A perpetuity is an annuity whose payments go on forever—an infinite stream of equal cash flows received at regular intervals over time. A constant growth perpetuity also has payments that never end, but the payments increase at a constant rate over time. Although true perpetuities are rare, some cash flow streams can be treated as a constant growth perpetuity, such as the cash dividend payment stream of dividend-paying companies. A cash dividend payment stream of a dividend-paying company is sometimes treated as a constant growth perpetuity if the payment of dividends is constant and reliable, and if the amount paid in the form of a dividend generally grows over time. Can you think of other examples of perpetuities?

Present Value of No-Growth Perpetuities

Because a perpetuity is a stream of cash flows that continues forever, you will not be able to find its future value after the last cash flow has occurred; by definition, this will never happen. The future value of a perpetuity at some intermediate future date is the normal future value of an annuity (FVA) for cash flows up to that date.

Consider how you would go about computing the present value of an unending stream of constant payments (i.e., a no-growth perpetuity). You could use the formula for present value of an annuity for larger and larger values of N, the number of periods, until the PVA stopped increasing. As you might recall from the Future Value topic in "Time Value of Money," the marginal increases in payoff diminish as the compounding frequency increases. Since discounting is the opposite of compounding, the same holds true. The marginal decrease in the present value of cash flows will diminish as payments extend further and further in the future.

Perpetuities View animation

You can test this theory mathematically, using the formula for the present value of an ordinary annuity.

In the PVA formula, C is the payment amount; r is the periodic discount rate; and N is the total number of annuity payments. What happens as N goes to infinity? Examine the term (1 + r)^–N. Remember that

As N gets larger, the term (1 + r)^–N gets smaller. As N approaches infinity, the term (1 + r)^-N approaches zero. Subsititute zero for the term (1 + r)^-N in the PVA formula to get the present value of a perpetuity (PVP).

This equation can be simplified.

Generally speaking, the present value of a perpetuity is simply the perpetuity's cash flow at the end of period 1, C, divided by the periodic discount rate, r. The formula for the PV of a perpetuity (= C ÷ r) implies that the first payment occurs one period from today.

Example
What would you be willing to pay for an infinite stream of $37 annual payments (cash inflows) beginning one year from today if the interest rate is 8 percent?

You are asked to value a time line that goes on forever with equally spaced cash flows of $37 occurring at the end of each year. Although this problem seems like it will take a lot of work, it is actually quite easy. Use the formula for the present value of a no-growth perpetuity.

	C = the constant payment amount of $37 r = the annual interest rate of 8 percent

You may construct a time line to help you visualize the cash flow stream.

Substitute the given values into the formula for the present value of a no-growth perpetuity, and solve.

So you would pay $462.50 today in exchange for an infinite stream of $37 annual cash inflows. Notice that the $37 annual payment is the interest earned on the $462.50 principal over one year: $462.50 x .08 = $37. At the end of each year you receive $37 as a cash inflow. Unlike finite investments, you never get the principal investment back as a cash inflow.

More often than not, a perpetuity assumes the first payment occurs at the end of the first period. Unless otherwise specified, you should assume this. However, the first payment can occur immediately.

Present Value of a Perpetuity Due

Would your answer change if the first $37 payment occurred today (i.e., an annuity-due form of cash flows)? It certainly would, because you now receive one extra payment today.

To solve this problem, you already know the lump-sum present value of the $37 payments occurring at periods 1 through , so you have only to add to this amount the additional $37 payment already received at 0.

Construct a time line.

Substitute the given values into the formula for the present value of a perpetuity, and add one extra payment.

If you received an additional $37 payment today, you would pay $499.50. Here again, you are simply stripping away an 8 percent annual interest payment of $37 each period, starting today.

Present Value of a Constant-Growth Perpetuity

Company stocks are commonly treated as growth perpetuities. One method used to compute an appropriate value an investor is willing to pay for a company's stock uses the dividend payment stream, which represents real cash flow to the investor and usually does not fluctuate as much as earnings. In addition, investors may assign a growth rate to the dividends, based on an expectation of growth of the company's dividend payments. Why is this necessary?

