Business case

Harvard Business School 9-173-003 Rev. September 28, 2000

This note was prepared by Associate Professor John S. Hammond III as the basis for class discussion.

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Introduction to Accumulated Value, Present Value and Internal Rate of Return

In business decisions—such as planning, capital budgeting, acquisitions, real estate development or new product introduction—to mention a few—money may be invested “now” in the hopes of yielding future returns. For example, a lumber yard might consider investing $40,000 in a new fork lift truck, because it would save (net of operating costs) $10,000 per year in labor for the next five years.

Making such decisions is difficult for a number of reasons, perhaps the most significant of which is predicting the future returns. However, even if the future returns could be forecasted with certainty, choosing among alternative investments is not without its difficulties. The problem is that the timing of the returns associated with each alternative may be different. For example, in one alternative the returns may extend over a number of years whereas in another they may be of shorter duration but of greater magnitude. In another case, while the durations may be the same, one alternative may pay more earlier on, whereas the other may pay off handsomely near the end of its life. In such situations, how can the decision maker know which is “best”? Dealing with this problem is the topic of this note.

A Simple Example

The problem referred to above can be highlighted and clarified by considering a very simple example. Imagine the following alternative investments, having three things in common; each requires an initial outlay of $50,000, has returns lasting just three years into the future, and these returns are certain to occur. However, one returns $20,000 per year at the end of each of the next three years, whereas the second pays $40,000 a year from now, and $9,000 per year at the end of the second and third years. We can show these future patterns of returns and initial investment graphically (see Figure 1).

Which one of these investments do you prefer? You probably notice first that Investment 1 pays back $60,000 whereas Investment 2 pays back only $58,000. Based on this you may find yourself leaning toward Investment 1. that is, until you notice that Investment 2 pays $20,000 more in the first year. You say to yourself, “I could do something with that extra $20,000. At the very least I could get—say—5% from an insured savings account. If I’m clever, I can do even better.”

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Figure 1 The Cash Flow from Two Alternate Investments

1 2 3 Years from now

$20,000 inflow

$20,000 inflow

$20,000 inflow

Investment 1

0

$50,000 outflow

1 2 3 Years from now

$40,000 inflow

$9,000 inflow

$9,000 inflow

Investment 2

0

$50,000 outflow

Now the issue isn’t so clear; maybe it isn’t sufficient to add up the future cash flows and compare. When you get the money is important as well as how much you get. So what should you do?

The fact that you said, “I could do something with that money,” is the clue. The question is what would you do? Suppose you had plenty of opportunities in which to invest any extra funds you might have at a sure 10% per year. (We’ll refer to this rate in what follows as your “opportunity rate.”) Let’s take account of these opportunities in comparing the investments, by seeing how much money you would have at the end of the three years in both cases. This is done in Table 1, where the receipts from the two investments go to work at 10% the moment they are received.

The calculations shown in Table 1 are similar to those computing the balance of an interest- bearing bank account. For example, in Investment 1, consider the $20,000 amount available at the end of year 1. In the second year, when this sum is invested into other opportunities paying a sure 10%, it will earn $2,000. Hence, when we add in the $20,000 inflow at the end of year 2, the total value of this sum is $42,000. In a similar fashion, the $42,000 from year 2 earns $4,200 and, when the final $20,000 is added, totals $66,200 at the end of year 3. We can also apply the same type of reasoning to Investment 2. For each investment the total value for the stream of cash flows is shown underlined in the lower right corners. The results indicate that Investment 2 leaves you better off if you have 10% opportunities available.

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Table 1 Comparing Investments 1 and 2 When the Investor Has a 10% Opportunity Rate*

INVESTMENT 1

Year 1 Year 2 Year 3

Beginning of year balance $ 0 $20,000 $42,000

Earnings on balance at 10% 0 2,000 4,200

Inflow at end of year from Investment 1 20,000 20,000 20,000

Total amount available for following year $20,000 $42,000 $66,200

INVESTMENT 2

Year 1 Year 2 Year 3

Beginning of year balance $ 0 $40,000 $53,000

Earnings on balance at 10% 0 4,000 5,300

Inflow at end of year from Investment 2 40,000 9,000 9,000

Total amount available for following year $40,000 $53,000 $67,300

*These calculations are exactly equivalent to those done on a savings account. If we consider the inflows as deposits and the earnings as interest during the year on these deposits, then the underlined value in the lower right-hand corner will be the resultant balance in that account at the end of the third year.

