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 nogrowth 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 nogrowth 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 annuitydue form of cash flows)? It certainly would, because
you now receive one extra payment today.

To solve this problem, you already know the lumpsum 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 ConstantGrowth 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 dividendpaying 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 constantgrowth 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 constantgrowth 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 PVP_{g} 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 constantgrowth 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 constantgrowth 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 constantgrowth 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
