SYSTEMATIC RISK OF HIGHER-ORDER MOMENTS AND ASSET PRICING

Transcription

1 SYSTEMATIC RISK OF HIGHER-ORDER MOMENTS AND ASSET PRICING Aybike Gürbüz Yapı Kredi Bank, Credit Risk Control İstanbul, Turkey and Middle East Technical University Institute of Applied Mathematics M.Sc. in Financial Mathematics Program Seza Danışoğlu* Middle East Technical University Department of Business Administration and Institute of Applied Mathematics Dumlupınar Bulvarı No:1, Ankara, Turkey Phone: December 2017 *: Corresponding author

2 1. Introduction The efficient allocation of capital in an economy is possible only if the price mechanism reflects the true value of assets in that economy. The modeling of the price process has long attracted the attention of economists and different frameworks have been developed for this purpose since the late 1950s. It would be fair to say that most of these models are built on the mean-variance framework of Sharpe-Lintner-Mossin's Capital Asset Pricing Model (CAPM). CAPM's empirical validity is often questioned since it makes some restrictive assumptions regarding the equilibrium in financial markets. More specifically, CAPM first assumes that investors make decisions while trying to maximize their utility that behaves according to a quadratic utility function. Second, CAPM assumes that asset returns are normally distributed. This assumption makes it possible to model the decision-making process of investors by simply referring to the mean and variance of the return distribution since, under the normality assumption, the first two moments of the distribution are sufficient to describe the behavior of asset returns. The vast empirical literature on the CAPM shows that the normality assumption does not seem to adequately address the stylized facts regarding the distribution of returns (Jurczenko and Maillet, 2006) and asset returns typically are shown to be skewed and leptokurtic. Such deviations from normality may be more severe in emerging financial markets where extreme outcomes and high volatility are frequently observed. One way of addressing non-normality is to include the higher moments of the return distribution as additional parameters in asset pricing models. In this study, the third and fourth moments of the return distribution are used to develop additional measures of systematic risk in financial markets. A linear model is used to describe the equilibrium relationship between the expected rate of return on an asset and several systematic risk measures, including the higherorder measures developed in this study. In order to provide evidence that the proposed risk factors based on higher-order moments are systematically priced in the market, it is necessary to show that these factors explain returns when assets are classified into portfolios based on several different characteristics. In this study, prices of individual stocks traded on Borsa Istanbul between January 1989 and June 2013 and the associated company financials are used to calculate the measures of systematic risk. In addition, excess returns are calculated for portfolios that are formed according to beta, size, book-to-market, momentum, skewness and kurtosis factors. The Fama-French three-factor model is used as the base model and skewness and kurtosis factors are added to the base model separately. The incremental effect of skewness and kurtosis factors over the Fama- French factors are examined with time series regressions. According to the time series regression estimations of different portfolio combinations, the skewness and kurtosis factors do not have an additional effect on asset returns over and above the Fama-French factors. It is argued in the literature that risk averse investors may be more concerned with the characteristics of the return distribution that make losses more likely. As a result, as an alternative to skewness and kurtosis, negative skewness and positive kurtosis factors are included in the models. Empirical results show that negative skewness is indeed priced in the market with a positive and significant coefficient. This factor has a significant marginal contribution for explaining portfolio excess returns and its inclusion increases the adjusted R 2 of the models. This finding suggests that investors in Borsa Istanbul require a higher rate of return as a compensation for the risk of a greater probability of extreme outcomes. In addition, when negative skewness is added to the Fama-French model, the intercept term becomes statistically insignificant, regardless of how portfolios are formed. This indicates that most of the variation in returns is explained by the combination of Fama French factors and negative skewness. Similarly, when the positive kurtosis factor is added to the Fama-French three-factor model, it is observed that this factor also has a positive and significant coefficient, implying that investors at Borsa in BIST require higher rates of return because of the higher probability of extreme observations and large fluctuations. As before, the inclusion of positive kurtosis also increases the explanatory power (R 2 ) of the models.

