lio managers is in the sophisticated security analysis that underlies their choices. The rule of

GIGO (garbage in“garbage out) applies fully to portfolio selection. If the quality of the secu-

rity analysis is poor, a passive portfolio such as a market index fund can yield better results

than an active portfolio tilted toward seemingly favorable securities.

>

5. Two portfolio managers work for competing investment management houses. Each

Concept

employs security analysts to prepare input data for the construction of the optimal

CHECK portfolio. When all is completed, the efficient frontier obtained by manager A dom-

inates that of manager B in that A™s optimal risky portfolio lies northwest of B™s.

Is the more attractive efficient frontier asserted by manager A evidence that she

really employs better security analysts?

6.5 A SINGLE-FACTOR ASSET MARKET

factor model

Statistical model to We started this chapter with the distinction between systematic and firm-specific risk. Sys-

measure the firm-

tematic risk is largely macroeconomic, affecting all securities, while firm-specific risk factors

specific versus

affect only one particular firm or, perhaps, its industry. Factor models are statistical models

systematic risk of a

designed to estimate these two components of risk for a particular security or portfolio. The

stock™s rate of return.

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6 Efficient Diversification

first to use a factor model to explain the benefits of diversification was another Nobel Prize

winner, William S. Sharpe (1963). We will introduce his major work (the capital asset pricing

model) in the next chapter.

The popularity of factor models is due to their practicality. To construct the efficient fron-

tier from a universe of 100 securities, we would need to estimate 100 expected returns, 100

variances, and 100 99/2 4,950 covariances. And a universe of 100 securities is actually

quite small. A universe of 1,000 securities would require estimates of 1,000 999/2

499,500 covariances, as well as 1,000 expected returns and variances. We will see shortly that

the assumption that one common factor is responsible for all the covariability of stock returns,

with all other variability due to firm-specific factors, dramatically simplifies the analysis.

Let us use Ri to denote the excess return on a security, that is, the rate of return in excess

of the risk-free rate: Ri ri rf. Then we can express the distinction between macroeconomic excess return

and firm-specific factors by decomposing this excess return in some holding period into three Rate of return

components in excess of the

risk-free rate.

Ri E(Ri ) iM ei (6.5)

In Equation 6.5, E(Ri) is the expected excess holding-period return (HPR) at the start of the

holding period. The next two terms reflect the impact of two sources of uncertainty. M quan-

tifies the market or macroeconomic surprises (with zero meaning that there is “no surprise”)

during the holding period. i is the sensitivity of the security to the macroeconomic factor.

Finally, ei is the impact of unanticipated firm-specific events.

Both M and ei have zero expected values because each represents the impact of unantici-

pated events, which by definition must average out to zero. The beta ( i) denotes the respon-

siveness of security i to macroeconomic events; this sensitivity will be different for different beta

securities. The sensitivity of a

As an example of a factor model, suppose that the excess return on Dell stock is expected security™s returns to

to be 9% in the coming holding period. However, on average, for every unanticipated increase the systematic or

market factor.

of 1% in the vitality of the general economy, which we take as the macroeconomic factor M,

Dell™s stock return will be enhanced by 1.2%. Dell™s is therefore 1.2. Finally, Dell is affected

by firm-specific surprises as well. Therefore, we can write the realized excess return on Dell

stock as follows

RD 9% 1.2M ei

If the economy outperforms expectations by 2%, then we would revise upward our expecta-

tions of Dell™s excess return by 1.2 2%, or 2.4%, resulting in a new expected excess return

of 11.4%. Finally, the effects of Dell™s firm-specific news during the holding period must be

added to arrive at the actual holding-period return on Dell stock.

Equation 6.5 describes a factor model for stock returns. This is a simplification of reality;

a more realistic decomposition of security returns would require more than one factor in

Equation 6.5. We treat this issue in the next chapter, but for now, let us examine the single-

factor case.

Specification of a Single-Index Model of Security Returns

A factor model description of security returns is of little use if we cannot specify a way to

measure the factor that we say affects security returns. One reasonable approach is to use the

rate of return on a broad index of securities, such as the S&P 500, as a proxy for the common

macro factor. With this assumption, we can use the excess return on the market index, RM , to

measure the direction of macro shocks in any period.

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The index model separates the realized rate of return on a security into macro (systematic)

index model

and micro (firm-specific) components much like Equation 6.5. The excess rate of return on

A model of stock

each security is the sum of three components:

returns using a

market index such

as the S&P 500 to

Symbol

represent common

or systematic

1. The stock™s excess return if the market factor is neutral, that is, if the market™s

risk factors.

excess return is zero. i

2. The component of return due to movements in the overall market (as represented

by the index RM); i is the security™s responsiveness to the market. iRM

3. The component attributable to unexpected events that are relevant only to this

ei

security (firm-specific).

