Market Risk Premium Violations in Asset Pricing Models – A Higher Order Moments Approach

Market Risk Premium Violations in Asset Pricing Models – A Higher Order Moments Approach Pankaj Kumar Gupta Centre for Management Studies, JMI University New Delhi, India Prabhat Mitt

Market Risk Premium Violations in Asset Pricing Models

– A Higher Order Moments Approach

Pankaj Kumar Gupta

Centre for Management Studies, JMI University New Delhi, India

Prabhat Mittal

University of Delhi, India

Nabeel Hasan

Centre for Management Studies, JMI University New Delhi, India

Abstract

Conventional asset pricing models like Capital Asset Pricing Model (CAPM) are not efficient in estimating return on traded assets in various emerging markets including India Non-normality of returns distributions coupled with investors desire to maximize returns in volatile markets has accentuated the need for modeling portfolios based on higher order moments like skewness and kurtosis We examine the relevance of higher moments in selection of portfolios in Indian stock markets using weekly returns of 100 stocks listed on Bombay Stock Exchange for the period April, 2012 to March, 2017 that includes the volatile periods and captures major fundamental events Results of the optimization and higher moments regression models indicate that investors expect a high return to compensate them for additional risk of holding equities and place negative market risk premium for systemic variance The investors in Indian stock market are demanding negative risk premiums for market risk in terms of variance while they demand positive (negative) risk premium for positive (negative) skewness Our results are therefore opposite to the basic propositions of Modern Portfolio Theory (MPT) We also establish that Indian investors are highly risk averse to the effect of systematic kurtosis

Keywords: Portfolio Optimization, Higher Order Moments, CAPM, Skewness, Kurtosis

JEL Classification: G11, D53, C10

1 Introduction

Harry Markowitz in his landmark theory (1952) established a relationship between risk and return preferences among the investors Markowitz theory was further extended by Sharpe (1965) and Linter (1966), which established a linear relationship between the market risk and return contributed by individual security

or portfolio In recent years the Capital Asset Pricing Model has been finding inconsistent with several empirical models Banz (1981) shows an inverse relationship between the size of the firm and return, likewise Fama and French (1992) established the relationship between expected returns with the ratio of book to market value

The effects of skewness and kurtosis on the pricing of assets have been analyzed in several studies Ingersoll (1975), Kraus and Litzenberger (1976), Brocket and Kahane (1992), Campbell and Siddiqui (2000) incorporated the effect of higher moments by extending the Capital Asset Pricing Model (CAPM)

Several studies have been conducted in developing countries to study the impact of higher moments Javid (2009), Hasan, Kamil, Mustafa and Baten (2013), Tang and Shum (2003) The Sharpe-Linter (CAPM) has been come up with mixed findings done by several researches in the past Several studies like Friend and Blume

(1970), Black et al (1972), Fama and Macbeth (1973) find inconsistency in their empirical analysis of traditional

Sharpe- Linter model It is seen that in these studies the intercept has been on a higher side and slope lower than expected in capital asset pricing model

Kraus and Litzenberger (1976) analyzed a three moment asset pricing model in which coskewness and covariance explains the expected returns for market risk They find that there is a significant relationship between the coskewness and covariance and expected returns and the overall model explain the risk and return relationship better than two moments CAPM Similarly, Fang and Lai (1997) further extended the model

to four momemt They found that the investors are rewarded with excess return for taking systematic kurtosis risk in the market

The results for higher moment asset pricing model in developing world are mixed Javid (2009) found that higher moments perform well in explaining risk and return relationship in Pakistan stock market but higher moments have marginal role in explaining asset price It is seen that conventional asset pricing models like Capital Asset Pricing Model (CAPM) are not efficient in estimating return on traded assets in various emerging markets including India Non-normality of returns distributions coupled with investors desire to maximize returns in volatile markets has accentuated the need for modeling portfolios based on higher order moments

like skewness and kurtosis Hasan et al (2013) also find that coskewness and cokurtosis risk is rewarded in

emerging markets like Bangladesh In an Indian context, there are few studies conducted that primarily relate

to periods before the financial crisis

We find motivation to investigate if there is any impact of systematic skewness and systematic kurtosis on the price of traded assets Since, skewness is concerned with the degree of symmetry of an asset returns around its mean value Investors prefer assets with positive skewness Kurtosis explains the relative peakedness of an asset returns Investors are averse to extreme deviations and therefore avoid high kurtosis

