Empirical tests of multi-factor models for asset return prediction

Apple Boston

2022-02-25 00:00:00 Fri ET

Empirical tests of multi-factor models for asset return prediction 

The capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965), and Black (1972) marks the birth of asset pricing theory. The CAPM and its cousins, such as the Fama-French three-factor model and the Carhart model with momentum, continue to be ubiquitous in modern applications such as mutual fund performance evaluation, equity cost estimation, and corporate event assessment. The CAPM builds on Markowitz’s (1959) model of portfolio choice by assuming that risk-averse investors care only about the mean and variance of their investment return in the selection of asset portfolios. As a result, investors choose mean-variance efficient portfolios in the sense that these portfolios either minimize the variance of portfolio return for a given mean return or maximize the mean return for a given portfolio variance (cf. second-degree stochastic dominance). Through two-fund monetary separation, a linear combination of mean-variance efficient frontier portfolio returns stochastically dominates any other portfolio return. Markowitz’s portfolio model provides an algebraic condition on risky asset weights in mean-variance efficient portfolios. The CAPM turns this algebraic statement into a central prediction about the relation between risk and return by identifying a mean-variance efficient market portfolio when asset prices clear the market for all risky assets.


Since the market beta of a given asset is the slope in the regression of its return on the market return, a conventional interpretation of beta is that this risk measure reflects the sensitivity of the asset’s return to the variation in the market return. Another interpretation pertains to the variance of the market return. In particular, the riskiness of the market portfolio is a weighted average of the covariance risks of the risky assets in the market portfolio. The CAPM formula depicts a linear relationship between the excess return on a given asset and the corresponding risk premium. Thus, the mean return on any asset is the risk-free rate plus a risk premium, which is the asset’s beta times the market premium per unit of beta risk.


The early cross-sectional regression tests focus on the CAPM’s predictions about the intercept and slope in the relation between return and beta. The approach is to regress a cross-section of average asset returns on beta estimates. The model predicts that the intercept is the risk-free interest rate while the coefficient on beta is the market risk premium. At least a couple of econometric problems arise from this context. First, the beta estimates for individual assets are imprecise and thus create a measurement error problem in the regression of asset returns. To improve the precision of beta estimates, Friend and Blume (1970) and Black, Jensen, and Scholes (1972) make use of portfolio returns rather than individual stock returns. The use of portfolios in the cross-sectional regressions of average returns on betas helps reduce the errors-in-variables issue. Second, the residual terms have common sources of variation, such as industry effects in average returns. Positive correlation in these residual terms produces a downward bias in the usual ordinary least squares estimates of the standard errors of the cross-sectional regression slopes. Fama and MacBeth (1973) propose a two-stage procedure to alleviate this concern. The first step is to run month-by-month cross-sectional regressions of average monthly returns on betas. Then the second step involves using the time-series intercept and slope estimates to test whether the mean premium for beta is positive.


Fama and French (2004) point out that the early tests reject the empirical validity of the CAPM. Although there is a positive relation between beta and average return, this relation is too “flat”. The returns on the low beta portfolios are too high, and the returns on the high beta portfolios are too low. In other words, the coefficient on beta is less than the average excess market return (Black, Jensen, and Scholes, 1972; Blume and Friend, 1973; Fama and MacBeth, 1973; Fama and French, 1992). In addition to the cross-sectional view, Jensen (1968) suggests that the CAPM predicts a time-series relation between the excess returns on the individual and market portfolios: the intercept in the time-series CAPM regression should be zero for each asset, i.e. the null hypothesis states that Jensen’s alpha should not be too far from zero. To test the hypothesis that market betas suffice to explain the variation in returns, one estimates the time-series regression for a set of assets or portfolios and then jointly tests the vector of regression intercepts against zero. Gibbons, Ross, and Shanken (1989) design an econometric F-test for this purpose. This test examines whether the distance between the squared Sharpe ratios for the individual and market portfolios is statistically significant. A subsequent strand of asset pricing literature by Fama and French (1993, 1995, 1996, and 1998) implements Gibbons et al’s F-test to assess whether Jensen’s alpha vanishes toward nil when one augments the CAPM with the size and book-to-market portfolio return differences.


