash Jlov; from the acquisitiort and/or

cash

,levelopntent )-ears. Pa1,66r1, can be calcurated either from the start of a project or from the start of production For the calculation of payback period, positive cash flow is generally considered to flow uniformly during a year rather than at end, middle or beginning of a year. Sometimes payback period

438


calculations are based on discounted cash flow at a specified discount rate such as l)vo, l2va or some other rate that often represents the minimum DCFROR.

The basic economic anarysis phirosophy behind the usoof ;;;;; period as an evaluation technique is that the faster you get investment doliars back from project cash flow, the better the economici of an investment. However, this often is not the case. payback can be very misleading as an indicator of economic differences in investment projects t..u,rr" it neglects what happens after payback. For example, a project with a z ye* payback period and a 3 yezu life may be economically inferior to a project wittr a4 year payback period and aZ0 year life. As a measure of risk, or financial analysis rather than economic analysis reasons, payback period calculations can be useful.

If an investor

is considering a

foreign investment in an underdeveloped country associated with high political instability and the investor cannot recover initial investments within a year or two, he rnay elect to forego investing even though long-term investment calculation resuhs look very good. similarly, a company in a tight financial situation and needing money to meet current obligations such as for operating expenses and debt repayment may elect to invest in projects with short payback-periods to help

t:

*

r $

meet short-term cash flow needs, even though these projects have much poorer economic potential than other potential investments with longer payback periods.

Another application of payback relates to the writing of 3oint venture

contracts. Working interests, carried interests and related reveriionary interests often change after payback. Because different joint venture partners

often arg in different financial and tax situations, before-tax payback is almost ahvays the basis for payback used in legal contracts. Tix holidays for state or provincial mineral severence or excise taxes often involve a

before-tax payback calculation. The following two exampres ilustrate the mechanics of carculating payback in four different ways and show how the project with the shortesl payback rnay not be the best economic choice.

EXAMPLE 9-1A payback Catcutations To illustrate the mechanics of the different methods for determining payback, calculate the undiscounted and discounted payback for a 12.0% minimum rate of return from both the start of the project (time zero), and from the start of production (beginning of year 2, or the e1d of year 1) for the foilowing project with ail ittet-iax cash ftow (ATCF) dollar values in thousands:

$ 1:

Ciapter g: Aiter-Tax lnvestment Decision Methods and Appiicai rns

Year ATCF

-'100

-200

150

200

439

250

Solution: This project has a DCFROR of 32.85% and an NPV @ 12.0oi" ot $142.24, both of which indicate a.cceptable economics. For Payback, assume the time zero cdsh flow is a discreie sum at thai poirrt and thai tne subsequent cash flows in periods 1 through 4 flow continuousiy during each corresponding year. Payback From the Start of the Project, (Time Zero):

______>_r

2i

Year

ATCF CUM ATCF

-100 -200 150 -100 -300 -150

lzoo 50

250 300

2 Years + (150 I 200) = 2.75 "Years The year 2 cumulative after-tax cash flow of -l S0 represents the cumulative cost yet to be recovered from the positive cash flow. lts absolute value forms the basis for the numerator in the paltback calculation. Rounding up, this could be described as a 3 year payback. lf an investor considers time zero as a full year, rather than a discrete point, the payback could be described as either 3.75 or 4 years.

Payback from the Start of Production (Year One): Year ATCF CUM ATCF

0 1 2 13 150 200 -100 -200 -100 -300 -150 50

4

250 300

I 2OO) = 1 .75 Years While the point in time where payback occurs does not change,

1 Year + (150

the stariing reference point does. So looking back from the payback point to the start of production is 1.75 years.

-

Discounted Payback from the Start of the Proiect (Time Zerol: Discounted payback involves making the same calculations, but utilizing the discounted after-tax cash flows, (DATCF) for each year as illustrated:

440


Year -.: 0 ATCF -1 00 DATCF -100 CUM DATCF -100

2

1

3 200 142

4 250

-17

142

159

Where: -200(P/Fp%.i = -179 150(PlFpy",2) = 120, etc... 3 Years + (17 I 159) = 3.1 Years Discounted Payback was illustrated graphicatty eartier in the development of the cumufative Npv Diagram in chapter 3. The point where cumulative NPV equals zero is the graphicat apptroach to discounted payback as illustrated : Cumulative NPV