3 In summary, the empirical findings of this study imply that higher-order moments of the return process seem to be systematically priced in the market and these risks should be included as additional factors in the asset pricing models. The remainder of the study is organized as follows. Section 2 presents a brief summary of the literature on multi-factor asset pricing models where the third and fourth moments of the return distribution are used as systematic risk factors. Section 3 provides information about the data and methodology used in the study. Section 4 presents the empirical results, and Section 5 offers the conclusions. 2. Literature Review Kraus and Litzenberger (1976) extend the capital asset pricing model by adding a systematic skewness on asset pricing in the US market between 1926 and 1935 are used. The results of their analysis indicate that investors have an aversion to variance and a preference for positive skewness. Also, the addition of the skewness factor increases the explanatory power of the CAPM according to the findings of this study. Following the study of Kraus and Litzenberger (KL), a new line of research started where several studies tested the possibility of including the higher moments of the return distribution as a systematic risk factor. First, Friend and Westerfield (1980) repeat the KL tests by using different market portfolio proxies and grouping methods in their analyses. In their study, estimation results for the model that is created by including coskewness in the CAPM seem to be change based on the different market indexes, sample periods, and testing and estimation methods used. Friend and Westerfield conclude that the 1976 KL results are sample specific. Following these findings, Sears and Wei (1988) try to explain why the market risk premiums affect the empirical tests of higher moment pricing models. When the additional explanatory variable (skewness) is included in the model created based on the two-fund separation theory, the pricing model includes the market risk premium in nonlinear form. Sears and Wei show that if this nonlinearity is not taken into account, the sign of the parameter estimate can be misinterpreted. After Sears and Wei, Lim (1989) tests the KL three-moment capital asset pricing model by using Hansen s (1982) generalized method-of-moments (GMM). The GMM is a convenient method to test the KL model since it does not have strong distributional assumptions about the returns of asset and it also prevents measuring error problems and obtains more efficient estimators. Lim uses the US monthly stock returns between 1933 and The results of the study show that coskewness of a stock with the market is preferred by investors when market returns have a positive skewness. Otherwise, investors do not seem to prefer coskewness with the market. In general, the analyses provide some evidence that skewness is priced in the market. In a similar study, Lee, Moy and Lee (1996) also try to understand the role of coskewness in asset pricing and use the multivariate testing method created by Gibbons (1982). Their findings indicate that covariance and coskewness risks are both statistically significant. Moreover, when the size of the coefficients are compared, coskewness risk has a smaller effect on asset prices compared to covariance. In a more recent study, Harvey and Siddique (2000) examine the effect of including conditional skewness in the model in order to explain the cross sectional variation in asset returns. They use a nonlinear multi-factor model and analyze the monthly US stock between July 1963 and December They form several portfolios based on industry, size, book-to-market ratio, and momentum. Furthermore, beta coefficients are calculated for stocks that are placed into one of three portfolios based on their past coskewness ranks, with the S portfolio including bottom 30 percent of stocks with the most negative skewness, the S + portfolio including top 30 percent of stocks with the most positive skewness, and the S 0 portfolio including the middle 40 percent of stocks. Tests of the Fama-French model with the additional coskewness factors show that SMB and HML have similar affects with skewness to capture information. Besides, the addition of the skewness factor to the three factor model reduces the Gibbons-Ross-Shanken (1989) F-statistic which is used to test the significance of the intercept term. As a result, there is some evidence that coskewness is important for

4 explaining the cross section of asset returns. It is also seen that the momentum effect has a relation with the systematic skewness. In addition, the winner portfolio has a lower skewness compared to that of the loser portfolio. In 2003, Harvey and Siddique's study is repeated for the Taiwan market by Lin and Wang. Their portfolios are also formed based on industry, size, book-to-market, and momentum. Lin and Wang results are very similar to the Harvey and Siddique results since when the skewness factor is added to the three-factor model, the Gibbons-Ross-Shanken F- test statistic decreases and the R 2 increases. It is also seen that the systematic skewness is related to the size effect and the winner portfolio seems to have lower skewness compared to the loser portfolio. Following these initial series of studies, Adesi, Gagliardini, and Urga (2004) and Smith (2007) also provide evidence supporting the Harvey and Siddique (2000) findings. In 2009, the Harvey and Siddique study is repeated for Borsa Istanbul. Mısırlı and Alper (2009) show that coskewness has a significant effect on the CAPM, especially for size portfolios. Coskewness also has an explanatory power over and above the single-factor CAPM for industry portfolios but its effect is not as high as the size portfolios. Moreover, the coskewness factor does not have a significant effect over and aove Fama-French three-factor model. After the extension of the CAPM with the inclusion of a skewness factor, Fang and Lai (1997) added the kurtosis factor to the three-moment CAPM and formed the four-moment CAPM. Their results show that returns are negatively correlated with skewness and investors demand higher expected rates of return for assets that are negatively skewed in order to compensate for their risk. Furthermore, investors seem to demand higher expected rates of return for higher systematic variance and systematic kurtosis. The Hung, Shackleton and Xu (2004) and Kostakis, Muhammad, and Siganos (2012) studies for the UK stock returns, the Lajili (2005) study for the French stock returns, the Messis, Iatridis, and Blanas (2007) study for the Greek stock returns, the Iqbal, Brooks, and Galagedera (2007) and the Javid (2008) studies for the Pakistani stock returns, the Doan, Lin and Zurbruegg (2010) study for the Australian stock returns, the Hasan, Kamil, Mustafa and Baten (2013) study for the Bangladeshi stock returns, and the Heaney, Lan and Treepongkaruna (2011) and Lambert and Hübner (2013) studies for the US stock returns all present evidence that skewness and kurtosis seem to systematic risk factors that are priced by investors in the market. 3. Data and Methodology 3.1 Data Sample Period, Frequency and Sources of Data - In this study, the dataset of the stocks listed on Borsa Istanbul (BIST) is used. Monthly returns are calculated based on the end-of-month stock prices that are adjusted for dividends and splits between January 1990 and June Financial institutions (banks, insurance companies, leasing and factoring companies, investment companies, investment trusts, and real estate investment trusts) are excluded from the sample. The monthly risk-free rate of return is calculated based on the compounded interest of the Treasury bill with a 90-day (or, closest to 90 days) maturity. The interest rates are obtained from the daily bulletins of Borsa Istanbul. Moreover, the BIST-100 index is used as the market proxy. All stock market data are obtained directly from Borsa Istanbul. 3.2 Methodology The primary aim of the study is to examine the effect of skewness and kurtosis on asset prices in Borsa Istanbul. We use the Sharpe-Lintner-Mossin single-index CAPM and the Fama-French three-factor models as a starting point and add the third and fourth return moments in order to test whether these properties of the return distribution are priced in the Turkish stock market.