The excess return on the stock now can be stated as

Ri i RM ei (6.6)

i

Equation 6.6 specifies two sources of security risk: market or systematic risk ( i RM),

attributable to the security™s sensitivity (as measured by beta) to movements in the overall

market, and firm-specific risk (ei ), which is the part of uncertainty independent of the market

factor. Because the firm-specific component of the firm™s return is uncorrelated with the mar-

ket return, we can write the variance of the excess return of the stock as4

Variance (Ri ) Variance ( i RM ei )

i

Variance ( i RM) Variance (ei )

22 2

(ei )

iM

Systematic risk Firm-specific risk (6.7)

Therefore, the total variability of the rate of return of each security depends on two

components:

1. The variance attributable to the uncertainty common to the entire market. This systematic

risk is attributable to the uncertainty in RM . Notice that the systematic risk of each

2

stock depends on both the volatility in RM (that is, M) and the sensitivity of the stock

to fluctuations in RM . That sensitivity is measured by i .

2. The variance attributable to firm-specific risk factors, the effects of which are measured

by ei . This is the variance in the part of the stock™s return that is independent of market

performance.

This single-index model is convenient. It relates security returns to a market index that in-

vestors follow. Moreover, as we soon shall see, its usefulness goes beyond mere convenience.

Statistical and Graphical Representation

of the Single-Index Model

Equation 6.6, Ri i RM ei , may be interpreted as a single-variable regression equa-

i

tion of Ri on the market excess return RM . The excess return on the security (Ri ) is the

dependent variable that is to be explained by the regression. On the right-hand side of

the equation are the intercept i; the regression (or slope) coefficient beta, i , multiplying the

independent (or explanatory) variable RM; and the security residual (unexplained) return, ei .

4

Notice that because is a constant, it has no bearing on the variance of Ri .

i

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6 Efficient Diversification

We can plot this regression relationship as in Figure 6.11, which shows a possible scatter

diagram for Dell Computer Corporation™s excess return against the excess return of the

market index.

The horizontal axis of the scatter diagram measures the explanatory variable, here the mar-

ket excess return, RM. The vertical axis measures the dependent variable, here Dell™s excess re-

turn, RD. Each point on the scatter diagram represents a sample pair of returns (RM, RD) that

might be observed for a particular holding period. Point T, for instance, describes a holding pe-

riod when the excess return was 17% on the market index and 27% on Dell.

Regression analysis lets us use the sample of historical returns to estimate a relationship be-

tween the dependent variable and the explanatory variable. The regression line in Figure 6.11

is drawn so as to minimize the sum of all the squared deviations around it. Hence, we say

the regression line “best fits” the data in the scatter diagram. The line is called the security

characteristic line, or SCL. security

The regression intercept ( D) is measured from the origin to the intersection of the regres- characteristic line

sion line with the vertical axis. Any point on the vertical axis represents zero market excess re- Plot of a security™s

turn, so the intercept gives us the expected excess return on Dell during the sample period excess return as a

when market performance was neutral. The intercept in Figure 6.11 is about 4.5%. function of the excess

return of the market.

The slope of the regression line can be measured by dividing the rise of the line by its run.

It also is expressed by the number multiplying the explanatory variable, which is called the re-

gression coefficient or the slope coefficient or simply the beta. The regression beta is a natu-

ral measure of systematic risk since it measures the typical response of the security return to

market fluctuations.

The regression line does not represent the actual returns: that is, the points on the scatter

diagram almost never lie on the regression line, although the actual returns are used to cal-

culate the regression coefficients. Rather, the line represents average tendencies; it shows the

effect of the index return on our expectation of RD. The algebraic representation of the regres-

sion line is

E(RD RM) D RM (6.8)

D

which reads: The expectation of RD given a value of RM equals the intercept plus the slope

coefficient times the given value of RM .

F I G U R E 6.11

Dell™s excess return (%)

Scatter diagram

RD

30 for Dell

T

20

10

±D

RM

10 20 30 40

Market excess return (%)

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Because the regression line represents expectations, and because these expectations may

not be realized in any or all of the actual returns (as the scatter diagram shows), the actual

security returns also include a residual, the firm-specific surprise, ei. This surprise (at point T,

for example) is measured by the vertical distance between the point of the scatter diagram and

the regression line. For example, the expected return on Dell, given a market return of 17%,

would have been 4.5% 1.4 17% 28.3%. The actual return was only 27%, so point T

falls below the regression line by 1.3%.

Equation 6.7 shows that the greater the beta of the security, that is, the greater the slope of

the regression, the greater the security™s systematic risk ( 2 M), as well as its total variance

2

D

2

( D). The average security has a slope coefficient (beta) of 1.0: Because the market is com-

posed of all securities, the typical response to a market movement must be one for one. An

“aggressive” investment will have a beta higher than 1.0; that is, the security has above-

average market risk.5 In Figure 6.11, Dell™s beta is 1.4. Conversely, securities with betas lower

than 1.0 are called defensive.

A security may have a negative beta. Its regression line will then slope downward, mean-

ing that, for more favorable macro events (higher RM), we would expect a lower return, and

vice versa. The latter means that when the macro economy goes bad (negative RM) and secu-

rities with positive beta are expected to have negative excess returns, the negative-beta se-

curity will shine. The result is that a negative-beta security has negative systematic risk, that

is, it provides a hedge against systematic risk.

The dispersion of the scatter of actual returns about the regression line is determined by the

residual variance 2(eD), which measures the effects of firm-specific events. The magnitude

of firm-specific risk varies across securities. One way to measure the relative importance of

systematic risk is to measure the ratio of systematic variance to total variance.

Systematic (or explained) variance

2

Total variance

2 2 2 2

D M D M

(6.9)

2 2 2 2