Introduction to Corporate Finance

PART 1: INTRODUCTION

3-6e FINDING THE PRESENT VALUE OF A GROWING PERPETUITY

By definition, perpetuities pay a constant periodic amount forever. However, few aspects of modern life

are constant, and most of the cash flows we care about have a tendency to grow over time. This is true

for items of income such as wages and salaries, many dividend payments from corporations, and pension

payments from the Australian government.^9 Inflation is only one factor that drives increasing cash flows.

Because of this tendency for cash flows to grow over time, we must determine how to adjust the present

value of a perpetuity formula to account for expected growth in future cash flows.

Suppose we want to calculate the present value (PV) of a stream of cash flows growing forever

(n = ∞) at rate g. Given a discount rate of r, the present value of the growing perpetuity is given by the

following equation, which is sometimes called the Gordon growth model:^10

Eq. 3.11 PV

CF

rg

=>^1 rg

−

Note that the numerator in Equation 3.11 is CF 1 , the first year’s cash flow that occurs exactly one

year from today. This cash flow is expected to grow at a constant annual rate (g) from now to the end of

time. We can determine the cash flow for any specific future year (t) by applying the growth rate (g) as

follows:

CFt = CF^1 × (1 + g)t – 1

9 Unfortunately, this is also true for expense items such as rent and utility expenses, car prices and tuition payments.

10 For this formula to work, the discount rate must be greater than the growth rate. When cash flows grow at a rate equal to or greater than the

discount rate, the present value of the stream is infinite.

Imagine that two years and 364 days have gone by,

and Big Mining Pty Ltd has announced that it will pay

the one-time $12 dividend tomorrow. After that, the

company expects the annual $4 preferred dividends

to resume. What is the present value now of the

expected dividend stream?

As before, we can break the dividend stream into

two parts. First, there is the $12 dividend, which will

be paid immediately. Clearly, its present value is $12.

Second, there is the perpetual $4 dividend stream

that starts in one year. From Equation 3.10, we know

that the value of this perpetuity is $50 ($4 ÷ 0.08).

Just add these two components together to find the

value of BM’s shares today:

Value of each preferred share = $12 + $50 = $62

Now consider the position of a BM preferred

shareholder who purchased the shares three

years ago, when the company suspended its

dividends. In the previous example we determined

that the value of BM preferred share at that time

was $49.22, and we have just discovered that its

value today is $62. What rate of return did the

investor earn over this period? Again, we can apply

Equation 3.1:

FV PV r

r

=+n

=+

()

$$.()

1

62 49221 3

We solved a problem like this in section 3-4. To

find the answer, you can use Excel’s ‘rate’ function,

use a financial calculator, or solve algebraically as

follows:

r=









−= =

$

$.

.%

62

4922

1008 8

1

3

An investor who purchased BM shares for $49.22

three years ago and held them until they recently

reached $62 would have earned 8% per year, exactly

the required return. The return comes entirely

from price appreciation in the shares, because no

dividends were paid during this period.

example

growing perpetuity

A cash flow stream that grows

each period at a constant rate

and continues forever

Gordon growth model

The valuation model that

views cash flows as a growing

perpetuity

Some companies (such as

IBM) have issued bonds that

are perpetuities. What sort

of information do you think

the companies have to tell

investors in the market about the

perpetuities to convince them to

buy them?

thinking cap
question