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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 1
^{st} January 2007 to 31
^{st} 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.

In financial economics literature, CAPM (Capital Assets Pricing Model) is one of the most vital advancements. CAPM was introduced by Sharpe [

Researcher utilized different technique to analysis the CAPM for different equity markets in the different region of the world. The studies conducted by Black, Jensen and Scholes [

In real the returns are asymmetric or fat tail distribution, this information is motivated us to used higher moment (skewness and kurtosis) in finding the risk return relationship. Doan [

There were few work related to higher moment CAPM. Fang and Lai [

In this research, the effects of unconditional skewness and unconditional kurtosis will be examined in case of Karachi Stock Exchange 100 index firm. The degree of asymmetry of distribution is shown by skewness, where positive (negative) skewness represent distribution with asymmetric tail extending towards more positive (negative) values. If we ignore skewness risk in designing portfolio causes CAPM model devalue. Kurtosis depicts the relative peakness or flatness in return distribution. Kurtosis greater than 3 indicates that distribution is more flat compared to normal distribution. According to Hood, John, Nofsinger and Kenneth [

After brief introduction and review literature of higher moment CAPM, next we discuss research methodology and data description in Section 2. Result and discussion in Section 3 and last but not the least conclusion of the research study.

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 individual stock return rather than portfolios for taking analysis Kim [

It has been observed that most of the economics and finance time series data has not normally distributed Brown and Matysiak [

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.

According to CAPM, which developed by Sharpe and Linter [

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

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.

_{i} is white noise error term,

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 [

where the parameter β denotes the co-variance,

The slope coefficient of above first pass equation (cubic CAPM) or time series equation is used in second pass equation.

Company | Mean (%) | Standard Deviation (%) | Skewness | Kurtosis | Jarque Bera (normality test) |
---|---|---|---|---|---|

