# 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 hypothesis for the constant term is not rejected. Moreover, the systematic risk coefficient is positive and statistically significant. Finally, the null hypothesis for the residual squared coefficient is not rejected. Hence, he concludes that none of the three necessary conditions for a valid model were rejected at the 95% level.

So as to estimate for the validity of the CAPM, Don Galagedera (2005) works on a Sample period from January 1995 to December 2004. This sample contains monthly data from emerging markets represented by 27 countries which are 10 Asian, 7 Latin American and 10 African, Middle-Eastern and European (Argentina, Brazil, Chile, China, Columbia, Czech, Egypt, Hungary, India, Indonesia, Israel, Jordan, Korea, Malaysia, Mexico, Morocco, Pakistan, Peru, Philippines, Poland, Russia, South Africa, Sri Lanka, Taiwan, Thailand, Turkey, and Venezuela ).

To proxy for the market index, the world index available in the MSCI database is used and the proxy for the risk-free rate is the 10-year US Treasury bond rate. The results indicate that the CAPM, compared to the downside beta CAPM, offers roughly the same estimates and the same performance.

Fama and French (2005) investigate the ability of the CAPM to explain the value premium in the US context during the period of 1926 to 2004 and include in their sample the NYSE, the AMEX and the NASDAQ stocks. They construct portfolios on the basis of the size, and the book-to-market. The size premium is the simple average of the returns on the three small stock portfolios minus the average of the returns on the three big stock portfolios. The value premium is the simple average of the returns on the two value portfolios minus the average of the returns on the two growth portfolios. They afterwards divide the whole period into two sub-periods which are respectively; 1926-1963 and 1963-2004.

Then, at the end of June of each year, the authors form 25 portfolios through the intersection of independent sorts of NYSE, AMEX, and NASDAQ stocks into five size groups and five book-to-market groups or Earning to price groups. Finally the size premium or the value premium of each of the six portfolios sorted by size and book-to-market is regressed against the market excess return.

The results show that for the size portfolios, when considering the whole period, the F-statistics shows that the intercepts are jointly different from zero. Moreover, it's found that the market premium is for all portfolios positive and statistically significant except. However, the explanatory power is relatively too low. Then, for the first sub-period the intercepts are not statistically significant which means that the intercepts are close to zero.

Moreover the beta coefficient is for all portfolios positive and statistically significant with a relatively higher explanatory power. Finally, for the last sub-period, the results are not supportive for the CAPM. In fact, for almost all portfolios, the beta is found to be negative and statistically significant. For further rejection, all the intercepts from the regression are statistically different from zero.

Erie Febrian and Aldrin Herwany (2007) test the validity of the CAPM in Jakarta Stock Exhange. They work involve three different periods which are respectively; the pre-crisis period (1992-1997), the crisis period (1997-2001), and the post-crisis period (2001-2007), and contains monthly data of all listed stocks in the Indonesian stock exchange. To proxy for the risk free rate, the authors make use of the 1-month Indonesian central bank rate (BI rate).

In order to estimate for the model, the authors use two approaches documented in the literature; it is bout particularly the times series regression and the cross sectional regression. The first one tests the relationship between variables in one time period, and the latter focuses on the observation of the relationship in various periods. To check the validity of the CAPM, they use manifold models used by; Sharpe and Linter (1965), Black Jensen and scholes (1972), Fama and Macbeth (1973), Fama and French (1992), and Cheng Roll and Ross (1986).

Using the Fama and French approach (1992) together with the cross sectional regression, the authors find that for all the three periods studied tests of the CAPM lead to the same conclusions. Indeed, the results indicate that the intercept from the cross sectional regression is negative and statistically significant which is a violation to the CAPM's assumption which pretends that the risk premium is the only risk factor. Nevertheless, the beta coefficient is found to be positive and statistically significant and that the R-squared is relatively high, which is a great support for the CAPM.

However, when using the Fama and Macbeth method (1973), the results are somehow different. In fact, the intercept is almost always not significant, unless for the latter period which stems for a negative and a significant intercept. The associated beta coefficient is only significant during the last period and the related sign is negative which contradicts the CAPM.

