1. Introduction
Markowtiz (1952) did the ground improve the CAPM (Capital Asset Pricing Model). From the study of the early theories we realize that the chance of your underlying security is measured by the typical deviation of its pay off or return. Therefore, for a more substantial risk we will have higher standard deviation of the respective security return. Markowtiz argued that the typical deviations of security returns for any two securities aren't additive if they're combined together unless the returns of those two assets are correctly positively correlated. He also observed that the standard deviation of security return of a portfolio is significantly 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 choose the portfolio which is most suited to them. Technically, an investor will hold a mean-variance efficient portfolio that will return the best pay back to them with confirmed 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 index where the asset return is related with the normal index. This common index can be any variable which includes influence on the asset return. We can apply this single index model to the portfolio as well since the expected return of the portfolio is the weighted average of the expected returns of the constituents of the portfolio.
When we need to analyze the chance of a person security, we must consider the other securities of the portfolio as well. Because, we want about the excess risk being put into the portfolio when one addition security is added to the portfolio. Thus the idea of risk share of an individual security to the portfolio differs from the chance of this security itself. An investor faces two types of risks. One is named the systematic risk and the other is known as unsystematic risk. Unsystematic risk is some sort of risk which can be minimized or eliminated by increasing how big is 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 entire movement of the market and the financial condition of the whole economy. By diversifying the portfolio, we can not eliminate the systematic risk.
Theoretically CAPM offers very commanding predictions about how exactly to measure risk and return relationship. However, the empirical evidence of CAPM is not so encouraging. You can conclude that these failings are rooted in poor construction of the model but once can argue that failing arises as a result of difficulties of building comprehensive and valid test model. The estimation strategy of CAPM is not free from the data-snooping bias. Due to the non-experimental nature of monetary theory we can not avoid this problem. Moreover a lot of investigations already have been done to check the validity of the CAPM. Thus, no attempt has been made in this paper to check the validity of the model. Within this paper we will critically examine some literatures on CAPM testing. We will get started with understanding the model. We will briefly outline some mathematics necessary to understand the underlying assumptions of the model. Then we will give attention to the single and multi-factor CAPM models to investigate the model assumptions and restrictions required to hold these models to be true.
2. The Capital Asset Pricing Model Explained
In 1959 Markowitz introduced the idea 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 task of Markowitz. Within their work it's been showed that if the investors' expectations are homogeneous so when the contain the mean-variance efficient portfolio then in the nonexistence of market friction the marketplace portfolio will be a mean-variance efficient portfolio.
There are two basic blocks to derive the CAPM: the first is the administrative centre market line (CML) and the other an example may be the security market line (SML). In CAPM the securities are priced in a way where in fact 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 examine the basic notions of CML and SML.
The capital market line (CML) conveys the return of an investor for his portfolio. As we have already mentioned, there's a linear relationship exists between the risk and return on the efficient portfolio that may be written the following:
On the Other hand the SML specifies the return what a person expects in terms of the risk-free rate and the relative threat of a portfolio. The SML with security i could be represented as follows:
Here the Beta is interpreted as the amount of non-diversifiable risk intrinsic in the security in accordance with the risk of the efficient market portfolio.
The utility function of the marketplace agent is either quadratic or normal
All the diversifiable risks are eliminated
The efficient market portfolio and the risk-free assets dominate the chance set of the risky asset.
We can use the security market line may 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's true for minimum variance portfolio, must be true for the marketplace portfolio as well. With all 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 marketplace 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 means that the asset return is not correlated with the market return. The next term of the equation represents the chance premium. Here the beta measures how sensitive the asset return has been the variation in the market return. Sharpe and Lintner centered on three important implications. They are really: 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. Within their model the beta becomes zero when the the covariance of a asset return offsets the variance of the other assets' returns. If the borrowing and lending is risk free so when the asset return is not correlated with the marketplace return then your asset return equals the chance free rate. In the Sharpe-Lintner model the partnership between your asset return and the beta is represented by the next equation:
However, this assumption of riskless borrowing and lending is unrealistic. Black (1972) developed a CAPM model where he didn't 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 conditions of the. Black observed that should be less than the expected market return which allows the premium for the marketplace beta to be positive. In the Sharpe-Lintner model the expect return was the chance free interest. The assumption that Black made about short selling is not realistic either. Because, if there is no risky asset (Sharpe-Lintner version) and when there is unrestricted short selling of the risky asset (Black version) then the efficient portfolio is in fact not efficient and there does not exist any relation between market beta and CAPM (Fama and French: 2003). So, the CAPM models are designed on some extreme assumptions. To testify the validity of the models researchers have tested the model against the market data. Within this paper we will investigate some of those empirical researches.
