# Capital Asset Pricing Model

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### 1. Introduction

Markowtiz (1952) did the ground work for the CAPM (Capital Asset Pricing Model). From the study of the early theories we know that the risk of an underlying security is measured by the standard deviation of its pay off or return. Therefore, for a larger risk we will have higher standard deviation of the respective security return. Markowtiz argued that the standard deviations of security returns for any two securities are not additive if they are combined together unless the returns of those two assets are perfectly positively correlated. He also observed that the standard deviation of security return of a portfolio is less than the sum of the standard deviation of those assets constituted the portfolio. Markowitz developed the efficient frontier of portfolio, the efficient set from where the investors select the portfolio which is most suitable for them. Technically, an investor will hold a mean-variance efficient portfolio which will return the highest pay off to them with a given level of variance. Markowitz’s computation of risk reduction is very rigorous and tedious. Sharpe (1964) developed the single index model which is computationally efficient. He derived a common index where the asset return is related with the common index. This common index can be any variable which has influence on the asset return. We can apply this single index model to the portfolio as well since the expected return of a portfolio is the weighted average of the expected returns of the constituents of the portfolio.

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When we need to analyze the risk of an individual security, we have to consider the other securities of the portfolio as well. Because, we are interested about the additional risk being added to the portfolio when one addition security is added to the portfolio. Thus the concept of risk share of an individual security to the portfolio is different from the risk of that security itself. An investor faces two kinds of risks. One is called the systematic risk and the other is known as unsystematic risk. Unsystematic risk is a kind of risk which can be minimized or eliminated by increasing the size of the portfolio, namely, by increasing the diversity of the portfolio. The systematic risk is well known as the market risk. Because, it depends on the overall movement of the market and the financial condition of the whole economy. By diversifying the portfolio, we cannot eliminate the systematic risk.

Theoretically CAPM offers very commanding predictions about how to measure risk and return relationship. However, the empirical evidence of CAPM is not very encouraging. One may conclude that these failings are rooted in poor construction of the model but once can argue that this failing arises because of the difficulties of building comprehensive and valid test model. The estimation strategy of CAPM is not free from the data-snooping bias. Because of the non-experimental nature of economic theory we cannot avoid this problem. Moreover a lot of investigations already have been done to test the validity of the CAPM. Thus, no attempt has been made in this paper to test the validity of the model. Here in this paper we will critically examine some literatures on CAPM testing. We will begin with understanding the model. We will briefly outline some mathematics required to understand the underlying assumptions of the model. Then we will focus on the single and multi-factor CAPM models to analyze the model assumptions and restrictions required to hold these models to be true.

### 2. The capital Asset pricing Model

In 1959 Markowitz introduced the notion of mean-variance efficient portfolio. According to him it is optimal for an investor to hold a mean-variance efficient portfolio. The mean-variance efficient portfolio is a portfolio for an investor where he minimizes the portfolio return, given the expected return and maximizes expected return, given the variance. Later Sharpe (1964) and Lintner (1965b) further developed the work of Markowitz. In their work it has been showed that if the investors’ expectations are homogeneous and when the hold the mean-variance efficient portfolio then in the nonexistence of market friction the market portfolio will be a mean-variance efficient portfolio.

There are two basic building blocks to derive the CAPM: one is the capital market line (CML) and the other one is the security market line (SML). In CAPM the securities are priced in a way where the expected risks are compensated by the expected returns. As we will be investigating different form of CAPM in this work it is worthy to review the basic notions of CML and SML.

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The capital market line (CML) conveys the return of an investor for his portfolio. As we have already mentioned, there is a linear relationship exists between the risk and return on the efficient portfolio that can be written as follows:

On the Other hand the SML specifies the return what an individual expects in terms of a risk-free rate and the relative risk of a portfolio. The SML with security i can be represented as follows:

Here the Beta is interpreted as the amount of non-diversifiable risk intrinsic in the security relative to the risk of the efficient market portfolio.

The utility function of the market agent is either quadratic or normal

All the diversifiable risks are eliminated

The efficient market portfolio and the risk-free assets dominate the opportunity set of the risky asset.

We can use the security market line can be used to test whether the securities are fairly priced.