Furthermore, the coefficient from the square of beta is neither positive nor significant at all but in the last period (negative coefficient) which means that the relation between the risk premium and the excess return is, in a way, linear. Finally, the results indicate that the coefficient from the residuals is negative and statistically significant for the first and for the last period. Hence, basing on the above results, it is difficult to infer whether the CAPM is valid or not since the results are very sensitive to the method used for the estimate of the model.

Ivo Welch (2007) in his test of the CAPM works on daily data during the period of 1962-2007. The Federal Funds overnight rate is used as the risk free rate, the short term rate divided by 255 is used as the daily rate of return, the market portfolio is estimated from the daily S&P500 data, and The Long-Term Rate is the 10-year Treasury bond.

Using the time series regression, the difference between the 10 years treasuries and the overnight return is regressed against the market excess return. The results indicate that the intercept is close to zero. In addition to that, the beta coefficient is positive and statistically different from zero. These results are consistent with the CAPM.

Working on daily data of stocks listed on the thirty components of the Dow Jones Index and the SP 500 index for five years, one year, or a half year (180 days) of daily returns. While using the cross section regression, the authors find that for all periods used, the results are far from rejecting the CAPM. In fact, the R-squared is 0.28, which indicates, subsequently that the marker beta is a good estimator for the expected return.

Michael. D (2008) studies the ability of the CAPM to explain the reward-risk relationship in the Australian context. The sample of the study covers monthly data on stocks listed in the Australian stock exchange for the period of January 1974- December 2004. The equally value weighted ASE indices is used to proxy for the market portfolio. He forms portfolios on the basis of the betas, i.e. stocks are ranged each month on the basis of their estimated betas.

The main interesting result from this study is that which show that when the highest beta portfolio and the lowest beta portfolio are removed, it's found that portfolios' returns tend to increase with beta. Hence, the author concludes that the beta is an appropriate measure of risk in the Australian stock exchange.

Simon.G.M and Ashley Olson (XXXX) try to investigate the reliability of the CAPM's beta as an indicator of risk. For this end, they work on a one year sample data from November 2005 to November 2006. Their sample includes 288 publicly traded companies and the S&P 500 index (used as the market portfolio). In order to look into the risk return relationship, the authors group stocks into three portfolios on the basis of their betas, i.e. portfolios respectively; with low beta (about 0.5), market beta (about 1), and high beta (around 2).

The results point to the rejection of the CAPM. In fact, it was found that the assumption according to which the beta is an appropriate measure of risk s rejected. For example, according to the results obtained, if an investor takes the highest beta portfolio, the chance that this risk be rewarded is about only 11%.

In order to test for the validity of the CAPM, Arduino Cagnetti (XXXX) works on monthly returns (end of the month returns) of 30 shares on the Italian stock market for during the period of January 1990 to June 2001. To proxy for the market portfolio, he uses the Mib30 market index and the Italian official discount rate as a proxy for the risk-free rate. Then, he divided the whole period into two sub-periods of 5 years each one.

In order to test for the CAPM, the author uses two step procedures. In fact, the first step consists of employing the time series to estimate for the betas of the shares. While the second step involves the regression of the sample mean return on the betas. Then he runs a cross sectional regression while using the average returns for each period as dependant variable and the estimated betas as independent variables.

The results indicate that for the second sub-period the beta displays a high significance in explaining returns with a relatively strong explanatory power. However, for the first sub-period and during the whole 10 years, this variable is not significant at all.

In their study, Pablo and Roberto (2007)examine the validity of the CAPM with reference to the reward beta model and the three factor model. It is praiseworthy to note, in this level, that our investigation for the article is limited only to the study of the CAPM.

To reach their objective, the authors work on a sample covering the period that goes from July 1967 and ends in December 2006 and consists of all data on North American stock markets. That is the NYSE, AMEX, and the NASDAQ stock exchanges with all available monthly data on stocks. The one month Treasury bond is used to proxy for the risk free rate and the CRSP index as a proxy for the market portfolio (portfolio built with all the NYSE, AMEX and NASDAQ stocks weighted with the market value). The methodology pursued along this study is the classic two step methodology, i.e. the times series regression to estimate for ex-ante CAPM's beta, and then the cross section regression using the betas and the sensibilities of the estimated factors as explanatory variables in explaining expected return. To examine the validity of the CAPM, tests were run on the Fama and French's portfolios (6 portfolios formed on the basis of the size and the book to market).

