Companies invest in lots of things. Some are tangible assets—that is, assets you can kick, like factories, machinery, and offices. Others are intangible assets, such as patents or trademarks. In each case, the company lays out some money now in the hope of receiving even more money later.
Individuals also make investments. For example, your college education may cost you $40,000 per year. That is an investment you hope will pay off in the form of a higher salary later in life. You are sowing now and expecting to reap later.
Companies pay for their investments by raising money and, in the process, assuming liabilities. For example, they may borrow money from a bank and promise to repay it with interest later. You also may have financed your investment in a college education by borrowing money that you plan to pay back in the future out of that fat salary.
All these financial decisions require comparisons of cash payments at different dates. Will your future salary be sufficient to justify the current expenditure on college tuition? How much will you have to repay the bank if you borrow to finance your degree?
In this chapter, we take the first steps toward understanding the relationship between the values of dollars today and dollars in the future. We start by looking at how money invested at a specific interest rate will grow over time. We next ask how much you would need to invest today to produce a specified future sum of money, and we describe some shortcuts for working out the value of a series of cash payments.
The term interest rate sounds straightforward enough, but rates can be quoted in different ways. We, therefore, conclude the chapter by explaining the difference between the quoted rate and the true or effective interest rate.
Once you have learned how to value cash flows that occur at different points in time, we can move on in the next two chapters to look at how bonds and stocks are valued. After that, we will tackle capital investment decisions at a practical level of detail.
For simplicity, every problem in this chapter is set out in dollars, but the concepts and calculations are identical in euros, Japanese yen, or Mongolian tugrik.
2-1Future Values and Present Values
Calculating Future Values
Money can be invested to earn interest. So, if you are offered the choice between $100 today and $100 next year, you naturally take the money now to get a year’s interest. Financial managers make the same point when they say that money has a time value or when they quote the most basic principle of finance: A dollar today is worth more than a dollar tomorrow.
Suppose you invest $100 in a bank account that pays interest of r = 7% a year. In the first year, you will earn interest of .07 × $100 = $7 and the value of your investment will grow to $107:
Value of investment after 1 year = $100 × (1 + r) = 100 × 1.07 = $107
By investing, you give up the opportunity to spend $100 today, but you gain the chance to spend $107 next year.
If you leave your money in the bank for a second year, you earn interest of .07 × $107 = $7.49 and your investment will grow to $114.49:
Value of investment after 2 years = $107 × 1.07 = $100 × 1.072 = $114.49
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Today |
Year 2 |
|
|
|
||
|
$100 |
× 1.072 |
$114.49 |
Notice that in the second year you earn interest on both your initial investment ($100) and the previous year’s interest ($7). Thus your wealth grows at a compound rate and the interest that you earn is called compound interest.
If you invest your $100 for t years, your investment will continue to grow at a 7% compound rate to $100 × (1.07)t. For any interest rate r, the future value of your $100 investment will be
Future value of $100 = $100 × (1 + r)t
The higher the interest rate, the faster your savings will grow. Figure 2.1 shows that a few percentage points added to the interest rate can do wonders for your future wealth. For example, by the end of 20 years, $100 invested at 10% will grow to $100 × (1.10)20 = $672.75. If it is invested at 5%, it will grow to only $100 × (1.05)20 = $265.33.
FIGURE 2.1 How an investment of $100 grows with compound interest at different interest rates
Calculating Present Values
We have seen that $100 invested for two years at 7% will grow to a future value of 100 × 1.072 = $114.49. Let’s turn this around and ask how much you need to invest today to produce $114.49 at the end of the second year. In other words, what is the present value (PV) of the $114.49 payoff?
You already know that the answer is $100. But, if you didn’t know or you forgot, you can just run the future value calculation in reverse and divide the future payoff by (1.07)2:
|
Today |
Year 2 |
|
|
|
||
|
$100 |
÷ 1.072 |
$114.49 |
In general, suppose that you will receive a cash flow of Ct dollars at the end of year t. The present value of this future payment is
The rate, r, in the formula is called the discount rate, and the present value is the discounted value of the cash flow, Ct. You sometimes see this present value formula written differently. Instead of dividing the future payment by (1 + r)t, you can equally well multiply the payment by 1/(1 + r)t. The expression 1/(1 + r)t is called the discount factor. It measures the present value of one dollar received in year t. For example, with an interest rate of 7% the two-year discount factor is
DF2 = 1 / (1.07)2 = .8734
Investors are willing to pay $.8734 today for delivery of $1 at the end of two years. If each dollar received in year 2 is worth $.8734 today, then the present value of your payment of $114.49 in year 2 must be
Present value = DF2 × C2 = .8734 × 114.49 = $100
The longer you have to wait for your money, the lower its present value. This is illustrated in Figure 2.2. Notice how small variations in the interest rate can have a powerful effect on the present value of distant cash flows. At an interest rate of 5%, a payment of $100 in year 20 is worth $37.69 today. If the interest rate increases to 10%, the value of the future payment falls by about 60% to $14.86.
FIGURE 2.2 Present value of a future cash flow of $100. Notice that the longer you have to wait for your money, the less it is worth today.
Valuing an Investment Opportunity
How do you decide whether an investment opportunity is worth undertaking? Suppose you own a small company that is contemplating construction of a suburban office block. The cost of buying the land and constructing the building is $700,000. Your company has cash in the bank to finance construction. Your real estate adviser forecasts a shortage of office space and predicts that you will be able to sell next year for $800,000. For simplicity, we will assume initially that this $800,000 is a sure thing.
The rate of return on this one-period project is easy to calculate. Divide the expected profit ($800,000 − 700,000 = $100,000) by the required investment ($700,000). The result is 100,000/700,000 = .143, or 14.3%.
Figure 2.3 summarizes your choices. (Note the resemblance to Figure 1.2 in the previous chapter.) You can invest in the project or pay cash out to shareholders, who can invest on their own. We assume that they can earn a 7% profit by investing for one year in safe assets (U.S. Treasury debt securities, for example). Or they can invest in the stock market, which is risky but offers an average return of 12%.
FIGURE 2.3 Your company can either invest $700,000 in an office block and sell it after 1 year for $800,000, or it can return the $700,000 to shareholders to invest in the financial markets
What is the opportunity cost of capital, 7% or 12%? The answer is 7%: That’s the rate of return that your company’s shareholders could get by investing on their own at the same level of risk as the proposed project. Here the level of risk is zero. (Remember, we are assuming for now that the future value of the office block is known with certainty.) Your shareholders would vote unanimously for the investment project because the project offers a safe return of 14% versus a safe return of only 7% in financial markets.
The office-block project is therefore a “go,” but how much is it worth and how much will the investment add to your wealth? The project produces a cash flow at the end of one year. To find its present value we discount that cash flow by the opportunity cost of capital:
Suppose that as soon as you have bought the land and paid for the construction, you decide to sell your project. How much could you sell it for? That is an easy question. If the venture will return a surefire $800,000, then your property ought to be worth its PV of $747,664 today. That is what investors in the financial markets would need to pay to get the same future payoff. If you tried to sell it for more than $747,664, there would be no takers because the property would then offer an expected rate of return lower than the 7% available on government securities. Of course, you could always sell your property for less, but why sell for less than the market will bear? The $747,664 present value is the only feasible price that satisfies both buyer and seller. Therefore, the present value of the property is also its market price.
