Asset pricing with higher-order co-moments and alternative factor models: The case of an emerging market

Transcription

1 Asset pricing with higher-order co-moments and alternative factor models: The case of an emerging market Javed Iqbal Robert D. Brooks Don U.A. Galagedera Department of Econometrics and Business Statistics, Monash University Abstract For emerging market returns there is strong evidence that the departure from normality is primarily driven by kurtosis and not skewness. This paper investigates the empirical validity of a return generating process that includes quadratic and cubic market returns as factors of pricing for an emerging market. Following Barone-Adesi et al. (004) a multivariate test of a three-moment pricing model is developed. The empirical evidence in the market returns support the stylized facts typical for an emerging market and reveal that any return generating process that includes only a quadratic term (coskewness) may be misspecified. The paper also investigates the Fama-French factor model as an alternative. The pricing model with higher co-moments does not appear to be superior to the model with the Fama-French factors. 1. Introduction The failure of the conventional capital asset pricing model (CAPM) to adequately explain cross-sectional variation of risky asset return has spurred alternative explanations of asset pricing. The Arbitrage Pricing Theory (APT) of Ross (1976) is one such alternative. APT stipulates that under no arbitrage the expected returns of assets are expressed as a linear function of certain common factors though the theory does not specify the factors themselves. Many studies follow this lead. For example, based primarily on statistical considerations, Fama and French (199) advocate inclusion of two factors mimicking the size and book-to-market value of the assets besides the systematic risk; the beta. Kraus and Litzenberger (1976) and Dittmar 1

2 (00) emphasize inclusion of higher co-moments namely coskewness and cokurtosis as explanatory variables of expected returns. Considering the investor sentiment that an investment is judged risky if the returns deviate below a certain target rate of return, Bawa and Lindenberg (1977) develop the lower partial moment based asset pricing model. Another stream of literature questions the constancy of expected return and the risk parameter over time and in that context Bollerslev, Engle, and Wooldridge (1988) develop a time varying version of the CAPM. The unknown form of the return distribution is unlikely to be described by the first two moments only. See for example Chung, Johnson and Schill (006) for more emphasis on this issue. Therefore incorporation of the higher-order co-moments such as coskewness and cokurtosis 1 besides the beta as additional risk factors is an important issue. The dynamics underlying higher order co-moments is consistent with the investor preference. Risk aversion is a common assumption but the wide spread prevalence of lotteries and ever increasing popularity of prize schemes implies that investors are also concerned about the tail of the return distribution. Kraus and Litzenberger (1976) incorporate systematic skewness in the investor preference set and present an extension of the CAPM to include skewness preference. Dittmar (00) extend the conventional asset pricing framework to cokurtosis. A simple mathematical explanation of the direction of preference of moments of any order is available in Scott and Horvath (1980). They reveal that the investor prefers odd order 1 The covariance of an asset with the market returns measure to the contribution of the asset to the variability of a well diversified portfolio. In a similar way coskewness and cokurtosis measure the contribution of the asset to the overall skewness and kurtosis of the portfolio. An asset with positive coskewness with the market diminishes the sensitivity of a portfolio to large absolute market returns, (Barone-Adesi et al., 004). Therefore ceteres paribus investors like assets with positive coskewness. Kurtosis here refers to the tail thickness of the return distribution. A higher cokurtosis therefore indicates higher likelihood that extreme returns jointly occur in the given asset and the market.

