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4.3Three Factor Model of Fama and French - Table of Figures

4.3Three Factor Model of Fama and French


Fama’s contribution is crowned by his work with his colleague French8 with whom he devised the three-factor model that extends the single market premium factor of traditional asset pricing theories. In their work, they show that sensitivity to size and value provides an adequate model for share price movements. The first factor is denoted as SMB (small minus big) which is the difference between the returns on diversified portfolios of small capitalisation stocks and a portfolio of large stocks constructed to be neutral with respect to book equity to market equity (BE/ME). The second factor is HML (high minus low) is the difference between the returns on diversified portfolios of high and low book equity to market equity shares constructed irrespective to size. The betas are evidently slopes in the regression. Their model is described in the equation below:



Ri is the return on asset i, Rf is the risk-free interest rate, and Rm is the return on the value-weight market portfolio.

Table 4 shows the summary statistics of the three factors’ regression on the July 1929 to June 1997 time-series. Fama and French split the sample on July 1963 to test whether the later period is unusual. The two sub periods are equal in length, 34 years. (Source: Fama and French 2000 p. 6)



Table 3: Summary Statistics for Monthly Percent Three-Factor Explanatory Returns

Rf is the one-month Treasury bill rate from Ibbotson Associates. Ri is the value-weight return on all NYSE, AMEX, and NASDAQ stocks with book equity data for the previous calendar year. At the end of June of each year t (1926 to 1996), stocks are allocated to two groups (small or big) based on their June market capitalisation, ME (market equity worked out by stock price times shares outstanding), is below or above the median for NYSE stocks. Stocks are allocated to three book-to-market equity (BE/ME) groups (L, M, or H) based on breakpoints for the bottom 30 percent, middle 40 percent, and top 30 percent of the values of BE/ME for the NYSE stocks in their sample. Six portfolios (S/L, S/M, S/H, B/L, B/M, and B/H) are formed as the intersections of the two size and the three BE/ME groups. Value-weight monthly returns on the portfolios are calculated from July of year t to June of t+1.

The average value of the market premium (Rm-Rf) for the full 68-year sample period is 0.67 percent per month (t-statistic = 3.34) which is about 2.4 standard errors from zero. This strong market premium in returns in not surprising. There is also a reliable value premium in returns. The average HML return for the full July 1929 to June 1997 is 0.46 percent per month (t-statistic = 4.24). The size effect however is modest in comparison with the previous results. The average SMB return for July 1929 to June 1997 is 0.20 percent per month (t-statistic = 1.78). Perhaps the reason for this is that SMB is neutral with respect to BE/ME because small stocks tend to have higher BE/ME than big stocks, and a size premium that is not neutral with respect to BE/ME in part reflects the value premium in returns.

4.4Characteristics Model of Daniel and Titman


The three factor model confirms that firm sizes and book-to-market equity ratios are both highly correlated with the average returns of stock market instruments. The Fama and French explanation of their model associates the size and value characteristics with returns. They explain that the characteristics are proxies for non-diversifiable factor risk, and that the sensitivities of underlying stocks with the factors (factor betas) directly influence returns. Daniel and Titman rebut this idea. In their article in 1997, they provide evidence that the return premia on small capitalisation and high book-to-market stocks does not arise because of factor betas. They voice the role of the characteristics rather than the covariance structure of returns that appear to explain the cross-sectional variation in stock returns.

Abundant evidence exist that cross-sectional pattern of stock returns can be explained by characteristics such as size, book-to-market ratios, leverage, past returns, and dividend-yield among others. Fama and French’s examination of these variables simultaneously concludes that the cross-sectional dispersion in expected returns can be satisfactorily explained by two of these characteristics, namely: size and book-to-market ratio. Beta, the risk measure in the traditional capital asset pricing model explains very little of the cross-sectional variation in expected returns once size is taken into account. In order to determine expected returns, Daniel and Titman suggest a model of the “Return Generating Process Firm characteristics” (Daniel et. al. 1997) rather than factor loadings.

Their characteristic-based pricing model, in contrast to the factor pricing model, assumes that high book-to-market stocks realise a return premium that is unrelated to the underlying covariance structure. As in Fama and French’s model, covariances are stationary over time and can be described by a factor structure. Here, a time-invariant J-factor describes the variance-covariance matrix of returns is assumed. (Source Daniel and Titman 1997 p9)



where is the loading of firm i on factor j and is the return on factor j at time t. However, in contrast to the previous models, factor loadings do not describe expected returns. Instead, it is assumed that expected returns are a function of the observable, slowly varying firm attribute or characteristic: (source Daniel and Titman 1997 p10)



As in the factor model, the innovations in are negatively correlated with the returns on the stock, but is not directly related to the loadings on the distressed factors. What is unique about the characteristics model is that firms exist that load on the distressed factors but which are not themselves distressed, and therefore have a low and thus low returns and vice versa. If the characteristics model holds, then a series of negative shocks to a specific factor may be followed by some stocks that, despite their high loadings on that factor, are still not distressed. The factor model suggests that these firms should still earn the distress premium, because they behave like other distressed firms. In contrast, the characteristics model suggests their returns behaviour does not matter: if they are not distressed they will not earn the premium. This model implies that a clever investor can earn the book-to-market return premium without loading on any common factors.

The regression results for the characteristic-balanced portfolios of Daniel and Titman support these findings. Table 5 presents each of the coefficients and t-statistics from the following time-series regression of the zero-investment portfolio returns, described below, on the excess-market, SMB and HML portfolio returns:



The regressions are over the period July 1973 to December 1993. The left hand side portfolios are formed based on size (SZ), book-to-market (BM), and pre-formation HML factor loadings; their returns are calculated as follows. From the resulting forty-five returns series, a zero-investment returns series is generated from each of the nine size and book-to-market categories. These portfolios are formed, in each category, by subtracting the sum of the returns on the 4th and 5th quintile factor-loading portfolios from the sum of the returns on 1st and 2nd factor-loading portfolios. The first nine rows of the table present the t-statistics for the characteristic-balanced portfolio that has a long position in the low expected factor loading portfolios and a short position in the high expected factor loading portfolios that have the same size and book-to-market rankings. The bottom row of the table provides the coefficient estimates as well as the t-statistics for this regression for a combined portfolio that consists of an equally-weighted combination of the above nine zero-investment portfolios. (Table below: source Daniel et. al. 1997 p18)


Table 4: Regression Results for the Characteristic-Balanced Portfolios

These are "characteristic-balanced" portfolios, since both the long and short positions in the portfolios are constructed to have approximately equal book-to market ratios and capitalisations. In the model, where no transaction costs are assumed, such a portfolio construction costs nothing initially.

The characteristic-based model predicts that the average return from these zero cost characteristic-balanced portfolios should be indistinguishable from zero. The results reported in Table 5 reveals that all but one of the as from the time-series regressions of the nine individual characteristic-balanced portfolio returns on the factor returns are positive, and three of the nine have t-statistics above two which is consistent with the characteristic- based pricing model but inconsistent with the factor pricing models.

The latest study on this topic by Daniel and Titman (2001) replicated the methodology of their 1997 tests on the Japanese market from 1975 to 1997. The authors maintain that Japanese stock returns are even more closely related to their book-to-market ratios than their U.S. counterparts. The tests reject the Fama and French three-factor model, but fail to reject the characteristic model. The results, however, may be subject to sample bias and might not be applicable to other markets. Daniel et. al. (2001) conclude that this is a question that stimulates further research.
2014-07-19 18:44
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