Net Present Value
The office building is worth $747,664 today, but that does not mean you are $747,664 better off. You invested $700,000, so the net present value (NPV) is $47,664. Net present value equals present value minus the required investment:
NPV = PV − investment = 747,664 − 700,000 = $47,664
In other words, your office development is worth more than it costs. It makes a net contribution to value and increases your wealth. The formula for calculating the NPV of your project can be written as:
NPV = C0 + C1 / (1 + r)
Remember that C0, the cash flow at time 0 (that is, today) is usually a negative number. In other words, C0 is an investment and therefore a cash outflow. In our example, C0 = −$700,000.
When cash flows occur at different points in time, it is often helpful to draw a timeline showing the date and value of each cash flow. Figure 2.4 shows a timeline for your office development. It sets out the net present value calculation assuming that the discount rate r is 7%.1
FIGURE 2.4 Calculation showing the NPV of the office development
Risk and Present Value
We made one unrealistic assumption in our discussion of the office development: Your real estate adviser cannot be certain about the profitability of an office building. Those future cash flows represent the best forecast, but they are not a sure thing.
If the cash flows are uncertain, your calculation of NPV is wrong. Investors could achieve those cash flows with certainty by buying $747,664 worth of U.S. government securities, so they would not buy your building for that amount. You would have to cut your asking price to attract investors’ interest.
Here we can invoke a second basic financial principle: A safe dollar is worth more than a risky dollar. Most investors dislike risky ventures and won’t invest in them unless they see the prospect of a higher return. However, the concepts of present value and the opportunity cost of capital still make sense for risky investments. It is still proper to discount the payoff by the rate of return offered by a risk-equivalent investment in financial markets. But we have to think of expected payoffs and the expected rates of return on other investments.2
Not all investments are equally risky. The office development is more risky than a government security but less risky than a start-up biotech venture. Suppose you believe the project is as risky as investment in the stock market and that stocks are expected to provide a 12% return. Then 12% is the opportunity cost of capital for your project. That is what you are giving up by investing in the office building and not investing in equally risky securities.
Now recompute NPV with r = .12:
The office building still makes a net contribution to value, but the increase in your wealth is smaller than in our first calculation, which assumed that the cash flows from the project were risk-free.
The value of the office building depends, therefore, on the timing of the cash flows and their risk. The $800,000 payoff would be worth just that if you could get it today. If the office building is as risk-free as government securities, the delay in the cash flow reduces value by $52,336 to $747,664. If the building is as risky as investment in the stock market, then the risk further reduces value by $33,378 to $714,286.
Unfortunately, adjusting asset values for both time and risk is often more complicated than our example suggests. Therefore, we take the two effects separately. For the most part, we dodge the problem of risk in Chapters 2 through 6, either by treating all cash flows as if they were known with certainty or by talking about expected cash flows and expected rates of return without worrying how risk is defined or measured. Then in Chapter 7 we turn to the problem of understanding how financial markets cope with risk.
Present Values and Rates of Return
We have decided that constructing the office building is a smart thing to do since it is worth more than it costs. To discover how much it is worth, we asked how much you would need to invest directly in securities to achieve the same payoff. That is why we discounted the project’s future payoff by the rate of return offered by these equivalent-risk securities—the overall stock market in our example.
We can state our decision rule in another way: Your real estate venture is worth undertaking because its rate of return exceeds the opportunity cost of capital. The rate of return is simply the profit as a proportion of the initial outlay:
The cost of capital is once again the return foregone by not investing in financial markets. If the office building is as risky as investing in the stock market, the return foregone is 12%. Since the 14.3% return on the office building exceeds the 12% opportunity cost, you should go ahead with the project.
Building the office block is a smart thing to do, even if the payoff is just as risky as the stock market. We can justify the investment by either one of the following two rules:3
· Net present value rule. Accept investments that have positive net present values.
· Rate of return rule. Accept investments that offer rates of return in excess of their opportunity costs of capital.
Properly applied, both rules give the same answer, although we will encounter some cases in Chapter 5 where the rate of return rule is easily misused. In those cases, it is safest to use the net present value rule.
Calculating Present Values When There Are Multiple Cash Flows
One of the nice things about present values is that they are all expressed in current dollars—so you can add them up. In other words, the present value of cash flow (A + B) is equal to the present value of cash flow A plus the present value of cash flow B.
Suppose that you wish to value a stream of cash flows extending over a number of years. Our rule for adding present values tells us that the total present value is:
This is called the discounted cash flow (or DCF) formula. A shorthand way to write it is
where Σ refers to the sum of the series of discounted cash flows. To find the net present value (NPV) we add the (usually negative) initial cash flow:
EXAMPLE 2.1 
Present Values with Multiple Cash Flows
Your real estate adviser has come back with some revised forecasts. He suggests that you rent out the building for two years at $30,000 a year, and predicts that at the end of that time you will be able to sell the building for $840,000. Thus there are now two future cash flows—a cash flow of C1 = $30,000 at the end of one year and a further cash flow of C2 = (30,000 + 840,000) = $870,000 at the end of the second year.
The present value of your property development is equal to the present value of C1 plus the present value of C2. Figure 2.5 shows that the value of the first year’s cash flow is C1/(1 + r) = 30,000/1.12 = $26,786 and the value of the second year’s flow is C2/(1 + r)2 = 870,000/1.122 = $693,559. Therefore our rule for adding present values tells us that the total present value of your investment is:
FIGURE 2.5 Calculation showing the NPV of the revised office project
It looks as if you should take your adviser’s suggestion. NPV is higher than if you sell in year 1:
NPV = $720,344 − $700,000 = $20,344
Your two-period calculations in Example 2.1 required just a few keystrokes on a calculator. Real problems can be much more complicated, so financial managers usually turn to financial calculators especially programmed for present value calculations or to computer spreadsheet programs. A box near the end of the chapter introduces you to some useful Excel functions that can be used to solve discounting problems.
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The Opportunity Cost of Capital
By investing in the office building you are giving up the opportunity to earn an expected return of 12% in the stock market. The opportunity cost of capital is therefore 12%. When you discount the expected cash flows by the opportunity cost of capital, you are asking how much investors in the financial markets are prepared to pay for a security that produces a similar stream of future cash flows. Your calculations showed that these investors would need to pay $720,344 for an investment that produces cash flows of $30,000 at year 1 and $870,000 at year 2. Therefore, they won’t pay any more than that for your office building.
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Introduction to Excel
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Confusion sometimes sneaks into discussions of the cost of capital. Suppose a banker approaches. “Your company is a fine and safe business with few debts,” she says. “My bank will lend you the $700,000 that you need for the office block at 8%.” Does this mean that the cost of capital is 8%? If so, the project would be even more worthwhile. At an 8% cost of capital, PV would be 30,000/1.08 + 870,000/1.082 = $773,663 and NPV = $773,663 − $700,000 = +$73,663.