The results were represented through two aspects whether or not the intercept is taken into account. On the one hand, when considering the CAPM with intercept, it is found that the latter is statistically different from zero which is a violation to the CAPM's assumptions. Moreover, results indicate that the risk premium is negative which is in contradiction with the Sharpe and Linter model. On the other hand, when removing the intercept, the results are rather supportive for the model. In fact, the risk premium appears to be positive and highly significant which goes in line with the CAPM. Nevertheless, the explanatory power in both cases is too low.

In his comparative study, Bornholt (2007) investigates the validity of three assets pricing models, it is about particularly, the CAPM, the three factor model and the reward beta model. It is trustworthy to mention again that in this literature we will be interested in the CAPM.

Working on monthly portfolios' return constructed according to the Fama and French methodology (1992) for the period that spans from July 1963 to December 2003. The one month Treasury bill is used as a proxy for the risk free rate and the GRSP value weighted index of all NYSE, AMEX, and NASDAQ stocks as a proxy for the market portfolio. The author uses the time series regression in order to estimate for the portfolios' betas (from 1963 to 1990) and then the cross section regression using the betas estimated as explanatory variables in expected return (1991 to 2003).

The results are interpreted from two sides depending on whether the estimation is done with or without the intercept. From the one side, when the intercept is included, the latter is found to be closed to zero which is in accordance with the CAPM. However, the beta coefficient is negative and not significant at all, which is in contradiction with the CAPM. This latter conclusion is enhanced further with the weak and the negative value of the R squared.

From the other side, the removal of the intercept does improve the beta coefficient which is in this case positive and statistically significant. Nevertheless, the explanatory power is getting worse. Hence, this study cannot claim to the acceptance or the rejection of the CAPM, since the results are foggy and far from allowing a rigorous conclusion.

Working on the French context, Najet et al. (2007) try to investigate the validity of the capital asset pricing model at different time scales. Their study sample entails daily data on 26 stocks in the CAC 40 index and covers the period that spans from January 2002 to December 2005. The CAC40 index is used as the market portfolio and the EUROBOR as the risk free rate. They consider six scales for the estimation of the CAPM which are; 2-4 days, 4-8 days dynamics, 8-16 days, 16-32 days, 32-64 days, 64-128 days, and 128-256 days.

The results from the OLS regression show that the relationship between the excess return of each stock is positive and statistically significant at all scales. The explanatory power moves up when the scales get widened. Which means that the higher the frequency is the stronger the relationship will be.

To further investigate the changes of the relationship over different scales, the authors use also the following intervals; 2-6, 6-12, 12-24, 24-48, 48-96, and 96-192. The results indicate that the relationship between the two variables of the model is becoming stronger as the scale increases. Nevertheless, they claim that the results obtained cannot draw conclusion about the linearity which remains ambiguous.

Within the Turkish context, Gürsoy and Rejepova (2007) check the validity of the CAPM. Their sample covers the period that spans from January 1995 to December 2004 and consists of weekly data on all stocks traded in the Istanbul Stock Exchange. The whole period was divided into five six-year sub-periods. The sub-periods are divided, in their turn, into 3 sub-periods of 2 years each, which correspond to respectively; portfolio formation, beta estimation and testing periods. All periods are considered with one overlapping year. They afterwards, form 20 portfolios of 10 stocks each one on the basis of their pre-ranking betas. Then for the regression equation they use two different approaches which are; the Fama and MacBeth's traditional approach, and Pettengil et al's conditional approach (1995).

With reference to the first approach, the CAPM is rejected in all directions. In fact, the results indicate that the intercept is statistically different from zero for all sub-periods except for one. In addition to that, the beta coefficient is found to be significant different from zero only in 2 sub-periods and almost in all cases is negative. Finally the R-squared is only significant in two sub-periods and beyond that it's too weak. Hence, this approach claims to the rejection of the CAPM.

As for the second approach, the beta is almost always found to be significant but either positive in up-market or negative in down- market. Furthermore, the results point to a very high explanatory power for both cases. However, the assumption according to which the intercept must be close to zero is not verified for both cases, which means that the risk premium is not the only risk factor in the CAPM. The authors conclude, on the basis of the results, that the validity of the CAPM remains a questionable issue in the Turkish stock market.

