FAMA AND FRENCH FACTORS AND
THEIR REGRESSION TENDENCIES:
IMPROVING THE ACCURACY OF RISK
ASSESSMENTS
TIM DROST (1472739) University of Groningen, Master’s Thesis in the field of Finance
Supervisor: Dr. A. Plantinga April 2012
ABSTRACT
The Fama and French three-factor model aids investors in making investment decisions. Historical data is often used to extrapolate past trends into the future. But what if historical estimates are not accurate predictors for future values? Blume (1971) found that market betas in a CAPM tend to regress to the mean, i.e. the overall market beta of one. Similarly, this study tests to what extent beta, , and coefficients in a Fama and French three-factor model are subject to mean reversion. Using a 1990-2010 U.K. sample, I demonstrate that all three coefficients regress to the mean. In addition, while the regression rates of beta and coefficients are fairly stable over time, for coefficients these are highly volatile. The results are robust for composing portfolios based on median BE/ME splits, including negative-BE firms, and analyzing longer periods with lower frequency data.
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I. INTRODUCTION
Quoting from Peter L. Bernstein’s book, Against the Gods: The Remarkable Story of Risk: “The revolutionary idea that defines the boundary between modern times and the past is the mastery of risk: the notion that the future is more than a whim of the gods and that men and women are not passive before nature” (1996: 1). Since Markowitz (1952) laid the groundwork for modern portfolio theory by quantifying investment risk, mastery of risk has possibly become one of the most important concepts in finance; academics and practitioners have become increasingly interested in modeling and forecasting stock market volatility.
But what defines risk? And how can we assess risk properly so that we make the right investment decisions? It is “[s]till of controversy … what constitutes risk and how it should be measured” (Blume, 1971). This quote still holds, since it is exactly this quest that, roughly sixty years after Markowitz’ (1952) insight, still drives researchers to develop new models and redefine existing ones. This study involves the latter. In particular, the main subject of the study is the three-factor model developed by Fama and French (1993). It is regarded as one of the most influential extensions of the Capital Asset Pricing Model (CAPM); since its origination the model has won in support and it is widely used ever since.
Most asset pricing models are estimated with historical data, although investors recognize that a significant margin of error surrounds the results of such an approach. The problem is that we often erroneously assume – beyond the domain of the theoretical model – that past firm performance is representative of future performance. Blume (1971) exposed the ramifications of this extrapolation bias when he examined betas in succeeding periods: he found that historical betas are not accurate predictors of corresponding betas in a subsequent period. Oversensitivity to estimation error makes the results even more tentative and is the single most important limitation of mean-variance optimization (Bernstein, 1996: 258; Michaud and Michaud, 2008). Hence, not taking into account the margin of error when extrapolating historical estimates will lead portfolio managers to form suboptimal portfolios with poor out-of-sample performance.
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what Blume (1971) discovered for betas in a single-factor equilibrium model? Blume’s (1971) research as well as most important formal tests are dated, but these techniques are still being used on a widespread basis. Therefore, it seems appropriate to reexamine the theories. Third, is it possible to correct assessments and consequently improve the accuracy of forecasts? Along these lines this paper tries to extend the existing literature on the topic, since to the best of my knowledge there are no published papers that pursue a similar line of investigation.
The data set used in this paper consists of all U.K. quoted firms over the years 1990 to 2010. By combining the methodologies employed by Fama and French (1993) and Blume (1971), a two-step procedure is followed: in the first stage, time-series regressions are run for each individual security to estimate Fama and French coefficients; in the second stage, the coefficients’ rates of regression are determined through running cross-sectional regressions. The results show a tendency for all three coefficients to regress towards the mean over time. Correcting for this regression tendency by using historical rates of regression results in considerably more accurate assessments of future values. This even holds for coefficients, despite of the highly volatile rate of regression of coefficients.
The remainder of this paper is structured as follows. The next section briefly explores the preceding developments with regard to the assessment of risk, before discussing Blume’s adjusted beta, the Fama and French three-factor model, and regression to the mean. After discussing the methodology in Section III and the data set and its basic properties in Section IV, Section V describes the results of the research. The last section states the conclusions of the study and discusses the results.
II. LITERATURE REVIEW
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variance of return, the “undesirable thing” that investors try to minimize (Bernstein, 1996: 252). Studying mutual correlations between stock price developments Markowitz’s key insight was the strategic role of diversification – by taking into account correlations, it is possible to build portfolios that generate an optimal level of return for a given level of risk. Hence, the concept of efficient portfolio was introduced. The practical application of Markowitz’ mean-variance optimization was established by Sharpe (1964), Lintner (1965) and Mossin (1966), who are considered to be the founding fathers of the Capital Asset Pricing Model (CAPM). The model relates the variance of securities to the market as a whole – in the limit, beta i.e. systematic risk is the only relevant risk measure – and is based on the axioms of mean-variance optimization, market efficiency, and homogenous expectations. As a result, at equilibrium a portfolio has the same composition of risky assets for all investors; individual portfolios can differ in the allocation of the risk-free asset and the risky assets.
The CAPM was the terminus a quo for the studies of Blume (1971) and Fama and French (1993). Blume (1971) tested the practical implementability of the model by measuring the accuracy of simple extrapolations of historical betas; and Fama and French (1993) examined whether beta is indeed the only relevant risk measure. The following sub-sections will go deeper into these studies.
Blume’s adjusted beta
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or small portfolios are far less precise. A second test shows that there actually is a tendency for betas to change gradually over time; lower risk portfolios get riskier in the subsequent period and higher risk portfolios become less risky. The risk coefficients tend to regress towards the means, i.e. the overall market beta of one. The larger the departure from the mean, the more fierce the mean reversion.
Regressing estimated betas of one period on the values estimated in a previous period, Blume (1971) found the following relationship1 which is now used by many investors to modify historical betas and obtain more accurate forecasts:
where is the beta on stock i in the later period and is the beta on stock i for the earlier
period. The tendency of high betas to overpredict and low betas to underpredict future betas, is overcome by Blume’s adjusted beta.
Intuition – Why might observed future betas differ from historical betas? There are two possible explanations. The first is that the risk of the security or portfolio changes through time, because for example the risk of existing projects that companies undertake may tend to become less extreme over time, or because new projects may tend to have less extreme risk characteristics than existing projects (Blume, 1975). The second explanation is that betas are estimated with a random error, and the larger the random error, the less predictive power betas possess. Specifically, an estimated beta is a function of the true underlying beta and a function of sampling error. If we compute a very high (low) beta for a stock, there is an increased probability that we have a positive (negative) sampling error. Since there is no reason to expect positive (negative) sampling errors for stocks to be followed by positive (negative) sampling errors in a next period, we should find that betas on average tend to converge to 1 in successive time periods (Elton, Gruber, Brown, and Goetzmann, 2010). In a later work (1975), Blume responded to this issue and demonstrated with empirical analyses that part of the observed regression tendency represents real non-stationarities in the betas, and part stems from the sampling error and order bias, although the latter is of no overwhelming importance. Vasicek (1973) proposed a different method by introducing a Bayesian estimation technique. This technique includes a weighting procedure, where observations are adjusted dependent on
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the size of the uncertainty. The weight placed on a stock’s beta is inversely related to the stock’s standard error of beta. The larger the sampling error, the greater the adjustment: observations with large standard errors are adjusted further toward the mean than observations with small standard errors. The weights reflects the relative confidence of the researcher in each of the estimates.
Both Blume’s (1971) and Vasicek’s (1973) technique suffer from its own potential source of bias. Blume’s technique results in a continued extrapolation of a upward or downward trend in average betas observed in earlier periods: using historical rates of regression to adjust assessments, if the average beta increased over the previous periods, it is also assumed that average betas will increase over the next period. On the other hand, the technique of Vasicek tends to pull betas in a downward direction; this is because high-beta stocks have larger standard errors of beta than low-beta stocks and, as a result, have their betas adjusted by a greater percentage. Unless there are predictable trends in average betas or correlation coefficients, the effect of these biases on the forecast accuracy of both techniques will be random from period to period. The evidence on the choice between the Blume and Bayesian adjustment is therefore mixed. With respect to other estimation techniques, empirical evidence indicates that either the Bayesian adjustment or the Blume adjustment is preferred to forecast future betas, since they have a higher forecasting power than unadjusted betas and the historical correlation matrix (Elton, Gruber, and Urich, 1978; Klemkosky and Martin, 1975). However, although most important formal tests as well as Blume’s (1971) own research are dated, these techniques are still being widely-used.