But that can’t be right. First, the interest rate on the loan has nothing to do with the risk of the project: it reflects the good health of your existing business. Second, whether you take the loan or not, you still face the choice between the office building and an equally risky investment in the stock market. A financial manager who borrows $700,000 at 8% and invests in an office building is not smart, but stupid, if the company or its shareholders can borrow at 8% and make an equally risky investment in the stock market offering an even higher return. That is why the 12% expected return on the stock market is the opportunity cost of capital for your project.
2-2Looking for Shortcuts—Perpetuities and Annuities
How to Value Perpetuities
Sometimes there are shortcuts that make it easy to calculate present values. Let us look at some examples.
On occasion, the British and the French have been known to disagree and sometimes even to fight wars. At the end of some of these wars the British consolidated the debt they had issued during the war. The securities issued in such cases were called consols. Consols are perpetuities. They are bonds that the government is under no obligation to repay but that offer a fixed income for each year to perpetuity. The British government is still paying interest on consols issued all those years ago. The annual rate of return on a perpetuity is equal to the promised annual payment divided by the present value:
We can obviously twist this around and find the present value of a perpetuity given the discount rate r and the cash payment C:4
The year is 2030. You have been fabulously successful and are now a billionaire many times over. It was fortunate indeed that you took that finance course all those years ago. You have decided to follow in the footsteps of two of your philanthropic heroes, Bill Gates and Warren Buffett. Malaria is still a scourge and you want to help eradicate it and other infectious diseases by endowing a foundation to combat these diseases. You aim to provide $1 billion a year in perpetuity, starting next year. So, if the interest rate is 10%, you need to write a check today for
Two warnings about the perpetuity formula. First, at a quick glance, you can easily confuse the formula with the present value of a single payment. A payment of $1 at the end of one year has a present value of 1/(1 + r). The perpetuity has a value of 1/r. These are quite different.
Second, the perpetuity formula tells us the value of a regular stream of payments starting one period from now. Thus your $10 billion endowment would provide the foundation with its first payment in one year’s time. If you also want to provide an up-front sum, you will need to lay out an extra $1 billion.
Sometimes you may need to calculate the value of a perpetuity that does not start to make payments for several years. For example, suppose that you decide to provide $1 billion a year with the first payment four years from now. Figure 2.6 provides a timeline of these payments. Think first about how much they will be worth in year 3. At that point the endowment will be an ordinary perpetuity with the first payment due at the end of the year. So our perpetuity formula tells us that in year 3 the endowment will be worth $1/r = $1/.1 = $10 billion. But it is not worth that much now. To find today’s value we need to multiply by the three-year discount factor 1/(1 + r)3 = 1/(1.1)3 = .751. Thus, the “delayed” perpetuity is worth $10 billion × .751 = $7.51 billion. The full calculation is:
FIGURE 2.6 This perpetuity makes a series of payments of $1 billion a year starting in year 4
How to Value Annuities
An annuity is an asset that pays a fixed sum each year for a specified number of years. The equal-payment house mortgage or installment credit agreement are common examples of annuities. So are interest payments on most bonds, as we shall see in the next chapter.
You can always value an annuity by calculating the value of each cash flow and finding the total. However, it is often quicker to use a simple formula that states that if the interest rate is r, then the present value of an annuity that pays $C a period for each of t periods is:
The expression in brackets shows the present value of $1 a year for each of t years. It is generally known as the t-year annuity factor.
If you are wondering where this formula comes from, look at Figure 2.7. It shows the payments and values of three investments.
FIGURE 2.7 An annuity that makes payments in each of years 1 through 3 is equal to the difference between two perpetuities
Row 1 The investment in the first row provides a perpetual stream of $1 starting at the end of the first year. We have already seen that this perpetuity has a present value of 1/r.
Row 2 Now look at the investment shown in the second row of Figure 2.7. It also provides a perpetual stream of $1 payments, but these payments don’t start until year 4. This stream of payments is identical to the payments in row 1, except that they are delayed for an additional three years. In year 3, the investment will be an ordinary perpetuity with payments starting in one year and will therefore be worth 1/r in year 3. To find the value today, we simply multiply this figure by the three-year discount factor. Thus, as we saw earlier
Row 3 Finally, look at the investment shown in the third row of Figure 2.7. This provides a level payment of $1 a year for each of three years. In other words, it is a three-year annuity. You can also see that, taken together, the investments in rows 2 and 3 provide exactly the same cash payments as the investment in row 1. Thus the value of our annuity (row 3) must be equal to the value of the row 1 perpetuity less the value of the delayed row 2 perpetuity:
Remembering formulas is about as difficult as remembering other people’s birthdays. But as long as you bear in mind that an annuity is equivalent to the difference between an immediate and a delayed perpetuity, you shouldn’t have any difficulty.5
EXAMPLE 2.2 Costing an Installment Plan
Most installment plans call for level streams of payments. Suppose that Tiburon Autos offers an “easy payment” scheme on a new Toyota of $5,000 a year, paid at the end of each of the next five years, with no cash down. What is the car really costing you?
First let us do the calculations the slow way, to show that if the interest rate is 7%, the present value of these payments is $20,501. The timeline in Figure 2.8 shows the value of each cash flow and the total present value. The annuity formula, however, is generally quicker; you simply need to multiply the $5,000 cash flow by the annuity factor:
FIGURE 2.8 Calculations showing the year-by-year present value of the installment payments
Valuing Annuities Due
When we costed the installment plan we assumed that the first payment was made at the end of the year. Suppose instead that the first of the five yearly payments is due immediately. How does this change the cost of the car?
If we discount each cash flow by one less year, the present value is increased by the multiple (1 + r). In the case of the car purchase the present value of the payments becomes 20,501 × (1 + r) = 20,501 × 1.07 = $21,936.
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A level stream of payments starting immediately is called an annuity due. An annuity due is worth (1 + r) times the value of an ordinary annuity.
Calculating Annual Payments
Annuity problems can be confusing on first acquaintance, but you will find that with practice they are generally straightforward. For example, here is a case where you need to use the annuity formula to find the amount of the payment given the present value.
EXAMPLE 2.3 Paying Off a Bank Loan
Bank loans are paid off in equal installments. Suppose that you take out a four-year loan of $1,000. The bank requires you to repay the loan evenly over the four years. It must therefore set the four annual payments so that they have a present value of $1,000. Thus,
PV = annual loan payment × 4-year annuity factor = $1,000
Annual loan payment = $1,000 / 4-year annuity factor
Suppose that the interest rate is 10% a year. Then
and
Annual loan payment = 1,000 / 3.170 = $315.47
Let’s check that this annual payment is sufficient to repay the loan. Table 2.1 provides the calculations. At the end of the first year, the interest charge is 10% of $1,000, or $100. So $100 of the first payment is absorbed by interest, and the remaining $215.47 is used to reduce the loan balance to $784.53.