Robert and Janmaat (2009) examine the ability of the cross sectional and the multivariate tests of the CAPM under ideal conditions. They work on a sample made of monthly return on portfolios provided by the Kenneth French's data library. While examining the intercepts, the slopes and the R-squared, the authors reveal that these parameters are unable to inform whether the CAPM holds or not. Moreover, they claim that the positive and the statistically significant value of the beta coefficient, doesn't indicate that the CAPM is valid at all. The results indicate, also, that the value of the tested parameters, i.e. the intercepts and the slopes, is roughly the same independently whether the CAPM is true or false.

In addition to that, the results from the cross sectional regression have two different implications. From the one side, they indicate that tests of the hypothesis that the slope is equal to zero are rejected in only 10% of 2000 replications, which means, subsequently, that the CAPM is dead. From the other side, it's found that tests of the hypothesis that the intercept is equal to zero are rejected in 9%. As consequent, one may think that the CAPM is well alive. As for the R squared, its value is relatively lower particularly when the CAPM is true. The authors find, also, that the tests of the hypothesis that the intercept and the slope are equal to zero differ on whether they make use of the market or the equal weighted portfolio which confirms the Roll's critique.

### 5. IS THE CAPM DEAD OR ALIVE? SOME RESCUE ATTEMPTS

After reviewing the literature on the CAPM, it is difficult if not impossible to reach a clear conclusion about whether the CAPM is still valid or not. The assaults that tackle the CAPM's assumptions are far from being standards and the researchers versed in this field are still between defenders and offenders.

Actually, while Fama and French (1992) have announced daringly the dead of the CAPM and its bare foundation, some others (see for example, Black, 1993; Kothari, Shanken, and Sloan, 1995; MacKinlay, 1995 and Conrad, Cooper, and Kaul, 2003) have attributed the findings of these authors as a result of the data mining (or snooping), the survivorship bias, and the beta estimation. Hence, the response to the above question remains a debate and one may think that the reports of the CAPM's death are somehow exaggerated since the empirical literature is very mixed. Nevertheless, this challenge in asset pricing has opened a fertile era to derive other versions of the CAPM and to test the ability of these new models in explaining returns.

Consequently, three main classes of the CAPM's extensions have appeared and turn around the following approaches; the Conditional CAPM, the Downside CAPM, and the Higher-Order Co-Moment Based CAPM.

### 5.1. The Conditional CAPM

The academic literature has mentioned two main approaches around the modeling of the conditional beta. The first approach stems for a conditional beta by allowing this latter to depend linearly on a set of pre-specified conditioning variables documented in the economic theory (see for example Shanken, 1990). There have been several evidences that go on this road of research form whose we mention explicitly among others (Jagannathan and Wang, 1996; Lewellen, 1999; Ferson and Harvey, 1999; Lettau and Ludvigson, 2001 and Avramov and Chordia, 2006).

In spite of its revolutionary idea, this approach suffers from noisy estimates when applied to a large number of stocks since many parameters need to be estimated (see Ghysels, 1998). Furthermore, this approach may lead to many pricing errors even bigger than those generated by the unconditional versions (Ghysels and Jacquier, 2006). These limits are further enhanced by the fact that the set of the conditioning information is unobservable.

The second non parametric approach to model the dynamic of betas is that based on purely data-driven filters. The approaches in this category include the modeling of betas as a latent autoregressive process (see Jostova and Philipov, 2005; Ang and Chen, 2007), or estimating short-window regressions (Lewellen and Nagel, 2006), or also estimating rolling regressions (Fama and French, 1997). Even if these approaches sustain to the need to specify conditioning variables, it is not clear enough through the literature how many factors are they in the cross-sectional and time variation in the market beta.

In order to model the beta variation, studies have tried different modeling strategies. For instance, Jagannathan and Wang (1996) and Lettau and Ludivigson (2001) treat beta as a function of several economic state variables in a conditional CAPM. Engle, Bollerslev, and Wooldridge (1988) model the beta variation in a GARCH model. Adrian and Franzoni (2004, 2005) suggest a time-varying parameter linear regression model and use the Kalman filter to estimate the model. The following section treats these findings one by one while showing their main results.

### 5.1.1. Conditional Beta on Economic States

‘'If one were to take seriously the criticism that the real world is inherently dynamic, then it may be necessary to model explicitly what is missing in a static model", Jagannathan and Wang (1996), p.36.