Fama and French’s three-factor model
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practitioners are able to apply the model as long as the proxies are mean-variance efficient, although the danger is that it is unlikely to have the same empirical fit in the future.
In the course of time, various anomalies of the CAPM have arisen and a host of studies revealed that the model seemed incapable of capturing all relevant risk factors. For example, Fama and MacBeth (1973) – considered to be the empirical breakthrough of the CAPM – already found that there are variables in addition to beta that systematically affect returns. These omitted variables appeared to be related to non-diversifiable or firm-specific risk, as illustrated by the residuals in the regression. During the late eighties of the previous century the literature again picked up on this issue of idiosyncratic risk, firstly initiated by Graham and Dodd (1937) but largely in the background during the subsequent rise of the portfolio theory. Various researchers developed models incorporating firm-specific variables, that rely on for example historical dividends, firm size, earnings and/or book-to-market values (e.g. Banz and Breen, 1986; Basu, 1983; Campbell and Schiller, 1988; Fama and French, 1988, 1992; Hodrick, 1992; and Stattman, 1980). This led Fama and French (1993) to develop a multi-index model based on the overall market factor beta, size, and book-to-market equity. Conducting their research on monthly stock returns for stocks traded on the NYSE, Amex, and NASDAQ from 1963 to 1990, they found that a cross section of average returns is negatively related to size and positively related to book-to-market equity – implying that small cap and value portfolios experience higher expected returns than respectively large cap and growth portfolios. Furthermore, including the two additional factors improved the explanatory power of the model from roughly 70% to 90%. In short, the results show that size and book-to-market equity capture shared variation in stock returns that is missed by the market.
Fama and French (1993) construct two portfolios to test whether size and book-to-market equity affect returns, portfolios and , which have returns that mimic the impact of the variables.2 The rationale behind the construction of these mimicking portfolios, instead of using size and book-to-market equity as direct inputs into the regression, is twofold. First, in this way a set of variables that cannot be observed at frequent intervals is converted into a set of traded assets that have prices and returns that can be observed at any moment of time (Elton, Gruber, Brown, and Goetzmann, 2010). Second, in the sense of the arbitrage pricing theory (APT) factor loadings have a clear interpretation as risk-factor sensitivities (Fama and French, 1993).
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An additional finding, especially relevant for this study, is that “adding and to the regressions collapses the [market] βs for stocks toward 1.0: low βs move up toward 1.0 and high βs move down” (Fama and French, 1993). They attribute this effect to the correlation between the market and or . Do and alleviate the tendency of beta to regress to the mean? In other words, is the regression tendency of beta (partly) overcome by extending the CAPM to the three-factor model, eliminating some of the random noise surrounding beta and thereby reducing the sampling error?
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investors extrapolate past experience too far into the future. This statement underlines the practical relevance of this study. According to Bernstein (1996: 170), “[r]egression to the mean motivates almost every variety of risk-taking and forecasting.” Over- and undervaluation simply mean that fear or greed has driven the stock’s price away from an intrinsic value to which it is certain to return.
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Regression to the mean
How should regression to the mean be rationalized? Blume (1975) speaks of “some unstated economic or behavioral reasons”. From an economic perspective, it presumably depends on the degree of competitive rivalry within an industry. Fast-growing, profitable firms find that competition increases, which slows their rate of growth and erodes their profits. With respect to book-to-market equity for example, growth stocks are associated with persistently high earnings (Fama and French, 1993). Economic theories suggest that most forms of competitive advantage cannot be sustained indefinitely, because earning excessive profits invites competition. Hence, in time growth stocks find themselves surpassed by value stocks. A behavioral explanation is, as discussed above, that cognitive biases cause stocks to be mispriced. When sufficient investors eventually recognize the over- or undervaluations and correct the accumulated valuation errors, regression to the mean takes over. Finally, mean reversion is essentially a driving force of nature – normal distributions are present all around us. Consider for example the length of human beings or animals. When regression to the mean were not at work “the world would consist of nothing but midgets and giants” and would go “completely haywire or running out to extremes we cannot even conceive of” (Bernstein, 1996: 167).3
“A theoretical model… should not be judged by the accuracy of its assumptions but rather by the accuracy of its predictions” (Blume, 1971). The preceding sub-sections examined the progress made in theory and research on Blume’s adjusted beta and the Fama and French three-factor model. The goal of this study is not to expand on the theoretical justification of the theories, but rather on the practical implementability. In other words, the study aims to provide a prescriptive theory, by offering advice on how people can improve their decision making and get closer to the normative ideal. Specifically, this paper examines how accurate the Fama and French three-factor model is in its predictions. To quote Blume (1971) once more, “[n]o economic variable including the beta coefficient is constant over time.” Blume’s adjusted beta is often used as a means to improve beta estimates, but what about
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market equity and size? Given the dynamics of economic variables, do these two variables also have to be adjusted for regression tendencies?
III. METHODOLOGY
This research focuses on the Fama and French three-factor model:(1)
where is the excess stock or portfolio return; the excess market return; the portfolio mimicking the risk factor in returns related to size; and the portfolio mimicking the risk factor in returns related to book-to-market equity.
The main goal is to find out whether the property found by Blume (1971) – in the context of a one-factor model CAPM betas have a tendency to regress toward the mean – also holds for the coefficients of the three-factor model developed by Fama and French (1993). Thus, the methodology employed throughout this research largely follows the procedures of Fama and French (1993) and Blume (1971), although some minor modifications or additions were made (when I deviate from the original methods I will specifically mention it). The research consists roughly of two parts. The first part tests how well the model fits the data employed in this paper. The second part investigates the mean reversion hypotheses. Both involve estimating model (1), so firstly I will elaborate on the design of the independent variables.
Construction of the independent variables
and – Each year stocks are ranked based on size, measured at June 30st of year ,
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of year . Here I depart from the methods deployed by Fama and French (1993) in two ways. First, I employ weekly instead of monthly returns. Further on, testing the presence of mean reversion, weekly data allows using shorter time periods while preserving the number of observations. By doing this, more time periods can be compared, thus increasing the quantity of the results. Moreover, shorter time periods comprise more live stocks and thus enlarge the sample size. Second, where Fama and French (1993) work with linear returns and calculate value-weighted portfolio returns, I compute portfolio returns with equally weighted compounded stock returns.
The six portfolios described above are used to calculate the two independent variables (small minus big) and (high minus low). Portfolio is the difference, each week, between the simple average of the returns on the three small-stock portfolios (S/L, S/M, and S/H) and the simple average of the returns on the three big-stock portfolios (B/L, B/M, and B/H). Portfolio is the difference, each week, between the simple average of the returns on the two high BE/ME portfolios (S/H and B/H) and the average of the returns on the two low-BE/ME portfolios (S/L and B/L). and are meant to mimic the risk factors in returns related to size and book-to-market equity, respectively.
– Finally, the excess market return is the third independent variable: is the equal-weighted weekly return on all stocks4 included in the sample period; and is the weekly nominal yield on a 10-year U.K. gilt.5
First part of the research
To test the fitness of the model and explanatory power of the independent variables, 25 portfolios are formed in a similar way as the previously discussed six size-BE/ME portfolios. Each year stocks are ranked independently on size and book-to-market equity and split in five size quintiles and five BE/ME quintiles. Again, negative-BE firms are excluded. From the intersections 25 portfolios are formed. The weekly excess portfolio returns serve as dependent variables and each of the 25 portfolios is subjected to a separate OLS time series regression of the form of equation (1). The fitness of the model is discussed in the Data section.