This method of valuing stock (often called the dividend discount model) was extremely popular a number of years ago, when the majority of companies used dividends as a primary method to return value to shareholders. It remains a popular method used to value traditional dividend-paying stocks issued by banks, utilities, and industrial companies.

The formula for the present value of a perpetuity assumes the periodic payment remains constant. But dividends can grow over time with the growth of a company. What is an example of this?

If you assume a constant growth rate for dividends, then the formula can be changed to

Here, C is the amount of the first payment at the end of period 1, r is the periodic discount rate, and g is the periodic constant rate of growth. The rate r must be greater than the rate g. Why?

Example
What would you pay for a share of stock, given a required annual rate of return of 13 percent compounded quarterly and the following information? The next quarterly dividend will be $1.67, and it is the company's policy to increase the dividend by 3 percent each quarter.

Recognize that this is a constant-growth perpetuity problem, because the dividends continue forever and grow at a constant rate of 3 percent. You can use this equation to decide how much you would pay for the stock. What is C? Since this a constant-growth perpetuity, the dividends will increase over time, and C is the first dividend to occur in the growth stream. You expect that the next dividend will be $1.67 and that each subsequent dividend will grow 3 percent, so C is $1.67. What does the stream of dividends look like? and so on until you get to

What would these cash flows look like on a time line? Set up a complete time line.

The values in the PVPg formula are first payment, C = $1.67, the growth rate g = 3 percent, or .03, and the discount rate r = 13 percent ÷ 4, or .0325. Use these values in the formula for the present value of a constant-growth perpetuity.

Clearly, a stream of payments that gets larger every quarter is worth more than a stream of constant payments. You are willing to pay $668.00 for this constant-growth perpetuity. How much would you pay for it if it did not grow?

How many decimal places should you use?

1. You are thinking of buying some stock in a company listed on the New York Stock Exchange (NYSE). Before you buy any stock, you should compute a price based on the dividends you expect a stock to pay in the future. This company has paid a $1.15 dividend every quarter for the past 12 years, and you expect this trend to continue to infinity. What do you believe the price of this stock should be if you require a 13 percent quarterly compounded return?

Solution 1

2. Since graduating from college, you have grown rich. As a gesture of good will (with a tax benefit), you have decided to endow your alma mater with a research grant in finance. The endowment calls for the grant to be a constant-growth amount paid once a year in perpetuity. The first payment will be $10,000 and is to be made in exactly one year, with subsequent payments growing at a rate of 4.5 percent annually. If you are able to secure a 9 percent annual rate of return on the endowment fund, how much should you put into the endowment fund today?

Solution 2

3. A thriving multinational magazine publisher wants to issue some new equity to its stockholders. You have been hired to price its stock based on its current dividend policy and its stockholders' required rate of return. The company has told you that over the last three years, it paid stock dividends of $1.13, $1.16, and $1.19, respectively. The publisher has also assured you that this growth rate in its dividends will continue indefinitely. If the company will pay its next dividend in exactly one year and believes that its stockholders require a 12.85 percent rate of return, what should be the price of its stock? Use an average of the growth rate of dividend payments over the last three years and the formula for the present value of a constant growth dividend to compute the stock price.

Solution 3

4. A major automobile manufacturer wants to ascertain its investors' required rate of return based on the fact that its stock is currently trading for $45. If the company pays a constant dividend of $2.27 semiannually, what is the investors' implied required rate of return, assuming that this situation occurs to infinity?

Solution 4

5. Assume a company has total annual revenues of $100 million, and pays $2 million in dividends. Its total market capitalization is $200 million. What is the implied growth rate if the market normally requires a 15 percent return?

Solution 5

6. Using the definition of an annuity, add a periodic growth factor, g, after the first period's cash flow, take the number of periods to infinity, and solve for the formula of a constant growth perpetuity.
Hint: Use (1 + g)⁰ on the first period's cash flow.

Solution 6