### 3. The Logic of the Model:

To understand the logic of CAPM, let us consider a portfolio M. To clear the asset market this portfolio must be on the efficient frontier. Thus the underlying concept that is true for minimum variance portfolio, must be true for the market portfolio as well. With the minimum variance condition for portfolio M when there are N risky assets, we can write the minimum variance condition by the following equation:

Where is the expected return on the asset i and . The market beta for the asset is derived by dividing the covariance of the market return and individual asset return by the variance of the market return,

In the minimum variance condition stands for the expected asset return whose market beta is zero which implies that the asset return is not correlated with the market return. The second term of the equation represents the risk premium. Here the beta measures how sensitive the asset return is with the variation in the market return. Sharpe and Lintner focused on three important implications. They are: 1)the intercept is zero; 2) Beta can completely capture the cross sectional variation of expected access asset return; and, 3)The market risk premium is positive.

Sharpe and Lintner in their CAPM model assumed that the pay off from a risky asset is uncorrelated with the market return. In their model the beta becomes zero when the the covariance of a asset return offsets the variance of the other assets’ returns. When the borrowing and lending is risk free and when the asset return is not correlated with the market return then the asset return equals the risk free rate. In the Sharpe-Lintner model the relationship between the asset return and the beta is represented by the following equation:

However, this assumption of riskless borrowing and lending is unrealistic. Black (1972) developed a CAPM model where he did not make this extreme assumption. He showed that the mean variance efficient portfolio can be obtained by allowing the short selling of the risky assets. The Black and Sharpe-Lintner model differ in terms of the . Black observed that has to be less than the expected market return which allows the premium for the market beta to be positive. In the Sharpe-Lintner model the expect return was the risk free interest rate. The assumption that Black made about short selling is not realistic either. Because, if there is no risky asset (Sharpe-Lintner version) and if there is unrestricted short selling of the risky asset (Black version) then the efficient portfolio is actually not efficient and there does not exist any relation between market beta and CAPM (Fama and French: 2003). So, the CAPM models are built on some extreme assumptions. To testify the validity of these models researchers have tested the model against the market data. In this paper we will investigate some of those empirical researches.

### 4. Early literatures on CAPM testing

There are three relationships between expected return and market beta which is implied by the model. First, the expected returns on all the underlying assets are linearly related to their respective betas. Second, the premium for beta is positive which implies that the expected return on the market portfolio exceeds the expected return on assets. Moreover, the returns of these assets are uncorrelated with the expected return of market portfolio. Third, in the Sharpe-Lintner model we see that the underlying assets which are uncorrelated with the market portfolio have the expected returns which are equal to the risk neutral interest rate. In that model, if we subtract the risk free rate from the expected market return, we get the beta premium. Conventionally the tests of CAPM are based on those three implications mentioned above.

### 4.1 Tests on Risk Premiums

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Examples of our workMost of the previous cross-section regression tests primarily focus on the Sharpe-Lintner model’s findings about the concept and the slope term which studies the relationship between expected return and the market beta. In that model they regressed the mean asset returns on the estimated asset betas. The model suggests that the constant term in the cross-section regression stands for the risk free interest rate and the slope term stands for the difference between market interest rate and risk free interest rate.

There are some demerits of the study. First of all, the estimated betas for individual assets are imprecise which creates the measurement error when we use them to explain average returns. Secondly, the error term in the regression has some common sources of variation which produces positive correlation among the residuals. Thus the regression has the downward bias in the usual OLS estimate. Blume (1970) and Black, Scholes and Jensen (1972) worked on overcoming the shortcomings of Sharpe-Lintner model. Instead of working on the individual securities they worked on the portfolios. They combined the expected returns and market beta in a same way that if the CAPM can explain the security return, it can also explain portfolio return. As the econometric theory suggests, the estimated beta for diversified portfolios are more accurate than the estimated beta for the individual security. Therefore, if we use the market portfolio in the regression of average return on betas, it lessens the critical problem. However, grouping shrinks the range of estimated betas and shrinks the statistical power as well. To tackle this researchers sort securities to create two portfolios. The first one contains securities with the lowest beta and it moves up to the highest beta.

We know that when there exists a correlation among the residuals of the regression model, we cannot draw accurate inference from that. Fama and Macbeth (1973) suggested a method to address this inference problem. They ran the regression of returns on beta based on the monthly data rather than estimating a single cross-section regression of the average returns on beta. In this approach the standard error of the means and the time series means can be used to check whether the average premium for beta is positive and whether the return on the asset is equal to the average risk free interest rate.

