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 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 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 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 xi of his wealth in the ith risky asset, and - Σxi in the risk free asset The moments are � ′ (�̅ − �� ), � ′��, � [� ′ (� − �̅ )/√� ′ ��] √� ′ ��� [�′(� − �̅ )/ ′ where � = (x1, x2, x3,…, xn) is N x 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- 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 ith asset βi is the systematic variance of ith asset γi is the systematic skewness of ith security δi is systematic kurtosis of the ith 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 - m + δiR3mt + εit ; i = 1, 2, .n and t = 1,2, T w βi, γi , and δi are multiple regression γiR2 coefficients identical to the parameter in equation According to utility theory b1 > as higher variance is connected with higher probability of uncertain outcome b2 has opposite sign of market skewness b3 > 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 m Rit = αi + βiRmt + m + δiR3 + εit γiR2 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 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 timeseries for analysis is divided into three periods using the structural breaks method in order to avoid time varying effect in our analysis 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 – Structural Breaks Analysis Figure – 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 Rweekly = Rf /52 Figure 3- Derived Risk Free Rate using GOI Bond Yields 9.5 8.5 7.5 6.5 1-Apr-12 1-Apr-13 1-Apr-14 1-Apr-15 1-Apr-16 We derive the value for higher moments as follows Table – Higher Order Moments (April 2012 to May 2014) Estimator ��� − �� = �0 + �1�� + ��� − �� = �0 + �1�� + Kurtosis 2.591 Skewness 0.0631 ��� − �� = �0 + �1�� �� �� �� �� �� 0.296 -0.103 1.131 -0.027 0.180 0.299 -0.180 0.129 * 0.145 0.303 -0.147 * * 0.138 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 – Higher Order Moments (May, 2014 - July, 2016) Estimator ��� − �� = �0 + �1�� + ��� − �� = �0 + �1�� + Kurtosis 3.700 Skewness -0.166 ��� − �� = �0 + �1�� �� �� �� �� �� 0.369 -0.126 -0.021 0.009 0.619 0.371 -0.096 -0.016 * 0.442 0.366 -0.147 * * 0.408 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 – Higher Order Moments (July 2016 – March, 2017) Estimator ��� − �� = �0 + �1�� + ��� − �� = �0 + �1�� + ��� − �� = �0 + �1�� Kurtosis 2.386 Skewness -0.373 �� �� �� �� 0.016 0.323 -0.231 0.013 -0.129 -0.303 0.230 * 0.863 0.767 0.433 -0.090 * * 0.712 �� 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 – Higher Order Moments (Full Period April 2012- March, 2017) Estimator ��� − �� = �0 + �1�� + ��� − �� = �0 + �1�� + ��� − �� = �0 + �1�� Kurtosis 3.254 Skewness -0.139 �� �� �� �� 0.445 -0.308 -0.236 0.045 0.432 0.457 -0.164 -0.248 * 0.261 0.419 -0.175 * * 0.251 �� In Table 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) 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 Campbell, R Harvey and Siddiue, Akhtar (2000), "Conditional Skewness in Asset