This report surveys three-factor asset pricing models introduced in series of works by Eugene Fama and Kenneth French: Fama and French (1992), Fama and French (1993) and Fama and French (1995). Fama and French’s empirical works suggest that size and book-to-market equity possess decent explanatory power in cross-sectional variation of stocks returns. This report also conducts an empirical test using Fama and French two pass regression with monthly return of 20 securities from 2010 to 2014. This report proceeds with literature review, methodology, empirical analysis, and conclusion and implications.

**Literature Review**

Markowitz is considered the father of modern portfolio theory. His model (1952, 1959) assumes that investors gather information available at time t-1 in order to form portfolios delivering a return at time t. Markowitz assumes that 1) investors are risk-averse and maximize terminal wealth for a single period, 2) capital markets are perfect, and 3) asset return distribution is multivariate normal. Combining these assumptions means that investors form efficient portfolios that maximize the expected return (mean) for a given level of risk (variance). Markowitz’ analysis leads to the formation of the

*minimum variance frontier*.

Sharpe (1964) and Lintner (1965) add two more assumptions to the Markowitz analysis. First, investors have homogeneous expectations about the assets’ return distributions. Next, investors are able to borrow and lend unlimited amounts at some risk-free rate. Adding these two assumptions to the Markowitz analysis leads directly to the CAPM model. Because investors have homogeneous expectations, they all choose to hold the same portfolio. This ‘market’ portfolio is identified by a line that 1) passes through the risk-free rate and is 2) tangent to the minimum variance frontier, as this yields the line with the steepest slope. Black (1972) changes the final assumption of Sharpe-Lintner by proposing that investors are able to short sell unlimited quantities of assets. This assumption leads to the finding that all investors hold efficient portfolios and that the ‘market’ portfolio is simply the weighted average of these portfolios. The Sharpe-Lintner and Black versions of the CAPM differ in their conclusion about the intercept of the linear model. Sharpe-Lintner argue that this intercept must equal the risk-free rate; Black claims that the return on the zero-beta asset must simply be lower that the return on the market portfolio. Regardless of the final assumption, the model leads to the following (theoretically) testable implications:

1) There is a positive risk premium associated with the market portfolio.

2) Asset returns are linear in their exposure to market risk.

3) No other risks explain asset returns.

Fama and MacBeth (1973) provides a test of the two-parameter model and theories about market efficiency using the monthly returns on New York Stock Exchange stocks from January 1926 through June 1986. They specifically test the three hypotheses detailed above as well as the Sharpe-Lintner hypothesis that the model’s intercept is equal to the risk-free rate. For each version of the model, Fama and MacBeth (1973) find a statistically significant positive risk premium for each version of the model. Finally, the authors cite their results about the auto-correlation of these estimates as evidence in support of an efficient capital market. If auto-correlations are low, then this suggests that risk premiums are a “fair game.” Roll (1977) writes an essay criticizing the existing empirical tests of the two-parameter model. His basic critique makes two points. First and foremost, Roll points out that the analysis leading to the original two-parameter model is based on a true market portfolio that includes positive quantities of all assets that exist, both traded and non-traded. Roll (1977) claims that this difficulty in identifying the ‘market’ portfolio not only invalidates the current tests, but also makes it unlikely that any researcher will truly be able to test the CAPM. Stambaugh (1982) is a direct response to the Roll (1977) Critique of the empirical tests of the CAPM by using various testing assets and market portfolios. The conclusions are the CAPM testing is not sensitive to the specifications of market portfolios, but sensitive to different testing assets.

Starting from late 1970s, there booms a literature on market anomalies. Anomalies are empirical findings that seem to be inconsistent with classic theories of asset pricing. Their existence may imply either market inefficiency or possible flaws in the benchmark set by asset-pricing models, say CAPM. Basu (1975) and Basu (1977) document that portfolios with high (low) earnings’ yield (E/P) appear to have higher (lower) risk-adjusted rates of return. Banz (1981) and Reinganum (1981) find stocks of small firms, on average, have higher risk-adjusted returns than the common stocks of large firms. Keim (1983) further finds that the size effect has an interesting

*turn-of-the-year*feature, that is, nearly half of the average abnormal return associated with size effect is due to anomalous January abnormal returns. Reiganum (1983) shows that the January effect seems to be consistent with tax-loss selling. However, small firms least likely to be sold for tax reasons also exhibit large January effect, which implies tax loss selling may not be the only reason behind. In an interesting work by Blume and Stambaugh (1983), the authors support the existence of January effect, but argue that the use of quoted closing prices could cause upward bias in equally-weighted portfolio returns. The return on a buy-and-hold portfolio could largely avoid this bias and they find that after the correction the full-year size effect is only half as large as previously reported.