4. Literature 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 the 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 hold the expected returns that are equal to the chance neutral interest. In that model, if we subtract the chance free rate from the expected market return, we get the beta premium. Conventionally the tests of CAPM derive from those three implications mentioned above.
4. 1 Tests on Risk Premiums
Most of the previous cross-section regression tests mainly concentrate on the Sharpe-Lintner model's findings about the idea and the slope term which studies the relationship between expected return and the marketplace 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 and the slope term means the difference between market interest and risk free interest rate.
There are some demerits of the study. To begin with, the estimated betas for individual assets are imprecise which creates the measurement error whenever we use them to describe average returns. Secondly, the error term in the regression has some common sources of variation which produces positive correlation one of the residuals. Thus the regression has the downward bias in the most common OLS estimate. Blume (1970) and Black, Scholes and Jensen (1972) worked on overcoming the shortcomings of Sharpe-Lintner model. Instead of working on the average person securities they worked on the portfolios. They combined the expected returns and market beta in a same manner that if the CAPM can make clear the security return, it can also describe portfolio return. As the econometric theory suggests, the estimated beta for diversified portfolios tend to be more accurate than the estimated beta for the individual security. Therefore, if we use the marketplace portfolio in the regression of average return on betas, it lessens the critical problem. However, grouping shrinks the number of estimated betas and shrinks the statistical power as well. To tackle this researchers sort securities to set-up 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 list of 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 predicated on the monthly data rather than estimating an individual cross-section regression of the average returns on beta. In this process 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 without risk interest rate.
Jensen (1968) noted that Sharpe-Lintner model also implies a period series regression test. According to Sharpe-Lintner model, the average realized CAPM risk premium explains the average value associated with an asset's excess return. The intercept term in the regression entails that "Jensen's alpha". Enough 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 means the chance free rate and the slope term indicates the expected market return in access of the chance neutral rate. For the reason 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 resistant to the restrictions imposed by the model.
In past several studies it has been confirmed that the partnership among average return and beta is too flat (Blume: 1970 and Stambaugh: 1982). With the low betas the frequent 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 marketplace rate. Fama and French (2004) observed that risk premium for beta (per unit) is leaner than the Sharpe-Lintner model and the partnership 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 an essential role in testing the validity of the CAPM.
In the analysis by Fama and Macbeth (1973), we can truly add pre-determined explanatory variables to the month wise cross section regressions of asset return on the market beta. Provided that all the dissimilarities in expected return are explained by the betas, the coefficients of any additional variable shouldn't be dependably not the same as zero. So, in the cross-section analysis the important thing is to carefully choose the additional variable. In this regard we can take the exemplory case of the study by Fama and MacBeth (1973). For the reason that work the excess variables are squared betas. These variables haven't any impact in explaining the common asset return.
By using the time series regression we can also test the hypothesis that market betas completely describe expected asset return. As we have mentioned previously that in enough time series regression analysis, the regular term is the difference between the asset's average return and the surplus return predicted by the Sharpe-Lintner model. We can not group assets in portfolios where in fact the regular term is dependably different from zero which applies only the model holds true. For instance, for a portfolio, the constant term for a higher earning to price ratio and low earning to price ratio should be zero. Therefore, in order to check 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 types of tests to see whether the constant term in the time-series regression is zero. However, it's very debatable to conclude about the best small sample properties of the test. Gibbons, Shanken and Ross (1989) developed an F-test for the regular term that gets 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 common value of any asset's excess return. Then we can test if the efficient set and the chance free asset is superior to that certain 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 describe the expected returns. The statistical test what's conventionally done is if the explanatory variables can identify the returns which are not explained by the marketplace betas. We are able to use the market proxy and the left hand side of the regression we can construct a test to see if the marketplace 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 may be constructed from it and the left hand side of that time period series regression found in the statistical test. Its noteworthy here that the left hand side of that time period series regression will not include all marketable assets and it is really very difficult to get the marketplace portfolio data (Roll, 1977). So, many researchers figured the chance of testing the validity of CAPM is not very encouraging.
