6.3MultifactorModels

The one-factor market model provides a simple description of stock returns, but unfor-

tunately, it is unrealistic. For example, when stocks are sensitive to interest rate risk as

well as to market risk and firm-specific risks, the interest rate risk generates correla-

tion between the market model residuals, implying that more than one common factor

generates stock returns.

The Multifactor Model Equation

The algebraic representation of a multifactormodel, that is, a factor model with more

than one common factor, is given in equation (6.4) below

˜ F˜F˜. . .F˜˜
r	(6.4)
iii11i22iKKi

The assumption behind equation (6.4) is that securities returns are generated by a

relatively small number of common factors, each factor symbolized by a subscripted F,

for which different stocks have different sensitivities, or s, along with uncorrelated firm-

specific components, the s, which contribute negligible variance in well-diversified

portfolios.

Interpreting Common Factors

The Fs in equation (6.4) can be thought of as proxies for newinformation about macro-

economic variables, such as industrial production, inflation, interest rates, oil prices,

and stock price volatility. Because Fs represent new information, they are generally

scaled to have means of zero, which also has the convenient benefit of allowing the s

to be interpreted as the mean (or expected) returns of securities.

Investors and other market participants obtain information about these macro-

economic factors from a variety of sources, including economic announcements, such

as the employment report announced one Friday each month by the U.S. Labor Depart-

ment. Other important regularly scheduled announcements are the merchandise trade

deficit, the producer price index, money supply growth, and the weekly jobless claims

report. Irregular news events like the testimony or press releases of important policy-

making officials or actions taken by central banks also can provide information about

important macroeconomic forces.

Sometimes a seemingly nonfinancial event such as the Iraqi invasion of Kuwait

will affect these fundamental forces. When Saddam Hussein’s army invaded Kuwait in

1990, oil prices shot up while bond and stock prices fell dramatically. This drop con-

tinued for several weeks as it became more apparent each day that Iraqi troops were

not going to be dislodged from Kuwait quickly or easily.

Several factor model “interpretations” explain why stock prices fell at the time

of Kuwait’s invasion. One interpretation is that higher expected import prices for oil

may have lowered corporate earnings because oil prices are an important cost of

production. Alternatively, these political events may have changed investor expec-

tations about the inflation factor or altered the risk of most securities because of an

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184Part IIValuing Financial Assets

increase in uncertainty about future inflation, interest rates, and the gross domestic

product (GDP), a measure of the value of an economy’s production of goods and

services.

In sum, a common factor is an economic variable (or portfolio return that acts as

a proxy for an economic variable) that has a significant effect on the returns of broad

market indexes rather than individual securities alone. The common factors affect

returns by changing discount rates or earnings prospects, or both.

6.4	Estimating the Factors There are three ways to estimate the common factors in a factor model.

1.Use a statistical procedure, such as factor analysis,to determine factor

portfolios, which are portfolios of securities designed to mimic the factors.

2.Use macroeconomic variables like interest rate changes and changes in

economic activity as proxies for the factors.

3.Use firm characteristics, like firm size, to form portfolios that act as proxies

for the factors.

Exhibit 6.4 illustrates the advantages and disadvantages of the various methods. We

now elaborate on each of these factor-estimation techniques individually.

Using Factor Analysis to Generate Factor Portfolios

Factoranalysisis a statistical estimation technique based on the idea that the covari-

ances between stock returns provide information that can be used to determine the com-

mon factors that generate the returns. Factor structures determine the covariance

between stock returns (see Section 6.6). Factor analysis starts with the covariances and

discovers which factor structure best explains it.¹²

With only two stocks, it is possible to construct a single factor and a pair of fac-

tor betas that explains their covariance perfectly. However, when there are a large num-

ber of stocks, the best factor analysis estimates can only approximately explain the

covariances, measured from historical returns, between all pairs of stocks. The factors

and factor betas generated by factor analysis are those that provide the best possible

explanation of the covariances estimated from historical stock returns.

The advantage of forming factors based on factor analysis is that the factors

selected are those that best explain the covariances between all stocks. In theory, if

covariances do not change over time the factors derived from factor analysis are exactly

the factors desired. However, the downside to factor analysis is that it sheds little insight

into which economic variables the factors are linked. Corporate managers would like

to be able to relate the riskiness of their firm to factors like interest rate changes and

exchange rate changes, but it is difficult to “name” the factors with the purely statisti-

cal factor analysis approach. The second problem is that the technique assumes that the

return covariances do not change over time. As a result, the technique is severely dis-

advantaged in its ability to explain, for example, why the return variability of a stock

increases following a large decline in its stock price.

¹²The

application of this procedure to stock returns was pioneered in Roll and Ross (1980).