Jensen (1968) noted that Sharpe-Lintner model also implies a time series regression test. According to Sharpe-Lintner model, the average realized CAPM risk premium explains the average value of an asset’s excess return. The intercept term in the regression entails that “Jensen’s alpha”. The time series regression takes the following form:

In early studies we reject Sharpe-Lintner model for CAPM. Although there exists a positive relation between average return and beta, it’s too flat. In Sharpe-Lintner model the intercept stands for the risk free rate and the slope term indicates the expected market return in access of the risk neutral rate. In that regression model the intercept is greater than the risk neutral rate and the coefficient on beta is less than . In Jensen’s study the p value for the thirty years period is 0.02 only which indicates that the null hypothesis is rejected at 5% significance level. The five and ten year sub-period demonstrates the strongest evidence against the restrictions imposed by the model.

In past several studies it has been confirmed that the relationship in between average return and beta is too flat (Blume: 1970 and Stambaugh: 1982). With the low betas the constant term in the time series regression of excess asset return on excess market return are positive and it becomes negative for the high betas of the underlying assets.

In the Sharpe-Linter model, it has been predicted that portfolios are plotted along a straight line where the intercept equals the risk free rate, , and the slope equals to the expected excess return on the market rate . Fama and French (2004) observed that risk premium for beta (per unit) is lower than the Sharpe-Lintner model and the relationship between asset return and beta is linear. The Black version of CAPM also observes the same where it predicts only the beta premium is positive.

### 4.2 Testing the ability of market betas of explaining expected returns

Both the Sharpe-Lintner and Black model predict that market portfolio is mean-variance efficient. The mean-variance efficiency implies that the difference in market beta explains the difference in expected return of the securities and portfolios. This prediction plays a very important role in testing the validity of the CAPM.

In the study by Fama and Macbeth (1973), we can add pre-determined explanatory variables to the month wise cross section regressions of asset return on the market beta. Provided that all the differences in expected return are explained by the betas, the coefficients of any additional variable should not be dependably different from zero. So, in the cross-section analysis the important thing is to carefully choose the additional variable. In this regard we can take the example of the study by Fama and MacBeth (1973). In that work the additional variables are squared betas. These variables have no impact in explaining the average asset return.

By using the time series regression we can also test the hypothesis that market betas completely explain expected asset return. As we have already mentioned that in the time series regression analysis, the constant term is the difference between the asset’s average return and the excess return predicted by the Sharpe-Lintner model. We cannot group assets in portfolios where the constant term is dependably different from zero and this applies only the model holds true. For example, for a portfolio, the constant term for a high earning to price ratio and low earning to price ratio should be zero. Therefore, in order to test the hypothesis that betas suffice to explain expected returns, we can estimate the time-series regression for the portfolios and then test the joint hypothesis for the intercepts against zero. In this kind of approach we have to choose the form of the portfolio in a way which will depict any limitation of the CAPM prediction.

In past literatures, researchers tend to follow different kinds of tests to see whether the constant term in the time-series regression is zero. However, it is very debatable to conclude about the best small sample properties of the test. Gibbons, Shanken and Ross (1989) came up with an F-test for the constant term that has the exact-small sample properties and which is asymptotically efficient as well.

For the tangency portfolio, this F-test builds an entrant by combining the market proxy and the average value of an asset’s excess return. Then we can test if the efficient set and the risk free asset is superior to that one obtained by combining the market proxy and risk free asset alone. From the study of Gibbons, Ross, and Shanken (1989) we can also test whether market betas are sufficient enough to explain the expected returns. The statistical test what is conventionally done is if the explanatory variables can identify the returns which are not explained by the market betas. We can use the market proxy and the left hand side of the regression we can construct a test to see if the market proxy lies on the minimum variance frontier.

All these early tests really do not test the CAPM. These tests actually tested if market proxy is efficient which can be constructed from it and the left hand side of the time series regression used in the statistical test. Its noteworthy here that the left hand side of the time series regression does not include all marketable assets and it is really very difficult to get the market portfolio data (Roll, 1977). So, many researchers concluded that the prospect of testing the validity of CAPM is not very encouraging.

From the early literatures, we can conclude that the market betas are sufficient enough to explain expected returns which we see from the Black version of CAPM. That model also predicts that the respective risk premium for beta is positive also holds true. But at the same time the prediction made by Sharpe and Lintner that the risk premium beta is derived from subtracting the risk free interest rate from the expected return is rejected. The attractive part of the black model is, it is easily tractable and very appealing for empirical testing.