From the first literatures, we can conclude that the market betas are sufficient enough to describe 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 created by Sharpe and Lintner that the risk premium beta comes from subtracting the risk free interest rate from the expected return is rejected. The attractive area of the black model is, it is easily tractable and incredibly 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 effect on the marketplace 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 your future returns on high earning-price ratios are significantly greater than the return in CAPM. Instead of sorting the stocks by E/P, if we sort it by market capitalization then your mean returns on small stocks are higher than the main one in CAPM (Banz, 1981) if we do the same by book-to-market equity ratios then the group of stocks with higher ratio gives higher average return (Statman and Rosenberg, 1980).
The ratios have been found in the above mentioned literatures associate the stock prices which involves the information about expected returns that are not captured by the market betas. The price of the stock will not solely be based upon the cash flows, rather it depends upon the present discounted value of the money flow. So, the several kind of ratios discussed above play a crucial role in analyzing the CAPM. Consistent with this Fama and French (1992) empirically analyzed the failure of the CAPM and figured all these ratios have impact on stock return which is provided by the betas. In a period series regression analysis they concluded a similar thing. In addition they observed that the partnership between the average return and the beta is even flatter following 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 constant with the findings of Fama and French (1992) implies that the contradictions of the CAPM associated with price ratios aren't sample specific.
5. Efficient Group 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 main 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 a few of the useful results from it (Roll, 1977).
Here we assume that there are N risky assets with a mean vector Ој and a covariance matrix О. Furthermore 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 perfect solution is to the next constrained optimization:
We solve this minimization problem by setting the Lagrangian function. Let's define the next:
The efficient frontier can be produced from any two minimum variance portfolios. Why don't we 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've the next:
The covariance of the asset return of the global minimum portfolio g and another portfolio as defined as a is as follows:
For a multiple regression of the return of a secured 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 effect. As. The result is obvious. So, we just need to show that
and. Why don't we assume that r be the minimum variance portfolio with expected return. From the minimization problem we can write the next:
Portfolio a can be expressed as a combo of portfolio r and an arbitrage portfolio which comprises 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 next minimization problem:
The treatment for the optimization problem is c=0. Another solution will contradict q from being the minimum variance.
Since, , thus taking the derivative provides following expression:
Setting the derivative add up to zero and by substituting in the answer 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 marketplace portfolio is the tangency portfolio then the intercept of the surplus 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 firmly taking the unconditional expectation we get,
As we've showed above, the weight vector of the marketplace 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 marketplace 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 clear 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 surplus return on some efficient market index at time t. The SCL can be written as follows:
Here is the regular term which represents the asset return (constant) and can be an estimated value of. We use this estimated value as an explanatory variable in the following cross-sectional regression:
Conventionally this regression is utilized to check for a confident risk return trade off. The coefficient of is significantly not the same as zero and it is assumed to be positive in order to hold the CAPM to be true. This also represents the marketplace price of risk. Whenever we test the validity of CAPM we test if is true estimate of. We also test if 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's important to assume that the returns are IID and jointly multivariate normal. The CAPM is very helpful in predicting stock return. We also assume that investors can borrow and lend at a risk free rate. Inside the Black version of CAPM we assume that zero-beta portfolio is unobservable and therefore 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 centered on individual security returns. The empirical email address details are off-putting. Miler and Scholes (1972) found some statistical setback faced when working with individual securities in analyzing the validity of the CAPM. Although, a few of the studies have overcome the issues by using portfolio returns. In the study by Black, Jensen and Scholes (1972) on New York stock exchange data, portfolios have been formed and reported a linear relationship between your beta and average excess portfolio return. The intercept approaches to be negative (Positive) for the beta higher 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 data of a larger intercept than the chance neutral rate. In addition they discovered that a linear relationship exists between the average returns and the beta. It has also been observed that this linear relation becomes more powerful when we utilize a dataset for an extended period. However, other subsequent studies provide weak empirical proof this zero beta version.
We have mixed findings about the asset return and beta relationship predicated on days gone by empirical research. In the event the portfolio used as market proxy is inefficient then the single factor CAPM is rejected. This is especially true if the proxy portfolio is inefficient by a little margin (Roll: 1977, Ross: 1977). Moreover, there is survivorship bias in the info 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 problems with the model specifications too. Amihud, Christen and Mendelson (1993) observed that we now have errors in variables and these errors have impact on the final outcome of the empirical research.
