6.11Estimating FactorRisk Premiums and FactorBetas

Factor risk premiums are needed to implement the APT. Typically, these have been

estimated from the average historical risk premiums of factor portfolios, that is, their

historical average returns in excess of the risk-free return. Since historical averages

are not true means, considerable statistical error is associated with the factor risk

premiums.²³

To estimate factor betas when factors are prespecified, either as macroeconomic

factors or as portfolios tied to firm characteristics, one must use a regression of the his-

torical returns of the security against the historical factor realizations. The slope coef-

ficients in this regression are the factor betas. With a factor analysis implementation of

a factor model, both the factor portfolios and the factor betas are generated directly as

outputs of the factor analysis procedure. However, the resulting factor betas in this case

are still identical to multiple regression slope coefficients from a regression of histori-

cal returns against historical factor realizations.

6.12Empirical Tests of the Arbitrage Pricing Theory

As Chapter 5 noted, some researchers have suggested that market prices do not reflect

fundamental long-term values because characteristics like size, market-to-book, and

momentum better explain average stock returns than the CAPM beta. As a result,

investors can realize superior performance by buying the stocks of small capitalization

companies with low market-to-book ratios which have performed well over the past 3

to 12 months. If this were true, it would have important implications for the way cor-

porations make financing decisions as well as for how investors select their portfolios.

In an irrational market, for example, corporations may be able to lower their costs of

capital by timing their share issues to correspond to periods when the shares are over-

priced.

However, not all researchers share this view of financial markets. Others have

argued that the return premiums associated with these characteristics arise because

stocks with these characteristics are exposed to systematic risk factors. Since this risk

is not reflected in the CAPM betas of the stocks, a multifactor model like the APTis

needed to explain the returns.

²³We

believe that just as beta estimates can be improved with a statistical adjustment, so can factor

risk premiums. The insights of the last section, which combine the CAPM with the APT, imply that the

risk premiums of the factor portfolios should be their CAPM betas times the risk premium of the market

portfolio. To the extent that investors have some confidence in the CAPM risk-expected return relation,

they should make some comparison between the risk premiums for the factors that would exist if the

CAPM were true and the risk premiums estimated by averaging historical data. For those factors with

historical risk premiums that deviate substantially from their CAPM-predicted risk premiums, it is

possible to improve the estimated risk premium by taking a weighted average of the historical risk

premium and the CAPM-predicted risk premium. The weighting would depend on the relative confidence

one has in the historical estimate over the CAPM estimate. To the extent that the factor has been

extremely volatile, one has less confidence in the historical estimate of the factor risk premium. To the

extent that the CAPM predictions for all risk premiums seem to bear little relation to the historical

averages, one has less confidence in the CAPM estimates of the factor risk premiums.

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Chapter 6

Factor Models and the Arbitrage Pricing Theory

207

Some researchers have argued that low market-to-book stocks are more sensitive

to swings in the business cycle and changes in credit conditions because the compa-

nies are more likely to have their financing cut off during an economic downturn, and

that the added sensitivity to these economic factors is not reflected in their covariation

with the S&P500. In other words, returns are generated by multiple factors, as

described by the APT, and small capitalization and low market-to-book stocks may have

high betas on factors underrepresented by the S&P500.

Unfortunately, the empirical literature on the multifactor APTis not as well devel-

oped as the empirical literature on the CAPM, and the results are less conclusive. As

a consequence, the debate about whether these effects are driven by psychological

behavior or by the sensitivity of stocks to risk factors that researchers have ignored is

not yet resolved. With this caveat in mind, the remainder of this section explores what

is known about the APT.

Empirical Implications of the APT

Tests of the APTexamine the following three implications:

1.The expected return of any portfolio with factor betas that are all equal to

zero is the risk-free rate.

2.The expected returns of securities increase linearly with increases in a given

factor beta.

3.No other characteristics of stocks, other than factor betas, determine expected

returns.

Evidence from Factor Analysis Studies

Roll and Ross (1980) published one of the first APTtests using factor analysis. Because

of the computational limitations of standard statistical factor analysis programs, they

were forced to estimate factors on small numbers of stocks. In a test of 42 groups of

30 securities each, they found that in 88.1 percent of the groups there was at least one

factor with a nonzero risk premium; in 57.1 percent of the groups, at least two with

nonzero risk premiums; and in about one-third of the groups, at least three factors with

nonzero risk premiums. Roll and Ross concluded that at least three factors are impor-

tant for the APT’s risk-expected return relation, but probably no more than four are

important.

