The Security Market Line (SML)

Figure 28-3

Required Rate SML: r i = rRF + (rM “ rRF) bi

of Return (%) = 6% + (11% “ 6%) bi

= 6% + (5%) bi

rHigh = 16

Relatively

Market Risk Risky Stock™s

rM = rA = 11

Premium: 5%. Risk Premium: 10%

Applies Also to

an Average Stock,

rLow = 8.5 and Is the Slope

Safe Stock™s

Coefficient in the

Risk

SML Equation

Premium: 2.5%

rRF= 6

Risk-Free

Rate, rRF

0 0.5 1.0 1.5 2.0 Risk, b i

28-18 Basic Financial Tools: A Review

Chapter 28

combinations of the market portfolio and the risk-free asset. It shows the

best available set of portfolios, based on risk and return, available to

investors. The SML, on the other hand, shows the relationship between the

required rate of return on individual stocks and their market risk as mea-

sured by beta.14

The Characteristic Line: Calculating Betas

Before we can use the SML to estimate a stock™s required rate of return, we

need to estimate the stock™s beta coef¬cient. Recall that beta is a measure of

how the stock tends to move with the market. Therefore, we can use the his-

torical relationship between the stock™s return and the market™s return to cal-

culate beta. First, note the following de¬nitions:

“ ˆ

rJ historical (realized) rate of return on Stock J. (Recall that rJ and rJ

are de¬ned as Stock J™s expected and required returns, respectively.)

“

rM historical (realized) rate of return on the market.

aJ vertical axis intercept term for Stock J.

bJ slope, or beta coef¬cient, for Stock J.

eJ random error, re¬‚ecting the difference between the actual return on

Stock J in a given year and the return as predicted by the regression

line. This error arises because of unique conditions that affect Stock

J but not most other stocks during a particular year.

The points on Figure 28-4 show the historical returns for Stock J plotted

against historical market returns. The returns themselves are shown in the

bottom half of the ¬gure. The slope of the regression line that best ¬ts these

points measures the overall sensitivity of Stock J™s return to the market

return, and it is the beta estimate for Stock J. The equation for the regres-

sion line can be obtained by ordinary least squares analysis, using either a

calculator with statistical functions or a computer with a regression software

package such as a spreadsheet™s regression function. In his 1964 article

which ¬rst described the CAPM, Sharpe called this regression line the stock™s

characteristic line. Thus, a stock™s beta is the slope of its characteristic line.

Figure 28-4 shows that the regression equation for Stock J is as follows:

rJ 8.9 1.6 r M eJ. (28-17)

This equation gives the predicted future return on Stock J, given the mar-

ket™s performance in a future year.15 So, if the market™s return happens to be

20 percent next year, the regression equation predicts that Stock J™s return

will be 8.9 1.6(20) 23.1%.

We can also use Equation 28-16 to determine the required rate of return

on Stock J, given the required rate of return on the market and the risk-free

rate as shown on the Security Market Line:

rJ rRF 1.6(rM rRF).

14

Required rates of return are equal to expected rates of return as seen by the marginal investor if markets are ef¬cient

and in equilibrium. However, investors may disagree about the investment potential and risk of assets, in which case the

required rate of return may differ from the expected rate of return as seen by an individual investor.

15

This assumes no change in either the risk-free rate or the market risk premium.

28-19

Risk and Return

Calculating Beta Coefficients

Figure 28-4

Realized Returns

_

on Stock J, r J (%)

Year 1

40 Year 5

30 _ _

r J = a J + b J r M + eJ

_

= “ 8.9 + 1.6r M + e J

20

Year 3

10

Year 4

7.1

“10 0 10 20 30 Realized Returns

_

on the Market, r M (%)

a J = Intercept = “ 8.9% _

∆ rJ = 8.9% + 7.1% = 16%

“10

_

∆r

_ Rise 16

∆ r M = 10% = _J =

bJ = = 1.6

∆r M

Run 10

“ 20

Year 2

STOCK J ( “J ) MARKET ( “M )

YEAR r r

1 38.6% 23.8%

2 (24.7) (7.2)

3 12.3 6.6

4 8.2 20.5

5 40.1 30.6

“

Average r 14.9% 14.9%

σr 26.5% 15.1%

“

Thus, if the risk-free rate is 8 percent and the required rate of return on the

market is 13 percent, then the required rate of return on Stock J is 8%

1.6(13% 8%) 16%.