The conditional CAPM is that in which betas are allowed to vary and to be non stationary through time. This version is often used to measure risk and to predict return when the risk can change. The conditional CAPM asserts that the expected return is associated to its sensitivity to a set of changes in the state of the economy. For each state there is a market premium or a premium per unit of beta. These price factors are often the business cycle variables.

The authors, who are interested in the conditional version of the CAPM, demonstrate that stocks can show large pricing errors compared to unconditional asset pricing models even when a conditional version of the CAPM holds perfectly. In what follows, a summary of the main studies versed in this field of research is presented.

Jagannathan and Wang (1996) were the first to introduce the conditional CAPM in its original version. They claim that the static CAPM has been founded on two unrealistic assumptions. The first one is that according to which the betas are constant over time. While the second one is that in which the portfolio, containing all stocks, is assumed to be a good proxy for the market portfolio. The authors assert that it would be completely reasonable to allow for the betas to vary over time since the firm's betas may change depending on the economic states. Moreover, they state that the market portfolio must involve the human capital.

Consequently, their model includes three different betas; the ordinary beta, the premium beta based on the market risk premium which allows for conditionality, and the labor premium which is based on the growth in labor income. Their study includes stocks of non financial firms listed in the NYSE and AMEX during the period that spans from July 1963 to December 1990. They, after that, group stocks into size portfolios and use the GRSP value weighted index as proxy for the market portfolio.

For each size decile, they estimate the beta and, afterwards, class stocks into beta deciles on the basis of their pre-ranking betas. Hence, 100 portfolios are formed and their equivalent equally weighted return is calculated. The regressions are made using the Fama and Macbeth procedure (1973).

The authors find that the static version of the CAPM does not hold at all. In fact, the results show a small evidence against the beta which appears to be too weak and not statistically different from zero. Moreover, the R-squared is only about 1.35%. Then, the estimation of the CAPM when taking into account the beta variation shows that the beta premium is significantly different from zero. Furthermore, the R-squared moves up to nearly 30%. Nevertheless, the estimation of the conditional CAPM with human capital indicates that this variable improves the regression. Indeed, the results show a positive and a statistically significant beta labor. In addition to that, the beta premium remains significant and the adjusted R-squared goes up to reach 60%. This is a great support for the conditional version of the CAPM.

Meanwhile, one of the most important theoretical results which contributes to the survival of the CAPM is that of Dybvig and Ross (1985) and Hansen and Richard (1987) who find that the conditional version of the CAPM is adequate even if the static one is blamed.

Pettengill et al. (1995) find that the conditional CAPM can explain the weak relationship between the expected return and the beta in the US stock market. They argue that the relationship between the beta and the return is conditional and must be positive during up markets and negative during down markets.

These findings are further supported by the study of Fletcher (2000) who investigates the conditional relationship between beta and return in international stock markets. He finds that the relationship between the two variables is significantly positive during up markets months and significantly negative during down markets months.

Another study performed by Hodoshima, Gomez and Kunimura (2000) and supports the conditional relationship between the beta and the return in the Nikkei stock market.

Jean-Jacques. L, Helene Rainelli. L M, and Yannick. G (XXXX) run the same study as Jagannathan and Wang (1996) but in an international context. They work on monthly data for the period which goes from January 1995 to December 2004 related to six countries which are; Germany, Italy, France, Great Britain, United States, and Japan. For each country the MSCI index is retained as a proxy for the market portfolio and the 3 month Treasury bond as a proxy for the risk free rate.

Following the same methodology as for Fama and French (1992), the authors form portfolios on the basis of the size, the book to market. Hence, for each country 12 portfolios are formed and regressed on the explanatory variables using the cross-sectional regression.

The results indicate that the market beta is statistically significant only in one country (the Great Britain). However, the beta premium is found to be significant in four countries, i.e. Italy, Japan, Great Britain and the US. As for the premium labor, the results exhibit significance only in three cases (Germany, Great Britain and France). Moreover, the results stem to a very high explanatory power for all countries which is on average beyond 40%.

Meanwhile, Durack et al. (2004) run the same study as Jagannathan & Wang (1996) in the Australian context. Their sample contains monthly data of all listed Australian stocks over the period of January 1980- December 2001.

