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Shortly after the ground-breaking work of Markowitz on modern portfolio theory (1952) a new branch in Finance developed trying to explain the expected return on any financial asset. Soon the model with probably largest impact on the financial industry was born, the Capital Asset Pricing Model. Even after many different studies questioning the validity of the model, it is still the most used by practitioners. A lot of other models were subsequently developed on the same reasoning. Fama & French Three-Factor Model is considered one of the most promising and consistent.
We start this paper briefly explaining the CAPM and its shortcomings. On those grounds, we explain the Fama & French model. Then, we test both models in US data from 1967 till now. Different portfolios were used, testing for the impact of size, book-to-market ratios, and the specific industry.
We end up drawing conclusions on the results found.
The Capital Asset Pricing Model (henceforth CAPM) has a very curious history, being built independently by Jack Treynor (1965), William Sharpe (1964), John Lintner (1965) and Jan Mossin (1966), all in the same time span of the early sixties. This work was based on the earlier revolutionary theory of Markowitz and also on Tobin’s Separation Theorem.
The CAPM has several strong assumptions inherited from the said projects of mean variance efficiency that essentially create a perfect market environment. Investors are rational and risk averse, can borrow and lend unlimited amounts at the risk-free rate and have homogenous expectations and information about all assets returns. There are no taxes, inflation, transaction costs, no short selling restrictions and all assets are infinitely divisible and perfectly liquid.
The assumptions constrain the setting for the CAPM world. They set a stage that only non-diversifiable risks are rewarded with extra returns, and since each additional asset introduced into a portfolio further diversifies the portfolio, the optimal portfolio must comprise every asset with each asset value-weighted. All such optimal portfolios comprise the efficient frontier. This makes the expected return of any asset or portfolio to vary linearly with the returns of the market portfolio, according to the following formula:
Beta is the key measure as it gives the sensitivity of the excess returns of an asset or portfolio compared to the excess returns of the market portfolio. Since the unsystematic risk is diversifiable, the risk of a portfolio can be viewed as beta.
The CAPM is best described by Sharpe (1988) as a “simple, yet powerful description of the relationship between risk and return in an efficient market”. This is a very intuitive thought process. The level of returns one expects to get is directly related to the exposure to market volatility. Stock specific error is diversified away when choosing the efficient portfolio, and as such the only source of return comes from choosing the relation your portfolio has with the market.
The CAPM is so important that the standard deviation of a stock return no longer was the normally used risk measure, but rather its relation to the market returns. It is also the number one tool to find discount rates for company valuation and for portfolio management. However it has not been free of criticism.
After it was proposed, empirical tests were executed normally running the following regression:
Where a proxy of excess market returns is used and regressed against a certain asset return. The Alpha of the regression indicates the excess return (either positive or negative) that is not explained by the CAPM. According to CAPM, as the correlation with market should completely explain its return, the apha of the previous regression should be 0.
First of all, the use of a market proxy leads to Richard Roll’s critique (1976). It is quite simple but revealing and it simply states the CAPM can never be tested as the exact composition of the market portfolio is not known. All proxies used might be mean variance efficient but the market might not, leading to all tests being inherently biased. Besides, the interpretation of Beta using market proxies leads to relative measures of risk, as the Beta obtained depends on the market proxy used.
Besides Roll’s opinion on the theory, a number of anomalies were found on the model. Characteristics such as size, earnings/price, Cash flow/price, book-to-market-equity, past sales growth had effects on average returns of stocks. These are called anomalies as they are not explained by CAPM, leading to the idea that risk is multidimensional and as such the CAPM is fundamentally wrong in its core conclusion.
Eugene Fama and Kenneth French (1996) made the greatest stride, when stating that anomaly variables include a risk premium contained in the characteristics of these variables. These anomalies are mainly divided by two main factors. Size, which they explain theoretically, and relative distress, passing through the E/P and book to market as measures.