### 4.3 Recent Tests on CAPM

Recent investigations started in the late 1970s have also challenged the success of the Black version of the CAPM. In recent empirical literatures we see that there are other sources are variation in expected returns which do not have any significant impact on the market betas. In this regard Basu’s (1977) work is very significant. He shows that if we sort the stocks according to earning-price ratios, then the future returns on high earning-price ratios are significantly higher than the return in CAPM. Instead of sorting the stocks by E/P, if we sort it by market capitalization then the mean returns on small stocks are higher than the one in CAPM (Banz, 1981) and if we do the same by book-to-market equity ratios then the set of stocks with higher ratio gives higher average return (Statman and Rosenberg, 1980).

The ratios have been used in the above mentioned literatures associate the stock prices which involves the information about expected returns which are not captured by the market betas. The price of the stock does not solely depend on the cash flows, rather it depends on the present discounted value of the cash flow. So, the different kind of ratios discussed above play a crucial role in analyzing the CAPM. In line with this Fama and French (1992) empirically analyzed the failure of the CAPM and concluded that the above mentioned ratios have impact on stock return which is provided by the betas. In a time series regression analysis they concluded the same thing. They also observed that the relationship between the average return and the beta is even flatter after the sample periods on which early CAPM studies were done. Chan, Hamao, and Lakonishok (1991) observed a strong significant relationship between book-to-market equity and asset return for Japanese data which is consistent with the findings of Fama and French (1992) implies that the contradictions of the CAPM associated with price ratios are not sample specific.

### 5. Efficient Set of Mathematics

The mathematics of mean-variance efficient set is known as the efficient set of mathematics. To test the validity of the CAPM, one of the most important parts is to test the mean-variance efficiency of the model. Thus, it is very important to understand the underlying mathematics of the model. Here, we will discuss some of the useful results of it (Roll, 1977).

Here we assume that there are N risky assets with a mean vector μ and a covariance matrix Ω. In addition we also assume that the covariance matrix is of full rank. is vector of the portfolio weight. This portfolio has the average return; and variance. Portfolio p is the minimum variance portfolio with the mean return if its portfolio weight vector is the solution to the following constrained optimization:

We solve this minimization problem by setting the Lagrangian function. Let’s define the following:

The efficient frontier can be generated from any two minimum variance portfolios. Let us assume that p and r be any two minimum variance portfolio. The covariance of these two portfolios is as follows:

For a global minimum-variance portfolio g we have the following:

The covariance of the asset return of the global minimum portfolio g and any other portfolio as defined as a is as follows:

For a multiple regression of the return of an asset or portfolio on any minimum variance portfolio except the global minimum variance portfolio and underlying zero-beta portfolio we have the following:

The above mentioned result deserves some more attention. Here we will prove the result. As . The result is obvious. So, we just need to show that

and . Let us assume that r be the minimum variance portfolio with expected return . From the minimization problem we can write the following:

Portfolio a can be expressed as a combination of portfolio r and an arbitrage portfolio which is composed of portfolio a minus portfolio . The return of is expressed as:

Since , the expected return of is zero. Because, as mentioned earlier that it is an arbitrage portfolio with an expected return of zero, for a minimum variance portfolio q. We have the following minimization problem:

The solution to the optimization problem is c=0. Any other solution will contradict q from being the minimum variance.

Since, , thus taking the derivative gives the following expression:

Setting the derivative equal to zero and by substituting in the solution c=0 gives:

Thus the return of is uncorrelated with the return of all other minimum variance portfolio.

Another important assumption of the CAPM is if the market portfolio is the tangency portfolio then the intercept of the excess return market model is zero. Here we will prove the result. Let us consider the following model with the IID assumptions of the error term:

Now by taking the unconditional expectation we get,

As we have showed above, the weight vector of the market portfolio is,

Using this weight vector, we can calculate the covariance matrix of asset and portfolio returns, the expected excess return and the variance of the market return,

Combining these results provide,

Now, by combining the expression for beta and the expression for the expected excess return give,

Therefore, the immediate result is

### 6. Single-factor CAP

In practice, to check the validity of the CAPM we test the SML. Although CAPM is a single period ex-ante model, we rely on the realised returns. The reason being the ex ante returns are unobservable. So, the question which becomes so obvious to ask is: does the past security return conform to the theoretical CAPM?