3 moments but demand a premium for even order moments. Utility based empirical testing of higher moment asset pricing models are considered in Kraus and Litzenberger (1976), Sears and Wei (1985, 1988), Homaifar and Graddy (1988), Tan (1991) and Fang and Lai (1997) among others. This approach is criticized for being subject to errors-in-variable bias and less efficient due to multicollinearity problems. Moreover Brockett and Kahane (199) demonstrate by giving counter examples that ascertaining direction of preference for moments on the basis of the sign of the derivatives of the utility function is not justified. The utility based higher moments asset pricing is derived assuming both two-fund separation and a representative agent. The validity of these assumptions for emerging markets is questioned by Hwang and Satchell (1999). Barone-Adesi (1985) and Barone-Adesi, Gagliardini and Urga (004) (henceforth referred to as BA and BAGU respectively) provide an arbitrage based approach to test the restriction imposed by the APT on the system of a multivariate quadratic model. The modelling and econometric testing framework adopted by BA and BAGU is important since it recasts the covariance-coskewness CAPM as the APT restriction on the system of quadratic market model for which the Gibbons (198) multivariate methodology is readily applicable. Thus they reconcile the two important alternative explanations of expected returns. Their testing approach also avoids the errors-invariables and multicollinearity problems of utility based asset pricing and makes better use of available information by employing the contemporaneous covariance among the asset returns in a multivariate setting. This approach of APT testing is also efficient. Some approaches involve pre-specified macroeconomic variables. BA and BAGU approach uses the information on the return on the stocks and the market portfolio only thereby being less dependent on external macroeconomic data. Despite 3

4 its intuitive appeal there is no application of the BA and BAGU methodology on emerging markets. Relevance of higher moments and their likely impact on expected returns are established to be different in emerging and developed markets. For example, Aggarwal el al. (1999) observe that generally the skewness in the return distribution of emerging market indices is negative while it is positive for developed markets. Moreover there has been evidence to suggest that for emerging market returns the departure from normality is primarily driven by kurtosis and not skewness. In a group of 17 emerging markets including Pakistan, Hwang and Satchell (1999) show that cokurtosis has at least as much explanatory power as coskewness. Further, the wide spread evidence of outliers in emerging market returns suggest that the extreme outcomes have a higher probability of occurrence in emerging markets. BA and BAGU point out the possibility of a missing systematic factor in their pricing model. They did not consider cokurtosis as a potential explanatory variable of asset returns. According to the stylized facts of emerging markets returns it appears that cokurtosis might be a useful factor for such markets. The purpose of this paper is to extend the multivariate methodology of BAGU to incorporate cokurtosis and provide empirical evidence from an emerging market. BAGU adopt a Pseudo- maximum likelihood approach to estimate the model parameter and test the model through an asymptotic least square methodology that does not rely on the normality assumption. Nonnormality of returns is an important consideration when modelling emerging market returns as the microstructure and relatively turbulent political and economic environment makes the normality assumption difficult to justify. The specification test for the mean return equation in BAGU did not support a cubic market return factor. 4

5 The APT does not prescribe the factors that should be included in the factor space 3. Fama and French (1993, 1996) suggest size and book-to-market portfolios returns as potential APT factors. For developed capital markets several authors have compared the Fama-French factors and higher order co-moments in explaining asset returns. For example, in the US market using CRISP portfolios Chung, Johnson and Schill (006) find that Fama-French factors ceased to be effective in explaining asset returns when the first 10 co-moments are incorporated. Therefore they conclude that Fama-French factors may proxy for higher order co-moments. Using Fama-French size portfolios, BAGU report that the size factor anomaly appears to be resolved by incorporating coskewness in the pricing model. On the other hand for a sample of UK data Hung, Shackleton and Xu (004) provide limited evidence in favour of higher order market factors associated with coskewness and cokurtosis compared to the Fama-French factors. To our knowledge no study has investigated relative performance of different APT factors in emerging markets. In this paper we investigate whether the Fama- French factors or the higher co-moment market factors explain portfolio returns better for an emerging market. We compare the performance of alternative factor models statistically and on the basis of economic significance. In the empirical analysis we consider the Karachi stock market which is the largest stock exchange in Pakistan 4. This market has received considerable attention in recent years when in 00 it was declared as the best performing stock market in the world in terms of the percentage increase in the local market index value. We investigate whether an asset pricing model with higher co-moments is able to explain risk-return relation in this emerging market. 3 The BA and BAGU provide a heuristic approach of linking the quadratic term as an APT factor. 4 Karachi Stock Exchange is the largest of the three stock markets in Pakistan. On April 17, 006 the market capitalization was US$ 57 billion which is 46 percent of Pakistan s GDP for the Fiscal Year (Ref: Pakistan Economic Survey, ). 5