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Positive- and negative-BE stocks
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Second part of the research
The mean reversion conjecture is tested on all three coefficients. The method described below is done in threefold – either for , , or . It is important to emphasize however that for a given period the same securities are included when testing for either one of the three coefficients. Moreover, the risk coefficients of equation (1) are estimated simultaneously. The difference between the three tests only occurs afterwards when the securities are ranked – this is the part of mean-reversion testing that is actually done in threefold. I will describe the method for , but the same steps are applied in case of and .
Each time two successive periods of two years are compared.6 For every individual security, coefficients of equation (1) are estimated for both periods separately by performing OLS time-series regressions.7 Subsequently, estimates of (i.e. betas estimated over the first two years) are ranked in ascending order. Portfolios are formed of N securities (where N = 1, 5, 10, 20, 50, or 100) by the following principle: the N securities with the N smallest ’s form the first portfolio; the following N securities with the next N smallest ’s form the second portfolio; and this process continues until the remaining n<N securities fill the last portfolio. This stratified sampling method is straightforward and easy to implement. However, one have to take into account that it may lead to potential biases, considering that the split offs of N are not based on any theoretical grounds. In particular the last portfolio with n<N securities could bias the results, especially if n is small, since this results in an undiversified portfolio.8
Estimates of are compared to the respective estimates of (i.e. betas estimated over the second two years): by investigating how the two are correlated; by examining actual estimates of portfolios of 100 securities; and by performing Chow tests for portfolios of 100 securities. In addition, executing the process independently for all three coefficients, the following cross-sectional regressions are run:
(2)
(3)
, (4)
6 Only positive-BE stocks with complete data over all four years are included. 7 N.B. halfway a two-year period the portfolios and
are recomposed; the securities are rearranged by ranking on the most recent values of size and book-to-market equity.
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where , , and are risk measures calculated using the first two years of data; and , , and are the risk measures calculated using the subsequent two years of data. Where the former might be regarded as an assessment of the future risk, the latter can be regarded as the realized risk. Finally, I test whether forecasts can be improved by applying historical rates of regression to correct assessments. The outcomes of the second part are discussed in the Results section.
IV. DATA
The data set used in this paper consists of all U.K. quoted firms over the years 1990 to 2010. The sample is taken from Thomson Reuters Datastream, a financial database accessible at the University of Groningen. To construct the data set, U.K. firms were extracted from the live constituents list of the FTSE All Share Index (LFTALLSH),9
plus, in order to mitigate survivorship bias, from the dead U.K. quoted equity list (DEADUK).10
Return indexes including dividends and capital gains (RI) are collected for these files. In addition, size is measured by share price times the number of ordinary shares in issue (MV). Finally, book-to-market equity is defined by the common shareholders' investment in a company (WC03501) plus deferred taxes and investment tax credit (WC03263), divided by market equity (MV).
Descriptives – Table 111 shows that the composition of my 1990-2010 U.K. sample differs from the 1963-1991 U.S. sample of Fama and French (1993) in three ways. First, the number of firms are more evenly distributed over the size and book-to-market quintiles. Second, the range in average firm size is substantially wider: the average firm sizes of the 5 small-size portfolios are approximately 2 to 3 times smaller than the 5 small-size portfolios of Fama and
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The terms in capitals in this section refer to the Datastream list/mnemonic. 10
The Datastream DEADUK list comprises companies which have been, but are no longer, listed in the U.K. The way dead stock is processed in Datastream creates a problem. That is, when the stock goes ‘dead’ (i.e. the company delists) the last reported return index value (RI) – or any other variable – keeps showing up at any other interval after that, instead of a customary ‘NA’ input. I dealt with this by checking the last seven (weekly) RI values for each period. When these are all equal to each other, I interpret this as a delisted stock and it is deleted from the sample. This process is done automatically because of time/effort considerations, and the choice to compare the last seven values is rather arbitrary. I am well aware that this method involves a margin of error: firms that delist six weeks or less before the ending of a period stay incorrectly in the sample, and firms that ‘accidently’ have seven equal RI values in a row at the end of the period are wrongfully deleted.
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French (1993). Similarly, the average firm sizes of the 5 big-size portfolios exceed the respective ones of Fama and French (1993) with a factor 12 to as high as 83. Hence, the gap between small and big firms has greatly expanded over the last decades. As a result, the data exhibits a very skewed distribution of value toward the biggest size quintiles, evident given the fact that the 5 big-size portfolios capture 97% of the average combined value of stocks in the 25 portfolios. Third, average book-to-market ratios support this higher degree of dispersion, but what is especially striking are the three extremely high BE/ME ratios in the high book-to-market quintile. Interestingly, these extreme values are caused merely by DEADUK firms. Several of these firms have been traded on the exchange for less than a fraction of its book value, sometimes during multiple years. Normally such outliers could have a significant effect on OLS regressions (e.g. causing non-normality). However, since BE/ME values are only used for ranking purposes, and not as direct regression variables, the effect of these large outliers is insignificant. That is, the size of the ratio does not influence the percentile splits. In addition, this study focuses on precisely these outer values.
TABLE 1
Descriptive Statistics for 25 Stock Portfolios Formed on Size and Book-to-Market Equity: 07/1990 to 06/2010
Size quin-tiles
Book-to-market equity quintiles
Low 2 3 4 High Low 2 3 4 High Average of annual averages of
firm size (in millions)
Average of annual BE/ME ratios for portfolios Small 8.0 8.2 8.1 7.7 6.6 0.17 0.42 0.70 1.08 39.64 2 38.8 37.8 37.6 39.3 38.6 0.18 0.42 0.69 1.08 3.65 3 138.0 140.6 132.1 132.8 129.9 0.19 0.42 0.69 1.08 3.74 4 634.2 643.9 636.7 606.1 703.4 0.18 0.41 0.68 1.08 23.15 Big 82,587.1 245,092.8 200,982.0 92,947.0 27,502.2 0.18 0.42 0.68 1.06 91.05
Average of annual % of market value in portfolio
16 TABLE 2
Summary Statistics for the Weekly Dependent and Explanatory Returns in the Regressions of Tables 3 to 8: 07/1990 to 06/2010, 1043 observations
Panel A: Explanatory returns
in % Autocorr
1 lag
Correlations Variable Mean StDev Median Min Max Skew Kurt RM-RF SMB HML RM-RF -0.18 1.84 -0.01 -17.00 6.06 -1.62 10.03 0.33 1.00
SMB -0.22 1.40 -0.15 -11.60 6.21 -0.90 7.70 0.02 -0.33 1.00
HML 0.24 1.03 0.19 -5.29 4.56 0.32 2.97 0.30 -0.15 -0.25 1.00
Panel B: Dependent variables - Excess returns on 25 portfolios formed on size and book-to-market equity
Size quintiles
Book-to-market equity quintiles
Low 2 3 4 High Low 2 3 4 High Mean (in %) Standard deviation (in %)
Small -0.61 -0.47 -0.43 -0.32 -0.19 2.70 2.36 2.23 2.04 1.78 2 -0.46 -0.31 -0.29 -0.12 -0.06 2.49 2.22 1.98 1.83 2.01 3 -0.29 -0.11 -0.08 -0.02 -0.02 2.49 2.15 2.02 1.97 2.56 4 -0.17 -0.13 -0.13 0.01 -0.04 2.54 2.43 2.63 2.39 2.61 Big -0.12 -0.06 -0.03 0.03 0.04 2.56 2.44 2.63 2.45 2.93
Median (in %) Min (in %)
Small -0.54 -0.36 -0.37 -0.22 -0.14 -13.69 -14.81 -10.93 -12.49 -14.31 2 -0.28 -0.15 -0.18 -0.02 0.10 -18.08 -15.72 -21.06 -13.14 -15.99 3 -0.08 0.10 0.08 0.12 0.23 -17.83 -15.61 -14.85 -19.61 -20.30 4 0.03 0.02 0.04 0.10 0.14 -18.30 -17.79 -18.75 -20.19 -23.05 Big 0.05 0.15 0.08 0.20 0.19 -19.59 -18.94 -24.74 -21.53 -21.70
Max (in %) Skew
Small 11.63 7.89 11.86 10.32 6.98 -0.12 -0.73 -0.23 -0.40 -0.77 2 8.74 8.60 12.53 7.03 6.84 -1.03 -0.97 -1.48 -0.92 -1.18 3 8.58 8.14 7.22 6.58 12.68 -0.96 -1.03 -1.17 -1.55 -0.94 4 8.97 11.44 12.51 9.29 10.21 -0.96 -0.88 -1.26 -1.12 -1.27 Big 13.05 15.28 17.05 10.72 22.10 -0.88 -0.54 -1.11 -1.08 -0.87
Kurt Autocorr. (1 lag)
17 Summary Statistics – Table 2 describes the summary statistics and reveals the unfavorable overall market conditions of the last twenty years. I found negative returns for 22 of the 25 portfolios and an average excess market return of -9.3% per year,12 so an investor would have been far better off by holding 10-year U.K. gilts over this period – obviously to a great extent due to the occurrence of two major crises. Negative skewness, which is displayed by all 25 portfolios and defined by the propensity to generate negative returns with greater probability than positive returns, support these observations. In particular small stocks and low book-to-market equity stocks performed badly, considering the almost monotonically increasing relationship between size or book-to-market equity on the one hand and excess portfolio return on the other hand. The positive relationship between size and return is in contrast with the premise of Fama and French (1993), i.e. that small stocks tend to perform better than the market as a whole. This contradicting finding is also reflected in the negative average return, which is consistent with other studies (e.g. Dimson and Marsh, 1999; Gompers and Metrick, 2001; and Gustafson and Miller, 1999).
