Equivalent Annual Annuity
The equivalent annual annuity formula is used in c apital budgeting to show the net present value of an investment as a series of equal cash flows for the le ngth of the investment. The net present value(NPV) formula shows the present value of an investment that has uneven cash flows. When comparing two different investments using the net present value method, the length of the investment ( n) is not taken into consideration. An investment with a 15 year term may show a higher NPV than an investment with a 4 year term. By showing the NPV as a series of cash flows, the equivalent annual annuity formula provides a way to factor in the length of an investment.
How is the Equivalent Annual Annuity Formula Useful? An example of how the equivalent annual annuity formula may be useful is comparing two new projects where one project has a 15 year term and the other has a 4 year term. Assume that both projects have the same NPV. The 4 year project will receive the return sooner so it will show a higher cash flow when using the equivalent annual annuity formula. In real life, comparing two investments will not always be so obvious a nd the formula should be applied. Another way of explaining the usefulness of the equivalent a nnual annuity formula is that an investment with a shorter life span can be reinvested and the earnings on the reinvestment is not taken into consideration when using the N PV formula. The equivalent annual annuity formula provides a comparison relative to time which eliminates the need for considering reinvestment with the same earnings as the current investment.
How is the Equivalent Annual Annuity Formula Derived? The equivalent annual annuity formula uses the annuity payment formula for when present value is given. Net present value replaces present value to give relevance to the use of the equivalent annual annuity formula.
Example of the Equivalent Annual Annuity Formula Using the prior example of comparing one project with a 4 year term and another project with a 15 year term, the NPV of the 4 year project is $100,000 and the NPV of the 15 year
project is $150,000. The rate used for both is 8%. Putting the variables of the 4 year project in the equivalent annual annuity formula shows
which returns an equivalent annual annuity of $30,192.08. Putting the variables of the 15 year project into the equivalent annual annuity formula shows
which returns an equivalent annual annuity of $17,524.43. Comparing these two projects, the 4 year project will return a higher amount relative to the time of the investment. Although the 15 year project has a higher NPV, the 4 year project can be reinvested and have additional earnings for the 11 years that remain on the 15 year project.
Equivalent annual cost From Wikipedia, the free encyclopedia
In finance the equivalent annual cost (EAC) is the cost per year of owning and operating an asset over its entire lifespan. EAC is often used as a decision making tool in capital budgeting when comparing investment projects of unequal lifespans. For example if project A has an expected lifetime of 7 years, and project B has an expected lifetime of 11 years it would be improper to simply compare the net present values (NPVs) of the two projects, unless neither project could be repeated. EAC is calculated by dividing the NPV of a project by the present value of an annuity factor. Equivalently, the NPV of the project may be multiplied by the loan repayment factor .
EAC= The use of the EAC method implies that the project will be replaced by an identical project.
A practical example A manager must decide on which machine to purchase: Machine A Investment cost $50,000 Expected lifetime 3 years Annual maintenance $13,000 Machine B Investment cost $150,000 Expected lifetime 8 years Annual maintenance $7,500 The cost of capital is 5%. The EAC for machine A is: ($50,000/ A3,5)+$13,000=$31,360 The EAC for machine B is: ($150,000/ A8,5)+$7,500=$30,708 The conclusion is to invest in machine B s ince it has a lower EAC. Note: The loan repayment factors (A values) are for t years (3 or 8 years) and 5% cost of capital. A3,5 is given
by
= 2.723 and A8,5 is given by
= 6.463. (See ordinary annuity formulae
for a derivation.) The larger an A value is, the greater the present value is on a succession of future annuity payments, thus contributing to a smaller annual cost.
Alternative method: The manager calculates the NPV of the machines: Machine A EAC=$85,400/A3,5=$31,360 Machine B EAC=$198,474/A8,5=$30,708 Note: To get the numerators add the present value of the annual maintenance to the purchase price. For example, for Machine A: 50,000 + 13,000/1.05 + 13,000/(1.05)^2 + 13,000/(1.05)^3 = 85,402. The result is the same, although the first method is easier it is essential that the annual maintenance cost is the same each year. Alternatively the manager can use the NPV method under the assumption that the machines will be replaced with the same cost of investment each time. This is known as the chain method since 8 repetitions of machine A are chained together and 3 repetitions of machine B are chained together. Since the time horizon used in the NPV comparison must be set to 24 years (3*8=24) in order to compare projects of equal length, this method can be slightly more complicated than calculating the EAC. In addition, the assumption of the same cost of investment for each link in the chain is essentially an assumption of zero inflation, so a real interest rate rather than a nominal interest rate is commonly used in the calculations.
