The Capital Asset Pricing Model - Essay Example

First publication on CAPM Capital asset pricing model (CAPM) was first widely introduced in a fundamental work of William Sharpe (1964). It was presented as an extension of Markowitz’s (1952) portfolio theory, according to which investors should consider overall risk-return rates of a portfolio instead of constructing portfolio from securities with best individual risk-return characteristics. Using the definition of Markowitz’s market portfolio (it consists of every asset weighted proportionally to its total market value) Sharpe added several assumptions, such as: 1) there are no taxes or transaction costs; 2) all investors share the same market opportunities; and 3) all investors have the same information on expected returns, volatilities, and correlations of securities available. It was found that under these assumptions Tobin’s (1958) super efficient portfolio (it consists of the risk-free asset added to Markowitz’s portfolio on the efficient frontier) must also be market portfolio. Further on, Sharpe (1964) divided portfolio risk into systematic and specific. While systematic risk affects every asset of a portfolio (as the market moves, each individual asset is more or less affected), specific risks are unique to individual assets (it represents the component of an assets return which is uncorrelated with general market moves) and thus can be diversified in the context of a whole portfolio. In other words, the expected rate of return of a portfolio depends not on specific risks of assets, but on the systematic risk of a portfolio. This conclusion was mathematically expressed in Security Market Line (SML) equation: ERi=Rf + (ERm – Rf) β, where ERi is the expected rate of return on asset i, Rf is a risk-free rate, ERm is the expected rate of return of the market portfolio, and β is systematic risk. As can be seen from the SML equation, excess return depends on beta alone and not on systematic risk plus specific risk. Moreover, the connection between rate of return and beta is linear for portfolios. Obviously, CAPM was designed as a way to determine prices of assets in market portfolios. Indeed, given a systematic risk value and asset’s expected rate of return investor can adjust the price of an asset using the SML formula. However, because of its ‘ideal’ nature CAPM is often seen only as a theoretical tool. In practice its main assumptions are not true, and all investors have different information on risk-return characteristics of assets. Empirical Tests of CAPM Since CAPM introduction to nowadays SML equation became a topic of wide academic discussion. Studies performed to assess the validity of CAPM can be divided into three general groups: supporting CAPM (e.g. Black, Jensen, and Scholes 1972; Fama and MacBeth 1973), criticizing CAPM (e.g. Fama and French 1992), and criticizing the critiques of CAPM (e.g. Roll 1977). Abundant empirical data provided on CAPM and SML equation is best reflected with several classical studies. All of the reviewed empirical tests concentrated on whether beta alone can explain historical average returns on portfolios. They took a representative value-weighted index as a market portfolio and examined correlation between the results of SML equation and historical average returns on securities. Since neither expected returns nor betas were unknown all of them had to plot the return and beta data against each other. These fundamental studies are reviewed below in chronological order. Study by Black, Jensen and Scholes (1972) Black, Jensen and Scholes (1972) used all of the stocks on New York Stock Exchange (NYSE) during 1926-1965 as 10 market portfolios. They sorted assets into portfolios basing on historical betas, which allowed them to acquire portfolios with different historical beta estimates, increasing the sustainability of the test. One of the first findings produced by Black-Jensen-Scholes test using linear regression of average monthly ecess returns on beta was the significant difference between the slope and intercept of historical regression line and theoretical values produced by CAPM. In fact, while the average monthly excess return on the market proxy used in the study is 1.42 per cent, and the estimated slope for regression line is 1.08 instead of 1.42 per cent predicted by CAPM. The picture is the same with intercept: 0.519 (hisstorical) instead of 0 (CAPM). Thus, regression line values differ from theoretically predicted results significantly. Yet, this does not mean the test does not support CAPM. Indeed, authors came to an obvious explanation of such differences: if no risk-free asset exists, then the CAPM does not predict the intercept of 0. Therefore the data provided by the empirical test is consistent with CAPM modification introduced by Black (1972), where is no risk-free asset available. Therefore, the prediction of linearity of SML equation is not rejected in this study. Black, Jensen and Scholes have found a remarkably close relationship between beta and monthly returns. Study by Fama and MacBeth (1973) Another test supportive of CAPM was produced by Fama and MacBeth (1973). It was also based on the traded stocks on the NYSE between 1926 and 1965 and took equally weighted portfolio of all NYSE stocks as their proxy for market portfolio. They examined whether there is a positive linear relation between average return and beta. Additionally, the study tried to find whether the expected return is purely determined by portfolio’s beta and not by residual variance of the portfolio. In other words, they tried to find whether the squared value of beta and