r)2

1 r (1

(1 t 1

Here the Dt terms represent the dividend expected in each year t, and r is the

stock™s required rate of return as determined in the preceding section. There

is no maturity date, so the present value must be for all expected dividend

payments extending out forever.

Equation 28-19 for stocks differs from Equation 28-6 for bonds in that

Dt represents an expected but uncertain, and nonconstant, dividend rather

than a ¬xed, known coupon or principal payment, and also because the

summation goes out to in¬nity. The required rate of return on the stock, r,

could either be determined by the CAPM or just estimated subjectively by

the investor.

Dividend Growth Model

Rather than explicitly projecting dividends forever, under certain assump-

tions we can simplify the process. In particular, if our best guess is that the

¬rm™s earnings and dividends will grow at a constant rate g on into the fore-

seeable future, then we can use the dividend growth model. For example,

suppose a company has just paid a dividend of $1.50, and its dividends are

projected to grow at a rate of 6 percent per year. The expected dividend at

the end of the current year will be $1.50(1.06) $1.59. The dividend

2

expected during the second year will be $1.50(1.06) $1.6854, and the

dividend expected during the t™th year will be $1.50(1.06)t.

g)t in Equation 28-19, in which case we

We can replace Dt with D0(1

have a power series that can be solved to give the following formula:16

D0(1 g) D1

ˆ (28-20)

Price now P0 .

rg r g

This equation is called the constant growth model, or the Gordon model

after Myron J. Gordon, who did much to develop and popularize it. In our

example, D0 $1.50 and g 0.06. If the required rate of return on the

stock is 13 percent, then the present value of all expected future dividends

will be 1.50(1.06)/(0.13 0.06) $22.71, and this is the value of the stock

according to the model.

16

See Chapter 5 for the derivation of Equation 28-20.

28-22 Basic Financial Tools: A Review

Chapter 28

Equation 28-20 also provides an alternative to the CAPM for estimating

the required rate of return on a stock. First, we transform Equation 28-20

to form Equation 28-21:

ˆ

rs D1/P0 g. (28-21)

ˆ

Here we write the rate of return variable as rs rather than r to indicate that

it is an expected as opposed to a required rate of return. The equation indi-

cates that the expected rate of return on a stock whose dividend is growing

at a constant rate, g, is the sum of its expected dividend yield and its con-

stant growth rate. In equilibrium, this is also the required rate of return.

Both the current price and the most recent dividend can be readily deter-

mined, and analysts make and publish growth rate estimates. Thus, given the

ˆ

input data, we can solve for rs. Here is the situation for our illustrative stock:

rs D1/P0 g $1.50(1.06)/$22.71 0.06 0.13 13%.

ˆ

In Chapter 10 we discuss the types of ¬rms for which this analysis is

appropriate. We also provide other simpli¬ed models in the next two sec-

tions for use in situations where the assumption of constant growth is not

appropriate.

Perpetuities

A second simpli¬ed version of the general stock valuation model as

expressed in Equation 28-19 can be used to value perpetuities, which are

securities that are expected to pay a constant amount each period forever.

Preferred stock is an example of a perpetuity. Many preferred stocks have no

maturity date, and they pay a constant dividend. In this case g 0 in

Equation 28-20, and the expression reduces to P D/r. A share of $3 pre-

ferred stock (i.e., the stock pays $3 every year) with a required rate of return

of 9 percent will sell for $3.00/0.09 $33.33. The perpetuity formula also

makes it easy to ¬nd the expected yield on a security, given the dividend and

ˆ

the current price: rs D/P. So, if a share of preferred stock that pays an

annual dividend of $4 per share trades for $65 per share, the stock™s yield is

$4/$65 6.15%.

Nonconstant Growth Model

Often it is inappropriate to assume that a company™s dividends will grow at

a constant rate forever. Most ¬rms start out small and pay no dividends dur-

ing their initial growth phase. Later, when they are able, they start paying a

small dividend, and they then increase this dividend relatively rapidly, as

earnings continue to grow. Finally, as the ¬rm begins to mature, its dividend

growth rate declines to the overall growth rate of the company or the indus-

try. Thus, the growth rate is initially zero, then it becomes positive and large,

and ¬nally it declines and approaches a constant rate. In this situation, the

dividend growth model given in Equation 28-20 is not appropriate, and an

attempt to use it would give inaccurate and misleading results.

The best way to deal with a changing dividend growth rate is to use the

nonconstant growth model. This involves several steps:

Estimate each dividend during the nonconstant growth period.

