### 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.

### Keywords:

CAPM, CAPM's variants, circumstances, literature survey.

### 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 markets which supports the conditional relationship for all the period studied. Moreover, it is shown that the conditional relationship is better in the down market than in the up market.

In order to look at the ability of the conditional CAPM, M. Lettau and S. Ludvigson (2001) study the consumption CAPM within a conditional framework. The study sample includes the returns of 25 portfolios formed according to Fama and French (1992, 1993). These portfolios are considered as the value weighted returns for the intersection of five size portfolios and five books to market portfolios on the NYSE, AMEX, and NASDAQ stocks in COMPUSTAT. The period of study goes from July 1963 to June 1998 and contains quarterly data of all portfolios.

For the estimate of the conditional CAPM, the GRSP value weighted return is considered as proxy for the market portfolio, and the cross-sectional regression methodology proposed in Fama and MacBeth (1973) is used. The results indicate that the static CAPM fails in explaining the cross-sectional of stocks return motivating by a small R-squared that is only of 1%. However, for the conditional CAPM, it was found that the scaled variable is positive and statistically significant. Moreover, the adjusted R-squared moves up to reach the 31%. Subsequently, this means that the conditional version of the CAPM performs well the static version.

Goezmann et.al (2007), expand the results of the Jagannathan and Wang's study (1996). In their study, the authors assume that the market beta and the premium beta as well as the premium its self can vary across time. Their idea is based on the assumption according to which the expected return is conditional on the economic states. Furthermore, these economic states are determined through the investor expectations about the future prospect of the economy. Hence, they suppose that the expected real Gross Domestic Product (GDP) growth rate is considered as a predictive instrument to know the state of the economy.

To reach their objective, the authors form 25 portfolios from the intersection of 5 five book-to-market portfolios and five size portfolios. They, afterwards, measure the beta instability risk directly through the bad and good states of the GDP. Using a two beta model and the general method of moments, the authors find that the risk premium is increasing with the book-to-marker portfolios ranges.

In fact, the results indicate that the conditional market risk premium is not priced at all. Besides, when considering the conditional beta (on bad and good states), it is found that the good time sensitivity are positively associated to book to market portfolios. In addition to that, it's found that stocks whose returns are positively related to the GDP earn negative premium. Finally, the results point to a positive beta premium for value stocks which turns into negative for growth stocks.

In order to test for the risk-return relationship in the Karachi stock exchange Javid A and Ahmad E (2008) study the conditional CAPM. Their sample study includes daily and monthly return of 49 companies and the KSE 100 index during the period of July 1993 to December 2004 which is divided into five overlapping intervals (1993-1997, 1994-1998, 1995-1999, 1996-2000, and 1997-2001).

Using the time series regression (whole period) together with the cross section regression (sub-periods), the results point to weak evidence towards the unconditional version of the CAPM. In fact, it was found that the positive relationship between the expected return and the risk premium doesn't hold in any sub-period. Moreover, the intercepts are found to be statistically indifferent from zero in almost all cases. This weakness is further enhanced by the fact that the residues play a significant role in explaining the return. They, hence, assert that the return distribution must vary over time.

Having this in mind, the authors allow for the risk premium to vary along with macroeconomic variables which are supposed to contain the business cycle information. These conditioning variables are the following; market return, call money rate, term structure, inflation rate, foreign exchange rate, growth in industrial production, growth in real consumption, and growth in oil prices.

Applying the conditional version of the CAPM, the authors find that the risk premium is positive for roughly all sub-periods (1993-1995, 1993-1998, 1999-2004, 2002-2004), and for the overall sample period 1993-2004. In addition to that, beta coefficient is statistically significant which indicate that investors get a compensation for bearing risk. Nevertheless, the results show that for all sub-periods and for the whole period the intercepts are statistically different from zero.

Within an international context, Fujimoto A, and Watanabe M (2005) study the time variation in the risk premium of the value stocks and the growth ones as well. The authors assume that the CAPM holds each period allowing for the beta and for the market premium to vary through time.

Their sample includes data set of 13 countries which represent, roughly, 85% of the world's market capitalization. These countries are the following; Australia, Belgium, Canada, France, Germany, Italy, Japan, Netherlands, Singapore, Sweden, Switzerland, UK and the US. For each country, monthly stock returns, market capitalizations, market-to-book values, as well as value-weighted market returns for the non-US are used for the period that ends in 2004 and begin in 1963 for the US, 1965 for the UK, 1982 for Sweden, and in 1973 for all the other countries.

They, firstly, form portfolios as the intersection of the book-to-market portfolios and those of size. Then, the authors suppose that the market premium and the betas of the portfolios are presumed to be a function of the dividend yield, the short rate, the term spread, and the default spread. The cross section regression indicates, firstly, that the beta sensitivity is positive and statistically significant in 9 countries for value stocks.

Furthermore, it is found also that growth stocks exhibit negative beta premium sensitivity. Moreover, for the long-short portfolio and the size/book-to market portfolios the beta sensitivity is for almost all countries positive and statistically significant. Finally, it is found that the beta premium sensitivity cannot explain the whole variation in value premium in international markets.

From their part, Michael R. Gibbons and Wayne.Ferson (1985), also relax the assumption related to the stagnation of the risk premium. Hence, the expected return is conditional on a set of information variables. They use daily return of common stocks composing the Dow Jones 30 for the period that goes from1962 to 1980.

So the daily stocks returns are regressed against the lagged stock index, the Monday dummy, and an intercept. The results show that the Monday dummy and the lagged GRSP value-weighted index, are highly significant. Nevertheless, the coefficient of determination is beyond 5%. Consequently, they conclude that their study is robust to missing information.

Ferson and Harvey (1999), try to look into the conditioning variables and their impact on the cross section of stock returns. The sample of the study covers the period of 1963-1994 and contains monthly data of the US common stock portfolios. The lagged instrumental variables used in here are respectively; the difference between the one-month lagged returns of a three-month and a one-month treasury bills, the dividend yield of the standard and Poors 500 index, the spread between Moody's Baa and Aaa corporate bond yields, the spread between ten-year and one-year treasury bond yield.

The regression produces significant values for roughly all variables. The conditional version implies, also, that the intercepts are time varying which means that they are not zero. They conclude, hence, that the conditional version is not valid.

Wayne E. Ferson and Andrew F. Siegel (2007) in their trial to investigate the portfolio efficiency with conditioning information, test the conditional version of the CAPM. For this objective, they use a standard set of lagged variables to model the conditioning information and a sample that ranges from 1963 to 1994.

The instrumental variables used are the following; the lagged value of a one-month Treasury bill yield, the dividend yield of the market index, the spread between Moody's Baa and Aaa corporate bond yields, the spread between ten-year and one-year constant maturity Treasury bond yields, and the difference between the one-month lagged returns of a three-month and a one-month Treasury bill.

They, after that, group stocks into two classes. The former is a sorting according to the Twenty five value-weighted industry portfolios. The latter, is a classification with reference to their prior equity market capitalization, and separately into five groups on the basis of their ratios of book value to market value. The results indicate that the conditioning variables do not improve very well the estimation. Nevertheless, during all periods these variables exhibit high and statistically significant coefficients.