An important lesson from this discussion is that the Jensen or Fama-MacBeth regressions do not directly test the CAPM. What is literally tested in the above regressions is whether a specific proxy for the market portfolio is mean-variance efficient (Roll, 1977). The market portfolio is elusive in both theoretical and empirical terms. It is not theoretically clear which assets (for example, human capital) can legitimately be excluded from the market portfolio, and data availability substantially limits the scope of assets that are included in the market portfolio. As a consequence, empirical tests of the CAPM require the use of proxies for the market portfolio and so in effect only help test whether such proxies are on the minimum variance frontier. Roll’s critique thus suggests that because the tests use at best proxies but not the true market portfolio, we learn little about the CAPM.


Nobel Laureate Eugene Fama and his long-time coauthor Kenneth French propose a multi-factor model to address various anomalies in empirical asset return prediction.

Some more recent tests lead to a theme in the contradictions of the CAPM. Stock price ratios convey key information about the variation in stock returns that market betas cannot adequately capture. For instance, Basu (1977) reports evidence in favor of portfolio sorts based on earnings-price ratios. Future returns on stocks with high earnings-price ratios exceed the average returns that the CAPM predicts. Banz (1981) finds a size effect. When stocks are sorted on market capitalization, average returns on small stocks are higher than the average returns that the CAPM predicts. Also, Bhandari (1988) reports that high debt-equity ratios as a measure of financial leverage are closely associated with stock returns that are too high relative to their betas. Lastly, Statman (1980) and Rosenberg, Reid, and Lanstein (1985) document that stocks with high book-to-market equity ratios yield high returns that their betas cannot readily capture.


Fama and French (1992) update and synthesize the evidence on the above empirical failures of the CAPM. Using the Fama-MacBeth cross-sectional regressions, Fama and French confirm that size, earnings-price ratios, debt-equity ratios, and book-to-market equity ratios help explain the cross-section of stock returns. The combination of size and book-to-market equity subsumes the explanatory power of both earnings-price and debt-equity ratios. However, the CAPM relation between beta and return does not hold in the same cross-section. If betas do not suffice to explain the variation in stock returns, the various proxies for the market portfolio are not mean-variance efficient in light of Roll’s critique.


Fama and French (1993, 1995, and 1996) seek to design a three-factor model to modify the CAPM with risk factors that help capture the joint size and book-to-market phenomenon. Carhart (1997) augments the Fama-French model with a momentum factor in his study of mutual fund performance. This modifi-cation serves as a response to the empirical fact that the Fama-French three-factor model cannot readily capture the short-term momentum in stock returns, i.e. the recent winners continue to produce high excess returns at the one-year horizon while the reverse holds true for the recent losers (Jegadeesh and Titman, 1993; Jegadeesh, Lakonishok, and Titman, 1996; Jegadeesh and Titman, 2001).


Fama and French attempt to connect their three-factor model to Merton’s (1973) intertemporal capital asset pricing model (ICAPM). The ICAPM begins with a different assumption about investor objectives. In the CAPM, investors care only about the wealth that their portfolio produces at the end of the current period. In the ICAPM, investors are concerned not only with their terminal payoff but also the opportu-nities that such investors will have to consume or to invest the payoff. ICAPM investors consider how their current wealth might vary with future state variables, such as labor income, the prices of consump-tion goods, and the nature of portfolio opportunities, and various expectations about the labor income, consumption, and investment opportunities available after the present time. Fama and French’s (1993) brute force constructs for the joint size and book-to-market phenomenon serve as useful proxies for state variables in the ICAPM (Fama, 1996; Fama and French 1996; Petkova, 2006). Thereby, the Fama-French stock portfolios are multifactor mean-variance efficient. These stock portfolios yield the largest possible average returns for a given set of return variances and covariances with the relevant state variables (Fama, 1996).


