Copyright 1997 by Campbell R. Harvey and Stephen Gray. All rights reserved. No part of this lecture may be reproduced without the permission of the authors.
Latest Revision: January 5, 1997
3.0 Overview:
This class provides an overview of capital budgeting  determining which investments a firm should undertake. The net present value (NPV) rule, which is widely used in practice, is developed and illustrated with several examples. A number of alternative evaluation techniques including internal rate of return and payback period are also illustrated, highlighting potential problems with their use. The NPV technique is illustrated in the context of choosing between mutually exclusive projects and projects with different lives.
3.1 Objectives:
After completing this class, you should be able to:
3.2 Net Present Value (NPV)
The net present value (NPV) of a project is defined as the present value of all future cash flows produced by an investment, less the initial cost of the investment:
where n is the number of cash flows generated by the investment and r_{p} is the required return on the investment project.
3.3 The NPV Decision rule
In determining whether to accept or reject a particular projected, the NPV decision rule is
3.4 NPV Computations
In order to illustrate the computation of Net Present Values, we consider a series of examples.
Example 3.5
Recall that we have previously considered the following investment proposal:
Assuming that the required rate of return for this project is r_{p} =10%, is this a worthwhile investment? Applying the NPV rule here requires the calculation of the present value of the future cash flows followed by a comparison with the investment cost of $100 million. Since NPV < 0 we reject this proposal. 
Example 3.6
Consider now the example from the previous section on the determination of cash flows:
Assuming that the required rate of return for this project is r_{p}= 10%, is this a worthwhile investment? Once again, to apply the NPV rule we must find the present value of the future cash flows and compare them with the investment cost in period zero. Since NPV>0, we must accept this proposal. 
NPV analysis relies upon an evaluation of cash flows resulting from a project. There are four basic rules for calculating net cash flows:
3.7 Mutually Exclusive Projects
In many cases, a firm will be faced with a choice of between mutually exclusive investment projects. These are cases in which the firm can undertake only one of the potential projects. For example, a firm may be considering whether to construct an office building or a shopping mall on a parcel of land, or deciding whether to refurbish an old apartment building or turn it into a parking garage. In this case, the NPV rule is to undertake the project with the largest NPV, so long as it is positive.
Example 3.8
A manufacturer is considering purchasing one of two machines, A and B. The cash flows of each of the projects are represented below on a time line. The project’s required rate of return is 10 percent. Since these projects are mutually exclusive, which proposal (if any) should the manufacturer choose? Project A
Project B
The NPV computations are: Since these are mutually exclusive projects and both have NPV > 0, we take the project with the highest NPV. Project A is thus the preferred alternative. 
3.9 Justification of the NPV approach
The NPV rule makes intuitive sense, is easy to use, and can be justified as the appropriate evaluation technique since it leads to the maximization of shareholder wealth. To prove that this is the case, consider the simple economy described below. We assume that:
We use the following notation:
E_{0}^{0} 
Market value of old shareholders common stock before investment. 
E_{1}^{0} 
Market value of old shareholders common stock after investment. 
D_{0}^{0} 
Market value of old bondholders bonds before investment. 
D_{1}^{0} 
Market value of old bondholders bonds after investment. 
V_{0} 
Total market value of the firm before investment. 
V_{1} 
Total market value of the firm after investment. 
p_{0} 
Price per share of common stock before investment. 
p_{1} 
Price per share of common stock after investment. 
n_{0} 
Number of shares of common stock before investment. 
n_{1} 
Number of shares of common stock after investment. 
X_{0} 
Firm’s net cash flow before investment. 
X_{1} 
Firm’s net cash flow after investment. 
E_{1}^{n} 
Market value of new common stock issued to finance the investment. 
D_{1}^{n} 
Market value of new bonds issued to finance the investment. 
C 
Coupon payment on perpetual bonds. 