We experience less favourable evidence for CAPM in the late 1970s in the so called anomalies literature. We are able to think the anomalies as the farm characteristics that can be used to group assets to be able to truly have a high ex post Sharpe ratio in accordance with the ratio of the market proxy for the tangency portfolio. These characteristics provide explanatory power for the cross-section of the common mean returns beyond the beta of the CAPM which really is a contradiction to the prediction of CAPM.
We have previously mentioned that the first anomalies include the size effect and P/E ratio as we've mentioned previously. Basu (1977) observed that the portfolio formed on the basis of P/E ratio is better than the portfolio formed in line with the mean-variance efficiency. With a lesser P/E firms have higher sample average return and with high P/E ratio have lower mean return than will be the case if the market portfolio is mean-variance efficient. Alternatively the size effect implies that low market capitalization businesses have higher sample return than would be likely if the marketplace portfolio was mean-variance efficient.
Fama and French (1992, 1993) observed that beta cannot describe the difference between the portfolio formed predicated on ratio of book value of equity to the market value of equity. Firm has higher average return for higher book market ratio than actually predicted by the CAPM. However, these results signal economically deviations from CAPM. In these anomalies literatures, there are almost no motivations to review the farm characteristics. Thus there's a likelihood of overstating the evidence up against the CAPM since there are sample selection bias problem in estimating the model and also there is a issue of data snooping bias. This a kind of bias identifies the biases in drawing the statistical inference that comes 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 continues to be being widely used in finance. Addititionally there is the controversy exists about how exactly we ought to interpret the data against the CAPM. Some researchers often argue that CAPM should be replaced with multifactor model with different resources 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 lots of studies we have discovered that market data alone cannot make clear 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 essential 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 element in cross sectional analysis of the equity returns. Chung, Johnson and Schill (2001) discovered that the SMB and HML become statistically insignificant if higher order co-moments are contained in the cross sectional portfolio return analysis. We can infer from here that the SMB and HML can be viewed as as good proxies for the higher order co-moments. Ferson and Harvey (1999) made a point that lots of econometric model specifications are rejected because they may have the tendency of ignoring conditioning information.
Now we will show one of the very important results of the multifactor model. Let us look at 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 the regression will be zero and this factor regression coefficients for just about any asset will sum to unity. Let the number of the portfolios in the set be K which is the (Kx1) vector of time frame t of asset returns. For just about any value of the constant Ој, there exists a blend of portfolio and assets. Let us consider Ој 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 components of,
Then for the K portfolios we have,
By rearranging, we get the next,
Substituting this value into Ој returns the following:
Now let us look at 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 your econometric theory we have,
Again by substituting these expressions into Ој give,
That is the factor regression coefficients for each asset sum to unity. If then and we've a=o which indicates that the regression intercept will be zero for many assets including asset a.
As we have already proved that the intercept of the multifactor style of expected return is zero. For instance, let us consider the next regression,
Fama and French (1993) using the aforementioned model observed that the model captures most of the variations of portfolio returns when these portfolios derive from B/M equity and other price ratios that cause problems for CAPM. The estimated intercept term of the aforementioned regression can be used to calibrate the sensitivity of the stock prices to the new information. In 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 their state variables. The three factor model is not clear of the momentum effect either. Stocks that did well in past almost a year are historically tend to do well in future month too. We can certainly distinguish this momentum effect from the value effect captured by B/M equity and other ratios. The three factor model cannot make clear this momentum effect. Its noteworthy here that CAPM also does not do that.