5 The study has three major steps. First, the sensitivity to market risk (β), small-minus-big (SMB), high-minus-low (HML), winner-minus-loser (WML), systemic skewness, systematic kurtosis, coskewness, and cokurtosis factors are calculated. Second, the returns of the portfolios, which are formed according to the sorted beta, size, book to market, momentum, skewness and kurtosis factors, are calculated and used as the dependent variable. In the final part of the study, time series regressions are estimated to observe the additional explanatory power of the skewness and kurtosis factors in the CAPM and Fama French models Calculation of Risk Factors Excess market return (R mt R ft ) The excess market return is calculated as the difference between the monthly return on the BIST-100 index and the monthly return on the 90-day Treasury security. Small minus Big (SMB) and High minus Low (HML) The Fama and French (1993) risk factors are calculated following their methodology. First, all stocks that have available data on the last trading day of June of year t (t=1990, ) are divided into two groups based on their market value of common equity. The stocks that have a market value higher than the median market value are labeled as Big Stocks and the remaining stocks are labeled as Small Stocks. Second, independent of the former grouping, all stocks that have available data on the last day of year t-1 are divided into three groups based on their ranked Book-to-Market Value of Equity (B/M) values. The High class includes those stocks from the highest 3 deciles (highest 30%), the Medium class includes stocks from the 4 th to 7 th deciles (medium 40%) and the Low class includes the remaining stocks (lowest 30%). The intersection of the size and B/M independent sorts creates six portfolios: Big-High, Big-Medium, Big-Low, Small- High, Small-Medium and Small-Low. The monthly value-weighted portfolio returns are calculated for each of these portfolios between July of year t and June of year t+1. The portfolios are reformed as described above in June of year t+1. The SMB factor is calculated as the difference between the arithmetic average of the returns on the three Small portfolios and the three Big portfolios. The HML factor is calculated as the difference between the arithmetic average of the returns on the two Low portfolios and the two High portfolios. Time series for the SMB and HML factors are constructed between July 1990 and June Winner minus Loser (WML) The Carhart (1997) risk factor is calculated by replicating his methodology. First, stocks are ranked based on their past 11-month returns where there is a one-month lag between the last day of the 11-month period and the day of ranking. After the calculation of the returns and ranking of the stocks based on these returns, stocks are grouped into two categories. The Winner (Loser) group includes those stocks that have the highest (lowest) 30% of the ranked returns. The WML factor is calculated as the difference between the equally-weighted Winner and the equally-weighted Loser portfolio returns. A time series is constructed for the WML factor between July 1990 and June Skewness and Kurtosis Factors Skewness and kurtosis are statistical measures to define the shape of the probability distribution of a random variable. Skewness is the third standardized moment of the probability distribution and it measures the degree of the asymmetry of the distribution around its mean. The normal distribution is symmetric and its skewness is equal to 0. If skewness is greater (less) than 0, the distribution is positively (negatively) skewed. Positively skewed return distributions have a long right tail and frequent small losses and few extreme gains are expected with such distributions. On the other hand, negatively skewed return distributions have a long left tail and frequent small gains and few extreme losses are expected with such distributions. In other words, with a negatively skewed return distribution, there is a higher risk of extreme negative outcomes. For this reason, investors typically do not prefer negative skewness and demand higher risk premiums for the negatively skewed stocks. The comparisons between a

