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Traditional Capital Assets Pricing Model

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Results from its application throughout a narrative literature review. Second the paper 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 it’s 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.

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keywords: CAPM, CAPM’s variants, Assumptions, Literature Survey.

JEL Classification: G11.

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 paper 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 quarrel on 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, this paper is organized as follows: the first section presents the theoretical bases of the model and the problematic issues within the model. The third section presents a literature survey on the classic version of the model. The fourth 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 and Problematic Issues

2.1 Theoretical Bases

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 (1952) formulated this intuition by resorting to the diversification’s mathematics. Indeed, he claims that the investors must in general choose the portfolios while basing 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 if its systematic risk is known.

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).

In fact, 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. 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.

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

i. The markets are perfect and there are neither taxes nor expenses or commissions of any kind; ii. All the investors are risk averse and maximize the mean-variance criterion;

iii. The investors have homogeneous anticipations concerning the distributions of the returns’

probabilities (Gaussian distribution); and

iv. 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 βasset ; – The market return noted E (RM ) ; – The risk free rate (Treasury bills), noted RF

This model is the following:

E (Rasset ) = RF + βasset
E (RM − RF )

Where:
E (RM − RF ) ; 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; βasset ; 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.

β = cov (RM − Rasset )
asset
var (RM )

βasset
= ∂σ M • σ asset
∂σ asset σ M

Where:
σ M ; represents the standard deviation of the market return (market risk), and βasset ; 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 β 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. Conversely, the 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 (discussed in the following section).

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

2.2 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.

2.2.1 The Mean-Variance Market Portfolio

As we have mentioned above; the CAPM is build upon some simplifying assumptions that guarantee its validity. However, the most important one that raises much debate among researchers is that related to the mean-variance efficiency of the market portfolio. The CAPM assumes that investors are risk-averse and, when they choose between various portfolios, they care only about the mean and the variance of their one period investment strategy return. Consequently, investors should choose the portfolio that either minimizes the variance for a given level of return or maximizes the return for a given variance. From where, the CAPM is often called the mean-variance model

Roll (1977) and Ross (1977) assert that testing the validity of the CAPM is equivalent to test the mean variance efficiency of the market portfolio. They claim that, for a portfolio p to be

efficient there would be a constant
y0 p
such as the vector of securities return
r1 , r2 ,….., rn
is an

exact linear function of the vectors of all securities betas on the expected return of the market

portfolio. We have:

ri = y0 p + β i (rp − y0 p )

Where;

rp , is the expected return on the market portfolio.

Note that the betas are the slope from the time series regression of securities return on the market portfolio. We write:

Rit
= α i + β i R pt

In addition to that, the market portfolio is mean variance efficient if and only if rp > y0 p so that

the expected return is an increasing linear function of beta. The beta can be interpreted as the marginal contribution of asset i to the total market risk (variance). Hence, the hypothesis that guarantees the efficient of the market portfolio is that α i = 0 . Matthew Pollard (2008) presents a proof of Roll’s (1977) that the mean-variance efficiency and

the equation of the CAPM are equivalent. However, many researchers have examined the mean variance efficiency of diverse market portfolio proxies and found that all of these proxies were inefficient, hence, far from the efficient frontier. Many assaults were gathered against the mean- variance efficiency of the market portfolio (see for example; Black and Litterman, 1992, Best and Grauer, 1991, among others). Best and Grauer (1991), show that the efficient portfolios are extremely sensitive to changes in asset means. They find that, a small change in the asset’s mean drives roughly half the securities from an equally weighted mean-variance efficient portfolio.

A.Graig Mackinlay and Mattew P.Richardson (1991) while using returns from size based portfolios for the period that spans from 1926 to 19688, show
that the mean variance efficiency of the market portfolio is sensitive to the test considered. They employ the Generalized Method

of Moments (GMM) to test for the mean variance efficiency of the market portfolio. They find that misspecified distributional assumptions can have a strong adverse effect on statistical inference.