This is not the case if you have only 5% opportunities available as shown in Table 2; Investment 1 looks better. Clearly, then, the attractiveness of a given pattern of future cash flows depends in part on what use you can make of the receipts. Further, we have seen that the timing of receipt of cash flows is also important—the earlier the better; for example, with a 10% opportunity rate, the money earned on the earlier receipts of Investment 2 overcame the disadvantage that the total receipts over the 3 years from Investment 2 was $2,000 less than from Investment 1. (This is why we say money has a time value.) Finally, note that the “earlier-the-better” rule is stronger the more you can earn on the receipts; when the return dropped from 10% to 5% the relative attractiveness of the two investments reversed.

Table 2 Comparing Investments 1 and 2 When the Investor Has a 5% Opportunity Rate

INVESTMENT 1

Year 1 Year 2 Year 3

Beginning of year balance $ 0 $20,000 $41,000

Earnings on balance at 5% 0 1,000 2,050

Inflow at end of year from Investment 1 20,000 20,000 20,000

Total amount available for following year $20,000 $41,000 $63,050

INVESTMENT 2

Year 1 Year 2 Year 3

Beginning of year balance $ 0 $40,000 $51,000

Earnings on balance at 5% 0 2,000 2,550

Inflow at end of year from Investment 2 40,000 9,000 9,000

Total amount available for following year $40,000 $51,000 $62,550

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What we have done so far is to compare the two investments with each other and have shown which is more attractive under two different assumptions about how the receipts would be used. We could make the comparison between the investments, without considering the magnitude of the initial outlay, because both had the same initial outlay. Still unanswered is the question of whether either investment is worth putting $50,000 into it. Again, alternative use of the money becomes important. Suppose, once more, that there were 10% opportunities around; then the approach used in Tables 1 and 2 can be used to show how well off you would be if the $50,000 was directly invested at 10%. This is done in Table 3, and shows that you would end up with $66,650 3 years hence as opposed to $66,200 for Investment 1 and $67,300 for Investment 2, which we derived in Table 1. In other words, Investment 1 doesn’t do quite as well as $50,000 invested at 10% for 3 years, whereas Investment 2 does a little better. A similar calculation (not reproduced here) for $50,000 for 3 years at 5% yields $57,356.25, indicating that both Investments 1 and 2 (especially 1) are preferable to investment of the $50,000 at 5%, since they yield (see Table 2) $63,050 and $62,550, respectively.

Table 3 The $50,000 Invested at 10%

Year 1 Year 2 Year 3

Beginning of year balance $50,000 $55,000 $60,500

Earnings on balance at 10% 5,000 5,500 6,050

Inflow at end of year 0 0

Total amount available for following year $55,000 $60,500 $66,550

To summarize, it is possible to compare different patterns of future cash flows by computing what the funds generated by each alternative would accumulate to at the investor’s opportunity rate. The resultant number in each case is called the accumulated value of the pattern of cash flows.

Present Value

Using accumulated value has great intuitive appeal in comparing patterns of future cash flows, not only because it properly takes account of the investor’s opportunity rate, and the timing and amount of cash flows, but also because the computations required are like interest calculations. These calculations are part of our everyday experience and are easy to carry out.

Unfortunately, there is an inconvenience in using accumulated value for making many business decisions which has nothing to do with its correctness or ease of calculation. You have already encountered the inconvenience (perhaps without realizing it) in the example used in the last section. It arose when we attempted to answer the question, “Is either investment worth the initial $50,000 outlay?” To get the answer we had to find the accumulated value of the $50,000 3 years hence, using the appropriate opportunity rate and compare it with the accumulated value of the investment. In other words, we were required to make the comparison as of some future point in time—“when all is said and done” with respect to the investment—in this case 3 years from now.