TABLE 2.1 An example of an amortizing loan. If you borrow $1,000 at an interest rate of 10%, you would need to make an annual payment of $315.47 over four years to repay that loan with interest.
Next year, the outstanding balance is lower, so the interest charge is only $78.45. Therefore $315.47 − $78.45 = $237.02 can be applied to paying off the loan. Because the loan is progressively paid off, the fraction of each payment devoted to interest steadily falls over time, while the fraction used to reduce the loan increases. By the end of year 4, the amortization is just enough to reduce the balance of the loan to zero.
Loans that involve a series of level payments are known as amortizing loans. “Amortizing” means that part of the regular payment is used to pay interest on the loan and part is used to pay off or amortize the loan.
EXAMPLE 2.4 Calculating Mortgage Payments
Most mortgages are amortizing loans. For example, suppose that you take out a $250,000 house mortgage from your local savings bank when the interest rate is 12%. The bank requires you to repay the mortgage in equal annual installments over the next 30 years.
Thus,
and
Annual mortgage payment = 250,000 / 8.055 = $31,036
Figure 2.9 shows that in the early years, almost all of the mortgage payment is eaten up by interest and only a small fraction is used to reduce the amount of the loan. Even after 15 years, the bulk of the annual payment goes to pay the interest on the loan. From then on, the amount of the loan begins to decline rapidly.
FIGURE 2.9 Mortgage amortization. This figure shows the breakdown of mortgage payments between interest and amortization.
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Future Value of an Annuity
Sometimes you need to calculate the future value of a level stream of payments.
EXAMPLE 2.5 Saving to Buy a Sailboat
Perhaps your ambition is to buy a sailboat; something like a 40-foot Beneteau would fit the bill very well. But that means some serious saving. You estimate that, once you start work, you could save $20,000 a year out of your income and earn a return of 8% on these savings. How much will you be able to spend after five years?
We are looking here at a level stream of cash flows—an annuity. We have seen that there is a shortcut formula to calculate the present value of an annuity. So there ought to be a similar formula for calculating the future value of a level stream of cash flows.
Think first how much your savings are worth today. You will set aside $20,000 in each of the next five years. The present value of this five-year annuity is therefore equal to
Once you know today’s value of the stream of cash flows, it is easy to work out its value in the future. Just multiply by (1.08)5:
Value at end of year 5 = $79,854 × 1. 085 = $117,332
You should be able to buy yourself a nice boat for $117,000.
In Example 2.5, we calculate the future value of an annuity by first calculating its present value and then multiplying by (1 + r)t. The general formula for the future value of a level stream of cash flows of $1 a year for t years is, therefore,
There is a general point here. If you can find the present value of any series of cash flows, you can always calculate future value by multiplying by (1 + r)t:
Future value at the end of year t = present value × (1 + r) t
2-3More Shortcuts—Growing Perpetuities and Annuities
Growing Perpetuities
You now know how to value level streams of cash flows, but you often need to value a stream of cash flows that grows at a constant rate. For example, think back to your plans to donate $10 billion to fight malaria and other infectious diseases. Unfortunately, you made no allowance for the growth in salaries and other costs, which will probably average about 4% a year starting in year 1. Therefore, instead of providing $1 billion a year in perpetuity, you must provide $1 billion in year 1, 1.04 × $1 billion in year 2, and so on. If we call the growth rate in costs g, we can write down the present value of this stream of cash flows as follows:
Fortunately, there is a simple formula for the sum of this geometric series.6 If we assume that r is greater than g, our clumsy-looking calculation simplifies to
Therefore, if you want to provide a perpetual stream of income that keeps pace with the growth rate in costs, the amount that you must set aside today is
You will meet this perpetual-growth formula again in Chapter 4, where we use it to value the stocks of mature, slowly growing companies.
Growing Annuities
EXAMPLE 2.6 Winning Big at the Lottery
In August 2017, a Massachusetts woman invested in a Powerball lottery ticket and won a record $758.7 million. We suspect that she received unsolicited congratulations, good wishes, and requests for money from dozens of more or less worthy charities, relations, and newly devoted friends. In response, she could fairly point out that the prize wasn’t really worth $758.7 million. That sum was to be paid in 30 annual installments. The payment in the first year was only $11.42 million, but it then increased each year by 5% so that the final payment was $47.00 million. The total amount paid out was $758.7 million, but the winner had to wait to get it.
If the interest rate was 2.7%, what was that $758.7 prize really worth? Suppose that the first payment occurs at the end of year 1, so that C1 = $11.42 million. If the payments then grow at the rate of g = .05 each year, the payment in year 2 is 11.42 × 1.05, and in year 3 it is 11.42 × 1.052. Of course, you could calculate each of the 30 cash flows and discount them at 2.7%. The alternative is to use the following formula for the present value of a growing annuity:7
In the case of our lottery, the present value of the growing stream of payments is
Thus, the present value of a growing stream of payments starting at the end of the first year is $468 million. In practice, the news is not quite that bad because the lottery winner receives the first payment immediately (in year 0) and the last one is received in year 29 rather than in year 30. Therefore, we need to increase our estimate of present value by 1 + r. So the present value of the prize is 468 × 1.027 = $481 million.
If the total Powerball prize money was paid out immediately, it would be worth $757.8 million. Paying out this money over the next 29 years reduces the value of the prize to $481 million, much below the well-trumpeted prize but still not a bad day’s haul.
For winners with big spending plans, lottery operators generally make arrangements so that they may take an equivalent lump sum. In our example, the winner could either take the $758.7 million spread over 30 years or receive $481 million up front. Both arrangements had the same present value. Too many formulas are bad for the digestions. So we will stop at this point and spare you any more of them. The formulas discussed so far appear in Table 2.2.
TABLE 2.2 Some useful shortcut formulas
Note: a. The growing perpetuity formula works only if the discount rate r is greater than the growth rate g.
b. The growing annuity formula blows up if r = g . In this case, the value of the growing annuity is C × t/(1 + r ).
2-4How Interest Is Paid and Quoted
In our examples we have assumed that cash flows occur only at the end of each year. This is sometimes the case. For example, in France and Germany, the government pays interest on its bonds annually. However, in the United States and Britain, government bonds pay interest semiannually. So if a U.S. government bond promises to pay interest of 10% a year, the investor in practice receives interest of 5% every six months.
If the first interest payment is made at the end of six months, you can earn an additional six months’ interest on this payment. For example, if you invest $100 in a bond that pays interest of 10% compounded semiannually, your wealth will grow to 1.05 × $100 = $105 by the end of six months and to 1.05 × $105 = $110.25 by the end of the year. In other words, an interest rate of 10% compounded semiannually is equivalent to 10.25% compounded annually. The effective annual interest rate on the bond is 10.25%.