Note that since all new investments are financed with either new common stock (equity) or new bonds (debt),
(a)
Now suppose the firm undertakes the proposed investment projected. The total market value of the firm after the investment is:
(b)
where the value of the firm before the investment is
(c)
Subtracting Equation (c) from Equation (b), therefore, gives us the incremental value of the firm as a result of undertaking the investment project,
(d) V_{1}  V_{0} = (E_{1}^{0}  E_{0}^{0}) + (D_{1}^{0}  D_{0}^{0}) +1
where the identity in Equation (a) was applied to Equation (d) for simplification.
Since we are interested in identifying whether the shareholders existing before the investment (i.e., the “old” shareholders) benefit from the firm undertaking the investment project, we rearrange Equation (d) to yield:
(e) E_{1}^{0}  E_{0}^{0} = (V_{1}  V_{0})  (D_{1}^{0}  D_{0}^{0})  1
The second term on the righthandside of Equation (e) is the change in the value of the old bonds as a result of undertaking the investment. As long as the investment does not affect the probability of default, the market value of the old bonds before the investment,
(f)
is the same as the value of the old bonds after the investment,
(g)
Since D_{0}^{0 }= D_{1}^{0}, Equation (e) simplifies to:
(h) E_{1}^{0}  E_{0}^{0} = (V_{1}  V_{0})  1
Focusing on the term V_{1}V_{0}, first recognize that the firm’s value may be expressed as the present value of the firm’s cash flows both before and after the investment
(i)
and
(j)
Subtracting Equation (i) from Equation (j) yields:
Substituting Equation (k) into Equation (h), we find that the NPV of the project may be interpreted as the increment to shareholder wealth,
3.10 Alternative Evaluation Techniques
This section outlines several alternatives to the NPV rule. These evaluation techniques include:
3.11 Internal Rate of Return (IRR)
The internal rate of return, IRR, of a project is the rate of return which equates the net present value of the project’s cash flows to zero; or equivalently the rate of return which equates the present value of inflows to the present value of cash outflows. The internal rate of return (IRR) solves the following equation:
In determining whether to accept or reject a particular project, the IRR decision rule is
where r_{p }is the required return on the project. We illustrate the use of the IRR rule, and some of the pitfalls of this approach via a series of examples.
Example 3.12
Suppose a firm whose required rate of return is 10% is considering a project with the following cash flows:
Is this a worthwhile investment? The internal rate of return of this project is the rate of return which solves Note that we have to interpolate or use an iterative technique such
as Excel's Solver to find the IRR in this case. The internal
rate of return of this project turns out to be 21.86%. Applying the decision
rules above, we would accept the project since 
3.13 Problems with IRR
However, the IRR method has a number of potential difficulties which are outlined below:
3.14 Problem 1  Timing of Inflows and Outflows
Consider the desirability of the following 2 investment proposals for a firm whose required rate of return is 10%.
Both projects have an IRR of 50%, yet clearly project A is preferred to project B. Notice that the NPV rule correctly identifies the profitable alternative. The IRR rule fails in this case because it ignores the ordering of the inflows and outflows. Project A is a situation in which we are investing 1000 now and it accumulates to 1500 in 1 period. This is a very good return on our investment. In project B we are borrowing 1000 and having to repay 1500 in 1 period. This is an extremely high interest rate which we have to pay. However, the IRR does not take account of whether we are borrowing or lending and reports the same rate for both projects. If we followed the IRR rule and accepted project B, the firm would be worse of by $363.64. That is, the value of the firm would immediately fall by $363.64. Thus, we see that the IRR rule can yield results inconsistent with the objective of maximizing shareholder wealth. 