Fama French Three-Factor Model
Eugene Fama and Kenneth French since expanded the CAPM to the Fama-French (FF) tri-factor model (1992), which adds two variables to capture the cross-sectional variation in average stock returns associated with market: Beta, size, leverage, book to market and earnings-price ratio. This creates the following model:
, which can be transformed into
Where the factors added to the CAPM are the SMB (Small minus Big), a measure of the historic excess return of small caps over big caps, and HML (High minus Low), the same difference for returns of value stocks over growth stocks. This model is not as widely used as the CAPM, but we will test empirically if it performs better than the original one-factor model.
After introducing the theoretical bases of these models, we will explain the methodology we used on our tests. We used data from Kenneth French´s website, consisting of market excess returns from NYSE, AMEX, and NASDAQ firms and the values of returns from all those companies divided into size and book to market quintiles and also divided into five sections of industries – Consumer Goods, Manufacturing (energy and utilities), High-tech, Healthcare and Service industry. The data is monthly from 1967 to 2010. Our variables of interest comprise the alphas of each regression (i.e., returns unexplained by the model) and the adjusted, which adjusts for the number of explanatory terms in a model – unlike the regular, the adjusted increases only if new variables improve the model. We used all this data to run the normal empirical test regression expressed in (2).
We will ignore Roll’s critique in the tests and use a certain market proxy as in our opinion data on returns of a certain index representative of the country where investors negotiate is quite representative of market returns, as that data is amply divulged and influences all assets related.
These are the results in regression form and the values of the alphas obtained with double standard error bands:
Table 1 – Regression Results from Size Portfolios
(Values in parenthesis refer to the t-stat of the variable above)
Looking at the alpha values of the regressions under the CAPM, the 4th quintile is the only one significant on a 95% confidence interval. All the beta values are significant and different than zero. Alpha values decrease as we go from portfolios of smaller to bigger companies, as does the of the regressions. As for the Fama French model, the values of factors are significant in all the size regressions, and the alpha value is only significant in the 5th quintile of biggest companies. The follows a similar behavior.
These results seem to favour the tri-factor approach, as including the SMB variable seems to improve the quality of fit of smaller companies. The difference in the adjusted of the lowest 20% quintile between the CAPM and Fama French models is a whopping 30%, indicating that some unsystematic risks, captured by the difference between big and small firms, affect returns. In other words, these results favor Fama and French´s model in explaining returns over the CAPM.
Chart 1 – Plot of CAPM alpha with double standard error band
Chart 2 – Plot of FF alpha with double standard error band
These charts tell a more interesting story. The alpha values of the CAPM diminish a lot when going from small cap quintiles to large cap ones, from relatively high alphas to close to zero. Everything changes when using FF three-factor model where the alpha values are negative for small caps and go to positive when moving to bigger companies.
The larger range of alphas in the CAPM over FF, especially in smaller companies, again indicates that returns are not fully captured by measuring only correlation with the market. Accordingly, by adding SMB this range is considerably reduced, especially in the portfolios based on the lowest 20% companies in size.
Table 2 – Regression Results from Book-to-Market Portfolios
(Values in parenthesis refer to the t-stat of the variable above)
Here the Betas of all regressions are significant. The fourth and fifth quintiles on the CAPM present a high alpha rejecting the null hypothesis that they are not significant, with a 95% confidence level. On the other hand, the FF model rejects only the lowest 20% B/M portfolios, and by the tiniest of margins.
These results show evidence that Fama and French were indeed correct by considering the HML factor in their regression. In fact, the existence of significant alphas in the two highest quintiles in the CAPM, combined with the substantial differences in the adjusted – 13% for the 4th quintile, almost 20% in the 5th – again demonstrate that CAPM is not considering important variables in determining returns.
Chart 3 – Plot of CAPM alpha with double standard error band
Chart 4 – Plot of FF alpha with double standard error band
As the B/M values increase, CAPM’s results are ever worse regarding alpha. By adding double standard error bands, CAPM’s portfolios based on the highest 20% value have alphas ranging from 0.1 and 0.6, very substantial values. FF performs much better, with alphas not moving far away from 0.