6 Several studies have investigated the Karachi stock market in the recent times. Iqbal and Brooks (007) investigate the alternative risk estimators and their applicability on the asset pricing tests through the Fama-MacBeth procedure using 6 years of data on 89 actively trading stocks from 1999 to 005. In their study Iqbal and Brooks (007) report that certain risk variables including the skewness explain individual stock returns with daily and weekly data. This is confirmed in a broader sample of 13 years ranging from 199 to 006 in Iqbal and Brooks (008). The traditional CAPM is supported only in the most recent 5 year sub-period. However, in a sample of stock returns observed in the period 1999 to 005, a multivariate Gibbons (198) type of Wald and GMM test rejects the Black-CAPM restriction (Iqbal and Brooks, 006). Thus the earlier research on asset pricing on this market is largely inconclusive. This can be partly explained by the microstructure issues that adversely affect asset pricing efficiency. Using the broker-level data from the Karachi stock market Khawaja and Mian (005) highlight some of these issues. The rest of the paper is organized as follows: Section describes estimation and inference for a higher-order co-moment model. Section 3 discusses the framework for comparing the higher-order co-moment model with the Fama French alternative. Section 4 discusses the data. Empirical results are analysed in section 5 and section 6 provides some concluding remarks.. Framework for estimation and inference for higher co-moments model We consider a specification of the return generating process with a quadratic and a cubic market return factor 5. Let R denote an N 1 vector of N asset returns at time t t 5 The framework that we outline here is an extension of the BAGU approach on a return generating process with a quadratic term. 6

7 and Rmt and R ft represent the return of the market portfolio and the risk free rate respectively. The cubic market model can be expressed as: r t = α + β r + γ q + δ c + ε (1) mt mt mt t where r t = Rt R ft is the vector of excess returns, rmt Rmt R ft =, qmt R mt R ft = and c mt 3 mt = R R. The N intercepts are collected in vector α and each of β, γ ft and δ is N 1vector of sensitivities. The ε t assumed to satisfy is the vector of error term which is E( ε ) = 0 and E ε ε ' I ) = Σ () t I t ( t t t The information set I t includes all current and past lagged values of R m and Although γ and δ do not exactly correspond to the usual definition of coskewness and cokurtosis, BAGU argue that they are good proxies for these measures. Further, the multivariate methodology used in the estimation of the cubic market model avoids the problems of error-in-variable in measuring coskewness and cokurtosis and also avoids multicollinearity. Although the cubic market model is only a statistical description of the return generating process, following the arguments of the Kraus and Litzenberger (1976) and BA it can be stipulated that the cubic market model (1) is consistent with the four-moment CAPM. The APT approach of BA involves minimal assumptions about the investor s utility function which can be exploited for modelling the emerging markets risk-return relationship. According to BA the expected asset returns under APT is given by the following linear specification 1 3 R f. E ( ) = β λ + γ λ + δ λ (3) r t Where λ 1, λ and λ 3 are expected excess return on portfolios whose return are perfectly correlated with r m, q m and m c respectively. It is obvious that λ = E( r ). 1 m Assets with a higher value of coskewness decrease the risk of a portfolio with respect 7

8 to large absolute market returns. If skewness of the market returns is positive then λ < 0. Following the arguments of Scott and Horvath (1980) we have λ 0. Applying expectations to (1) and equating with (3) results in the following APT imposed restriction on the coefficients of the cubic market model, 1 ν 3 > α = γ ν + δ (4) where ν = λ E( q )] and ν = λ E( c )]. Therefore the arbitrage equilibrium 1 [ m [ 3 m consistent with coskewness and cokurtosis results in the following expected returns: E ( r ) r + γ q + δ c + ν γ + ν δ (5) t = β mt mt mt 1 As in BAGU a Quasi-Maximal Likelihood (QML) approach for testing (4) can be invoked. In the present context the essential idea of the QML approach is that consistent and asymptotically normally distributed estimators of the parameters can be obtained by correctly specifying the first two moments of the error distribution given in (). The normal log likelihood function for the restricted model can then be constructed to estimate the parameters and perform the inference. The consistency and asymptotic normality is guaranteed even if the likelihood is misspecified; see for example Mittelhammer et al. (000). Thus this approach does not rely upon the assumption of normality of the errors. The wide spread evidence of non-normality of the returns and the compelling reasons to include the higher moments dictate the importance of this normality-robust feature in estimation and inference in an emerging market context. Let B ˆ = [ ˆ α ˆ β ˆ γ ˆ δ ] be the N 4 matrix of the estimates of the parameters. Then QML implies that under assumption () T ˆ d B B) N(0, Σ E( F F ') (6) ( t t 8

