Whether it is to evaluate investment decisions or to value corporate transactions, determining the fair price of an asset is crucial to every market participant. The opportunity to consider which model is the more accurate to compute such a value might occur on several occasions to market players. In this paper, we will be considering the most popular asset pricing model, the CAPM developed by W. Sharpe (1964), J. Treynor (1965), J. Linter (1965) and I. Mossin (1966) which has radically modified the way of thinking in financial markets and remains the most important paradigm despite the critics formulated by Roll (1977), Basu (1977), Banz (1981) and Bhandari (1988) that we will be trying to highlight. Finally, we will extend asset pricing methodologies to recent developments in the finance literature such as the three-factor Fama and French model (1992) and the Arbitrage Pricing Theory model developed by R. Roll and S. Ross (1980).
Capital Asset Pricing Model
The original model
In the Capital Asset Pricing Model, W. Sharpe (1964), J. Treynor (1965), J. Linter (1965) and I. Mossin (1966) argue that for every financial asset, there is a growing linear relation between the risk and the return of this asset. To determine the relation between the expected return and the risk of a financial asset, the model assumes that investors constitute their portfolios according to the Markowitz () mean-variance criteria and formulates the hypothesis H0 that the market portfolio, noted M and defined as the portfolio including all the risky assets existing in the economy and no risk-free asset, is efficient. If a risk-free asset with a return r exists, then the H0 hypothesis is true if and only if the expected return of any asset or portfolio i is equal to:
µi = r + βi (µM - r)
The key element of the model is that it separates the risk affecting an asset's return into unsystematic and systematic risk which is directly correlated to the general uncertainty of the market. As shown in the equation, the CAPM model shows that the return of an asset should be equal to the yield on a risk-free asset (r), usually a government bond, plus a premium (µM - r) proportional to the systematic risk of the asset, (βi).
The Roll critique and the market proxy problem
In his article "A critique of the asset pricing theory's tests Part I: On past and potential testability of the theory", Richard Roll (1977) argues that the Capital Asset Pricing Model cannot be tested because the proxy used for the market portfolio return is mis-measured. Indeed, the market portfolio should include every single risky asset such as stocks, bonds, precious metals, human capital, real estate and so forth. Such return is not calculable and instead, the market portfolio proxy tends to use only US stocks and particularly marketable assets only thus occulting the non-marketable assets. The positive linear relation between the risk, measured by the Beta, and the expected return is derived from the assumption of the proxy's efficiency. Therefore, the Betas calculated from an approximated market portfolio could result in the absence of linearity between the expected return µi and the risk βi.
Other critiques of the CAPM
In addition to the Roll critique (1977) regarding the market portfolio inaccuracy, several inconsistencies have emerged from the CAPM which stress out that the beta is not sufficient to explain stock returns. Basu (1977) showed that stocks with a higher earnings-price ratios leads to returns higher than those calculated by the CAPM. Banz (1981) is the first author to explore the size-effect and documents that the difference of average return between small and large stocks is higher than predicted with the CAPM. Statman (1980) and Rosenberg, Reid and Lanstein (1985) illustrate that stocks with high book-to-market equity ratios tend to have high average returns that are not perceived in their betas. Bhandari (1988) demonstrates that leverage, meaning high debt-equity-ratio, are correlated to returns which are too high compared with their market betas. Finally, Fama and French (1992) illustrate that stocks with higher book-to-market ratios can expect high average returns which is not captured by the market betas. All those results mentioned from the literature tend to demonstrate that the market beta is not sufficient to explain stock returns and that some alternative explanatory variables should be investigated in order to develop more accurate pricing models as Fama and French will do when developing their three-factor model.
Fama and French three-factor model
The foundation of the three-factor model
In their article "The Cross-Section of Expected Stock Returns", Fama and French (1992) questioned the mono-beta model and demonstrated that in the U.S. between 1928 and 1990, the beta did not seem to play a role in explaining the average returns of a stock rather than the book-to-market ratio. This ratio allows to distinguish value stocks, with a high book-to-market ratio, from growth stocks with a smaller book-to-market ratio. Using the cross-sectional regression approach of Fama and MacBeth (1973), Fama and French (1992) showed the importance of the book-to-market equity and the size effect in explaining the average return of a stock. Banz (1981) demonstrated a strong negative relation between firm size and the average return of its stock and Ball (1978) and Keim (1988) state that size, E/P, leverage and book-to-market equity can be used to extract information from a stock price this Fama and French concluded that the size and book-to-market ratio can capture the cross-sectional variation in average stock returns.
Later, Fama and French (1993, 1996) found that three factors are in fact necessary to explain the observed returns: the book-to-market ratio, the size (measured by the market capitalization) and the market itself. Based on their empirical findings, Fama and French (1992, 1993, 1996) state that the following three-factor factor model applies to any portfolio i:
µi - rf = βi (μm - rf ) + si μSMB + hi μHML
where (μm - rf) indicates the market risk premium, μSMB equals the mean return on a "size" factor (the difference in the return of stocks of big and small companies), and μHML represents the mean return on a "book-to-market" factor (the difference in the returns of stocks with a high and a small book-to-market ratio). This three-factor model expected return equation implies that the intercept αi in the time-series regression is zero for all assets.
µi - rf = αi + βi (μm - rf ) + si μSMB + hi μHML + εi
This three-factor model is now broadly used as a model of expected returns and one of its merits is that the estimations of αi are now used, e.g. by Loughran and Ritter (1995) and Mitchell and Stafford (2000) to evaluate the stock prices quickness to react to new information.
Shortcomings attributed to the Fama and French model
Firstly, the empirical motivation of the three-factor model remains its biggest shortcoming. Indeed, Fama and French (2004) reckon that the SMB and HML returns that compose the explanatory variables of the model are not motivated by predictions of variables watched by investors rather than by concepts constructed to capture the patterns already discovered from previous work that explain how stock returns vary with book-to-market ratio and size.
In addition, Fama and French identify in their model, a risk premium called "distress premium". This distress premium can be interpreted as a risk premium expected by investors to invest in smaller stocks with high book-to-market ratios which have performed poorly in the past are more likely to enter financial distress. According to Fama and French (1992, 1993), the book-to-market factor captures the notion of "distress" and identify a risk premium called "distress premium". Indeed, firms with high book-to-market ratios are more likely to have had previously low earnings resulting in low market prices. Consequently, they are more prone to be near financial distress and therefore, they may not progress as the market rallies, and constitute an additional source of risk.
Finally, the most serious and still unresolved weakness pointed out by Fama and French (1996) to the three-factor model is that the price "momentum" effect, described by Jegadeesh and Titman (1993), is distinct from the value effect captured by the book-to-market ratio. The price momentum strategies remains, among all the anomalies observed by Fama and French (1996), the only one unexplained by the Fama and French (1993) three-factor model.
Arbitrage Pricing Theory
Ross and Roll model
Critiques
Conclusion
On the one hand, even if new models have proved the CAPM wrong and are gaining recognition in portfolio management, the CAPM despite its shortcomings is still the most used model in the financial industry. As mentioned by Fama and French (2004), the attraction of the CAPM is that it is quite a straightforward model that offers powerful and intuitive predictions to measure risk and the relation between expected return and risk. On the other hand, no one can tell whether the problems in asset pricing are due to irrational pricing or a bad asset pricing model.