FIGURE 1
Yearly Excess Market Returns, SMB Returns, and HML Returns
By illustrating the yearly returns of , , and , Figure 1 can provide more insight on the subject. Yearly portfolio returns are mostly negative, whereas
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portfolio returns are mostly positive. It is also evident that, on a yearly basis, is highly correlated with the market portfolio – the two lines move considerately parallel in the graph and the yearly correlation between and is 0.70. On the other hand, the line frequently moves in the opposite way and is negatively correlated with both and (-0.47 and -0.40, respectively). The most striking result is the good performance of the portfolio during or closely after crises (periods 2000-2002 and 2007-2009). Value stocks are valuable, especially in declining markets. Suppose one invested a priori in the portfolio on 30 June 1990, i.e. went long in high book-to-market equity stocks and short in low book-to-market equity stocks (following the strategy of Fama and French (1993) and reallocating its investment every year on the 30th of June), he or she would have made a return of 255% by 30 June 2010. In contradiction, pursuing a similar investment strategy for portfolio would have produced a return of -225%.13
Others characteristics of the data are (see Table 2): most variables exhibit a leptokurtic distribution, which is very common to financial data; the first-order autocorrelation coefficients (one-week lag) are higher for small firms than for large firms, but not disturbingly high; and there are no complications due to multicollinearity. The correlation of -0.25 between the mimicking returns for the size and book-to-market factors shows that the portfolio is largely free of the size factor in returns, and vice versa, although the correlation is not as low as what Fama and French (1993) computed (-0.08).
Model fit – Finally, Table 3 examines how well the Fama and French three-factor model fits the 1990-2010 U.K. data. In coherence with the original research, I find that the three stock-market factors capture strong common variation in stock returns. Several findings support this inference. First, all coefficient estimates are significant, except for one of the coefficients (t( ) = -1.094) and two of the coefficients (t( ) = 1.413 and t( ) = 0.359). Notice that the corresponding coefficients are close to zero, which explains the low t-values. Second, the way the factor loadings are structured resembles the range in factor loadings found by Fama and French (1993): the slopes on are related to size and the slopes on are related to book-to-market equity. Specifically, in every book-to-market quintile the slopes on decrease monotonically from smaller- to bigger-size quintiles, and in every size quintile the slopes on increase monotonically from lower- to higher-BE/ME quintiles. There are however two differences. The first difference is with regard to the distribution. Because I
19 TABLE 3
Regressions of Excess Stock Returns on the Excess Market Return and the Mimicking Returns for the Size and Book-to-Market Equity Factors: 07/1990 to 06/2010, 1043 weeks
Dependent variable: Excess returns on 25 portfolios formed on size and book-to-market equity
Size quin-tiles
Book-to-market equity quintiles
Low 2 3 4 High Low 2 3 4 High
20
employ log-price relatives instead of linear returns, the values are normally distributed around a mean of zero.14 Nonetheless, the implications are the same. The second difference involves the coefficients. Although the distribution of factor loadings on concur with Fama and French (1993), the implications are not quite the same. Recall from Table 2 and Figure 1 that portfolio experienced a negative average return over the sample period. Consequently, in this case the higher factor loadings of small stocks imply that these stocks are more sensitive to the negative returns of , thus resulting in higher losses. The negative factor loadings of big stocks reflect opposite behavior – big stocks will have positive returns when portfolio returns are negative. The third finding that supports the appropriate specification of the three-factor model are the values for . In well-specified asset-pricing models, the intercepts in excess-return regressions (in this case, combined with zero-investment portfolios and ) should be indistinguishable from zero. In other words, such a model captures all relevant risk premiums. For the examined stock portfolios, the intercepts are close to zero: varies between -0.1% and 0.1%, while 10 out of 25 intercepts
are insignificant indicating that they are not different from zero. Finally, the model fits the data very well considering the R2-values of 0.51 to 0.90.15
In summary, although the data used in this study differs in several aspects from the data used by Fama and French (1993) (e.g. wider distribution of size and book-to-market values, mostly negative portfolio returns due to harsh market conditions, and a negative average portfolio return), the three-factor model persists to do a good job explaining the cross-section of average stock returns. The market return, firm size and book-to-market equity indeed proxy for sensitivity to common risk factors in stock returns.
V. RESULTS
As discussed in the Methodology section, to test the mean reversion conjecture Fama and French coefficients are estimated for periods of two years. The total U.K. sample of 1990-2010 is split into ten equal time periods and each time two successive two-year periods are
14 Calculating and with simple returns, the corresponding coefficients and take values on a scale of roughly 0 to 1: = 1 = small cap and = 0 = large cap; and = 1 = high BE/ME and = 0 = low BE/ME. Calculating and with log returns, the corresponding coefficients and take values on a scale of roughly –1 to 1: = 1 = small cap and = –1 = large cap; and = 1 = high BE/ME and = –1 = low BE/ME. 15
21
compared. Although different properties are examined in Tables 4 to 7, they all involve the same successive sub-periods. Table 8 evaluates the effectiveness of using historical rates of regression to correct risk assessments.
Mean reversion testing
Correlations – Table 4 presents product moment and rank order correlation coefficients between estimated coefficients in two subsequent periods, where the estimate for the first period represents the assessment of future risk and the estimate for the second period the realized risk. The table demonstrates the power of diversification: while for single securities the correlation is often quite weak, especially in case of , correlations for portfolios of 100 securities are mostly between 0.9 and 1.0. For example, for assessments based on data from July 1990 to June 1992 and evaluated against data from July 1992 to June 1994, product moment correlations increase from 0.52 for single securities to 0.99 for portfolios of 100 securities (see Panel A). In the same manner, correlations increase from 0.57 to 0.94 (see Panel B) and correlations from 0.21 to 0.96 (see Panel C). More diversified portfolios thus enable to explain a higher degree of variation in Fama and French’s risk measures by assessing previous data than single securities. Note however that and perform better in this respect than .
Although the effect of diversification is present in all three coefficients and across most periods, there are some anomalies. While some are caused by biases in the method of forming portfolios,16 this does not apply to the strikingly weak correlations of estimates between July 2000 to June 2002 and July 2002 to June 2004. Extrapolations of July 2000 – June 2002 book-to-market equity values turned out to be very inaccurate for the subsequent two years. This outcome reflects the impact of the Internet bubble. During this speculative bubble firms in the internet sector and related fields experienced a huge increase in their equity value. However, in 2000-2001 this bubble bursted and a large correction on equity values took place. The weak correlations are thus a representation of the large correction that occurred due to the aftermath of the Internet bubble. (Also note how stands in sharp contrast to and during these periods, which exhibit similar behavior compared to other periods.)
