Abstract: Studies have documented several anomalies in average returns that cannot be explained by the single risk factor in the CAPM. Consistent with the APT, Fama and French (1993, 1996) propose a three-factor model including risk factors of market, size, and BtM. The model is claimed to capture much of the cross-sectional variation in average returns, as well as absorb most of the CAPM anomalies. Still, there are other academics seeking to further improve this model. Fama and French interpret their model as evidence for a distress premium. In contrast to the risk-based explanation, behavioralists explain the size and BtM anomalies as a result of irrational pricing. Criticisms of the three-factor model are centred on data mining and survivorship bias. These hypotheses can be tested by using different time periods, different countries, or a holdout sample. Nevertheless, a number of empirical studies have shown favourable evidence to support the three-factor model.
Keywords: CAPM anomalies, three-factor model, risk-based, irrational pricing, data biases
The aim of this research is to review the work that has been done regarding the three-factor model of Fama and French (1993, 1996), in attempt to examine whether it can explain CAPM anomalies. This paper comprises a broad range of literature study, including both theoretical and empirical works.
Asset pricing is an important topic to study. Researchers have documented a number of anomalies in average returns that cannot be explained by the CAPM. The influential CAPM is challenged by the three-factor model which is claimed to capture much of the cross-sectional variation in average returns, as well as absorb most of the CAPM anomalies. If this claim is true, the three-factor model can thereby be legitimately used in any application that requires estimate of expected returns. Hence, it is necessary to examine the validity and usefulness of the model.
Rather than a single risk factor underlying the CAPM, the three-factor model is consistent with the multifactor arbitrage pricing theory (APT), and it includes three risk factors of market, size, and book-to-market ratio (BtM). Based on numerous studies that support Fama and French's results, most researchers have reached the consensus that stocks with small capitalisation and high BtM can earn higher returns (Davis, 2001). Fama and French (1993, 1996) interpret the empirical success of their model as evidence for a distress premium - investors require higher returns for distressed stocks because of more riskiness. In contrast to the risk-based explanation, behavioralists explain the size and BtM anomalies as a result of mispricing.
Criticisms of the three-factor model are centred on data mining and survivorship bias. However, the three-factor model is again validated by a number of empirical studies showing evidence over different time periods, in different countries, and in holdout samples.
Finally, it is important to note that empirical tests of data mining and survivorship bias cannot solve the critical issue of whether size and BtM are proxies for common risk factors, or irrational pricing (Barber and Lyon, 1997). It suggests that future research should seek to solve this relation with more sophisticated study designs.
The capital asset pricing model (CAPM) and the arbitrage pricing theory (APT) are two important equilibrium asset pricing models that enable us to price risky assets (Copelan et al 2005). Building on the portfolio theory of Markowitz (1959), Sharpe (1964), Lintner (1965) and Black (1972) independently developed the CAPM model, which is appreciated as one of the most important developments in modern financial theory, and makes "the birth of asset pricing theory" (Fama and French 2004:25). The CAPM shows that the equilibrium rates of return on all risky assets are a positive linear function of their covariance with a mean-variance-efficient market portfolio. The market portfolio is the only source of systematic risk that is measured by beta coefficients. In contrast, the arbitrage pricing theory (APT) as developed by Ross (1976) is more general than the CAPM because it allows various risk factors (not just the market portfolio) to explain asset returns. As the APT does not specify the number of risk factors, nor does it identify the factors, a variety of multifactor asset pricing models have been developed building on the APT framework. Though the three-factor model of Fama and French (1993, 1996) is not the first multifactor asset pricing model, it might be the most prominent one which has also created huge amounts of academic debate.
Starting in the late 1970s, empirical studies have documented numerous deviations from the CAPM. Several factors regarding firm characteristics have been identified that apparently provide more power other than the CAPM betas in explaining average stock returns. Fama and French (1993, 1996) refer to these factors as anomalies. It is argued by proponents of multifactor pricing models that with only one risk factor, the CAPM does not capture all the factors that affect stock returns.
The three-factor model has been subjected to considerable theoretical investigation and empirical research. The aim of this research is to review the work that has been done regarding the three-factor model in explaining CAPM anomalies. Particularly, the study addresses the question of whether Fama and French's three-factor model can explain CAPM anomalies. If it can, what are possible explanations? Otherwise, what shortcomings underlying the model that prohibit its success? Studying asset pricing is valuable for making investment decisions, such as selecting portfolios, evaluating the performance of managed funds, measuring abnormal returns in event studies, and estimating the cost of capital for firms (Fama and French, 1993). It is therefore necessary to review the development of the CAPM and assess the validity of the three-factor model as a typical multifactor model.
