Elan Guides Formula Sheet CFA 2013 Level 2

We started Elan Guides with a mission to provide CFA® candidates with the most effective and efficient study solutions to help them take their careers to the next level. At the same time, we wanted to make highquality study materials more affordable for CFA candidates around the globe.

WHY USE ELAN GUIDES? At Élan Guides, we offer a complete portfolio of study products that help you understand, retain, review and master the CFA® Program curriculum. We are committed to your success , and together we can turn this seemingly intimidating journey into an exciting and productive experience. With us, you will be able to: Get a complete understanding of difficult concepts through more than 80 hours of comprehensive lecture videos for each level; Get in-depth explanations and examples from the 5 volumes of our study guides; Expedite your revision through our intensive review seminars; Nail difficult concepts through thousands of practice questions across our quizzes and mock exams; and Interact with our content experts and teaching assistants through our Q&A forum. We believe that our learning solutions are unparalleled in the CFA prep industry. That is why we offer all our Quantitative Methods study materials (study guide readings, lecture videos and quizzes) for both Levels I and II for free. If you are serious about passing these exams on the first try, sign up now (no credit card info required) and let us partner par tner you on your way to success on the CFA exams.

OUR INSTRUCTORS Basit Shajani, CFA

Basit graduated Magna Cum Laude from the world-renowned Wharton Wharton School of Business at the University of Pennsylvania with majors in Finance and Legal Studies. After graduating, Basit ran his own private wealth management firm. He started teaching CFA courses more than five years ago, and upon discovering how much he enjoyed teaching, he founded Elan Guides with a view to providing CFA candidates all around the globe access to efficient and effective CFA study materials at affordable prices. Basit remains an avid follower of equity, commodities and real estate markets and thoroughly enjoys using his knowledge and real-world finance experience to bring theory to life. Peter Olinto, CPA, JD

Peter has taught CPA and CFA Exam Review courses for the past ten years and is a real ‘celebrity’ ‘celebrity’ in the CPA and CFA prep industries. Previously he worked as an auditor for Deloitte & Touche, was a tax attorney for Ernst and Young, and later spent nearly ten years teaching law, accounting, financial statement analysis, and tax at both the graduate and undergraduate levels at Fordham University’ss business school. He graduated Magna Cum laude from Pace University and went on to University’ earn his JD degree from Fordham University School of Law.

© 2013 ELAN GUIDES

QUANTITATIVE METHODS

CORRELATION AND R EGRESSION EGRESSION n

Sample covariance = Cov (X,Y) =

 (X  X)(Y  Y)/(n  1) i

i

i = 1

where: n = sample size Xi = ith observation of Variable X X = mean observation of Variable X Yi = ith observation of Variable Y Y = mean observation of Variable Y

Sample correlation coefficient = r = =

Cov (X,Y) sXsY

n

Sample variance

2 = s X =

 (X  X) /(n  1) i

2

i = 1

Sample standard deviation = s X =

2

s X

Test statistic

r

n  2

Test-stat = t = = 1  r 2 Where: n = Number of observations r = Sample correlation

Linear Regression with One Independent Variable

Regression model equation = Y i = b0 + b1 X i + i, i = 1,...., n

 b1 and b0 are the regression coefficients.  b1 is the slope coefficient.  b0 is the intercept term.   is the error term that represents the variation in the dependent variable that is not explained by the independent variable.

© 2013 ELAN GUIDES


CORRELATION AND R EGRESSION EGRESSION n

Sample covariance = Cov (X,Y) =

 (X  X)(Y  Y)/(n  1) i

i

i = 1

where: n = sample size Xi = ith observation of Variable X X = mean observation of Variable X Yi = ith observation of Variable Y Y = mean observation of Variable Y

Sample correlation coefficient = r = =

Cov (X,Y) sXsY

n

Sample variance

2 = s X =

 (X  X) /(n  1) i

2

i = 1

Sample standard deviation = s X =

2

s X

Test statistic

r

n  2

Test-stat = t = = 1  r 2 Where: n = Number of observations r = Sample correlation

Linear Regression with One Independent Variable

Regression model equation = Y i = b0 + b1 X i + i, i = 1,...., n

 b1 and b0 are the regression coefficients.  b1 is the slope coefficient.  b0 is the intercept term.   is the error term that represents the variation in the dependent variable that is not explained by the independent variable.

© 2013 ELAN GUIDES


ˆ = bˆ + bˆ X , i = 1,...., n Regression line equation = Y 0 1 i i Regression Residuals n

 [Y  (bˆ  bˆ X )] 0

i

1 i

2

i = 1

where: Y i = Actual value of the dependent variable bˆ 0 + bˆ 1 X i = Predicted value of dependent variable The Standard Error of Estimate

(

SEE =

n



(Y i  bˆ 0  bˆ 1 X i)2

i = 1

) ( 1/2

=

n  2

n



(ˆ i)2

i = 1

n  2

)

1/2

=

(

SSE n  2

)

1/2

The Coefficient of Determination

Total variation = Unexplained variation + Explained variation R 2 =

Explained variation Total variation

=

Total variation  Unexplained variation Total variation

Unexplained variation

=1

Total variation

Hypothesis Tests on Regression Coefficients

CAPM: R ABC = R F + ABC(R M – R F) R ABC – R F =  + ABC(R M – R F) + 

 

The intercept term for the regression, r egression, b0, is . The slope coefficient for the regression, b1, is ABC

The regression sum of squares (RSS) n

RSS =

 (Y ^  Y )

2

i

 Explained

variation

i = 1

The sum of squared errors or residuals (SSE) n

SSE =

 (Y  Y ^ ) i

i

2

 Unexplained

variation

i = 1

© 2013 ELAN GUIDES


ANOVA Table Source of Variation

Degrees of Freedom

Sum of Squares

k

RSS

n  k + 1)

SSE

n1

SST

Regression (explained)

Error (unexplained)

Total

k = the number of slope coefficients in the regression.

Prediction Intervals

2 s f

= s

2

^  t s Y c f

© 2013 ELAN GUIDES

[

1

1 n



(X  X)2 2

(n  1) s x

]

Mean Sum of Squares

MSR =

RSS k

=

MSE =

RSS 1 SSE n2

= RSS

MULTIPLE REGRESSION AND ISSUES IN REGRESSION

MULTIPLE R EGRESSION AND ISSUES IN R EGRESSION ANALYSIS Multiple regression equation

Multiple regression equation = Y i = b0 + b1 X 1i + b2 X 2i + . . .+ bk X ki + i, i = 1,2, . . . , n Y i X ji b0 b1, . . . , bk

i

n

= the ith observation of the dependent variable Y = the ith observation of the independent variable X j , j = 1,2, . . . , k = the intercept of the equation = the slope coefficients for each of the independent variables = the error term for the ith observation = the number of observations

Residual Term ˆ = Y  (bˆ + bˆ X + bˆ X + . . .+ bˆ X ) ˆi = Y i  Y 0 1 1i 2 2i k k i i i

Confidence Intervals

bˆ j ± (t c  sbˆ j)

estimated regression coefficient ± (critical t -value)(coefficient standard error)

F -statistic F-stat =

MSR MSE

=

RSS/k SSE/[n  k + 1)]

R 2 and Adjusted R 2

R 2 =

Total variation  Unexplained variation Total variation

Adjusted R2 = R2 = 1 

(

n1 n  k  1

)

=

SST  SSE SST

=

RSS SST

(1  R2)

Testing for Heteroskedasticity- The Breusch-Pagan (BP) Test

2 = nR 2 with k degrees of freedom n = Number of observations R 2 = Coefficient of determination of the second regression (the regression when the squared residuals of the original regression are regressed on the independent variables). k = Number of independent variables

© 2013 ELAN GUIDES

MULTIPLE REGRESSION AND ISSUES IN REGRESSION

Testing for Serial Correlation- The Durban-Watson (DW) Test DW  2(1 – r ); where r is the sample correlation between squared residuals from one period and those from the previous period. Value of Durbin-Watson Statistic (H0: No serial correlation) Reject H0, conclude Positive Serial Correlation 0

Do not Reject H0

Inconclusive d l

Inconclusive

4  d u

d u

Reject H0, conclude Negative Serial Correlation 4  d l

4

Problems in Linear Regression and Solutions Problem

Effect

Solution

Heteroskedasticity

Incorrect standard errors

Use robust standard errors (corrected for conditional heteroskedasticity)

Serial correlation

Incorrect standard errors (additional Use robust standard errors problems if a lagged value of the (corrected for serial correlation) dependent variable is used as an independent variable)

Multicollinearity

High R 2 and low t-statistics

Remove one or more independent variables; often no solution based in theory

Model Specification Errors

Y i = b0 + b1lnX 1i + b2 X 2i + 

Linear Trend Models

yt = b0 + b1t + t ,

t = 1, 2, . . . , T

where: yt = the value of the time series at time t (value of the dependent variable) b0 = the y-intercept term b1 = the slope coefficient/ trend coefficient t = time, the independent or explanatory variable t = a random-error term

© 2013 ELAN GUIDES

TIME SERIES ANALYSIS

TIME-SERIES ANALYSIS Linear Trend Models

yt = b0 + b1t + t ,

t = 1, 2, . . . , T

where: yt = the value of the time series at time t (value of the dependent variable) b0 = the y-intercept term b1 = the slope coefficient/ trend coefficient t = time, the independent or explanatory variable t = a random-error term Log-Linear Trend Models

A series that grows exponentially can be described using the following equation: yt = eb0 + b1t

where: yt = the value of the time series at time t (value of the dependent variable) b0 = the y-intercept term b1 = the slope coefficient t = time = 1, 2, 3 ... T We take the natural logarithm of both sides of the equation to arrive at the equation for the loglinear model: ln yt = b0 + b1t + t ,

t = 1,2, . . . , T

AUTOREGRESSIVE (AR) TIME-SERIES MODELS

xt = b0 + b1 xt  1 + t

A pth order autoregressive model is represented as: xt = b0 + b1 xt  1 + b2 xt  2+ . . . + b p xt  p + t Detecting Serially Correlated Errors in an AR Model

t-stat =

Residual autocorrelation for lag Standard error of residual autocorrelation

where: Standard error of residual autocorrelation = 1/ T T = Number of observations in the time series

© 2013 ELAN GUIDES


Mean Reversion

xt =

b0

1  b1

Multiperiod Forecasts and the Chain Rule of Forecasting

^ = ^b + ^b x x 0 1 t t+1

Random Walks

xt = xt  1 + t , E(t ) = 0, E( t 2) = 2, E(t  s) = 0 if t  s

The first difference of the random walk equation is given as: 2 2 yt = xt  xt  1 = xt  1 + t  xt  1= t , E(t ) = 0, E(t ) =  , E(t  s) = 0 for t  s

Random Walk with a Drift

xt = b0 + b1 xt  1 + t b1 = 1, b0  0, or xt = b0 + xt  1 + t , E(t ) = 0

The first-difference of the random walk with a drift equation is given as: yt = xt  xt  1 , yt = b0 + t , b0  0

The Unit Root Test of Nonstationarity

xt  b0 + b1 xt  1 + t xt  xt  1  b0 + b1 xt  1  xt  1 + t xt  xt  1  b0 + (b1  1) xt  1 + t xt  xt  1  b0 + g 1 xt  1 + t

© 2013 ELAN GUIDES


Autoregressive Moving Average (ARMA) Models

xt = b0 + b1 xt  1 + . . . + b p xt  p + t + 1t  1 +. . . + qt  q

E(t ) = 0, E(t 2) = 2, E(t  s) = 0 for t  s Autoregressive Conditional Heteroskedasticity Models (ARCH Models)

^ 2 = a + a ^ 2 t

0

1 t  1

+ ut

The error in period t +1 can then be predicted using the following formula: ^ 2 = a^ + a ^ ^ 2 0 1 t t+1

© 2013 ELAN GUIDES

CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING

CURRENCY EXCHANGE R ATES: DETERMINATION AND FORECASTING Currency Cross Rates

For example, given the USD/EUR and JPY/USD exchange rates, we can calculate the cross rate between the JPY and the EUR, JPY/EUR as follows: JPY JPY = EUR USD

 USD

EUR

Cross Rate Calculations with Bid-Ask Spreads USD/EUR ask = 1.3806

USD/EUR bid = 1.3802

 Represents the price of EUR  An investor can buy EUR with

 Represents the price of EUR (base 

currency). An investor can sell EUR for USD at this price (as it is the bid price quoted by the dealer).

USD at this price.

