The multifactor model that occupies center stage these days is the three-factor model introduced by Fama and French. 14 The systematic factors in the Fama-French model are firm size and book-to-market ratio as well as the market index. These additional factors are empirically motivated by the observations, documented in Chapter 11, that historical- average returns on stocks of small firms and on stocks with high ratios of book equity to market equity (B/M) are higher than predicted by the security market line of the CAPM.
These observations suggest that size or the book-to-market ratio may be proxies for expo- sures to sources of systematic risk not captured by the CAPM beta and thus result in the return premiums we see associated with these factors.
How can we make the Fama-French model operational? Fama and French propose mea- suring the size factor in each period as the differential return on small firms versus large firms. This factor is usually called SMB (for “small minus big”). Similarly, the other extra- market factor is typically measured as the return on firms with high book-to-market ratios minus that on firms with low ratios, or HML (for “high minus low”). Therefore, the Fama- French three-factor asset-pricing model is 15
E(ri)2rf5ai1bi3E(rM)2rf41siE3SMB41hiE3HML4 (13.8)
14 Eugene F. Fama and Kenneth R. French, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics 33 (1993), pp. 3–56.
15 You may wonder why we subtract the risk-free rate from the return on the market portfolio, but not from the SMB and HML returns. The reason is that the SMB and HML factors already are differences in returns between two assets. They are return premiums of one portfolio relative to another (small minus big or high minus low), just as the market risk premium is the excess return of the index relative to the risk-free asset.
The coefficients b i , s i , and h i are the betas of the stock on each of the three factors, often called the factor loadings. According to the arbitrage pricing model, if these are the rel- evant factors, excess returns should be fully explained by risk premiums due to these factor loadings. In other words, if these factors fully explain asset returns, the intercept of the equation should be zero.
To create portfolios that track the size and book-to-market factors, Davis, Fama, and French 16 sort industrial firms by size (market capitalization or market “cap”) and by book-to- market (B/M) ratio. Their size premium, SMB, is constructed as the difference in returns between the smallest and largest third of firms. Similarly, HML in each period is the dif- ference in returns between high and low book-to-market firms. They use a broad market index, the value-weighted return on all stocks traded on U.S. national exchanges (NYSE, AMEX, and NASDAQ) to compute the excess return on the market portfolio relative to the risk-free rate, taken to be the return on 1-month Treasury bills.
To test the three-factor model, Davis, Fama, and French form nine portfolios with a range of sensitivities to each factor. They construct the portfolios by sorting firms into three size groups (small, medium, and big; or S, M, B) and three book-to-market groups (high, medium, and low; or H, M, L). The nine portfolios thus formed are labeled in the following matrix; for example, the S/M portfolio is comprised of stocks in the smallest third of firms and the middle third of book-to-market ratio.
Size
Book-to-Market Ratio Small Medium Big
High S/H M/H B/H
Medium S/M M/M B/M
Low S/L M/L B/L
For each of these nine portfolios, Davis, Fama, and French estimate Equation 13.8 as a first-pass regression over the 816 months between 1929 and 1997 by using the regression model
ri2rf5ai1bi(rM2rf)1siSMB1hiHML1ei (13.9) Table 13.5 presents some of their results. Notice that the intercepts of the regressions (the estimates of a i for each portfolio) are in fact small and generally (except for the S/L portfolio) statistically insignificant, with t -statistics below 2. The large R -square statistics, all in excess of .91, show that returns are well explained by the three-factor portfolios, and the large t -statistics on the size and value loadings show that these factors contribute sig- nificantly to explanatory power.
How should we interpret these tests of the three-factor model and, more generally, the association of the Fama-French factors with average returns? One possibility is that size and relative value (as measured by the B/M ratio) proxy for risks not fully captured by the CAPM beta. This explanation is consistent with the APT in that it implies that size and value are priced risk factors. Another explanation attributes these premiums to some sort of investor irrationality or behavioral biases.
16 James L. Davis, Eugene F. Fama, and Kenneth R. French, “Characteristics, Covariances, and Average Returns, 1929 to 1997,” Journal of Finance 55, no. 1 (2000), pp. 389–406.