We need to estimate the security characteristic line (SCL) in order to investigate the beta. Here the SCL considers the excess return on a specific security j to the excess return on some efficient market index at time t. The SCL can be written as follows:

Here is the constant term which represents the asset return (constant) and is an estimated value of . We use this estimated value as an explanatory variable in the following cross-sectional regression:

Conventionally this regression is used to test for a positive risk return trade off. The coefficient of is significantly different from zero and is assumed to be positive in order to hold the CAPM to be true. This also represents the market price of risk. When we test the validity of CAPM we test if is true estimate of . We also test whether the model specification of CAPM is correct.

The CAPM is single period model and they do not have any time dimension into the model. So, it is important to assume that the returns are IID and jointly multivariate normal. The CAPM is very useful in predicting stock return. We also assume that investors can borrow and lend at a risk free rate. In the Black version of CAPM we assume that zero-beta portfolio is unobservable and thus becomes an unknown parameter. In the Black model the unconstrained model is the real-return market model. Here we also have the IID assumptions and the joint normality return.

Many early studies (e.g. Lintner, 1965; Douglas, 1969) on CAPM focused on individual security returns. The empirical results are off-putting. Miler and Scholes (1972) found some statistical setback faced when using individual securities in analyzing the validity of the CAPM. Although, some of the studies have overcome the problems by using portfolio returns. In the study by Black,Jensen and Scholes (1972) on New York stock exchange data, portfolios had been formed and reported a linear relationship between the beta and average excess portfolio return. The intercept approaches to be negative (Positive) for the beta greater than one (less than one). Thus a zero beta version was developed of the CAPM model. The model was developed in a model where the intercept term is allowed to take different values in different period. Fama and Mcbeth (1973) extended the work of Black et al (1972). They showed the evidence of a larger intercept than the risk neutral rate. They also found that a linear relationship exists between the average returns and the beta. It has also been observed that this linear relation becomes stronger when we work with a dataset for a long period. However, other subsequent studies provide weak empirical evidence of this zero beta version.

We have mixed findings about the asset return and beta relationship based on the past empirical research. If the portfolio used as a market proxy is inefficient then the single factor CAPM is rejected. This is also true if the proxy portfolio is inefficient by a little margin (Roll: 1977, Ross: 1977). Moreover, there exists survivorship bias in the data used in testing the validity of CAPM (Sloan, 1995). Bos and Newbold (1984) observed that beta is not stable for a period of time. Moreover, there are issues with the model specifications too. Amihud, Christen and Mendelson (1993) observed that there are errors in variables and these errors have impact on the conclusion of the empirical research.

We experience less favourable evidence for CAPM in the late 1970s in the so called anomalies literature. We can think the anomalies as the farm characteristics which can be used to group assets in order to have a high ex post Sharpe ratio relative to the ratio of the market proxy for the tangency portfolio. These characteristics provide explanatory power for the cross-section of the average mean returns beyond the beta of the CAPM which is a contradiction to the prediction of CAPM.

We have already mentioned that the early anomalies include the size effect and P/E ratio as we have already mentioned. Basu (1977) observed that the portfolio formed on the basis of P/E ratio is more efficient than the portfolio formed according to the mean-variance efficiency. With a lower P/E firms have higher sample average return and with high P/E ratio have lower mean return than would be the case if the market portfolio is mean-variance efficient. On the other hand the size effect shows that low market capitalization firms have higher sample return than would be expected if the market portfolio was mean-variance efficient.

Fama and French (1992,1993) observed that beta cannot explain the difference between the portfolio formed based on ratio of book value of equity to the market value of equity. Firm has higher average return for higher book market ratio than originally predicted by the CAPM. However, these results signal economically deviations from CAPM. In these anomalies literatures, there are hardly any motivations to study the farm characteristics. Thus there is a possibility of overstating the evidence against the CAPM since there are sample selection bias problem in estimating the model and also there is a problem of data snooping bias. This a kind of bias refers to the biases in drawing the statistical inference that arises from data to conduct subsequent research with the same or related kind of data. Sample selection bias is rooted if we exclude certain sample of stocks from our analysis. Sloan (1995) argued that data requirements for the study of book market ratios lead to failing stocks being excluded which results the survivorship bias.