It would be nice to avoid this and instead find out the “worth” of the future cash flows, measured in today’s dollars, and then compare this “present value” directly with the initial investment. After all, the decision is made today, and it is easier to think in terms of today’s dollars than dollars at some time in the future. Further, “when all is said and done” may be many years in the future. In fact, it could even be for practical purposes “forever” as in the case of an acquisition or a long-lived project.

Fortunately, putting things in terms of today’s dollars isn’t very difficult. It involves “discounting,” the reverse of accumulating. Several examples will illustrate what we mean by

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“discounting” and “present value.” At a 10% opportunity rate, $10,000 today will accumulate to $11,000 a year from now. Consequently, we say that $10,000 is the present value of $11,000 one year hence when the opportunity rate is 10%. Another illustration is Table 3 where $50,000 accumulated in 3 years at 10% to $66,550. As shown in Table 4, by reversing the arrows we can say that $50,000 is the present value of $66,550 3 years hence at 10%.

Table 4

(a) Accumulated Value of $50,000 in three years at 10%

Present Year 1 Year 2 Year 3

Inflow $50,000

$66,550

(b) Present Value of $66,550 three years hence at 10%

Present Year 1 Year 2 Year 3

Inflow $66,550

$50,000

Thus it can be seen that present value answers the question “How much do I have to have today to yield a given amount, a given number of years in the future at a given opportunity rate?” Said another way, the present value is the amount that would make the investor indifferent between a sum of money at some point in the future or its present value now. It’s the amount that would allow him to say, “I don’t care whether I get $50,000 now or $66,550 in 3 years—it’s all the same to me.”

The above simple examples serve to illustrate that discounting and accumulating are the inverse of one another, and that there is a unique relationship between one value and the other. This was illustrated by reversing the direction of the arrows as shown in Table 4.

Mechanics of Discounting

Having seen what present value is, how it relates to accumulated value, and how useful it is to state things in today’s dollars, it remains to show how to determine the present value of a pattern of future cash flows. Our purpose in doing so is to enhance your understanding rather than make you a “pro” at computing the result. Fortunately computer spreadsheets can handle the mechanics (see page 8).

We shall see that you get the present value of a pattern of future cash flows by getting the present value of each of the individual future cash flows and adding them together. So we shall have to start out by showing how to obtain the present value of the individual cash flows.

Let’s take advantage of the fact that discounting is the inverse of accumulating to see how to do discounting. Referring back to Table 3, we see that in year 1, $50,000 becomes $55,000 by applying the calculation

“Principal” + “Interest” = “New Amount” or 50,000 + 50,000 × .10 =

50,000 × 1.10 = 55,000.

In year 2, we accumulate the $55,000 further as follows:

55,000 + 55,000 × .10 = 55,000 + 1.10 = 60,500.

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Going back to what happened in the first year, we can restate the accumulated value in year 2 as:

(55,000 × 1.10) × 1.10 = 50,000 × (1.10)2 = 60,500.

Similarly, we can obtain in year 3:

50,000 × (1.10)3 = 66,500.

As you can see, a pattern emerges; this example illustrates a more general rule, namely if

i = discount rate (in this case .10) p = initial amount ($50,000), and A = new (accumulated) amount n years later

then

p (1 + i)n = A.

This is in fact the implicit rule we used in determining the accumulated values for our investments. Table 5a illustrates this for Investment 2 at 10%. Note that, as expected, we obtain the same accumulated value as in Table 1.

Table 5a Accumulated Value of Investment 2 at 10% after 3 Years

Year 1 Year 2 Year 3 Accumulated

Value in Year 3

$40,000 $9,000 $9,000 $ 9,000

9,000 × 1.1 = 9,900

40,000 × (1.1) 2 = 48,400

$67,300

Now to get the present value of an amount A, we need to solve for p as follows:

P = Α

(1 +i) n

From this we have a rule for getting the present value of a cash flow n years from now. Simply divide the future cash flow by one plus the discount rate (expressed as a decimal) raised to the nth power.

As mentioned earlier, the present value of a pattern of future flows is the sum of the present values of the individual flows. An example, let’s apply this to Investment 2, using a 10% opportunity rate. (When computing present values, many people refer to the opportunity rate as the “discount rate.”) Table 5b shows the calculations.

