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 Mittal U

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)

62

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

′

3

′

]

√ ′ ]4 where ′ = (x 1 , x 2 , x 3,…, x n ) 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 because of the relative percentage invested in different assets, the portfolio can be re scaled Increase in asset mean and skewness of terminal wealth increases investors utility and increase in kurtosis of terminal wealth corresponds to increase in the probability of extreme deviations of terminal wealth which can result in either extreme gain or loss to investor Therefore kurtosis has negative impact

on the utility of the investor We wish to

{ ′ ( − ), [ ′ ( − )] 3 , [ ′ ( − )] 4 − λ[ ′ − 1]}

where λ is a langrangian multiplier for unit variance constant A separation theorem which all investors holds same probability beliefs and has identical wealth coefficients is employed (Cox, Ingersoll and Ross, 1985) The asset pricing model with skewness and kurtosis can thus

be derived as

follows-63

R - Rf = Φ1Cov(Rm, R) + Φ2 Cov(Rm 2, R) + Φ3Cov(Rm 3, 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 for 10 year Government bond yield between periods April

2012 to March 2017(Figure 3)

1 R weekly = R f /52

65

Figure 3- Derived Risk Free Rate using GOI Bond Yields

9.5 9 8.5 8 7.5 7 6.5 6

We derive the value for higher moments as follows

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

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)

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)

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

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 Javid, Attiya Yasmin (2009), “Test of Higher Moment Capital Asset Pricing Model in Case of Pakistani Equity Market, European Journal of Economics, Finance and Administrative Studies, Vol No 15.

Kraus, Alan and Litzenberger, Robert (1976), "Skewness Preference and the Valuation of Risk Assets", The Journal of Finance, Vol XXXI, No 4.

Lintner, J (1965), “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budget,” Review of Economics and Statistics, Vol 47, No 1, pp.13-37.

Markowitz, H (1952), “Portfolio Selection,” Journal of Finance, Vol.7, No 1, pp 77-91.

Sharpe, W F (1964) “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,” Journal of Finance,

Vol 19, No 3, pp.425-442.

Tang, G and W Shum (2003), “The Conditional Relationship between Beta and Returns: Recent Evidence from

International Stock Markets,” International Business Review, Vol.12, No.1, pp.109-126.

67

Appendix A

Moment Value of Individual BSE 100 Stocks

ADANIPORTS_BO 0.374974967 30.96674815 -0.065646993 4.049022272

APOLLOTYRE_BO 0.369218471 31.11679249 -1.190136973 8.779627671

BAJAJ_AUTO_BO 0.253859993 10.26189033 -0.138877241 2.992001777 BAJAJELEC_BO 0.189138371 29.17083735 -0.569501837 5.117954376 BALRAMCHIN_BO 0.389479027 33.15624892 0.239057279 3.738660682 BANKBARODA_BO 0.214211309 30.87454662 -0.039901552 3.873528584 BANKINDIA_BO -0.336363023 35.64228339 0.080235865 3.957558938

BHARATFORG_BO 0.502525687 18.75896681 0.175130977 3.554918273 BHARTIARTL_BO 0.030639156 15.58509812 0.224180133 3.731084213

BLUESTARCO_BO 0.511908078 20.15791096 0.573031937 4.449421496

CENTURYPLY_BO 0.563136329 42.60634787 0.269795888 5.473909888

INDUSINDBK_BO 0.567116342 15.21957671 0.083329796 3.974389115

JAICORPLTD_NS -0.051440625 48.07833557 0.188821798 4.144239821

JINDALSTEL_BO -0.560776078 46.40203587 0.039335069 4.784064952 JSWENERGY_BO 0.052483102 35.93926977 -0.128064872 3.431756369

68

JSWSTEEL_BO 0.588729224 21.9259236 0.700873963 4.03549925

LICHSGFIN_BO 0.359313264 18.85260988 -0.198908945 3.506383089

NATCOPHARM_BO 0.984211687 38.53290552 1.017457683 8.393531656

NETWORK18_BO -0.034617572 36.20376308 0.902214503 5.286789667

PIDILITIND_BO 0.571070473 13.03015142 0.435377215 4.080213183 POWERGRID_BO 0.259554038 9.376286357 -0.543930112 7.878002611

RAJESHEXPO_BO 0.599279429 35.97358095 0.765357793 6.181437681

SRTRANSFIN_BO 0.241644527 24.57597453 -0.223761757 4.336271647

TATAMOTORS_BO 0.213924132 22.15998992 0.056541218 4.500278272

Overall Moments