In order to estimate for the premium labor model, they use the value weighted stock index as the market portfolio and extract the macroeconomic data from the bureau of statistics in order to measure the beta premium and the beta labor (two variables; the first one captures the premium resulting from the change in the market premium and the second one captures the premium of the human capital). They, afterwards, sort the stocks into seven size portfolios, then, into seven further beta portfolios. Finally, 49 portfolios are formed and used for the estimate of the conditional CAPM.

Using the OLS technique, the authors find that the conditional CAPM does a great job in the Australian stock market. In fact, in both cases, i.e. conditional CAPM with and without human capital, the model accounts for nearly 70% of the explanatory power. Nevertheless, the results report a little evidence towards the beta premium which is found to be, in all cases, positive but not statistically significant at any significance level.

Furthermore, the beta of the premium variation is negative and statistically significant. But, unlike Jagannathan and Wang (1996), the authors find that the human capital does not improve the beta estimate which remains insignificant. Finally, they find that the intercept is found to be significantly different from zero which is a violation to the CAMP's assumptions.

Campbell R. Harvey (1989), tests the CAPM while assuming that both expected return and covariances are time varying. The author uses monthly data of the New York Stock Exchange from September 1941 to December 1987. Ten portfolios are sorted by market value and rebalanced each year on the basis of this criterion. The risk free rate is the return on Treasury bill that is closed to 30 days at the end of the year t-1 and the conditional information includes the first lag on the equally weighted NYSE portfolio, the junk bond premium, a dividend yield measure, a term premium, a constant and a dummy variable for January are included in the model as well.

The results indicate that the conditional covariance change over time. Moreover, it is found that the higher the return is the more important the conditional covariance will be. Nevertheless, it is found that the model with a time-varying reward to risk appears to be worse than the model with a fixed parameter. In fact, the intercepts vary so high to be able to explain the variance of the beta. Consequently, this rejects the CAPM even in this general formula.

Jonathan Lewelen and Stefan Nagel (2003) study the validity of the conditional CAPM in the US context. The authors work on a sample including the period that spans from 1964 to 2001, and contains different data frequencies, i.e. daily, weekly and monthly. The data includes all NYSE, and AMEX stocks on GRSP Compustat sorted on three portfolios' type on the basis of the size, the book-to-market, and the momentum effects.

The regression test is run on the excess return of all portfolios on the one month Treasury bill rate. The results indicate that the beta varies over time but that this variation is not enough to explain the pricing errors. In fact, the results show that beta cannot covary with the risk premium sufficiently in a way that can explain the alphas of the portfolios. Indeed, the alphas are found to be are high and statistically significant which is a violation to the CAPM.

Treerapot Kongtoranin (2007) applies the conditional CAPM to the stock exchange of Thailand. He works on monthly data of 170 individual stocks in SET during 2000 to 2006. The author uses the SET index and the three-month Treasury bill to proxy, respectively for the market portfolio and the risk free rate.

Before testing the validity of the conditional CAPM, the author classify the stocks into 10 portfolios of 17 stocks each one on the basis of their average return. They afterwards estimate for the conditional CAPM using the cross sectional regression. The beta is calculated using the ratio of covariance between the individual portfolio and the market portfolio, and the variance of the market portfolio. The covariance is determined by using the ARMA model and the variance of the market portfolio is determined by the GARCH (1, 1) model.

The results indicate that the relationship between the beta and the return is negative and not statistically significant. Meanwhile, when a year period is considered, it is found that in 2000, 2004, and 2005 the beta premium is negative and statistically significant. Consequently, these results reject the CAPM which assume that the risk premium is positive.

Abd.Ghafar Ismaila and Mohd Saharudin Shakranib (2003) study the ability of the conditional CAPM to generate the returns of the Islamic unit trusts in the Malaysian stock exchange. They work on weekly price data of 12 Islamic unit roots and the Sharjah index for the period that spans from 1 May 1999 until 31 July 2001.

They, firstly, estimate for each unit trust the equivalent beta is estimated for the whole sample and for the two following sub-samples; 1 May 1999 - 23 June 2000, and 24 June 2000 - 31 July 2001. Then, they estimate the average beta using the conditional CAPM. The results indicate that the betas coefficients are significant and have a positive value in up markets and negative value in down m

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