5.1.2. The Conditional CAPM: data-driven filters

Unlike the first approach which is based on pre-specified conditioning information, the data-driven filters approach is based on purely empirical bases. In fact, the data used is the source of factors and it is the only responsible of the beta variation. This means that one do not require a well defined variables, rather one allows for the data to define these variables. Some papers interested in this approach are discussed next.

Always in the same field of research, Nagel S and Singelton K (2009) try to investigate the conditional version of assets pricing models. Their methodology is somehow different from the others as for the moment when they use the Stochastic Discount Factor (SDF) as a conditionally affine function of a set of priced risk factors.

Their sample study includes quarterly data on three instruments variables over the period of 1952-2006. These variables are; the consumption-wealth ratio of Lettau and Ludvigson (2001a), the corporate bond spread as in Jagannathan and Wang (1996) or the labor income-consumption ratio of Santos and Veronesi (2006). They, next, form portfolios sorted by size and book-to-market ratio as well.

Applying the time varying SDF, the authors find that when the two conditioning information, i.e. the consumption-wealth ratio, and the corporate bond spread, is incorporated in the estimation; the model fails in explaining the cross-sectional of stocks returns. They conclude, hence, that the conditional asset pricing models do not play a good role in improving the pricing accuracy.

In the US context, Huang P and Hueng J (XXXX) investigate the risk-return relationship in a time-varying beta model according to the view of Pettengill, Sundaram, and Mathur (1995). For this objective, they use a sample which includes daily returns of all stocks listed in the S&P 500 index over the period of November 1987-December 2003. For the estimation of the time-varying beta model, they make use of the Adaptive Least Squares with Kalman Foundations (ALSKF) proposed by McCulloch (2006).

The results support the Pettengil et al. model (1995). In fact, it is found that the beta premium is positive and statistically significant in up markets, whereas the risk-return relationship is found to be negative and statistically significant in down markets. Moreover, the results exhibit that none of the intercept is statistically different from zero. Finally, while using the ALSKF, the authors find that the estimation is more precise than that obtained via the OLS regression.

The study of Demos A and Pariss S (1998) aims at investigating the validity of the validity of the conditional CAPM within the Athens Stock Exchange. For this end, the authors work on a sample covering fortnightly returns of the nine Athens Stock Exchange (ASE) sectorial indices and the Value weighted index from January 1985 to June 1997.

Then, to model the idiosyncratic conditional variances, the ARCH type process is used. The results from the OLS regression indicate that the static version of the CAPM is doing a good job. In fact, the beta coefficient is in all cases positive and statistically significant. This result does not depend on whether an intercept is included in the regression. This is because the intercepts only in three out of nine cases are found to be statistically different from zero.

They after that, consider that Value Weighted Index follows a GQARCH(1,1)-M process, the authors find similar results to the static CAPM. In fact, all the estimated betas are found to be positive and statistically different from zero. Nevertheless, the authors find that within the CAPM in both static and dynamic versions, the idiosyncratic risk is priced. Moreover, it's found that the intercepts are jointly statistically different from zero. Basing on the pre-mentioned result the authors claim to the invalidity of the model. They consider that the potential cause of failure is the use of the value weighted index rather than the equally weighted one.

Consequently, they repeat the same procedure while using the equally weighted index. Firstly, with respect to the static version, the results indicate that all intercepts are statistically indifferent from zero except for the bank sector and are jointly different from zero. Secondly, as for the dynamic version of the CAPM, the results point to a very supportive result. Indeed, the betas coefficients not only have the right sign but also are highly significant. Finally, the most supportive result is that which shows that the idiosyncratic risk is not priced.

In their study, Basu D and Stremme A (2007), study the conditional CAPM while assuming time variation in risk premium. This time variation is captured by a non linear function of asset of variables related to the business cycle.

In order to model the time-varying factor risk premium, the authors use the approach of Hansen and Richard (1987). This approach permits to construct a candidate stochastic discount factor given as an affine transformation of the market risk factor. Then, the coefficients from this transformation are non-linear of the conditioning variables sensed to capture the time-variation of the risk premium.

They work on monthly data of different kinds of portfolios, i.e. the portfolios sorted on CAPM beta (stocks relative to the S&P 500) over the period of 1980-2004, the momentum portfolios over the period of 1961-1999, the book-to-market portfolios over the 1961-1999 period, and finally the 25 and 100 portfolios sorted by size and book-to-market over the 1963-2004 and the 1963-1990 periods. As for the conditioning variables, the authors make use of the following instruments; the one-month Treasury Bill rate, the term spread (the difference in yield between the 10-year and the one-year Treasury bond), the credit spread (the difference in 10-year yield between the AAA-rated corporate and the corresponding government bond), and the convexity the of the yield curve (the 5-year yield minus the sum of the 10-year and 1-year yields).

The results point to a weak evidence towards the static CAPM. In fact, it is found that this latter account only for 1.6% of the returns cross sectional variation. Moreover, the expected return for all studied portfolios has the U-shape which, subsequently, contradicts the CAPM's assumptions.

However, when the conditional version is set into play, the results are somehow surprising. Definitely, the scaled version of the CAPM captures 60% of the cross-sectional variation. Furthermore, this model predicts relatively better the expected return, since it accounts for lower errors for the extreme as well as middle portfolios. Finally, the scaled version explains better the risk premium of all portfolios with comparison to the static version.

Tobias A and Franzoni F (2008) introduce the unobservable long-run changes to the conditional CAPM as a risk factor. They, hence, propose to model the conditional betas using the Kalman filter since investors are supposed to learn the long level of factor loading through the observation of the realized return. For this reason, they suppose that the betas change overtime following a mean-reverting process.

For this aim, they work on quarterly data of all stocks listed in the NYSE, AMEX, and NASDAQ for the period that spans from 1963 to 2004. They, afterwards, group stocks into 25portfolios basing on the size and the book-to-market criterion. As for the conditioning variables, they make use of value-weighted market portfolio, the term spread, the value spread, and the CAY variable documented Lettau and Ludvigson (2001). Then, in order to extract for the filtered betas, they apply the Kalman Filter. The times series test on size and book-to-market portfolios indicates that the introduction of the learning process into the conditional CAPM contributes to the decrease of the pricing errors. Moreover, when the learning is not included in the conditional CAPM, the results point to a rejection.

In the interim, Jon A. Christopherson, Wayne E. Ferson and Andrew L. Turner (1999), try to find out the effect of the conditional alphas and betas on the performance evaluation of portfolios. They assert that the betas and alphas move together with a set of conditioning information variables. They find that the excess returns are partially predictable through the information variables. The results point, also, to statistically insignificant alphas which is consistent with the CAPM's predictions.

In the same path of research, Wayne E. Ferson, Shmuel Kandel, and Robert F. Stambaugh (1987), apply the conditional version of the CAPM. They assume that the market beta and the risk premium vary overtime. They work on weekly data from the NYSE over the period of 1963-1982 which includes the return of ten common portfolios sorted on equity capitalization. The results suggest that the single factor model is not rejected when the risk premium is allowed to vary overtime and when the risk related to that risk premium is not constrained to be equal to market betas.

In the same way, Paskalis Glabadanidis (2008), studies the dynamic asset pricing models while assuming that both the factor loading and the idiosyncratic risk are time varying. In order to model the time variation in the idiosyncratic risk of the one factor model, he uses the multivariate GARCH model.