6 normal distribution and skewed distributions are shown below in the Figure 1. In the Figure, the bold shapes indicate the normal distribution. Figure 1. Comparison of normal and skewed distributions Kurtosis is the fourth standardized moment of a probability distribution and it measures the peakedness of the distribution. The kurtosis value of the normal distribution is equal to 3 (a mesokurtic distribution). If kurtosis is greater (less) than 3, then the distribution is leptokurtic (platykurtic). A leptokurtic distribution has a sharper peak than the normal distribution and has fatter tails. The fat tails indicate that there is a risk of outliers and extreme observations are much more probable to occur compared to the normal distribution. Investors typically try to avoid this type of a risk and therefore risk averse investors demand higher rates of return from stocks with a leptokurtic return distribution. On the other hand, a platykurtic distribution has a lower peak and thinner tails, and therefore extreme observations are less likely to occur compared to the normal distribution. For this reason, risk averse investors typically prefer stocks with a platykurtic return distribution. Figure 2 represents the comparison of leptokurtic, mesokurtic (normal) and platykurtic distributions and the bold shape indicates the normal distribution. Figure 2. Comparison of normal and lepto- and platy-kurtic distributions The skewness and kurtosis of a return distribution are calculated as follows: Standardized Skewness = E(R i R i) 3 σ 3 i (1) Standardized Kurtosis = E(R i R i) 4 σ 4 i (2) In Equations (1) and (2), R i. R i and σ i represent asset i s returns, mean of asset i s returns and standard deviation of asset i s returns, respectively. As can be seen in the formulations, the return on the market index is not taken into account while calculating the skewness and kurtosis of a stock s return distribution. Kraus and Litzenberger (1976) argue that in order to test whether skewness and kurtosis are priced risk factors, it is necessary to calculate the skewness and

7 kurtosis of a stock s return distribution relative to the skewness and kurtosis of the market portfolio returns. For this reason, a formula defining the relation between an asset s skewness (kurtosis) and the market portfolio s skewness (kurtosis) is needed. In the literature, there is a complication about definition of such a formula. The complication arises from the fact that not only are there different formulas offered to measure the relative skewness and kurtosis but also the same formulas are given different names in different studies. For this reason, in this study, we use two sets of the most widely used formulas. The first group measures the so called systematic skewness and systematic kurtosis and the second group measures coskewness and cokurtosis. As explained above, Kraus and Litzenberger (1976) (KL) advocate using a systematic instead of a total skewness measure within the context of asset pricing. According to KL, the non-diversifiable (systematic) measure of skewness should be defined in a similar fashion as systematic risk (beta). KL s formulation of systematic skewness is as follows: Systematic Skewness(S i ) = E[(R i R i)(r m R m) 2 ] E[(R m R m) 3 ] = Cov(R i, R m 2 ) E[(R m R m) 3 ] (3) Following KL, Fang and Lai (1997) define systematic kurtosis as follows: Systematic Kurtosis (K i ) = E[(R i R i)(r m R m) 3 ] E[(R m R m) 4 ] = Cov(R i, R m 3 ) E[(R m R m) 4 ] (4) In Equations (3) and (4), R i, R m, R i, R m indicate asset i s return, market return, mean of asset i s return and mean of market return, respectively. Since systematic skewness and kurtosis are defined in a fashion similar to systematic market risk, higher values represent higher risk. For instance, higher systematic skewness means that the likelihood of observing extreme losses is higher, or, higher systematic kurtosis means that the stock s returns are more leptokurtic compared to the market portfolio returns. As a result, risk averse investors would prefer lower systematic skewness or kurtosis and demand larger risk premiums for higher systematic skewness or kurtosis. In a more recent study, Harvey and Siddique (2000) (HS) revisit the issue of skewness as a risk factor in asset pricing and define coskewness as the contribution of a stock to the skewness of the market portfolio. In the HS setting, negative coskewness means that the asset adds negative skewness to the market portfolio returns and increases the likelihood of obtaining extremely low returns. On the other hand, positive coskewness means that the asset adds positive skewness to the market portfolio returns and makes it more likely to obtain extremely high returns. In light of these definitions, risk averse investors would want to avoid negatively coskewed stocks and would prefer positively coskewed stocks. The HS formulation of standardized unconditional coskewness is as follows: Standardized Unconditional Coskewness (CSK i ) = E(ε 2 i,t+1ε M,t+1 ) 2 2 E(ε i,t+1 )E(ε M,t+1 ) (5) In Equation (5), ε i,t+1 = r i,t+1 α i β i (r M,t+1 ) (the residual from the regression of CAPM), ε M,t is the difference between excess market return in month t and the average market return over the window of returns t-60 to t. r i,t, r m,t are the return of asset i and market return at time t and α i, β i are the intercept term and the beta coefficient, respectively. Kostakis, Muhammad, and Siganos (2012) develop the methodology of HS and define cokurtosis as a risk factor within the asset pricing context. Cokurtosis is a statistical measure that compares the peakedness of an asset s return distribution with that of the market portfolio and measures the contribution of the stock to the kurtosis of the market portfolio returns. Higher (lower) cokurtosis means that the stock makes the market portfolio returns more (less)