Let’s take another example. Suppose a bank offers you an automobile loan at an annual percentage rate, or APR, of 12% with interest to be paid monthly. By this the bank means that each month you need to pay one-twelfth of the annual rate, that is, 12/12 = 1% a month. Thus the bank is quoting a rate of 12%, but the effective annual interest rate on your loan is 1.0112 – 1 = .1268 or 12.68%.8
Our examples illustrate that you need to distinguish between the quoted annual interest rate and the effective annual rate. The quoted annual rate is usually calculated as the total annual payment divided by the number of payments in the year. When interest is paid once a year, the quoted and effective rates are the same. When interest is paid more frequently, the effective interest rate is higher than the quoted rate.
In general, if you invest $1 at a rate of r per year compounded m times a year, your investment at the end of the year will be worth [1 + (r/m)]m and the effective interest rate is [1 + (r/m)]m – 1. In our automobile loan example r = .12 and m = 12. So the effective annual interest rate was [1 + .12/12]12 – 1 = .1268, or 12.68%.
Continuous Compounding
Instead of compounding interest monthly or semiannually, the rate could be compounded weekly (m = 52) or daily (m = 365). In fact, there is no limit to how frequently interest could be paid. One can imagine a situation where the payments are spread evenly and continuously throughout the year, so the interest rate is continuously compounded.9 In this case m is infinite.
It turns out that there are many occasions in finance when continuous compounding is useful. For example, one important application is in option pricing models, such as the Black–Scholes model that we introduce in Chapter 21. These are continuous time models. So you will find that most computer programs for calculating option values ask for the continuously compounded interest rate.
It may seem that a lot of calculations would be needed to find a continuously compounded interest rate. However, think back to your high school algebra. You may recall that as m approaches infinity [1 + (r/m)]m approaches (2.718)r . The figure 2.718—or e, as it is called—is the base for natural logarithms. Therefore, $1 invested at a continuously compounded rate of r will grow to er = (2.718)r by the end of the first year. By the end of t years it will grow to ert = (2.718)rt.
Example 1 Suppose you invest $1 at a continuously compounded rate of 11% (r = .11) for one year (t = 1). The end-year value is e.11, or $1.116. In other words, investing at 11% a year continuously compounded is exactly the same as investing at 11.6% a year annually compounded.
Example 2 Suppose you invest $1 at a continuously compounded rate of 11% (r = .11) for two years (t = 2). The final value of the investment is ert = e.22, or $1.246.
Sometimes it may be more reasonable to assume that the cash flows from a project are spread evenly over the year rather than occurring at the year’s end. It is easy to adapt our previous formulas to handle this. For example, suppose that we wish to compute the present value of a perpetuity of C dollars a year. We already know that if the payment is made at the end of the year, we divide the payment by the annually compounded rate of r:
If the same total payment is made in an even stream throughout the year, we use the same formula but substitute the continuously compounded rate.
Suppose the annually compounded rate is 18.5%. The present value of a $100 perpetuity, with each cash flow received at the end of the year, is 100/.185 = $540.54. If the cash flow is received continuously, we must divide $100 by 17%, because 17% continuously compounded is equivalent to 18.5% annually compounded (e.17 = 1.185). The present value of the continuous cash flow stream is 100/.17 = $588.24. Investors are prepared to pay more for the continuous cash payments because the cash starts to flow in immediately.
Example 3 After you have retired, you plan to spend $200,000 a year for 20 years. The annually compounded interest rate is 10%. How much must you save by the time you retire to support this spending plan?
USEFUL SPREADSHEET FUNCTIONS
Discounting Cash Flows
Spreadsheet programs such as Excel provide built-in functions to solve discounted cash flow (DCF) problems. You can find these functions by pressing fx on the Excel toolbar. If you then click on the function that you wish to use, Excel asks you for the inputs that it needs. At the bottom left of the function box there is a Help facility with an example of how the function is used.
Here is a list of useful functions for DCF problems and some points to remember when entering data:
· FV: Future value of single investment or annuity.
· PV: Present value of single future cash flow or annuity.
· RATE: Interest rate (or rate of return) needed to produce given future value or annuity.
· NPER: Number of periods (e.g., years) that it takes an investment to reach a given future value.
· PMT: Amount of annuity payment with a given present or future value.
· NPV: Calculates the value of a stream of negative and positive cash flows. (When using this function, note the warning below.)
· EFFECT: The effective annual interest rate, given the quoted rate (APR) and number of interest payments in a year.
· NOMINAL: The quoted interest rate (APR) given the effective annual interest rate.
All the inputs in these functions can be entered directly as numbers or as the addresses of cells that contain the numbers.
Three warnings:
1. PV is the amount that needs to be invested today to produce a given future value. It should therefore be entered as a negative number. Entering both PV and FV with the same sign when solving for RATE results in an error message.
2. Always enter the interest or discount rate as a decimal value (for example, .05 rather than 5%).
3. Use the NPV function with care. Better still, don’t use it at all. It gives the value of the cash flows one period before the first cash flow and not the value at the date of the first cash flow.
Spreadsheet Questions
The following questions provide opportunities to practice each of the Excel functions.
1. (FV) In 1880, five aboriginal trackers were each promised the equivalent of 100 Australian dollars for helping to capture the notorious outlaw Ned Kelly. One hundred and thirteen years later, the granddaughters of two of the trackers claimed that this reward had not been paid. If the interest rate over this period averaged about 4.5%, how much would the A$100 have accumulated to?
2. (PV) Your adviser has produced revised figures for your office building. It is forecasted to produce a cash flow of $40,000 in year 1, but only $850,000 in year 2, when you come to sell it. If the cost of capital is 12%, what is the value of the building?
3. (PV) Your company can lease a truck for $10,000 a year (paid at the end of the year) for six years, or it can buy the truck today for $50,000. At the end of the six years the truck will be worthless. If the interest rate is 6%, what is the present value of the lease payments? Is the lease worthwhile?
4. (RATE) Ford Motor stock was one of the victims of the 2008 credit crisis. In June 2007, Ford stock price stood at $9.42. Eighteen months later it was $2.72. What was the annual rate of return over this period to an investor in Ford stock?
5. (NPER) An investment adviser has promised to double your money. If the interest rate is 7% a year, how many years will she take to do so?
6. (PMT) You need to take out a home mortgage for $200,000. If payments are made annually over 30 years and the interest rate is 8%, what is the amount of the annual payment?
7. (EFFECT) First National Bank pays 6.2% interest compounded annually. Second National Bank pays 6% interest compounded monthly. Which bank offers the higher effective annual interest rate?
8. (NOMINAL) What monthly compounded interest rate would Second National Bank need to pay on savings deposits to provide an effective rate of 6.2%?
Let us first do the calculations assuming that you spend the cash at the end of each year. In this case we can use the simple annuity formula that we derived earlier:
Thus, you will need to have saved $1.7 million by the time you retire.
Instead of waiting until the end of each year before you spend any cash, it is more reasonable to assume that your expenditure will be spread evenly over the year. In this case, instead of using the annually compounded rate of 10%, we must use the continuously compounded rate of r = 9.53% (e.0953 = 1.10). Therefore, to cover a steady stream of expenditure, you need to set aside the following sum:10
To support a steady stream of outgoings, you must save an additional $83,600.