This paper comprises a literature study and attempts to examine the three-factor model from both theoretical rationale and empirical evidence. The remainder of the paper is structured as follows. Section II reviews some important empirical contradictions of the CAPM; and then introduces the influential three-factor model, as well as its development with regard of two new multifactor models of Carhart (1997) and Chen and Long (2009). Section III discusses competing explanations for the size and BtM effects that are captured by the three-factor model. Section IV examines criticisms of the model. Section V presents empirical evidence that against those criticisms. Section VI concludes and proposes possible recommendations for future studies.
Empirical Contradictions of the CAPM and the Development of the Three-Factor Model
The CAPM implies that market betas suffice to describe the differences in expected returns across assets. However, since the late 1970s, the CAPM appears to be challenged by a number of studies, which provide evidence to show that much of the variation in expected returns is unrelated to betas (Fama and French, 2004). Banz (1981) reveals a size effect: the average returns on stocks of firms with small market capitalizations are higher than predicted by the CAPM. Basu (1977, 1983) identifies that firms with high (low) earnings-price ratios (P/E) have lower (higher) returns than expected. DeBondt and Thaler (1985) find that stocks with low returns during the past three to five years would experience long-term return reversals. The momentum effect as found by Jegadeesh and Titman (1993) shows that stocks with high returns during the past three to twelve months tend to yield excess returns in the future. Bhandari (1988) finds that firms with high leverage are associated with abnormally higher returns in terms of their betas. Rosenberg, Reid and Lanstein (1985) maintain that stocks with high book-to-market equity ratios (BtM) have high average returns that are not captured by betas. All studies discussed above cast doubt on the ability of the CAPM to explain the cross-section of expected returns. According to the CAPM, above risk variables should not be able to explain average returns better than betas (Davis, 2001).
The empirical contradictions of the CAPM motivate Fama and French (1992) to synthetically evaluate the explanatory power of beta, size, E/P, leverage, and BtM in the cross-section of average returns on NYSE, AMEX, and NASDAQ stocks. The return files of non financial firms from the Centre for Research in Security Prices (CRSP) and COMPUSTAT database are used for the period of 1963-1990. They conclude that size and BtM capture the cross-sectional variation in average stock returns, whereas market betas reveal little information about average returns (ibid.).
By using the times-series regression approach, Fama and French (1993) test 25 stock portfolios formed on size and BtM of value-weighted NYSE, AMEX, and NASDAQ stocks over the 1963-1993 period. The empirical results confirm that factors related to size and BtM can explain the differences in the average cross-section stocks returns. However, the differences between the average stock returns and one-month T-bill rates cannot be explained by these two factors alone; the inclusion of the market factor can explain this as the market risk premium links the average returns on stocks and T-bills (ibid.). Therefore, they conclude that the common variation in stock returns is largely captured by three factors of market, size and BtM.
Based on this evidence, Fama and French (1996, p. 55) developed a three-factor model (it is a three-factor version of the APT in nature). Specifically, the expected return on portfolio i in excess of the risk-free rate is,
E(Ri) - Rf = bi[E(RM) - Rf] + siE(SMB) +hiE(HML)
where E(RM) - Rf is the excess returns on a broad market portfolio. SMB (small minus big) is the difference between the return on a diversified portfolio of small and large stocks. Similarly, HML (high minus low) is the difference between the returns on a diversified portfolio of high BtM and low BtM stocks. The bi, si, and hi coefficients measure the sensitivity of the portfolio's return to three factors (ibid.).
Based on empirical successes in their 1993, 1995 and 1996 studies, Fama and French (1996, p. 50) assert that except for the short-term return momentum effect, the three-factor model "captures most of the average-return anomalies of the CAPM". As the momentum effect is left unexplained, Carhart (1997) develops a four-factor model by adding one momentum factor to the original three-factor model. He claims that with the additional momentum factor, the four-factor model substantially improves on the average pricing errors of both the CAPM and the Fama-French model (ibid.). However, it is important to note that the four-factor model shares the same risk-based intuition with the three-factor model; both models cannot ward off scepticism from the behavioralists in terms of irrational pricing, which are discussed in the next section.