Arbitrage pricing theory (APT) by Ross (1976) implies that we don't need the condition of man-variance optimization for any vendors. APT is more general than CAPM since it allows multiple risk factors. Moreover, APT will 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 are in need of some additional assumptions in order to check the validity of the idea. In APT we assume that markets are competitive and frictionless. The asset return producing process is as follows:
Ross (1976) implies that the absence of the arbitrage opportunity in large economy implies that,
Where Ој is the (Nx1) vector of the expected asset return, is zero-beat parameter which 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 merely an specification. To make it testable we need more restrictions on it. To obtain those restrictions we have to have additional structure to make the approximation exact. Among the restricts is the marketplace portfolio must be well diversified and the factors have to be pervasive. We can consider the marketplace portfolio be diversified if the proportion of any asset to overall economy asset is very insignificant. Certain requirements of factors be pervasive allows the investors to diversify their idiosyncratic risk without restricting the decision 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 precise 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. These are,
1) Factors are the portfolios of the underlying traded assets and there is a riskfree asset
2) Factors are the portfolios of the underlying traded assets and there is no risk free asset
3) Factors will be the portfolios of the traded assets
4) Factors will be 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 aid the precise factor pricing model is fixed. Among the strongest evidences originates from the testing by using dependent portfolios formed on market value of equity and B/M ratios. Furthermore, the multifactor model cannot clarify the size effect and B/M effect properly. However, the portfolios derive from 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 rather than three. They figured for stocks we need three factors but for bond portfolios we have to include five factors. Lehmann and Modest (1988) analyzed the sensitivity of the amount of the dependent variables included in the model. With fewer portfolios the p-values were low in the result which can be an issue of the power of test. By reducing the amount of portfolios and without deviating from the null hypothesis, we can boost the power of the test. This is true because now we require to check fewer restrictions than before.
The multifactor models give a very alternative to the single-factor CAPM but researchers using these models need to be alert to two pitfalls arise when factors are chosen to match the info without taking into consideration the economic theory. First of all, due to data snooping bias the model may over fit the data and if that is the case then your 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 nevertheless they will imply Sharpe ratios for factor portfolios that are too much for being consistent with an acceptable underlying market equilibrium model. However, we must await sufficient amount of new data become open 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 aren't sufficient enough to characterise the distribution completely. Thus it encourages the researchers to look for the bigger order co-moments. In practice we estimate the skewness (third moment) and kurtosis (fourth moment). In lots of studies researchers taken notice of 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 a protracted version of CAPM. Because 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 is becoming very helpful in measuring the value vulnerable. 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 identifying 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 donate to the chance 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 desire a positive premium in order to accept the downside risk. Alternatively a negative premium corresponds with the up market beta. This up market beta is considered as a more appropriate way of measuring 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 common risk neutral rate and zero. They specified three measures to recognize what constitute an along market. Those months for which the marketplace return exceeds that the average market return so when it is above the chance free rate or higher 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. Within 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) figured 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 your beta-expected return relationship becomes depending on the excess market return. From that study we that there exists a positive relationship between beta and ecpected return throughout a up market. In line with their study, Crombez and Vennet (2000) studied the conditional relationship between asset return and beta. They figured beta is a reliable 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 match 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 relates to the systematic co-skewness.
9. CAPM: Depending on time-varying volatility
Engle (1982) introduced the ARCH/GARCH process with which we can test enough time varying volatility of stock return. This approach has drawn a considerable focus on the recent CAPM literature. Although nearly convincing, this approach provides much better 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 surplus return is negative they allowed for a negative relationship of risk and return. They figured 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 sort of model allows the market volatility and asymmetrical response for beta to start to see the bad and the good news impact on the asset or portfolio returns. Galagedera and Faff (2003) did a report over a conditional three-beta model. They modeled that one as GARCH (1, 1) process and with regards to the size and nature of the volatilities, they defined three state of volatilities. From that study we see that for most of the marketplace portfolios, the betas aren't significantly not the same as zero.
Conclusion
The empirical evidences of Sharpe (1964) and Lintner (1965) version of CAPM have never been encouraging. However, allowing for the flatter trade from average asset return for market beta achieved some success but researches after late 1970s observed various other important variables such as different kind of price ratios, the momentum, size, etc having crucial effect on asset return. Thus, CAPM hold well conditionally. The textbook model often identifies estimate the price tag on the equity capital. This kind of model suggests to estimate market beta and also to combine with the chance premium to get an estimate of the price of the equity. But, we've already seen that the beta and average return relationship is a lot flatter than the Sharpe-Lintner version of CAPM. Which means 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 carry the CAPM to be true and when 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 make clear the asset return distribution. To make clear the non-normal asset return distribution we need the bigger order moments. Again, the empirical evidence in favour of the higher order moments is not unarguable. As the marketplace beta didn't clarify the cross-sectional variation in security returns, the multifactor model came in front. As we have mentioned previously, these models incorporated some important variables such as different price ratios, size, etc. Some authors argue that CAPM is overstated due to issue 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 idea of asset pricing theory, empirical evidences make its use in application arguable.
