The Capital Asset Pricing Model (CAPM)

University Degree Business and Administrative studies

Introduction

The Capital Asset Pricing Model (CAPM) was originally developed by Harry Markowitz in 1959 and further published by others, including William Sharpe, over a decade later. CAPM describes the relationship between risk and expected return, and serves as a model for the pricing of risky securities. It says that the expected return of a security or a portfolio equals the rate on a risk-free security plus a risk premium. If this expected return does not meet or beat our required return then the investment should not be undertaken. The commonly used formula to describe the CAPM relationship is as follows:

E(R) = Rf +[E(Rm) – E(Rf)]

where E(R) is the expected return on an asset or portfolio uncorrelated with the market and 1 is defined as Cov(R, Rm) /2(Rm). Beta measures the volatility of the security, relative to the asset class. The formula can be thinking as predicting a security’s behaviour as a function of beta: CAPM says that if investors know a security’s beta then investors know the value of r that expects it to have.

CAPM decomposes a portfolio’s risk into systematic and specific risk. Systematic risk is the risk of holding the market portfolio. As the market moves, each individual asset is more or less affected. To the extent that any asset participates in such general market moves, that asset entails systematic risk. Specific risk is the risk which is unique to an individual asset. It represents the component of an asset’s return which is with general market moves. According to CAPM, the marketplace compensates investors for taking systematic risk but not for taking specific risk. This is because specific risk can be away. When an investor holds the market portfolio, each individual asset in that portfolio entails specific risk, but through diversification, the investor's net exposure is just the systematic risk of the market portfolio.

The figure below describes portfolio opportunities.

The horizontal axis shows portfolio risk, measured by the standard deviation of portfolio return; the vertical axis shows expected return. The curve abc, which is called the minimum variance frontier, traces combinations of expected return and risk for portfolios of risky assets that minimize return variance at different levels of expected return. The tradeoff between risk and expected return for minimum variance portfolios is apparent. For example, an investor who wants a high expected return, perhaps at point a, must accept high volatility. At point T, the investor can have an intermediate expected return with lower volatility. If there is no risk-free borrowing or lending, only portfolios above b along abc are mean-variance-efficient, since these portfolios also maximize expected return, given their return variances.

The assumptions under CAPM

The CAPM is important as it quantifies and prices systematic risk and expresses it relative to the market portfolio. Thus CAPM provides us with the expected return of any asset or portfolio based on its risk as measured by beta, the risk premium of the market, and the risk free rate. This model is constructed in a hypothetical world based on several rigorous assumptions, but like in any economic model, these simplifications are essential in developing a workable model in a complex and diverse financial world. The assumptions of CAPM and deriving the CAPM through a simple proof, as pointed out in Elton and Gruber are as follows. The first of all assumption to take is there are no transaction costs. This assumption implies frictionless markets; however in reality transaction costs play a varied part in investment decisions. If transaction costs were present, the return from any asset would be a function of whether or not the investor owned it before the decision period. Thus transaction costs play an important role in reality as in most cases as it is not instantaneous shifting one portfolio of assets to another, and also the delivery costs may dissuade an investor even though it might be an included to form an optimum portfolio.

The second assumption behind the CAPM is that assets are infinitely divisible. This implies that an investor can buy any portion of an investment regardless of its size. In reality this is a highly unrealistic assumption as nearly all assets have a unit price.

The third one is the absence of personal income tax. Under this assumption, taxes such as Capital Gains Taxes are absent. Thus the derivation of CAPM is based on a simplified real world based on numerous assumptions.

The fourth assumption is unlimited lending and borrowing at a risk free rate. This implies ...

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The fourth assumption is unlimited lending and borrowing at a risk free rate. This implies that the investor can borrow and lend unlimited quantities at a rate equal to that of the riskless security. In reality it is impossible to borrow unlimited funds at a riskless rate; however it is not unrealistic to lend unlimited quantities at a riskless rate.

The fifth assumption is that an individual cannot affect the price of a stock by his buying or selling action. This implies perfect competition and this is a plausible real world assumption as there are many examples where perfect competition exists, where the share of a player is so small that it cannot affect the overall market.

The sixth one is that individuals only make their investment decisions based on standard deviations and expected value. Under this assumption, all investors are assumed to make all their investment decisions in terms of expected values and standard deviations of returns. Thus this does not allow for company fundamentals in their investment decisions, nor does it allow for tastes or trends.

The seventh assumption is unlimited short sales are allowed. This means that an investor can sell any security that he does not own and use the funds to buy another security. This is not a necessary assumption as CAPM is in equilibrium it implies that no investor sells any security short.

The eighth and ninth assumptions deal with the homogeneity of expectations. Investors are assumed to be concerned with the mean and variance of returns and assumed to have identical expectations with respect to the necessary inputs to the portfolio decision. This assumption is highly unrealistic as real world investors have definite heterogeneity with regards to expectations and interpretation of information.

The last assumption is that all assets can be bought on the open market. This assumption implies that the market portfolio exists and that all assets including human capital can be bought and sold in the market. It is physically impossible to market all global assets, let alone quantify all assets.

Current debates of CAPM

Until 1990’s, empirical tests of CAPM supported this model. However, Fama and French, who also tested the CAPM, inserted that the CAPM is useless for precisely what it was developed to do. From then on, numbers of researchers have been scrambling to figure out just what was going on. Among the studies, some are that support the CAPM (including Black; Black, Jensen, and Scholes; Fama and MacBenth), some are that challenge the model (Banz; Fama and French are included), others are that oppugn the challenges (Amihud, Christensen, and Mendelson; Black; Jagannathan and Wang; Kothari, Shanken, and Sloan).