Abbot Laboratory | −0.21 | 3.67 | −1.05 | 207.18 | 47176.65 |

Al Abbas Suger Mills | 1.11 | 3.01 | −0.77 | 246.17 | 64811.01 |

Al Gazi Tractor | 0.08 | 2.35 | −1.71 | 176.15 | 26240.24 |

Atlas Battery | 0.68 | 4.76 | −1.31 | 140.71 | 25616.36 |

Atlas Honda | 0.02 | 1.24 | −1.01 | 150.1 | 21899.60 |

Attock Cement | 0.02 | 2.99 | −0.23 | 137.97 | 19385.41 |

Attock Petroleum | 0.07 | 3.59 | −20.32 | 762.13 | 615009.94 |

BATA | 0.12 | 5.37 | −6.60 | 363.55 | 138524.67 |

Buxly Paint | −0.04 | 3.37 | −11.90 | 338.88 | 120653.54 |

D.G khan Cement | 0.00 | 2.35 | −2.42 | 41.46 | 1599.21 |

Dewan cement | −0.07 | 4.70 | −0.21 | 79.03 | 6151.19 |

Dewan sugar | −0.06 | 5.25 | 0.15 | 44.37 | 1821.80 |

General Tyre | −0.02 | 3.29 | −0.03 | 132.52 | 17850.98 |

Gillite Pakistan | 0.01 | 3.47 | −0.19 | 181.52 | 33916.07 |

Glaxo Smith | −0.02 | 2.12 | −2.61 | 51.51 | 2533.42 |

Gul Ahmed | −0.03 | 3.15 | −0.78 | 214.92 | 47792.32 |

Habib Sugar | 0.04 | 4.12 | −0.23 | 372.11 | 144983.03 |

Hino Pak Motor | 0.02 | 4.65 | −10.51 | 385.53 | 156187.66 |

Honda Atlas Car | −0.05 | 4.45 | −1.83 | 149.75 | 22932.77 |

ICI Pakistan | 0.03 | 2.24 | −6.33 | 160.27 | 26490.82 |

Indus Motor Ltd. | 0.02 | 2.18 | −4.16 | 93.89 | 8864.82 |

Ittehad Chemical | 0.01 | 2.85 | −1.81 | 114.48 | 13239.69 |

Japan Power | −0.02 | 4.78 | 0.06 | 47.86 | 2141.34 |

Johanson & Philips | 0.04 | 6.16 | −4.37 | 262.29 | 71623.85 |

Kohinoor Textile | −0.04 | 5.85 | 0.38 | 149.35 | 22791.87 |

Lakson Tobbaco | 0.03 | 4.36 | 0.19 | 272.96 | 77552.51 |

Mitchall Fruits | 0.00 | 2.82 | −14.37 | 483.39 | 246459.16 |

Milat Tractor | 0.04 | 2.00 | −0.36 | 4.27 | 2.25 |

Nishat Mills | 0.05 | 3.22 | −1.80 | 261.01 | 70855.75 |

Nestle Pakistan | 0.05 | 1.78 | 0.01 | 3.98 | 1.03 |

OGDC | 0.00 | 1.91 | −0.19 | 5.50 | 6.81 |

Pak Suzuki | 0.04 | 1.64 | −0.47 | 10.11 | 54.69 |

Pak Petroleum Ltd | −0.08 | 4.85 | 0.30 | 119.08 | 14340.57 |

PIA | −0.02 | 1.90 | −0.03 | 3.41 | 0.18 |

PTCL | 0.00 | 1.74 | −2.10 | 45.33 | 1925.53 |

Shall Pakistan | −0.01 | 3.77 | −4.40 | 118.50 | 14278.85 |

Singer Pakistan | −0.06 | 3.99 | 0.32 | 51.61 | 2515.39 |

Southern Electric | 0.02 | 9.40 | −0.25 | 269.35 | 75495.33 |

Samin Textile | 0.05 | 2.79 | −1.77 | 72.25 | 5116.99 |

Siemens | −0.01 | 1.95 | −0.04 | 3.78 | 0.66 |

SNGC | −0.02 | 1.99 | 0.08 | 5.27 | 5.50 |

SSGC | 0.02 | 1.23 | 0.12 | 2.32 | 23.45 |

Significance level 5%.

daily returns are varied from −0.08% (Pakistan Petroleum) to 1.11% Adam Sugar mills. The skewness ranger from −20.3 (Attock Petroleum) to 0.32 (Singer Pakistan). Excess Kurtosis could be as high as 762 (attock Petroleum) ranging from 3.41 (PIA). 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 [

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.

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.

Variable | Coefficients | Standard Error | t statistics | Adjusted R-Square |
---|---|---|---|---|

Constant | 0.032 | 0.00384 | 8.33 | 0.021 |

Market Beta | 0.064 | 0.0325 | 1.97 |

Significance level 5%.

Variable | Coefficients | Standard Error | t statistics | Adjusted R-Square |
---|---|---|---|---|

Constant | 0.021 | 0.00394 | 5.33 | 0.106 |

Market Beta | 0.045 | 0.413 | 0.11 | |

Skewness | 0.048 | 0.018 | 2.67 |

Significance level 5%.

Variable | Coefficients | Standard Error | t statistics | Adjusted R-Square |
---|---|---|---|---|

Constant | 0.02 | 0.0016 | 12.47 | 0.09 |

Market Beta | 0.037 | 0.1721 | 0.215 | |

Kurtosis | 0.021 | 0.0057 | 3.67 |

Significance level 5%.

Variable | Coefficients | Standard Error | t statistics | Adjusted R-Square |
---|---|---|---|---|

Constant | −0.254 | 0.0181 | 7.89 | 0.167 |

Market Beta | 0.014 | 0.1312 | 2.09 | |

Skewness | 0.256 | 0.0670 | 3.21 | |

Kurtosis | 0.012 | 0.0154 | 3.99 |

Significance level 5%.

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

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.

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.

Irfan Lal,Muhammad Mubeen,Adnan Hussain,Muhammad Zubair, (2016) An Empirical Analysis of Higher Moment Capital Asset Pricing Model for Karachi Stock Exchange (KSE). Open Journal of Social Sciences,04,53-60. doi: 10.4236/jss.2016.46006