Despite an ample amount of evidences against CAPM, it is still being widely used in finance. There is also the controversy exists about how we should interpret the evidence against the CAPM. Some researchers often argue that CAPM should be replaced with multifactor model with different sources of risks. In the following section we will analyze the multifactor model.

### 7. Multifactor Models

So far we have not talked anything about the cross sectional variation. In many studies we have found that market data alone cannot explain the cross sectional variation in average security returns. In the analysis of CAPM, some variables like, ratio of book-to-market value, price-earning ratio, macroeconomic variables, etc are treated as the fundamental variables. The presence of these variables account for the cross-sectional variation in expected returns. Theoretical arguments also signal that more than one factor are required.

Fama and French (1995), in their study showed that the difference between the return of small stock and big stock portfolio (SMB) and the difference between high and low book-to-market stock portfolio (HML) become useful factor in cross sectional analysis of the equity returns. Chung, Johnson and Schill (2001) found that the SMB and HML become statistically insignificant if higher order co-moments are included in the cross sectional portfolio return analysis. We can infer from here that the SMB and HML can be considered as good proxies for the higher order co-moments. Ferson and Harvey (1999) made a point that many econometric model specifications are rejected because they have the tendency of ignoring conditioning information.

Now we will show one of the very important results of the multifactor model. Let us consider a regression of portfolio on the returns of any set of portfolios from which the entire minimum variance boundary can be generated. We will show that the intercept of this regression will be zero and that factor regression coefficients for any asset will sum to unity. Let the number of the portfolios in the set be K and is the (Kx1) vector of time period t of asset returns. For any value of the constant μ, there exists a combination of portfolio and assets. Let us consider μ be the global minimum variance portfolio and we denote the portfolio as op. Corresponding to op is minimum variance portfolio p which is uncorrelated with the return of op. As long as p and op are efficient portfolios in terms of the minimum variance their returns are the linear combinations of the elements of ,

where and are (Kx1) vectors of portfolio weights. As p and op are minimum variance portfolio their returns are linear combinations of the elements of ,

Then for the K portfolios we have,

By rearranging, we get the following,

Substituting this value into μ returns the following:

Now let us consider a multivariate regression of N assets on K factor portfolios,

where a is the (Nx1) intercept vector, B is (NxK) matrix of the coefficients of the regression. From the econometric theory we have,

Again by substituting these expressions into μ give,

That is the factor regression coefficients for each asset sum to unity. If then and we have a=o which indicates that the regression intercept will be zero for all assets including asset a.

As we have already proved that the intercept of a multifactor model of expected return is zero. For example, let us consider the following regression,

Fama and French (1993) using the above model observed that the model captures most of the variations of portfolio returns when these portfolios are based on B/M equity and other price ratios that cause problems for CAPM. The estimated intercept term of the above regression is used to calibrate the sensitivity of the stock prices to the new information. From the theoretical perspective the discouraging part of the three factor model is its empirical motivation. The SMB and HML explanatory returns are not motivated by forecasts on the state variables. The three factor model is not free from the momentum effect either. Stocks that did well in past several months are historically tend to do well in future month too. We can easily distinguish this momentum effect from the value effect captured by B/M equity and other ratios. The three factor model cannot explain this momentum effect. Its noteworthy here that CAPM also fails to do that.

Arbitrage pricing theory (APT) by Ross (1976) shows that we do not need the condition of man-variance optimization for all vendors. APT is more general than CAPM since it allows multiple risk factors. Moreover, APT does not require the market portfolio identification as it requires in the CAPM. However, this structure is not above any argument. APT gives us an approximate nexus for expected asset return with some unknown unidentified factors. Unless arbitrage opportunity exists, we just simply cannot rule out the theory. Therefore we need some additional assumptions in order to test the validity of the theory. In APT we assume that markets are competitive and frictionless. The asset return generating process is as follows:

Ross (1976) shows that the absence of the arbitrage opportunity in large economy implies that,

Where μ is the (Nx1) vector of the expected asset return, is zero-beat parameter and is the (Nx1) vector for the factor risk premium. The coefficient of the zero-beta parameter is the conforming vectors of ones. Again, this is just an specification. To make it testable we need more restrictions on it. To obtain those restrictions we need to have additional structure to make the approximation exact. One of the restricts is the market portfolio has to be well diversified and the factors have to be pervasive. We can consider the market portfolio be diversified if the proportion of any asset to overall economy asset is very insignificant. The requirements of factors be pervasive allows the investors to diversify their idiosyncratic risk without restricting the choice of factor risk revelation. Dybvig (1985) investigated the influence of the deviations from the exact factor pricing based on preference revealed by an investor. He observed that if the parameters of the economy are reasonably specified then the effect of deviating from the exact factor pricing is negligible.