Several studies show that except for the momentum effect, the Fama-French model still leaves many CAPM anomalies unexplained, such as earnings surprises, financial distress, net stock issues, and asset growth (Chen and Zhang, 2009). Based on q-theory, Chen and Zhang (2009) update the Fama-French model and propose a new three-factor model including the market factor, a low-minus-high investment factor, and a high-minus-low return-on-assets (ROA) factor. They test this model empirically for a broad sample of stocks over the period of 1972 - 2006. Results show that the new model substantially outperforms the Fama-French model as it can explain many patterns anomalous to the original model (ibid.). However, with similar shortcomings like the Carhart (1997) model, the new three-factor model is silent on factor effects regarding compensation for risk and behavioural mispricing explanations.
Theoretical Explanations: Rational Risk or Irrational Pricing?
The economic interpretation of the Fama and French's three-factor model is debatable. The first story is the risk-based explanation. The studies of Fama and French (1993, 1996) show that there exists covariation in returns related to size and BtM, which are captured by loadings on SMB and HML factors, and beyond the covariation is explained by the market return. It suggests that the three factors in the model capture much of the common variation in portfolio returns that is missed by univariate risk factor (betas) in the CAPM. The CAPM anomalies reflect the fact that size and BtM are proxies for distress. Small stocks and high BtM (value) stocks have high average returns because they are risky, for which investors require a positive risk premium.
In contrast to the risk-based explanation, behavioralists explain the size and BtM anomalies as a result of irrational pricing. Lakonishok, Shleifer and Vishny (1994) suggest that higher returns associated with small stocks and value stocks are due to mispricing. They interpret the BtM anomaly as investors tend to extrapolate firms' past performance into the future. Therefore, prices of growth stocks (low BtM) are usually too high as a result of over-optimistic expectations. Nevertheless, these pricing errors will eventually be corrected, resulting low returns for growth stocks. Similarly, distressed stocks are undervalued hence have high returns. Other proponents of this view include DeBondt and Thaler (1987), and Haugen (1995). The mispricing explanation implies that investors can increase returns without bearing additional risks, simply by buying value stocks and selling growth stocks (Davis, 2003). It clearly contradicts the Fama and French's argument because the average HML return in the three-factor model that based on rational-pricing is interpreted as a risk premium for distress.
Daniel and Titman (1997, 1998) reject the risk-based explanation by arguing that it is characteristics (e.g., high versus low BtM) rather than covariances (e.g., factor sensitivities) that determine stock returns. In support of this argument, they provide evidence that for the 1973-1993 period, expected returns do not appear to be positively related to the loadings on the three factors in the Fama-French model after controlling for firm characteristics. It follows that there is no return premium associated with any of the three factors as proposed by Fama and French (Daniel and Titman, 1998). However, their characteristics explanation is argued to be specific to their rather short sample period. As a longer 1929-1997 period is examined in Davis, Fama and French (2000), covariances show more explanatory power than characteristics. Therefore, Davis, Fama and French (2000) refute the hypothesis that the BtM characteristic is compensated irrespective of risk loadings. Instead, the three-factor model better explains the value premium because expected returns compensate risk loadings (ibid.).
Criticisms: Data Mining and Survivorship Bias
Criticism of the three-factor model has centred on data mining (or data snooping) and survivorship bias. Black (1993) and Mackinlay (1995) contend that CAPM anomalies could be the result of data mining. The data mining story predicts that the size and BtM effects would disappear in out-of-sample tests. In other words, when analyze another time period or another data source, the three-factor model will reduce to the CAPM and the three factors will be completed explained by CAPM betas if the data mining story holds (Fama and French, 1996).
The studies of Fama and French are further criticised for the COMPUSTAT database that they used. As many stocks with high BtM and small size do not survive and are excluded from the COMPUSTAT, their results suffer from a sample-selection bias. Kothari, Shanken and Sloan (1995) re-examine the work of Fama and French (1993) in order to determine whether beta and BtM capture the cross-sectional variation in average returns. Using an alternative Standard & Poor (S&P) industry-level data from 1947 to 1987, they find little relation between BtM and average returns but substantial compensation for beta risk when measured by annual rather than monthly data (ibid). They attribute the contradictions to the survivorship bias. They argue that average returns on high BtM stocks are overstated by the three-factor model because the COMPUSTAT includes distressed firms that have survived and excludes distressed firms that have failed.
Furthermore, the CAPM is rejected by the three-factor model based on the evidence that size and BtM are capturing cross-sectional variation in average returns that cannot be explained by betas (Fama and French 1996). Nevertheless, it is argued by Levy (1997, p. 120) that in rejecting the CAPM, Fama and French use "historical betas as measures of betas based on future returns". As the CAPM is concerned about ex ante expectations and not ex post returns, their empirical findings could not refute the CAPM (Laubscher, 2002).