Their main objective relies on the fact that the risk return relationship should contain the proper adjustment which may account for serial autocorrelations in volatility and time variation in the return distribution. His study is run on monthly return of 25 size and book-to-market portfolios and 30 industry ones as well for the period of 1963-1993. The results indicate that the dynamic CAPM could reduce the pricing errors. However, the results indicate that the null intercept hypothesis cannot be rejected at any significance level.

### 5.2. The Downside Approach

“A man who seeks advice about his actions will not be grateful for the suggestion that he maximize his expected utility.”

Roy (1952)

### 5.2.1. From the Mean-Variance to the Downside Approach

The mean-variance approach lies on the fact that the variance is an appropriate measure of risk. This latter assumption is founded upon at least one of the following conditions. Either, the investor's utility function is quadratic, or the portfolios' returns are jointly normally distributed. Subsequently, the optimal portfolio chosen based on the criterion of the mean variance would be the same as that which maximizes the investor's utility function.

Nevertheless, the adequacy of the quadratic utility function is tackled as for the moment when the investor's risk aversion would be an increasing function of his wealth, whereas the opposite is completely possible. Furthermore, the normal distribution of the return is criticized since the data may exhibit high frequency such as skewness (Leland, 1999; Harvey and Siddique, 2000 and Chen, Hong, and Stein, 2001), or kurtosis (see for example Bekaert, Erb, Harvey, and Viskanta, 1998 and Estrada, 2001c).

Levy and Markowitz (1979) find that the mean-variance behavior is a good approximation to the expected utility. In fact, they show that the integration of the skewness or the kurtosis or even the both worsens the approximation to the expected utility.

The credibility of the variance as a measure of risk is valid only in the case of symmetric distribution of the return. Then, it is reliable only in the case of normal distribution. Moreover, the beta which is the measure of risk according to the mean-variance approach suffers from diverse critics (discussed in the first and the second section of the chapter).

Meanwhile, Brunel (2004), states that the mean-variance criterion is not able to generate a successful allocation of wealth given that investors, in this case, do not consider the higher statistical moment issues. For that reason, choices have to be made on other parameters of the return distribution such as skewness or kurtosis. Skewness preference indicates that investors allocate more importance to downside risk than to upside risk.

From these critics, one may think that the failure of the traditional CAPM comes from its ignorance of the extra reward prime required by investors in the bear markets. In fact, by intuition investors would require higher return for holding assets positively correlated with the market in distress periods and a lower return for holding assets negatively correlated with the market in bear periods. Consequently, upside and downside periods are not treated symmetrically, from where the birth of the semi-deviation or also the semi-variance approach.

The concept of the semi-variance was firstly introduced by Markowitz (1959) and was later refined by Hogan and Warren (1974) and Bawa and Lindenberg (1977). This approach preserves the same characteristics as the regular CAPM with the only difference in the risk measures. In fact, while the former uses the semi-variance and the downside beta, the latter uses the variance and the regular beta. In the particular case where the returns are symmetrically distributed, the downside beta is equal to the regular beta.

However, for asymmetrical distribution the two models diverge largely. The standard deviation identifies the risk related to the volatility of the return, but it does not make a distinction between upside changes and downside changes. In practice, the separation between these two aspects is though important. In fact, if the investor is risk averse, then he will be averse to downside volatility and accept gladly the upside volatility. So the risk occurs when the wrong scenario is put into play.

The semi-variance is more plausible than the variance as a measure of risk. Indeed, the semi-deviation accounts for the downside risk that investors want to prevent contrary to the upside risk which has the welcome. In a nutshell, the semi-variance is an adequate measure of risk for a risk averse investor.

### 5.2.2. A Brief History on the Downside Risk Measures

In order to understand the contribution and the concept of the downside risk, it is important to study the history of its development. The purpose of this section is to review the measures of the downside risk and to clarify its major innovation

Along with the academic literature, there have been two major measures that have been commonly used; it is about the semi-variance and the lower partial moment. All of these measures were tools to develop the portfolio theory and to determine the efficient choice among portfolios for a risk averse investor.

The first paper appeared in the field of finance and which was interested in the downside risk theory was that of Markowitz (1952). This author develops a theory that uses the mean return and the variance and covariance to construct the efficient frontier in where one may find all portfolios that maximizes the return for a given risk level or that minimizes the risk for a given return level. Hence, the investor should make a risk-return tradeoff according to his utility function. Nevertheless, due to the human being nature which is not obvious, it's difficult to determine a common utility function.

The second paper published in this field of research was that of Roy (1952).this latter states that the creation of mathematical utility function for an investor is very difficult. Consequently, an investor would not be satisfied for simply maximizing his expected utility. He suggests, for that reason another measure of risk which he calls the safety first technique. This measure suggests that an investor would prefer the safety of his wealth and chooses some minimum acceptable return that conserves the principal.

The minimum acceptable return is called, according to Roy (1952), the disaster level or also the target return beyond which we would not accept the risk. That is why; the optimal choice of an investor will be that which has the smallest probability of going under the disaster level. He develops, thus, the reward to variability ratio which, for a given investor, minimizes the probability of the portfolio to go under the target return level.

After that, Markowitz (1959) has admitted the Roy's approach and the importance of the downside risk. He states that the downside risk is crucial for portfolio choice for at least two reasons; first because the return distribution is far form being normal. Second, because only the downside risk is pertinent for the investor. He proposes, therefore, two measures of the downside risk; the below-mean semi variance and the below-target semi variance. Both of them use only the return below the mean or the target return.

Since that, many researchers have explored the downside measures in their study and have demonstrated the superiority of these risk measures over the variance (see for example; Quirk and Saposnik, 1962; Mao, 1970; Balzer, 1994 and Sortino and Price, 1994 among others). Meanwhile, Klemkosky (1973) and Ang and Chua (1979) have demonstrated the plausibility of the below-target semi variance approach as a tool in evaluating the mutual fund performance. Hogan and Warren (1974) have developed a below- target capital asset pricing model which is useful in the case where the return distribution is non normal and asymmetric.

The development of the downside risk is pulled along with the emergence of the low partial moment measure introduced by Bawa (1975) and Fishburn (1977).Then, Nantell and Price (1979), and Harlow and Rao (1989) have suggested another version of the downside CAPM known since as the lower partial moment CAPM. The lower partial moment as a risk measure developed for the first time by Bawa (1975) and Fishburn (1977) which moves the constraint from having only one utility function and provide a whole rainbow of utility functions. Moreover, it describes the risk in terms of risk tolerance.

Another support for the downside risk provided by Roy (1952), Markowitz (1959), Swalm (1966), and Mao (1970) who demonstrated that the investors are not concerned with the above-target returns. From this, the semi variance is more practical in evaluating risk in investment and financial strategies.

### 5.2.3. Background on the Downside CAPM

Over the last decade, extensive empirical literature had been carried out to investigate the downside approach as a risk measure. Indeed, taking as a starting point the failure of the CAPM's beta in representing risk, several researchers have tried to improve the relationship risk return and to fill the gaps of its limitations with reference to the market model. In order to obviate these limitations, Hogan and Warren (1974) and Bawa and Linderberg (1977) put forward the use of the downside risk rather than the variance as a risk measure and developed a MLPM-CAPM, which is a model that does not rely on the CAPM's assumptions.