Determining the EUR/USD bid cross rate: EUR/USD bid = 1/(USD/EUR ask ) Determining the EUR/USDask cross rate: EUR/USDask = 1 / (USD/EUR bid) Forward exchange rates (F) - One year Horizom

FFC/DC = SFC/DC 

(1 + iFC) (1 + iDC)

FPC/BC = SPC/BC 

Forward exchange rates (F) - Any Investment Horizom

FFC/DC = SFC/DC 

1 + (iFC  Actual 360) 1 + (iDC  Actual 360)

FPC/BC = SPC/BC 

1 + (iPC  Actual 360) 1 + (iBC  Actual 360)

Currencies Trading at a Forward Premium/Discount

FFC/DC  SFC/DC = SFC/DC

FPC/BC  SPC/BC = SPC/BC

© 2013 ELAN GUIDES

( (

) )

 iDC)  Actual 360 1 + (iDC  Actual 360)

(iFC

 iBC)  Actual 360 1 + (iBC  Actual 360)

(iPC

(1 + iPC) (1 + iBC)


Covered Interest Rate Parity

FPC/BC = SPC/BC 

1 + (iPC  Actual 360) 1 + (iBC  Actual 360)

The forward premium (discount) on the base currency can be expressed as a percentage as: FPC/BC  SPC/BC

Forward premium (discount) as a % =

SPC/BC

The forward premium (discount) on the base currency can be estimated as: Forward premium (discount) as a %  FPC/BC  SPC/BC  iPC  iBC Uncovered Interest Rate Parity

Expected future spot exchange rate:

SeFC/DC = SFC/DC 

(1 + iFC) (1 + iDC)

The expected percentage change in the spot exchange rate can be calculated as: Expected % change in spot exchange rate = 

SePC/BC =

SePC/BC – SPC/BC SPC/BC

The expected percentage change in the spot exchange rate can be estimated as: Expected % change in spot exchange rate  SePC/BC

 iPC  iBC

Purchasing Power Parity (PPP)

Law of one price: PXFC = PXDC  SFC/DC Law of one price: PXPC = PXBC  SPC/BC Absolute Purchasing Power Parity (Absolute PPP) SFC/DC = GPLFC / GPLDC SPC/BC = GPLPC / GPLBC Relative Purchasing Power Parity (Relative PPP) Relative PPP: E(STFC/DC) = S0FC/DC

(

1  FC 1 + DC

T

)

Ex Ante Version of PPP Ex ante PPP: %SeFC/DC Ex ante PPP: %SePC/BC

 eFC  eDC  ePC  eBC

© 2013 ELAN GUIDES


Real Exchange Rates

The real exchange rate (qFC/DC) equals the ratio of the domestic price level expressed in the foreign currency to the foreign price level. qFC/DC =

PDC in terms of FC PFC

=

PDC  SFC/DC PFC

= SFC/DC

( ) PDC PFC

The Fisher Effect

Fischer Effect: i = r + e International Fisher effect: (iFC – iDC) = (eFC – eDC) Figure 1: Spot Exchange Rates, Forward Exchange Rates, and Interest Rates

Ex Ante PPP

Foreign-Domestic Expected Inflation Differential

eFC  eDC

International Fisher Effect

Expected change in Spot Exchange Rate %SeFC/DC

Forward Rate as an Unbiased Predictor

Uncovered Interest Rate Parity

Foreign-Domestic Interest rate Differential iFC  iDC

Forward Discount FFC/DC  SFC/DC SFC/DC

Covered Interest Rate Parity

Balance of Payment

Current account + Capital account + Financial account = 0 Real Interest Rate Differentials, Capital Flows and the Exchange Rate

qL/H – qL/H = (iH – iL) – (eH – eL) – (H – L)

© 2013 ELAN GUIDES


The Taylor rule

i = r n +  +     y  y*) where i = the Taylor rule prescribed central bank policy rate r n = the neutral real policy rate  = the current inflation rate * = the central bank’s target inflation rate y = the log of the current level of output y* = the log of the economy’s potential/sustainable level of output qPC/BC = qPC/BC + ( r nBC  r nPC) + BC  BC  PC  PC

 yBC  y*BC)  yPC  y*PC)]  BC  PC)

© 2013 ELAN GUIDES

ECONOMIC GROWTH AND THE INVESTMENT DECISION

ECONOMIC GROWTH AND THE INVESTMENT DECISION Relationship between economic growth and stock prices

P = GDP

( )( ) E GDP

P E

P = Aggregate price or value of earnings. E = Aggregate earnings This equation can also be expressed in terms of growth rates:

P = (GDP) + (E/GDP) + (P/E) Production Function

Y = AK L1- Y = Level of aggregate output in the economy L = Quantity of labor K = Quantity of capital A = Total factor productivity. Total factor productivity (TFP) reflects the general level of productivity or technology in the economy. TFP is a scale factor i.e., an increase in TFP implies a proportionate increase in output for any combination of inputs.  = Share of GDP paid out to capital 1   = Share of GDP paid out to labor y = Y/L = A(K/L)(L/L)1- = Ak  y = Y/L = Output per worker or labor productivity. k = K/L = Capital per worker or capital-labor ratio

Cobb-Douglas production function

Y/Y =A/A + K/K + (1   )L/L Potential GDP

Growth rate in potential GDP = Long-term growth rate of labor force + Long-term growth rate in labor productivity

Labor Supply

Total number of hours available for work = Labor force  Average hours worked per worker

© 2013 ELAN GUIDES

ECONOMIC GROWTH AND THE INVESTMENT DECISION

Neoclassical Model (Solow’s Model)

( )[( )

Y 1 = K s

]

 ++n  (1-)

s = Fraction of income that is saved  = Growth rate of TFP  = Elasticity of output with respect to capital y = Y/L or income per worker k = K/L or capital-labor ratio  = Constant rate of depreciation on physical stock n = Labor supply growth rate. Savings/Investment Equation:

sy =

[(

]

 +  + n k (1  )

)

Growth rates of Output Per Capita and the Capital-Labor Ratio

y y

k k

=

 Y  + s (1) K

=

 Y +s  (1) K

( ) (

)

Production Function in the Endogenous Growth Model

ye = f(k e) = ck e

© 2013 ELAN GUIDES

INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENT AND RATIOS

INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENTS AND R ATIOS Ending Inventory = Opening Inventory + Purchases - Cost of goods sold

LIFO and FIFO Comparison with Rising Prices and Stable Inventory Levels LIFO

FIFO

COGS

Higher

Lower

Income before taxes

Lower

Higher

Income taxes

Lower

Higher

Net income

Lower

Higher

Total cash flow

Higher

Lower

EI

Lower

Higher

Working capital

Lower

Higher

LIFO versus FIFO with Rising Prices and Stable Inventory Levels Type of Ratio

Effect on

Effect on

Numerator

Denominator

Effect on Ratio

Income is lower under LIFO because COGS is higher

Sales are the same under both

Lower under LIFO

Same debt levels

Lower equity and assets under LIFO

Higher under LIFO

Profitability ratios

NP and GP margins

Solvency ratios

Debt-to-equity and debt ratio Liquidity ratios

Current ratio

Current assets are lower under LIFO because EI is lower

Current liabilities are the same.

Lower under LIFO

Quick ratio

Quick assets are higher under LIFO as a result of lower taxes paid

Current liabilities are the same

Higher under LIFO

Inventory turnover

COGS is higher under LIFO

Average inventory is lower under LIFO

Higher under LIFO

Total asset turnover

Sales are the same

Lower total assets under LIFO

Higher under LIFO

Activity ratios

© 2013 ELAN GUIDES


LIFO reserve (LR)

EIFIFO = EILIFO + LR where LR = LIFO Reserve COGSFIFO is lower than COGSLIFO during periods of rising prices: COGSFIFO = COGSLIFO  (Change in LR during the year) Net Income after tax under FIFO will be greater than LIFO net income after tax by:

Change in LIFO Reserve  (1  Tax rate)

When converting from LIFO to FIFO assuming rising prices: Equity (retained earnings) increases by: LIFO Reserve  (1  Tax rate)

Liabilities (deferred taxes) increase by: LIFO Reserve  (Tax rate)

Current assets (inventory) increase by: LIFO Reserve

© 2013 ELAN GUIDES


Impact of an Inventory Write-Down on Various Financial Ratios Effect on Numerator

Effect on Denominator

COGS increases so profits fall

Sales remain the same

Lower (worsens)

Debt levels remain the same

Equity decreases (due to lower profits) and current assets decrease (due to lower inventory)

Higher (worsens)

Current assets decrease (due to lower inventory)

Current liabilities remain the same.

Lower (worsens)

Inventory turnover

COGS increases

Average inventory decreases

Higher (improves)

Total asset turnover

Sales remain the same

Total assets decrease

Higher (improves)

Type of Ratio

Effect on Ratio

Profitability ratios

NP and GP margins

Solvency ratios

Debt-to-equity and debt ratio

Liquidity ratios

Current ratio

Activity ratios

© 2013 ELAN GUIDES

LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS

LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND R ATIOS Effects of Capitalization Effects on Financial Statements

Initially when the cost is capitalized

 Noncurrent assets increase.  Cash flow from investing activities decreases.

In future periods when the asset is depreciated or amortized

   

Noncurrent assets decrease. Net income decreases. Retained earnings decrease. Equity decreases.

Effects of Expensing Effects on Financial Statements

When the item is expensed

 Net income decreases by the entire after-tax amount 

 

of the cost. No related asset is recorded on the balance sheet and therefore, no depreciation or amortization expense is charged in future periods. Operating cash flow decreases. Expensed costs have no financial statement impact in future years.

Financial Statement Effects of Capitalizing versus Expensing Capitalizing

Expensing

Higher Lower Higher Higher Higher Lower Lower Lower

Lower Higher Lower Lower Lower Higher Higher Higher

Net income (first year) Net income (future years) Total assets Shareholders’ equity Cash flow from operations activities Cash flow from investing activities Income variability Debt to equity ratio Straight Line Depreciation Depreciation expense =

Original cost  Salvage value Depreciable life

Accelerated Depreciation DDB depreciation in Year X =

2 Depreciable life

× Book value at the beginning of Year X

© 2013 ELAN GUIDES


Gross investment in fixed assets Annual depreciation expense

=

Estimated useful or depreciable life

The historical cost of an asset divided by its useful life equals annual depreciation expense under the straight line method. Therefore, the historical cost divided by annual depreciation expense equals the estimated useful life.

Accumulated depreciation Annual depreciation expense

+

Net investment in fixed assets Annual depreciation expense

Average age of asset

Remaining useful life

Annual depreciation expense times the number of years that the asset has been in use equals accumulated depreciation. Therefore, accumulated depreciation divided by annual depreciation equals the average age of the asset.

The book value of the asset divided by annual depreciation expense equals the number of years the asset has remaining in its useful life.

Income Statement Effects of Lease Classification Income Statement Item

Finance Lease

Operating Lease

Lower (Depreciation)

Higher (Lease payment)

Higher (Interest expense)

Lower (None)

EBIT (operating income)

Higher

Lower

Total expenses- early years

Higher

Lower

Total expenses- later years

Lower

Higher

Net income- early years

Lower

Higher

Net income- later years

Higher

Lower

Operating expenses Nonoperating expenses

Cash Flow Effects of Lease Classification CF Item

Finance Lease

Operating Lease

CFO

Higher

Lower

CFF

Lower

Higher

Total cash flow

Same

Same

© 2013 ELAN GUIDES


Table 9: Impact of Lease Classification on Financial Ratios

Ratio

Numerator under Finance Lease

Denominator under Finance Lease

Ratio Better or Worse under Effect on Ratio Finance Lease

Asset turnover

Sales- same

Assets- higher

Lower

Worse

Return on assets* Net income- lower Assets- higher

Lower

Worse

Current ratio

Current assetssame

Current liabilitieshigher

Lower

Worse

Leverage ratios (D/E and D/A**)

Debt- higher

Equity- same Assets- higher

Higher

Worse

Equity- same

Lower

Worse

Return on equity* Net income- lower

**Notice that both the numerator and the denominator for the D/A ratio are higher when classifying the lease as a finance lease. Beware of such exam questions. When the numerator and the denominator of any ratio are heading in the same direction (either increasing or decreasing), determine which of the two is changing more in percentage terms. If the percentage change in the numerator is greater than the percentage change in the denominator, the numerator effect will dominate. Firms usually have lower levels of total debt compared to total assets. The increase in both debt and assets by classifying the lease as a finance lease will lead to an increase in the debt to asset ratio because the percentage increase in the numerator is greater.

Financial Statement Effects of Lease Classification from Lessor’s Perspective Financing Lease Operating Lease

Total net income Net income (early years) Taxes (early years) Total CFO Total CFI Total cash flow

Same Higher Higher Lower Higher Same

Same Lower Lower Higher Lower Same

© 2013 ELAN GUIDES

INTERCORPORATE INVESTMENTS

INTERCORPORATE INVESTMENTS Summary of Accounting Treatment for Investments In Financial Assets

In Associates

Business Combinations In Joint Ventures

Influence

Not significant

Significant

Controlling

Typical percentage interest

Usually < 20%

Usually 20%  50% Usually > 50% Varies

Accounting Classified into one of Treatment four categories based on management intent and type of security.