Market versus Diversifiable Risk

Equation 28-17 can also be used to show how total, diversi¬able, and

market risk are related. The total risk for Stock J, 2, can be broken down

J

into market risk and diversi¬able risk:

Total risk Variance Market risk Diversifiable risk. (28-18)

2

b2 2 2

eJ.

J JM

28-20 Basic Financial Tools: A Review

Chapter 28

2

Here J is Stock J™s variance (or total risk), bJ is the stock™s beta coef¬cient,

2 2

M is the variance of the market, and eJ is the variance of Stock J™s regres-

sion error term. If all of the points in Figure 28-4 plotted exactly on the

regression line, then J would have zero diversi¬able risk and the variance of

2

the error term, eJ, would be zero. However, in our example all of the points

do not plot exactly on the regression line, and the decomposition of total

risk for J is thus

0.2652

Total risk Market risk Diversifiable risk

terms, the market risk is 20.0584 24.2%, and the diversi¬able risk is

(1.6)2(0.151)2 2

0.0702 eJ

20.0118 10.9%.

2

0.0702 0.0584 eJ.

2 2

Solving for eJ gives diversi¬able risk of eJ 0.0118. In standard deviation

Portfolio Betas

The beta of a portfolio can be calculated as the weighted average of the betas

of the individual assets in the portfolio. This is in sharp contrast to the dif-

¬cult calculation required in Equation 28-13 for ¬nding the standard devia-

tion of a portfolio. To illustrate the calculation, suppose an analyst has

determined the following information for a four-stock portfolio:

Stock Weight Beta Product

(1) (2) (3) (4) (2) (3)

I 0.40 0.6 0.24

J 0.20 1.0 0.20

K 0.30 1.3 0.39

L 0.10 2.1 0.21

1.00 Portfolio beta: 1.04

The beta is calculated as the sum of the product terms in the table. We could

also calculate it using this equation:

bp 0.4(0.6) 0.2(1.0) 0.3(1.3) 0.1(2.1) 1.04.

Here the ¬rst value in each term is the stock™s weight in the portfolio, the

second term is the stock™s beta, and the weighted average is the portfolio™s

beta, bp 1.04.

Now assume that the risk-free rate is 8 percent, and the required rate of

return on the market portfolio is 12 percent. The portfolio™s required return

can be found using Equation 28-16:

rp 8% 1.04(12% 8%) 12.16%.

How are the expected return and standard deviation calculated from a proba-

Self-Test Questions

bility distribution?

When is the coefficient of variation a useful measure of risk?

What is the difference between diversifiable risk and market risk?

Explain the following statement: “An asset held as part of a portfolio is gener-

ally less risky than the same asset held in isolation.”

What does it mean to say that beta is the theoretically correct measure of a

stock™s riskiness?

What is the difference between the CML and the SML?

How would you calculate a beta?

What is optimal about an “optimal portfolio”?

28-21

Stock Valuation

Stock Valuation

The techniques used earlier in this chapter to value bonds can also be used,

with a few modi¬cations, to value stocks. First, note that the cash ¬‚ows

from stocks that companies provide to investors are dividends rather than

coupon payments, and there is no maturity date on stocks. Moreover, divi-

dend payments are not contractual, and they typically are expected to grow

over time, not to remain constant as bond interest payments do. Here is the

general equation used to value common stocks:

Dt

D1 D2 Dn

a (1

Price PV . (28-19)