Gopal Basak, Ravi Jagannathan, and Guoqiang Sun (2000), examine the mean variance efficiency of the market portfolio using tests that allow for the possibility that short positions in the primitive assets may not be possible. They find that the value weighted return on stocks is mean-variance efficient with reference to the frontier of the 25 portfolios of Fama and French (1993). This is valid only when the possibility of short selling is relaxed.

Consequently, if the tests of the tests reject the CAPM one may not conclude straightforward that the CAPM doesn’t hold but rather that the theoretical bases are violated.

2.2.2 The Efficient Frontier

The Efficient frontier is a term introduced by Markowitz (1958) in developing the portfolio choice theory also called the Markowitz frontier. The representation of all possible risk-return combinations presents the market line. Subsequently, the combination along this line represents portfolios which have the lowest level of risk for a given level of return. Benninga (2000) claim that ‘’The efficient frontier represents that set of portfolios that has the maximum rate of return for every given level of risk or the minimum risk for every level of return.’’.

Mathematically speaking the efficient frontier is the intersection of the set of portfolios with the minimum risk and those with the maximum return. Thus a rational investor will only hold portfolios on the frontier because those which are under this frontier are suboptimal. However, the region above the efficient frontier is unachievable when holding only risky assets. It’s
achievable only when allowing for lending and borrowing with a unique risk free rate.

Many researchers (see for axample; Elton and Gruber, 1995; Sharpe, Alexander, and Bailey,

1999; Bodie, Kane, and Marcus ,2000; among others) while basing on the method devised by Elton, Gruber, and Padberg (1976), have constructed the capital market line (hereafter the CML). They first, develop the mean-variance portfolio frontier of risky assets standing on the Lagrange constrained optimization methods. They, afterwards identify the tangency portfolio, i.e. the

riskiest portfolio that is always on the CML. Finally, they try to span the CML using the tangency portfolio and the risk-free rate. Likewise, Ingersoll (1987) and Huang and Litzenberger (1988) derive the efficient frontier using the Lagrange constrained optimization methods. Merton (1972), develops a set of efficient frontiers under diverse conditions, i.e. both when only holding risky assets and a combination of riskless and risky assets as well. He argued that under certain conditions the graphical presentation of the efficient frontier is incorrect. Similarly, Perold (1984) developed a general technique to set the efficient frontier especially when the covariance matrix is nonnegative definite X. Y. Zhou and D. Li (2000), while using the stochastic linear quadratic control model, show that this latter is an appropriate tool to study the mean-variance problem. They, find that basing on this model, the efficient frontier is found to be closer to that derived from the original portfolio selection problem. Likewise, David Feldman and Haim Reisman (2002) provide a simple method of constructing efficient frontier. They show that the portfolio whose risky assets weights is given by the product of the inverse variance-covariance matrix of securities rate of return multiplied by the excess return vector is a CML portfolio.. Moreover, they find that basing on this, there’s an immediate construction of the efficient frontier of risky assets, the “tangency” portfolios, “reflection” portfolios, and a CAPM relationship.

2.2.3 The Equity Premium Puzzle

The CAPM is built upon the assumption according to which investor are compensated for bearing risk. The excess return of the individual securities’ return or the overall market return over the risk free rate is called the risk premium. This risk premium compensates investors for taking relatively higher risk and varies together with the risk of the assets considered.

Theoretically, investors holding risky assets earned a premium or an extra (with reference to a strategy when only riskless assets are held) return for taking risk. However, it’s shown recently (see for example; Mehra and Prescott, 1985; Hansen and Jagannathan,1991; among others) that the CAPM is unable to account for a risk premium as high as 4 percentage points. The CAPM predicts only an average risk premium of about 0.25 of a percentage point. This is a complete break between theory and practice is commonly known as ‘’the equity premium puzzle’’.