the volatilityof the return on an asset can explain the residual variation in average returns across assets that is not explained by beta alone. Fama and MacBeth first tested the effectiveness of CAPM in justifying observed cross-sectional variability of returns. Fama-MacBeth regression estimated the parameters in two steps: 1) regress each asset against risk factors determining beta particularly for certain asset and certain risk factor; 2) regress all asset returns for a fixed time period against estimated betas to determine risk premium for each factor. After estimating betas and historical average returns, following regressions were used: Rp=α0+α1β+α2β2+ε Rp= α0+α1β+α2β2+α3RV+ε, where RV is the average residual variance of each security included in portfolio p. Finally, Fama and French came to notion that if SML equation is valid, and the relation between average return and beta is linear, then α0 should be risk-free interest rate, α1 should be excess return on the market and α2 and α3 should be infinitesimal. The results obtained supported CAPM in both questions of the research: the relation between beta and average return was found linear, and beta was the only factor determining expected return, since α2 and α3 are equal to zero. Methodologies developed by Black, Jensen and Scholes (1972) and Fama-MacBeth (1973) were considered as breakthroughs. Most of the subsequent test of the CAPM used techniques developed within these frameworks, called a two-pass methodology. In general, the researcher at first implemented regression of securities or portfolio returns on their betas. However, since the beta is unknown, the first pass a time series regression of portfolio returns on the market return is run, providing estimates of portfolio betas. During the second pass cross-sectional regressions on a month-by-month basis are performed and then the time-series average of the estimated risk premium is taken (this modification was made by Fama-MacBeth of the initial Black-Jensen-Scholes regression in cross-section of the average returns on the estimated betas). Roll’s (1977) Critique The next significant work in CAPM development was the article of Richard Roll (1977). Since the SLM equation does not require us to make any assumption about supply-demand equality, therefore expected returns an equation holds behave quite independently of how securities are actually priced on the market. This fact has laid a base for Roll’s critique, that is: a test of any variable an equation holds does no tell you in practice anything about whether securities are priced in accordance with equilibrium in a mean-variance framework. Put another way, since the true market portfolio return is never observed, tests of CAPM are quite difficult to perform. The critique expressed by Richard Roll made all the previous tests of CAPM questionable. Indeed, as market portfolio is unobservable we have to use a proxy. On the one hand, if the proxy is not true you can reject the model even if CAPM is true. On the other hand, if CAPM is false and the proxy is mean-variance efficient you are unable to reject the model. Thus, any tests of CAPM cannot reveal the whole picture. Finally, Roll argues that tests of the CAPM are at best tests of the mean-variance efficiency of the portfolio that is taken as the market proxy. This critique was devastating. Roll demonstrates that the market, as defined in the theoretical CAPM, is not a single equity market, but an index of all wealth, and the market index must include bonds, property, foreign assets, human capital and anything else, tangible or intangible, that adds to the wealth of mankind. A number of authors have tried to tackle Roll’s critique. For example, Shanken (1987), and Kandel and Stambaugh (1987) both argue that, while stock market is in fact not the true market they still must be highly correlated with the true market portfolio. Alternative response to Roll’s critique was the use of proxies with broader sets of assets, including bonds and property. The problem with these tests was that CAPM was not holding to real historical performance according to them. Once again authors came to a problem outlined by Roll: tests, according to which CAPM was successful were based on inadequate proxies, and other tests criticized CAPM, while still being unable to reject it. Fama and French Study (1992) The important study of Fama and French (1992) provided evidences of other factors apart from beta that affect stock returns. In this study portfolio groups of similar betas and size were constructed from all non-financial stocks traded on NYSE, NASAQ, an AMEX between 1963 and 1990. Then these groups were compared to portfolios based on size alone. Surprisingly, CAPM performed well when working with size group (the latter one), but its performance was poor when working solely with size-and-betas-based groups. This has led authors to following conclusions: “(i)when one allows for variation in CAPM market βs that is unrelated to size, the univariate relation between β and average return for 1941-1990 is weak; (ii) β does not suffice to explain average return.” (Fama and French 1996, p. 1947). Other factors like firm size and other accounting ratios are considered to be better predictors of observed returns than beta. These evidences laid base for Fama-French three factor model. While it still complies with general CAPM philosophy, it uses three factors instead of one for more accurate predictions. Two additional factors are “small cap minus big” (SMB) and “high book-value-to-price minus low” (HML). Fama and French (1992) observed that two classes of stocks perform