8 leptokurtic. Accordingly, risk averse investors prefer stocks with lower cokurtosis. Kostakis et al. define cokurtosis as follows: Standardized Unconditional Cokurtosis (CKT i ) = E(ε 3 i,t+1ε M,t+1 ) 2 3 E(ε i,t+1 )E(ε M,t+1 ) (6) Variables used in Equation (6) are defined as before. In this study, both the KL-based systematic skewness and kurtosis and the HS-based coskewness and cokurtosis definitions are used for calculating the risk factors. First, for each stock, KL- or HS-based skewness (kurtosis) are values calculated based on monthly returns observed over rolling windows of 60 months between January 1990 and June Next, for the 61 st months, stocks are ranked according to their skewness (kurtosis) values and two sets of portfolios are formed. The first set of portfolios include 30% of stocks with high systematic skewness (kurtosis) or negative coskewness (high cokurtosis). The second set of portfolios include 30% of stocks with low systematic skewness (kurtosis) or positive coskewness (low cokurtosis). Value-weighted monthly returns are calculated for these portfolios throughout the sample period. Finally, the skewness risk factor is described as the difference between the returns of the high and low systematic skewness (or negative and positive coskewness) portfolios. Similarly, the kurtosis risk factor is described as the difference between the returns of the high and low systematic kurtosis (or high and low cokurtosis) portfolios Portfolio Formation Lewellen, Nagel and Shanken (2010) argue that in order to claim that the factors included on the right-hand-side of an asset pricing model are truly asset-pricing factors, it is necessary to show that these factors explain returns when assets are classified into portfolios based on several different characteristics, especially those that are not highly correlated with the size and B/M risks. This section describes how portfolios are formed based on different criteria and also presents the estimation results. Beta-Sorted Portfolios The Fama and MacBeth (1973) methodology is used to estimate the beta coefficients for the stocks in the sample. The first four years of sample data between January 1990 and December 1993 are used to calculate the individual stock beta coefficients as the ratio of the covariance between the stock and market portfolio returns to the variance of the market portfolio return. Next, stocks are sorted in ascending order based on their estimated beta coefficients and grouped into ten decile portfolios. After portfolio formation, the next four years of data between January 1994 and December 1997 are used to re-calculate the beta coefficient for each individual stock. Next, for the testing period from January 1998 to December 1998, each portfolio s beta coefficient is calculated as the arithmetic average of the constituent stocks re-calculated betas. The portfolio beta coefficients are re-calculated for each of the 12 months in the testing period in order to reflect any variation in the number of stocks that stay in the portfolios throughout the year. While the portfolio beta coefficients are re-calculated monthly, the individual stock betas are re-calculated annually by extending the initial estimation period one year at a time to include the most recent year. This process is reiterated until June Size and Book-to-Market Double-Sorted Portfolios Nine portfolios are formed following a method similar to the calculation of the SMB and HML factors. This time, the stocks that have the lowest 35% of market value ranks are classified as Small (S) stocks, the middle 30% as Medium (M) stocks, and the highest 35% as Big (B) stocks. Similarly, the group labeled High (H) is made up of stocks from the highest 35% of B/M ranks, Medium (M) is made up of stocks from the middle 30% and Low (L) is made up of stocks from the lowest 35% of B/M ranks. The reason for choosing these cutoffs is to ensure that a reasonable number of stocks remain

9 in the intersection portfolios. The intersection of the size and B/M independent sorts creates nine portfolios: S/L, S/M, S/H, M/L, M/M, M/H, B/L, B/M, B/H. Monthly equally-weighted portfolio returns are calculated for each of these portfolios between July 1990 and June Independently Sorted Size and Book-to-Market Portfolios In addition to the double-sorted portfolios from the previous section, value-weighted returns on portfolios sorted independently on size and book-to-market are calculated and tested in order to see the effect of these factors separately. Size and B/M cutoffs and portfolio classifications are the same as the previous section. Past-Return-Sorted Portfolios Decile portfolios based on past returns are formed by using a method similar to the formation of the WML factor. Originally, De Bondt and Thaler (1985) provide evidence of market overreaction and show that firms with low (high) returns over the past three to five years are likely to have high (low) future returns. Jegadeesh and Titman (1993) revisit these results and demonstrate that there is a momentum effect in short-term past returns and firms that were winners (losers) over the past year tend to be future winners (losers). Following Carhart s (1997) momentum factor definition, stocks are classified into decile portfolios based on their compound returns over the past 12, 24, 36, 48 and 60 months between February 1990 and June There is a one-month lag between the calculated past returns and the portfolio formation in order to avoid market microstructure effects such as price ticks and price limits. Portfolio 1 represents the lowest return decile and Portfolio 10 represents the highest return decile. Skewness and Kurtosis Portfolios Stocks are ranked in ascending order based on their systematic skewness (kurtosis) and coskewness (cokurtosis) values and grouped into ten portfolios. Value-weighted returns are calculated for these portfolios. As before, Portfolio 1 represents the lowest return decile and Portfolio 10 represents the highest return decile. 4. Empirical Results 4.1 Preliminary Analysis The aim of this study is to examine the impact of skewness and kurtosis in explaining the variation of excess returns on the portfolio groups which are sorted depending on beta, size, book to market, momentum, coskewness and cokurtosis factors for BIST. Before the time series regressions are estimated, summary statistics are calculated to determine whether there is any evidence regarding the role of the third and fourth moments of the return distributions in the pricing of stocks traded on Borsa Istanbul. Table 1 represents the skewness and kurtosis values, Kolmogorov-Smirnov statistics and associated p-values. Results presented in this table provide evidence about whether the returns of portfolio groups are normally distributed. The null hypothesis for the Kolmogorov-Smirnov test is normality of the portfolio returns. According to the test statistics presented in different panels of Table 1, both skewness and kurtosis values are larger than the levels that would indicate normally distributed portfolio returns. These results imply that the portfolios formed in this study have non-normal returns since there is evidence of skewness and kurtosis. Normality test results are the first indication that skewness and kurtosis may be risk factors that are priced in Borsa Istanbul. Table 2 presents the summary statistics of market, coskewness and cokurtosis factor loadings for the same portfolio groups in Table 1. The beta coefficients (factor loadings) for R m R f (market), S S + (coskewness), K + K (cokurtosis), S (negative coskewness) and K + (positive cokurtosis) factors are calculated with the univariate regressions of different portfolio returns 1. In Table 2, it is seen that there is enough preliminary evidence about the 1 Since the results with KL-based and HS-based skewness and kurtosis definitions are qualitatively the same, only the HS-based results are presented throughout the study. KL-based results are available from the authors upon request.