•

Determine the present value of each of the nonconstant growth dividends.

•

28-23

Stock Valuation

Use the constant growth model to ¬nd the discounted value of all of the

•

dividends expected once the nonconstant growth period has ended, which

is the expected stock price at the end of the nonconstant growth period.

Find the present value of the expected future price.

•

Sum the PVs of the nonconstant dividends and the PV of the expected

•

future price to ¬nd the value of the stock today.

For example, suppose the required rate of return on a stock is 12 percent.

A dividend of $0.25 has just been paid, and you expect the dividend to dou-

ble each year for four years. After Year 4, the ¬rm will reach a steady state

and have a dividend growth rate of 8 percent per year forever. The follow-

ing time line shows the cash ¬‚ows:

0 1 2 3 4 5

...∞

Growth rate 100% 100% 100% 100% 8% 8%

Dividend 0.50 1.00 2.00 4.00 4.32

To begin, let™s ¬nd the price at the end of the fourth year, at which point we

will have a constant growth stock. Someone buying the stock will receive D5

and the subsequent dividends, which will presumably grow at a constant

ˆ

rate of 8 percent forever. Equation 28-20 can be used to ¬nd P4:

D5

ˆ

Price at end of Year 4 P4 . (28-20a)

r g

This means that the constant growth rate model can be used as of Year 4,

and the stock price expected in that year is found as follows:

D5 4.32

ˆ

P4 $108. (28-20b)

r g 0.12 0.08

This gives us the following time line that stops at the end of Year 4:

0 1 2 3 4

Dividend 0.50 1.00 2.00 4.00

Stock price 108.00

Total cash ¬‚ow 0.50 1.00 2.00 112.00

The present value of these four cash ¬‚ows at the 12 percent required rate of

return is

ˆ

P0 Present value of cash flows

0.50 1.00 2.00 112.00

$73.85.

1.122 1.123 1.124

1.12

Thus, the stock should trade for $73.85 today, and it is expected to rise to

$108 four years from now.

For some ¬rms, especially startups and other small companies, it is more

reasonable to expect the company to be acquired by a larger company than

to continue forever as an independent operation. In such cases, we can mod-

ify the nonconstant growth model by using a terminal value based on the

expected acquisition price rather than a price based on the constant growth

28-24 Basic Financial Tools: A Review

Chapter 28

model. We discuss procedures for valuing such companies in Chapters 10

and 25.

Is a higher percentage of a stock™s value based on this year™s earnings and div-

Self-Test Questions

idends or on a forecast of long-term earnings and dividends?

Write out and explain the valuation model for a constant growth stock and for

a perpetuity.

How do you value a stock that is not expected to grow at a constant rate?

How can you use the constant growth model to find the required rate of return

on a stock?

Summary

The goal of this chapter was to review the fundamental tools of (1) time value of

money, (2) risk and return, and (3) valuation models for stocks and bonds. The

key concepts covered are listed below.

PV(1 i)n PV(FVIFi,n).

• The future value of a single payment is FVn

• The future value of an annuity is

...

i)n 1

i)n 2

Future value FVAn PMT(1 PMT(1 PMT(1 i) PMT

i)n

PMT c d

(1 1

i

PMT(FVIFAi,n).

i)n.

The present value of a single payment is PV FVn/(1

•

The present value of an annuity is

•

PMT PMT PMT PMT PMT

Present value PVAn ...

(1 i)n

(1 i)2 (1 i)3 (1 i)n 1

(1 i)

1

1

i)n

PMT £ §

(1

i

PMT(PVIFAi,n).

The effective annual rate for m compounding intervals per year at a nominal

•

inom m

a1 b

rate of inom per year is EAR (or EFF%) 1.0.

m

The effective annual rate for continuous compounding at a nominal rate of inom

•

per year is EARcontinuous einom 1.0.

If you know the cash ¬‚ows and the PV (or FV) of a cash ¬‚ow stream, you can

•

determine the interest rate using a ¬nancial calculator to solve for the interest rate.

The general valuation equation for a series of cash ¬‚ows is

•

n

CFt

CF1 CF2 CFn

...

a (1 i)t .

PV

(1 i)n

2

1i (1 i) t 1

The price of a bond is the present value of its coupon and principal payments:

•

N

Coupon Par

a

Price of bond .

rd)t (1 rd)N

t 1 (1

28-25

Questions

The riskiness of an asset™s cash ¬‚ows can be considered either on a stand-alone

•

basis, with each asset thought of as being held in isolation, or in a portfolio con-

text, where the investment is combined with other assets and its risk is reduced