Both studies sustained that the MLPM-CAPM model outperforms the CAPM at least on theoretical grounds. Harlow and Rao (1989) improved the MLPM-CAPM model and introduced a more general model, which is known as the Generalised Mean-Lower Partial Moment CAPM. This is a MLPM-CAPM model for any arbitrary benchmark return. Particularly, their empirical results suggest the use of the generalized MLPM-CAPM model, since no evidence goes in support for traditional CAPM. The other caveat from the latter study is that target return should equal to the mean of the assets' returns rather than the risk-free rate.

In this road of research, we note, particularly, Leland (1999) who investigates the risk and the performance measures for portfolios with asymmetrical return distribution. This author criticizes the plausibility of ‘'alpha'' and the ‘'Sharpe ratio'' to evaluate portfolios' performance, and suggest the use of the downside risk approach. He proposes, hence, another risk measure which differs from the CAPM beta, particularly, when the assets' return or that of the portfolio is assumed to be non linear in the market return.

Estrada (2002) in her seminal paper evaluates the mean-semi variance behavior in the sense that it yields a utility level similar to the investor's expected utility. She divided the whole world on three markets; all markets, developed markets and the emerging markets. She finds that the standard deviation is an implausible risk measure and suggests in turn the semi-deviation as a better alternative. The results indicate, also, that the mean-semi variance behavior outperforms the mean-variance behavior in emerging markets and in the whole sample of all markets.

Then, using the J-test of Davidson and MacKinnon (1981), the author finds that none of the two approximations does better than the other in explaining the variability of the expected utility. However, it is shown that the mean-semivariance approach outperforms the other in the case of the negative exponential utility function. The author reports, also, that MSB is not only consistent with the maximization of expected utility but also with the maximization of the utility of expected compound return.

She ended, finally, her article with an inquiry of whether the downside beta can be used in a one factor model as an alternative to the traditional beta. She prepares, hence, the area to the next paper treating the CAPM within a downside framework.

In the same year, Estrada investigates the downside CAPM within the emerging markets context. The downside CAPM replaces the original beta by the downside beta. This beta is defined as the ratio of the cosemivariance to the market's semivariance.

She works on a sample that covers the entire Morgan Stanley Capital Indices database of emerging markets. The data contains monthly returns on 27 emerging markets for various sample periods, some of the data begins at January 1988 and some others start later. The author demonstrates, while making use of the average monthly return for the whole sample, that both the beta and the downside beta are significant in generating returns), and that the latter explains better the average return witnessed by a high explanatory power. However, when considering the two risk measures into one model, only the downside beta is found to be significant (cross section regression). The results indicate, in addition to that, that the returns are more sensitive to the changes in the downside risk.

Then, the author compares the performance of the CAPM and the downside CAPM. The results support the downside CAPM since the downside beta explains roughly 55% of the returns variability in emerging markets. As a conclusion, the author mentions the plausibility of the mean-semivariance behavior in explaining the return on the sample of markets studied.

For the meantime, Thierry Post and Pim Vliet (2004) investigate the downside risk and the CAPM. In order to assign weights to the market's portfolios return, the authors use the pricing Kernel. Their samples includes the ordinary common US stocks listed on the New York Stock Exchange (NYSE), American Stock Exchange (AMEX), and NASDAQ markets and uses a monthly frequency for the period of 1926-2002. Portfolios are sorted according to their betas and downside betas for the previous 60 months and the average return is then computed for the following next 12 months.

For further discussion, the authors control also for the size and the momentum (portfolios formed on the basis of the size as in Fama and French (1992) and the momentum). The results indicate that the downside betas are higher than the regular betas. Furthermore, it is found that the downside betas decrease the pricing errors.

They, afterwards, apply a double sorting routine, in order to distinguish the effect of the two betas. Hence, they first sort stocks on quintile portfolios based on regular beta and then divide each of those portfolios into five portfolios based on downside beta and the opposite procedure is done also. Consequently, 25 portfolios is constructed and then regressed separately against the regular beta and the downside beta.

The results from the regression indicate that the average return is positively associated to the downside beta within each regular beta quintile. Nevertheless, the relation between the average return and the regular beta tends to fade up. Finally, Thierry Post and Pim Vliet (2004) assert that the mean-semivariance CAPM strongly outstrips the mean-variance CAPM and that the downside risk is a better risk measure both theoretically and empirically.

In the same way, Thierry Post, Pim Vliet, and Simon Lansdorp (2009) look into the downside beta and its ability to derive return. They use monthly stock returns from the GRSP at the University of Chicago and select the ordinary common US stocks listed on the NYSE, AMEX, and NASDAQ for the period that spans from January 1926 to December 2007. They, moreover, investigate the role of the downside beta for various sub-samples which are; 1931-1949, 1950-1969, 1970-1988, and 1989-2007 and use the regular beta portfolio and the downside beta portfolio as a benchmark.

After that, the authors carry out several portfolios' classification, first on the basis of the regular beta, then on downside beta, on regular beta first and then on three definitions of downside betas (semivariance beta, ARM beta, and downside covariance beta.), and even on one downside beta then on the other definitions of downside betas. They obtain in sum 12 sets of double sorted beta portfolios. They, also, control for the size, the book-to-market, the momentum, the co-skewness and the total volatility.

Thierry P et al. (2009) find that the downside beta is more pertinent to investors than regular beta and that the downside beta measured by the semivariance is more plausible than that determined through the other definitions. Furthermore, it is shown that the downside covariance beta does not yield a better performance than the regular beta (mean spread of only 5 or 6 basis points).

As for the other characteristics per se the size, the value and the momentum, the authors make use of only the semivariance measure since it has shown a great dominance over the other measures. In here, the results indicate that the difference between the regular beta and downside beta cannot account for the omitted stock characteristics. In a nutshell, the authors sustain, while including the other characteristics into the regression, that the significance of the downside beta remains higher and outperforms all the other betas in each of the sub-sample and notably in most recently years. This latter result corroborates the importance of the semi-variance as a risk measure and the superiority of the downside CAPM over the traditional one.

Ang, Chen and Xing (2005) in their attempt to investigate the downside risk, use data from the Center for Research in Security Prices (CRSP) to construct portfolios of stocks sorted by various characteristics of returns. It is about, particularly, ordinary common stocks listed on NYSE, AMEX and NASDAQ during the period that spans from July 3rd, 1962 to December 31st, 2001.

Using the Fama and Macbeth (1973) regression, the authors examine separately the downside and the upside component of beta and find that the downside risk is priced. In fact, the downside risk coefficient is found to be positive and highly significant. Nevertheless, the upside beta coefficient is negative which goes in the same current as the previous literature investigating the availability of the CAPM's beta. They show that the downside risk premium is always positive roughly about 6% per annum and statistically significant.

They find also that this positive significant premium remains even when controlling for other firm characteristics and risk characteristics. In opposite, the upside premium changes its sign to turn into negative when considering the other characteristics.

Likewise, Olmo (2007) finds, while performing his study on a number of UK sectoral indices, that both the CAPM beta and the downside beta are pricing factors for risky assets. He finds, also, that stocks which co-vary with the market in downturn periods generate higher return than that predicted by the CAPM. Contrary, stocks that are negatively correlated with the market in bad times are found to have lower return. The major result of this study is that the sectors which are indifferent to bad states changes and belong generally to the safe sectors seem to be not priced by downside beta.