200 150

i

100

o

OrscountedPayDa"k,3.,

years

50

s o

0

B

-50

E

t_

o

-1 50

-200

-250 -300

Years

Figure 9-1 cumulative NpV lllustrating Discounted payback

Discounted Payback from the start of production (year one): Year ATCF

01

DATCF -100 CUM DATCF -1OO

_179 _279

2 Years + (17 /159) = 2.1 years

2

3

150

200 142

250

-17

142

120 -1 59

159

Ci)apter g: Arter-Tax lnvestment Decision Methods a:d Applicaticrs

441

EXAMPLE 9-18 payback calcutations and Flesults compared to Discounted Cash Flow Analysis Results lf projects "A" and "8" are mutualiy exciusive investments, which is ir:ij:cateci to be preferacle using undiscounted payback period, disccunted payback period, Npv, pvR anci DCFRon. rnL minimum ra:e of re iurn is 1Z/". Cash Flow = -$100 Project A) Cash Flow =-$100 Project B)

$28s.6 4 years

1

2

3

$46.2

$46.2

$46.2

$46.2 4 years

3

Soiution: Assurning revenue is realized uniformly over year 4: Project "A", Undiscounted payback 3 + (100/2g5.6) = = 3.35 years lf the project "A" revenue is assumed to be a lump sum revenue such as from the sale of real estate or common stock, then the project "A", Undiscounted payback is 4 years. Project "B", Undiscounted payback 2 + (1 oa-g2.4)/46.2 = = 2.16 years undiscounied Payback indicates select project ,,8,, with the smaller pa;,'back.

Diagrams for Discounted Cash Flow Discounted cash Frow = -$100

A)orzs4year

at

- - -

12yo

$2g5.6(p/F1e,4)=$1gt.5

Project A Discounted payback 3 + (1OO/1g1 .5) 3.55 years = = lf the project A revenue is assumed to ire rump r.r"nue, then Discounted Payback = 4 years. Discounted Cash Ftow = -$lgg_l1f .2S $36.83 $32.88 $29.36 Q\ B) 4 years Where year 1 discounted cash flow equals a6.2(P1F12,1) and so 'orth for years 2,3 and 4. )roject B Discounted payback = z + (100-7g.0g)lg2.gg = 2.67 years )iscounted Payback lndicates select project,,B',, the smaller payback.

,u,

012

3

442


Net Present Value Analysis (Mutually Exclusive lnvestments)

- 100 = +$81.5 Select Largest NPV, "A'

NPVA = 285.6(P/Ft2,4)

NPV3 = 46.2(Pl A12,4)

-

1

00 = +$40.3

Present Value Ratio Analysis

PVR4=81.5/100=+0.815 PVRB=40.3/100=+0.403 Since both ratios relate to the same investment dollars, select "A",

whether the alternatives are mutually exclusive or non-mutually exclusive.

Discounted Cash Flow Hate of Return Trial and error analysis gives DCFROR4 = 3Oo/o and DCFRORB = 30%. Because of differences in the distribution of positive cash flow on the project "A" and "B" diagrams, these 30% DCFROR results have very different meaning after the first year. lncremental analysis must be made for a valid economic decision with DCFROR. Make the incremental analysis so that incremental cost (negative cash flow) is followed by revenue (positive cash flow). lncremental Cash Flow For Project A-B) PW Eq: 0 = -46.2(P/A;,3) +

-$+0.2 -$46.2 -$46.2

3

+$239.4 4 years

n9.af lFi,a)

Trial and Error, i = lncremental "A-8" DCFROR = 2g.g"/" >

i

= 12/"

Select Project "A"

All discounted cash flow analysis techniques indicate "select A". Payback period on either a discounted or undiscounted basis indicates "Select 8," showing the inconsistency in 'payback' as a method for selecting investments that will maximize profitability from available investment dollars.

)h?'r:er 9r a.fterTax lnvestment Decisiorr I'r,4ethods and Aoolicaticns

).3

443

Savings are Analogous to Income

l'i ecoulrmic evaluation work we verY frequently find otrrselves con'rollr'(i u'ith analysis of determining the most economic way to provide a .,rr'\ti-e. -fitis is the most conl)ton type of evaluation probleur of all and in - rr il,l:er -] rve ciiscusscd the fivc basic economic analysi. approaches to ana.2,: iiris rlpe of problem. The |r'e basic approaches are (11 comp:trison of r. ..1.tnt ui.rth CrlStS, (2) corlperison of annuAi worth C0SIS, (3) Coinparison .i'iuiiire.,r.orth cttsts, (:1) increniental ROR or Net Value Analysis or (5) :,r'.';rl,-ei'e n analysis such as service life break-el'en analysis' Of these flve t-rL-ill()(ls increntental ROR arid Net Value analysis are very popular because