9 where F = 1 r q c ]. The joint statistical significance of the coefficients in the t [ m m m unrestricted system (1) can be carried out using a Wald test. For example, the test of the hypothesis H : δ 0 results in the following test statistic: 0 = 1 ˆ 1 ˆ' ˆ d W ( / 3/ ) 1 = T N δ Σ δ χ ( N) ˆ 33 Σ f (7) Here Σˆ f 33 represents the (3,3) element in the inverse of the covariance matrix of f = t r q c [ mt mt mt ]'. The Wald test statistic is adjusted with a finite sample correction suggested in Jobson and Korkie (198). The constrained model (3) involves cross-equation restrictions. The restricted parameters are: [ ˆ' ˆ' ˆ']' = [ T T β γ δ r Hˆ '][ Hˆ Hˆ '] (8) t t t= 1 t= 1 H ˆ [ r q +ν ˆ c + ˆ ]', Z ˆ = [ ˆ γ ˆ δ ] (9) t = mt mt 1 mt ν t t [ ˆ ν ˆ ν ]' = ( Zˆ' Σˆ Zˆ) Zˆ' Σˆ ˆ β r ˆ γ q ˆ δ c ( rt mt mt mt ) (10) The parameters can be iteratively estimated with starting values provided by their unrestricted counterparts. Alternatively the parameters in the restricted model can be estimated by non-linear Feasible Generalized Least Square as discussed in Gallant (1987) and Srivastava and Giles (1987). This approach is also robust to normality. The restriction (4) is tested using an asymptotic least square statistics 6 ~ ~ ˆ 1 ( ˆ α ν ˆ ˆ)' ( ˆ ~ ˆ ~ 1γ ν δ Σ α ν 1γ ν ˆ) δ d W ( / 3/ ) = T N ~ ~ χ ( N ) 1 ' ˆ 1 + λ Σ λ f (11) ~ where λ = ˆ μ + [0 ~ ν ~ 1 ν ]', ˆ μ = [ r q c ]' and [ ~ ~ ν ˆ 1 1 ]' ( ˆ' ˆ) ˆ' ˆ 1 v = Z Σ Z Z Σ ˆ α. mt mt mt In the QML based approach the moments must be correctly specified. This essentially translates into specifying the return generating process correctly. Therefore we consider two other alternative specifications of the return generating process; one that 6 The unrestricted system has 4N parameters and the restricted system has 3N+ parameters. The APT therefore imposes N- restrictions which are employed as the degrees of freedom. 1 9

10 considers only the coskewness and another only the cokurtosis in addition to systematic beta risk. To select the most appropriate return generating specification a joint Wald test on the parameters of unrestricted system (1) is performed. In the literature, the following three-factor Fama and French (199) model is advocated as an alternative to the CAPM where the size (SMB) and the book-tomarket (HML) factors are stipulated as another possible set of APT factors. r t = + β1 rmt + β SMBt + β3 β 0 HML + ε (1) t t 3. Comparison of the factor models To compare the relative performance of the higher order market factors and the Fama- French factors, we consider a J-type non-nested model testing approach originally adopted by Chen (1983) for comparing CAPM and APT and applied by Faff (199) in a multivariate context. Consider an artificial regression it = i rˆ it, HM + (1 αi ) rˆ it, FF r α + e, i = 1,... N and t = 1,... T (13) it The independent variables are the predicted returns from the higher moment (HM) and the Fama French (FF) models respectively. A t-test of the hypothesis that α i is zero can be performed for each portfolio. A rejection of this hypothesis then favours the Fama-French alternative. Similarly the rejection of the hypothesis that α i equals one would favour the higher moment alternative. A more powerful multivariate test as employed by Faff (199) is as follows. The null hypothesis that favours the higher moment factor is H α α =... α 1 (14) 0 : 1 = N = This hypothesis can be tested through a Likelihood Ratio test given by 10