3.15 Problem 2  Ambiguous Results
Another problem with the IRR rule is that there can be multiple internal rates of return. Suppose a firm whose required rate of return is 10% is contemplating an investment project whose incremental cash flows are as follows:
The internal rate of return is the value of IRR that solves the following equation: Dividing both sides by 1,000, we get: Now, multiply both sides by (1+IRR)^{2} to get: 3.2(1+IRR)  2.4  (1+IRR)^{2} = 0 Rearranging the terms, and letting x = (1+IRR) and we have: x^{2}  3.2x  2.4 = 0 which is a quadratic equation with a wellknown solution: Recall that x = (1+IRR). Hence, the project has two possible internal rates of return, 100% or 20%. Assuming the marginal cost of capital is 10%, the project appears to be acceptable no matter which value of IRR we use. The NPV of the project, however, indicates that project acceptance will cause shareholders to be worse off: The inconsistency revealed through this illustration is referred to as the multiple root problem. For every change in sign of the cash flows through time, there can exist an additional internal rate of return. In this example, there were two changes in sign, from 1,000 to +3,200 and from +3,200 to 2,400, and there are two internal rates of return. Both of these IRRs exceed the required return of the firm, but acceptance of the project will cause share price to fall (the project has a negative NPV). Note that this example is not extreme by any means. Many investments (e.g., the clean up stage in mining operations) require large negative cash flows in the final period of the project’s life. The graph below depicts the projects NPV at various discount rates. Note that the parabola intersects the xaxis at both of its roots. 
3.16 Problem 3  Scale of Investment
A manufacturer is considering the purchase of one of two machines, A and B. The machine will be installed in the manufacturer’s factory and will produce items for sale. Both machines take exactly the same input and produce exactly the same final product. The only difference between the machines is that one is more efficient and therefore has lower operating costs, resulting in higher net cash flows. Assume that the required rate of return is 10% and that the firm has excess debt capacity (i.e., it can borrow as much as it likes to finance any profitable investment). The cash flows of each of the projects are as follows:
The internal rates of return of projects A and B are 19.86% and 22.11%, respectively. Thus, it appears that project B should be chosen. However,
so project A is actually superior to project B from a shareholder wealth maximization standpoint. The inconsistency arises from a scale of investment problem. The IRR method gives only a measure of the profitability of the project per dollar invested and does not measure absolute profitability. 
3.17 NonUniform Term Structure
Now we consider the possibility of a nonuniform term structure. Suppose we are faced with the following cash flows and one period interest rates:
Period 
Interest Rate 
Cash Flow 
Discount 
Present Value 
0 
1,000 
1 
1,000 

1 
20% 
80 
1.2 
66.67 
2 
10% 
80 
(1.2)(1.1) 
60.61 
3 
4% 
80 
(1.2)(1.1)(1.04) 
58.28 
4 
4% 
80 
(1.2)(1.1)(1.04)^{2} 
56.03 
5 
4% 
1,080 
(1.2)(1.1)(1.04)^{3} 
727.36 
This example just replicates the cash flows from purchasing a five year bond that pays an 8% annual coupon. The internal rate of return is 8%. The final two rows of the table calculate the present value of each cash flow. These results are shown below. Although the IRR of this investment is 8%, the NPV of the investment is clearly negative. In the case of a nonuniform term structure, the IRR is not that meaningful a measure.
Present Value of Inflows 
968.94 
Investment 
1,000 
Net Present Value 
31.03 
3.18 Undefined IRR
Consider the following cash flows:
Period 
0 
1 
2 
Cash Flow 
100 
200 
150 
To solve for the IRR, use the quadratic equation as above. The result comes out to:
We cannot solve the square foot of a negative number with a real number. The solutions to the IRR involve using imaginary numbers. This does not help us evaluate the worth of the project. We cannot provide a graphical representation of the differences between NPV and IRR here unless we use the complex plane. Note in this particular case that the NPV is greater than zero for all discount rates.
3.19 What use is the IRR?
The IRR has one strong attraction: it provides a rate of return which is easier to interpret than the NPV. Hence, are there any applications where we would be able to use it? The answer is: very few. Essentially, we have to be careful that none of the above problems occurs.
An example would be a mortgage. A mortgage is a financing, where one cash inflow is followed by a sequence of cash outflows. Hence, the cash flow pattern has only one sign change, so the IRR is unique (avoids problem 2). Moreover, you need to compare mortgages with the same repayment horizon in order to avoid problem 1, and for the same amount in order avoid problem 3. Then you can use the rule that the mortgage with the lower rate (i. e. the lower IRR) is better. Hence, the moral is that you can use the IRR for some stylized situations, but not for the general capital budgeting problem, where the NPV is the dominant criterion that is robust to the problems listed above.