better than others: small caps and stocks with a high book-value-to-price ratio. SMB and HML indicators measure the historic excess returns of small caps and "value" stocks over the market as a whole. Thus, Fama and French three-factor model is quite contrary to CAPM, it uses linear relation between return and three factors instead of one. Despite, the fact that Fama and French provided evidences of superior performance of their model over CAPM (Fama and French 1996), it is still under a question whether one model is unambiguously more accurate than the other. For instance, Bruner et al (1998) and Graham and Harvey (2001) prefer to use CAPM for estimating the cost of equity. While most of the academic sources advice to use three-factor model CAPM advocates argue that it does provide additional accuracy, but provides more complex calculations. Ang and Chen Study (2007) The most modern of all the studies reviewed, Ang and Chen (2007) focuses on conditional variation of CAPM with time-varied beta. Their initial assumption was that book-to-market effect, which caused Fama and French to criticize traditional CAPM, because of its inability to explain and consider this effect (stocks with high book-to market ratio have higher average returns than CAPM expects), can be overcome with time-varied factors and dynamic risk premia. Indeed, their research indicates that betas of stocks with high book-to-market ratios vary from 2.5 during 1940s to -0.5 at the end of 2001 (Ang and Chen 2007, p.2). Therefore authors propose a conditional CAPM with time-varying betas, time-varying market risk premia and stochastic systematic volatility to use in order to explain the book-to-market anomaly. Despite the fact that several other authors used conditional CAPM to explain size effects (e.g. Zhang 2005), their results are debated and still induce doubts. For instance, Lewellen and Nagel (2006) question whether the conditional CAPM can truly address asset-pricing anomalies. They argue, first if the conditional CAPM was truly able to explain book-to-market anomalies only small deviations from the unconditional CAPM should be expected. Observed pricing errors seem to be simply too large to be explained by time variation in beta. In order to support this view they provide results of empirical test, finding that betas vary significantly over time, with relatively high-frequency changes from year to year. However, these changes are not enough to generate significant unconditional pricing errors. While Ang and Chen (2007) study seems to produce some light into the book-to-market phenomena, there is still not enough evidences and criticisms too strong to definitely support the notion that book-to-market anomalies are caused by beta variance over time. Conclusion Since its development CAPM has come a long historical path from total support in academic literature to criticisms of empirical tests supporting it, and further search for alternatives. From practical viewpoint even now the traditional CAPM seem to be of little value. However, its theoretical development has revealed the following important notions which cannot be denied. First of all, it was found that expected returns are linearly related to risk factors. Second, Roll’s (1977) critique provided important knowledge on testing CAPM. Since market portfolio cannot be properly approximated, we should not rely on a single test but rather find similar evidences in different ones. Third, Fama and French (1992) revealed the anomalies that cannot be explained by unconditional CAPM. They also introduced alternative three-factor model. Finally, the debate over conditional CAPM’s ability to resolve the stated issues continues, and requires further evidences for supporting or rejecting this hypothesis. References Ang, A., Chen, J. (2007). “CAPM over the long run: 1926-2001”, Journal of Empirical Finance, 1-40. Black, F. (1972). “Capital market equilibrium with restricted borrowing”, Journal of Business, 45 (7), 444–55. Bruner, R. F., Eades K., Harris R. and Higgins R. (1998), “Best Practices in Estimating the Cost of Capital: Survey and Synthesis”, Financial Practice and Education, Vol. 8(1), 13-28. Fama, E.F., French, K.R. (1992). “The Cross-Section of Expected Stock Returns”, Journal of Finance, 47 (2), 427-465. Fama, E.F., French, K.R. (1996). “CAPM is Wanted, Dead or Alive”, The Journal of Finance, 51(5), 1947-1958. Fama, E., MacBeth, J. (1973). “Risk, return and equilibrium: Empirical tests”, Journal of Political Economy, 71, 607-636. Graham, J.R. and Harvey, C.R. (2001). “The Theory and Practice of Corporate Finance: Evidence from the Field”, Journal of Financial Economics, 60(1-2), 187-243. Kandel S., Stambaugh, R.F. (1987). “On correlations and inferences about mean-variance efficiency”, Journal of Financial Economics, 18(1), 61-90. Lewellen, J., Nagel, S. (2006). “The Conditional CAPM Does Not Explain Asset-Pricing Anomalies”, Journal of Financial Economics, 82(2), 289-314. Markowitz, H.M. (1952). “Portfolio selection”, Journal of Finance, 7 (1), 77-91. Roll, R. (1977). “A critique of the asset pricing theorys tests”, Journal of Financial Economics, 4, 129-176 Shanken, J. (1987). “Multivariate proxies and asset pricing relations: Living with the Roll Critique”, Journal of Financial Economics, 18(1), 91-110 Sharpe, W.F. (1964). “Capital asset prices: A theory of market equilibrium under conditions of risk”, Journal of Finance, 19 (3), 425-442. Tobin, J. (1958). “Liquidity preference as behavior towards risk”, The Review of Economic Studies, 25, 65-86. Zhang, L. (2005). “The value premium”, Journal of Finance, 60, 67-103. Read More