# Analysis of the Capital Assets Pricing Model

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### Chapter I

### The QUARREL ON THE CAPM: A LITERATURE SURVEY

### Abstract

The current chapter has attempted to do three things. First it presents an overview on the capital asset pricing model and the results from its application throughout a narrative literature review. Second the chapter has argued that to claim whether the CAPM is dead or alive, some improvements on the model must be considered. Rather than take the view that one theory is right and the other is wrong, it is probably more accurate to say that each applies in somewhat different circumstances (assumptions). Finally the chapter has argued that even the examination of the CAPM’s variants is unable to solve the debate into the model. Rather than asserting the death or the survival of the CAPM, we conclude that there is no consensus in the literature as to what suitable measure of risk is, and consequently as to what extent the model is valid or not since the evidence is very mixed. So the debate on the validity of the CAPM remains a questionable issue.

### 1. INTRODUCTION

The traditional capital assets pricing model (CAPM), always the most widespread model of the financial theory, was prone to harsh criticisms not only by the academicians but also by the experts in finance. Indeed, in the last few decades an enormous body of empirical researches has gathered evidences against the model. These evidences tackle directly the model’s assumptions and suggest the dead of the beta (Fama and French, 1992); the systematic risk of the CAPM.

If the world does not obey to the model’s predictions, it is maybe because the model needs some improvements. It is maybe because also the world is wrong, or that some shares are not correctly priced. Perhaps and most notably the parameters that determine the prices are not observed such as information or even the returns’ distribution. Of course the theory, the evidence and even the unexplained movements have all been subject to much debate. But the cumulative effect has been to put a new look on asset pricing. Financial Researchers have provided both theory and evidence which suggest from where the deviations of securities’ prices from fundamentals are likely to come, and why could not be explained by the traditional CAPM.

Understanding security valuation is a parsimonious as well as a lucrative end in its self. Nevertheless, research on valuation has many additional benefits. Among them the crucial and relatively neglected issues have to do with the real consequences of the model’s failure. How are securities priced? What are the pricing factors and when? Once it is recognized that the model’s failure has real consequences, important issues arise. For instance the conception of an adequate pricing model that accounts for all the missing aspects.

The objective of this chapter is to look at different approaches to the CAPM, how these have arisen, and the importance of recognizing that there’s no single ‘’right model” which is adequate for all shares and for all circumstances, i.e. assumptions. We will, so move on to explore the research task, discuss the goodness and the weakness of the CAPM, and look at how different versions are introduced and developed in the literature. We will, finally, go on to explore whether these recent developments on the CAPM could solve the quarrel behind its failure.

For this end, the recent chapter is organized as follows: the second section presents the theoretical bases of the model. The third one discusses the problematic issues on the model. The fourth section presents a literature survey on the classic version of the model. The five section sheds light on the recent developments of the CAPM together with a literature review on these versions. The next one raises the quarrel on the model and its modified versions. Section seven concludes the paper.

### 2. THEORETICAL BASES OF THE CAPITAL ASSET PRICING MODEL

In the field of finance, the CAPM is used to determine, theoretically, the required return of an asset; if this asset is associated to a well diversified market portfolio while taking into account the non diversified risk of the asset its self. This model, introduced by Jack Treynor, William Sharpe and Jan Mossin (1964, 1965) took its roots of the Harry Markowitz’s work (1952) which is interested in diversification and the modern theory of the portfolio. The modern theory of portfolio was introduced by Harry Markowitz in his article entitled “Portfolio Selection”, appeared in 1952 in the Journal of Finance.

Well before the work of Markowitz, the investors, for the construction of their portfolios, are interested in the risk and the return. Thus, the standard advice of the investment decision was to choose the stocks that offer the best return with the minimum of risk, and by there, they build their portfolios.

On the basis of this point, Markowitz formulated this intuition by resorting to the diversification’s mathematics. Indeed, he claims that the investors must in general choose the portfolios while getting based on the risk criterion rather than to choose those made up only of stocks which offer each one the best risk-reward criterion. In other words, the investors must choose portfolios rather than individual stocks. Thus, the modern theory of portfolio explains how rational investors use diversification to optimize their portfolio and what should be the price of an asset while knowing its systematic risk.