Fama and French (1993 and 1996) implement Gibbons, Ross, and Shanken’s F-test to ascertain whether the intercept terms are jointly zero across all the relevant stock portfolios. The Fama-French three-factor model captures much of the variation in average returns for stock portfolios that are sorted on size, book-to-market equity and other price ratios that cause embarrassment for the CAPM. Furthermore, Fama and French (1998) find that an international version of the three-factor model performs better than an inter-national version of the CAPM in explaining the variation in stock portfolio returns in 13 major markets. This strand of asset pricing literature by Fama and French minimizes the likelihood of data-mining bias. Although Fama and French find evidence in support of the view that Jensen’s alpha is jointly zero across the various size and book-to-market portfolios, some subsequent papers shed skeptical light on the F-test on Jensen’s alpha when one designs and carries out behavioral investment strategies such as momentum, value, and style investment strategies (Jegadeesh and Titman, 1993 and 2001; Jegadeesh, Lakonishok, and Titman, 1996; Lakonishok, Shleifer, and Vishny, 1994; Haugen and Baker, 1996; Daniel and Titman, 1997; Titman, Wei, and Xie, 2004; Pontiff and Woodgate, 2008; Lam and Wei, 2014).


There is a long-standing conflict between the behavioral mispricing story and the rational risk story for the empirical failures of the (I)CAPM. This conflict leaves us at a timeworn impasse and reminds us of Fama’s (1970) emphasis on the dual-hypothesis issue. The famous thesis on efficient capital markets that prices properly reflect available information must be tested in the context of an asset pricing model. To test whether prices reflect changes in rational investor sentiment, one must take a stand on how the market sets prices. When empirical tests reject the validity of the CAPM or some other asset pricing model, one might infer whether the problem is with the asset pricing model or the notion of market efficiency. This dual-hypothesis issue highlights the empirical difficulties in testing the relation between risk and return in the context of the CAPM or its extensions.


The behavioral interpretation deserves some merit. Lakonishok, Shleifer, and Vishny (1994) assert that sorting stocks on book-to-market equity ratios exposes investor over-reaction to both good and bad times. Investors tend to over-extrapolate past performance such as past corporate income or recent sales growth, and this extrapolation results in stock prices that are too high for glamour stocks with low book-to-market ratios or too low for value stocks with high book-to-market ratios. The eventual correction of such over-reaction then gives rise to high returns for value stocks and low returns for glamour stocks (DeBondt and Thaler, 1987; Lakonishok, Shleifer, and Vishny, 1994; Haugen and Baker, 1996). Intuitive forecasts are often non-regressive because people tend to make extreme predictions on the basis of information whose reliability and predictive validity are known to be low (Kahneman and Tversky, 1982). In order to exploit the flaw of intuitive forecasts, contrarian investors short stocks with high past growth and high expected future growth and also long stocks with low past growth and low expected future growth. Prices of these stocks are most likely to reflect the failure of investors to impose mean reversion on financial forecasts. La Porta (1996) bolsters this behavioral view by analyzing the systematic expectational errors in analysts’ growth forecasts. Value strategies earn superior returns because analysts’ forecasts about future earnings growth are too extreme. In addition to short-run momentum investment strategies, contrarian strategies help exploit the predictable profitability of long-term stock return reversals (Hong and Stein, 1999).


Fama (1998) reassesses the various asset pricing anomalies and discusses their implications for the long-term notion of market efficiency vis-à-vis behavioral finance. His thesis suggests that if there is a random split between market over-reaction and under-reaction at different time horizons, the capital market may still be efficient while the asset return process remains a random walk. Campbell, Lo, and MacKinlay’s (1997) synthesis of recent literature on stock return predictability suggests an alternative view. This view takes into account an important property of the random walk hypothesis that the variance of random walk increments must be a linear function of the time interval. The standard test entails the use of the general q-period variance ratio, which is a linear combination of the first k–1 autocorrelation coefficients of the returns with linearly declining weights.


While the autocorrelation coefficients may be non-zero, a weighted average of these autocorrelation co-efficients can be equal to zero. For instance, a variance ratio for q=3 can be VR(3)=1+2*{2/3ρ(1)+1/3ρ(2)} while the autocorrelation coefficients are non-zero: ρ(1)=1/3 and ρ(2)=–2/3. The former autocorrelation suggests short-term return momentum, and the latter autocorrelation suggests long-term return reversal. This standard variance ratio test has low power. There is in fact predictable variation in stock returns, but the evidence does not reject the random walk hypothesis. This baseline case thus serves as an objection to Fama’s (1998) conventional wisdom.  