In fact, Mehra and Prescott (1985), show that the stocks are not sufficiently riskier than the risk free rate (treasury bills rate) to explain the spread of their returns. Philippe Weil (1989) advances another explanation to the equity premium puzzle. He claims that the relatively higher risk premium implies that investors are highly risk averse. Basing on this high risk aversion models of preferences should in turn suppose that they don’t like growth very much. Sanford Grossman and Robert Shiller (1981) advance that asset prices fluctuate too much to be explained by the changes in dividends or in per capita consumption. Robet Hall (1988) asserts that expected returns vary also so high with reference to the changes in consumption. Cochrane and Hansen (1992) find out a possible explanation to the risk premium puzzle involving the so- called default premium and term structure puzzles. Another view of the equity premium puzzle is represented by Grossman and Shiller (1981) who use a representative agent model with rational predictions in the market. Narayana R. Kocherlakota (1995) while examining the literature tries to explain both the risk premium and the risk-free rate puzzles. They come out with the conclusion that these puzzles
will confront any asset pricing model which relies on the three following assumptions: preferences have a particular parametric form, asset markets are complete, and asset trade is frictionless. The author concludes that relaxing these assumptions reveals many possible explanations for the risk-free rate puzzle. However, the equity premium puzzle remains questionable.

3. Background on the CAPM

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 supporting 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.

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.

Black et al. (1972) 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.

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. On the one hand, it is found that the relation between the mean excess return and beta is linear which is consistent with the CAPM. On the other hand, 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. Basing on these results, Black, et al. (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. 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, it’s found that the higher the risk is, the higher the return should be.

Blume. M. and Friend .I. (1973) try to examine theoretically and empirically the reasons beyond the failure of the market line to explain excess return on financial assets. They point out that the failure of the capital assets pricing model in explaining returns maybe due to the simplifying assumption that 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. 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 considering, 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.

S.P.Kothari, J. 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. In fact, 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). They find 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. This is quite surprising since S.P Kothari and J. Shanken (1999), find that the annual betas perform well as they are significantly associated to the average stock returns. Moreover, the ability of the beta to predict return with reference to the size and the book to market is relatively higher.

Fama and French (2004), and in agreement with their previous studies, confirms that the relation between the expected return on assets and their betas is much flatter than the prediction of the CAPM. However, Fama and French (2006) find that the market premium is for all portfolios positive and statistically significant. But, the explanatory power is relatively too low and all the intercepts from the regressions are statistically different from zero. Conversely, the study of Ivo Welch (2007) indicates that the intercept is close to zero. In addition to that, the beta coefficient is positive and statistically different from zero. He claims that whether we use the time series or the cross section regression, the results are far from rejecting the CAPM.

In the Australian context, Michael. D (2008) 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. Robert and Janmaat (2009) examine the ability of the cross sectional and the multivariate tests of the CAPM under ideal conditions. 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.

4. Is the CAPM Dead or Alive? Some Rescue Attempts

After a literature review 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.

4.1 The Conditional CAPM

‘’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 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.

4.1.1 Conditional Beta on Economic States

The conditional CAPM is that in which betas are allowed to vary and to be non stationary over 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 fact, Jagannathan and Wang (1996) claim that the static CAPM is founded on two unrealistic assumptions. The first one is that 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. They 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. The authors find that the static version of the CAPM does not hold at all. However, the estimation of the
CAPM when taking into account the beta variation shows that the beta premium is significantly different from zero. Furthermore, the estimation of the conditional CAPM with human capital indicates that this variable improves the regression.

Meanwhile, Durack et al. (2004) run the same study as Jagannathan & Wang (1996) in the Australian context. 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. But, unlike Jagannathan and Wang (1996), the authors find that the human capital does not improve the beta estimate which remains insignificant.

Campbell R. Harvey (1989), tests the CAPM while assuming that both expected return and covariances are time varying. 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. Similarly, Jonathan Lewelen and Stefan Nagel (2006) find 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.

Goezmann et.al (2007), expand the results of the Jagannathan and Wang’s study (1996). They find that the conditional market risk premium is not priced at all. From their part, Michael R. Gibbons and Wayne .Ferson (1985), relax the assumption related to the stagnation of the risk premium. Hence, the expected return is conditional on a set of information variables. The results show that the lagged GRSP value-weighted index is 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) find that the conditional version implies that the intercepts are time varying which means that they are not zero. They conclude, hence, that the conditional version is not valid. Likewise, Wayne E. Ferson and Andrew F. Siegel (2009) find that the conditioning variables do not improve very well the estimation.

4.1.2 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.