Such investors are so-called to meet only one source of risk inherent to the total performance of the market; more clearly, they support only the market risk. Thus, the return on a risky asset is determined by its systematic risk. Consequently, an investor who chooses a less diversified portfolio, generally, supports the market risk together with the uncertainty’s risk which is not related to the market and which would remain even if the market return is known.

Sharpe (1964) and Linter (1965), while basing on the work of Harry Markowitz (1952), suggest, in their model, that the value of an asset depends on the investors’ anticipations. They claim, in their model that if the investors have homogeneous anticipations (their optimal behavior is summarized in the fact of having an efficient portfolio based on the mean-variance criterion), the market portfolio will have to be the efficient one while referring to the mean-variance criterion (Hawawini 1984, Campbell, Lo and MacKinlay 1997).

The CAPM offer an estimate of a financial asset on the market. Indeed, it tries to explain this value while taking into account the risk aversion, more particularly; this model supposes that the investors seek, either to maximize their profit for a given level of risk, or to minimize the risk taking into account a given level of profit.

The simplest mean-variance model (CAPM) concludes that in equilibrium, the investors choose a combination of the market portfolio and to lend or to borrow with proportions determined by their capacity to support the risk with an aim of obtaining a higher return.

### 2.1. Tested Hypothesis

The CAPM is based on a certain number of simplifying assumptions making it applicable. These assumptions are presented as follows:

– The markets are perfect and there are neither taxes nor expenses or commissions of any kind;

– All the investors are risk averse and maximize the mean-variance criterion;

– The investors have homogeneous anticipations concerning the distributions of the returns’ probabilities (Gaussian distribution); and

– The investors can lend and borrow unlimited sums with the same interest rate (the risk free rate).

The aphorism behind this model is as follows: the return of an asset is equal to the risk free rate raised with a risk premium which is the risk premium average multiplied by the systematic risk coefficient of the considered asset. Thus the expression is a function of:

– The systematic risk coefficient which is noted as;

– The market return noted;

– The risk free rate (Treasury bills), noted

This model is the following:

Where:

; represents the risk premium, in other words it represents the return required by the investors when they rather place their money on the market than in a risk free asset, and;

; corresponds to the systematic risk coefficient of the asset considered.

From a mathematical point of view, this one corresponds to the ratio of the covariance of the asset’s return and that of the market return and the variance of the market return.

Where:

; represents the standard deviation of the market return (market risk), and

; is the standard deviation of the asset’s return. Subsequently, if an asset has the same characteristics as those of the market (representative asset), then, its equivalent will be equal to 1. Conversely, for a risk free asset, this coefficient will be equal to 0.

The beta coefficient is the back bone of the CAPM. Indeed, the beta is an indicator of profitability since it is the relationship between the asset’s volatility and that of the market, and volatility is related to the return’s variations which are an essential element of profitability. Moreover, it is an indicator of risk, since if this asset has a beta coefficient which is higher than 1, this means that if the market is in recession, the return on the asset drops more than that of the market and less than it if this coefficient is lower than 1.

The portfolio risk includes the systematic risk or also the non diversified risk as well as the non systematic risk which is known also under the name of diversified risk. The systematic risk is a risk which is common for all stocks, in other words it is the market risk. However the non systematic risk is the risk related to each asset. This risk can be reduced by integrating a significant number of stocks in the market portfolio, i.e. by diversifying well in advantage (Markowitz, 1985). Thus, a rational investor should not take a diversified risk since it is only the non diversified risk (risk of the market) which is rewarded in this model. This is equivalent to say that the market beta is the factor which rewards the investor’s exposure to the risk.

In fact, the CAPM supposes that the market risk can be optimized i.e. can be minimized the maximum. Thus, an optimal portfolio implies the weakest risk for a given level of return. Moreover, since the inclusion of stocks diversifies in advantage the portfolio, the optimal one must contain the whole stocks on the market, with the equivalent proportions so as to achieve this goal of optimization. All these optimal portfolios, each one for a given level of return, build the efficient frontier. Here is the graph of the efficient frontier:

The (Markowitz) efficient frontier

The efficient frontier

Lastly, since the non systematic risk is diversifiable, the total risk of the portfolio can be regarded as being the beta (the market risk).