Following the series of documented anomalies, which imply the failure of simple one-factor CAPM, Fama and French (1992) aim to evaluate the joint-effect of well-documented stock return determinants (β, size, E/P, leverage, book-to-market equity) in explaining the cross-sectional returns on NYSE, AMEX and NASDAQ stocks. Fama and French (1993) formally proposes the three-factor model, where three factors are market factor, size factor and book-to-market factor. Fama and French (1995) attempts to give an economic interpretation to size and book-to-market equity factors. Further details of the three-factor model is elaborated in the following methodology section. The contribution of the Fama-French three-factor model is, first, it formalizes size and book-to-market factor as two other risk factors besides the market factor. Second, their model significantly improves the explanatory power of the cross-sectional variations in asset returns.

Fama and French (1992) is a ground-breaking yet controversial discovery in asset pricing. Fisher Black in Black (1993a) and Black (1993b) argue that the results obtained by Fama and French lack theoretical underpinning. In other words, these results could be merely artifactos of data mining. Kothari et al. (1995) questions the data set and β estimation of Fama and French’s results. They argue that COMPUSTAT data suffers from a survivorship bias. On the other side, Jaganathan and Wang (1996) introduces human capital and default spread as two extra factors into the static CAPM. They show that their model dramatically improves the performance. They also put size into their model for a horserace comparison. And in most of their specifications, size is not priced. Carhart (1997) further develops a four-factor model, adding momentum as the fourth factor, to explain anomalies that is not accounted in three-factor model. Daniel and Titman (1997) argues that it is the firm characteristics determines the risk premium, rather than the factor loadings.

**Methodology**

Fama-French three-factor model is:

Where the variables are defined as:

: monthly return for the stock;

risk-free rate, the T-bill return of that month;

monthly market return, the CRSP value-weighted market index;

SMB: the return of a factor portfolio that buys small cap stocks and sells large cap stocks;

HML: the return of a factor portfolio that buys high BE/ME stocks and sells low BE/ME stocks

The two-pass regression approach is first estimating βs using time-series data for each individual stocks, and then estimating a cross-sectional regression of average return on estimated βs from the first step. Mathematically, the first stage regression is:

The variables are defined in the above three-factor model. And this is estimated for each stock

*j*to get estimated , , and Estimated βs suffers from estimation errors. This problem will affect the estimation results in the second stage equation, which is known as “errors-in-variables” problem. This further affects our hypothesis testing, which is described in the following part. So a precise estimation of βs is very critique to the testing of asset pricing models. Portfolio beta, constructed as average of βs of underlying individual assets, shall be more precisely estimated as long as the errors in estimated individual βs are not perfectly correlated. This is a reason why most empirical work are conducted based on portfolios rather than individual assets.

The second stage cross-sectional regression is:

Where is the average stock return, is average risk-free rate. This step gives estimations for , , , and .

With Fama-French two-pass regressions, the following testing hypothesis can be tested.

*H1:*There is no other priced factor. Statistically, equals 0, or not significantly different from 0.

*H2:*All three factors are priced. Statistically, , , and are significantly different from zero. They are risk premiums associated with these risk factors. So normally, we expect them to be positive.

*H3:*To test if idiosyncratic risk is priced or not, the variance of residuals estimated from the first stage is added to the second stage regression. Then the second-stage regression becomes:

Where is the variance of residuals for each stock

*j*from the first-stage regression. Fama-French three-factor model implies that there is no other priced factor besides market, size and book-to-market factors. Thus, we expect equals 0. Idiosyncratic risk does not have a pricing role in the asset pricing theory with a belief that idiosyncratic risk can be diversified away in a portfolio. However, recently there are papers claiming idiosyncratic risk matters! Goyal and santa-Clara (2003) find a significant positive relation between average stock variance (largely idiosyncratic) and the return on the market. Ang, Hodrick, Xing, and Zhang (2006) find that monthly stock returns are negatively related to the one-month lagged idiosyncratic volatilities. They try to rationalize this using option theory, taking equity as an option on the company’s asset. Further, Fu (2009) shows that idiosyncratic volatilities are time-varying and finds a significantly positive relation between the estimated conditional idiosyncratic volatilities and expected returns.