Similarly, Diana Abu-Ghunmi (2008) explores the downside risk in a conditional framework within the U.K context. Her sample includes all common stocks traded on the London Stock Exchange and the FTSE index from July 1981 to December 2005. They also use the coincident index to split their sample into expansion and recession periods. The portfolio's formation is done upon monthly return and based on three main risk measures per se; the beta, the upside beta, and the downside beta. They, afterwards, run the Fama-MacBeth (1973) cross sectional regression of excess returns on realized betas to examine the downside risk- return relation using individual stocks.

The results point to a positive and a significant premium between the expected return and the unconditional downside risk. They note, as well, that conditioning the risk return relationship on the state of the world contributes to increase monotonically the relationship between the expected return and the downside beta during expansion periods.

However, in recession periods they find no evidence between the return and the downside beta or the CAPM's beta. They conclude that downside beta plays a major role in pricing in pricing small and value stocks but not large and growth stocks. But, although the downside approach has been basis for many academic papers and has had significant impact on academic and non academic financial community, it is still subject to severe critics.

### 5.2.4. The High Order Moment CAPM

### 5.2.4.1. Evidence from the existence of the skewness and the kurtosis in the returns' distribution

Literature on the CAPM has shown several evidences on favor of non normality and asymmetrical returns distribution. The attractive attribute of the CAPM is its pleasing and powerful explanation with a well built theoretical background about the risk-return relationship. This model is built on the basis of some assumptions but a critical one which imposes normality on the return distribution, so that the first two moments (mean and variance) are largely sufficient to describe the distribution.

Nevertheless, this latter assumption is far from being satisfied as demonstrated by Fama (1965), Arditti (1971), Singleton and Wingender (1986), and more recently, Chung, Johnson, and Schill (2006). These studies point that the higher moments of return distribution are crucial for the investors and, from that, must not be neglected. They suggest, hence, that not only the mean and the variance but also higher moments such as skewness and kurtosis should be included in the pricing function.

Consequently these attacks have led to the rejection of the CAPM within the Sharpe and Linter version and lead the way to the development of asset pricing models with higher moment than the variance like for instance; Fang and Lai (1997), Hwang and Satchell (1998) and Adcock and Shutes (1999) who introduced the kurtosis coefficient in the pricing function or also Kraus and Lizenberg (1974) who introduce the Skewness coefficient.

The skewness coefficient is a measure of the asymmetry in the distribution. Particularly it is a tool to check that the distribution does not look to be the same to the left and the right with reference to a center point. The negative value of the skewness indicates that the distribution is concentrated on the right or skewed left. However, the positive value of the skewness indicates analogically that the data are skewed right.

For more precision, we mean by skewed left that the left tail is longer than the right one and vice versa. Subsequently, for a normal distribution the skewness must be near to zero. The formula of the skewness is given by the ratio of the third moment around the mean divided by the third power of the standard deviation.

Similarly the kurtosis coefficient is a measure to check whether the data are peaked or flat with reference to a normal distribution. This means, subsequently, that data with high kurtosis tend to have a different peak around the mean, decreases rather speedily and have heavy tails. However, data with low kurtosis tend to have rather a flat top near the mean. From this, the negative kurtosis indicates that the distribution is flat contrary the positive kurtosis indicates peaked distribution. For a normal distribution the kurtosis must be equal to zero. The formula of the kurtosis is given by the fourth moment around the mean divided by the square of the variance minus three which one calls also the excess kurtosis with reference to the normal distribution.

The distribution that has zero excess Kurtosis is called “mesokurtotic” which is the case of all the normal distribution family. However, the distribution with positive excess Kurtosis is called “leptokurtotic” and means that this distribution has more than normal of values near the mean and a higher probability than normal of values in extreme (fatter tails). Finally, the distribution with negative excess Kurtosis is called “platykurtotic” and indicates that there is a lower probability than the normal to find values near to the mean and a lower probability than the normal to find extreme values (thinner tails).

The French mathematician Benoit Mandelbrot (2004) in his book entitled ‘'The Misbehavior of Markets: A Fractal view of risk, ruin and reward'' concluded that the failure of any model (for example the option model of Black and Sholes or also the CAPM of Sharpe and Linter) or any investment theory in the modern finance can be due to the wide reliance on the normal distribution assumption.

Generally speaking, investors who maximize their utility function have preferences which cannot be explained only as a straightforward comparison between the first two moments of the returns' distribution. In fact, the expected utility function of a given investor uses all the available information relating to the assets' returns and can be somehow linked to the other moments.

It is not strange then to see authors like Arditti (1967), Levy (1969), Arditti and Levy (1975) and Kraus and Litzenberger (1976) extending the standard version of the CAPM to incorporate the skewness in the pricing function. Or even to see others incorporating the kurtosis coefficient, we name among others Dittmar (2002) who extends the three moments CAPM and examines the co-kurtosis coefficient. All these works stem to the necessity of introducing the high moment to the distribution in order to ameliorate the assets pricing if the restriction of normality is moved away.

For example Dittmar(2002) finds that investors dislike co-kurtosis and prefer stocks with lower probability mass in the tails of the distribution rather than stocks with higher probability mass in tails of the distribution. He concludes, hence, that assets that increase the portfolios' kurtosis must earn higher return. Likewise, assets that decrease the portfolios' kurtosis should have lower expected return.

As for Arditti (1967), Levy (1969), Arditti and Levy (1975) and Kraus and Litzenberger (1976), their results imply a preference for a positive skewness. They find that investors prefer stocks that are right skewed to those which are left skewed. Hence, assets that decrease the portfolios' skewness are more risky and must earn higher return comparing to those which increase the portfolios' skewness. These findings are further supported by studies like that of Fisher and Lorie (1970) or also that of Ibbotson and Sinquefield (1976) who find that the return distribution is skewed to the right.

This has led Sears and Wei (1988) to derive the elasticity of substitution between the systematic risk and the systematic skewness. More recently, Harvey and Siddique (2000) show that this systematic skewness is highly significant with a positive premium of about 3.60 percent per year and therefore must be well admitted in pricing assets.

The empirical evidence on the high moment CAPM is very mixed and very rich not only by its contribution but also by the methodologies used and the moments introduced. That's why the following section is devoted to further understand these approaches and to summarize the most important papers that explore this model in their studies in a purely narrative review.

### 5.2.4.2. Literature review on the high order moments CAPM

Kraus & Litzenberger (1976) were the first to suggest that the higher co-moments should be priced. They claim that in the case where the return's distribution is not normal, investors are concerned about the skewness or the kurtosis.

Just like Kraus and Litzenberger (1976), Harvey and Siddique(2000), have studied non-normal asset pricing models related to co-skewness. The study sample covers the period of 1963-1993 and contains the NYSE, AMEX, and NASDAQ equity data. They define the expected return as a function of covariance and co-skewness with the market portfolio in a three-moment CAPM and find that this model is better in explaining return and report that coskewness is significant and commands on average a risk premium of 3.6 percent per annum.