22 TABLE 4
Product Moment and Rank Order Correlation Coefficients of the Fama and French Factors in Two Successive Periods for Portfolios of N Securities PANEL A: Correlation Coefficients of
# of Sec./ Port. 7/90-6/92 and 7/92-6/94 7/92-6/94 and 7/94-6/96 7/94-6/96 and 7/96-6/98 7/96-6/98 and 7/98-6/00 7/98-6/00 and 7/00-6/02 7/00-6/02 and 7/02-6/04 7/02-6/04 and 7/04-6/06 7/04-6/06 and 7/06-6/08 7/06-6/08 and 7/08-6/10 P.M. Rank P.M. Rank P.M. Rank P.M. Rank P.M. Rank P.M. Rank P.M. Rank P.M. Rank P.M. Rank 1 0.52 0.53 0.29 0.31 0.32 0.21 0.27 0.33 0.23 0.31 0.32 0.38 0.33 0.42 0.33 0.38 0.41 0.42 5 0.80 0.81 0.56 0.50 0.60 0.41 0.50 0.62 0.44 0.53 0.59 0.62 0.58 0.63 0.65 0.67 0.70 0.68 10 0.89 0.89 0.75 0.66 0.71 0.50 0.62 0.71 0.55 0.65 0.66 0.78 0.66 0.75 0.72 0.76 0.81 0.77 20 0.95 0.95 0.84 0.74 0.79 0.66 0.71 0.78 0.62 0.69 0.85 0.91 0.69 0.84 0.85 0.89 0.89 0.86 50 0.98 0.99 0.88 0.87 0.85 0.77 0.93 0.93 0.51 0.71 0.93 0.97 0.77 0.88 0.90 0.97 0.95 0.95 100 0.99 1.00 0.94 0.92 0.87 0.73 0.97 0.98 0.32 0.55 0.99 0.98 0.91 0.98 0.97 1.00 0.98 1.00
PANEL B: Correlation Coefficients of
23 TABLE 4 CONTINUED
Product Moment and Rank Order Correlation Coefficients of the Fama and French Factors in Two Successive Periods for Portfolios of N Securities PANEL C: Correlation Coefficients of
24
Overall it can be stated that extrapolations of historical risk estimates become more accurate when the size of portfolios increase. The next two tests will therefore examine more diversified portfolios, i.e. portfolios of 100 securities.
Stationarity of estimated coefficients in successive periods – Table 5 compares actual Fama and French coefficients of portfolios of 100 securities estimated for two successive periods. In Panel A, portfolios are formed by ranking : the 100 securities with the 100 smallest s form the first portfolio; the following 100 securities with the next 100 smallest ’s form the second portfolio; and so on. estimates are then compared to the respective estimates, i.e. estimates for portfolios of similar compositions. In Panel B and Panel C the same process is executed independently for and , respectively.
While there are in total 27 different sets of portfolios (nine sets for each risk parameter), the rank order correlations between successive estimates are only 1.00 for eight sets of portfolios: three times for and five times for (see also Table 4). This could be due to the relatively short time periods, increasing the impact of volatility.17 While ‘perfect ranking’ only occurs in some cases, there is obviously a tendency for the estimated values of all three risk parameters to regress to the mean. The range of , , and estimates is considerately smaller each time than the range of the respective , , and estimates. The mean reversion tendency is more pronounced when risk parameters become more extreme, i.e. the lowest and highest risk portfolios.18
There are some cases where mean reversion is not present, but these portfolios are mostly located in the middle, i.e. the average risk portfolios.
Furthermore, assessments of and are more accurate than assessments of . In Panel C, there is a lot of variation between the different sets of portfolios. In some periods assessments are fairly accurate (July 1992 - June 1994 and July 1998 - June 2000), while for other periods there seems to be hardly any forecasting power at all (July 1996 - June 1998 and July 2000 - June 2002). The results for July 2000 to June 2002 concur with the weak correlations found above: risk assessments calculated with July 2000 - June 2002 data do not match the realized risk values of the subsequent period, considering that all of these future coefficients are insignificant. Again, this is evidence of the bad market predictions of book-to-market values made during the Internet bubble; in the aftermath of the bubble a large correction took place
17 Indeed, in one of the sensitivity analyses, using monthly data and longer time periods drastically improves rank order correlations.
18
25 TABLE 5
Estimated Fama and French Coefficients for Portfolios of 100 Securities in Two Successive Periods PANEL A: Estimated Coefficients for Portfolios of 100 Securities in Two Successive Periods
Port- folio 7/90- 6/92 7/92- 6/94 7/92- 6/94 7/94- 6/96 7/94- 6/96 7/96- 6/98 7/96- 6/98 7/98- 6/00 7/98- 6/00 7/00- 6/02 7/00- 6/02 7/02- 6/04 7/02- 6/04 7/04- 6/06 7/04- 6/06 7/06- 6/08 7/06- 6/08 7/08- 6/10 1 0.14** 0.44** 0.09** 0.50** -0.10** 0.68** -0.10** 0.69** 0.04** 0.50** 0.00** 0.59** 0.13** 0.61** 0.13** 0.59** 0.38** 0.70** 2 0.46** 0.68** 0.37** 0.78** 0.42** 0.96** 0.39** 0.80** 0.39** 0.73** 0.37** 0.77** 0.41** 0.70** 0.60** 0.90** 0.75** 0.84** 3 0.63** 0.77** 0.54** 0.80** 0.59** 0.84** 0.64** 0.80** 0.61** 1.00** 0.58** 0.82** 0.55** 0.95** 0.81** 0.92** 0.91** 0.86** 4 0.76** 0.82** 0.68** 0.96** 0.72** 0.88** 0.84** 0.93** 0.76** 1.04** 0.77** 0.96** 0.70** 0.86** 0.98** 1.04** 1.07** 1.07** 5 0.89** 0.92** 0.80** 1.12** 0.86** 0.80** 1.02** 0.97** 0.91** 0.99** 0.95** 1.01** 0.85** 1.08** 1.15** 1.08** 1.28** 1.19** 6 1.03** 1.10** 0.90** 0.93** 1.01** 0.99** 1.18** 1.02** 1.06** 0.99** 1.12** 1.07** 0.98** 1.05** 1.40** 1.13** 1.75** 1.36** 7 1.18** 1.14** 1.03** 0.99** 1.14** 1.10** 1.36** 1.20** 1.25** 1.14** 1.32** 1.05** 1.14** 1.09** 2.08** 1.33** 8 1.41** 1.27** 1.18** 0.98** 1.30** 1.15** 1.61** 1.09** 1.50** 1.28** 1.59** 1.21** 1.34** 1.15** 9 1.85** 1.38** 1.37** 1.16** 1.54** 0.93** 2.34** 1.44** 1.98** 1.26** 2.27** 1.48** 1.74** 1.35** 10 2.62** 2.14** 1.63** 1.20** 2.40** 1.62** 3.23** 0.79** 2.60** 1.37** 11 2.32** 1.49** AVG 0.96 0.98 0.99 0.98 0.99 0.99 0.99 1.01 1.00 N 916 1084 998 870 918 894 964 694 590
Note: * and ** denote significance at the 5% and 1% levels, respectively.
