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=

(I + r / 2 p 2

(1+ d2)'

EXAMPLE 7.12 Assume the same data as in Example 7.11, except the interest is paid semiannually.

179

RISK, RETURN, AND VALUATION

CHAP. 71

Common Stock Valuation

Like bonds, the value of a common stock is the present value of all future cash inflows expected to

be received by the investor. The cash inflows expected to be received are dividends and the future price

at the time of the sale of the stock. For an investor holding a common stock for only 1year, the value

of the stock would be the present value of both the expected cash dividend to be received in 1year ( D l )

and the expected market price per share of the stock at year-end ( P I ) .If r represents an investor's

required rate of return, the value of common stock ( P O ) ould be:

w

D1 +- p 1

P =-

O

(1+ r)l ( 1 + r)l

EXAMPLE 7.13 Assume an investor is considering the purchase of stock A at the beginning of the year. The

dividend at year-end is expected to be $1.50, and the market price by the end of the year is expected to be $40. If

the investor's required rate of return is 15 percent, the value of the stock would be:

P

I

D1 $1.50 $40

P =- +-= = $1.50(0.870) + $40(0.870) = $1.31 + $34.80 = $36.11

O

+

(1 + r)' (1 + r)' (1 + 0.15) (1+ 0.15)

Since common stock has no maturity date and is held for many years, a more general, multiperiod

model is needed. The general common stock valuation model is defined as follows:

D,

OD

r=l

There are three cases of growth in dividends. They are (1)zero growth; (2) constant growth; and

(3) nonconstant, or supernormal, growth.

In the case of zero growth, if

DO= D1 = . = D,

S .

then the valuation model

reduces to the formula:

O D1

P =-

r

EXAMPLE 7.14 Assuming D equals $2.50 and r equals 10 percent, then the value of the stock is:

P O = $2.50

-= $25

0.1

In the case of constant growth, if we assume that dividends grow at a constant rate of g every year

[i.e., D,= D o ( l + g)'], then the above model is simplified to:

This formula is known as the Gordon growth model.

EXAMPLE 7.15 Consider a common stock that paid a $3 dividend per share at the end of the last year and is

expected to pay a cash dividend every year at a growth rate of 10 percent. Assume the investor's required rate of

180 RISK, RETURN, AND VALUATION [CHAP. 7

return is 12 percent. The value of the stock would be:

D, = Do(l + g ) = $3(1 + 0.10) = $3.30

$3.30

PO=-=

D1

$165

=

0.12 - 0.10

r -g

Finally, consider the case of nonconstant, or supernormal, growth. Firms typically go through life

cycles, during part of which their growth is faster than that of the economy and then falls sharply. The

value of stock during such supernormal growth can be found by taking the following steps: (1)Compute

the dividends during the period of supernormal growth and find their present value; (2) find the price

of the stock at the end of the supernormal growth period and compute its present value; and (3) add

these two PV figures to find the value (PO)f the common stock.

o

EXAMPLE 7.16 Consider a common stock whose dividends are expected to grow at a 25 percent rate for 2 years,

after which the growth rate is expected to fall to 5 percent. The dividend paid last period was $2. The investor desires

a 12 percent return. To find the value of this stock, take the following steps:

1. Compute the dividends during the supernormal growth period and find their present value. Assuming Do

is $2, g is 15 percent, and r is 12 percent:

D1 = Do(1 + g ) = $2(1 + 0.25) = $2.50

= Do(1 +g)â€™ = $2(1.563) = $3.125

02

= DI(1 + g ) =

or $2.50(1.25) = $3.125

02

D1 D2 - $2.50 $3.125

PV of dividends = - - + (1 + r)2 - (1 + 0.12) (1 + 0.12)2

+

(1 + r)l

+

= $2.50(PVIF12%,1) $3.125(PVIF12%,2)

= $2.50(0.8929) + $3.125(0.7972) = $2.23 + $2.49 = $4.72

2. Find the price of the stock at the end of the supernormal growth period. The dividend for the third

year is:

D3 = D2(1+ gâ€™), where gâ€™ = 5%

= $3.125(1 + 0.05) = $3.28

The price of the stock is therefore:

â€˜3â€˜28

p2=-= 3 = $46.86

0

r - gâ€™ 0.12 - 0.05

PV of stock price = $46.86(PVIF12yo,2) $46.86(0.7972) = $37.36

=

3. Add the two PV figures obtained in steps 1 and 2 to find the value of the stock.

PO = $4.72 + $37.36 = $42.08

Expected Rate of Return on a Bond: Yield to Maturity

The expected rate of return on a bond, better known as the bondâ€™s yield to maturity, is computed

by solving the following equation (the bond valuation model) for r:

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