Equity method

Consolidation

Debt only:  Held-to-maturity (amortized cost, changes in value ignored unless deemed as impaired) Debt and Equity:  Held for trading (fair value, changes in value recognized in profit or loss)  Available-for-sale (fair value, changes in value recognized in equity)  Designated at fair value (fair value, changes in value recognized in profit or loss)

Combination

Description

Merger Acquisition Consolidation

Company A + Company B = Company A Company A + Company B = (Company A + Company B) Company A + Company B = Company C

© 2013 ELAN GUIDES

Shared Control

IFRS: Equity method or proportionate consolidation U.S. GAAP: Equity method (except for unincorporated ventures in specialized industries)

INTERCORPORATE INVESTMENTS

Adjusted Values Upon Reclassification of Sale of Receivables:

CFO CFF Total cash flow Current assets Current liabilities Current ratio (Assuming it was greater than 1)

Lower Higher Same Higher Higher Lower

Difference between QSPE and SPE Securitized Transaction: Qualified Special

Securitized Transaction: Special

Purpose Entity

Purpose Entity

 Originator of receivables sells financial  

assets to an SPE. The originator does not own or hold or expect to receive beneficial interest. SFAS 140 (before 2008 revision) allowed seller to derecognize the sold assets if transferred assets have been isolated from the transferor and are beyond the reach of bankruptcy, and are financial assets.

 Originator of receivables sells financial     

assets to an SPE. Seller is primary beneficiary; absorbs risks and rewards. Seller maintains some level of control. Seller is required to consolidate. Seller’s balance sheet would still show receivables as an asset. Debt of SPE would appear on seller’s balance sheet.

Impact of Different Accounting Methods on Financial Ratios Equity Method

Proportionate Consolidation

Acquisition Method

Better (lower) as liabilities are lower and equity is the same

In-between

Worse (higher) as liabilities are higher and equity is the same

Net Profit Margin

Better (higher) as sales are lower and net income is the same

In-between

Worse (lower) as sales are higher and net income is the same

ROE

Better (higher) as equity is lower Same as under the and net income is the same equity method

Worse (lower) as equity is higher and net income is the same

ROA

Better (higher) as net income is the same and assets are lower

Worse (lower) as net income is the same and assets are higher

Leverage

In-between

© 2013 ELAN GUIDES

EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED

EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED Final year’s salary = Current salary × [(1 + Annual compensation increase)years until retirement] Estimated annual payment = (Estimated final salary × Benefit formula) × Years of service Annual unit credit = Value at retirement / Years of service Types of Post-Employment Benefits Amount of PostType of Benefit

Sponsoring Company’s

Employment Benefit to

Obligation of Sponsoring

Pre-funding of Its Future

Employee

Company

Obligation

Defined contribution Amount of future benefit is pension plan not defined. Actual future benefit will depend on investment performance of plan assets. Investment risk is borne by employee.

Amount of the company’s Not applicable. obligation (contribution) is defined in each period. The contribution, if any, is typically made on a periodic basis with no additional future obligation.

Defined benefit pension plan

Amount of future benefit is defined, based on the plan’s formula (often a function of length of service and final year’s compensation). Investment risk is borne by company.

Amount of the future obligation, based on the plan’s formula, must be estimated in the current period.

Companies typically prefund the DB plans by contributing funds to a pension trust. Regulatory requirements to pre-fund vary by country.

Other postemployment benefits (e.g., retirees’ health care)

Amount of future benefit depends on plan specifications and type of benefit.

Eventual benefits are specified. The amount of the future obligation must be estimated in the current period.

Companies typically do not pre-fund other postemployment benefit obligations.

A company’s pension obligation will increase as a result of:  Current service costs.  Interest costs.  Past service costs.  Actuarial losses. A company’s pension obligation will decrease as a result of:  Actuarial gains.  Benefits paid.

© 2013 ELAN GUIDES


Reconciliation of the Pension Obligation: Pension obligation at the beginning of the period

+ Current service costs + Interest costs + Past service costs + Actuarial losses – Actuarial gains – Benefits paid Pension obligation at the end of the period

The fair value of assets held in the pension trust (plan) will increase as a result of:  A positive actual dollar return earned on plan assets; and  Contributions made by the employer to the plan. The fair value of plan assets will decrease as a result of:  Benefits paid to employees. Reconciliation of the Fair Value of Plan Assets: Fair value of plan assets at the beginning of the period

+ Actual return on plan assets + Contributions made by the employer to the plan  Benefits paid to employees Fair value of plan assets at the end of the period Balance Sheet Presentation of Defined Benefit Pension Plans

Funded status = Pension obligation  Fair value of plan assets

 

If Pension obligation > Fair value of plan assets: Plan is underfunded  Positive funded status  Net pension liability. If Pension obligation < Fair value of plan assets: Plan is overfunded  Negative funded status  Net pension asset.

Calculating Periodic Pension Cost

Priodic pension cost = Ending funded status  Beginning funded status + Employer contributions

Periodic pension cost = Current service costs + Interest costs + Past service costs + Actuarial losses  Actuarial gains  Actual return on plan assets

Under the corridor method, if the net cumulative amount of unrecognized actuarial gains and losses at the beginning of the reporting period exceeds 10% of the greater of (1) the defined benefit obligation or (2) the fair value of plan assets, then the excess is amortized over the expected average remaining working lives of the employees participating in the plan and included as a component of periodic pension expense on the P&L.

© 2013 ELAN GUIDES


Components of a Company’s Defined Benefit Pension Periodic Costs IFRS Component IFRS Recognition

U.S. GAAP Component

U.S. GAAP Recognition

Service costs

Recognized in P&L.

Current service costs Past service costs

Recognized in P&L. Recognized in OCI and subsequently amortized to P&L over the service life of employees.

Net interest income/ expense

Recognized in P&L as the following amount: Net pension liability or asset × interest rate(a)

Interest expense on pension obligation Expected return on plan assets

Recognized in P&L.

Actuarial gains and losses including differences between the actual and expected returns on plan assets

Recognized immediately in P&L or, more commonly, recognized in OCI and subsequently amortized to P&L using the corridor or faster recognition method.(b)  Difference between expected and actual return on assets = Actual return  (Plan assets × Expected return).  Actuarial gains and losses = Changes in a company’s pension obligation arising from changes in actuarial assumptions.

Remeasurements: Recognized in OCI and Net return on plan not subsequently assets and actuarial amortized to P&L. gains and losses

 Net return on plan assets = Actual return

 (Plan assets × Interest rate).

 Actuarial gains and losses = Changes in a company’s pension obligation arising from changes in actuarial assumptions.

Recognized in P&L as the following amount: Plan assets × expected return.

(a) The interest rate used is equal to the discount rate used to measure the pension liability (the yield on highquality corporate bonds.) (b) If the cumulative amount of unrecognized actuarial gains and losses exceeds 10 percent of the greater of the value of the plan assets or of the present value of the DB obligation (under U.S. GAAP, the projected benefit obligation), the difference must be amortized over the service lives of the employees.

© 2013 ELAN GUIDES


Impact of Key Assumptions on Net Pension Liability and Periodic Pension Cost Impact of Assumption on Net

Impact of Assumption on Periodic

Assumption

Pension Liability (Asset)

Pension Cost and Pension Expense

Higher discount rate

Lower obligation

Pension cost and pension expense will both typically be lower because of lower opening obligation and lower service costs

Higher rate of compensation increase

Higher obligation

Higher service and interest costs will increase periodic pension cost and pension expense.

Higher expected return on plan assets

No effect, because fair value of plan assets are used on balance sheet

Not applicable for IFRS No effect on periodic pension cost under U.S. GAAP Lower periodic pension expense under U.S. GAAP

© 2013 ELAN GUIDES

MULTINATIONAL OPERATIONS

MULTINATIONAL OPERATIONS 

The presentation currency (PC) is the currency in which the parent company reports its financial statements. It is typically the currency of the country where the parent is located. For example, U.S. companies are required to present their financial results in USD, German companies in EUR, Japanese companies in JPY, and so on.



The functional currency (FC) is the currency of the primary business environment in which an entity operates. It is usually the currency in which the entity primarily generates and expends cash.



The local currency (LC) is the currency of the country where the subsidiary operates.

Table 1 Foreign Currency Transaction

Type of Exposure

Export sale Import purchase

Asset (account receivable) Liability (account payable)

Strengthens

Weakens

Gain Loss

Loss Gain

Methods for Translating Foreign Currency Financial Statements of Subsidiaries Current Rate/ Temporal Method

Local Currency

T

Functional Currency

CR

Presentation Currency

Temporal Method

Local Currency

T

Functional Currency

=


Current Rate Method

Local Currency

=

Functional Currency

CR


  

The current rate is the exchange rate that exists on the balance sheet date. The average rate is the average exchange rate over the reporting period. The historical rate is the actual exchange rate that existed on the original transaction date.

© 2013 ELAN GUIDES


Rules for Foreign Currency Translation Current Rate Method FC = LC Income Statement Component

Sales Cost of goods sold Selling expenses Depreciation expense Amortization expense Interest expense Income tax Net income before translation gain (loss) Translation gain (loss) Net income Less: Dividends Change in retained earnings

Balance Sheet Component

Temporal Method FC = PC

Exchange Rate Used

Average rate Average rate Average rate Average rate Average rate Average rate Average rate

Average rate Historical rate Average rate Historical rate Historical rate Average rate Average rate Computed as Rev – Exp

N/A Computed as Rev – Exp Historical rate Computed as NI – Dividends Used as input for translated B/S

Plug in Number Computed as RE + Dividends Historical rate From B/S

Exchange Rate Used

Cash Accounts receivable Monetary assets Inventory Nonmonetary assets measured at current value Property, plant and equipment Less: Accumulated depreciation Nonmonetary assets measured at historical cost

Current rate Current rate Current rate Current rate Current rate

Current rate Current rate Current rate Historical rate Current rate

Current rate Current rate Current rate

Historical rate Historical rate Historical rate

Accounts payable Long-term notes payable Monetary liabilities Nonmonetary liabilities: Measured at current value Measured at historical cost Capital stock Retained earnings



Current rate Current rate Historical rate From I/S

Cumulative translation adjustment

Plug in Number

Current rate Historical rate Historical rate To balance Used as input for translated I/S N/A

© 2013 ELAN GUIDES


Balance Sheet Exposure

Foreign Currency (FC)

Strengthens Weakens Positive translation adjustment Negative translation adjustment Negative translation adjustment Positive translation adjustment

Balance Sheet Exposure

Net asset Net liability

Effects of Exchange Rate Movements on Financial Statements Temporal Method,

Temporal Method,

Net Monetary

Net Monetary Asset

Liability Exposure

Exposure

Foreign currency Revenues Revenues Assets Assets strengthens Liabilities Liabilities relative to parent’s  Net income  Net income presentation  Shareholders’ equity Shareholders’ equity currency Translation loss Translation gain

Foreign currency weakens relative to parent’s presentation currency

Current Rate Method Revenues Assets Liabilities  Net

income Shareholders’ equity Positive translation adjustment

 Revenues

 Revenues

 Revenues

 Assets

 Assets

 Assets

 Liabilities

 Liabilities

 Liabilities

 Net

 Net

income Shareholders’ equity Translation gain

 Net income income  Shareholders’ equity  Shareholders’ equity Translation loss Negative translation adjustment

Measuring Earnings Quality

Aggregate accruals = Accrual-basis earnings – Cash earnings Balance Sheet Approach

Net Operating Assets (NOA) NOAt = [(Total assetst  Casht)  (Total liabilitiest  Total debtt)] Aggregate Accruals Aggregate accrualst b/s = NOAt  NOAt1 Aggregate Ratio Accruals ratiot b/s =

© 2013 ELAN GUIDES

(NOAt  NOAt1) (NOAt + NOAt1)/2

INTEGRATION OF FINANCIAL STATEMENT ANALYSIS TECHNIQUES

INTEGRATION OF FINANCIAL STATEMENT ANALYSIS TECHNIQUES A Financial Statement Analysis Framework : Phase

Sources of Information

Examples of Output

1. Define the purpose and context of the analysis.

 The nature of the analyst’s

 Statement of the purpose or

 

function, such as evaluating an equity or debt investment or issuing a credit rating. Communication with client or supervisor on needs and concerns. Institutional guidelines related to developing specific work product.

 Financial statements, other

2. Collect input data.

 

financial data, questionnaires, and industry/economic data. Discussions with management, suppliers, customers, and competitors. Company site visits (e.g., to production facilities or retail stores)

  

objective of analysis. A list (written or unwritten) of specific questions to be answered by the analysis. Nature and content of report to be provided. Timetable and budgeted resources for completion.

 Organized financial statements.  Financial data tables.  Completed questionnaires, if applicable.