**Empirical Analysis**

Monthly adjusted stock prices of 20 US stocks are obtained from Yahoo Finance. A list of these 20 stocks is included in the appendix. Return is calculated as , where is adjusted price after dividends and splits from Yahoo Finance. Monthly three factors and risk-free rate of return are all from French’s website

Table 1 gives the estimation of first-stage regression, including estimated , , and and variance of individual errors. Table 2 presents the results for second-stage regression, including , , , and . The testing results for hypothesis H1 and H2 can be drawn from this table by examining t-statistics. First, is barely significant. This accepts hypothesis H1 that there is no other priced factors. Second, none of the risk premiums associated to the three factors are significant. And the sign for size premium and value premium are negative, although not significant. These observations violate the hypothesis H2, where it predicts a significantly positive risk premiums for the risk factors. Table 3 further tests for hypothesis H3, whether idiosyncratic risk is priced or not, by adding the variance of individual errors from first-stage into the second-stage regression. Again, from the t-statistics associated with , we can conclude that the idiosyncratic risk is not priced since its risk premium is not significantly different from zero.

**Conclusion and Implications**

The empirical results tend to conclude that using five-year of monthly returns for these randomly picked 20 securities, Fama and French three-factor model seems not explaining the cross-sectional variations in these 20 stocks very well. A few reasons that I can think of are, first, as mentioned in the β estimation part, the estimated β is not the true β, but rather the true β plus an error. This “errors-in-variables” problem introduced in the second stage regression affects the hypothesis testing in the second-stage. Generally, it leads to a less likely significant coefficients estimation in the second stage regression. Furthermore, as noted in the literature, the three-factor model is a static model, which assumes a constant factor loadings and constant risk premiums for all factors. This is not necessarily true. As being shown in the empirical work for CAPM, conditional model, which takes the time-varying β and risk premium into account, always performs better than static CAPM. But exploring a conditional three-factor model is beyond the scope of this essay. Lastly, with static model, it’s a practical issue when it comes to the question that how long of the historical data we should use to estimate the factor loadings. Empirical, researchers use five-year of monthly data, 60 data points. But the persistence of the true β is a problem that can not be observed or detected. From this point of view, rolling regressions may do a better job here.

Overall, this empirical exercise reminds me of the cautions I should use while reading the research papers. These documented facts could be specific to certain period of time or empirical designs. But we can’t easily conclude here with using just 20 stocks and recent five-year of data. And we can never avoid the empirical issue that these models are tested using limited available data.

**References**

Ang A., Hodrick R., Xing Y., and Zhang X., 2006, The Cross-Section of Volatility and Expected Returns, Journal of Finance 61, 259-299

Basu S., 1983, The Relationship between Earnings’ Yield, Market Value and Return for NYSE Common Stocks: Further Evidence, Journal of Financial Economics 12, 129-156

Black F., 1993a, Estimating Expected Return, Financial Analysts Journal 49, 36-38

Black F., 1993b, Return and the Beta, Journal of Portfolio Management 20, 8-18

Banz R., 1981, The Relationship between Return and Market Value of Common Stocks, Journal of Financial Economics 9

Carhart M., 1997, On Persistence in Mutual Fund Performance, The Journal of Finance 52, 57-82

Fama E., and French K., 1992, The Cross-Section of Expected Stock Returns, The Journal of Finance 47, 427-465

Fama E., and French K., 1993, Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics 33, 3-56

Fama E., and French K., 1995, Size and Book-to-Market Factors in Earnings and Returns, The Journal of Finance 50, 131-155

Fu F., 2009, Idiosyncratic Risk and the Cross-section of Expected Stock Returns, Journal of Financial Economics 91, 24-37

Keim D., 1983, Size-related Anomalies and Stock Return Seasonality: Further Empirical Evidence, Journal of Financial Economics 12, 13-32

Reinganum M. R., 1983, The Anomalous Stock Market Behavior of Small Firms in January, Journal of Financial Economics 12, 89-104

Roll R., 1977, A Critique of the Asset Pricing Theory’s Test, Journal of Financial Economics 4, 129-176