Another place where IRR is frequently used is in the pricing of bonds. The yield of a bond is just its IRR. Like the mortgage, there is a unique solution when calculating a yield. Investors can run into trouble when they compare the yields of two bonds, if the bonds are very different. But if two bonds have similar characteristics, then comparing the yield is a good way to compare the values.
The payback period, PP, is the length of time it takes to recover the initial investment of the project. To apply the payback period criterion, it is necessary for management to establish a maximum acceptable payback value PP*. In practice, PP* is usually between 2 and 4 years. In determining whether to accept or reject a particular project, the payback period decision rule is:
Example 3.21
Suppose a firm is considering two mutually exclusive projects, C and D, where the firm’s required rate of return is 10% and the projects’ cash flows are:
The payback method dictates that project C should be accepted, however the NPV indicates that if C is accepted, the share price will fall. It appears that the payback method is not consistent with the goal of shareholder wealth maximization. The problems with the payback method are that:

3.22 Payback Period: accounting for money at risk
One of the attractions of the payback period is that it provides some measure of the "money at risk". At the start of the project we are presented with a lot of uncertainty about future cash flows, and the economic environment and our cash flows may turn out more or less favorable than we initially anticipated, with uncertainty being larger for those cash flows in the more distant future. However, the payback criterion is the wrong method to account for that. There are two tools for analyzing the risk associated with more distant cash flows. The first concerns the setting of discount rates. We shall see later (lectures 9 and 10) that discount rates can be decomposed into a riskfree rate, which is a compensation for the time value of money, and a risk premium, which rewards investors for risk. Hence, the discount rate is:
Discount rate = Risk free rate + Risk premium
Suppose the risk free rate is 10% and the required risk premium is 5% (we will discuss how to determine risk premiums later in the course). Then we obtain the following relationship between the discount rates and the time horizon:
Period 1 
Period 2 
Period 3 
Period 4 

Discount factor at 10% 
0.91 
0.83 
0.75 
0.68 
Discount factor at 15% 
0.87 
0.76 
0.66 
0.57 
Difference 
0.0395 
0.0703 
0.0938 
0.1113 
% Difference 
4.35 
8.51 
12.48 
16.29 
The picture emerges quite clearly: the risk premium reduces the value of one dollar at the end of period 1 from $0.91 to $0.87, or by 3.95 cents or 4.35%. However, dollars at the end of period 2 are reduced much more substantially by 7.03 cents or 8.51%, and the reduction increases with the time horizon. Hence, our NPV criterion, with appropriately set discount rates already accounts for the fact that risk increases with the time horizon.
In addition to this, we know much less about the more distant future than the immediate future, and we would typically change the design of a project if circumstances change in the future. Hence, we need to reflect the fact that one project commits our money for a short period of time and another one for a long period of time in our analysis, because projects with longer time horizons give us somewhat less flexibility. We shall see later in the course that this argument has also some merit, because flexibility has economic value. However, the appropriate tool for analyzing flexibility is decision tree analysis or option analysis (socalled "real options"). We shall discuss these tools in lecture 11 and see that they extend NPV analysis and allow us to quantify the value of flexibility and of "money at risk". Using payback period is an illegitimate shortcut.
3.23 Profitability Index
Another capital budgeting technique, the profitability index, is used when firms have only a limited supply of capital with which to invest in positive NPV projects. This type of problem is referred to as a capital rationing problem. Given that the objective is to maximize shareholder wealth, the objective in the capital rationing problem is to identify that subset of projects that collectively have the highest aggregate net present value. To assist in that evaluation, this method requires that we compute each project’s profitability index PI.
We then rank the project’s PI from highest to lowest, and then select from the top of the list until the capital budget is exhausted. The idea behind the profitability index method is that this will provide the subset of projects that maximize the aggregate net present value. However, this is not always the case (as the examples below show).
Example 3.24
Suppose a firm is considering the following investment projects and only has $12,000 available to invest.