3. Process input data, as required, into analytically useful data.

 Data from the previous phase.

 Adjusted financial statements.  Common-size statements.  Forecasts.

4. Analyze/interpret the data.

 Input data and processed data

 Analytical results

5. Develop and communicate conclusions and recommendations (e.g., with an analysis report).

 Analytical results and previous

 Analytical report answering

6. Follow-up.

 Information gathered by



reports Institutional guidelines for published reports



questions posed in Phase 1 Recommendations regarding the purpose of the analysis, such as whether to make an investment or grant credit.

 Update reports and

periodically repeating above steps as necessary to determine whether changes to holdings or recommendations are necessary

recommendations

DuPont Analysis

ROE = Tax Burden × Interest burden × EBIT margin × Total asset turnover × Financial leverage ROE =

NI EBT

×

EBT EBIT

×

EBIT Revenue

×

Revenue Average Asset

×

Average Asset Average Equity

© 2013 ELAN GUIDES

CAPITAL BUDGETING

CAPITAL BUDGETING Expansion Project

Initial investment outlay for a new investment= FCInv + NWCInv NWCInv =  Non-cash current assets –  Non-debt current liabilities Annual after-tax operating cash flows (CF) CF = (S – C – D) (1 – t) + D

or

CF = (S – C) (1 – t) + tD

Terminal year after-tax non-operating cash flow (TNOCF): TNOCF = SalT + NWCInv – t(SalT – BVT)

Replacement Project

Investment outlays: Initial investment for a replacement project = FCInv + NWCInv – Sal0 + t(Sal0 – BV0) Annual after-tax operating cash flow:

CF = (S – C) (1 – t) + tD Terminal year after-tax non-operating cash flow: TNOCF = SalT + NWCInv – t(SalT – BT)

Mutually Exclusive Projects with Unequal Lives



Least Common Multiple of Lives Approach In this approach, both projects are repeated until their ‘chains’ extend over the same time horizon. Given equal time horizons, the NPVs of the two project chains are compared and the project with the higher chain NPV is chosen.

 Equivalent Annual Annuity Approach (EAA) This approach calculates the annuity payment (equal annual payment) over the project’s life that is equivalent in present value (PV) to the project’s NPV. The project with the higher EAA is chosen.

SML

R i = R F + ßi[E(R M) – R F] R i = Required return for project or asset i R F = Risk-free rate of return ßi = Beta of project or asset i [E(R M) – R F] = Market risk premium © 2013 ELAN GUIDES

CAPITAL BUDGETING

Economic Income

Economic income = After-tax operating cash flow + Increase in market value Economic income = After-tax operating cash flow + (Ending market value – Beginning market value) Economic income = After-tax operating cash flow – (Beginning market value – Ending market value) Economic income = After-tax cash flows – Economic depreciation

Economic Profit

Economic profit = [EBIT (1 – Tax rate)] – $WACC Economic profit = NOPAT – $WACC NOPAT = Net operating profit after tax $WACC = Dollar cost of capital = Cost of capital (%) × Invested capital Under this approach, a project’s NPV is calculated as the sum of the present values of economic profit earned over its life discounted at the cost of capital. 

NPV = MVA =

 (1 + WACC) EPt

t

Residual Income

Residual income = Net income for the period – Equity charge for the period Equity charge for the period = Required return on equity × Beginning-of-period book value of equity The RI approach calculates value from the perspective of equity holders only. Therefore, future residual income is discounted at the required rate of return on equity to calculate NPV. 

NPV =

 (1 + r ) RIt

E

t

Claims Valuation

  

First, we separate the cash flows available to debt and equity holders Then we discount them at their respective required rates of return. o Cash flows available to debt holders are discounted at the cost of debt, o Cash flows available to equity holders are discounted at the cost of equity. The present values of the two cash flow streams are added to calculate the total value of the company/asset.

© 2013 ELAN GUIDES

CAPITAL STRUCTURE

CAPITAL STRUCTURE The Capital Structure Decision

r D(1  t) +

r WACC =

rE

r D = Marginal cost of debt r E = Marginal cost of equity t = Marginal tax rate D = Market value of the company’s outstanding debt E = Market value of shareholders’ equity V = D + E = Value of the company MM Proposition II without Taxes: Higher Financial Leverage Raises the Cost of Equity

r WACC =

( ) ( ) r D +

r E = r 0

Company’s cost of equity (r E) under MM Proposition II without taxs is calculated as: Intercept

Independent variable

r E = r 0 + (r 0  r D)

Depende nt vari able

Slope

The total value of the company is calculated as: V=

Interest EBIT  Interest + r D r E

The systematic risk (ß) of the company’s assets can be expressed as the weighted average of the systematic risk of the company’s debt and equity.

A =

() ( ) +

This formula can also be expressed as:

E =

+ (A  D)

© 2013 ELAN GUIDES

()

CAPITAL STRUCTURE

Relaxing the Assumption of no Taxes

=

+ tD

The WACC is then calculated as:

r WACC =

r D(1  t) +

rE

And the cost of equity is calculated as:

r E = r 0 + (r 0  r D) (1  t)

Modigilani and Miller Propositions Without Taxes

Proposition I Proposition II

With Taxes

= r E = r 0 + (r 0  r D)

=

+ tD

r E = r 0 + (r 0  r D) (1  t)

The Optimal Capital Structure: The Static Trade-Off Theory

VL = VU + tD – PV(Costs of financial distress)

© 2013 ELAN GUIDES

DIVIDENDS AND SHARE REPURCHASE

DIVIDENDS AND SHARE R EPURCHASE The expected decrease in share price when it goes ex-dividend can be calculated using the following equation:

PW

PX =

D

Pw = Share price with the right to receive the dividend PX = Share price without the right to receive the dividend D = Amount of dividend TD = Tax rate on dividends TCG = Tax rate on capital gains

Double Taxation System

ETR = CTR + [(1 – CTR) × MTR D] ETR = Effective tax rate CTR = Corporate tax rate MTR D = Investor’s marginal tax rate on dividends

Split-Rate Tax System

ETR = CTR D + [(1 – CTR D) × MTR D] CTR D = Corporate tax rate on earnings distributed as dividends.

Stable Dividend Policy

The expected increase in dividends is calculated as: Expected dividend increase = Increase in earnings × Target payout ratio × Adjustment factor Adjustment factor = 1/N N = Number of years over which the adjustment is expected to occur

Analysis of Dividend Safety

Dividend payout ratio = (dividends / net income) Dividend coverage ratio = (net income / dividends) FCFE coverage ratio = FCFE / [Dividends + Share repurchases]

© 2013 ELAN GUIDES

MERGERS AND ACQUISITION

MERGERS AND ACQUISITION Mergers and the Industry Life Cycle Industry Life Cycle Stage

Industry Description

Pioneering development

 Low but slowly 

increasing sales growth. Substantial development costs.

Rapid accelerating growth

 High profit

Mature growth

 Decrease in the



 Stabilization and market maturity

Motives for Merger

 Younger, smaller companies may sell  Conglomerate themselves to larger firms in mature  Horizontal 

 To meet substantial capital

margins. Low competition.

requirements for expansion.

 Horizontal savings, and operational efficiencies.  Vertical

entry of new competitors. Growth potential remains.

constraints Increasing competition.

 Overcapacity.  Eroding profit margins.

 Conglomerate  Horizontal

 To achieve economies of scale,



Deceleration of growth and decline

or declining industries to enter into a new growth industry. Young companies may merge with firms that allow them to pool management and capital resources.

 Increasing capacity  To achieve economies of scale in 

Types of Merger

 Horizontal

research, production, and marketing to match low costs and prices of competitors. Large companies may buy smaller companies to improve management and provide a broader financial base.

 Horizontal mergers to ensure survival.  Horizontal  Vertical mergers to increase efficiency  Vertical and profit margins.  Conglomerate  Conglomerate mergers to exploit 

synergy. Companies in the industry may acquire companies in young industries.

Source: Adapted from J. Fred Weston, Kwang S. Chung, and Susan E. Hoag, Mergers, Restructuring, and Corporate Control (New York: Prentice Hall, 1990, p.102) and Bruno Solnik and Dennis McLeavy, International Investments, 5th edition (Boston: Addison Wesley, 2004, p. 264 – 265).

© 2013 ELAN GUIDES


Major Differences of Stock versus Asset Purchases Stock Purchase

Payment

Asset Purchase

Target shareholders receive compensation in exchange for their shares. Shareholder approval required.

Payment is made to the selling company rather than directly to shareholders. Approval Shareholder approval might not be required. Tax: Corporate No corporate-level taxes. Target company pays taxes on any capital gains. Tax: Shareholder Target company’s shareholders No direct tax consequence for target are taxed on their capital gain. company’s shareholders. Liabilities Acquirer assumes the target’s Acquirer generally avoids the liabilities. assumption of liabilities. Herfindahl-Hirschman Index (HHI) n

 i

(

Sales or output of firm i Total sales or output of market

 100

)

2

HHI Concentration Levels and Possible Government Response Post-Merger HHI

Concentration

Change in HHI Government Action

Less than 1,000

Not concentrated

Any amount

No action

Between 1,000 and 1,800 Moderately concentrated 100 or more

Possible challenge

More than 1,800

Challenge

Highly concentrated

50 or more

FCFF is estimated by:

+

Net income Net interest after tax

= +

Unlevered income Changes in deferred taxes

= NOPLAT (net operating profit less adjusted taxes) + Net noncash charges – Change in net working capital – Capital expenditures (capex) Free cash flow to the firm (FCFF) Net interest after tax = (Interest expense – Interest income) (1 – tax rate) Working capital = Current assets (excl. cash and equivalents) – Current liabilities (excl. short-term debt)

© 2013 ELAN GUIDES


Comparable Company Analysis

TP =

(DP  SP) SP

TP = Takeover premium DP = Deal price per share SP = Target’s stock price per share Bid Evaluation

Target shareholders’ gain = Takeover premium = PT – VT Acquirer’s gain = Synergies – Premium = S – (PT – VT) S = Synergies created by the merger transaction The post-merger value of the combined company is composed of the pre-merger value of the acquirer, the pre-merger value of the target, and the synergies created by the merger. These sources of value are adjusted for the cash paid to target shareholders to determine the value of the combined post-merger company. VA* = VA + VT + S – C VA* = Value of combined company C = Cash paid to target shareholders

© 2013 ELAN GUIDES

EQUITY VALUATION: APPLICATIONS AND PROCESSES

EQUITY VALUATION: APPLICATIONS AND PROCESSES Perceived mispricing: Perceived mispricing = True mispricing + Error in the estimate of intrinsic value. VE – P = (V – P) + (VE – V) VE = Estimate of intrinsic value P = Market price V = True (unobservable) intrinsic value

© 2013 ELAN GUIDES

RETURN CONCEPTS

R ETURN CONCEPTS Holding Period Return

Holding period return =

PH – P0 + DH P0

PH = Price at the end of the holding period P0 = Price at the beginning of the period DH = Dividend

Required Return

 

The difference between an asset’s expected return and its required return is known as expected alpha, ex ante alpha or expected abnormal return. o Expected alpha = Expected return – Required return The difference between the actual (realized) return on an asset and its required return is known as realized alpha or ex post alpha. Realized alpha = Actual HPR – Required return for the period o

When the investor’s estimate of intrinsic value (V0) is different from the current market price (P0), the investor’s expected return has two components: 1. 2.

The required return (r T) earned on the asset’s current market price; and The return from convergence of price to value [(V0 – P0)/P0].

Internal Rate of Return

Intrinsic Value =

V0 =

Next year’s expected dividend Required return – Expected dividend growth rate D1 k e – g

If the asset is assumed to be efficiently-priced (i.e. the market price equals its intrinsic value), the IRR would equal the required return on equity. Therefore, the IRR can be estimated as:

Required return (IRR) =

k e (IRR) =

Next year’s dividend

D1 P0

Market price

+ Expected dividend growth rate

+g

© 2013 ELAN GUIDES

RETURN CONCEPTS

Equity Risk Premium

The required rate of return on a particular stock can be computed using either of the following two approaches. Both these approaches require the equity risk premium to be estimated first. 1.

Required return on share i = Current expected risk-free return + ßi(Equity risk premium)

 A beta greater (lower) than 1 indicates that the security has greater-than-average (lower-thanaverage) systematic risk. 2.

