6.1The Market Model:The First FactorModel

The simplest possible factor model is a one-factormodel, which is a factor model with

only one common factor. It is often convenient to think of this one factor as the mar-

ket factor and to refer to the model as the market model. Intuition for the CAPM is

often based on the properties of the market model. However, as this section shows, the

CAPM is not necessarily linked to the market model; thus, this intuition for the CAPM

is often wrong.

The Market Model Regression

To understand the market model, consider the regression used to estimate market betas

in Chapter 5. There we estimated beta as the slope coefficient in a regression of the

return of Dell’s stock on the return of the S&P500 and pictured the regression as the

line of best fit for the points in Exhibit 5.7 on page 160. The algebraic expression for

the regression is simply equation (5.6), on page 158, applied specifically to Dell

˜ R˜˜	(6.1)
r
DELLDELLDELLS&PDELL

With quarterly data from 1990 through 1999, the estimates are

regression intercept .18

DELL

regression slope coefficient (Dell’s market beta) 1.56

DELL

˜ regression residual, which is constructed to have a mean of zero

DELL

By the properties of regression, ˜and R˜

DELLS&Pare uncorrelated.

Ignoring the constant, , equation (6.1) decomposes the uncertain return of

DELL

Dell into two components:

•	Acomponent that can be explained by movements in the market factor. This component is the product of the beta and the S&Preturn.

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178	Part IIValuing Financial Assets

•	Acomponent that is not the result of market movements, the regressionresidual, ˜. DELL

The Market Model Variance Decomposition

Because ˜and R˜

are uncorrelated, and because is a constant that does

DELLS&PDELL

not affect variances, the variance of the return on Dell stock can be broken down into

a corresponding set of two terms:⁴

2	˜˜)var(˜)
	var(R
DELL	DELLS&PDELL
	2˜)var(˜)	(6.2)
	var(R
	DELLS&PDELL

AGlossary of Risk Terms.The first term on the right-hand side of equation (6.2),

2˜

var(R), is referred to variously as Dell’s “systematic,” “market,” or “nondi-

DELLS&P

versifiable” risk. The remaining term, var(˜), is referred to as its “unsystematic,”

DELL

“nonmarket,” or “diversifiable” risk.⁵

We prefer to use systematicand unsystematic risk

when referring to these terms; referring to these terms as diversifiable and nondiversi-

fiable is misleading in most instances, as this chapter will show shortly. The following

definitions are more precise.

1.The systematic (market) riskof a security is the portion of the security’s

return variance that is explained by market movements. The unsystematic

(nonmarket) riskis the portion of return variance that cannot be explained

by market movements.

2.Diversifiable riskis virtually eliminated by holding portfolios with small

weights on every security (lest investors put most of their eggs in one basket).

Since the weights have to sum to 1, this means that such portfolios, known as

well-diversified portfolios, contain large numbers of securities.

Nondiversifiable riskcannot generally be eliminated, even approximately, in

portfolios with small weights on large numbers of securities.

Regression R-squared and Variance Decomposition.Acommonly used statistic from

the regression in equation (6.1), known as the R-squared,⁶

measures the fraction of the

return variance due to systematic risk. First, generalize the regression in equation (6.1) to

an arbitrary stock (stock i) and an arbitrary market index with return ˜.This yields

R

M

˜˜˜	(6.3)
r R
iiiMi

Exhibits 6.1 and 6.2 graph data points for two such regressions: one for a company

with mostly systematic risk (high R-squared in Exhibit 6.1), the other for a company

with mostly unsystematic risk (low R-squared in Exhibit 6.2). The horizontal axis in

both exhibits describes the value of the regression’s independent variable, which is the

⁴This is based on the portfolio variance formula, equation (4.9a) on page 108 in Chapter 4, with a

covariance term of zero.

⁵Other terms that are synonymous with “diversifiable risk” are “unique risk” and “firm-specific risk.”

We will elaborate on the latter term and diversification shortly.

⁶In

the case of Dell, one measures R-squared as the ratio of the first term on the right-hand side of

equation (6.2) to the sum of the two terms on the right-hand side. This ratio is a number between 0 and 1.

In addition to the interpretation given here, one often refers to R-squared as a measure of how close the

regression fits historical data. R-squared is also the square of the correlation coefficient between ˜and ˜

rR

M.