Often in finance you need only a ballpark estimate of present value. An error of 5% in a present value calculation may be perfectly acceptable. In such cases it doesn’t usually matter whether you assume that cash flows occur at the end of the year or in a continuous stream. At other times precision matters, and you do need to worry about the exact frequency of the cash flows.
SUMMARY
Firms can best help their shareholders by accepting all projects that are worth more than they cost. In other words, they need to seek out projects with positive net present values. To find net present value we first calculate present value. Just discount future cash flows by an appropriate rate r, usually called the discount rate, hurdle rate, or opportunity cost of capital:
Net present value is present value plus any immediate cash flow:
Net present value (NPV) = C0 + PV
Remember that C0 is negative if the immediate cash flow is an investment, that is, if it is a cash outflow.
The discount rate r is determined by rates of return prevailing in financial markets. If the future cash flow is absolutely safe, then the discount rate is the interest rate on safe securities such as U.S. government debt. If the future cash flow is uncertain, then the expected cash flow should be discounted at the expected rate of return offered by equivalent-risk securities. (We talk more about risk and the cost of capital in Chapters 7 to 9.)
Cash flows are discounted for two simple reasons: because (1) a dollar today is worth more than a dollar tomorrow and (2) a safe dollar is worth more than a risky one. Formulas for PV and NPV are numerical expressions of these ideas.
Financial markets, including the bond and stock markets, are the markets where safe and risky future cash flows are traded and valued. That is why we look to rates of return prevailing in the financial markets to determine how much to discount for time and risk. By calculating the present value of an asset, we are estimating how much people will pay for it if they have the alternative of investing in the financial markets.
You can always work out any present value using the basic formula, but shortcut formulas can reduce the tedium. We showed how to value an investment that makes a level stream of cash flows forever (a perpetuity) and one that produces a level stream for a limited period (an annuity). We also showed how to value investments that produce growing streams of cash flows.
When someone offers to lend you a dollar at a quoted interest rate, you should always check how frequently the interest is to be paid. For example, suppose that a $100 loan requires six monthly payments of $3. The total yearly interest payment is $6 and the interest will be quoted as a rate of 6% compounded semiannually. The equivalent annually compounded rate is (1.03)2 – 1 = .06 or 6.1%. Sometimes it is convenient to assume that interest is paid evenly over the year so that interest is quoted as a continuously compounded rate.
PROBLEM SETS
Select problems are available in McGraw-Hill’s Connect. Answers to questions with an “*” are found in the Appendix.
1. Opportunity cost of capital Which of the following statements are true? The opportunity cost of capital:
a. Equals the interest rate at which the company can borrow.
b. Depends on the risk of the cash flows to be valued.
c. Depends on the rates of return that shareholders can expect to earn by investing on their own.
d. Equals zero if the firm has excess cash in its bank account and the bank account pays no interest.
2. Opportunity cost of capital Explain why we refer to the opportunity cost of capital, instead of just “cost of capital” or “discount rate.” While you’re at it, also explain the following statement: “The opportunity cost of capital depends on the proposed use of cash, not the source of financing.”
3. Compound interest Old Time Savings Bank pays 4% interest on its savings account. If you deposit $1,000 in the bank and leave it there:
a. How much interest will you earn in the first year?
b. How much interest will you earn in the second year?
c. How much interest will you earn in the tenth year?
4. Compound interest New Savings Bank pays 4% interest on deposits. If you deposit $1,000 in the bank and leave it there, will it take more or less than 25 years for your investment to double? You should be able to answer this without a calculator.
5. Compound interest In 2017, Leonardo da Vinci’s painting Salvator Mundi sold for a record $450.3 million. In 1958, it sold for $125, equivalent in purchasing power to about $1,060 at 2017 prices. The painting was originally commissioned by King Louis XII of France in about 1500. The Wall Street Journal guesstimated that the king may have paid Leonardo the equivalent of $575,000 in 1519.11
a. What was the annual rate of appreciation in the price of the painting between 1958 and 2017 adjusted for inflation?
b. What was the annual estimated rate of appreciation in the price of the painting between 1519 and 2017 adjusted for inflation.
6. Future values If you invest $100 at an interest rate of 15%, how much will you have at the end of eight years?
7. Future values* Compute the future value of a $100 investment for the following combinations of rates and times.
a. r = 6%, t = 10 years
b. r = 6%, t = 20 years
c. r = 4%, t = 10 years
d. r = 4%, t = 20 years
8. Future values In the five years preceding the end of 2016, the price of Amazon shares rose by 34% a year. If you had invested $100 in Amazon at the beginning of this period, how much would you have by the end of the period?
9. Discount factors
a. If the present value of $139 is $125, what is the discount factor?
b. If that $139 is received in year 5, what is the interest rate?
10. Present values If the cost of capital is 9%, what is the PV of $374 paid in year 9?
11. Present values A project produces a cash flow of $432 in year 1, $137 in year 2, and $797 in year 3. If the cost of capital is 15%, what is the project’s PV? If the project requires an investment of $1,200, what is its NPV?
12. Present values What is the PV of $100 received in:
a. Year 10 (at a discount rate of 1%)?
b. Year 10 (at a discount rate of 13%)?
c. Year 15 (at a discount rate of 25%)?
d. Each of years 1 through 3 (at a discount rate of 12%)?
13. Present values* Lofting Snodbury is considering investing in a new boring machine. It costs $380,000 and is expected to produce the following cash flows:
If the cost of capital is 12%, what is the machine’s NPV?
14. Present values A factory costs $800,000. You reckon that it will produce an inflow after operating costs of $170,000 a year for 10 years. If the opportunity cost of capital is 14%, what is the net present value of the factory? What will the factory be worth at the end of five years?
15. Present values Recalculate the NPV of the office building venture in Example 2.1 at interest rates of 5, 10, and 15%. Plot the points on a graph with NPV on the vertical axis and the discount rates on the horizontal axis. At what discount rate (approximately) would the project have zero NPV? Check your answer.
16. Present values and opportunity cost of capital Halcyon Lines is considering the purchase of a new bulk carrier for $8 million. The forecasted revenues are $5 million a year and operating costs are $4 million. A major refit costing $2 million will be required after both the fifth and tenth years. After 15 years, the ship is expected to be sold for scrap at $1.5 million.
a. What is the NPV if the opportunity cost of capital is 8%?
b. Halcyon could finance the ship by borrowing the entire investment at an interest rate of 4.5%. How does this borrowing opportunity affect your calculation of NPV?
17. Perpetuities* An investment costs $1,548 and pays $138 in perpetuity. If the interest rate is 9%, what is the NPV?
18. Perpetuities You have just read an advertisement stating, “Pay us $100 a year for 10 years and we will pay you $100 a year thereafter in perpetuity.” If this is a fair deal, what is the rate of interest?
19. Growing perpetuities A common stock will pay a cash dividend of $4 next year. After that, the dividends are expected to increase indefinitely at 4% per year. If the discount rate is 14%, what is the PV of the stream of dividend payments?