Other papers used procedures that allow researchers to generate factor portfo-

lios from much larger data sets. These include works by Chen (1983), Connor and

Korajczyk (1988), and Lehmann and Modest (1988), which were all particularly

interested in whether the factors explain the size effect.²⁴

Chen claimed that his fac-

tors explain the size effect: After controlling for differences in factor sensitivities

between large and small firms, the return premium for size becomes negligible.

However, Lehmann and Modest argued that there is still a size effect, even after

controlling for these differences.

Evidence from Studies with Macroeconomic Factors

Chen, Roll, and Ross (1986), who analyzed a number of macroeconomic factors, found

that the factors representing the growth rate in GDP, the yield spread between long-

²⁴The

market-to-book ratio and the momentum effect had not attracted much academic attention at the

time these papers were written.

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208Part IIValuing Financial Assets

and short-term government bonds and changes in default spreads had significant effects

on risk premiums. The two inflation factors had a weaker but still significant effect.

They also performed tests using consumption and oil prices as factors and found that

neither affected the expected returns of common stocks. In addition, when added to the

regression, the return of the market index could not explain expected returns.

Chan, Chen, and Hsieh (1985) examined the size effect in the context of the Chen,

Roll, and Ross model. They created 20 size-ranked portfolios and estimated the factor

sensitivities of each portfolio to the five Chen, Roll, and Ross factors as well as the

equal-weighted NYSE portfolio. They found that the difference in residuals between the

portfolio of smallest firms and that of the largest is positive, but not statistically signif-

icant. They also conducted a test using the logarithm of firm size as an independent vari-

able and found its coefficient to be insignificantly different from zero in the multifactor

model. The authors concluded that the multifactor model explains the size anomaly.²⁵

Jagannathan and Wang (1996) use some of the Chen, Roll, and Ross macro fac-

tors to predict time series changes in the risk premiums associated with the factors and

add an additional macro factor, aggregate labor income, to explain the average stock

returns. Anumber of interesting observations come from the Chan, Chen, and Hsieh

(1985) and Jagannathan and Wang (1996) papers, which provide some insights about

the small firm effect:

•	Small company stock returns appear to be highly correlated with changes inthe spread between Baa and default-free bonds.
••	The spread seems to be a fairly good predictor of future market returns.Small companies have higher market betas when the spread is higher.
•	Small company stock returns seem to covary more with per capita laborincome than do the returns of large company stocks.

The first and last points imply that it is possible, at least in part, to explain the

small-firm effect using a standard APT-type model that identifies the default spread and

labor income as systematic factors associated with positive risk premiums. The middle

two observations suggest that small firms, in essence, successfully “time the market.”

In other words, small firms have higher betas when the market risk premium is high-

est. The explanation of the small firm effect that comes from these studies is that small

cap stocks have higher returns than large cap stocks because they are riskier in two

aspects: (1) Their returns are more sensitive to short-term business cycle and credit

movements that seem to be captured by the spread between high- and low-grade bonds

and changes in aggregate labor income; (2) small cap stocks are especially sensitive to

movements in the overall market when the market is the most risky. This means that the

CAPM beta of the stock of a typical small company underestimates the stock’s true risk.

Evidence from Studies That Use Firm Characteristics

Fama and French (1993) suggested a three-factor model composed of the following

three zero-cost (that is, self-financing) portfolios.

•	Along position in the value-weighted index portfolio and a short position in T-bills—the difference between the realized return of the value-weightedmarket index and the return of Treasury bills.

²⁵Fama

and French (1993) dispute this conclusion. They argue that the Chen, Roll, and Ross

macroeconomic factors cannot explain the firm size effect.

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Chapter 6

Factor Models and the Arbitrage Pricing Theory

209

•	Along position in a portfolio of low market-to-book stocks and a shortposition in high market-to-book stocks.
•	Along position in a portfolio of small capitalization stocks and a short positionin a portfolio of large capitalization stocks.

Fama and French (1993, 1996a) asserted that the three factors explain most of the risk

premiums of stocks, including those that cannot be accounted for by the CAPM. A

notable exception is the momentum effect.²⁶