##### Study Session 8

### Corporate Finance

##### Reading 22

### Capital Budgeting

Learning Outcome Statements

##### c. evaluate capital projects and determine the optimal capital project in situations of 1) mutually exclusive projects with unequal lives, using either the least common multiple of lives approach or the equivalent annual annuity approach, and 2) capital rationing;

Exam: June 2015 Level 2 > Study Session 8. Corporate Finance >
Reading 22. Capital Budgeting > Subject 2. Comparing projects with unequal lives.

### Subject 2. Comparing projects with unequal lives.

*CFA Program Curriculum (2015), Volume 3, page 39.*

If mutually exclusive projects have unequal lives, it may be necessary to adjust the analysis to put the projects on an equal life basis. This can be done using either one of the following approaches. They should lead to the same decisions if consistent assumptions are used.

**Least Common Multiple of Lives Approach**: It assumes that each project can be repeated as many times as necessary to reach a common life span; the NPVs over this life span are then compared, and the project with the*higher*common life NPV is chosen. This approach is easier (than EAA) to explain to decision makers. However, the drawback is that the arithmetic may be too complex.**Equivalent Annual Annuity (EAA) Approach**: It calculates the annual payments a project would provide if it were an annuity. When comparing projects of unequal lives, the one with the*higher*equivalent annual annuity should be chosen. This approach is easier to apply than the first one.

Note that the unequal life issue does not arise for independent projects. Suppose XYZ Inc. is planning to modernize its production facilities. It can purchase either a Model A or Model B system. The figure below shows both the expected net cash flows, NPVs and IRRs for these two mutually exclusive projects.

**The Least Common Multiple of Lives Approach**XYZ Inc. can buy another Model B system in at the end of year 3 to extend the overall combined life of the combined projects to 6 years. We would find the NPV of project B over a six-year period, assuming there's no change in annual cash flows and the cost of capital.

**The Equivalent Annual Annuity Approach**

__Step 1. Find each project's NPV:__NPVA = $6,491.

NPVB = $5,155.

__Step 2. For each project, find a constant annuity cash flow (the equivalent annual annuity) that has the same present value of the project's NPV:__For example, EAA

_{B}= $2,146. This level of cash flow stream, when discounted back three years at 12%, has a present value equal to project B's original NPV. EAA

_{A}= $1,579.

__Step 3. The project with a higher EAA should be chosen:__In this example, project B should be chosen.

**Capital Rationing**

- In the ideal situation, the budget is adequate to invest in all profitable (positive NPV) projects.
- A common case is that the company has more profitable projects than it can choose. Sometimes the company can choose to invest in the most profitable ones available, but other times it may not be able to due to the size constraint.

Under

**hard capital rationing**, the budget is fixed. In the case of hard rationing, managers use trial and error and sometimes mathematical programming to find the optimal set of projects. In that situation, it is best to use the NPV or PI valuation methods. Under

**soft capital rationing**, managers may be allowed to over-spend their budgets if they argue effectively that the additional funds will be deployed profitably.

#### User Comments

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kalps: This is usually used in replacement analysis therefore the EAA approach should read if the EAA is lower then accept this projectsekib: Text on pg 439 of Brigham & Houstonproject w/ higher EAA will always have the higher NPV when extended out to any common life therefore chose project w/ higher EAA

sarath: Higher EAA is chosen always...first calculate the NPV and then find the corresponding EAA over the time period...RichardWang: How to use BA II to calculate EAA from NPV?adeelj: You can calculate the Annuity Factors using the annuity formula : (1-(1+i)^t)/iwhere i is the interest rate(discount rate) and t is the time period.

Hence in the example above we see the annuity factor = (1-(1.12)^(-3))/.12 = 2.40183

Hence EAA equals 5,155 / 2.40183 = 2,146

GIJCFA: Using BA II calculator: For Project B,N=3, I/Y=12, FV=0, PV=NPV=5155, CPT PMT = 2146.28

jmcarr02: PI = NPV/InvestmentShagunSingh: Well explained.