26 TABLE 5 CONTINUED
Estimated Fama and French Coefficients for Portfolios of 100 Securities in Two Successive Periods PANEL B: Estimated Coefficients for Portfolios of 100 Securities in Two Successive Periods
Port- folio 7/90- 6/92 7/92- 6/94 7/92- 6/94 7/94- 6/96 7/94- 6/96 7/96- 6/98 7/96- 6/98 7/98- 6/00 7/98- 6/00 7/00- 6/02 7/00- 6/02 7/02- 6/04 7/02- 6/04 7/04- 6/06 7/04- 6/06 7/06- 6/08 7/06- 6/08 7/08- 6/10 1 -1.74** -1.19** -1.48** -0.93** -1.88** -0.92** -2.26** -0.66** -1.65** -0.76** -1.87** -0.61** -1.41** -0.77** -1.15** -0.60** -1.01** -0.47** 2 -1.00** -0.77** -0.99** -0.67** -0.95** -0.53** -0.84** -0.67** -0.83** -0.31** -0.95** -0.44** -0.80** -0.51** -0.67** -0.49** -0.52** -0.41** 3 -0.53** -0.30** -0.70** -0.33** -0.55** -0.46** -0.36** -0.26** -0.53** -0.28** -0.55** -0.25** -0.50** -0.51** -0.39** -0.34** -0.15** -0.18** 4 -0.15** 0.01 -0.41** -0.25** -0.26** -0.10 -0.05 -0.03 -0.25** -0.02 -0.25** -0.20** -0.26** -0.17** -0.15** -0.03 0.17** 0.11* 5 0.09** 0.29** -0.17** 0.01 -0.04 0.17* 0.17** 0.00 -0.03 0.04 0.01** 0.01 -0.10* -0.02 0.13* 0.21** 0.51** 0.39** 6 0.27** 0.29** 0.03 -0.17 0.13* 0.27** 0.38** 0.25** 0.16** 0.07 0.24** 0.03 0.06* 0.14* 0.53** 0.35** 1.11** 0.57** 7 0.47** 0.29** 0.19** 0.33** 0.29** 0.01 0.62** 0.26** 0.37** 0.24** 0.53** 0.20** 0.21** 0.08 1.46** 0.70** 8 0.71** 0.38** 0.36** 0.27** 0.51** 0.09 0.93** 0.32** 0.62** 0.32** 0.85** 0.29** 0.42** 0.49** 9 1.25** 0.56** 0.55** 0.48** 0.84** 0.37** 1.68** 0.69** 1.42** 0.39** 1.74** 0.67** 0.78** 0.47** 10 2.38** 0.77** 0.86** 0.32** 1.62** 0.87** 3.20** 0.70** 1.68** 0.74** 11 1.61** 0.65** AVG -0.03 -0.04 -0.03 -0.03 -0.02 -0.04 -0.05 -0.04 0.00 N 916 1084 998 870 918 894 964 694 590
Note: * and ** denote significance at the 5% and 1% levels, respectively.
27 TABLE 5 CONTINUED
Estimated Fama and French Coefficients for Portfolios of 100 Securities in Two Successive Periods PANEL C: Estimated Coefficients for Portfolios of 100 Securities in Two Successive Periods
Port- folio 7/90- 6/92 7/92- 6/94 7/92- 6/94 7/94- 6/96 7/94- 6/96 7/96- 6/98 7/96- 6/98 7/98- 6/00 7/98- 6/00 7/00- 6/02 7/00- 6/02 7/02- 6/04 7/02- 6/04 7/04- 6/06 7/04- 6/06 7/06- 6/08 7/06- 6/08 7/08- 6/10 1 -1.69** -0.19* -1.35** -0.24* -1.79** -0.53** -1.94** -0.30** -2.07** -1.22** -2.12** -0.06 -1.70** -0.34** -1.19** -0.38** -1.26** -0.16 2 -0.81** -0.28** -0.74** -0.18* -0.67** -0.20* -0.71** -0.16 -0.69** -0.30** -0.54** -0.04 -0.79** -0.18* -0.40** -0.11 -0.44** -0.10 3 -0.47** -0.18** -0.50** -0.33** -0.39** -0.32** -0.36** -0.05 -0.27** 0.07 -0.14** -0.01 -0.47** -0.04 -0.14** -0.20** -0.04 -0.10* 4 -0.24** -0.06 -0.32** -0.15* -0.22** -0.02 -0.10** 0.05 -0.04** 0.17** 0.05** 0.10 -0.22** 0.10 0.07** 0.11 0.29** 0.18** 5 -0.05** 0.01 -0.18** -0.08 -0.09** -0.07 0.10** -0.06 0.13** 0.26** 0.19** 0.01 -0.03** 0.10 0.27** 0.14* 0.62** 0.15* 6 0.11** -0.05 -0.06** -0.06 0.05** 0.01 0.27** 0.18** 0.32** 0.20** 0.34** 0.06 0.14** 0.03 0.51** 0.35** 1.14** 0.18* 7 0.32** 0.05 0.07** -0.39** 0.19** -0.14 0.48** 0.08 0.49** 0.34** 0.49** -0.02 0.32** -0.01 1.11** 0.13 8 0.61** -0.03 0.22** -0.03 0.36** 0.01 0.83** 0.20** 0.76** 0.25** 0.72** -0.01 0.57** 0.16* 9 1.34** 0.55** 0.42** 0.20** 0.66** 0.11 2.29** 0.44** 1.32** 0.47** 1.27** 0.00 0.97** 0.11 10 4.68** 1.69** 0.74** 0.24* 2.16** 1.29** 2.84** 0.65** 2.15** 0.09 11 2.19** 1.42** AVG 0.00 0.02 0.01 0.03 0.04 0.02 0.01 0.02 0.02 N 916 1084 998 870 918 894 964 694 590
Note: * and ** denote significance at the 5% and 1% levels, respectively.
28
of unrealistic high valuations. This is in accordance with the description of Sharpe (1990) of wealth in- and decreases in respectively bull and bear markets. Changes in risk aversion – wealth is negatively related to risk aversion – tend to run bull and bear markets to extremes. Ultimately, regression to the mean takes over when sufficient investors recognize the overvaluations and correct the accumulated valuation errors.
Table 5 exposes another issue for – some of the high rank order correlations of in Table 4 are misleading. The high correlations could be interpret as an indication that the assessments are quite accurate. However, examining for example the first set of portfolios for
(see Panel C), it appears that five coefficients from July 1992 to June 1994 are insignificant; for the same portfolios those coefficients range from -0.24 to 0.32 in the preceding period.
OLS time series regressions embody the implicit assumption that parameters are constant – within the entire sample period as well as for any subsequent period used in the construction of forecasts. However, Table 4 and 5 already showed that this does not hold for the estimated Fama and French coefficients in this research. Another way to test this implicit assumption is by using parameter stability tests. The one employed in this paper and described in Table 6 is the Chow test. This test compares the residual sum of squares of three models – two estimated for sub-periods and one for all the data. In this case, the two sub-periods are two successive periods of two years and the third model includes all four years.
29 TABLE 6
Chow Tests for Portfolios of 100 Securities
PANEL A: Chow Test Statistics for Portfolios of 100 Securities Formed by Ranking
Port- folio 7/90-6/92 and 7/92-6/94 7/92-6/94 and 7/94-6/96 7/94-6/96 and 7/96-6/98 7/96-6/98 and 7/98-6/00 7/98-6/00 and 7/00-6/02 7/00-6/02 and 7/02-6/04 7/02-6/04 and 7/04-6/06 7/04-6/06 and 7/06-6/08 7/06-6/08 and 7/08-6/10 1 20.20* 10.36* 12.12* 19.44* 12.49* 22.49* 17.94* 16.60* 7.41* 2 8.02* 9.53* 12.52* 6.60* 13.51* 15.72* 8.12* 9.32* 3.49* 3 3.97* 4.23* 8.19* 2.65* 9.53* 5.99* 15.26* 4.13* 0.65 4 1.60 7.61* 1.84 0.99 6.55* 4.15* 3.70* 1.77 0.05 5 0.87 6.58* 0.90 2.34 1.53 2.38 7.23* 0.87 1.57 6 4.48* 0.23 0.73 2.15 4.09* 2.20 1.99 7.50* 10.24* 7 1.52 1.40 1.83 2.51* 3.21* 9.35* 2.74* 16.39* 8 1.73 2.79* 1.64 13.77* 3.58* 12.48* 4.91* 9 16.50* 1.77 10.85* 11.71* 11.60* 18.63* 5.50* 10 1.34 7.58* 5.13* 19.98* 19.82* 11 7.98*
PANEL B: Chow Test Statistics for Portfolios of 100 Securities Formed by Ranking
Port- folio 7/90-6/92 and 7/92-6/94 7/92-6/94 and 7/94-6/96 7/94-6/96 and 7/96-6/98 7/96-6/98 and 7/98-6/00 7/98-6/00 and 7/00-6/02 7/00-6/02 and 7/02-6/04 7/02-6/04 and 7/04-6/06 7/04-6/06 and 7/06-6/08 7/06-6/08 and 7/08-6/10 1 8.71* 8.44* 9.61* 21.59* 10.67* 18.47* 13.64* 15.55* 7.85* 2 4.15* 5.08* 3.30* 1.01 10.94* 8.65* 7.13* 2.89* 1.25 3 4.63* 6.66* 0.57 1.64 5.59* 5.02* 2.04 2.16 0.51 4 1.49 3.90* 3.21* 0.53 3.36* 1.73 1.86 4.64* 2.01 5 2.87* 3.49* 1.39 4.96* 0.39 1.34 5.78* 1.02 1.77 6 0.37 1.23 0.47 3.82* 0.84 2.64* 13.72* 3.34* 14.77* 7 2.67* 0.98 1.82 6.31* 3.18* 3.93* 1.18 10.58* 8 7.30* 1.86 4.41* 11.74* 5.32* 8.85* 0.42 9 14.44* 1.10 6.28* 14.17* 20.42* 22.79* 4.01* 10 8.93* 12.95* 6.91* 11.20* 11.08* 11 15.89*
Note: * denotes significance at the 5% level.