B/M Size
Excess
Return a b s h t ( a ) t ( b ) t ( s ) t ( h ) R 2 S/L 0.55 22.39 0.61 20.42 1.06 1.39 0.09 24.34 30.78 19.23 1.73 0.91 S/M 1.11 22.15 1.05 20.01 0.97 1.16 0.37 20.18 53.55 19.49 9.96 0.96 S/H 2.83 19.05 1.24 20.03 1.03 1.12 0.77 20.73 67.32 39.21 26.97 0.98 M/L 0.53 55.85 0.70 20.06 1.04 0.59 20.12 21.29 55.83 18.01 24.30 0.96 M/M 1.07 55.06 0.95 20.01 1.05 0.47 0.34 20.15 32.98 17.50 9.50 0.96 M/H 2.18 53.21 1.13 20.04 1.08 0.53 0.73 20.90 47.85 8.99 11.12 0.97 B/L 0.43 94.65 0.58 0.02 1.02 20.10 20.23 0.88 148.09 26.88 213.52 0.98 B/M 1.04 92.06 0.72 20.09 1.01 20.14 0.34 21.76 61.61 24.96 13.66 0.95 B/H 1.87 89.53 1.00 20.09 1.06 20.07 0.84 21.40 52.12 20.86 21.02 0.93 Table 13.5
Three-factor regressions for portfolios formed from sorts on size and book-to-market ratio (B/M)
Source: James L. Davis, Eugene F. Fama, and Kenneth R. French, “Characteristics, Covariances, and Average Returns, 1929 to 1997,”
Journal of Finance 55, no. 1 (2000), p. 396. Reprinted by the permission of the publisher, Blackwell Publishing, Inc.
Risk-Based Interpretations
Liew and Vassalou 17 show that returns on style portfolios (HML or SMB) seem to predict GDP growth, and thus may in fact capture some aspects of business cycle risk. Each bar in Figure 13.1 is the average difference in the return on the HML or SMB portfolio in years before good GDP growth versus in years with poor GDP growth. Positive values mean the portfolio does better in years prior to good macroeconomic performance. The predominance of positive values leads them to conclude that the returns on the HML and SMB portfolios are positively related to future growth in the macroeconomy, and so may be proxies for business cycle risk. Thus, at least part of the size and value premiums may reflect rational rewards for greater risk exposure.
Petkova and Zhang 18 also try to tie the average return premium on value portfolios to risk premiums. Their approach uses a conditional CAPM. In the conventional CAPM, we treat both the market risk premium and firm betas as given parameters. In contrast, as we noted earlier in the chapter, the conditional CAPM allows both of these terms to vary over time, and possibly to co-vary. If a stock’s beta is high when the market risk premium is high, this positive association leads to a “synergy” in its risk premium, which is the prod- uct of its beta and market risk premium.
What might lead to such an association between beta and the market risk premium?
Zhang 19 focuses on irreversible investments. He notes that firms classified as value firms (with high book-to-market ratios) on average will have greater amounts of tangible capital.
Investment irreversibility puts such firms more at risk for economic downturns because in a severe recession, they will suffer from excess capacity from assets already in place.
(In contrast, growth firms are better able to deal with a downturn by deferring investment
17 J. Liew and M. Vassalou, “Can Book-to-Market, Size and Momentum Be Risk Factors That Predict Economic Growth?” Journal of Financial Economics 57 (2000), pp. 221–45.
18 Ralitsa Petkova and Lu Zhang, “Is Value Riskier than Growth?” Journal of Financial Economics 78 (2005), pp. 187–202.
19 Lu Zhang, “The Value Premium,” Journal of Finance 60 (2005), pp. 67–103.
plans.) The greater exposure of high book-to-market firms to recessions will result in higher down-market betas. Moreover, some evidence suggests that the market risk premium also is higher in down markets, when investors are feeling more economic pressure and anxiety.
The combination of these two factors might impart a positive correlation between the beta of high B/M firms and the market risk premium.