### 3. Problematic issues on the CAPM

Since its conception as a model to value assets by Sharpe (1964), the CAPM has been prone to several discussions by both academicians and experts. Among them the most known issues concerning the mean variance market portfolio, the efficient frontier, and the risk premium puzzle.

### 3.1 The mean-variance market portfolio

The modern portfolio theory was introduced for the first time by Harry Markowitz (1952). The contribution of Markowitz constitutes an epistemological shatter with the traditional finance. Indeed, it constitutes a passageway from an intuitive finance which is limited to advices related to financial balance or to tax and legal nature advices, to a positive science which is based on coherent and fundamental theories. One allots to Markowitz the first rigorous treatment of the investor dilemma, namely how obtaining larger profits while minimizing the risks.

### 3.2 The efficient frontier

### 3.3 The equity premium puzzle

### 4. Background on the CAPM

“The attraction of the CAPM is that it offers powerful and intuitively pleasing predictions about how to measure risk and the relation between expected return and risk. Unfortunately, the empirical record of the model is poor – poor enough to invalidate the way it is used in applications. The CAPM’s empirical problems may reflect theoretical failings, the result of many simplifying assumptions.”

Fama and French, 2003, “The Capital Asset Pricing Model: Theory and Evidence”, Tuck Business School, Working Paper No. 03-26

Being a theory, the CAPM found the welcome thanks to its circumspect elegance and its concept of good sense which supposes that a risk averse investor would require a higher return to compensate for supported the back-up risk. It seems that a more pragmatic approach carries out to conclude that there are enough limits resulting from the empirical tests of the CAPM.

Tests of the CAPM were based, mainly, on three various implications of the relation between the expected return and the market beta. Firstly, the expected return on any asset is linearly associated to its beta, and no other variable will be able to contribute to the increase of the explanatory power. Secondly, the beta premium is positive which means that the market expected return exceeds that of individual stocks, whose return is not correlated with that of the market. Lastly, according to the Sharpe and Lintner model (1964, 1965), stocks whose return is not correlated with that of the market, have an expected return equal to the risk free rate and a risk premium equal to the difference between the market return and the risk free rate return. In what follows, we are going to examine whether the CAPM’s assumptions are respected or not through the empirical literature.

Starting with Jensen (1968), this author wants to test for the relationship between the securities’ expected return and the market beta. For this reason, he uses the time series regression to estimate for the CAPM´ s coefficients. The results reject the CAPM as for the moment when the relationship between the expected return on assets is positive but that this relation is too flat. In fact, Jensen (1968) finds that the intercept in the time series regression is higher than the risk free rate. Furthermore, the results indicate that the beta coefficient is lower than the average excess return on the market portfolio.

In order to test for the CAPM, Black et al. (1972) work on a sample made of all securities listed on the New York Stock Exchange for the period of 1926-1966. The authors classify the securities into ten portfolios on the basis of their betas.They claim that grouping the securities with reference to their betas may offer biased estimates of the portfolio beta which may lead to a selection bias into the tests. Hence, so as to get rid of this bias, they use an instrumental variable which consists of taking the previous period’s estimated beta to select a security’s portfolio grouping for the next year.

For the estimate of the equation, the authors use the time series regression. The results indicate, firstly, that the securities associated to high beta had significantly negative intercepts, whereas those with low beta had significantly positive intercepts. It was proved, also, that this effect persists overtime. Hence, these evidences reject the traditional CAPM. Secondly, it is found that the relation between the mean excess return and beta is linear which is consistent with the CAPM.

Nevertheless, the results point out that the slopes and intercepts in the regression are not reliable. In fact, during the prewar period, the slope was sharper than that predicted by the CAPM for the first sub period, and it was flatter during the second sub period. However, after that, the slope was flatter. Basing on these results, Black, Fischer, Michael C. Jensen and Myron Scholes (1972) conclude that the traditional CAPM is inconsistent with the data.