The present value ($50,564) turns out to be more than the amount of the initial investment ($50,000), which says that the cash flows generated by the investment (stated in today’s dollars) are worth just a bit more than the $50,000 it takes to “buy” them. Hence, Investment 2 is worth the initial outlay of $50,000. Alternatively, since the present value is greater than $50,000, you make just over 10% on your money with Investment 2. This conclusion isn’t surprising, given our earlier observation

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that the accumulated value (at 10%) of the cash flows due to Investment 2 ($67,300) was just a little more than the accumulated value of $50,000 ($66,550).1

Table 5b Present Value of Investment 2 at 10%

Present Value Year 1 Year 2 Year 3

$40,000 $9,000 $9,000

$36,364 = 40,000/ 1.1

7,438 = 9,000/(1.1)2

6,762 = 9,000/(1.1)3

$50,564

Another interesting observation from Table 5b is that the $9,000 received in the third year contributes less to the present value ($6,762) than the $9,000 received in the second year does ($7,438). This illustrates the earlier-the-better rule: the amount received later is worth less in today’s dollars.

The comparison of the initial outlay with the present value of the future flows is facilitated by the concept of net present value, the difference between the present value of the future flows and the initial outlay. In the example shown in Table 5b, it is positive ($50,564 − $50,000 = $564), indicating, as we have said before, that the investors are making more than 10% on their money.

The Discount Rate

In the simple example it was convenient to talk about an opportunity rate for the individual faced with the choice of investments. In the more usual case of a firm faced with capital budget alternatives the appropriate rate is not as obvious. In some situations the board of directors of a company may state as a policy that they are unwilling to invest in a project unless it will generate at least a given percentage return after taxes. Such a minimum acceptable rate is referred to as a hurdle rate. They may set the hurdle rate because they believe that they have a large number of other investments available to them that would yield that rate. In such a situation their hurdle rate would correspond to an opportunity rate.

In still other situations, the capital available for projects may be obtainable from either equity capital or loans or a combination of the two. In such a case the board may set a hurdle rate based on the rate of return they believe is needed to attract the equity capital or to obtain the loan or get some combination of the two, the so-called “Cost of Capital.” In still other cases a board may believe that stockholders would have individual uses for funds which they would prefer to projects which yielded less than a given return after taxes. Again, they could specify such a rate as a hurdle rate to be used in the analysis of alternatives.

However, our purpose is not to describe the procedures by which the policy-making group in an organization may arrive at an appropriate hurdle rate. Rather, it is to use a hurdle rate, once set, to help in choosing among alternatives.

1 As a check of your understanding of the mechanics of discounting and of the statement that there is a unique relationship between present and accumulated value, you might wish to show that the present value of $67,300 three years hence at 10% is $50,564. You might also wish to show that the present value of Investment 1 at 10% is $49,737.

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Internal Rate of Return

There is another question often asked by people who invest money now on the promise of future returns, and it is so closely related to the one we have been addressing that it is worth our attention. The question is, “What am I making on my money?” This question is asked regularly by bond investors, for example, when they want to know a bond’s yield.

Let’s use Investment 2 as an example, to illustrate how the question is answered. First, take stock of what we know. The net present value of this investment at 10% is positive, that is, the present value of the future cash flows are “worth” more than the initial outlay. That’s another way of saying that the investment yields a rate of return greater than 10%. But how much more?

Try calculating the present value of Investment 2 using an 11% discount rate; this is done in Table 6 which shows a result of $49,922. Now the situation is reversed; the net present value has gone negative, which says that the investment yields less than 11%. So we know the rate of return is somewhere between 10% and 11%. By a process of trial and error we can try values between 10% and 11%, until a discount rate that results in a net present value of zero is found. This occurs at 10.88% (as shown in Table 7), which turns out to be the internal rate of return for the investment.

To summarize, the internal rate of return of an initial investment followed by a pattern of future cash flows is the discount rate that causes the net present value to be zero.2

Doing the Calculations

Fortunately, computer spreadsheet software can easily calculate the present value or internal rate of return of given patterns of future cash flows. 3 Also in many accounting and financial texts there are tables available4 of 1/(1+i)n (the so-called “discount factor”) which facilitate hand calculations such as those shown in Table 5b.