From his part Dittmar (2002) uses the cubic function as a discount factor in a Stochastic Discount Factor framework. The model is a cubic function of return on the NYSE value-weighted stock index, and labor growth following Jagannathan and Wang (1996). He finds that the co-kurtosis must be included with labor growth so as to arrive to an admissible pricing kernel.

Within the American context, Giovanni Barone Adesi, Patrick Gagliardini, and Giovanni Urga (2002) investigate the co-skewness in a quadratic market model. The study sample includes monthly return of 10 stock portfolios of the NYSE, AMEX, and NASDAQ formed by size from July 1963 to December 2000. The results from the OLS regression indicate that the extension of the return generating process to the skewness is praiseworthy. In fact, it is found that portfolios of small firms have negative co-skewness with the market. The results show also that there's an additional component in portfolios' return which not explained neither by the covariance nor by the co-skewness.

Daniel Chi-Hsiou Hung, Mark Shackleton and Xinzhong Xu (2003) investigate the plausibility of the high co-moments CAPM (co-skewness and co-kurtosis) in explaining the cross section of stock returns in the UK context. They work on a 26 year period from January 1975 to December 2000 and incorporate monthly data of all listed stocks. For the portfolios formed on the basis of the beta, the authors find that the higher beta portfolio has the highest total skewness and kurtosis. Moreover, the higher co-moments show little significance in explaining cross section returns and do not increase the explanatory power of the model. However, the intercepts are found to be all insignificant.

For the size sorted portfolios, the higher co-moments seem to be somehow significant and increase the explanatory power of the model. Moreover, for these portfolios the betas are all insignificant. Overall, Hung et al. (2003) find that the beta is very significant in every model and the addition of the higher order co-moment terms do not improve the explanatory power of the model which remains roughly unchanged. Furthermore, in all these models, the intercepts are insignificant.

They conclude, finally, that in the UK stock exchange there exists a little evidence on favor of non-linear market models, since the higher order moments are found to be too weak.

From their part, Rocky Roland and George Xiang (2004) suggest an asset pricing model with higher moments than the variance and extend the traditional version to a three-moment CAPM and a four-moment CAPM. They conclude, through their theoretical study, that further tests must be conducted to check the accuracy of the model even if some of research is already done.

Angelo Ranaldo, and Laurent Favre (2005) put forward the extension of the two moments CAPM to a four moments one including the co-skewness and the co-kurtosis in pricing the hedge funds' returns. Their study is run on monthly returns 0f 60 hedge fund indices which are equally weighted. The market portfolio is constructed upon 70% of the Russell 3000 index and 30% of the Lehman US aggregate bond index and the risk free rate is the US 1 month Certificate of Deposit.

Including the skewness and considering the quadratic model (time series regression), the authors find, on the one hand, that the adjusted R squared increases with more than half comparing to that of the two-moment CAPM. On the other hand, it is found that the coefficient of the co-skewness is positive and statistically significant which supports the existence of the co-skewness. However, the results indicate that the betas from the quadratic model are smaller than those implied by the two-moment CAPM. This latter result may be explained by the fact that some of the explanatory power of the beta is taken away by the co-skewness.

Then, taking into account the co-kurtosis in a cubic model, Angelo Ranaldo, and Laurent Favre (2005) find that the additional coefficient has no major function in explaining the hedge fund return since it is only significant in four strategies. They point, hence, on the basis of their findings that the higher moments are more suited to represent the hedge fund industry return.

In 2007, Chi-Hsiou Hung investigates the ability of the higher co-moments in predicting returns of portfolios formed from combining stocks in international markets and portfolios invested locally in the UK and the US. His sample includes monthly US dollar denominated returns, market value of common shares and interest rates for the period that spans from January 1975 to December 2004 and contains 18 countries.

For this end, the author constructed portfolios on the basis of the momentum criterion. Hence, 10 equally weighted momentum sorted portfolios are obtained every six months based on past six months compounded returns. Then, portfolios are sorted on size every 12 months by ranking stocks on the basis of the market value at the time of the ranking. Finally, to dismantle the momentum effect and that of the size, the author builds 36 portfolios from the intersection of six size portfolios and six momentum portfolios. Overall, 100 equally weighted momentum, size and 36 double sorted momentum-size portfolios are obtained and regressed against the beta, the co-skewness and the co-kurtosis. He uses also a beta-gamma-delta sorts by firstly groups stocks into beta deciles, then into gamma and finally into delta deciles.

Applying the cross sectional regression and looking at the beta-gamma-delta sequential sorts, the author finds that the risk premium associated to the market, the skewness and the kurtosis are highly significant. This is further supported by a higher adjusted R-squared for the four moment model compared to the two moment model. As for the momentum portfolios, the beta of portfolios is found to be statistically significant and the co-skewness and the co-kurtosis are generally almost significant. In addition to that, the adjusted R-squared increases when including the third and the fourth moment.

For size portfolios, it is found that all portfolios' betas are negatively associated to the size and the co-skewness and co-kurtosis premiums are highly significant. However, unlike the other sorted portfolios, for the size portfolios all the intercepts are significant. Finally, for the two-way-sorts, it's found that the inclusion of the higher co-moments renders all intercepts insignificant and engenders a positive and a statistically significant premium for both the co-skewness and the co-kurtosis. For robustness check, Chi-Hsiou Hung (2007) investigates within a time series regression a cubic market model and finds that returns on the winner, loser and the smallest size deciles are cubic functions of the market model.

One year later, particularly in 2008, the same author i.e., Chi-Hsiou Hung, investigates the ability of the non linear market-models in predicting assets' returns. The sample of the study contains a set of nineteen countries including the Canada, the United States, Belgium, Denmark, Finland, France, Germany, Italy, Netherlands, Norway, Spain, Sweden, Switzerland, United Kingdom, Australia, Hong Kong, Japan, Singapore and Taiwan and covers the 954-weeks' returns from 22 September 1987 to 27 December 2005 for both listed and delisted firms. The analysis of the higher order moment models includes the three moments and the four moment framework and run on the momentum and size portfolios.

Using the times series regression, Chi-Hsiou Hung (2008) finds that the beta coefficient is highly significant in every model for both the winner and the loser portfolios. For the winner portfolios, adding the co-skewness to the CAPM increases the adjusted R-squared as the coefficient of the co-skewness is negative and statistically significant. However, for the loser portfolios, the addition of this coefficient does not contribute to the improvement of the explanatory power which is enhanced by an insignificant value of the coefficient. As for the fourth moment, the results do not provide any support on its favor. Though, in all models and for both the winner and the loser portfolios, the intercepts are found to be positive and highly significant which weakens the accuracy of these models in predicting returns.

Concerning the size portfolios, the results indicate in sum there major conclusions. First, the standard CAPM explains well the return on the biggest size portfolios. Then, adding the squared of the markets contributes obviously to the improvement of the estimation for the smallest size portfolios. Nevertheless, the inclusion of the cubed market term does not have any utility since the coefficient is insignificant in all models and for both the studied portfolios.