Fama and MacBeth (1973) propose another regression method so as to overcome the problem related to the residues correlation in a simple linear regression. Indeed, instead of estimating only one regression for the monthly average returns on the betas, they propose to estimate regressions of these returns month by month on the betas. They include all common stocks traded in NYSE from 1926 to 1968 in their analysis.

The monthly averages of the slopes and intercepts, with the standard errors of the averages, thus, are used to check, initially, if the beta premium is positive, then to test if the averages return of assets which are not correlated with the market return is from now on equal to the average of the risk free rate. In this way, the errors observed on the slopes and intercepts are directly given by the variation for each month of the regression coefficients, which detect the effects of the residues correlation over the variation of the regression.

Their study led to three main results. At first, the relationship between assets return and their betas in an efficient portfolio is linear. At second, the beta coefficient is an appropriate measure of the security’s risk and no other measure of risk can be a better estimator. Finally, the higher the risk is, the higher the return should be.

Blume and Friend (1973) in their paper try to examine theoretically and empirically the reasons beyond the failure of the market line to explain excess return on financial assets. The authors estimate the beta coefficients for each common stock listed in the New York Stock Exchange over the period of January 1950 to December 1954. Then, they form 12 portfolios on the basis of their estimated beta. They afterwards, calculate the monthly return for each portfolio. Third, they calculate the monthly average return for portfolios from 1955 to 1959. These averaged returns were regressed to obtain the value of the beta portfolios. Finally, these arithmetic average returns were regressed on the beta coefficient and the square of beta as well.

Through, this study, the authors point out that the failure of the capital assets pricing model in explaining returns maybe due to the simplifying assumption according to which the functioning of the short-selling mechanism is perfect. They defend their point of view while resorting to the fact that, generally, in short sales the seller cannot use the profits for purchasing other securities.

Moreover, they state that the seller should make a margin of roughly 65% of the sales market value unless the securities he owns had a value three times higher than the cash margin. This makes a severe constraint on his short sales. In addition to that, the authors reveal that it is more appropriate and theoretically more possible to remove the restriction on the short sales than that of the risk free rate assumption (i.e., to borrow and to lend on a unique risk free rate).

The results show that the relationship between the average realized returns of the NYSE listed common stocks and their correspondent betas is almost linear which is consistent with the CAPM assumptions. Nevertheless, they advance that the capital assets pricing model is more adequate for the estimates of the NYSE stocks rather than other financial assets. They mention that this latter conclusion is may be owed to the fact that the market of common stocks is well segmented from markets of other assets such as bonds.

Finally, the authors come out with the two following conclusions: Firstly, the tests of the CAPM suggest the segmentation of the markets between stocks and bonds. Secondly, in absence of this segmentation, the best way to estimate the risk return tradeoff is to do it over the class of assets and the period of interest.

The study of Stambaugh (1982) is interested in testing the CAPM while taking into account, in addition to the US common stocks, other assets such as, corporate and government bonds, preferred stocks, real estate, and other consumer durables. The results indicate that testing the CAPM is independent on whether we expand or not the market portfolio to these additional assets.

Kothari Shanken and Sloan (1995), show that the annual betas are statistically significant for a variety of portfolios. These results were astonishing since not very early, Fama and French (1992), found that the monthly and the annual betas are nearly the same and are not statistically significant. The authors work on a sample which covers all AMEX firms for the period 1927-1990. Portfolios are formed in five different ways. Firstly, they from 20 portfolios while basing only on beta. Secondly, 20 portfolios by grouping on size alone. Thirdly, they take the intersection of 10 independent beta or size to obtain 100 portfolios. Then, they classify stocks into 10 portfolios on beta, and after that into 10 portfolios on size within each beta group. They, finally, classify stocks into 10 portfolios on size and then into 10 portfolios on beta within each size group. They use the GRSP equal weighted portfolio as a proxy for the whole market return market.

The cross-sectional regression of monthly return on beta and size has led to the following conclusions: On the one hand, when taking into account only the beta, it is found that the parameter coefficient is positive and statistically significant for both the sub periods studied. On the other hand, it is demonstrated that the ability of beta and size to explain cross sectional variation of the returns on the 100 portfolios ranked on beta given the size, is statistically significant. However, the incremental economic benefit of size given beta is relatively small.