The conjecture of mean reversion is tested using the Chow test. This parameter stability test splits data into sub-periods and estimates three models – two for each of the sub-parts and one for all the data. Subsequently, the residual sum of squares (RSS) of each of the models is compared. The test statistic is given by
30 TABLE 6 CONTINUED
Chow Tests for Portfolios of 100 Securities
PANEL C: Chow Test Statistics for Portfolios of 100 Securities Formed by Ranking
Port- folio 7/90-6/92 and 7/92-6/94 7/92-6/94 and 7/94-6/96 7/94-6/96 and 7/96-6/98 7/96-6/98 and 7/98-6/00 7/98-6/00 and 7/00-6/02 7/00-6/02 and 7/02-6/04 7/02-6/04 and 7/04-6/06 7/04-6/06 and 7/06-6/08 7/06-6/08 and 7/08-6/10 1 27.21* 21.34* 14.45* 24.18* 15.09* 40.44* 16.78* 17.62* 14.34* 2 9.11* 10.71* 8.29* 9.25* 4.46* 6.28* 8.10* 2.86* 2.56* 3 3.70* 5.04* 1.66 3.75* 5.58* 7.72* 4.39* 1.09 1.95 4 1.62 2.80* 1.12 3.41* 3.13* 1.46 7.06* 4.47* 0.93 5 3.23* 3.24* 1.34 2.15 0.99 1.76 3.02* 1.94 4.99* 6 0.72 0.78 2.16 2.11 2.71* 6.09* 1.74 1.22 11.65* 7 3.32* 9.10* 3.01* 5.81* 4.30* 14.26* 4.93* 12.25* 8 11.46* 3.96* 4.42* 9.80* 14.30* 17.75* 4.93* 9 10.46* 3.25* 5.12* 24.20* 21.10* 48.71* 14.72* 10 5.77* 6.92* 3.85* 8.39* 22.09* 11 5.18*
Note: * denotes significance at the 5% level.
The conjecture of mean reversion is tested using the Chow test. This parameter stability test splits data into sub-periods and estimates three models – two for each of the sub-parts and one for all the data. Subsequently, the residual sum of squares (RSS) of each of the models is compared. The test statistic is given by
31 TABLE 7
Measurement of Regression Tendency of Estimated Fama and French Coefficients for Individual Securities
Regression Tendency
Implied Between Periods = a + b = a + b = a + b
7/92-6/94 and 7/90-6/92 = 0.406** + 0.586** = -0.019 + 0.580** = 0.013 + 0.169** 7/94-6/96 and 7/92-6/94 = 0.601** + 0.395** = -0.017 + 0.528** = 0.009 + 0.613** 7/96-6/98 and 7/94-6/96 = 0.530** + 0.470** = -0.012 + 0.398** = 0.005 + 0.293** 7/98-6/00 and 7/96-6/98 = 0.754** + 0.227** = -0.029 + 0.199** = 0.030 - 0.023 7/00-6/02 and 7/98-6/00 = 0.734** + 0.257** = -0.015 + 0.350** = 0.015 + 0.478** 7/02-6/04 and 7/00-6/02 = 0.663** + 0.333** = -0.026 + 0.326** = 0.003 + 0.014 7/04-6/06 and 7/02-6/04 = 0.696** + 0.316** = -0.005 + 0.528** = -0.003 + 0.121** 7/06-6/08 and 7/04-6/06 = 0.662** + 0.329** = -0.009 + 0.514** = 0.001 + 0.124* 7/08-6/10 and 7/06-6/08 = 0.448** + 0.545** = -0.006 + 0.527** = 0.020 + 0.142**
Note: * and ** denote significance at the 5% and 1% levels, respectively.
32 Effectiveness of using historical rates of regression – In accordance with Blume´s (1971) adjusted beta, I regress the estimated values of , , and on the estimated values in a previous period ( , , and , respectively). By using these regression equations to modify assessments of future risk, I examine whether historical rates of regression can correct assessments and consequently improve the accuracy of risk assessments.
Table 7 presents the rates of regression for the two-year periods under investigation.19 For all three risk parameters the slopes are smaller than one, which is in agreement with the regression tendency. For , the magnitude of the intercept coefficients and slope coefficients seem to differ from that found by Blume (1971). Although the rate of regression varies over time, the intercepts lie mostly around 0.6-0.7 and the slopes around 0.2-0.3; in Blume’s research this was roughly the other way around. For , the slope coefficients vary from 0.2 to 0.6, although mostly situated around 0.5. For these two coefficients, the regression tendency is relatively stationary over time. On the other hand, in case of the rate of regression is much more volatile, with slopes ranging from 0.6 to as strong as 0 (a slope of 0 indicates that even the most extreme values regress to zero).
Table 8 compares adjusted assessments with unadjusted assessments. For example, the estimated Fama and French coefficients for the period from July 1992 to June 1994 are modified using the three equations in the first row of Table 7, which hold the regression tendencies implied between the ‘current’ period (July 1992 – June 1994) and the previous period (July 1990 – June 1992).20 Subsequently these adjusted assessments are compared to the respective unadjusted assessments, which were used in Tables 4, 5, and 6. The accuracy of the two alternative methods of assessment is compared through the mean squared errors of the assessments versus the estimated coefficients in the next period, July 1994 to June 1996. The same process is executed for the next seven periods using respectively the next seven equations in Table 7. Except for during the period July 1992 – June 1994, adjusted assessments are more accurate than unadjusted assessments. This holds for diversified portfolios as well as for single securities. Hence, correcting assessments for historical rates of regression improves the accuracy of forecasts even though the rate of regression over time is not strictly stationary. In fact, the rates of regression of vary substantially over time, but MSE almost never fails to improve when coefficients are adjusted for the previous rate of
19
The regression computations are performed for individual securities.
33 TABLE 8
Mean Square Errors Between Assessments and Future Estimated Values
# of Sec./ Port.
PANEL A: Assessments Based Upon
7/92-6/94 7/94-6/96 7/96-6/98 7/98-6/00 7/00-6/02 7/02-6/04 7/04-6/06 7/06-6/08 unadj adj unadj adj unadj adj unadj adj unadj adj unadj adj unadj adj unadj adj 1 0.72 0.61 1.30 1.14 0.61 0.35 0.72 0.48 0.62 0.43 0.54 0.35 0.48 0.32 0.34 0.30 5 0.26 0.14 0.44 0.36 0.35 0.10 0.36 0.11 0.30 0.09 0.28 0.08 0.22 0.06 0.10 0.07 10 0.19 0.07 0.26 0.11 0.30 0.07 0.30 0.06 0.31 0.05 0.29 0.05 0.22 0.04 0.07 0.04 20 0.19 0.05 0.22 0.05 0.33 0.06 0.28 0.04 0.23 0.02 0.35 0.04 0.18 0.02 0.07 0.02 50 0.17 0.03 0.21 0.02 0.30 0.03 0.44 0.05 0.20 0.01 0.37 0.03 0.17 0.01 0.05 0.02 100 0.14 0.02 0.20 0.02 0.22 0.02 0.71 0.06 0.16 0.01 0.23 0.01 0.14 0.00 0.04 0.01 # of Sec./ Port.