2 Methodology

We have used the four moment asset pricing model proposed by Fang and Lei (1997) We assume that there are N risky assets where R = A (N x 1) is a vector of returns of N risky assets; Re = A (N x 1) vector of expected returns The assets are assumed to have limited liability and returns are received in the form of capital gains We assume capital markets are perfectly competitive with absence of taxes and transactions cost The investors are assumed to be maximizing their utilities defined by the moments - mean, variance, skewness and kurtosis

of the terminal wealth subject to budget constraints An investor invests x i of his wealth in the i th risky asset,

and 1 - Σx i in the risk free asset The moments are 𝑋′(𝑅̅ − 𝑅𝑓), 𝑋′𝑉𝑋, 𝐸 [𝑋′(𝑅 − 𝑅̅)/√𝑋′𝑉𝑋]3 𝑎𝑛𝑑 [𝑋′(𝑅 − 𝑅̅)/

√𝑋′𝑉𝑋]4 where 𝑋′= (x1, x2, x3,…, xn) is N x 1 vector of holding in risky assets They argue that the investor’s performance can be defined as the function the mean, variance, skewness and kurtosis subject to unit variance

R̅ - Rf = Φ1Cov(Rm, R) + Φ2 Cov(Rm2, R) + Φ3Cov(Rm3, R)

Fang and Lai (1997) rearrange the equations to make linear empirical version of four moments CAPM as

Rei - Rf = b1βi + b2γi + b3δ, i = 1,2, n ,

Where

Rei is the expected rate of return on the i th asset

βi is the systematic variance of i th asset

γi is the systematic skewness of i th security

δi is systematic kurtosis of the i th asset

Parameters b1, b2, b3 are market premiums for respective risks The cubic market model equation which is consistent with four moment CAPM is

Rit = αi + βiRmt - γiR2mt + δiR3mt + εit ; i = 1, 2, n and t = 1,2, T w βi, γi , and δi are multiple regression coefficients identical to the parameter in equation According to utility theory

b 1 > 0 as higher variance is connected with higher probability of uncertain outcome

b 2 has opposite sign of market skewness

b 3 > 0 as positive kurtosis can increase extreme outcomes

We have applied the Fama Macbeth two step regression models to calculate the risk premium from exposure to higher moments The regression follows two steps – First, stock returns are regressed against market returns wherein factor exposures βi, γi , and δi are estimated using t regressions

Rit = αi + βiRmt + γiR2mt + δiR3mt + εit

Second, the T cross sectional regression is run for each time period to calculate risk premium

Rei - Rf = b1βi + b2γi + b3δ

The coefficients b1 , b2 , b3 are thus obtained

The data set consist of One hundred securities listed on Bombay Stock Exchange and come from all diversified sectors The data used in the analysis consist of weekly returns for 5 years from April, 2012 to March, 2017 The security prices were obtained from Yahoo Finance We have used R programming framework to develop the necessary algorithms for analysis of large scale data representing the weekly returns

of 100 selected stocks The time-series for analysis is divided into three periods using the structural breaks method in order to avoid time varying effect in our analysis

3 Results and Discussion

We have conducted an analysis of the whole sample period from April 2012 to March 2017 broken into sub period based on the structural breaks (Figure 1) The derived sub-periods are (a) April, 2012 to May, 2014, (b) May 2014 to July 2016 and (c) July 2016 to March 2017 In these periods the Residual sum of Square is quite low The break points were not chosen to be more than two because more breakpoints will divide the data into highly unequal time periods that were unfavorable for performing analysis

Figure 1 – Structural Breaks Analysis

Figure 2 – Observed RSS

The higher moments of data of hundred stocks is given in Appendix A In our data, the mean return vary between -0.56 to 1.35 The mean returns were found to be 0.37 for 100 securities The variance of the security varies between 8.55 to 76.37 (excluding the effect of outliers The mean variance for the data found to be 88.887 The negative skewness in the data varies between -1.19 to -0.0019 while the positive skewness varies between 0.018 to 4.39 The mean skewness for the data is 0.4 The kurtosis varies between 2.992 to 12.799 excluding outliers The overall moments values are given in Appendix B