11 Here is the determinant and Σˆ r and T N k Σˆ r LRT = 1 ~ F( N, T N k) (15) N Σˆ u Σˆ u are restricted and unrestricted residual covariance matrices from the multivariate regression (1) respectively and k represent the number of restrictions per equation which is one in the present case. One way of assessing economic significance alternative models is to compare their pricing errors from the equilibrium model. We compare the mispricing of the two competing models for the three set of portfolios constructed according the size, beta and the industry sorts. Finally the model s goodness of fit are compared through univariate and multivariate system based methods. 4. Description of the Data The tests discussed in section III are applied to portfolios formed from a sample of stocks listed on the Karachi Stock Exchange (KSE). The sample period spans 13 ½ years from September 199 to April 006 and includes 16 monthly observations. The data consist of monthly closing prices of 101 stocks and the Karachi Stock Exchange 100 index (KSE-100) and are collected from the DataStream database. The criteria for stocks selection was based on the availability of time series data on active stocks for which the prices have been adjusted for dividend, stock split, merger and other corporate actions. The KSE-100 is a market capitalization weighted index. It comprises top companies from each sector of KSE in terms of their market capitalization. The rest of the companies are picked on the basis of market capitalization without considering their sector. We consider the KSE-100 as a proxy for the market portfolio. The 101 stocks in the sample account for approximately eighty per cent of the market in terms of capitalization. Market capitalization data is not routinely available for all firms in the database. However the financial daily, the 11

12 Business Recorder 7 report information on firms over the recent past. We selected the market capitalization of all selected stocks at the beginning of July 1999 which corresponds roughly to the middle of the sample period considered in the study. We use monthly data and compute raw returns assuming continuous compounding. The 30 day repurchase option rate is used as a proxy for the risk-free rate. To investigate robustness in the empirical results we consider portfolios based on both beta and industry in addition to the size portfolio considered in BAGU. We construct seventeen equally weighted size portfolios. First the stocks are ranked on market capitalization in ascending order. The first portfolio consists of the first five stocks while the rest comprise of six stocks each. The portfolio return is calculated as the equally weighted average return of the stocks in the portfolio. We also construct seventeen beta portfolios which are based on ranking of the stocks on the CAPM beta estimated through the market model. For the industry portfolios the stocks are classified into sixteen major industrial sectors. The sector sizes range from two stocks in the transport sector and thirteen stocks in the communication sector. 8 These sectors serve as natural portfolios. Construction of the Fama French factors requires firm level data on shareholder equity, number of outstanding stocks and market capitalization. The State Bank of Pakistan s document Balance sheet analysis of joint stock companies comprise of annual data on balance sheet items for non-financial firms. For financial firms the data is obtained from the State Bank 9. The market related data on the capitalization and the The industry sectors employed are Auto and allied, Chemicals, Commercial Banks, Food products, Industrial Engineering, Insurance, Oil and Gas, Investment banks and other financial companies, Paper and board, Pharmacy, Power and utility, Synthetic and Rayon, Textile, Textile Spinning and Weaving, Transport and communication and Other /Miscellaneous firms that include tobacco, metal and building material companies. 9 We are thankful to Mazhar Khan for his helpful cooperation in the data access. 1