To quantify these notions, Petkova and Zhang attempt to fit both beta and the market risk premium to a set of “state variables,” that is, variables that summarize the state of the economy. These are:
DIV 5 Market dividend yield.
DEFLT 5 Default spread on corporate bonds (Baa – Aaa rates).
TERM 5 Term structure spread (10-year–1-year Treasury rates).
TB 5 1-month T-bill rate.
They estimate a first-pass regression, but first substitute these state variables for beta as follows:
rHML5a1brMt1ei
5a13b01b1DIVt1b2DEFLTt1b3TERMt1b4TBt4rMt1ei 5bt da time-varying beta
Figure 13.1 Difference in return to factor portfolios in year prior to above-average versus below- average GDP growth. Both SMB and HML portfolio returns tend to be higher in years preceding better GDP growth.
Source: J. Liew and M. Vassalou, “Can Book-to-Market, Size and Momentum Be Risk Factors That Predict Economic Growth?” Journal of Financial Economics 57 (2000), pp. 221–45. © 2000 with permission from Elsevier Science.
−20
−15
−10
−5 0 5 10 15 20 25 30 35
Australia Canada France Germany Italy Japan Netherlands Switzerland U.K. U.S.
Past Year Return (%)
HML SMB
The strategy is to estimate parameters b 0 through b 4 and then fit beta using the parameter estimates and the values at each date of the four state variables. In this way, they can esti- mate beta in each period.
Similarly, one can estimate the determinants of a time-varying market risk premium, using the same set of state variables:
rMkt,t2rft5c01c1DIVt1c2DEFLTt1c3TERMt1c4TBt1et
We can estimate the expected market risk premium for each period using the regression parameter estimates and the values of the state variables for that period. The fitted value from this regression is the estimate of the market risk premium.
Finally, Petkova and Zhang examine the relationship between beta and the market risk premium. They define the state of economy by the size of the premium. A peak is defined as the periods with the 10% lowest risk premiums; a trough has the 10% highest risk premiums. The results, presented in Figure 13.2 , support the notion of a countercyclical value beta: the beta of the HML portfolio is negative in good economies, meaning that the beta of value stocks (high book-to-market) is less than that of growth stocks (low B/M), but the reverse is true in recessions. While the covariance between the HML beta and the market risk premium is not sufficient to explain by itself the average return premium on value portfolios, it does suggest that at least part of the explanation may be a rational risk premium.
Behavioral Explanations
On the other side of the debate, several authors make the case that the value premium is a manifestation of market irrationality. The essence of the argument is that analysts tend to extrapolate recent performance too far out into the future, and thus tend to overestimate the value of firms with good recent performance. When the market realizes its mistake, the
Figure 13.2 HML beta in different economic states. The beta of the HML portfolio is higher when the market risk premium is higher.
Source: Ralitsa Petkova and Lu Zhang, “Is Value Riskier than Growth?” Journal of Financial Economics 78 (2005), pp. 187–202. © 2005 with permission from Elsevier Science.
.40
.05
−.33
−.15
−.40
−.30
−.20
−.10 .00 .10 .20 .30 .40 .50
Beta of HML portfolio
Peak Expansion Value beta < Growth beta
Trough Recession
Value beta > Growth beta
prices of these firms fall. Thus, on average, “glamour firms,” which are characterized by recent good performance, high prices, and lower book-to-market ratios, tend to underper- form “value firms” because their high prices reflect excessive optimism relative to those lower book-to-market firms.
Figure 13.3 , from a study by Chan, Karceski, and Lakonishok, 20 makes the case for overreaction. Firms are sorted into deciles based on income growth in the past 5 years. By construction, the growth rates uniformly increase from the first through the tenth decile.
The book-to-market ratio for each decile at the end of the 5-year period (the dashed line) tracks recent growth very well. B/M falls steadily with growth over past 5 years. This is evidence that past growth is extrapolated and then impounded in price. High past growth leads to higher prices and lower B/M ratios.
