An Empirical Analysis of Higher Moment Capital Asset Pricing Model for Karachi Stock Exchange (KSE)

The purpose behind this study is to explore the relationship between expected return and risk of portfolios. It is observed that standard CAPM is inappropriate, so we introduce higher moment in model. For this purpose, the study takes data of 60 listed companies of Karachi Stock Exchange 100 index. The data are inspected for the period of 1st January 2007 to 31st December 2013. From the empirical analysis, it is observed that the intercept term and higher moments coefficients (skewness and kurtosis) are highly significant and different from zero. When higher moment is introduced in the model, the adjusted R square is increased. The higher moment CAPM performs cooperatively perform well


Introduction
In financial economics literature, CAPM (Capital Assets Pricing Model) is one of the most vital advancements. CAPM was introduced by Sharpe [1], Lintner [2] and Mossin [3]. It was first development of meanvariance CAPM, which identified the expected return on portfolio is linearly related to market based or systematic risk.

Data Description
The data utilized in this study consist of 60 non financial firms for the period of 1 st January 2007 to 31 st December 2013 (daily data).
The rate of return of each stock or equity was calculated as follow ( ) where P t is closing price at period t, P t−1 is closing price at period t − 1, ln is natural log. In this study we use in-dividual stock return rather than portfolios for taking analysis Kim [29]. For proxy of market portfolio KSE-100 index return used. The proxy of risk free return is 3 months T-Bills of government of Pakistan.

Normality Test of Returns
It has been observed that most of the economics and finance time series data has not normally distributed Brown and Matysiak [30]. In the same line most of the stock return are observed fat tails more peak than normal distribution Bekaret and Harvey [31]. The causes of non normal distribution of stock return is that due to illiquidity, lack of divisibility and low information of transparency Ranaldo and Favre [18].
To check the normality of a sample's distribution, the prominent test: Jarque-Bera test was considered in this research. The Jarque-Bera test for normality is now presented by considering the following null hypothesis To analysis the normality in data of stock return, the study use Jarque-Bera test, which most prominent test of normality. The Jarque-Bera test for normality is set following hypothesis. Ho = Return follows the normal distribution. H1 = Return do not follows the normal distribution.
where n is number of observation. S is the Skewness and K is the excess kurtosis. The test follow the chi square distribution with two degree of freedom.

Estimation of Mean Variance CAPM
According to CAPM, which developed by Sharpe and Linter [1] return can be elucidate as follows where R it is the rate of return of i th firm at time t, R ft is a risk free rate of return at time t. R mt is the rate of return on the market index at time t and i β is firm beta of company, which is co-variance of market return and individual firm return divided by variance of market return. First of all we regress following equation to determine systematic risk. It is also known as first pass equation.

( )
where e it is the white noise error term in the above CAPM regression model at time t. Above equation is estimated by using OLS (ordinary least square) method. In second stage, we run second pass equation as follows.
r refer to average excess returns of individual firm over the sample period. Β is the estimate of the systematic risk or market risk of individual firm, which obtained from first pass equation. e i is white noise error term, 0 γ and 1 γ are parameter of second pass equation.

Estimation of Higher Moment CAPM
The result of JB normality test shows that stock returns are distributed asymmetric and leptokurtic, so the mean variance CAPM is inappropriate because it cannot capture co-skewness (third moment) and co-kurtosis (fourth moment) factors. As suggested by Kraus and Litzenberger [16], Homaifar & Graddy [32] and Hussain [33] the following equation used to capture higher moment.
where the parameter β denotes the co-variance, i δ shows co-skewness and i κ is co-kurtosis of stock i which are time series regression coefficient of first pass equation.
The slope coefficient of above first pass equation (cubic CAPM) or time series equation is used in second pass equation. Table 1 reported the first four moments of daily stock returns of 60 non financial firms. It is noted that average . In above table skewness shows that out 60 firms only 9 firms have positively skewed. The excess kurtosis column shows that the behavior of the firms is leptokurtic, which means that the curve was relatively more peaked than normal curve. These findings are consistent with the finding of Mandelbrot [34], Mandelbrot and Taylor [35], Campbell [36] and Md Zobear [37] as they identified that stock return exhibit fat tails distribution. The result of JB test shows that only 6 firms returns are normally distributed out of 60 firms. The main features of the KSE data are that returns were positive, volatile, asymmetric and fat tails. According CAPM model the intercept term or constant term insignificant and should not be difference from zero and there is positive relation or trade of between risk and return. Table 2 shows that constant term statistically significant, which indicated that important variables are missing. Also there are slightly positive relationship between market beta or market risk and return, our result are in line of Thomas [38] and Mecangni and Sourial [39] found positive relationship between risk and return. Hence based on the intercept criterion, the CAPM hypothesis is rejected in case of KSE.

Result and Discussion
To analysis the effects of higher moment of CAPM model, 3 rd and 4 th moment were incorporated in CAPM model. The results of higher CAPM model is reported in Tables 2-5.
The results show that the coefficient of variance, skewness and kurtosis are positive and significant. All investor are compensated in higher expected return for taking the systematic variance, skewness and kurtosis risk.     The coefficient of kurtosis is a positive investment incentive. A positive kurtosis coefficient means that the asset is adding kurtosis to the market portfolio or vice versa. The result of Table 4 and Table 5 shows that the risk premium for kurtosis was significant and shows expected sign as portfolio return are positive correlated with kurtosis. The finding of our research indicated that higher kurtosis is compensated by higher portfolio's returns.
The introducing of higher moment (skewness and kurtosis) as additional explanatory component in the regression of portfolio's returns. The finding suggest that CAPM model is not linear its non-linear. After introducing skewness and kurtosis, the adjusted R square was increase 0.021 to 0.167. The model with skewness was better than the model with kurtosis because it exhibited better performed.

Conclusion
The paper analyzes the importance of higher moment (skewness and kurtosis) of returns distribution in capturing the variation of average stock returns for companies listed in the KSE. The finding of the study shows that standard CAPM is unable to capture assets return efficiently. The JB test of normality shows that stock returns of KSE not normally distributed. The investor concerns about the higher moment of returns. Our study supports strongly the inclusion of terms represents skewness and kurtosis. The study also showed that after inclusion of higher moments in the model, the adjusted R square increased, which also supported higher moment in KSE. Therefore, we concluded that higher moment CAPM was more superior to Sharpe and Linter standard CAPM model. It is important for future research to design theoretical model which in-corporate higher moment in CAPM model.