In this case, the profitability index is successful since the combination of projects A, B, and C provides the highest aggregate net present value of $5,000. The second best alternative is projects B, C and D with an aggregate net present value of $4,800. Note, however, that if the firm had invested in all of the projects it would have an aggregate net present value of $5,900. Suppose a firm is considering the following investment projects and only has $12,000 available to invest
In this case, the profitability index does not work. The best alternative is BCD with an aggregate net present value of $5,100. The project ranked most highly by the PI (i.e., Project A) is not even included in the final set. Once again, note that if the firm had not imposed the capital rationing constraint of $12,000 it could have taken all of the projects and thus achieved an aggregate net present value of $6,200. Obviously the profitability index gives results inconsistent with the maximization of shareholder wealth and it should not be used. Indeed, the concept of capital rationing gives results inconsistent with the maximization of shareholder wealth. It is hard to believe that a firm with positive NPV projects can not go out to the capital market and borrow the required funds to finance these projects. 
3.25 Comparing Projects with Different Lives
The Annual Equivalent
Suppose a firm with a required rate of return of 10% is considering the acquisition of a new machine to produce its product. It is deciding between Machine A and Machine B. Machine A has a useful life of the 3 years and machine B has a useful life of 5 years. The net present values of the machines are as follows:
Machine 
NPV 
Machine Life 
A 
2000 
3 Years 
B 
3000 
5 Years 
Should the firm choose Machine A or Machine B? Machine B has a higher NPV, but it also has a useful life of 5 years versus the 3 year life of Machine A. Since the machine is going to be used to produce the output of the firm, it is reasonable to assume that Machine A will be replaced at the end of year 3 and thus its NPV above is understated. That is, assume that the machine will be used by the firm indefinitely. Hence, at the end of the machine’s useful life it will be replaced by another identical machine. Machines of type A are replaced on a 3year cycle and machines of type B will be replaced on a 5year cycle.
We need to restate the NPVs of the two alternatives in a way that will allow direct comparison. One way to do this is to compare the annual equivalent cash flows of the two alternative projects. Machine A has a NPV of $2,000, but at the required return of 10% we would be indifferent between $2,000 at the beginning of period zero and the annual equivalent (AE) of:
at the end of each of the 3 years. This is because
so that
Since we assume that the machine is going to be used indefinitely, this implies that we will receive the annual equivalent cash flow of AE_{A}=$804.25 indefinitely.
In general, the annual equivalent cash flow is given by:
The annual equivalent cash flow of machine B is thus given by:
Therefore, we can now compare the annual equivalent cash flows of the two proposals and the decision rule is to accept the proposal with the highest annual equivalent cash flow. Here AE_{A }> AE_{B} so the firm should accept project A.
3.26 The Decision Rule
The rule is that for mutually exclusive projects with different lives it is not appropriate to compare the NPVs directly. We should, instead, convert these NPVs to annual equivalent cash flows (AE) where:
and take the project with the highest AE. This applies to cases where the firm is considering one type of machine which is to be replaced indefinitely or an alternative type of machine that is to be replaced indefinitely.
Examples illustrating these concepts
Example 3.27
A corporation is considering replacing an existing machine with a new machine. The new machine costs $60,000 plus installation costs of $2,000. It will generate revenues of $155,000 annually and cash expenses annually of $100,000. It will be depreciated to a salvage of $6,000 over a sevenyear life using the straightline method. The old machine has a book value of $40,000 and a remaining useful life of 5 years. It can be sold immediately for $15,000. If retained, the machine will generate revenues of $150,000 and cash expenses annually of $110,000. Assuming the firm has a marginal cost of capital of 12% and is in the 34% marginal tax bracket, should it replace the existing machine? Assume that this is a oneoff decision  the choice is either keep the existing machine for five years or buy the new machine and run it for seven years. Step 1  Determine the Cash Flows Old Machine  First compute the tax expense.
Now, compute net cash flow:
New Machine  First, compute the tax expense
Notes: a. (60,000+2,0006,000)/7 Now, compute the net cash flows:
Notes: a. Old Machine Step 2  Determine Net Present Value Old Machine New Machine Step 3  Make the decision Since NPV_{n} > NPV_{0} we replace the machine 
Summary of Important Formulas