Fama and French published in 1992 a famous study putting into question the CAPM, called since then the “Beta is dead” paper (the article announcing the death of Beta). The authors use a sample which covers all the stocks of the non-financial firms of the NYSE, AMEX and NASDAQ for the period of the end of December 1962 until June 1990. For the estimate of the betas; they use the same test as that of Fama and Macbeth (1973) and the cross-sectional regression.

The results indicate that when paying attention only to the betas variations which are not related to the size, it is found that the relation between the betas and the expected return is too flat, and this even if the beta is the only explanatory variable. Moreover, they show that this relationship tend to disappear overtime.

In order to verify the validity of the CAPM in the Hungarian stock market, Andor et al. (1999) work on daily and monthly data on 17 Hungarian stocks between the end of July 1991 and the beginning of June 1999. To proxy for the market portfolio the authors use three different indexes which are the BUX index, the NYSE index, and the MSCI world index.

The regression of the stocks’ return against the different indexes’ return indicates that the CAPM holds. Indeed, in all cases it is found that the return is positively associated to the betas and that the R-squared value is not bad at all. They conclude, hence, that the CAPM is appropriate for the description of the Hungarian stock market.

For the aim of testing the validity of the CAPM, Kothari and Jay Shanken (1999), study the one factor model with reference to the size anomaly and the book to market anomaly. The sample used in their study contains annual return on portfolios from the GRSP universe of stocks. The portfolios are formed every July from 1927 to 1992. The formation procedure is the following; every year stocks are sorted on the basis of their market capitalization and then on their betas while regressing the past returns on the GRSP equal weighted index return. They obtain, hence, ten portfolios on the basis of the size. Then, the stocks in each size portfolio are grouped into ten portfolios based on their betas. They repeat the same procedure to obtain the book-to-market portfolios.

Using the Fama and MacBeth cross-sectional regression, the authors find those annual betas perform well since they are significantly associated to the average stock returns especially for the period 1941-1990 and 1927-1990. Moreover, the ability of the beta to predict return with reference to the size and the book to market is higher. In a conclusion, this study is a support for the traditional CAPM.

Khoon et al. (1999), while comparing two assets pricing models in the Malaysian stock exchange, examine the validity of the CAPM. The data contains monthly returns of 231 stocks listed in the Kuala Lumpur stock exchange over the period of September 1988 to June 1997. Using the cross section regression (two pass regression) and the market index as the market portfolio, the authors find that the beta coefficient is sometimes positive and some others negative, but they do not provide any further tests.

In order to extract the factors that may affect the returns of stocks listed in the Istanbul stock exchange, Akdeniz et al. (2000)make use of monthly return of all non financial firms listed in the up mentioned stock market for the period that spans from January 1992 to December 1998. They estimate the beta coefficient in two stages using the ISE composite index as the market portfolio.

First, they employ the OLS regression and estimate for the betas each month for each stock. Then, once the betas are estimated for the previous 24 months (time series regression), they rank the stocks into five equal groups on the basis of the pre-ranking betas and the average portfolio beta is attributed to each stock in the portfolio. They, afterwards, divide the whole sample into two equal sub-periods and the estimation procedure is done for each sub-period and the whole period as well.

The results from the cross sectional regression, indicate that the return has no significant relationship with the market beta. This variable does not appear to influence cross section variation in all the periods studied (1992-1998, 1992-1995, and 1995-1998).

In a relatively larger study, Estrada (2002) investigates the CAPM with reference to the downside CAPM. The author works on a monthly sample covering the period that spans from 1988 to 2001 (varied periods are considered) on stocks of 27 emerging markets.

Using simple regression, the authors find that the downside beta outperforms the traditional CAPM beta. Nevertheless, the results do not support the rejection of the CAPM from two aspects. Firstly, it was found that the intercept from the regression is not statistically different from zero. Secondly, the beta coefficient is positive and statistically significant and the explanatory power of the model is about 40%. This result stems for the conclusion according to which the CAPM is still alive within the set of countries studied.

