(Luận văn) 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

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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)

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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

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

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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

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

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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

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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

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)

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

is negative The risk premium for systematic kurtosis was positive

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

For sub period (Table 3) July 2016 to March 2017 the Multiple R squared value is again for four moments

6 6.5 7 7.5 8 8.5 9 9.5

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

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

Banz, R.W (1981), "The Relationship between Return and Market Value of Common Stocks", Journal of Financial Economics, Vol 9, pp

3-18

Brockett, Patrick L and Kahane, Yehuda (1992), "Risk, Return, Skewness and Preference", Management Science, Vol 6

Campbell, R Harvey and Siddiue, Akhtar (2000), "Conditional Skewness in Asset Pricing Tests", The Journal of Finance, Vol LV, No 3

Cox, John, Jonathan Ingersoll, and Stephen Ross “An Intertemporal General Equilibrium Model of Asset Prices.” Econometrica, Vol 53,

pp.363-384

F Black, M Jensen and M Scholes (1972), “The Capital Asset Pricing Model: Some Empirical Results," Studies in the Theory of Capital

Markets, M Jensen (ed.), New York: Praeger

Fama, E., and French, K R (1995), "The Cross-Section of Expected Stock Returns", Journal of Finance, Vol 47, No 2, p.427-465

Fama, Eugene F and James D MacBeth (1973), “Risk, Return and Equilibrium: Empirical Tests”, Journal of Political Economy, Vol 81,

No.3, pp 607–36

Fang, H and T Y Lai (1997), “Co-Kurtosis and Capital Asset Pricing”, The Financial Review, Vol 32, pp 293–307

I Friend and M Blume (1970), "Measurement of Portfolio Performance Under Uncertainty," American Economic Review

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Appendix A

Moment Value of Individual BSE 100 Stocks

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JSWSTEEL_BO 0.588729224 21.9259236 0.700873963 4.03549925

Overall Moments

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Appendix B

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Tiêu đề	Market risk premium violations in asset pricing models – a higher order moments approach
Tác giả	Pankaj Kumar Gupta, Prabhat Mittal, Nabeel Hasan
Trường học	Centre for Management Studies, JMI University
Chuyên ngành	Asset Pricing and Financial Markets
Thể loại	Thesis
Năm xuất bản	2017
Thành phố	New Delhi

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Số trang	10
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