For the empirical test of CAPM, neither expected return nor betas are known. Therefore both of them should be estimated. Black, Jensen and Scholes used the different betas to do the empirical test – time series test. They estimated betas by regressing historical returns on a proxy for the market portfolio; sort assets based on the historical betas; grouped assets into portfolios with increasing historical betas; held these portfolios for a period of years; changed the portfolio composition periodically. After their empirical study of CAPM, Black, Jensen and Scholes found that the data are consistent with the predictions of CAPM, and the CAPM is an approximation to reality as well as the other models.

Later, Fama and French used return data of NYSE during 1926 and 1968, so as to examine if there is a positive linear relation between average return and beta and if the squared value of beta and the volatility of return on an asset can explain by beta alone. As a result, they discovered that the data generally support the CAPM.

Nevertheless, the CAPM is not empirically conclusive, and its hypotheses are not absolute. The researchers, who grasped the failure of the CAPM model, argued that the empirical results were not always conclusive, because the market portfolio is unobservable. Therefore the CAPM is in a highly fragile position that “if the test including additional variable are justified, they show that the model is a failure”.

The first challenge of the benchmark CAPM model is the size of the firms. Banz tested the CAPM by examining whether the firm size can partly explain the residual variation in average returns across assets, which can not be represented by the CAPM betas. The test showed that the firm size does explain the cross – sectional variation in average returns on the particular assets better than the betas. He found that the small firs had a higher average return to stock than the large firms according to 1936 – 75 data. This is known as the size effect. Moreover, after the analysis of the data from July 1963 to December 1990, it had been discovered by Fama and French that the firm size is significant with or without beta. Thus, for a large collection of stocks, beta has no significant ability to explain the cross – sectional variation in average returns, whereas size has the explanatory power.

Furthermore, Fama and French emphasized the other factors that attributes to the cross – sectional variation. The main alternative to CAPM and the one academics recommend, at least, for estimation of portfolio returns, is the three-factor model suggested by Fama and French. The alternative is the use of more sophisticated estimation techniques to deal with problems such as errors in variables which arise when the simple techniques are used. In this model, size and book to market factors are included, in addition to a market index, as explanatory variables. As discussed above, this model is not popular among practitioners. The question is, why? In an attempt to answer this question, the performance of the three-factor model is compared with that of CAPM. Using 5 years of monthly data, it is found that the Fama - French model is at best able to explain, on average, 5% of differences in returns on individual stocks, independent of the index used. Such a small gain in explanatory power probably does not justify the extra work involved in including two more factors.

The evidence against the CAPM can be concluded as follows. For the sample periods, the relation between the average return and beta is completely flat. In addition, other explanatory variables such as firm size and book – to – market equity could better explain the cross – sectional variation in average asset returns.

Although the evidence against the CAPM is powerful, researchers still argue against the methodology of these challenges of CAPM model. Kothari, Shanken, and Sloan argued that a wide range of economically plausible risk premiums can not be rejected statistically. Amihud, Christensen, and Mendelson arose that the data are too noisy to invalidate the CAPM. They found that when a more efficient statistical method is used, the estimated relation between average return and beta is positive and significant. At the same time, Black suggested that the size effect noted by Banz could simply be a sample period effect. He claimed that the size affect appeared in some period but not in others. Meanwhile, even if there is a size effect, Jagannathan and Wang reported, there is a doubt about its importance given the relatively small value of small firms as a group used in these studies. Kothari, Shanken, and Sloan also considered the reason why the ratio of book – to – market equity is a powerful explanatory to average returns as a potential bias. They pointed that the firms that had a higher ratio of book – to – market equity early in the sample were less likely to survive, even though the survived ones presented a higher returns later. Therefore, Jagannathan and Wang developed a conditional CAPM as they thought that the lack of empirical support for CAPM should be due to the inappropriateness of some assumptions made to facilitate the empirical analysis of model.

With academics debating the value of the CAPM, obviously, capital budgeting decision were made before there was a CAPM, and they can be made again without it. However, the data seems suggested that those who chose to use CAPM now despite the academic debate will actually not be getting worthless advice.

Conclusion

The version of the CAPM developed by Sharpe (1964) and Lintner (1965) has never been an empirical success. In the late 1970s, research begins to uncover variables like size, various price ratios and momentum that add to the explanation of average returns provided by beta. The problems are serious enough to invalidate most applications of the CAPM. Finance textbooks often recommend using the Sharpe-Lintner CAPM risk-return relation to estimate the cost of equity capital. But empirical work, old and new, tells that relation between beta and average return is flatter than predicted by the Sharpe-Lintner version of the CAPM. The problem is that, because of the empirical failings of the CAPM, even passively managed stock portfolios produce abnormal returns if their investment strategies involve tilts toward CAPM problems. The CAPM, like Markowitz’s (1959) portfolio model on which it is built, is nevertheless a theoretical tour de force. Empirical failures of the CAPM could result from using bad proxies for the market portfolio. Although the true market may be mean–variance efficient, the proxies used in empirical tests may not. If the proxies are inefficient, then applications using them rely on flawed estimates of expected return. Thus, the CAPM is not a useful approximation of expected returns.

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