Nagel S and Singelton K (2009) use a methodology somehow different from the others as they use the Stochastic Discount Factor (SDF) as a conditionally affine function of a set of priced risk factors. 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.

The study of Demos. A and Pariss. S (1998) models the idiosyncratic conditional variances with the ARCH type process. 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. They consider that the potential cause of failure is the use of the value weighted index rather than the equally weighted one. They, hence, repeat the same procedure while using the equally weighted index. The results point to a very supportive result. Indeed, the betas coefficients not only have the right sign but also are highly significant. However, the most supportive result is that which shows
that the idiosyncratic risk is not priced.

Basu D and Stremme A (2007), model the beta variation in a non linear function of variables related to the business cycle. They find that when the conditional version is set into play, the results are somehow surprising from the static version. 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.

In the interim, Jon A. Christopherson, et al. (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, et al. (1987), find 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.

4.2 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)

4.2.1 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 since 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. 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.

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 for a risk-averse investor. Indeed, the semi-deviation accounts for the downside risk that investors want to prevent contrary to the upside risk which has the welcome.

4.2.2 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 (Mean Lower Partial Moment 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 one, 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. Another caveat from the latter study is that the 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 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. J (2002) in his seminal paper evaluates the mean-semi variance behavior in the sense that it yields a utility level similar to the investor’s expected utility. He asserts that that the mean-semivariance approach outperforms the other in the case of the negative exponential utility function. The author reports, also, that mean-semivariance is not only consistent with the maximization of expected utility but also with the maximization of the utility of expected compound return.

In (2004), 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. The results support the downside CAPM since the downside beta explains roughly 55% of the returns variability in emerging markets. Similarly, 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.

Ang, et al., (2006) show that the downside risk premium is always positive (it’s roughly about

6% per annum) and it’s statistically significant. They find also that this positive and 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.

But, although the downside approach has been the basis for many academic papers and has had significant impact on academic and non academic financial community, it is still subject to severe critics.

4.3 The High Order Moment CAPM

4.3.1 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. 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 by Chung, Peter Y., et al. (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. For instance; Fang and Lai (1997), Hwang and Satchell (1999) and Adcock and Shutes (1999) introduce the kurtosis coefficient in the pricing function and Kraus and Lizenberg (1974) 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 Robert F. 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 Robert F. 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.

4.3.2 Literature Review on the High Order Moments CAPM

Kraus and 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. They find that the three-moment CAPM 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 Robert F. Dittmar (2002) uses the cubic function as a discount factor in a Stochastic Discount Factor framework. He finds that the co-kurtosis must be included with labor growth so as to arrive to an admissible pricing kernel.

Barone-Adesi G., et al. (2002) investigate the co-skewness in a quadratic market model. They find 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. It’s found also that there’s an additional component in portfolios’ return which not explained neither by the covariance nor by the co-skewness.

Chi-Hsiou D.H, et al. (2004) 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 find that, the higher co-moments show little significance in explaining cross section returns and do not increase the explanatory power of the model. Roland. R and Xiang. G (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.

Ranaldo. A, and Favre. L (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. The results indicate that the coefficient of the co-skewness is positive and statistically significant which supports the existence of the co-skewness. However, it’s found that the co-kurtosis has no major function in explaining the hedge fund return. Smith. D. R (2007) finds 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.

Chi-Hsiou. D.H (2008) find that adding the co-skewness to the CAPM increases the adjusted R- squared as the coefficient of the co-skewness is negative and statistically significant. As for the fourth moment, the results do not provide any support on its favor.

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. Recently, Hurlin .C, et al. (2009) find that, when co-skewness is taken into account, portfolios characteristics have no explanatory power in explaining returns. Nevertheless, the ignorance of the co-skewness produces contradictory results. Similarly, Benoit Carmichael (2009) finds that the skewness market premium is proportional to the standard market risk premium of the CAPM. This result defends that standard market risk is the most important determinant of the cross-sectional variations of asset returns.

5. 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, et al., 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 gamblers and the casino players.

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, et al., 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 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.

The last extension of the CAPM 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, et al. (2003) 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, et al. (2006) 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.

6. 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 its self 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.

We conclude, hence, 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.

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