rf rlrc large nunlber of companies and individuals that use ROR and Net r'aluc analS'sis irs the primary decision making criterion for all types of proect analyses. Incremental analysis of alternatives that provide a service ii,,r.21,q generales ilcremental costs :tnd savings, and dtlllars saved are just ' r.:,jr' :.tls elrttcd ! For t$ter-tttx DCFRC)R, Net Value, or Rario anal-r*sis of alternatives 'ttvrtlving srrllrzgs 1'ou ntrtst CotNert savirtgs to after-ta.r cash flovt e.\QctLv 'Ite sone tis irtcrenrcntal itrcome rnust be converted to cash flow. If an increne:rfal int'esturent genelates sar inss b1' redLrcing operaiing costs bel0\' a :orrler lcvel. thc. lower operating costs will result in lou'er tirx deductiotls 'br operatiug costs which means you have morc taxable illcome in the sarrre titr{-}unI as il'the savings wcre incremental revenue. 'fhe follourng example illustrates DCFROR and NPV analysis of

an

ni estnrenl that Senerates savings.

EXAMPLE 9-2 DCFROR and NPV Analysis lnvolving Savings

A natural gas distribution company is evaluating the economic ootential of installing a neur compressor that costs $420,000 to satisfy Jas compression requirements and save 80,000 MCF per year of natural gas (where MCF equals thousand cubic feet) compared to the present operating costs. The gas saved could be sold to industrial oustomers {or $3.00 per MCF and compressor life is estimated to be 10 years with zero salvage value. The new compressor would be depreciated straight line over a 7 yeil life starting at year 0 with the half year Convention. Assume Compressor maintenance costs will be exactly offset by increased sales revenues due to increased gas Sell-

444


ing prices. The effective income tax rate is 40"/" and the minimum DCFROR is 12/". Use DCFROR and Np.v anqlysis.!g delgryine if it is economical to buy the n.* rorfressor. Assume other income exists against which to use deductions in any year.

Solution: All Values in Thousands, Except Selling price C=Capital Cost, OC - Operating Cost, 5 = Savings To illustrate the concept of incremental analysis that is required to make a proper economic evaluation of the eiample problem, consider the following. Let X equal the "new" compressor natural gas usage in thousands of MCF per year so X + go MCF per year represents "present" compressor natural gas use in thousanos of Mcrper year. The savings of $240 (80 MCF times $s.oo; per year frcm going to the new compressor must be converted to cash flow the same as incremental income would be converted to cash flow.

otd t

0 tew C=

F($3.00/|\4CF)

1....

......10 $.oo/McF)

0

New C=$420

S = $240

S = $240

-ord Year

1-6

Revenue

-Depreciation Taxable lnc.

-Tax

@ 4O"h

Net lncome

240 240 -30 -60 -30 -30 180 210 12 -72 -84

-96

-18 108 126

144

+Depreciation 30 -Capital Costs -420 Cash Flow

8-10

60

240 24A

30

-408 168 156

144

Chapter 9: Alter-Tax lnvestment Decision Methods and Applications

445

PW Eq: 0 = -408 + 168(P/A;,0) + t56(PiF;,7) + 144(PlAip)(PlFi,Z)

i = DCFROR=

39.2/o>i*

=12."/o,

new compressor is satisfactory

Npv

@

4.111 0.4523 i lzx = -408 + 168(piAi2,6) + 156(prF12,7) 2.402

0.4523

+ 1 44(Pl A12,9(Plr

12,7) = + $51 0 > 0,

satisfactory

The DCFBOFI of 39.2"/o, which is more than three times the minimum DCFROR of 127", indicates very satisfactory economics for the nev'/ compressor, consistent with positive NPV of $510,000 that is greater than tne cosi of $420,000 that generated the NPV. Whenever NPV is pasitive and similar in magnitude to the cost that generated it, project economics are very good and typically relate to a proiect with a DCFROR three or four times the minimum DCFROR. 9.'1 Sunk Costs and Opportunity Costs in Evaluations Srtttk costs are cost., that have already been incurred in the past and that n()llting v'e do now or in the.t'uture cen affect. Economic analysis studies for invesrrnenr decision making purposes deal with project costs and revenues artd iax ettrcts yet to be incurred now or in the future. Sunk costs are not r.'levont ro the anal ,-sis of either income or sertice producing investment alternativ,es except for remaining sale value and tax effects, yvhich are opporturtitl' cost considerations discussed in the next paragraph. Past commitments to expend money as well as past expenditures are sunk revenues and costs. Revenues realized in the past from a projerct are sunk revenues the same as past costs are sunk costs. Classic examples of sunk costs include the costs of equiprnent acquircd several years ago and now being considered for replacement, the costs for research or exploration work incurred in earlier years, an

Economic Evaluation and Investment Decision Methods

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