An example of the use of spreadsheets to evaluate an investment using net present value and internal rate of return is shown in Table 8. An investor drilled a gas well at a cost of $140,000. The well generated revenues (net of expenses) from the sale of gas for six years and was sold at the end of the sixth year for $32,000.

As the spreadsheet shows, the net present value is positive at 10% and at 15%, but negative at 20%. Therefore the internal rate of return must be between 15% and 20%. Indeed, it is 18.0%, as shown. You can conclude that the investment made sense at hurdle rates of 10% or 15% but not 20%. In addition you could say that the investor made a return of 18.0% on her money.

Looking at the right hand side of Table 8, you can see two generalizations about the influence of increasing the discount rates on present value. First, as the discount rate rises, the present value of all future cash flows drops. Second, the present value of cash flows further out in time (see for example year six) become a smaller and smaller portion of the total discounted value. (At 10% the $26,387 in Year 6

2 There are some circumstances, albeit rare ones, where an investment will have more than one internal rate of return. However, for such multiple rates to occur, one or more of the cash flows that follow the initial investment must be negative (that is, corresponding to a cash outflow in one or more of the years), and even under these conditions, there is usually only one internal rate. 3 To calculate the present value for a set of cash flows in a row or column of a spread sheet use the function NPV in Excel or @NPV in Lotus 1-2-3. The function for internal rate of return is IRR in Excel and @IRR in Lotus 1-2-3. 4 See, for example, Tables A and B in the Appendix of Robert N. Anthony, David F. Hawkins, and Kenneth A. Merchant, Accounting: Text and Cases, 10th Edition, Irwin McGraw-Hill, New York, 1999.

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represents 15.6% of the $169,005 future discounted value, whereas at 20% the $15,655 in Year 6 represents only 11.7% of the $134,136 future discounted value.)

Table 6 Present Value of Investment 2 at 11%

Present Value Year 1 Year 2 Year 3

$40,000 $9,000 $9,000

$36,036 = 40,000/ 1.11

7,305 = 9,000/(1.11)2

6,581 = 9,000/(1.11)3

$49,922

Table 7 The Net Present Value of Investment 2 at 10.88% Is Zero

Present Value Year 1 Year 2 Year 3

$40,000 $9,000 $9,000

$36,076 = 40,000/ 1.1088

7,321 = 9,000/(1.1088)2

6,603 = 9,000/(1.1088)3

$50,000 −50,000 Less Initial Investment

0 Net Present Value

Table 8 xxx Evaluation of Audry Thompson’s Gas Well Investment*

Cash Flows Discounted at:

Year Cash Flows 10% 15% 20%

0 Drilling and completing well $-140,000 -$140,000 -$140,000 -$140,000 1 Gas sales (net of expenses) 64,647 58,770 56,215 53,873 2 Gas sales (net of expenses) 42,160 34,843 31,879 29,278 3 Gas sales (net of expenses) 29,125 21,882 19,150 16,855 4 Gas sales (net of expenses) 22,926 15,659 13,108 11,056 5 Gas sales (net of expenses) 18,462 11,463 9,179 7,419 6 Gas sales (net) & sale of well 46,746 26,387 20,210 15,655

TOTAL $ 84,067 $ 29,005 $ 9,741 $ -$5,864

Net present value at 10% = $29,005 15% = $9,741 20% = -$5,864

Internal rate of return = 18.0%

*Only the portion to the left of the vertical line is necessary to compute present value and internal rate of return. The right hand portion was added to show the contribution of each year to the discounted value.

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Summary

We have introduced you to a way of comparing various future patterns of cash flows by reducing each to a single number, its present value. Our objective has been to further your understanding of present value and of internal rate of return, so that you will have a better “feel” for these numbers you encounter them in practice. The present value can be thought of as an endowment of the cash flows. In other words, if an amount equal to the present value was used to open (“endow”) a bank account, earning interest at the opportunity rate, and withdrawals were made equal to the future cash flows at appropriate times, the withdrawal corresponding to the last cash flow would precisely exhaust the remaining funds in the account.

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