Empirical Evidence Against Criticisms
According to Barber and Lyon (1997), the data snooping and sample-selection bias hypothesis can be tested by examining the Fama-French model using different time periods, different countries, or a holdout sample. In contrast to above criticisms for data snooping and sample-selection bias, Fama and French's model is also supported by a number of empirical studies.
Davis (1994) provides strong evidence to counter-argue the data mining and survivorship bias claims. He maintains that the distress premium is not specific to the period studied in Fama and French (1992, 1993). In support of this argument, he constructs a database of large US industrial firms for the period of 1940-1963, which is "pre-COMPUSTAT era" and also precedes the period studied by Fama and French. He finds a strong relation between BtM and cross-section of realised stock returns, which agrees with Fama and French's results. As Davis (1994) uses a different database to study independent time periods yet produces consistent results, he rejects the criticisms of the three-factor model in terms of data mining and survivorship bias.
Further evidence is provided by Chan, Jegadeesh and Lakonishok (1995), showing that sample-selection bias does not have a significant effect on Fama and French's results. They find that the proportion of CRSP companies missing from COMPUSTAT over the 1968-92 period is small; at most 3.1 percent can be broadly interpreted as financially distressed firms that are omitted from COMPUSTAT. In addition, the average return is not very different between them. They also construct a dataset of large firms which is free from any selection bias from back-filling data. They conclude that selection bias on COMPUSTAT is exaggerated, and the BtM effect in Fama and French (1992, 1993) is confirmed for the top 20 percent of NYSE-Amex stocks.
Barber and Lyon (1997) empirically test the data snooping issue by analyzing the relation between size, BtM ratios, and stock returns for financial firms, which are excluded in Fama and French (1992). By examining the 1973-1994 period, they find that the firm size and BtM patterns in returns are similar for both financial and nonfinancial firms. They also show evidence that survivorship bias on COMPUSTAT data does not significantly affect either size or BtM premiums (ibid.). Since they use a large holdout sample, their results apparently challenge data snooping and selection biases criticisms.
Other studies based on international evidence also refute the data bias criticisms. The early study of Chan, Hamao, and Lakonishok (1991) documents a significant cross-sectional relationship between BtM ratio and stock returns in the Japanese stock market from 1971 to 1988. Capaul, Rowley and Sharpe (1993) observe a similar BtM premium in Japan as well as four other developed countries during the 1981-1992 period. For a longer time interval during 1975 to 1995, Fama and French (1998) find pervasive BtM premium in twelve of thirteen major markets, as well as in emerging markets. In addition, they claim that with a risk factor for relative distress included, their model captures the BtM premium in country and global returns (ibid.). There is further evidence in emerging markets documented in more recent studies that tends to support the three-factor model, for example, tests of Connor and Sehgal (2001) in the Indian market, and tests of Tony and Veeraragavan (2005) in the Chinese market.
In conclusion, this research reviews literature regarding the three-factor model, in attempt to examine whether it can explain CAPM anomalies. Rather than a single risk factor underlying the CAPM, multifactor APT models allow various risk factors to explain asset returns. Studies have documented several CAPM anomalies. Consistent with the APT, the Fama-French model that includes three risk factors of market, size and BtM is claimed to capture much of the cross-sectional variation in average returns, and also it seems to absorb most of the CAPM anomalies. Fama and French (1993, 1996) interpret the empirical success of their model as evidence for a distress premium, since risk factors of SMB and HML capture independent sources of systematic risk which are missed by the CAPM. In contrast, behavioralists explain the size and BtM premium as a result of mispricing. They argue that investors always make wrong expectations and hence misprice distressed stocks. Once pricing errors are corrected, returns for these stocks would be high. Criticisms of the three-factor model are centred on data mining and survivorship bias. Nevertheless, these hypotheses can be tested by using different time periods, different countries, or a holdout sample. A large number of empirical studies have shown favourable evidence to support the three-factor model.
However, empirical tests of data mining and the survivorship bias cannot solve the critical issue of whether size and BtM are proxies for common risk factors, or irrational mispricing (Barber and Lyon, 1997). Future work should seek to solve this relation with more sophisticated study designs. Additionally, as Fama and French (1996) point out, their model might not explain all asset returns. Thus, future studies are suggested to look actively for a richer model that includes other possible risk factors. For example, liquidity seems to be a priced risk, which might be the possible direction for future study.
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