ñòð. 83 |

r, = the expected (or required) return on security j

where

rf = the risk-free security (such as a T-bill)

r,,, = the expected return on the market portfolio (such as Standard & Poorâ€™s 500 Stock

Composite Index or Dow Jones 30 Industrials)

b = beta, an index of nondiversifiable (noncontrollable, systematic) risk

The key component in the CAPM, beta (b), is a measure of the securityâ€™s volatility relative to that

of an average security. For example: b = 0.5 means the security is only half as volatile, or risky, as the

average security; b = 1.0 means the security is of average risk; and b = 2.0 means the security is twice

as risky as the average risk.

The whole term b(rm- r f ) represents the risk premium, the additional return required to

compensate investors for assuming a given level of risk.

Thus, in words, the CAPM (or SML) equation shows that the required (expected) rate of return on

a given security (rj) is equal to the return required for securities that have no risk ( r f )plus a risk premium

required by investors for assuming a given level of risk. The higher the degree of systematic risk (b),

the higher the return onâ€™a given security demanded by investors. Figure 7.1 graphically illustrates the

CAPM as the security market line.

177

CHAP. 7

1 RISK, RETURN, AND VALUATION

Security market line

I,!,,

0% I

I

0 0.5 2.0

1.5

1.o

Beta

Fig, 7-1 CAPM as security market line

EXAMPLE 7.9 Assuming that the risk-free rate is 8 percent, and the expected return for the market (r,J is

(I/)

12 percent, then if

+O(12Yo - 8%) = 8%

= 8%

b = 0 (risk-free security) rj

rj = 8% + 0.5(12% - 8%) = 10%

b = 0.5

ri = 8% + 1.0(12% - 8%) = 12%

b = 1.0 (market portfolio)

rj = 8% + 2.0(12% - 8%) = 16%

b = 2.0

The Arbitrage Pricing Model (APM)

The CAPM assumes that required rates of return depend only on one risk factor, the stockâ€™s beta.

The Arbitrage Pricing Model (APM)disputes this and includes any number of risk factors:

r = rf + bl RPl + b2RP2 + +b, RP,,

0

r = the expected return for a given stock or portfolio

where

rf = the risk-free rate

bi = the sensitivity (or reaction) of the returns of the stock to unexpected changes in economic

forces i (i = 1, . . .,n)

RPi = the market risk premium associated with an unexpected change in the ith economic

force

n = the number of relevant economic forces

Roll and Ross suggest the following five economic forces:

1. Changes in expected inflation

2. Unanticipated changes in inflation

Unanticipated changes in industrial production

3.

Unanticipated changes in the yield differential between low- and high-grade bonds (the default-

4.

risk premium)

5. Unanticipated changes in the yield differential between long-term and short-term bonds (the

term structure of interest rates)

[CHAP 7

RISK, RETURN, AND VALUATION

178

7.3 BOND AND STOCK VALUATION

The process of determining security valuation involves finding the present value of an asset's

expected future cash flows using the investor's required rate of return. Thus, the basic security valuation

model can be defined mathematically as follows:

V=C- c r

"

r=l (1+ r)'

V = intrinsic value or present value of an asset

where

C, = expected future cash flows in period f = 1,. . ., n

r = investor's required rate of return

Bond Valuation

The valuation process for a bond requires a knowledge of three basic elements: (1)the amount of

the cash flows to be received by the investor, which is equal to the periodic interests to be received and

the par value to be paid at maturity; (2) the maturity date of the loan; and (3) the investor's required

rate of return.

Incidentally, the periodic interest can be received annually or semiannually. The value of a bond is

simply the present value of these cash flows. Two versions of the bond valuation model are presented

below:

If the interest payments are made annually, then

M

" I

+ - I(PV1FA.n) + M(PVIFr,,)

V=C- =

(1 + r)' (1+ r)"

r=l

I = interest payment each year = coupon interest rate X par value

where

M = par value, or maturity value, typically $1,000

I = investor's required rate of return

n = number of years to maturity

PVIF'A = present value interest factor of an annuity of $1 (which can be found in Appendix D)

PVIF = present value interest factor of $1(which can be found in Appendix C )

EXAMPLE 7.11 Consider a bond, maturing in 10 years and having a coupon rate of 8 percent. The par value is

$1,000.Investors consider 10percent to be an appropriate required rate of return in view of the risk level associated

with this bond. The annual interest payment is $80(8% X $l,OOO). The present value of this bond is:

=$80(6.1446) + $1,000(0.3855) = $491.57 + $385.50 = $877.07

If the interest is paid Semiannually, then

2n

M

I/2 I

-(PVIFA,D,h) + M( PVIF,D,h)

V=C-

+

ñòð. 83 |