Daniel and Titman (1997) take the behavioral story one step further. Their empirical work suggests that although high book-to-market stocks do covary strongly with other high book-to-market stocks, such co-variances do not arise from the existence of distress risk. Rather, these covariances reflect the fact that these stocks operate in similar lines of businesses, in the same industries, or from the same regions. The same rationale holds for stocks with low market capitalization. After controlling for firm characteristics, there is no clear positive relation between stock returns and risk sensitivities on the Fama-French factors. In other words, Daniel and Titman present evidence in support of the behavioral story that firm charac-teristics but not covariances determine the variation in stock returns. This evidence disturbs the standard positive relation between risk and return that both the CAPM and its ICAPM cousins attempt to capture.


More recent empirical research relies on mutual fund flows to determine which asset pricing models most investors use in their investment decisions (Berk and van Binsbergen, 2013; Barber, Huang, and Odean, 2014). These authors find that the various competing asset pricing models do not serve as improvements over the CAPM. The novel idea can be traced back to Berk and Green’s (2004) seminal work on mutual fund performance. Berk and Green derive a model of how the capital market for mutual fund investment equilibrates. In perfectly competitive capital markets, all mutual funds must have enough assets under management such that these funds face diminishing returns to scale. When new information convinces investors that a particular mutual fund represents a positive net present value investment, investors react by investing more capital in this fund. This process continues until enough new capital is invested in this fund to eliminate the above opportunity. As a result, mutual fund flows react to past performance while future fund performance is unpredictable. Berk and van Binsbergen (2013) suggest that if an asset pricing model under consideration correctly prices risk, investors must be channeling their capital to the mutual funds that implement this model. In this light, mutual fund flows can reveal investors’ risk preferences. Berk and van Binsbergen find that the CAPM outperforms all of its extensions in attracting mutual fund flows. The various extensions to the CAPM, such as the empirical models specified by Fama and French (1993) and Carhart (1997) as well as the dynamic equilibrium models derived by Merton (1973), Breeden (1979), Campbell and Cochrane (1999), and Bansal and Yaron (2004), do not represent true progress toward a better asset pricing model.


In addition to Berk and van Binsbergen’s (2013) empirical work, Barber et al (2014) find greater capital flows to mutual funds with higher CAPM alpha ranks than flows to mutual funds with higher ranks that are based on the alternative Fama-French-Carhart models. The CAPM is the clear victor in this horserace. Investors rely most on the CAPM alpha when they evaluate mutual fund performance. Also, mutual fund investors heavily discount average stock returns that can be traced to market beta. Yet, these investors do not share the same risk perceptions of size, value, and momentum. Barber et al infer from this evidence that the CAPM outperforms the various competing models as mutual fund flows shed new light on mutual fund investors’ choice of the correct asset pricing model. Berk and van Binsbergen (2013) and Barber, Huang, and Odean (2014) both emphasize the use of mutual fund flows, rather than stock prices or returns, in running a new set of empirical tests on the CAPM and its extensions.


A novel insight for future research is to apply the Kalman filter to extract Jensen’s alpha and multi-factor betas that may vary over time. One plausible conjecture is that these time-varying multifactor betas move in tandem with macroeconomic fluctuations. In this case, one can test whether these time-varying multi-factor betas tend to cointegrate with macroeconomic series such as shifts in the term structure of interest rates. In addition, the direction of causation is important. One can carry out both Granger-causality tests and vector autoregressions to examine the potential lead-lag relationships among the various multifactor betas. This time-series analysis offers a direct test on whether the time-varying multifactor betas represent compensation for distress risk or some other macroeconomic considerations (Fama and French, 1996 and 1998; Petkova, 2006; Campbell, Hilscher, and Szilagyi, 2008). In essence, this analysis helps investigate whether the multifactor betas serve as proxies for financial risk at different parts of the macroeconomic cycle. For instance, distress risk may be a severe financial constraint in a downturn while procyclicality risk requires a subsequent risk capital buffer in an upturn. By relaxing the unduly restrictive assumption of invariant multifactor betas, we shine new light on Fama’s (1970 and 1998) notion of market efficiency at different junctures. Over time, the market adapts to changes in most investors’ expectations about the relation between risk and return (Lo, 2004 and 2005). As a consequence, investors adapt to shifts in the economic environment through the use of simple heuristics. Because the relation between risk and return varies over time (i.e. the multifactor betas may not be constant), investors can be more or less sensitive to changes in the ICAPM state variables. This idea serves as a contribution to the development of a new multifactor asset pricing model.


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