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	Chapter 6Factor Models and the Arbitrage Pricing Theory	179
EXHIBIT6.1	High R-Squared Regression
	Stock i’s return

Market return

EXHIBIT6.2Low R-Squared Regression

Stock i’s return

Market return

market return. The vertical axis describes the regression’s dependent variable, which is

the company’s stock return.

Diversifiable Risk and Fallacious CAPM Intuition

The intuition commonly provided for the CAPM risk-expected return relation is that

systematic risk is nondiversifiable. Thus, investors must be compensated with higher

expected rates of return for bearing such risk.⁷In contrast, one often hears unsystem-

atic risk referred to as being “diversifiable,” implying that additional expected returns

are not required for bearing unsystematic risk. Although this intuition is appealing, it

⁷Note that the market model regression, which uses realized returns, differs from the CAPM, which

uses mean returns. If the CAPM holds, (1 )rin equation (6.3). Exercise 6.9 asks you to prove

iif

this.

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180Part IIValuing Financial Assets

is somewhat misleading because, as shown below, some of the risk generated by the

market model residual is not necessarily diversifiable.

For example, risk from the residual in General Motors’market model regression is

not diversifiable because it is likely to pick up common factors to which GM is especially

sensitive. For example, an unanticipated increase in interest rates is likely to have a neg-

ative effect on most stocks. Interest rate risk is nondiversifiable because it is not elimi-

nated by holding well-diversified portfolios. Instead, interest rate risk is a common factor.

GM’s stock price is clearly affected by interest rate risk. New car sales plummet

when buyers find the rates on automobile loans prohibitively expensive. Indeed, inter-

est rate increases are much more likely to affect the return on GM’s stock than the

return on the market portfolio. Where does the interest rate effect show up in equation

(6.3)? Clearly, some of the effect of the increase in interest rates will be reflected in

the systematic component of GM’s return—GM’s beta times the market return—but

this is not enough to explain the additional decline in GM’s stock price relative to the

market. The rest of the interest rate effect has to show up in GM’s regression residual.

Since the change in interest rates, clearly a nondiversifiable risk factor, affects the

market model regression residual, all of the risk associated with the residual, ˜,can-

i

not be viewed as diversifiable. While it is true that one can construct portfolios with

specific weights that eliminate interest rate risk (with methods developed in this chap-

ter),⁸mostportfolios with small portfolio weights on large numbers of securities do not

eliminate this source of risk.

Residual Correlation and Factor Models

If the market model is to be useful for categorizing diversifiable and nondiversifiable

risk, the market portfolio’s return must be the only source of correlation between dif-

ferent securities. As discussed above, this generally will not be true. However, if it is

true, it must be the case that the return of security i can formally be written as

˜˜˜,

r R

iiiMi

where

˜

Ris the return on the market portfolio

M

˜˜

and Rare uncorrelated

iM

the ˜s of different securities have means of zero and

i

the˜sofdifferentsecurities are uncorrelatedwith each other⁹

i

The fact that the ˜s of the different stocks are all uncorrelated with each other is the

i

key distinction between the one-factor market model expressed above and the more

general “return generating process”—equation (6.3) without the uncorrelated ˜assump-

tion—used in discussions of the CAPM.

This “one-factor model” has only one common factor, the market factor, generat-

ing returns. Each stock’s residual return, ˜, is determined independently of the com-

i

mon factors. Because these ˜s are uncorrelated, each ˜represents a change in firm

ii

value that is truly firm specific. As the next section shows, firm-specific components

of this type have virtually no effect on the variability of the returns of a well-balanced

8

Factor risk, in general, can be eliminated with judicious portfolio weight choices, as will be noted shortly.

⁹With a finite number of assets, some negligible but nonzero correlation must exist between residuals

in the market model because the market portfolio-weighted average of the residuals is identically zero.

We do not address this issue because the effect is trivially small.

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Chapter 6

Factor Models and the Arbitrage Pricing Theory

181

portfolio of a large number of securities. Hence, in the one-factor model, return vari-

ability due to firm-specific components, that is, firm-specific risk, is diversifiable.

Even though the interest rate discussion above suggests that a one-factor market

model is unlikely to hold in reality, studying this model helps to clarify the meaning

of diversifiable and nondiversifiable risk. After a brief discussion of the mathematics

and practical implementation of diversification, this chapter turns to more realistic mul-

tifactor models, built upon the intuition of diversifiable versus nondiversifiable risk.