Required return on share i = Current expected risk-free return + Equity risk premium ± Other risk premia/discounts appropriate for i

 This method of estimating the required return is known as the build-up method. It is discussed later in the reading and is primarily used for valuations of private businesses. Gordon Growth Model (GGM) Estimates

GCM equity risk premium estimate =

D1 P0

+ g – r LTGD

Macroeconomic Model Estimates Equity risk premium = {[(1 + EINFL) (1 + EGREPS) (1 + EGPE) – 1] + EINC} – Expected RF Expected inflation =

1 + YTM of 20-year maturity T-bonds 1 + YTM of 20-year maturity TIPS

– 1

The Captial Asset Pricing Model (CAPM)

Required return on i = Expected risk-free rate + Betai (Equity risk premium) The Fama-French Model

imktRMRF + isizeSMB + ivalueHML

r i = RF + ßmkt = Market beta ßsize = Size beta ßvalue = Value beta

The Pastor-Stambaugh model (PSM)

r i = R F +

imktRMRF + isizeSMB + ivalueHML + iliqLIQ

ßliq = Liquidity beta

© 2013 ELAN GUIDES

RETURN CONCEPTS

BIRR model r i = T-bill rate + (Sensitivity to confidence risk × Confidence risk) + (Sensitivity to time horizon risk × Time horizon risk) + (Sensitivity to inflation risk × Inflation risk) + (Sensitivity to business cycle risk × Business cycle risk) + (Sensitivity to market timing risk × Market timing risk) Build-up method

r i = Risk-free rate + Equity risk premium + Size premium + Specific-company premium

For companies with publicly-traded debt , the bond-yield plus risk premium approach can be used to calculate the cost of equity: BYPRP cost of equity = YTM on the company’s long-term debt + Risk premium

Adjusting Beta for Beta Drift

Adjusted beta = (2/3) (Unadjusted beta) + (1/3) (1.0) Estimating the Asset Beta for the Comparable Publicly Traded Firm: BASSET reflects only business risk of the comparable company. Therefore it is used as a proxy for business risk of the project being studied.

1

ßASSET = ß EQUITY

(

1 + (1 - t)

D E

)

BEQUITY reflects business and financial risk of comparable company.

where: D/E = debt-to-equity ratio of the comparable company. t = marginal tax rate of the comparable company. To adjust the asset beta of the comparable for the capital structure (financial risk) of the project or company being evaluated, we use the following formula: BPROJECT reflects business and financial risk of the project.

ßPROJECT = ßASSET

D 1 + (1 - t) E

BASSET reflects business risk of project.

where: D/E = debt-to-equity ratio of the subject company. t = marginal tax rate of the subject company. Country Spread Model

ERP estimate = ERP for a developed market + Country premium

© 2013 ELAN GUIDES

RETURN CONCEPTS

Weighted Average Cost of Capital (WACC) WACC =

MVD MVD + MVCE

r d (1 – Tax rate ) +

MVCE MVD + MVCE

MVD = Market value of the company’s debt r d = Required rate of return on debt MVCE = Market value of the company’s common equity r = Required rate of return on equity

© 2013 ELAN GUIDES

r

DISCOUNTED DIVIDEND VALUATION

DISCOUNTED DIVIDEND VALUATION One-Period DDM

V0 =

D1 (1 + r)1

+

P1 (1 + r) 1

D1 + P1

=

(1 + r)1

V0 = The value of the stock today (t = 0) P1 = Expected price of the stock after one year (t = 1) D1 = Expected dividend for Year 1, assuming it will be paid at the end of Year 1 (t = 1) r = Required return on the stock Multiple-Period DDM

V0 =

D1

Dn Pn + + + ... (1 + r)1 (1 + r)n (1 + r)n n

V0 =

 (1 + r) Dt

t

+

t = 1

Pn (1 + r)n

Expression for calculating Value of a share of stock 

V0 =

 (1 + r) Dt

t

t = 1

Gordon Growth Model

V0 =

D0 (1 + g) (r – g)

, or V0 =

D1 (r – g)

Present value of Growth Opportunities

V0 =

E1 r

+ PVGO

P/E ratio

Justified leading P/E ratio =

Justified trailing P/E =

P0 E0

P0 E1

=

=

D1/E1

D1/E0 r  g

r  g

=

=

(1  b) r  g

D0 (1  g) / E0 r  g

=

(1  b)(1  g) r  g

© 2013 ELAN GUIDES

DISCOUNTED DIVI DEND VALUATION

Value of Fixed-Rate Perpetual Preferred Stock

D

V0 =

r

Two-Stage Dividend Discount Model n

V0 =



D0 (1 + gS)t (1 + r)t

t = 1

+

D0 (1 + gS)n(1 + gL) (1 + r)n(r – gL)

gS = Short term supernormal growth rate gL = Long-term sustainable growth rate r = required return n = Length of the supernormal growth period The H-Model

V0 =

D0 (1 + gL) r – gL

+

D0H (gs – gL) r – gL

gS = Short term high growth rate gL = Long-term sustainable growth rate r = required return H = Half-life = 0.5 times the length of the high growth period The H-model equation can be rearranged to calculate the required rate of return as follows:

r=

( )

D0 [(1 + gL) + H(gs – gL)] + gL P0

The Gordon growth formula can be rearranged to calculate the required rate of return given the other variables.

r=

D1 P0

+g

Sustainable growth rate (SGR)

g = b × ROE b = Earnings retention rate, calculated as 1 – Dividend payout ratio

© 2013 ELAN GUIDES

DISCOUNTED DIVIDEND VALUATION

ROE can be calculated as: ROE =

Net income Sales Total assets × × Sales Total assets Shareholders’ equity

PRAT model

g = Profit margin × Retention rate × Asset turnover × Financial l everage

g=

Net income - Dividends Net income Sales Total assets × × × Net income Sales Total assets Shareholders’ equity

© 2013 ELAN GUIDES

FREE CASH FLOW VALUATION

FREE CASH FLOW VALUATION FCFF/FCFE 

Firm Value =

FCFFt

 (1+WACC)

t

t =1

WACC =

MV(Debt) MV(Debt) + MV(Equity)

r d (1  Tax Rate) +

MV(Equity) MV(Debt) + MV(Equity)

r

Equity Value = Firm Value  Market value of debt 

Equity Value =

FCFEt

 (1 + r ) t =1

t

Computing FCFF from Net Income

FCFF = NI + NCC + Int(1  Tax Rate)  FCInv  WCInv Investment in fixed capital (FCInv) FCInv = Capital expenditures  Proceeds from sale of long-term assets Investment in working capital (WCInv) WCInv = Change in working capital over the year Working capital = Current assets (exc. cash)  Current liabilities (exc. short-term debt)

Table: Noncash Items and FCFF Adjustment to NI to Noncash Item

Arrive at FCFF

Depreciation Amortization and impairment of intangibles Restructuring charges (expense) Restructuring charges (income resulting from reversal) Losses Gains Amortization of long-term bond discounts Amortization of long-term bond premiums Deferred taxes

Added back Added back Added back Subtracted Added back Subtracted Added back Subtracted Added back but requires special attention

© 2013 ELAN GUIDES

FREE CASH FLOW VALUATION

Computing FCFF from CFO Table: IFRS versus U.S. GAAP Treatment of Interest and Dividends IFRS

U.S. GAAP

Interest received Interest paid

CFO or CFI CFO or CFF

CFO CFO

Dividend received Dividends paid

CFO or CFI CFO or CFF

CFO CFF

FCFF = CFO + Int(1  Tax rate)  FCInv

Computing FCFF from EBIT

FCFF = EBIT(1 – Tax rate) + Dep – FCInv – WCInv

Computing FCFF from EBITDA

FCFF = EBITDA(1 – Tax rate) + Dep(Tax rate) – FCInv – WCInv

Computing FCFE from FCFF

FCFE = FCFF – Int(1– Tax rate) + Net borrowing

Computing FCFE from Net Income

FCFE = NI + NCC – FCInv – WCInv + Net Borrowing

Computing FCFE from CFO

FCFE = CFO + FCInv – Net borrowing

Computing FCFE from EBIT

FCFE = EBIT(1 – Tax rate) – Int(1 – Tax rate) + Dep – FCInv – WCInv + Net borrowing

Computing FCFE from EBITDA

FCFE = EBITDA(1 – Tax rate) – Int(1 – Tax rate) + Dep(Tax rate) – FCInv – WCInv + Net borrowing

© 2013 ELAN GUIDES

FREE CASH C ASH FLOW VALUATION VALUATION

Uses of FCFF

Increases in cash balances Plus: Net payments to providers of debt capital + Interest expense (1 – tax rate) rate) + Repayment of principal  New borrowings Plus: Net payments to providers of equity capital + Cash dividends + Share repurchases  New equity issues = Uses of FCFF Uses of FCFE

Increases in cash balances Plus: Net payments to providers of equity capital c apital + Cash dividends + Share repurchases  New equity issues = Uses of FCFE Constant Growth FCFF Valuation Model

Value of the firm =

FCFF1 WACC - g

=

FCFF0 (1 + g) WACC - g

WACC = Weighted average cost of capital g = Long-term constant growth rate in FCFF Constant Growth FCFE Valuation Model

Value of equity =

FCFE1 FCFE0 (1 + g) = - g r - g r -

= Required rate of return on equity r = g = Long-term constant growth rate in FCFE An International Application of the Single-Stage Model

Value of equity =

© 2013 ELAN GUIDES

FCFE0 (1 + greal) r real  greal

FREE CASH FLOW VALUATION VALUATION

General expression for the two-stage FCFF model: n

Firm value =

FCFFt

 (1 + WACC)

t

t=1

+

FCFFn+1

1

(WACC  g) (1 + WACC)n

Firm value = PV of FCFF in Stage 1 + Terminal Terminal value × Discount Factor

General expression for the two-stage FCFE model: n

Equity value =

FCFEt

 (1 + r)

t

+

FCFFn+1

1

r  g

(1 + r)n

t=1

Equity value = PV of FCFE in Stage 1 + Terminal Terminal value × Discount Factor Determining Terminal Value

Terminal value in year n = Justified Trailing P/E × Forecasted Earnings in Year Year n Terminal value in year n = Justified Leading P/E × Forecasted Earnings in Year Year n + 1

Non-operating Assets and Firm Value

Value of the firm = Value of operating assets + Value of non-operating assets

© 2013 ELAN GUIDES

MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES

MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES Price to Earnings Ratio

Trailing P/E ratio =

Current Stock Price Last year’s EPS

Forward P/E ratio =

Current Stock Price Expected EPS

Price to Book Ratio

P/B ratio =

P/B ratio =

Market price per share Book value per share Market value of common shareholders’ equity Book value of common shareholders’ equity

Book value of equity = Common shareholders’ equity = Shareholders’ equity – Total value of equity claims that are senior to common stock Book value of equity = Total assets – Total liabilities – Preferred stock Price to Sales Ratio

P/S ratio =

Market price per share Sales per share

Relationship between the P/E ratio and the P/S ratio P/E × Net profit margin = (P / E) × (E / S) = P/S Price to Cash Ratio

P/CF ratio =

Market price per share Free cash flow per share

Dividend Yield

Justified trailing dividend yield Trailing dividend yield = Last year’s dividend / Current price per share Justified leading dividend yield Leading dividend yield = Next year’s year ’s dividend / Current price per share

© 2013 ELAN GUIDES


Justified P/E Multiple Based on Fundamentals

D1

V0 =

(r  g)

Justified leading P/E multiple

Justified leading P/E =

P0 E1

=

D1/E1 r  g

=

(1  b) r  g

(1 – b) is the payout ratio. Justified trailing P/E multiple

Justified trailing P/E =

P0 E0

=

D1/E0 r  g

=

D0 (1  g) / E0 r  g

=

(1  b)(1  g) r  g

Justified P/B Multiple Based on Fundamentals

P0 B0

=

ROE  g r  g

ROE = Return on equity r = required return on equity g = Sustainable growth rate Justified P/S Multiple Based on Fundamentals

P0 S0

=

(E0/S0)(1  b)(1  g) r  g

E0/S0 = Net profit margin 1 – b = Payout ratio Justified P/CF Multiple Based on Fundamentals

FCFE0 (1  g)

V0 =

(r  g)

Justified Dividend Yield

D0 P0

=

r  g 1  g

© 2013 ELAN GUIDES


P/E-to-growth (PEG) ratio

PEG =

P/E Growth (%)

Terminal price based on fundamentals

TVn = Justified leading P/E  Forecasted earningsn +1 TVn = Justified trailing P/E  Forecasted earningsn

Terminal price based on comparables

TVn = Benchmark leading P/E  Forecasted earningsn +1 TVn = Benchmark trailing P/E  Forecasted earningsn

EV/EBITDA Multiple

Enterprise value = Market value of common equity + Market value of preferred stock + Market value of debt – Value of cash and short-term investments EBITDA = Net income + Interest + Taxes + Depreciation and amortization Alternative Denominators in Enterprise Value Value Multiples

Free Cash Net plus minus plus plus less less Flow to the Income Interest Tax Savings Depreciation Amortization Investment in Investment in Firm = Expense on Interest Working Capital Fixed Capital EBITDA= Net plus plus Income Interest Taxes Expense EBITA =