20. Perpetuities and annuities The interest rate is 10%.
a. What is the PV of an asset that pays $1 a year in perpetuity?
b. The value of an asset that appreciates at 10% per annum approximately doubles in seven years. What is the approximate PV of an asset that pays $1 a year in perpetuity beginning in year 8?
c. What is the approximate PV of an asset that pays $1 a year for each of the next seven years?
d. A piece of land produces an income that grows by 5% per annum. If the first year’s income is $10,000, what is the value of the land?
21. Discount factors and annuity factors*
a. If the one-year discount factor is .905, what is the one-year interest rate?
b. If the two-year interest rate is 10.5%, what is the two-year discount factor?
c. Given these one- and two-year discount factors, calculate the two-year annuity factor.
d. If the PV of $10 a year for three years is $24.65, what is the three-year annuity factor?
e. From your answers to parts (c) and (d), calculate the three-year discount factor.
22. Annuities* Kangaroo Autos is offering free credit on a new $10,000 car. You pay $1,000 down and then $300 a month for the next 30 months. Turtle Motors next door does not offer free credit but will give you $1,000 off the list price. If the rate of interest is .83% a month, which company is offering the better deal?
23. Annuities David and Helen Zhang are saving to buy a boat at the end of five years. If the boat costs $20,000 and they can earn 10% a year on their savings, how much do they need to put aside at the end of years 1 through 5?
24. Annuities Siegfried Basset is 65 years of age and has a life expectancy of 12 more years. He wishes to invest $20,000 in an annuity that will make a level payment at the end of each year until his death. If the interest rate is 8%, what income can Mr. Basset expect to receive each year?
25. Annuities Several years ago, The Wall Street Journal reported that the winner of the Massa chusetts State Lottery prize had the misfortune to be both bankrupt and in prison for fraud. The prize was $9,420,713, to be paid in 19 equal annual installments. (There were 20 installments, but the winner had already received the first payment.) The bankruptcy court judge ruled that the prize should be sold off to the highest bidder and the proceeds used to pay off the creditors.
a. If the interest rate was 8%, how much would you have been prepared to bid for the prize?
b. Enhance Reinsurance Company was reported to have offered $4.2 million. Use Excel to find the return that the company was looking for.
26. Annuities The annually compounded discount rate is 5.5%. You are asked to calculate the present value of a 12-year annuity with payments of $50,000 per year. Calculate PV for each of the following cases.
a. The annuity payments arrive at one-year intervals. The first payment arrives one year from now.
b. The first payment arrives in six months. Following payments arrive at one-year intervals (i.e., at 18 months, 30 months, etc.).
27. Annuities Dear Financial Adviser,
My spouse and I are each 62 and hope to retire in three years. After retirement we will receive $7,500 per month after taxes from our employers’ pension plans and $1,500 per month after taxes from Social Security. Unfortunately our monthly living expenses are $15,000. Our social obligations preclude further economies.
We have $1,000,000 invested in a high-grade, tax-free municipal-bond mutual fund. The return on the fund is 3.5% per year. We plan to make annual withdrawals from the mutual fund to cover the difference between our pension and Social Security income and our living expenses. How many years before we run out of money?
Sincerely,
Luxury Challenged
Marblehead, MA
You can assume that the withdrawals (one per year) will sit in a checking account (no interest) until spent. The couple will use the account to cover the monthly shortfalls.
28. Perpetuities and annuities Refer to Sections 2-3 and 2-4. If the rate of interest is 8% rather than 10%, how much would you need to set aside to provide each of the following?
a. $1 billion at the end of each year in perpetuity.
b. A perpetuity that pays $1 billion at the end of the first year and that grows at 4% a year.
c. $1 billion at the end of each year for 20 years.
d. $1 billion a year spread evenly over 20 years.
29. Annuities due The $40 million lottery prize that you have just won actually pays out $2 million a year for 20 years. The interest rate is 8%.
a. If the first payment comes after 1 year, what is the present value of your winnings?
b. What is the present value if the first payment comes immediately?
30. Annuities due A store offers two payment plans. Under the installment plan, you pay 25% down and 25% of the purchase price in each of the next 3 years. If you pay the entire bill immediately, you can take a 10% discount from the purchase price.
a. Which is the better deal if the interest rate is 5%?
b. How will your answer change if the four payments on the installments do not start until the end of the year?
31. Amortizing loans* A bank loan requires you to pay $70,000 at the end of each of the next eight years. The interest rate is 8%.
a. What is the present value of these payments?
b. Calculate for each year the loan balance that remains outstanding, the interest payment on the loan, and the reduction in the loan balance.
32. Amortizing loans Suppose that you take out a $200,000, 20-year mortgage loan to buy a condo. The interest rate on the loan is 6%, and payments on the loan are made annually at the end of each year.
a. What is your annual payment on the loan?
b. Construct a mortgage amortization table in Excel similar to Table 2.1, showing the interest payment, the amortization of the loan, and the loan balance for each year.
c. What fraction of your initial loan payment is interest? What about the last payment? What fraction of the loan has been paid off after 10 years? Why is the fraction less than half?
33. Future values and annuities
a. The cost of a new automobile is $10,000. If the interest rate is 5%, how much would you have to set aside now to provide this sum in five years?
b. You have to pay $12,000 a year in school fees at the end of each of the next six years. If the interest rate is 8%, how much do you need to set aside today to cover these bills?
c. You have invested $60,476 at 8%. After paying the above school fees, how much would remain at the end of the six years?
34. Growing annuities You estimate that by the time you retire in 35 years, you will have accumulated savings of $2 million. If the interest rate is 8% and you live 15 years after retirement, what annual level of expenditure will those savings support?
Unfortunately, inflation will eat into the value of your retirement income. Assume a 4% inflation rate and work out a spending program for your $2 million in retirement savings that will allow you to increase your expenditure in line with inflation.
35. Growing annuities You are contemplating membership in the St. Swithin’s and Ancient Golf Club. The annual membership fee for the coming year is $5,000, but you can make a single payment today of $12,750, which will provide you with membership for the next three years. Suppose that the annual fee is payable at the end of each year and is expected to increase by 6% a year. The discount rate is 10%. Which is the better deal?
36. Growing perpetuities and annuities
As winner of a breakfast cereal competition, you can choose one of the following prizes:
a. $100,000 now.
b. $180,000 at the end of five years.
c. $11,400 a year forever.
d. $19,000 for each of 10 years.
e. $6,500 next year and increasing thereafter by 5% a year forever. If the interest rate is 12%, which is the most valuable prize?
37. Growing perpetuities and annuities Your firm’s geologists have discovered a small oil field in New York’s Westchester County. The field is forecasted to produce a cash flow of C1 = $2 million in the first year. You estimate that you could earn a return of r = 12% from investing in stocks with a similar degree of risk to your oil field. Therefore, 12% is the opportunity cost of capital. What is the present value? The answer, of course, depends on what happens to the cash flows after the first year. Calculate present value for the following cases: a. The cash flows are forecasted to continue forever, with no expected growth or decline. b. The cash flows are forecasted to continue for 20 years only, with no expected growth or decline during that period. c. The cash flows are forecasted to continue forever, increasing by 3% per year because of inflation. d. The cash flows are forecasted to continue for 20 years only, increasing by 3% per year because of inflation.