10 significant effect of skewness and kurtosis factors on the portfolio returns. Accordingly, in the next section, these factors are included in the Fama-French model and their significance is formally tested with the time series regressions. 4.2 Time Series Estimations Times series estimation results are presented in Tables 3 through 8. In all tables, it is seen that without the inclusion of the skewness or kurtosis factors, the market, size and book-to-market factors are significantly priced in Borsa Istanbul. More specifically, for all eight portfolio sorts (beta, independent size, independent book-to-market, double-sorted size and book-to-market, past returns, coskewness and cokurtosis), the intercepts are insignificant and the three factors of the Fama-French model all have positive and significant coefficients. The skewness and kurtosis factors are included in the models in the four different ways explained in the Descriptive Statistics section. Regardless of the portfolio sort, the S S + (coskewness) and K + K (cokurtosis) factors are not significantly priced in the market and the inclusion of these variables in the models do not change the significance of the existing variables or the explanatory power (R 2 ) of the models. Interestingly, when skewness is included as a negative coskewness (S ) factor, coefficients for this variable become positive and significant for the majority of the portfolios while the other risk factors maintain their significance as well. This implies that investors in BIST are mainly concerned about the risk of generating extreme losses since skewness seems to be priced only when it is defined in a manner to account for the loss potential of the return distribution. Similarly, when kurtosis is included as positive cokurtosis (K + ), the coefficient of this factor becomes positive and significant for the majority of the portfolios. When the stock returns are leptokurtic, investors face the risk of generating extreme returns on either side of the distribution and investors seem to demand a higher risk premium for facing such an uncertainty. In addition, when the positive cokurtosis (K + ) factor is included in the model, the intercept of the model becomes significant and negative. This result indicates two characteristics of the pricing process in BIST. First, the positive cokurtosis (K + ) factor may not be enough by itself to represent the non-normality in stock returns since the significance of the model implies that there may be an additional risk factor that is being left out of the model. Second, the result that the intercept becomes negative implies that stocks with a leptokurtic distribution generate lower returns compared to the other stocks and this may be due to the negatively skewed distributions observed in Table 1 for most of the stocks traded in BIST. 5. Conclusion This study aims to analyze whether the non-normality in stock returns is priced in an emerging stock market. Since emerging stock markets are typically characterized by more extreme price movements, it is a plausible to expect that these extreme movements lead to non-normal stock returns. Preliminary evidence presented in the study indeed confirms that stocks traded in Borsa Istanbul have non-normal returns and the return distributions are generally negatively skewed and leptokurtic. In light of this evidence, the skewness and kurtosis of stock returns is calculated by using different formulas proposed in the literature. Also, portfolios are formed by ranking BIST stocks based on their sensitivity to market, size, book-to-market, past return, skewness and kurtosis risks. The Fama-French three-factor model is augmented by including skewness and kurtosis factors. Results show that investors in Borsa Istanbul are concerned about the negatively skewed and leptokurtic distribution of stock returns since these risk factors have significant and positive coefficients in addition to the other significant factors included in the models. References Adesi, G. B., Gagliardini, P., & Urga, G. (2004). Testing asset pricing models with coskewness. Journal of Business& Economic Statistics, Carhart, M. M. (1997). On Persistence in Mutual Fund Performance. The Journal of Finance,