11.16 - Comparing Projects With Unequal Lives As mentioned previously, NPV and IRR can sometimes lead to conflicting results in the analysis of mutually exclusive projects. One reason for this potential problem is the timing of the cash flows of the mutually exclusive projects. As a result, we need to adjust for the timing issue in order to correct this problem. There are two methods used to make the adjustments: 1.Replacement-chain method 2.Equivalent annual annuity
Once again, assume Newco is planning to add new machinery to its current plant. There are two machines Newco is considering, with cash flows as follows: Figure 11.8: Discounted cash flows for Machine A and Machine B
Compare the two projects with unequal lives using both the replacement-chain method and the equivalent annual annuity (EAA) approach. 1. Replacement-Chain Method In this example, Machine A has an operating lifespan of six years. Machine B has an operating lifespan of three years. The cash flows for each project are discounted by Newco's calculated WACC of 8.4%. NPV of Machine A is equal to $2,926. NPV of Machine B is equal to $1,735. • •
The initial analysis indicates that Machine A, with the greater NPV, should be the project chosen. The IRR of Machine A is equal to 8.3%. The IRR of Machine B is equal to 15.5%. • •
This analysis indicates that Machine B, with the greater IRR, should be the project chosen. The NPV analysis and the IRR analysis have given us differing results. This is most likely due to the unequal lives of the two projects. As such, we need to analyze the two projects over a common life. For Machine A (project 1), the lifespan is six years. For Machine B (project 2), the lifespan is three years. Given that the lifespan of the longest project is six years, in order to measure both over a common life, we must adjust the lifespan of Machine B to six years. Because the lifespan of Machine B is three years, the lifespan of this project needs to b e doubled to equal the six-year lifespan of Machine A. This indicates that another Machine B would have to be purchased (to get two machines with a lifespan of three years each) to get to the six-year lifespan of Machine A - hence, the replacement-chain method. The new cash flows would be as follows:
Figure 11.9: Cash flows over a common life
• •
NPV of Machine A remains $2,926. NPV of Machine B is now $3,098 given the adjustment.
The initial analysis indicates that Machine B, with the greater NPV, should be the project chosen. Recall, this is different from our first analysis where Machine A was chosen given its greater NPV. • •
The IRR of Machine A remains 8.3%. The IRR of Machine B remains 15.5%.
Look Out! Note, while the NPV has changed given the additional cash flows, the IRR for the projects remain the same.
This analysis indicates that Machine B, with the greater IRR, should be the project chosen. Recall, this is the same as our first analysis, where Machine B was chosen given its greater IRR. With the cash flows adjusted with the replacement-chain method, both the NPV and the IRR arrive at the same conclusion. With this adjusted analysis, Machine B (project 2), should be the project accepted. 2. Equivalent-Annual-Annuity Approach While easy to understand, the replacement-chain method can be time consuming. A simpler approach is the equivalent-annual-annuity approach.
This is the procedure for determining EAA: 1) Determine the projects' NPVs. 2) Find each project's EAA, the expected payment over the project's life, where the future value of the project would equal zero. 3) Compare the EAA of each project and select the project with the highestEAA. From our example, the NPV of each project is as follows: -NPV of Machine A is equal to $2,926. -NPV of Machine B is equal to $1,735.
To determine each project's EAA, it is best to use your financial calculator. - For, Machine A (project 1), our assumptions are as follows: i = 8.4% (the company's WACC) n=6 PV = NPV = -2,926 FV = 0 Find for PMT For Machine A, the EAA (the calculated PMT) is $640.64. -For Machine B (project 2), our assumptions are as follows: i = 8.4% (the company's WACC) n=3 PV = NPV = -1,735 FV = 0 Find for PMT For Machine B, the EAA (the calculated PMT) is $678.10. Answer Machine B should be the project chosen as it has the highest EAA, which is $678.10, relative to Machine A whose EAA is $640.64. Inflation Effects on Capital Budgeting Analysis Inflation exists and should not be forgotten when making capital-budgeting decisions. It is important to build inflation expectations into the analysis. If inflation expectations are left out of the capital-budgeting analysis, the NPV calculated from the biased cash flows will be incorrect.
As an example, suppose Newco unintentionally leaves out its inflation expectations when determining the plant addition. Since inflation expectations are included in the WACC, and PV of each cash flow is discounted by the WACC, the NPV will be incorrect and have a downward bias.
Equivalent Annual Annuity Approach (EAA) Equivalent Annual Annuity approach (EAA) is another sophisticated technique used in capital budgeting decisions. EAA determines the annual cost of the project over its economic life. It is determined by dividing net present value (NP V) of the project by the present value interest factor for annuity (PVIFAr,n) for a specific period and at a specific discount rate. PVIFAr,n can be found in financial tables. EAA is helpful when a project has to be selected from mutually exclusive projects with unequal lives. The project with the highest Equivalent Annual Annuity (EAA) is more attractive. If two mutually exclusive projects have equal EAA than the project with the shorter economic life is more acceptable.
Profitability index (PI) Profitability Index (PI) is another sophisticated technique used in capital budgeting decisions. PI is related to NPV as it indicates the net present value (NPV) per each dollar invested. PI is calculated as follows: PI = Total present value (PV) of cash inflows divided by Initial investment. PI is an especially advantageous technique if the company operates under capital rationing. If the PI is greater than one, the project is acceptable.