38. Compounding intervals A leasing contract calls for an immediate payment of $100,000 and nine subsequent $100,000 semiannual payments at six-month intervals. What is the PV of these payments if the annual discount rate is 8%?
39. Compounding intervals* Which would you prefer?
a. An investment paying interest of 12% compounded annually.
b. An investment paying interest of 11.7% compounded semiannually.
c. An investment paying 11.5% compounded continuously.
Work out the value of each of these investments after 1, 5, and 20 years.
40. Compounding intervals You are quoted an interest rate of 6% on an investment of $10 million. What is the value of your investment after four years if interest is compounded:
a. Annually?
b. Monthly?
c. Continuously?
41. Perpetuities and continuous compounding If the interest rate is 7% compounded annually, what is the value of the following three investments?
a. An investment that offers you $100 a year in perpetuity with the payment at the end of each year.
b. A similar investment with the payment at the beginning of each year.
c. A similar investment with the payment spread evenly over each year.
42. Continuous compounding How much will you have at the end of 20 years if you invest $100 today at 15% annually compounded? How much will you have if you invest at 15% continuously compounded?
43. Continuous compounding The continuously compounded interest rate is 12%.
a. You invest $1,000 at this rate. What is the investment worth after five years?
b. What is the PV of $5 million to be received in eight years?
c. What is the PV of a continuous stream of cash flows, amounting to $2,000 per year, starting immediately and continuing for 15 years?
CHALLENGE PROBLEMS
44. Future values and continuous compounding Here are two useful rules of thumb. The “Rule of 72” says that with discrete compounding the time it takes for an investment to double in value is roughly 72/interest rate (in percent). The “Rule of 69” says that with continuous compounding the time that it takes to double is exactly 69.3/interest rate (in percent).
a. If the annually compounded interest rate is 12%, use the Rule of 72 to calculate roughly how long it takes before your money doubles. Now work it out exactly.
b. Can you prove the Rule of 69?
45. Annuities Use Excel to construct your own set of annuity tables showing the annuity factor for a selection of interest rates and years.
46. Declining perpetuities and annuities You own an oil pipeline that will generate a $2 million cash return over the coming year. The pipeline’s operating costs are negligible, and it is expected to last for a very long time. Unfortunately, the volume of oil shipped is declining, and cash flows are expected to decline by 4% per year. The discount rate is 10%.
a. What is the PV of the pipeline’s cash flows if its cash flows are assumed to last forever?
b. What is the PV of the cash flows if the pipeline is scrapped after 20 years?
FINANCE ON THE WEB
Finance.yahoo.com is a marvelous source of stock price data. You should get used to using it.
1. Go to finance.yahoo.com and look up “Analyst Estimates” for Apple (AAPL). You should find earnings per share (EPS) for the current year, the percentage annual growth rate of EPS for the past five years, and also a five-year EPS growth-rate forecast. What will be Apple’s EPS after five years if EPS grows at the five-year historical average rate? What will EPS be if it grows at the analysts’ forecasted rate? Try the same exercise for other stocks, for example Microsoft (MSFT), Merck (MRK), or the railroad CSX (CSX).
2. You need to have accumulated savings of $2 million by the time that you retire in 20 years. You currently have savings of $200,000. How much do you need to save each year to meet your goal if your savings earn a return of 10%? Find the savings calculator on www.msn.com/en-us/money/tools/retirementplanner to check your answer.
1You sometimes hear lay people refer to “net present value” when they mean “present value,” and vice versa. Just remember, present value is the value of the investment today; net present value is the addition that the investment makes to your wealth.
2We define “expected” more carefully in Chapter 9. For now think of expected payoff as a realistic forecast, neither optimistic nor pessimistic. Forecasts of expected payoffs are correct on average.
3You might check for yourself that these are equivalent rules. In other words, if the return of $100,000/$700,000 is greater than r, then the net present value –$700,000 + [$800,000/(1 + r)] must be greater than 0.
4You can check this by writing down the present value formula
Now let C/(1 + r) = a and 1/(1 + r) = x. Then we have (1) PV = a(1 + x + x2 + • • • ). Multiplying both sides by x, we have (2) PVx = a(x + x2 + • • •). Subtracting (2) from (1) gives us PV(1 − x) = a. Therefore, substituting for a and x,
Multiplying both sides by (1 + r) and rearranging gives
5Some people find the following equivalent formula more intuitive:
6We need to calculate the sum of an infinite geometric series PV = a(1 + x + x2 + • • •) where a = C1/(1 + r) and x = (1 + g)/(1 + r). In footnote 4 we showed that the sum of such a series is a/(1 − x). Substituting for a and x in this formula,
7We can derive the formula for a growing annuity by taking advantage of our earlier trick of finding the difference between the values of two perpetuities. Imagine three investments (A, B, and C) that make the following dollar payments:
Investments A and B are growing perpetuities; A makes its first payment of $1 in year 1, while B makes its first payment of $(1 + g)3 in year 4. C is a three-year growing annuity; its cash flows are equal to the difference between the cash flows of A and B. You know how to value growing perpetuities such as A and B. So you should be able to derive the formula for the value of growing annuities such as C:
So
If r = g, then the formula blows up. In that case, the cash flows grow at the same rate as the amount by which they are discounted. Therefore, each cash flow has a present value of C/(1 + r) and the total present value of the annuity equals t × C/(1 + r). If r < g, then this particular formula remains valid, though still treacherous.
8In the U.S., truth-in-lending laws oblige the company to quote an APR that is calculated by multiplying the payment each period by the number of payments in the year. APRs are calculated differently in other countries. For example, in the European Union, APRs must be expressed as annually compounded rates, so consumers know the effective interest rate that they are paying.
9When we talk about continuous payments, we are pretending that money can be dispensed in a continuous stream like water out of a faucet. One can never quite do this. For example, instead of paying out $1 billion every year to combat malaria, you could pay out about $1 million every 8 3/4 hours or $10,000 every 5 1/4 minutes or $10 every 3 1/6 seconds but you could not pay it out continuously. Financial managers pretend that payments are continuous rather than hourly, daily, or weekly because (1) it simplifies the calculations and (2) it gives a very close approximation to the NPV of frequent payments.
10Remember that an annuity is simply the difference between a perpetuity received today and a perpetuity received in year t. A continuous stream of C dollars a year in perpetuity is worth C/r, where r is the continuously compounded rate. Our annuity, then, is worth
Since r is the continuously compounded rate, C/r received in year t is worth (C/r) × (1/e rt) today. Our annuity formula is therefore
sometimes written as
11 See J. Zweig, “Is Da Vinci’s Salvator Mundi Worth $450 Million or $454,680?” The Wall Street Journal, November 16, 2017.