The multifactor model does not specify the number of factors and the identification of the model. Therefore, to estimate the model we need to determine these factors. Conventionally we consider four types of exact factor pricing model. They are,

1) Factors are the portfolios of the underlying traded assets and there exists a riskfree asset

2) Factors are the portfolios of the underlying traded assets and there is no risk free asset

3) Factors are the portfolios of the traded assets

4) Factors are the portfolios of the traded assets where the factor portfolios span the efficient frontier of the risky assets.

There are a number of empirical literatures on multifactor models. Chen, Roll and Ross (1986) observed that the empirical evidence to support the exact factor pricing model is fixed. One of the strongest evidences comes from the testing by using dependent portfolios formed on market value of equity and B/M ratios. In addition, the multifactor model cannot explain the size effect and B/M effect properly. However, the portfolios are based on dividend yield on own variance provide little evidence against exact factor pricing. On the other hand, Fama and French (1993) observed some encouraging result of multifactor model if five factors are using instead of three. They concluded that for stocks we need three factors but for bond portfolios we need to include five factors. Lehmann and Modest (1988) analyzed the sensitivity of the number of the dependent variables included in the model. With fewer portfolios the p-values were lower in the result which is an issue of the power of test. By reducing the number of portfolios and without deviating from the null hypothesis, we can increase the power of the test. This is true because now we require to test fewer restrictions than before.

The multifactor models provide a very alternative to the single-factor CAPM but researchers using these models have to be aware of two pitfalls arise when factors are chosen to fit the data without considering the economic theory. First of all, because of the data snooping bias the model may over fit the data and if that is the case then the model looses its ability to predict asset return in future. Secondly, the model may capture empirical regularities that arise because of market inefficiency and in this case it may fit the data but they will imply Sharpe ratios for factor portfolios that are too high for being consistent with a reasonable underlying market equilibrium model. However, we have to wait for sufficient amount of new data become available to test the usefulness of the multifactor models.

### 8. CAPM with higher-order co-moments

We know that the unconditional security return distribution is not normal. Moreover, the mean and variance of security returns are not sufficient enough to characterise the distribution completely. Thus it encourages the researchers to look for the higher order co-moments. In practice we estimate the skewness (third moment) and kurtosis (fourth moment). In many studies researchers paid attention to the validity of CAPM in the presence of the higher order co-moments and their effects on the asset pricing. In many studies skewness has been incorporated in the asset pricing models and it provided mixed results.

Some studies incorporated conditional skewness in their models. For example Harvey and Siddique (2000) investigated an extended version of CAPM. Since the conditional skewness confines the asymmetry in risk, this version of CAPM is usually preferred over the fundamental one. In recent times, this concept of conditional skewness has become very useful in measuring the value at risk. From the study of Harvey and Siddique (2000) we notice that the conditional skewness captures the variation in cross-sectional regression analysis of expected returns significantly. This also holds true when factors based on size and book-to-market are also considered.

In some studies we see that in determining the security valuations, the non-diversified skewness and kurtosis play an important role. Fang and Lai (1997) reported a four-moment CAPM and in their study they showed that systematic variance, systematic skewness and kurtosis contribute to the risk premium of the underlying asset.

### 9. Conditional asset pricing models

Levy (1974) suggested to estimate different betas for bull and bear markets. Following that suggestion, Fabozzi and Francis (1977) estimated the betas for bull and bear markets. However, they didn’t find any evidence for beta instability. However, in another work Fabozzi and Francis (1978) reported that investors need a positive premium in order to accept the downside risk. On the other hand a negative premium corresponds with the up market beta. This up market beta is considered as a more appropriate measure of portfolio risk.