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Chapter 6

Factor Models and the Arbitrage Pricing Theory

185

EXHIBIT6.4Summary of the Three Different Ways to Construct Factors

Estimation Method

Advantages

Disadvantages

FactorAnalysis
Apurely statistical procedure	Provides the best	The assumption that
for estimating factors and	estimates of the	covariances are constant is
the sensitivity of returns to	factors given its	crucial and is probably
them	assumptions	violated in reality; does not
		“name” the factors

Macroeconomic Variables
Uses macroeconomic time-	Provides the most	Implies that the appropriate
series that capture changes	intuitive interpretation	factors are the unanticipated
in productivity, interest	of the factors	changes in the macrovariables.
rates, and inflation to act as		Unanticipated changes in
proxies for the factors		variables such as aggregate
generating stock returns		productivity and inflation
		may be difficult to measure
		in practice

Firm Characteristics
Uses firm characteristics	such	More intuitive than the	Portfolios selected on the basis
as firm size, which are		factor analysis	of past return anomalies,
known to be related to		portfolios; formation	which are only factors
stock returns, to form		does not require	because they explain
factor portfolios		constant covariances	historical “accidents,” may
			not be good at explaining
			expected returns in the
			future

Using Macroeconomic Variables to Generate Factors

Asecond approach for estimating the factors and factor betas is to use macroeconomic

variables as proxies for the common factors. This approach takes a large set of macro-

economic variables such as changes in unemployment, inflation, interest rates, oil

prices, and so forth. It then limits the number of factors to, say, five, and then exam-

ines which five of a larger set of macroeconomic variables best explains the observed

pattern of stock returns.

An initial attempt to identify which common factors have the greatest influence on

U.S. stock prices was undertaken in empirical research by Chen, Roll, and Ross (1986)

and Chan, Chen, and Hsieh (1985). These authors found that the following five factors

best explain the correlations between stock returns:

1.Changes in the monthly growth rate of the gross domestic product (GDP),

which alters investor expectations about future industrial production and

corporate earnings.

2.Changes in the default risk premium, measured by the spread between the

yields of AAAand Baa bonds of similar maturity. As this spread widens,

investors become more concerned about default.

3.Changes in the spread between the yields of long-term and short-term

government bonds; that is, the average slope of the term structure of interest

rates as measured by the yields on U.S. Treasury notes and bonds. This would

affect the discount rates for obtaining present values of future cash flows.

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186Part IIValuing Financial Assets

4.Unexpected changes in the price level, as measured by the difference between

actual and expected inflation. Unexpectedly high or low inflation alters the

values of most contracts. These include contracts with suppliers and

distributors, and financial contracts such as a firm’s debt instruments.

5.Changes in expected inflation, as measured by changes in the short-term T-bill

yield. Changes in expected inflation affect government policy, consumer

confidence, and interest rate levels.

One of the main advantages of using the macroeconomic variables approach is that

it names the factors. As a result, corporate managers who want economic intuition about

the sources of risk that are most likely to affect their cost of capital tend to prefer this

approach.¹³

On the other hand, it may be difficult to measure the unexpectedchanges in the

macroeconomic variables, which are needed to act as proxies for the factors. As a result,

some variables with semipredictable movements, like oil prices, may not show up as

factors with this procedure when they really are factors.¹⁴

Another disadvantage is that

some potentially important factors may be extremely difficult to quantify. For exam-

ple, political changes, like the fall of the former Soviet Union, can have a potentially

enormous effect on stock returns. However, constructing an index that reflects changes

in the world’s political environment is difficult.¹⁵

Using Characteristic-Sorted Portfolios to Estimate the Factors

Factors also can be estimated by using portfolios formed on the basis of the charac-

teristics of firms. In theory, these characteristics should be selected on the basis of what

common characteristics make stocks move up and down together on a daily basis. Thus,

if growth firms tend to have similar returns and value firms have similar returns, then

the return of a value portfolio or a growth portfolio, or some combination of the two,

should be a factor. However, in practice, the growth versus value characteristic is

selected not because of the degree to which it makes groups of stocks move up or down

together, but because it is associated with high and low average stock returns. The

rationale behind using the characteristic-based proxies for the factors that are tied to

average returns is due to the APT’s link between risk premiums and factor sensitivi-

ties, as described later in this chapter. If the risk premium (expected return less the risk-

free rate) associated with a characteristic, such as size, represents compensation for a

specific kind of factor risk, then portfolios consisting of stocks grouped on the basis of

that characteristic are likely to be highly sensitive to that kind of factor risk.

Given that covariances may change over time, portfolios formed in this way may

provide better proxies for these common factors than portfolios formed with factor

analysis. They also have the advantage of using the returns of financial securities which,

being largely unpredictable, give this method an advantage over the macroeconomic

variables approach in measuring the unexpected changes the factors are supposed to

capture. On the other hand, if there is no link between the return premiums of stocks

and factor sensitivities, then this method is not picking up true factors that explain

covariances, but simply picking out portfolios consisting of stocks that the financial

markets appear to be mispricing.

13

See Chapter 11.

¹⁴A

remedy that was not available at the time of these studies is to use futures prices that can serve

as a proxy for common factors like unexpected changes in oil prices.

¹⁵See

Erb, Harvey, and Viskanta (1995) for a discussion of political risk and stock returns.

Grinblatt392Titman: Financial	II. Valuing Financial Assets	6. Factor Models and the	© The McGraw392Hill
Markets and Corporate		Arbitrage Pricing Theory	Companies, 2002
Strategy, Second Edition

Chapter 6

Factor Models and the Arbitrage Pricing Theory

187