Table 3 – Regression Results from Industry Portfolios
(Values in parenthesis refer to the t-stat of the variable above)
Chart 5 – Plot of CAPM alpha with double standard error band
Chart 6 – Plot of FF alpha with double standard error band
Contrary to the previous analysis, the three-factor model only displays marginal improvements in the adjusted to the single-factor model when dealing with industry-based portfolios. In fact, the FF model has significant alphas in two different industries – Health Care and Others – while the CAPM has none. Moreover, the SMB variable seems to be irrelevant in the Consumer Goods and in other industries. We should not be surprised by these results, as the FF model was built around two ideas: small companies and those with high B/M ratios were undervalued by the market. Thus, when analyzing portfolios based on different restraints (like industry) the model will not perform much better compared to the CAPM.
A note on Fama-French Three-Factor Model
The FF model is an extension of the CAPM model in the sense that it uses two extra factors: SMB and HML. The first one increases the modulation of different size portfolios. The second one addresses the difference in book values of companies included in different portfolios.
We suspect that SMB is in fact important whenever we are trying to predict the different performance of portfolios split using size as the criteria. The same reasoning can be used to portfolios split using book-to-market ratio as the criteria.
We decided to apply this idea to the data, computing the average contribution of each factor to the total excess return of each portfolio. The resulting table is presented below.
Table 4 – Factor Contribution to Excess Return
We can see that, as we suspected, SMB is in fact very relevant (19% on average) to explain the excess return of different portfolios split with market size criteria. That is even more critical when we are considering portfolios of smaller stocks. In those portfolios, the factor HML is not particularly important.
When we move to book-to-market value different portfolios, it is HML that contributes significantly (14%), especially to high book-to-value stocks, and SMB can be neglected.
Finally, when the criterion to split portfolios is neither size nor book-to-market, the two extra factors of the Fama-French model have no explanatory power on average. We can see the average weights are very close to standard CAPM. We can speculate on the difference across industries: for instance, hi-tech and health care stocks tend to have higher book-to-market ratios, and so the HML factor is relevant. It is possible that a factor like “high dividend yield less low dividend yield” might be robust to explain performance differences among portfolios split according to dividend yield level.
We are not questioning the applicability of the Fama-French model. What we are addressing here is that each factor does not have a generalized relevant contribution to explain excess returns. In certain situations, like small cap portfolios and growth stocks, each factor in turn becomes very important. Outside of these “native environments”, the factors do not contribute to explain or predict excess returns.
Throughout this work we have shown that Fama and French’s tri-factor model is superior to the CAPM in capturing some non-systematic anomalies not considered by the simple one-factor approach. These anomalies include the undervaluation of small firms and those with high B/M ratios. Adding variables that reflect this effect considerably improves the quality of fit of the model and eliminates loose ends as reflected by the significant alphas present in some portfolios using the CAPM. However, we must pay close attention to data, as performing a FF regression on data that does not reflect these variables, – as industry – does not improve the models.
References and Other Bibliography
Fama, E. F., & French, K. R. (1992). The Cross-Section of Expected Stock Returns. The Journal of Finance, 47(2), 427-465.
Fama, E. F., & MacBeth, J. D. (1973). Risk, Return, and Equilibrium: Empirical Tests. The Journal of Political Economy, 81(3), 607-636.
Lintner, J. (1965). The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. The Review of Economics and Statistics, 47(1), 13-37.
Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91.
Mossin, J. (1966). Equilibrium in a Capital Asset Market. Econometrica, 34(4), 768-783.
Roll, R. (1977). A critique of the asset pricing theory’s tests Part I: On past and potential testability of the theory. Journal of Financial Economics, 4(2), 129-176.
Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19(3), 425-442.
Treynor, Jack L. (1965). How to Rate Management of Investment Funds. Harvard Business Review, 43(1), 63-75.
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