But B/M at the beginning of a 5-year period shows little or even a positive association with subsequent growth (the solid colored line), implying that market capitalization today is inversely related to growth prospects. In other words, the firms with lower B/M (glamour firms) experience no better or even worse average future income growth than other firms.
The implication is that the market ignores evidence that past growth cannot be extrapolated far into the future. Book-to-market may reflect past growth better than future growth, con- sistent with extrapolation error.
More direct evidence supporting extrapolation error is provided by La Porta, Lakonishok, Shleifer, and Vishny, 21 who examine stock price performance when actual earnings are released to the public. Firms are classified as growth versus value stocks, and the stock price
20 L.K.C. Chan, J. Karceski, and J. Lakonishok, “The Level and Persistence of Growth Rates,” Journal of Finance 58 (April 2003), pp. 643–84.
21 R. La Porta, J. Lakonishok, A. Shleifer, and R.W. Vishny, “Good News for Value Stocks,” Journal of Finance 51 (1997), pp. 1715–42.
Figure 13.3 The book-to-market ratio reflects past growth, but not future growth prospects. B/M tends to fall with income growth experienced at the end of a 5-year period, but actually increases slightly with future income growth rates.
Source: L.K.C. Chan, J. Karceski, and J. Lakonishok, “The Level and Persistence of Growth Rates,” Journal of Finance 58 (April 2003), pp. 643–84. Reprinted by permission of the publisher, Blackwell Publishing, Inc.
Book/Market Ratio
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1.0 1.2
1 2 3 4 5 6 7 8 9 10
Growth Rate Decile
Median growth rate Beginning B/M Ending B/M
performance at earnings announcements for 4 years following the classification date is then examined. Figure 13.4 demonstrates that growth stocks underperform value stocks surround- ing these announcements. We conclude that when news of actual earnings is released to the public, the market is relatively disappointed in stocks it has been pricing as growth firms.
In the end, we would have to characterize the debate as unsettled. The Fama-French model is clearly a highly useful tool for comparing performance against a well-defined set of benchmarks. Whether the return premiums to those factors reflect fully rational risk pre- miums, market irrationality, or some of both still is a matter of considerable controversy.
Momentum: A Fourth Factor
Since the seminal Fama-French three-factor model was introduced, a fourth factor has come to be added to the standard controls for stock return behavior. This is a momen- tum factor. As we first saw in Chapter 11, Jegadeesh and Titman uncovered a tendency for good or bad performance of stocks to persist over several months, a sort of momen- tum property. 22 Carhart added this momentum effect to the three-factor model as a tool to evaluate mutual fund performance. 23 He found that much of what appeared to be the alpha of many mutual funds could in fact be explained as due to their loadings or sensitivities to market momentum. The original Fama-French model augmented with a momentum fac- tor has become a common four-factor model used to evaluate abnormal performance of a stock portfolio.
Of course, this additional factor presents further conundrums of interpretation. To char- acterize the original Fama-French factors as reflecting obvious sources of risk is already
22 Narasimhan Jegadeesh and Sheridan Titman, “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency,” Journal of Finance 48 (March 1993), pp. 65–91.
23Mark M. Carhart, “On Persistence in Mutual Fund Performance,” Journal of Finance 52 (March 1997), pp. 57–82.
Figure 13.4 Value minus glamour returns surrounding earnings announcements, 1971–1992. Announcement effects are measured for each of 4 years following classification as a value versus a growth firm.
Source: R. La Porta, J. Lakonishok, A. Shleifer, and R.W. Vishny, “Good News for Value Stocks,” Journal of Finance 51 (1997), pp. 1715–42. Reprinted by permission of the publisher, Blackwell Publishing, Inc.
3.22
2.26
1.18 1.60
2.79
0.0 1.0 2.0 3.0 4.0
1 2 3 4 5
Postformation Year
Difference in Returns (%)
a bit of a challenge. A momentum factor seems even harder to position as reflecting a risk–return trade-off. But as we saw in Chapter 9, recent work has resulted in a growing appreciation of the importance of liquidity, and particularly an illiquidity premium, in asset pricing. We will see in the next section that a good part of the momentum effect may be related to liquidity.