In order to check the validity of the CAPM, and the absence of anomalies that must be incorporated to the model, Andrew and Joseph (2003) try to investigate the ability of the model to predict book-to market portfolios. If it is the case, then the CAPM captures the Book-to-market anomaly and there’s no need to further incorporate it in the model.

For this intention, the authors work on a sample that covers the period of 1927-2001 and contains monthly data on stocks listed in the NYSE, AMEX, and NASDAQ. So as to form the book-to-market portfolios, they use, alike Fama and French (1992), the size and the book-to-market ratio criterion. To estimate for the market return, they use the return on the value weighted portfolios on stocks listed in the pre-cited stock exchanges and to proxy for the risk free rate; they employ the one-month Treasury bill rate from Ibbotson Associates. They, afterwards, divide the whole period into two laps of time; the first one goes from July 1927 to June 1963, and the other one span from July 1963 to the end of 2001.

Using asymptotic distribution the results indicate that the CAPM do a great job over the whole period, since the intercept is found to be closed to zero, but there is no evidence for a value premium. Hence, they conclude that the CAPM cannot be rejected. However, for the pre-1963 period the book to market premium is not significant at all, whereas for the post-1963 period this premium is relatively high and statistically significant. Nevertheless, when accounting for the sample size effect, the authors find that there is an overall risk premium for the post-1963 period. The authors conclude then that, taken as a whole, the study fails to reject the null that the CAPM holds. This study points to the necessity to take into account the small sample bias.

Fama and French (2004), estimate the betas of stocks provided by the CRSP (Center for Research in Security Prices of the University of Chicago) of the NYSE (1928-2003), the AMEX (1963-2003) and the NASDAQ (1972-2003). They form, thereafter, 10 portfolios on the basis of the estimated betas and calculate their return for the eleven months which follow. They repeat this process for each year of 1928 up to 2003.

They claim that, the Sharpe and Lintner model, suppose that the portfolios move according to a linear line with an intercept equal to risk free rate and a slope which is equal to the difference between the expected return on the market portfolio and that of the risk free rate. However, their study, and in agreement with the previous ones, confirms that the relation between the expected return on assets and their betas is much flatter than the prediction of the CAPM.

Indeed, the results indicate that the expected return of portfolios having relatively lower beta are too high whereas expected return of those with higher beta is too low. Moreover, these authors indicate that even if the risk premium is lower than what the CAPM predicts, the relation between the expected return and beta is almost linear. This latter result, confirms the CAPM of Black which assumes that only the beta premium is positive. This means, analogically, that only the market risk is rewarded by a higher return.

In order to test for the consistency of the CAPM with the economic reality, Thierry and Pim (2004) use monthly return of stocks from the NYSE, NASDAQ, and AMEX for the period that spans from 1926-2002. The one -month US Treasury bill is used as a proxy for the risk free rate, The CRPS total return index which is a value-weighted average of all US stocks included in this study is used as a proxy for the market portfolio.

They sort stocks into ten deciles portfolios on the basis of historical 60 months. They afterwards, calculate for the following 12 months their value weighted returns. They obtain, subsequently, 100 beta-size portfolios. The results from the time series regression indicate, firstly, that the intercepts are statistically indifferent from zero. Secondly, it is found that the betas’ coefficients are all positive. Furthermore, in order to check the robustness of the model, the authors split the whole sample into sub-samples of equal length (432 months). The results indicate, also, that for all the periods studied the intercepts are statistically not different from zero except for the last period.

In his empirical study, Blake T (2005) works on monthly stocks return on 20 stocks within the S&P 500 index during January 1995-December 2004. The S&P 500 index is used as the market portfolio and the 3-month Treasury bill in the Secondary Market as the risk free rate. His methodology can be summarized as follows; the excess return on each stock is regressed against the market excess return. The excess return is taken as the sample average of each stock and the market as well. After estimating of the betas, these values are used to verify the validity of the CAPM. The coefficient of beta is estimated by regressing estimated expected excess stock returns on the estimates of beta and the regression include intercept and the residual squared so as to measure the non systematic risk.

The results confirm the validity of the CAPM through its three major assumptions. In fact, the null hy

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