Net plus plus Income Interest Taxes Expense

EBIT =

Net plus plus Income Interest Taxes Expense

plus plus Depreciation Amortization

plus Amortization

Justified forward P/E after accounting for Inflation

P0 E1

=

1

  (1  ) I

 = The percentage of inflation in costs that the company can pass through to revenue.  = Real rate of return I = Rate of inflation

© 2013 ELAN GUIDES


Unexpected earnings (UE)

UEt = EPSt – E (EPS (EPSt) Standardized unexpected earnings (SUE)

SUEt =

EPSt  E (EPS (EPSt )

[EPSt  E (EPS (EPSt )]

EPSt = Actual EPS for time t (EPSt ) = Expected EPS for time t E (EPS

[EPSt  E (EPS (EPSt )] = Standard deviation of [EPSt  E (EPS (EPSt )]

© 2013 ELAN GUIDES

RESIDUAL INCOME IN COME VALUATION VALUATION

R ESIDUAL ESIDUAL INCOME VALUATION The Residual Income

Residual income = Net income – Equity charge Equity charge = Cost of equity capital × Equity capital Residual income = After-tax operating profit  Capital charge Capital charge = Equity charge + Debt charge Debt charge = Cost of debt × (1 – Tax rate) × Debt capital Economic Value Added

EVA = NOPAT – (C% × TC) NOPA NOPAT = Net Net operating profit after tax = EBIT (1 – Tax Tax rate) C% = Cost of capital (WACC) TC = Total capital Market Value Added

MVA = Market value of the company – Accounting book value of total capital Market value of company = Market value of debt + Market value of equity. The Residual Income Model

RIt = Et – (r × Bt-1) RIt = Residual income at time t Et = Earnings at time t r = Required rate of return on equity Bt-1 = Book value at time t-1 Intrinsic value of a stock: 

V 0 = B0 +

RIt

 (1 + r )

t

i = 1



E t  rBt-1

 (1 + r )

= B0 +

t

i = 1

V0 = Intrinsic value of the stock today B0 = Current book value per share of equity Bt = Expected book value per share of equity at any time t r = Required rate of return on equity Et = Expected EPS for period t RIt = Expected residual income per share

© 2013 ELAN GUIDES

RESIDUAL I NCOME VALUATION

Residual Income Model (Alternative Approach)

RIt = EPSt - (R × Bt-1) RIt = (ROE - r)Bt-1 

V0 = B0 +



(ROEt  r )Bt-1 (1 + r )t

t = 1

V0 = B0 +

ROE  r B0 r  g

Tobin’s q

Tobin’s q =

Market value of debt and equity Replacement cost of total assets

Multi-Stage Residual Income Valuation T

V0 = B0 +

 t=1

(Et  rBt 1) (1 + r)t

+

PT  BT (1 + r)T

When residual income fades over time as ROE declines towards the required return on equity, the intrinsic value of a stock is calculated using the following formula:

 (E(1 rB + r) T-1

V0 = B0 +

t

t 1) t

t=1

+

ET  rBT-1 (1 + r  )(1 + r)T1

 = Persistence factor. Implied Growth Rate

g=r

[

(ROE  r) × B0 V0  B0

]

© 2013 ELAN GUIDES

PRIVATE COMPANY VALUATION

PRIVATE COMPANY VALUATION The Capitalized Cash Flow Method

Vf =

FCFF1 WACC  gf

Vf = Value of the firm FCFF1 = Free cash flow to the firm for next twelve months WACC = Weighted average cost of capital gf = Sustainable growth rate of free cash flow to the firm

V=

FCFE1 r  g

V = Value of the equity FCFE1 = Free cash flow to the equity for next twelve months r = Required return on equity g = Sustainable growth rate of free cash flow to the equity Methods Used to Estimate the Required Rate of Return for a Private Company

Capital Asset Pricing Model Required return on equity = Risk-free rate + (Beta × Market risk premium) Expanded CAPM Required return on equity = Risk-free rate + (Beta × Market risk premium) + Small stock premium + Company-specific risk premium Build-Up Approach Required return on equity = Risk-free rate + Equity risk premium + Small stock premium + Company-specific risk premium + Industry risk premium Discount for Lack of Control (DLOC)

DLOC = 1 -

© 2013 ELAN GUIDES

1 1 + Control Premium

PRIVATE REAL ESTATE INVESTMENTS

PRIVATE R EAL ESTATE INVESTMENTS Net Operating Income

Rental income at full occupancy + Other income (such as parking) = Potential gross income (PGI)  Vacancy and collection loss = Effective gross income (EGI)  Operating expenses (OE) = Net operating income (NOI) The Direct Capitalization Method

Cap rate = Discount rate – Growth rate The cap rate can be defined as the current yield on an investment:

Capitalization rate =

NOI1 Value

Rearranging the above equation, we can estimate the value of a property by dividing its firstyear NOI by the cap rate.

Value =

NOI1 Cap rate

An estimate of the appropriate cap rate for a property can be obtained from the selling price of similar or comparable properties.

Cap rate =

NOI Sale price of comparable property

The cap rate derived by dividing rent by recent sales prices of comparables is known as theall risks yield (ARY). The value of a property is then calculated as:

Market value =

Rent1 ARY

Other Forms of the Income Approach Gross income multiplier =

Selling price Gross income

Value of subject property = Gross income multiplier  Gross income of subject property

© 2013 ELAN GUIDES

PRIVATE REAL ESTATE INVESTMENTS

The Discounted Cash Flow Method (DCF)

Value =

NOI1 (r – g)

The Terminal Capitalization Rate

Terminal value =

NOI for the first year of ownership for the next investor Terminal cap rate

Appraisal-Based Indices

Return =

NOI  Capital expenditures + (Ending market value  Beginning market value) Beginning market value

Loan to Value ratio

LTV ratio =

Loan amount Appraised value

Debt Service Coverage ratio

DSCR =

NOI Debt service

Equity dividend rate/Cash-on-cash return

Equity dividend rate =

First year cash flow Equity investment

© 2013 ELAN GUIDES

PUBLICLY TRADED REAL ESTATE SECURITIES

PUBLICLY TRADED R EAL ESTATE SECURITIES VALUATION: NET ASSET VALUE APPROACH Capitalization rate

Capitalization rate =

NOI of a comparable property Total value of comparable property

Net Asset Value per Share

NAVPS =

Net Asset Value Shares outstanding

VALUATION: RELATIVE VALUATION (PRICE MULTIPLE) APPROACH Funds from operations (FFO)

Accounting net earnings Add: Depreciation charges on real estate Add: Deferred tax charges Add (Less): Losses (gains) from sales of property and debt restructuring Funds from operations Adjusted funds from operations (AFFO)

Funds from operations Less: Non-cash rent Less: Maintenance-type capital expenditures and leasing costs Adjusted funds from operations

AFFO is preferred over FFO as it takes into account the capital expenditures necessary to maintain the economic income of a property portfolio.

© 2013 ELAN GUIDES

PRIVATE EQUITY VALUATION

PRIVATE EQUITY VALUATION Quantitative Measures of Return



PIC (paid in capital): Ratio of paid in capital to date to committed capital.



DPI (distributed to paid-in) or cash-on-cash return: Value of cumulative distributions paid to LPs as a proportion of cumulative invested capital. o (DPI = Cumulative distributions / PIC)



RVPI (residual value to paid-in): Value of LPs’ shareholdings held with the fund as a proportion of cumulative invested capital. RVPI = NAV after distributions / PIC o



TVPI (total value to paid-in): Value of portfolio companies’ distributed (realized) and undistributed (unrealized) value as a proportion of cumulative invested capital. TVPI = DPI + RVPI o

NAV before distributions = Prior year’s NAV after distributions + Capital called

down – Management Fees + Operating results NAV after distributions = NAV before distributions – Carried interest – Distributions

Total Exit Value Exit value = Initial cost + Earnings growth + Multiple expansion + Debt reduction

Post-money valuation (POST) POST = PRE + I Proportionate ownership of the VC investor = I / POST

Post-money value Post-money value =

Exit value (1 + Required rate of return ) Number of years to exists

Required wealth Required wealth = Investment  (1 + IRR) Number of years to exit Ownership propotion Ownership proportion = Required wealth / Exit value

© 2013 ELAN GUIDES

PRIVATE EQUITY VALUATION

Shares to be issued

Shares to be issued =

Proportion of venture capitalist investment  Shares held by company founders Proportion of investment of company founders

Price per share Price per share =

Amount of venture capital investment Number of shares issued to venture capital investment

Adjusted discount rate

Adjusted discount rate =

1 + r –1 1–q

r = Discount rate unadjusted for probability of failure. q = Probability of failure.

© 2013 ELAN GUIDES

FUNDAMENTALS OF CREDIT ANALYSIS

FUNDAMENTALS OF CREDIT ANALYSIS Expected Loss

Expected loss = Default probability  Loss severity given default

Yield on a corporate bond:

Yield on a corporate bond = Real risk-free interest rate + Expected inflation rate + Maturity premium + Liquidity premium + Credit spread

Yield Spread:

Yield spread = Liquidity premium + Credit spread

For small, instantaneous changes in the yield spread, the return impact (i.e. the percentage change in price, including accrued interest) can be estimated using the following formula: Return impact  – Modified duration × Spread For larger changes in the yield spread, we must also incorporate the (positive) impact of convexity into our estimate of the return impact: Return impact

© 2013 ELAN GUIDES

 – (MDur × Spread) + (1/2 × Convexity × Spread2)

TERM STRUCTURE AND VOLATILITY OF INTEREST RATES

TERM STRUCTURE AND VOLATILITY OF INTEREST R ATES Measuring Historical Yield Volatility

X t = 100  ln

yt

( ) yt1

where yt = yield on day t Annualizing the Standard Deviation

Annualized standard deviation = Daily standard deviation    of days in a year

Calculating Variance of Daily Yield Changes T

2

T1

Variance =

X t

... assigns an equal weight to all observations

t = 1

T

2

T1

Variance =

W t X t

... attaches a greater weight to more recent information

t = 1

where: W t = the weight assigned to each daily yield change observation such that the sum of the weights equals 1.

© 2013 ELAN GUIDES

VALUING BONDS WITH EMBEDDED OPTIONS

VALUING BONDS WITH EMBEDDED OPTIONS Treasury Market Benchmark Spread Measure

Benchmark

Reflects Compensation For

Nominal Zero-volatility Option-adjusted

Treasury yield curve Treasury spot rate curve Treasury spot rate curve

Credit risk, liquidity risk and option risk Credit risk, liquidity risk and option risk Credit risk and liquidity risk

Specific Bond Sector with a Given Credit Rating Benchmark Spread Measure

Benchmark



Sector yield curve Sector spot rate curve Sector spot rate curve

Credit risk, liquidity risk and option risk Credit risk, liquidity risk and option risk Credit risk and liquidity risk

Issuer-Specific Benchmark Spread Measure

Benchmark



Issuer yield curve Issuer spot rate curve Issuer spot rate curve

Liquidity risk and option risk Liquidity risk and option risk Liquidity risk

Summary of Relationships between Benchmark, OAS and Relative Value Benchmark

Negative OAS

Zero OAS

Positive OAS

Treasury market

Overpriced (rich) security

Overpriced (rich) security

Comparison must be made between security OAS and OAS of comparable securities (required OAS): If security OAS > required OAS, security is cheap If security OAS < required OAS, security is rich If security OAS = required OAS, security is fairly priced

Bond sector with a Overpriced (rich) security given credit rating (assumes credit rating higher (assumes credit rating than security being analyzed ) higher than security being analyzed )

© 2013 ELAN GUIDES

Overpriced (rich) Comparison must be made security between security OAS and OAS (assumes credit rating of comparable securities (required OAS): higher than security If security OAS > required OAS, being analyzed ) security is cheap If security OAS < required OAS, security is rich If security OAS = required OAS, security is fairly priced


Summary of Relationships between Benchmark, OAS and Relative Value (Contd.) Benchmark

Zero OAS

Negative OAS

Positive OAS

Bond sector with a Comparison must be made given credit rating between security OAS and OAS (assumes credit rating of comparable securities (required OAS): lower than security If security OAS > required OAS, being analyzed ) security is cheap If security OAS < required OAS, security is rich If security OAS = required OAS, security is fairly priced

Underpriced (cheap) Underpriced (cheap) security security (assumes credit rating lower than (assumes credit rating security being analyzed ) lower than security being analyzed )

Issuer’s own securities Overpriced (rich) security

Fairly valued

Under priced (cheap) security

Determining Bond Value at a Node Applying Backward Induction Bond's value in higher-rate state 1-year forward

1-year rate at the NHHL node at which we r are calculating the  3,HHL VHHL bond's value, VHHL

VHHHL  C r 4,HHHL NHHHL

Cash flow in higher rate state

NHHLL

r 4,HHLL VHHLL

 C

Cash flow in lower rate state

Bond's value in lower-rate state 1-year forward

The present values of the these two cash flows discounted at the 1-year rate (r 3,HHL) at Node NHHL are: 1.