It was impossible to observe real market portfolio Therefore a market portfolio proxy is assumed to be BSE

100 The data for BSE 100 consist of 260 observations of weekly returns The moments for market portfolio can

be observed in Appendix B The risk-free rate1 is calculated using data from Reserve Bank of India database

Figure 3- Derived Risk Free Rate using GOI Bond Yields

We derive the value for higher moments as follows

Table 1 – Higher Order Moments (April 2012 to May 2014)

𝑅𝑒𝑖− 𝑅𝑓 = 𝑏0+ 𝑏1𝛽𝑖+ 𝑏2𝛾𝑖+ 𝑏3𝛿

For sub period April 2012 to May 2014(Table 1) the R2 value for all moments show very poor results that can be attributed to extreme market movements in the given period The multiple R2 value is highest in the four moment model while lowest in two moment model The risk premium b1 for systematic variance found

to be negative while risk premium for systematic skewness were positive (it should be of opposite sign of market skewness) The kurtosis is found to have a positive premium

Table 2 – Higher Order Moments (May, 2014 - July, 2016)

𝑅𝑒𝑖− 𝑅𝑓 = 𝑏0+ 𝑏1𝛽𝑖+ 𝑏2𝛾𝑖+ 𝑏3𝛿

For sub period (Table 2) May, 2014 to July, 2016 the multiple R squared value is 0.619 for four moment model while Multiple R squared value is 0.442 and lowest for the two moment CAPM model which is around 0.408 The risk premium b1 for systematic variance is negative while risk premium for systematic skewness b2

is negative The risk premium for systematic kurtosis was positive

Table 3 – Higher Order Moments (July 2016 – March, 2017)

𝑅𝑒𝑖− 𝑅𝑓 = 𝑏0+ 𝑏1𝛽𝑖+ 𝑏2𝛾𝑖+ 𝑏3𝛿

For sub period (Table 3) July 2016 to March 2017 the Multiple R squared value is again for four moments CAPM while it is low for the two moment asset pricing model The risk premium for systematic variance b1 is negative and for systematic skewness b2 is also negative while systematic kurtosis b3 it is found to be positive

6 6.5 7 7.5 8 8.5 9 9.5

Table 4 – Higher Order Moments (Full Period April 2012- March, 2017)

𝑅𝑒𝑖− 𝑅𝑓 = 𝑏0+ 𝑏1𝛽𝑖+ 𝑏2𝛾𝑖+ 𝑏3𝛿

In Table 4 we can observe that the Multiple R squared value is highest for four moment asset pricing model while the Multiple R squared value for three moment asset pricing model is 0.261 and for two moment model

it is 0.251 From the result of overall period we find that the skewness marginally improve the asset pricing model but the once the effect of kurtosis is also incorporated the efficiency of asset pricing model increases dramatically Our findings are inconsistent with the findings of Kraus and Lichtenberger (1976) The investors

in Indian stock market are demanding negative risk premiums for market risk in terms of variance while they demand positive (negative) risk premium for positive (negative) skewness However, our findings for risk premium for systematic kurtosis are consistent with the finding of Fang and Lai (1997)

4 Conclusion

The two moments Capital Asset Pricing Model (CAPM) is inadequate for finding return in an asset The investor demand premium for higher moments The possible explanation for negative risk market risk premium for systematic variance can explain by the argument that during the period of our analysis India Stock Market boomed rapidly The equity investor expects rapid growth earning for the stock market to compensate them for additional risk of holding equities This would result in the bidding up for share prices and a consequent decline in the equity risk premium One of the unique findings in our research is that Indian investors are highly risk averse to the effect of systematic kurtosis Investor demands higher returns when the market shows extreme deviations in terms of market returns The phenomenon of skewness is still unexplained from our research and needs further in depth analysis to come up with an argument to explain

it

References

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Appendix A

Moment Value of Individual BSE 100 Stocks

JSWSTEEL_BO 0.588729224 21.9259236 0.700873963 4.03549925

Overall Moments

Appendix B