Gustavo M. de Athayde and Renato G. Flôres Jr (200?) extend the traditional version of the CAPM to include the higher moments and run tests on the Brazilian context. They use daily returns for the ten most liquid Brazilian stocks for the period that goes from January 2nd 1996 to October 23rd 1997. The results from the time series regression point toward the importance of the skewness coefficient. Indeed, adding the skewness to the classical CAPM generate a significant gain at the 1%level. Similarly, adding the skewness to the CAPM that contains already the kurtosis coefficient obviously contributes to the improvement of the latter since the coefficient is found to be significant. However, the addition of the kurtosis either to the classical CAPM or to that including skewness does not have any marginal gain. Gustavo M. and Renato G. (200?) conclude, therefore, that for the Brazilian context the addition of the skewness is appropriate while the gain in adding the kurtosis is irrelevant.

In a similar way, Daniel R. Smith (2007) tests whether the conditional co-skewness explains the cross section of expected return in the UK stock exchange. Their study is performed on 17 value-weighted industry portfolios, 25 portfolios formed by sorting stocks on their market capitalization and book-equity to market-equity ratio for the period that spans from July 1963 to December 1997. For the conditional version, he uses six conditioning information variables that are documented in the literature. The results indicate that the co-skewness is important determinant in explaining the cross section of equities' return. Moreover, it is found that the pricing relationship varies through time depending on whether the market is negatively or positively skewed. The author claims, through the results, that the conditional two-moment CAPM and the conditional three factor model are rejected. However, the inclusion of the co-skewness in both models cannot be rejected by the data at all.

Recently, Christophe Hurlin, Patrick Kouontchou, Bertrand Maillet(2009) try to include higher moments in the CAPM. As in Fama and French (1992), portfolios are formed with reference to their market capitalization and book-to-market ratio. The study sample consists of monthly data of listed stocks from the French stock market over the January 2002 through December 2006 sample period. Using three regression types, i.e. the cross section regression, the times series regression and the rolling regression, the authors find that, when co-skewness is taken into account, portfolios characteristics have no explanatory power in explaining returns, whereas, the ignorance of the co-skewness produces contradictory results.

The results point also to a relatively higher value of adjusted R-squared for the model with the skewness factor compared to the classical model and that the co-skewness is positively related to the size. In a sum, the authors assert that high frequency intra-day transaction prices for the studied portfolios and the underlying factor return produce more plausible measures and models of the realized co-variations.

Benoit Carmichael (2009) explores the effect of co-skewness and co-kurtosis on assets pricing. He finds that the skewness market premium is proportional to the standard market risk premium of the CAPM. This result supports that standard market risk is the most important determinant of the cross-sectional variations of asset returns.

Likewise, Jennifer Conrad, Robert F. Dittmar, and Eric Ghysels (2008) try to explore the effect of the volatility and the higher moments on the security returns. Using a sample made of option prices for the period that goes from 1996 to 2005, the authors estimate the risk moments for individual securities. The results point to a strong relationship between the third and the fourth moments and the subsequent returns. Indeed, it's found that the skewness is negatively associated to the subsequent returns. This is equivalent to say that stocks with a relatively lower negative or positive skewness earn lower return. It is found also that the kurtosis is positively and significantly associated to the returns. They claim also, that these relationships are robust when controlling for some firms' characteristics.

Then, using a stochastic discount factor and controlling for the higher co-moments, Conrad et al. (2008) find that idiosyncratic kurtosis is significant for short maturities whereas idiosyncratic skewness has significant residual predictive power for subsequent returns across maturities.

In the Pakistani context, Attiya Y. Javid (2009) looks at the extension of the CAPM to a mean-variance-skewness and a mean-variance-skewness-kurtosis model. He works on daily as well as monthly return of individual stocks traded in the Karachi stock exchange for the period from 1993 to 2004. Allowing for the covariance co-skewness and co-kurtosis to vary overtime, the results indicate that both the unconditional and the conditional three-moment CAPM performs relatively well compared to the classical version and the four moment model. Nevertheless, the results show that the systematic covariance and the systematic co-skewness have only insignificant role in explaining return.

### 6. THE QUARREL ON THE CAPM AND ITS MODIFIED VERSIONS

The CAPM developed by Sharpe (1964) and John Lintner (1965), and Mossin (1965) gave the birth to assets' valuation theories. For a long time, this model had always been the theoretical base of the financial assets valuation, the estimate of the cost of capital, and the evaluation of portfolios' performance.

Being a theory, the CAPM found the welcome thanks to its circumspect elegance and its concept of good sense which supposes that the risk-averse investors would require a higher return to compensate for supporting higher risk. It seems that a more pragmatic approach carries out to conclude than there are enough limits resulting from the empirical tests of the CAPM. In fact, since the CAPM is based on simplifying assumptions, it will be completely normal that the deviation from these assumptions generates, ineluctably, imperfections.

The most austere critic that was addressed to the CAPM, is that advanced by Roll (1977). In fact, in his paper, the author declares that the theory is not testable unless the market portfolio includes all assets in the market with the adequate proportions. Then, he blames the use of the market portfolio as a proxy, since the proxy should be mean/variance efficient even though the true market portfolio could not be. He, afterwards, passes judgment on the studies of both Fama and MacBeth (1973) and Blume and Friend (1973) in the sense that they present evidence of insignificant nonlinear beta terms.

In fact, he sustains that without verifying to how extent the proxy of the market portfolio is closer to the reality, these evidences won't serve to any conclusion at all. He concludes, then, that the most practical hypothesis in this theory is that the market portfolio is ex-ante efficient. He asserts, subsequently, that verifying whether the market proxy is a good estimator may allow verifying the testable hypothesis of the model.

With reference to Roll (1977), the results of empirical tests are dependent on the index chosen as a proxy of the market portfolio. If this portfolio is efficient, then we conclude that the CAPM is valid. If not, we will conclude that the model is not valid. But these tests do not allow us to ascertain whether the true market portfolio is really efficient.

The tests of the CAPM are based, mainly, on three various implications of the relationship between the return and the market beta. Initially, the expected return on any asset is linearly connected to its beta, and no other variable will be able to contribute to the increase of the explanatory power of the model.

Then, the premium related to beta is positive which means that the expected return of the market exceeds that of the individual stocks, whose return is not correlated with that of the market. Lastly, in the Sharpe and Lintner model (1964, 1965), the stocks whose returns are 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 that of the risk free rate.

Furthermore, the CAPM is based on the simplifying assumption that all investors behave in the same way, but this is not easily feasible. Indeed, this model is based on anticipations and since the individuals do not announce their beliefs concerning the future, the tests of the CAPM can only lead to the assumption that the future can present either less or more the past. As a conclusion, tests of the CAPM can be only partially conclusive.

Tests of the CAPM find evidences that are conflicting with the assumptions. For instance, many researchers (Jensen, 1968; Black, Jensen, and Scholes, 1972 among others) have found that the relationship between the beta and the expected return is weaker than the CAPM predicts. It is not bizarre, also, to find that low beta stocks earn higher returns than the CAPM suggests.

Moreover, the CAPM is based on the risk reward principle which says that investors who bear higher risk are compensated for higher return. However, sometimes the investors support higher risk but require only lower returns. It is, particularly, the case of the horse ... and the casino gamblers.

Several other assumptions are delicate tackling the validity of the model. For example, the CAPM assumes that the market beta is unchanged overtime. Yet, in a dynamic world this assumption remains a discussed issue. Since, the market is not static it would be preferable for a goodness of fit to model what is missing in a static model. In addition to that, the model supposes that the variance is an adequate measure of risk. Nevertheless, in the reality other risk measures such as the semi-variance may reflect more properly the investors' preferences.