2.

( (

VHHHL + C 1 + r 3,HHL VHHLL + C 1 + r 3,HHL

) )

 Present value in the higher one-year rate scenario

 Present value in the lower one-year rate scenario

© 2013 ELAN GUIDES


Finally, the expected value of the bond, VHHL at Node NHHLis calculated as: 1 2

(

VHHHL + C 1 + r 3,HHL

)( +

VHHLL + C 1 + r 3,HHL

)

Determining Call Option Value

Value of call option = Value of option-free bond – Value of callable bond. Determining Put Option Value

Value of put option = Value of putable bond  Value of option-free bond Effective Duration and Effective Convexity

Duration =

V  V+ 2V0 ( y)

Convexity =

V  V+  2V0 2V0 ( y)2

Traditional Analysis of a Convertible Security

Conversion value = Market price of common stock  Conversion ratio

Market conversion price =

Market price of convertible security Conversion ratio

Market conversion premium per share = Market conversion price  Current market price

Market conversion premium ratio =

Premium payback period =

Market price of common stock

Market conversion premium per share Favorable income differential per share

Favorable income differential per share =

Premium over straight value =

© 2013 ELAN GUIDES

Market conversion premium per share

Coupon interest  (Conversion ratio  Common stock dividend per share) Conversion ratio

Market price of convertible bond Straight value




An Option-Based Valuation Approach

Covertible security value = Straight value  Value of the call option on the stock

Covertible callable bond value = Straight value  Value of the call option on the stock  Value of the call option on the bond

Covertible callable and putable bond value = Straight value  Value of the call option on the stock  Value of the call option on the bond  Value of the put option on the bond

© 2013 ELAN GUIDES

MORTGAGE-BACKED SECTOR OF THE BOND MARKET

MORTGAGE-BACKED SECTOR OF THE BOND MARKET Single Monthly Mortality Rate (SMM)

Prepayment in month t

SMMt =

Beginning mortgage balance for month t  Scheduled principal payment in month t

Prepayment in month t = SMM × (Beginning mortgage balance for month t  Scheduled principal payment in month t )

Conditional Prepayment Rate (CPR)

CPR = 1  (1  SMM)12 Given the CPR, the SMM can be computed as: SMM = 1  (1  CPR )1/12

Average Life

Average life =

T

t  Projected principal recieved at time t

t = 1

12  Total principal



t = Number of months Distribution of Prepayment Risk in a Sequential-Pay CMO Tranche

Contraction Risk Extension Risk

A (sequential pay) B (sequential pay) C (sequential pay) Z (accrual pay)

HIGH

LOW





LOW

HIGH

Prepayment Risk in Different PAC Tranches Tranche

Prepayment Risk

PAC I - Senior PAC I - Junior PAC II Support

LOW

© 2013 ELAN GUIDES



HIGH

ASSET-BACKED SECTOR OF THE BOND MARKET

ASSET-BACKED SECTOR OF THE BOND MARKET Parties to the Securitization Party

Description

Party in Illustration

Seller

Originates the loans and sells loans to the SPV

ABC Company

Issuer/Trust

The SPV that buys the loans from the seller and issues the asset backed securities

SPV

Servicer

Services the loans

Servicer

Manufactured Housing-Backed Securities

SMM =

ABS =

ABS 1 – [ABS × (M – 1)] SMM 1 + [SMM × (M – 1)]

VALUING MORTGAGE-BACKED AND ASSET-BACKED SECURITIES Cash Flow Yield

ABS and MBS typically have monthly cash flows, so the cash flow yield on these securities is compared to the yield on Treasury coupon securities based on their bond equivalent yields. The bond equivalent yield for MBS/ABS is calculated as: 6

Bond equivalent yield = 2 [(1 + monthly cash flow yield) – 1] Option Cost

Option cost = Zero-volatility spread – Option-adjusted spread Duration Duration =

V  V+ 2V0 ( y)

© 2013 ELAN GUIDES

DERIVATES

DERIVATIVES

FORWARD MARKETS AND CONTRACTS Value of a Forward Contract Time

Long Position Value

Short Position Value

At initiation

Zero, as the contract is priced to prevent arbitrage

Zero, as the contract is priced to prevent arbitrage

During life of the contract

St



F(0,T)

F(0,T)

(1 + r)T-t

(1 + r)T-t

ST  F(0,T)

At expiration

 St

F(0,T)  ST

Price of an Equity Forward with Discrete Dividends n

PV(D,0,T) =

Di

 (1 + r)

ti

i=1

... Approach I

F(0,T) = [S0 – PV(D,0,T)] (1 + r)T n

FV(D,0,T) =

 D (1 + r)

Tti

i

... Approach II

i=1

T

F(0,T) = S0 (1 + r) – FV(D,0,T) Price of an Equity Forward with Continuous Dividends c

c

F(0,T) = (S0e T)er T F(0,T) = S0  e(r  )T c

c

r c = Continuously compounded risk-free rate

c = Continuously compounded dividend yield Value of an Equity Forward T–t

Vt(0,T) = [St – PV(D,t,T)] – [F(0,T) / (1 + r)

]

PV(D,t,T) = PV of dividends expected to be received over the remainder of the contract term (between t and T). Assuming continuous compounding, the value of a forward contract on a stock index or portfolio can be calculated as: Vt(0,T) = Ste – c(T – t) – F(0,T)e –rc(T – t) St

Vt(0,T) =

ec(T – t)

© 2013 ELAN GUIDES

–

F(0,T) erc(T – t)

DERIVATES

Calculating the No-Arbitrage Forward Price for a Forward Contract on a Coupon Bond

F(0,T) = [B0C(T+Y) – PV(CI,0,T)] × (1 + r)T Or F(0,T) = [B0C(T+Y)] (1 + r)T – FV(CI,0,T) BC = Price of coupon bond T = Time of forward contract expiration Y = Remaining maturity of bond upon forward contract expiration T+Y = Time to maturity of the bond at forward contract initiation. PV(CI,0,T) = Present value of coupon interest expected to be received between time 0 (contract initiation) and time T (contract expiration). FV(CI,0,T) = Future value of coupon interest expected to be received between time 0 (contract initiation) and time T (contract expiration).

Valuing a Forward Contract on a Coupon Bond

The value of the long position in a forward contract on a fixed income security prior to expiration can be calculated as: Vt(0,T) = BtC(T+Y) – PV(CI,t,T) – F(0,T) / (1 + r)T – t PV(CI,t,T) = Present value of coupon payments that are expected to be received between time t and time T. C Bt (T+Y) = Current value of coupon bond with time T+Y remaining until maturity Pricing a Forward Rate Agreement

1 + L0(h + m)

( ) ( ) ( ) h+m 360

 1

FRA(0,h,m) = 1 + L0( h )

h 360

360 m

FRA(0,h,m) = The annualized rate on an FRA initiated at Day 0, expiring on Day h, and based on m-day LIBOR. h = Number of days until FRA expiration m = Number of days in underlying hypothetical loan h+m = Number of days from FRA initiation until end of term of underlying hypothetical loan. L0 = (Unannualized) LIBOR rate today

© 2013 ELAN GUIDES

DERIVATES

FRA Payoff

FRA payoff =

NP × [(Market LIBOR – FRA rate) × No. of days in the loan term / 360] 1 + [Market LIBOR × (No. of days in the loan term / 360)]

Valuing FRA prior to expiration

NP × [(Current forward rate – FRA rate) × No. of days in the loan term / 360] 1 + {Current LIBOR × [(No. of days in loan term + No. of days till contract expiration) / 360]} Or: 1 + FRA(0,h,m)

1

Vg (0,h,m) =

1 + Lg (h  g)

(

h  g 360



)

1 + Lg (h + m  g)

(

( ) m 360

h + m  g 360

)

g = Number of days since FRA initiation. Pricing a Currency Forward Contracts

(1 + R DC)T

F(0,T) = S0 ×

(1 + R FC)T

F and S are quoted in terms of DC per unit of FC R DC = Domestic risk-free rate R FC = Foreign risk-free rate T = Length of the contract in years. Remember to use a 365-day basis to calculate T if the term is given in days. Valuing a Currency Forward Contract

The value of the long position in a currency forward contract at any time prior to maturity can be calculated as follows: Vt (0,T) =

St



(1 + R FC)(Tt)

F (0,T) (1 + R DC)(Tt)

Assuming continuous compounding, the price and value of a currency forward contract can be calculated by applying the formulas below: cFC × T

F(0,T) = (S0e – r

cFC × (T – t)

Vt(0,T) = [St / er

© 2013 ELAN GUIDES

cDC × T

) × er

cDC – r cFC) × T

or F(0,T) = S0 × e(r cDC × (T – t)

] – [F(0,T) / e r

]

c r here represents a continuously compounded riskfree rate in these formulas.

FUTURES MARKETS AND CONTRACTS

FUTURES MARKETS AND CONTRACTS If we ignore the effects of the mark-to-market adjustment on futures contracts, we can make the simplifying assumption that the futures price and forward price are the same. f 0(T) = F(0,T) = S0 × (1 + r)T f 0(T) = Futures price today of a futures contract that expires at time T. F(0,T) = Forward price of a forward contract that expires at time T. S0 = Spot price of underlying asset today r = Annual risk-free rate The Effect of Storage or Carrying Costs on the Futures Price

f 0(T) = S0 (1 + r)T + FV(SC,0,T) The Effect of Monetary Benefits on the Futures Price

f 0(T) = S0 (1 + r)T  FV(CF,0,T) The Effect of Non-Monetary Benefits on the Futures Price

FV(CB,0,T) = Costs of storage – Nonmonetary benefits (Convenience yield) If costs exceed benefits, FV(CB,0,T) is a positive number and is known ascost of carry. In this case, the general futures pricing formula is given as: f 0(T) = S0 (1 + r)T + FV(CB,0,T) Pricing Treasury Bond Futures

f 0(T) = B0C(T+Y) [(1 + r 0(T)]T – FV(CI,0,T) BC = Price of coupon bond T = Time of futures contract expiration Y = Remaining maturity of bond upon futures contract expiration T+Y = Time to maturity of the bond at futures contract initiation. r 0(T) = Interest rate at time 0 for period until time T. FV(CI,0,T) = Future value of coupon interest expected to be received between time 0 (contract initiation) and time T (contract expiration). The adjusted futures price of a t-bond futures contract is calculated as: f 0(T) =

B0C (T + Y) [1 + r 0 (T)]T  FV (CI,0,T) CF(T)

CF(T) = Conversion factor on CTD bond

© 2013 ELAN GUIDES

FUTURES MARKETS AND CONTRACTS

Pricing Stock Index Futures

f 0(T) = S0 (1 + r)T – FV(D,0,T) f 0(T) = S0  e(r –  )T c

c

Pricing Currency Futures

f 0(T) = S0 

(1 + r DC)T (1 + r FC)T

F and S are quoted in terms of DC/FC r DC = Domestic currency interest rate r FC = Foreign currency interest rate T = Length of the contract in years. Remember to use a 365-day year if maturity is given in days. If interest rates are assumed to be continuously compounded, then the no-arbitrage futures price is calculated as: cDC – r cFC)×T

f 0(T) = S0 × e(r

r c represents the continuously compounded risk-free rate.