11 De Bondt, W. F. M., & Thaler, R. (1985). Does the stock market overreact? The Journal of Finance 40(3), Doan, P., Lin, C., & Zurbruegg, R. (2010). Pricing assets with higher moments: Evidence from the Australian and US stock market. Journal of International Financial Markets, Institutions & Money, Fama, E. F., & French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, Fama, E. F., & MacBeth, J. D. (1973). Risk, Return and Equilibrium: Empirical Tests. Journal of Political Economy, Fang, H., & Lai, T. (1997). Co-kurtosis and Capital Asset Pricing. The Financial Review, Friend, I., & Westerfield, R. (1980). Co-skewness and Capital Asset Pricing. The Journal of Finance, Gibbons, M. R. (1982). Multivariate test of financial models: A new approach. The Journal of Financial Economics, Gibbons, M., Ross, S., & Shanken, J. (1989). A test of the effiency of a given portfolio. Econometrica, Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, Harvey, C. R., & Siddique, A. (2000). Condition Skewness in Asset Pricing Tests. The Journal of Finance, Hasan, Z., Kamil, A. A., Mustafa, A., & Baten, A. (2013). An empirical analysis of higher moment capital asset pricing model for Bangladesh Stock Market. Modern Applied Science, Heaney, R., Lan, Y., & Treepongkaruna, S. (2011). A test of co-skewness and co-kurtosis: The relevance of size, book to market, momentum in asset pricing. 24th Australasian Finance and Banking Conference. Sydney. Hung, D. C., Shackleton, M., & Xu, X. (2004). CAPM,higher co- moment and factor models of UK stock returns. Journal of Business Finance & Accounting, Iqbal, J., Brooks, R. D., & Galagedera, D. U. (2009). Asset pricing with higher-order co-moments and alternative factor models: The case of an emerging market. In Financial Innovations in Emerging Markets, edited by G. Gregoriu. London: Chapman Hall. Javid, A. Y., & Ahmad, E. (2008). Test of Multi-moment Capital Asset Pricing Model: Evidence from Karachi Stock Exchange. Pakistan Institute of Development Economics ISLAMABAD Working Paper. Jegadeesh, N., & Titman, S. (1993). Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency. The Journal of Finance, Jurczenko, Emmanuel, Bertrand Maillet, (2006). The Four-moment Capital Asset Pricing Model: Between Asset Pricing and Asset Allocation, in Emmanuel Jurczenko and Bertrand Maillet, ed: Multi-moment Asset Allocation and Pricing Models, Chapter 6, pp (Wiley: West Sussex) Kostakis, A., Muhammad, K., & Siganos, A. (2012). Higher co-moments and asset pricing on London Stock Exchange. The Journal of Banking and Finance, Kraus, A., & Litzenberger, R. H. (1976). Skewness Preference and the Valuation of Risk Assets. The Journal of Finance, Lajili, S. (2005). Size and book to market effects vs co-skewness and co-kurtosis in explaining stock returns. 32nd Annual Conference of the Northeast Business & Economics Association. Lambert, M., & Hübner, G. (2013). Comoment risk and stock returns. Journal of Empirical Finance, Lee, A., Moy, R. L., & Lee, C. F. (1996). A multivariate test of covariance-co-skewness restriction for the three moment CAPM. Journal of Economics and Business, Lewellen, J., Nagel, S., & Shanken, J. (2010). A skeptical appraisal of asset pricing tests, Journal of Financial Economics 96(2), Lim, K. (1989). A new test of the three-moment capital asset pricing model. The Journal of Financial and Quantitative, Lin, B., & Wang, J. M. C. (2003). Systematic Skewness in asset pricing: an emprical examination of the Taiwan stock market. Applied Economics, Lintner, J. (1965). Security Prices, Risk and Maximal Gains from Diversification. The Journal of Finance, Lintner, J. (1965). The Valuation of Risk Assets and The Selection of Risky Investments in Stock Portfolios and Capital Budgets. The Review of Economics and Statistics, Messis, P., Iatridis, G., & Blanas, G. (2007). CAPM and the efficacy of higher moment CAPM in the Athens Stock Market:An empirical approach. International Journal of Applied Economics, Mısırlı, E. U., & Alper, C. E. (2009). Drivers of expected returns in Istanbul Stock Exchange: Fama- French factors and coskewness. Applied Economics, Mossin, J. (1966). Equilibrium in a Capital Asset Market. Econometrica, Sears, S. R., & Wei, J. K.C. (1988). The Structure of skewness preferences in asset pricing models with higher moments:an empirical test. The Financial Review, Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk. The Journal of Finance, Smith, D. R. (2007). Conditional coskewness and asset pricing. Journal of Emprical Finance,

12 Table 1: Descriptive Statistics and Normality Tests Portfolio Skewness Kurtosis Kolmogorov-Smirnov P-Value Panel A: Beta-Sorted Portfolios Value-Weighted Returns <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** Panel B: Book to Market-Sorted Portfolios Value-Weighted Returns Low <0.010** Medium <0.010** High <0.010** Panel C: Size-Sorted Portfolios Value-Weighted Returns Small <0.010** Medium <0.010** Big <0.010** Panel D: Size and Book to Market-Sorted Portfolios Value-Weighted Returns S/L <0.010** S/M <0.010** S/H <0.010** M/L <0.010** M/M <0.010** M/H <0.010** B/L <0.010** B/M <0.010** B/H <0.010** Panel E: P11L1 Momentum Value-Weighted Returns <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010**

13 Table 1 Continued Portfolio Skewness Kurtosis Kolmogorov-Smirnov P-Value Panel F: P24L1 Momentum Value-Weighted Returns <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** Panel G: P36L1 Momentum Value-Weighted Returns <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** Panel H: P48L1 Momentum Value-Weighted Returns <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** Panel I: P60L1 Momentum Value-Weighted Returns <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010**

14 Table 1 Continued Portfolio Skewness Kurtosis Kolmogorov-Smirnov P-Value Panel J: Coskewness Sorted Value-Weighted Returns <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** Panel K: Cokurtosis Sorted Value-Weighted Returns <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** <0.010** ** and *indicate that the test is significant at the 5% and 10% levels, respectively.