Pricing Tests", The Journal of Finance, Vol LV, No 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 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 Appendix A Moment Value of Individual BSE 100 Stocks Company ABB_BO ACC_BO ADANIPORTS_BO AEGISLOG_BO AMBUJACEM_BO APOLLOTYRE_BO ASHOKLEY_BO AXISBANK_BO BAJAJ_AUTO_BO BAJAJELEC_BO BALRAMCHIN_BO BANKBARODA_BO BANKINDIA_BO BASF_BO BATAINDIA_BO BEL_BO BHARATFORG_BO BHARTIARTL_BO BHEL_BO BLUESTARCO_BO BPCL_BO CEATLTD_BO CENTURYPLY_BO COALINDIA_BO CONCOR_BO COROMANDEL_BO CROMPTON_BO DABUR_BO DALMIABHA_BO DLF_BO EICHERMOT_BO ELGIEQUIP_BO EMAMILTD_BO ESCORTS_BO ESSELPRO_BO EXIDEIND_BO FEDERALBNK_BO GAIL_BO GEPIL_BO GODREJCP_BO GRASIM_BO GREENPLY_BO GSKCONS_BO HDFC_BO HEROMOTOCO_BO HEXAWARE_BO HINDALCO_BO HINDPETRO_BO HSCL_BO IBREALEST_BO ICICIBANK_BO IDEA_BO INDUSINDBK_BO IOC_BO ITC_BO JAICORPLTD_NS JBFIND_BO JINDALSTEL_BO JSWENERGY_BO JSWHL_BO Mean 0.16973847 0.05963621 0.37497496 1.35416998 70.1614404 0.36921847 0.41621732 0.39213933 0.25385999 0.18913837 0.38947902 0.21421130 0.38387268 60.1520104 1.25076465 0.50252568 0.03063915 -6 0.15109353 0.51190807 0.68438865 1.02179388 0.56313632 0.07644063 0.31129494 0.0945414 0.58621798 0.38997152 1.01230314 -8 0.09742259 0.95563940 0.40327699 0.53820530 0.78063675 0.85657759 0.19754320 50.4984850 0.1754157 0.21425843 0.49195862 0.33905456 0.78189478 0.26757588 0.33921130 0.23540971 0.30909620 0.16780735 1.00691909 0.05504241 0.13626659 0.33715602 -7 0.05277618 0.56711634 0.52897534 0.28910632 -6 0.05144062 0.39231693 -6 0.56077607 0.05248310 0.25280562 Variance 21.8304610 11.5727906 30.9667481 49.4751922 14.9776581 31.1167924 27.7533064 22.7126222 10.2618903 29.1708373 33.1562489 30.8745466 35.6422833 24.4937637 13.8520740 45.6913420 18.7589668 15.5850981 36.1923056 20.1579109 20.63154 44.8093456 42.6063478 16.0282598 15.4730702 21.175820 58.2197002 8.55602620 37.8608815 46.5036981 21.6613488 14.4437657 918.523114 4992.25052 20.8835691 15.3214802 23.8740726 16.3091277 16.6412460 17.5613396 11.1807856 29.4939884 16.8704615 1373.94093 13.58765 21.9742622 29.0489845 31.4560768 66.1582830 52.4815604 21.5991624 27.9859126 15.2195767 19.4021891 10.0459985 48.0783355 32.3294416 46.4020358 35.9392697 33.5742258 Skewness 1.19922002 -0.10254192 0.06564699 0.80730670 -1 -0.20308808 1.19013697 0.17735608 0.04039077 -1 0.13887724 0.56950183 0.23905727 -9 0.08023586 1.2741438 0.01820222 3.60591889 0.17513097 0.22418013 -3 0.23049445 0.57303193 0.05884118 0.72210694 0.26979588 0.04671647 0.30837186 0.08548238 4.39249472 -6 0.11157117 0.76743085 -4 0.55643939 0.57638641 1.27532510 0.49325633 -3 0.02907166 0.60758265 0.09879418 0.71198115 0.53881247 0.52060545 0.02874715 0.02385259 0.05895471 0.68330816 0.04739522 0.33725681 -1 0.16142833 0.21985833 0.58089825 0.92031121 0.42712055 0.68558339 1.00869977 0.08332979 0.40269658 -8 0.36373271 0.18882179 1.09269589 0.03933506 -9 0.12806487 0.90854701 Kurtosis 6.09428952 5.18503224 4.04902227 4.85250457 3.69018993 8.77962767 14.3134128 3.17138361 2.99200177 5.11795437 3.73866068 3.87352858 3.95755893 9.08111943 