There are few other studies examined the randomness for beta. Kim and Zumwalt (1979) examined the variation in returns on portfolios in both up and down markets. They concluded that the up market comprises the months for which the market returns exceed the average market return, the average risk neutral rate and zero. They specified three measures to identify what make up an up and down market. Those months for which the market return exceeds that the average market return and when it is above the risk free rate or greater than zero constitute the up market. They observed that the respective betas of the down market is more accurate measure for the portfolio risk than the single beta we see in the conventional CAPM. In an investigation on risk-return relationship Chen (1982) allowed the beta to be non-stationary and observed that investor need compensation when they assume downside risk no matter whether the betas are constant or changing. The concluded the same about the down market risk that Kim (1979) did. Bhardwaj and Brooks (1993) concluded that the systematic risks are different in bull and bear time periods. The also classified the market as Kim and Zumwalt (1979) did but instead of comparing the market return with mean return they compared it with the median return.

Pettengill, Sundaram and Mathur (1995) observe that if we use the realized return then the beta-expected return relationship becomes conditional on the excess market return. From that study we that there exists a positive relationship between beta and ecpected return during a up market. In line with their study, Crombez and Vennet (2000) studied the conditional relationship between asset return and beta. They concluded that beta is a dependable meter in both bull (upward market) and bear market (downside risk). For different kind of specifications of the up and the down market this beta factor becomes robust and the investors can increase the expected asset return by considering the up and down market separately. Therefore different moments vary and correspond to the up and the down market. Galagedera and Silvapulle (2002) analyzed the asset return and the higher order co-moments in both bull and bear market and suggested that in the skewed market return distribution, the excess return is related to the systematic co-skewness.

### 9. CAPM: Conditional on time-varying volatility

Engle (1982) introduced the ARCH/GARCH process with which we can test the time varying volatility of stock return. This approach has drawn a considerable attention to the recent CAPM literature. Although not quite convincing, this approach provides much stronger evidence risk return relationship which we find in the conventional model of CAPM.

Fraser, Hoesli, Hamelink and Macgregor (2000) did a cross-sectional regression analysis of risk return relationship based on unconditional identification of the betas with the betas estimated from the ARCH and GARCH model. When the excess return is negative they allowed for a negative relationship of risk and return. They concluded that CAPM works better in a downward moving market than an upward one. They also observed that for a bear market beta as a proxy for risk measure works more accurately. Braun, Nelson and Sunier (1995) studied the leverage effects. By using EGARCH models they investigated the variation in beta. This kind of model allows the market volatility and asymmetrical response for beta to see the good and bad news impact on the asset or portfolio returns. Galagedera and Faff (2003) did a study on a conditional three-beta model. They modeled that one as GARCH (1,1) process and depending on the size and nature of the volatilities, they defined three state of volatilities. From that study we see that for most of the market portfolios, the betas are not significantly different from zero.

### Conclusion

The empirical evidences of Sharpe (1964) and Lintner (1965) version of CAPM have never been encouraging. However, allowing for the flatter trade off of average asset return for market beta achieved some success but researches after late 1970s observed some other important variables such as different kind of price ratios, the momentum, size, etc having crucial impact on asset return. Thus, CAPM hold well conditionally. The textbook model often refers to estimate the cost of the equity capital. This kind of model suggests to estimate market beta and to combine with the risk premium to get an estimate of the cost of the equity. But, we have already seen that the beta and average return relationship is much flatter than the Sharpe-Lintner version of CAPM. Therefore the estimates for the high-beta assets are too high and it is too low for the low-beta assets. CAPM is also used to measure the performance of the mutual funds. Here the problem is if the investment strategies incline to the CAPM problems then the mutual funds produce abnormal returns. For example any mutual or managed funds that concentrate on low beta stocks will tend to yields relative to the Sharpe-Lintner version of CAPM.

The normality assumption of asset return is very important to hold the CAPM to be true and if it holds to be true then only the betas should be priced but the problem is the stock return is non-normal in high frequency data. When the asset return is normal then mean and variance can explain the asset return distribution. To explain the non-normal asset return distribution we need the higher order moments. Again, the empirical evidence in favour of the higher order moments is not unarguable. As the market beta failed to explain the cross-sectional variation in security returns, the multifactor model came in front. As we have already mentioned, these models incorporated some important variables such as different price ratios, size, etc. Some authors argue that CAPM is overstated because of the problem of market proxy, negligence of conditioning information and data snooping bias and CAPM might hold in a dynamic equilibrium setting. The bottom line is despite having the simplicity and being the fundamental concept of asset pricing theory, empirical evidences make its use in application arguable.