What is Replacement Chain Analysis In Capital Budgeting, a financial analyst analyzes alternative and often mutually exclusive projects to make a selection. As it may happen, the alternative projects may differ in life span as one may be shorter in life as compared to the other. An example of this scenario would be, for example, when one has to select between two machines or pieces of equipments having different useful life. Since most machines will be obsolete after a given period of time thus it is likely that one of the machines is more duarble than the other. In such situtaions, one of the managerial options will be to replace the project after it expires. Here the task of evaluating a capital budgeting proposal with unequal or different lives is done with replacement chain analysis , method or approach in finance.
Projects with Unequal lives Take for example, a financial analyst who has to select from a choice of purchasing a Photocopier machine, one choice is to purchase XEROX made in America photocopier that can last for 9 years before becoming obsolete the other option is to purchase a cheap Chinese machine with a life time of 3 years. Here since a single replication is of 3 years time thus the option of replacement may be analyzed with finding the Net Present Value of the replacement chain analysis for both
options. We will look at the way to calculate NPV for replacement chain in a bit. Let us first show you the formula to find NPV with Replace Chain method or approach.
Replacement Chain NPV Formula
Here R is the number of replications, k is the weighted average cost of capital, NPVn is the net present value for the single Replication, n is the size of the replication and t is the time period
Replacement Chain NPV Example Let us show you by example how we can find the NPV for replacement chain method. The net present value with replacement chain is nothing more than the present value of the NPV of the replacement project. Let us say take the example of the decision of whether to choose an American made XEROX photocopier with 9 year life span or to accept the cheaper Chinese made photocopier machine. The cash flows for both copiers and the net savings from the machines are as follows. We will assume both machine will have a zero salvage value at the end of life spans. The question then is to find Replacement Chain NPV for both projects. Let us assume the cost of capital or the discount rate is 10% for both projects Chinese Photocopier
-1500 $1,000 $1,000 $1,000 XEROX Photocopier
-4000 $1,000 $1,000 $1,000 $1,000 $1,000 $1,000 $1,000 $1,000 $1,000
Replacement Chain NPV Calculation We need to find the Net Present Value of each of the projects individually, once we have ascertained the NPV for individual projects, we will try to find the Replacement Chain Net Present Value. As you can see the a single replication is of 3 years and it requires 3 replication of the Chinese project to have it set equal to the American XEROX project.
NPV for Chinese Photocopier at 10%
Year
Net Cash Flows
PVIF @ 10%
Present Value
0
$-1500
1.000
$-1,500
1
$1,000
0.909
$909
2
$1,000
0.826
$826
3
$1,000
0.751
$751
NPV
$986
PVIF @ 10%
Present Value
NPV for XEROX at 10%
Year
Net Cash Flows
0
-4000
1.000
$-4,000
1
$1,000
0.909
$909
2
$1,000
0.826
$826
3
$1,000
0.751
$751
4
$1,000
0.683
$683
5
$1,000
0.621
$621
6
$1,000
0.564
$564
7
$1,000
0.513
$513
8
$1,000
0.467
$467
9
$1,000
0.424
$424
NPV
$1,758
NPV comparison of XEROX and Chinese Photocopier Looking at the NPV results for the two projects one can see that NPV for the XEROX copier of $1,758 is higher than that of the Chinese photocopier which is $986. This differnce in NPV is deceptive since the two projects have unequal lives. So we can't just say that a project with 3 years life having a lower NPV is to be rejected when comparing it to a 9 years project with a higher NPV. This is where we need to find the Replacement Chain NPV for the two projects. The single replication for the Chinese project is 3, we need 3 replications of these to make it equal to the XEROX project. The discount rate is still 10% for both projects. Let us now find NPV with Replacement Chain analysis, method or approach in finance. NPVchain = ΣNPVn/(1+k)n(t-1) where n the size of replication is 3 t ranges from 1 to 3 the number of replications k is the discount rate NPVn is the NPV of the project n
NPV Chain for the Chinese photocopier NPVchain = 986/(1+10%)3(1-1) + 986/(1+10%)3(2-1) + 986/(1+10%)3(3-1) NPVchain = 986/(1.10)0 + 986/(1.10)3 + 986/(1.10)6 NPVchain = 986 + 986/1.331 + 986/1.771561 NPVchain = $2,270
Conclusion Thus with Replacement chain analyis we find that the Chinese Photocopier project with 3 year life span has a higher NPV than the 9 years XEROX project. Thus the financial analyst is likely to accept the machine with lower quality yet higher NPV. This just illustrates what went wrong with letting China do business as it does. When the Chinese with command economy are able to produce giffen products that leads to loss of market share of the American manufactured products simply because the end consumers sees benefits in saving money with a cheap product that makes him the most money. Now you replicate the behavior of the financial analyst, that had to choose from XEROX and the Chinese made photocopier, with that of million other business managers across America and you get the picture as to why we the United States have lost its domination in manufacturing sector. The bottom line seems to be the crucial decision making factor to businesses that rather save money than to keep America on top as the World's leading manufacturer.