© 2013 ELAN GUIDES

OPTION MARKETS AND CONTRACTS

OPTION MARKETS AND CONTRACTS Put-Call Parity

C0 +

X = P0 + S0 (1 + R F)T

Synthetic Securities Strategy

Consisting of

fiduciary call

long call + long bond

long call

long call

Value

C0 +

X T (1 + R F)

C0

Equals

Strategy

Consisting of

Value

=

Protective put

long put + long underlying asset

P0 + S0

=

long put + long Synthetic call underlying asset + short bond

P0 + S0 

X (1 + R F)T

=

long call + short Synthetic put underlying asset + long bond

C0  S0 +

X (1 + R F)T

Synthetic underlying asset

long call + long bond + short put

C0 +

Synthetic bond

long put + long underlying asset + short call

long put

long put

long underlying asset

long underlying asset

S0

=

long bond

long bond

X (1 + R F)T

=

P0

X (1 + R F)T

 P0

P0 + S0  C0

© 2013 ELAN GUIDES


One-Period Binomial Model

Computing the two possible values of the stock: S+ = Su S- = Sd Binomia Call Option Pricing

Call payoff = Max(0, S+ – X) Binomial Put Option Pricing

Put payoff = Max (0, X – ST) Compute the risk-neutral probabilities:

 =

(1 + r  d (u  d)

Calculating the value of the call option:

 c+ + (1 – ) c-

c=

1 + r

Calculating the value of the put option:

p =

 p+ + (1 – ) p1 + r

Hedge Ratio

n=

c+  cS+  S-

Intrinsic value of caplet at expiration:

Caplet value =

Max {0, [(One-year rate – Cap rate)  Notional principal]} 1 + One-year rate

Intrinsic value of floorlet at expiration:

Floorlet value =

© 2013 ELAN GUIDES

max {0, [(Floor rate – One-year rate)  Notional principal]} 1 + One-year rate


The Black-Scholes-Merton Formula c

c = S0 N(d1)  Xer T N(d2) c

p = Xe-r T[1  N(d2)]  S0[1  N(d1)] Where: d1 =

ln(S0 X) + [r c + ( 

 

d2 = d1  



= the annualized standard deviation of the continuously compounded return on the stock r c = the continuously compounded risk-free rate of return N(d1) = Cumulative normal probability of d1. Delta

Delta =

Change in option price Change in underlying price

Change in option price = Delta  Change in underlying price An approximate measure for option delta can be obtained from the BSM model:  N(d1) from the BSM model approximately equals call option delta.  N(d1) – 1 approximately equals put option delta. Therefore:  c  N(d1)   S  p  N(d1) – 1]   S Put-Call Parity for Forward Contracts Value at Expiration ST  X

Transaction

Current Value

Call and Bond Buy call Buy bond Total

c0 [X – F(0,T)]/(1 + r)T c0 + [X – F(0,T)]/(1 + r)T

0 X – F(0,T) X – F(0,T)

ST – X X – F(0,T) ST – F(0,T)

p

X – ST ST – F(0,T) X – F(0,T)

0 ST – F(0,T) ST – F(0,T)

Put and Forward Buy put Buy forward contract Total

0

0 p0

ST > X

© 2013 ELAN GUIDES


Put-call-forward parity c0 +

X – F(0,T) (1 + r)T

= p0

Forward Contract and Synthetic Forward Contract Value at Expiration Transaction

ST  X

Current Value

Forward Contract Long forward contract Synthetic Forward Contract Buy call Sell put Buy (or sell) bond Total

ST > X

0

ST – F(0,T)

ST – F(0,T)

c0 – p0

0 – ( X – ST) X – F(0,T) ST – F(0,T)

ST – X 0 X – F(0,T) ST – F(0,T)

T

[X – F(0,T)]/(1 + r) T c0 – p0 + [X – F(0,T)]/(1 + r)

The Black Model

The Black model is used to price European options on futures. c

c = er T [f 0(T)N(d1)  XN(d2)] c

p = er T (X[1  N(d2)]  f 0(T)[1  N(d1)]) Where: d1 =

ln(f 0(T) X) + ( 

d2 = d1  

  

f 0(T) = the futures price c

Notice that the Black model is similar to the BSM model except that er T f(T) is substituted for S0. In fact, the price of a European option on a forward or futures would be the same as the price of a European option on the underlying asset if the options and the forward/futures contract expire at the same point in time.

© 2013 ELAN GUIDES

SWAP MARKETS AND CONTRACTS

SWAP MARKETS AND CONTRACTS The Swap Fixed Rate

Swap fixed rate =

(

1  B0(N) B0(1) + B0(2) + B0(3) + ... + B0(N)

)

 100

Valuing a Swap

Value of pay-fixed side of plain-vanilla interest rate swap: Present value of floating-rate payments  Present value of fixed-rate payments Value of pay-floating side of plain-vanilla interest rate swap: Present value of fixed-rate payments  Present value of floating-rate payments

Valuing Equity Swaps

‘Pay a fixed rate and receive the return on equity’ swap [(1 + Return on equity)  Notional principal]  PV of the remaining fixed-rate payments ‘Pay a floating rate and receive the return on equity’ swap [(1 + Return on equity)  Notional principal]  PV (Next coupon payment + Par value) The value of a ‘pay the return on one equity instrument and receive the return on another equity instrument’ swap is calculated as the difference between the values of the two (hypothetical) equity portfolios: [(1 + Return on Index 2)  NP] – [(1 + Return on Index 1)  NP]

Payer swaption

(Market fixed-rate – Exercise rate) 

No. of days in the payment period 360

 Notional principal

Receiver swaption

(Exercise rate – Market fixed-rate) 

No. of days in the payment period 360

 Notional principal

© 2013 ELAN GUIDES

INTEREST RATE DERIVATIVE INSTRUMENTS

INTEREST R ATE DERIVATIVE INSTRUMENTS Table: Caps, Floors, Interest Rate Options, Bond Options and Interest Rates Security Benefits when…

Long cap (floor) Long call (put) option on interest rates Long call (put) option on a fixed income instrument

Interest rates rise (fall) Interest rates rise (fall) Interest rates fall (rise)

Payoff to the buyer of an interest rate cap

Payoff = Max [0,(Market interest-rate – Cap rate) 

No. of days 360

 Notional principal]

Payoff to the buyer of an interest rate floor

Payoff = Max [0,(Floor rate – Market interest-rate) 

© 2013 ELAN GUIDES

No. of days 360

 Notional principal]

PORTFOLIO CONCEPTS

PORTFOLIO CONCEPTS Expected return on Two-Asset Portfolio

E(R P) = w1E(R 1) + w2E(R 2) E(R 1) = expected return on Asset 1 E(R 2) = expected return on Asset 2 w1 = weight of Asset 1 in the portfolio w2 = weight of Asset 2 in the portfolio Variance of 2-asset portfolio:

2P = w1221 + w2222 + 2w1w2 12 1= the standard deviation of return on Asset 1 2= the standard deviation of return on Asset 2  = the correlation between the two assets’ returns Variance of 2-asset portfolio:

P2 = w1221 + w2222 + 2w1w2Cov1,2 Cov1,2 =  12 Expected Return and Standard Deviation for a Three-Asset Portfolio

Expected return on 3-asset portfolio: E(R P) = w1E(R 1) + w2E(R 2) + w3E(R 3) Variance of 3-asset portfolio:

P2 = w1212 + w2222 + w3223 + 2w1w2 12 + 2w1w3 13 + 2w2w3 23 Variance of 3-asset portfolio:

P2 = w1212 + w2222 + w3223 + 2w1w2Cov  + 2w1w3Cov  + 2w2w3Cov 

© 2013 ELAN GUIDES

PORTFOLIO CONCEPTS

Expected Return and Variance of the Portfolio

For a portfolio of n assets, the expected return on the portfolio is calculated as: n

E(R P) =

 w E(R ) j

j

j=1

The variance of the portfolio is calculated as: n



2 = P

n

  w w Cov(R ,R ) i j

i

j

i=1 j=1

Variance of an Equally-weighted Portfolio

P2 =

1 n

2P = 2

2 +

(

1 n

n  1 Cov n

+

)

Expected Return for a Portfolio Containing a Risky Asset and the Risk-Free Asset

E(R P) = RFR + P

[E(R i)  RFR]

i

Standard Deviation of a Portfolio Containing a Risky Asset and the Risk-Free Asset

P = wii

© 2013 ELAN GUIDES

PORTFOLIO CONCEPTS

CML

Expected return on portfolios that lie on CML: E(R P) = w1R f + (1  w1)E(R m) Variance of portfolios that lie on CML:

2 = w12f 2 + (1  w1)2m2 + 2w1(1  w1)Cov(R f , R m) Equation of CML: E(R P) = R f +

E(R m)  R f

m

 P

Calculation and Interpretation of Beta

i 

Cov(R i,R m)





2 m

i,mi,m    i,m i 2 m m

The Capital Asset Pricing Model

E(R i)  R f + i[E(R m) – R f ]

The Decision to Add an Investment to an Existing Portfolio

E(R new)  R F

new





E(R p)  R F

 p



Corr(R new,R p)

Market Model Estimates

R i = i + i R M + i R i = Return on asset i R M = Return on the market portfolio i = Average return on asset i unrelated to the market return i = Sensitivity of the return on asset i to the return on the market portfolio i = An error term



i is the slope in the market model. It represents the increase in the return on asset i if



the market return increases by one percentage point. i is the intercept term. It represents the predicted return on asset i if the return on the market equals 0.

Expected return on asset i E(R i) = i + iE(R M)

© 2013 ELAN GUIDES

PORTFOLIO CONCEPTS

Variance of the return on asset i 2 Var(R i) = 2i M + 2i

Covariance of the returns on asset i and asset j Cov(R i,R j) = i j2

M

Correlation of returns between assets i and j

Corr(R i,R j) =

i jM2 2 2 (2i M + 2 )1/2 ( j2M + 2 )1/2 i

j

Market Model Estimates: Adjusted Beta Adjusted beta = 0.333 + 0.667 (Historical beta)

Macroeconomic Factor Models

R i = ai + bi1FINT + bi2FGDP + i R i = the return to stock i ai = the expected return to stock i FINT = the surprise in interest rates FGDP = the surprise in GDP growth bi1 = the sensitivity of the return on stock i to surprises in interest rates. bi2 = the sensitivity of the return on stock i to surprises in GDP growth. i = an error term with a zero mean that represents the portion of the return to stock i that is not explained by the factor model. Fundamental Factor Models

R i = ai + bi1FDY + bi2FPE + i R i = the return to stock i ai = intercept FDY = return associated with the dividend yield factor FPE = return associated with the P-E factor bi1 = the sensitivity of the return on stock i to the dividend yield factor. bi2 = the sensitivity of the return on stock i to the P-E factor. i = an error term Standardized sensitivities are computed as follows: bij =

Assets i’s attribute value  Average attribute value Attribute values)

© 2013 ELAN GUIDES

PORTFOLIO CONCEPTS

Arbitrage Pricing Theory and the Factor Model

E(R P ) = R F + 1 p,      p,

E(R p) = Expected return on the portfolio p R F = Risk-free rate  j = Risk premium for factor j  p,j = Sensitivity of the portfolio to factor j K = Number of factors Active Risk

TE = s(R p  R B) Active risk squared = s2(R p  R B) Active risk squared = Active factor risk + Active specific risk n

Active specific risk =

w  i=1

a  i  i

Where: wai = The ith asset’s active weight in the portfolio (i.e., the difference between the asset’s weight in the portfolio and its weight in the benchmark). i = The residual risk of the ith asset (i.e., the variance of the ith asset’s returns that is not explained by the factors). Active factor risk = Active risk squared – Active specific risk.

Active Return

Active return = R p – R B Active return = Return from fctor tilts + Return from asset selection K

Active return =

[(Portfolio sensitivity)  (Benchmark sensitivity) ]  (Factor return) + Asset selection j

j

j

j=1

© 2013 ELAN GUIDES

PORTFOLIO CONCEPTS

Factor’s Marginal Contribution to Active Risk Squared (FMCAR) K

b ja

FMCAR j =

FMCAR j =

 b Cov(F ,F ) a i

j

i

i=1

Active risk squared Active factor risk Active risk squared

where: b ja = The portfolio’s active exposure to factor j K

b ja

 b Cov(F ,F ) = The active factor risk for factor j a i

j

i

i=1

The Information Ratio

IR =

R p  R B s(R p  R B)

© 2013 ELAN GUIDES

THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT

THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT Weight of security k in the active portfolio (Portfolio A) wk =

k / ek  n

i / ei  i=1

The expression for the optimal weight, w*, of the active portfolio (Portfolio A) in the optimal risky portfolio (Portfolio P) is given as:

 A

w* =

 A(1 A) + R M

2(e A) 2 M

Assuming (for simplicity) that the beta of Portfolio A equals 1, the optimal weight, w0, of Portfolio A in Portfolio P is calculated as:

 A

 A /2(e A) = w0 = 2 R M /2 M  (e A) R M

2 M If the beta of Portfolio A does not equal 1, we can use the following equation to determine the optimal weight, w*, of Portfolio A in Portfolio P. w* =

w0

  (1 A)w0

Evaluation of Performance

Sharpe Ratio The Sharpe ratio of the optimal risky portfolio (Portfolio P) can be separated into contributions from the market and active portfolio as follows: 2

 

2 A R M  A 2 2 S P = S M + 2 = +  (e A)  M (e A)



2

Information Ratio



 A (e A)

2

  n

=

i=1

i (ei)



2

© 2013 ELAN GUIDES

THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT

Imperfect Forecasts of Alpha Values

Actual (realized) alpha:   R  R M To measure the forecasting accuracy of the analyst, we can regress alpha forecasts ( f ) on realized alpha ().

 f  a  a   For simplicity, we assume that a and a equal 0 and 1 respectively. Given that forecast errors () are uncorrelated with true alpha () i.e., Cov, equals 0, the variance of the forecast is given as:

f   +  We can evaluate the quality of the analyst’s forecasts by calculating the coefficient of determination of the regression described above.

  R    f  +  

This estimate of R 2 is used as a shrinking factor to adjust the analyst’s forecasts of alpha.

© 2013 ELAN GUIDES

Elan Guides Formula Sheet CFA 2013 Level 2

Recommend Documents