3.53765809 29.2307187 3.55491827 3.73108421 4.45314048 4.44942149 5.13591577 4.47709207 5.47390988 3.44461587 3.96477671 4.59387009 51.0367493 4.73360676 64.5244821 6.24802553 3.93640665 9.43359320 53.8062603 12.7845436 4.09372439 3.73302422 5.13299973 5.78760603 4.03343545 3.05969053 4.54951934 5.53519738 10.4805586 127.803866 3.48327532 3.09270964 3.34884191 6.24163890 5.31935117 4.1781010 4.80548175 9.53064336 3.97438911 5.09144595 4.04227835 4.14423982 6.53889081 4.78406495 3.43175636 7.91981910 JSWSTEEL_BO KANSAINER_BO KOTAKBANK_BO LICHSGFIN_BO LT_BO M_MFIN_BO MARUTI_BO MCLEODRUSS_BO MINDAIND_BO MINDTREE_BO NATCOPHARM_BO NAUKRI_BO NETWORK18_BO NIITLTD_BO NILKAMAL_BO NLCINDIA_BO NMDC_BO NTPC_BO OBEROIRLTY_BO PFC_BO PGHH_BO PIDILITIND_BO POWERGRID_BO PRESTIGE_BO PVR_BO RAJESHEXPO_BO RECLTD_BO RELIANCE_BO SANOFI_BO SBIN_BO SHREECEM_BO SIEMENS_BO SOLARINDS_BO SRTRANSFIN_BO SUZLON_BO TATACHEM_BO TATAMOTORS_BO TATAPOWER_BO TATASTEEL_BO TVSMOTOR_BO 0.58872922 0.65657206 0.44663385 0.35931326 0.26065182 0.39154632 0.59569784 40.1585318 1.06182431 0.56111335 0.98421168 0.30738708 -1 0.03461757 0.24708773 0.87028207 0.17008432 0.04141548 0.04882833 0.12227977 0.31661948 0.48525144 0.57107047 0.25955403 0.28880148 80.8851726 0.59927942 0.37537484 0.26055452 0.31274856 0.34210600 0.66016539 0.19339670 0.66748239 0.24164452 0.10583444 0.26052946 0.21392413 0.00680870 0.05927195 60.9237279 21.925923 14.3297243 11.2862535 18.8526098 17.4538675 29.3409046 13.6380445 17.6973103 50.0344997 21.7337740 38.5329055 19.4328077 36.2037630 40.5620643 40.4390641 17.5132305 20.8569226 513.742190 25.5397095 434.959201 9.00845940 13.0301514 9.37628635 32.7197494 122.456041 35.9735809 28.6248842 10.8855707 910.531091 25.3476911 16.7332276 20.8558731 914.824241 24.5759745 76.3780218 11.7574690 22.1599899 15.63585 25.5401302 27.8454496 0.70087396 0.67689908 9- 0.0018796 0.19890894 0.08462236 7- 0.0464888 0.10825035 0.29749492 1.30311666 1.01745768 0.71244974 0.90221450 1.04609067 0.65192220 1.45540581 0.30665389 0.08766569 30.1536620 0.06952335 0.72054302 0.43537721 0.54393011 0.56046846 0.48215385 0.76535779 0.14979071 0.17105120 0.94144663 0.71683090 0.21718119 -8 0.08483149 0.94525383 1.08327510 0.14737955 0.05654121 0.23294078 0.54283998 0.24538235 4.0354992 4.68129547 3.30408966 3.50638308 4.08722815 5.15264343 3.51686510 54.0951477 6.94249566 4.83739030 8.39353165 4.12737835 5.28678966 5.69886012 4.71421761 8.69842830 5.15359600 7.11007489 3.57465729 4.89775903 3.92307575 4.08021318 7.87800261 4.53175380 4.73158889 6.18143768 3.32997023 3.29522515 4.64002171 54.9723263 4.01467152 4.07837862 7.00641058 4.33627164 9.47874812 3.72414629 4.50027827 3.76739619 4.12636741 3.99261748 Overall Moments Mean 0.2471174 0.1920187 0.3348942 0.2362115 Variance 3.75182 5.189568 2.675834 4.187638 Skewness 0.06335898 -0.203625 -0.3730007 -0.1396324 Kurtosis 2.591479 3.419489 2.38616 3.254541 Appendix B 71 ... 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. .. 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... 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