PANEL B: Assessments Based Upon
7/92-6/94 7/94-6/96 7/96-6/98 7/98-6/00 7/00-6/02 7/02-6/04 7/04-6/06 7/06-6/08 unadj adj unadj adj unadj adj unadj adj unadj adj unadj adj unadj adj unadj adj 1 1.52 1.36 3.55 3.02 1.95 0.95 1.35 0.97 1.16 0.68 0.72 0.61 0.63 0.47 0.53 0.42 5 0.46 0.30 1.17 0.99 1.02 0.26 0.61 0.20 0.69 0.19 0.27 0.16 0.27 0.10 0.20 0.09 10 0.34 0.16 0.54 0.30 0.83 0.16 0.50 0.10 0.66 0.11 0.25 0.10 0.27 0.07 0.16 0.05 20 0.34 0.11 0.36 0.13 0.74 0.10 0.45 0.07 0.55 0.05 0.31 0.07 0.20 0.03 0.15 0.03 50 0.23 0.05 0.27 0.05 0.61 0.05 0.61 0.04 0.48 0.03 0.29 0.06 0.15 0.01 0.13 0.01 100 0.17 0.02 0.22 0.02 0.46 0.02 0.86 0.03 0.39 0.00 0.15 0.05 0.14 0.01 0.10 0.01 # of Sec./ Port.
PANEL C: Assessments Based Upon
7/92-6/94 7/94-6/96 7/96-6/98 7/98-6/00 7/00-6/02 7/02-6/04 7/04-6/06 7/06-6/08 unadj adj unadj adj unadj adj unadj adj unadj adj unadj adj unadj adj unadj adj 1 3.18 3.22 6.41 4.53 3.14 1.29 1.18 1.13 2.14 1.39 1.73 0.95 1.73 1.28 1.05 0.53 5 0.92 0.94 1.34 0.69 1.57 0.29 0.52 0.43 1.15 0.42 1.06 0.23 0.77 0.34 0.61 0.12 10 0.52 0.60 0.64 0.49 1.20 0.14 0.41 0.36 1.07 0.30 1.17 0.12 0.87 0.29 0.55 0.07 20 0.39 0.46 0.49 0.32 1.25 0.08 0.36 0.33 0.99 0.26 1.57 0.08 0.42 0.07 0.57 0.03 50 0.28 0.33 0.35 0.12 1.17 0.03 0.48 0.30 0.88 0.20 1.31 0.03 0.32 0.03 0.47 0.01 100 0.25 0.12 0.32 0.06 0.79 0.02 0.69 0.25 0.78 0.17 0.78 0.02 0.25 0.03 0.41 0.01 The effectiveness of using historical rates of regression to correct assessments is evaluated here. For example, the estimated Fama and French coefficients for the period from 7/92 – 6/94 are adjusted using the three equations in the first row of Table 7, which hold the regression tendencies implied between the ‘current’ period (7/92 – 6/94) and the previous period (7/90 – 6/92). Subsequently, the accuracy of both the unadjusted and the adjusted assessments are tested by comparing the two alternative assessments with the estimated Fama and French coefficients in the next period (7/94 – 6/96). The same process is executed for the next seven periods using respectively the next seven equations in Table 7.
The accuracy of the two alternative methods of assessment is compared through the mean squared errors of the assessments versus the estimated coefficients in the next period. Specifically, in Panel A
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regression. This is even the case for the two slopes that are insignificant and thus treated as zeros, see periods July 1998 – June 2000 and July 2002 – June 2004.
In summary, above tests have shown that, for all three coefficients, there are mean reversion tendencies present. The historical rates of regression found for seem to differ from that found by Blume (1971); the regression tendencies are less strong (i.e. smaller slope coefficients). This effect could be caused by the two additional factors. Nonetheless, extending the CAPM to the three-factor model does not fully alleviate the mean reversion property of . But above all, the novelty of the results is manifested in the two other coefficients. Just like , and also exhibit regression tendencies. While for the rate of regression over time is fairly stationary, exhibits more volatility.
Sensitivity analyses
The power of the inferences drawn from this study is conditional on the validity of the employed methods and estimation techniques. In the first two sensitivity analyses two assumptions of Fama and French (1993) are brought into question. First, I determine the effect of making use of median book-to-market values to calculate the breakpoints and form the size-BE/ME portfolios. Second, I examine what the consequences are of including negative-BE firms when calculating the breakpoints and forming the portfolios. Both tests aim at improving the accuracy of assessments and I will show alternative results to Table 7 (rates of regression) and Table 8 (effectiveness of using historical rates of regression to correct assessments). The third sensitivity analysis assesses the robustness of the results to monthly data. For the this last sensitivity analysis I reestimate all models.
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differences between the analyzed periods and why the accuracy of assessments is relatively low compared to and .
Table 1A and 1B in the Appendix illustrate the effect of constructing the portfolio based on median book-to-market breakpoints. Table 1A shows that the differences in the rates of regression are very small. Moreover, the slope coefficients are insignificant in the same two periods. Table 1B demonstrates that using median BE/ME values is in fact inferior: 93 out of the 96 mean squared errors increased, so nearly all assessments, unadjusted as well as adjusted, worsened. The differences are especially large for unadjusted assessments and smaller portfolios.
Including negative-BE/ME firms – The main reason mentioned by Fama and French (1993) for not including negative-BE firms when calculating the breakpoints for BE/ME or when forming the size-BE/ME portfolios is that these firms are rare before 1980. They thus exclude negative BE-stocks assuming that their omission has an insignificant effect. However, Brown, Lajbcygier and Li (2008) state that since the late 1980s there exists a greater incidence of firm losses and consequently a greater number of firms with negative book equity values; in 2008 approximately 5% of all listed firms had negative book equities. Indeed, in the U.K. data set on average 3.8% of the firms have negative-BE values with percentages peaking during 1999-2001 and 2005-2008.21 Moreover, Brown et al. (2008) found that omitting negative-BE stocks is significant and they show that negative-BE stocks significantly enhance the premium. They argue that many of these stocks are in financial distress and that more default risk should lead to a higher default premium. Therefore, excluding negative-BE stocks not only reduces the investable universe and the amount of stocks in the regression analyses, but also the information contained in the factor.
Tables 2A and 2B in the Appendix illustrate the effect of including negative-BE firms when calculating the breakpoints for BE/ME or when forming the size-BE/ME portfolios. Again, the differences in the rates of regression are very small (see Table 2A). The intercept coefficients and the average and values converge closer to zero. Table 2B shows that including negative-BE firms does not positively influence the accuracy of assessments: 66 out of the 96 mean squared errors increased compared to Table 8. However, the differences
21
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are not as large as in the preceding sensitivity analysis, especially with regard to the unadjusted assessments.
Monthly U.K. data – Changing the frequency of observation is mainly on behalf of practical relevance, since investors usually work with monthly data. A further advantage of lower frequency data is that it is less prone to heteroscedasticity and volatility clustering. In addition, the length of the periods is altered to six years to further test the robustness of the results and to preserve the number of observations. Since the data set only comprises twenty years of data overlapping periods are selected,
Tables 3A till 3E in the Appendix illustrate the effect of using monthly data. The inferences of the results are the same as for weekly data. In addition, the correlations in Table 3A and the coefficient estimates in Table 3B demonstrate an even more clear picture of the mean reversion conjecture of and . However, the regression tendency of is still very volatile. Remarkably, in one period the slope in the rate of regression of is even greater than 1, indicating a diverging force (see Table 3D).
VI. CONCLUSION AND DISCUSSION
This paper examines the empirical behavior of the three Fama and French coefficients over time. Specifically, by examining the accuracy of extrapolations of historical Fama and French risk estimates, by studying the regression tendencies of the coefficients, and by testing if assessments can be corrected to improve the accuracy of forecasts.