Furthermore, while the CAPM assumes that the return's distribution is normal, it is though often observed that the returns in equities, hedge funds, and other markets are not normally distributed. It is even demonstrated that higher moments such as the skewness and the kurtosis occur in the market more frequently than the normal distribution assumption would expect. Consequently, one can find oscillations (deviations compared to the average) more perpetually than the predictions of the CAPM.

The reaction to these critics is converted into several attempts aiming at the conception of a well built pricing model. The Jagannathan and Wang (1996) conditional CAPM, for one, is an extension of the standard model. The conditional CAPM differ from the static CAPM in some assumptions about the market's state. In this model, the market is supposed to be conditioned on some state variables. Hence, the market beta is time varying reflecting the dynamic of the market. But, while the conditional CAPM is a good attempt to replace the static model, it had its limitations as well.

Indeed, within the conditional version there are various unanswered questions. Questions are of the type; how many conditioning variables must be included? Can we consider all information with the same weight? Should high quality information be heavily weighted? How can investors choose between all information available on the market?

For the first question, there is no consensus on the number of state variables included. Ghysels (1998) has criticized the conditional asset pricing models due to the fact that the incorporation of the conditioning information may lead to a great problem related to parameter instability. The problem is further enhanced when the model is used out-of-sample in corporate finance applications.

Then, since investors do not have the same investment perspectives, the set of information available on the markets is not treated in the same way by all of them. In fact, information may be judged as relevant by an investor and redundant by another. So, the former will attribute a great importance to it and subsequently assign it a heavy weight. The latter, whereas, neglect this information since it doesn't affect the decision making process.

To my own knowledge, the failure of the conditional CAPM may possibly come from the ignorance of the information weight. The beta of the model must be conditioning on the state of variables with the adequate weights. i.e., the contribution of each information variable in the market's risk must be proportional to its importance and relevancy in the decision making process for a given investor.

Furthermore, even investors are not certain about which information must be included and which is not. Investors are usually doubtful about the quality of these information sources. Shall they refer to announcements and disclosures, analyst reports, observed returns and so on? This is remains questionable since even the set of information is not observed and that investors are uncertain about these parameters.

A further limit associated to the conditional CAPM, is that they are prone to the underconditioning bias documented by Hansen and Richard (1987) and Jagannathan and Wang (1996). This means that there is a lack of the information included. Shanken (1990), and Lettau and Ludvigson (2001) suggest in order to overcome this problem to make the loadings depend on the observable state variables. Nevertheless, the knowledge of the ‘'real'' state variables clearly requires an expert.

With the intention of avoiding the use of ‘'unreal'' state variables, Lewellen and Nagel (2006) divide the whole sample into non-overlapping small windows (months, quarters, half-years) and estimate directly from the short window regressions the time series of the conditional alphas or betas. They find weak evidence for the conditional CAPM over the unconditional one. Nevertheless, the method of Lewellen and Nagel (2006) can lead to biases in alphas and betas known as the ‘'overconditioning bias'' (Boguth, Carlson, Fisher, and Simutin (2008)). This bias may occur when using a conditional risk proxy not fully included in the information set such as the contemporaneous realized betas.

The third contribution in the field of asset pricing models was the higher order moments CAPM. This model introduces the preferences about higher moments of asset return distributions such as skewness and kurtosis. The empirical literature highlighted a large discrepancy over the moments included in the model. For instance, Christie-David and Chaudry (2001) employ the four-moment CAPM on future markets. The result of their study indicates that the systematic co-skewness and co-kurtosis explain the return cross section variation.

Jurczenko and Maillet (2002), Galagedera, Henry and Silvapulle (2002) make use of the Cubic Model to test for coskewness and cokurtosis. Hwang and Satchell (1999) study the co-skewness and the co-kurtosis in emerging markets. They show that co-kurtosis is more plausible than the co-skewness in explaining the emerging markets return. Y. Peter Chung Herb Johnson Michael J. Schill (2004) show that adding a set of systematic co-moments of order 3 through 10 reduces the explanatory power of the Fama-French factors to insignificance in roughly every case.

Through these inconsistencies, one may think that modeling the non linear distribution suffers from the lack of a standard model to capture for the high moments. This problem is getting worse when cumulated together with the necessity to specify a utility function which is a faltering block in the application of higher moments CAPM-versions until now. Because only the investors utility function can determine their preferences.

The last extension of the CAPM is that related to the downside risk. The downside CAPM defines the investors risk as the risk to go below a defined goal. When calculating the downside risk, only a part of the return distribution is used and only the observations which are below he mean are considered, i.e. only losses. Hence the downside beta can be largely biased.

Moreover, the semi-variance is only useful when the return distribution is asymmetric. However, when the return distribution for a given portfolio is normal, then the semi-variance is only half the portfolio's variance. Consequently, the risk measure may be biased since the portfolio is mean-variance efficient.

Furthermore, the downside risk is always defined with reference to a target return such as the mean or the median or in some cases the risk-free rate which is supposed to be constant for a given lap of time. Nevertheless, investors change their objectives and preferences from time to time which modify, consequently, the accepted level of risk over time. So, a well defined downside risk should perhaps include a developing learning process about investors' accepted rate of risk.

### 7. CONCLUSION

The dispute over the CAPM has been for as much to answer the following question: ‘'is the CAPM dead or alive?''. Tests of the CAPM find evidences that are in some cases supportive and in some others aggressive. For instance, while Blume and Friend (1973), Fama and Macbeth (1973) accept the model, Jensen (1968), Black, Jensen, and Scholes (1972), and Fama and French (1992) reject it.

The above question raises various issues in asset pricing models that academicians must struggle in order to claim to the validity of the model or before drawing any conclusion. Issues are like for example; to know whether the CAPM's relationship is still valid or not? Or does this relationship change when the context of the study changes? Do statistical methods affect the validity of the model? Does the study sample impinge on the model?

Also since many improvements are joined to the model, the questions may be turned into types like for example; does conditional beta improve the CAPM? Does co-skewness or co-kurtosis or the both improve the risk-reward relationship? Does the Downside risk contribute to the survival of the CAPM?

So, to conclude whether the CAPM is dead or not, one may find various difficulties since the evidence is very mixed. Unfortunately, through the narrative literature review we do not come out with a clear conclusion about whether our answer is yes or not. It seems that this tool is inadequate for our study since a strong debate needs to be solved. In fact, to answer the question of interest, several issues, remain doubtful in the literature review, must be taken for granted. First, it is impossible to compare studies that do not have the same quality. Quality is measured through for example the statistical methods, the sample size, the data frequency …etc

Second, in order to reach a clear conclusion we must not rely only on studies that defend our point of view and neglect the opposite view. Hence, to defend the model, it is advisable to gather all positive studies, and to reject it is recommendable to accumulate only negative studies.

Finally, if many versions need to be examined, how can we draw conclusion about the validity of the version since in the version itself the evidence is mitigated. For instance, how shall we know whether the conditional version improves the model while the conditional version its self is contested.

In a nutshell, in order to move away the fogs on the CAPM and to brush away the hole inherent to the narrative literature review we recommend the use of the meta-analysis technique which is an instrument that affords accurate and pertinent conclusions.

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