15 Table 2: Factor loadings on the skewness and kurtosis factors Portfolio R m R f S S + S Panel A: Beta-Sorted Portfolios Value-Weighted Returns K + K ** ** ** ** ** ** ** * ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** Panel B: Book to Market-Sorted Portfolios Value-Weighted Returns Low ** ** ** Medium ** ** ** High ** * ** ** Panel C: Size-Sorted Portfolios Value-Weighted Returns Small ** ** ** ** Medium ** ** ** ** Big ** ** ** Panel D: Size and Book to Market-Sorted Portfolios Value-Weighted Returns S/L ** ** ** ** S/M ** ** ** ** S/H ** ** ** ** M/L ** ** ** ** M/M ** ** ** ** M/H ** ** ** ** B/L ** ** ** B/M ** ** ** B/H ** ** ** Panel E: P11L1 Momentum Value-Weighted Returns ** ** ** ** * ** ** ** ** ** ** ** * ** ** ** ** ** ** ** * ** ** ** ** ** ** ** ** ** ** ** ** ** ** * ** ** ** K +

16 Table 2 Continued Portfolio R m R f S S + S Panel F: P24L1 Momentum Value-Weighted Returns K + K ** ** ** ** ** ** ** * ** ** ** ** ** ** ** * ** ** ** * ** ** ** * ** ** ** * ** ** ** ** ** ** ** * ** ** Panel G: P36L1 Momentum Value-Weighted Returns ** ** ** ** * ** ** ** * ** ** ** ** ** ** ** ** ** ** ** ** ** * ** ** ** ** ** ** ** ** ** ** ** ** ** Panel H: P48L1 Momentum Value-Weighted Returns ** * ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** * ** ** Panel I: P60L1 Momentum Value-Weighted Returns ** ** ** ** ** * ** ** ** ** ** ** * ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** K +

17 Table 2 Continued Portfolio R m R f S S + S Panel J: Coskewness Sorted Value-Weighted Returns K + K ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** * ** ** ** ** ** ** ** ** ** ** ** ** ** ** Panel K: Cokurtosis Sorted Value-Weighted Returns ** ** ** ** ** ** ** ** ** ** ** ** ** * ** ** ** ** ** ** ** ** * ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** and *indicate that the test is significant at the 5% and 10% levels, respectively. K +

18 Table 3: Parameter estimates for beta-sorted portfolios Beta-Sorted Portfolios Value-Weighted Portfolio Returns Intercept (0.49) (0.60) (0.63) (0.29) (0.21) (0.55) (0.41) (0.37) (0.50) (0.17) R m R f SMB HML (0.02) (0.11) (0.00) (0.02) (0.19) (0.01) (0.00) (0.01) (0.05) (0.00) OLS R Intercept (0.42) (0.56) (0.90) (0.39) (0.72) (0.37) (0.38) (0.29) (0.54) (0.21) R m R f SMB HML (0.06) (0.18) (0.00) (0.01) (0.38) (0.05) (0.00) (0.03) (0.05) (0.00) S S (0.46) (0.68) (0.03) (0.25) (0.00) (0.07) (0.65) (0.30) (0.73) (0.66) OLS R Intercept (0.47) (0.61) (0.63) (0.30) (0.22) (0.57) (0.37) (0.38) (0.51) (0.18) R m R f SMB HML (0.03) (0.10) (0.00) (0.02) (0.25) (0.00) (0.00) (0.01) (0.04) (0.00) K + K (0.36) (0.64) (0.81) (0.32) (0.03) (0.07) (0.01) (0.51) (0.36) (0.56) OLS R Intercept (0.84) (0.54) (0.05) (0.30) (0.00) (0.80) (0.06) (0.68) (0.18) (0.81) R m R f SMB HML (0.01) (0.01) (0.00) (0.00) (0.02) (0.00) (0.00) (0.00) (0.00) (0.00) S (0.36) (0.02) (0.00) (0.00) (0.00) (0.08) (0.01) (0.40) (0.00) (0.00) OLS R Intercept (0.32) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.16) (0.00) (0.01) R m R f SMB HML (0.01) (0.01) (0.00) (0.00) (0.75) (0.00) (0.00) (0.00) (0.00) (0.00) K (0.01) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) OLS R