13 number of outstanding stocks are collected through the financial daily Business Recorder. As the market data are not available for the full sample period the data employed corresponds to roughly the middle of the sample period. The book value is obtained as the net assets of the firms excluding any preferred stocks. The mimicking portfolio of the size and book to market is constructed similar to the Fama and French (1993) methodology. The stocks are allocated to two size portfolios (small and large) depending on whether their market equity is above or below the median. A separate sorting of the stocks classifies them into three portfolios formed using the break points of the lowest 30 %, middle 40 % and the highest 30 % based on their book-to market value. From these independent sorting we construct six portfolios from the intersection of two size and three book to market portfolios (S/L, S/M, S/H, B/L, B/M, B/H). Equally weighted portfolios are constructed for the full sample range. The SMB factor is the return difference between the average returns on the three smallfirms portfolios; ( S / L + S / M + S / H ) / 3 and the average of the returns on three bigfirms portfolios; ( B / L + B / M + B / H ) / 3. In a similar way the HML factor is the return difference in each time period between the return of the two high book-tomarket portfolios; ( S / H + B / H ) / and the average of the returns on two low bookto-market portfolios; ( S / L + B / L) /. The construction in this way ensures that the two constructed factors represent independent dimensions in relation to the stock returns. Table 1 presents some descriptive statistics for excess returns on the size portfolios and the market portfolio. The last two columns report the Jarque-Bera normality test statistic and the associated p-value. The skewness of the market return is negative. This is a common feature in emerging markets (Aggarwal et al., 1999). The returns are very much volatile as observed by their standard deviations. It is generally observed that the source of non-normality is the excess kurtosis. 13

14 5. Results of the empirical analysis A. Higher-order co-moment model Table presents some goodness of fit measures of three alternative systems of unrestricted seemingly unrelated regression equations for the size, beta and industry portfolios. The results show that the model with cokurtosis (system ) has a higher overall average adjusted r-square compared to the model with coskewness only (system 1). This is observed in all of the three types of portfolios. The system with both coskewness and cokurtosis (system 3) has a slightly higher explanatory power than systems 1 and according to Glahn s (1969) squared composite correlation coefficient. Overall in terms of the goodness of fit system 3 is the preferred model. The bottom row of Table 3 reports Wald and F tests for the joint statistical significance of the set of coefficients in the unrestricted model of size portfolios. Jointly the beta coefficients are significantly different from zero. Further there is strong evidence to suggest that cokurtosis is jointly significant. Coskewness as a measure of risk is not jointly significant. Thus the initial diagnostic tests points toward the importance of cokurtosis relative to the coskewness in modelling portfolio returns of the emerging market under consideration. Table 3 presents the seemingly unrelated regression parameters estimates for the size portfolios from the unrestricted system of equation (1). The t-statistic of the coefficients for the individual equations is reported in the parenthesis. The intercept is significantly different from zero only in the largest size portfolio and at the 10 % level. In all but one portfolio the estimated systematic risk (beta) is highly significantly different from zero. The coefficient for the coskewness is not significant in any portfolio. However, the cokurtosis coefficient is significantly different from zero in 1 of the 17 portfolios at the 5 per cent level. This observation further highlights the importance of cokurtosis in explaining asset returns. Brooks and Faff 14

15 (1998) and Holmes and Faff (004) invoke the literature from market timing ability of managed funds to provide an interpretation of the sign of the coefficients in the higher order market model. They consider that the fund s time-varying beta is related to the market return and the squared market return. The gamma coefficient measures the market exposure when the market returns are higher and a lower market exposure when the market returns are lower. The funds with this positive market timing ability are therefore attractive. Similarly the delta coefficient measures the volatility timing ability of the funds. A negative delta implies that investors do not experience any return compensation during high volatility periods and fund managers should seek to avoid market exposure at these times. Interestingly in most cases in Table 3 the gamma coefficients are negative and delta coefficients are negative and significant too- a result that was found for a majority of funds in Holmes and Faff (004) and for a majority of countries in the international asset pricing study in Brooks and Faff (1998). The upper diagonal of Table 4 reports the residual covariance matrix for the unrestricted system (1) and the lower diagonal presents the residual correlation matrix. The pattern in the correlations in Table 4 are similar to that observed in some studies on US data such as Gibbons, Ross and Shanken (1989). Generally the correlation in the residuals among the smaller size portfolios is higher. The correlation between the residuals of the small portfolios with that of the two largest size portfolios are always small and in some cases the correlation is negative. The variances of the residuals are large in the small size portfolios compared to larger size ones. For medium size portfolios no pattern is apparent. Overall contemporaneous correlations are large therefore the multivariate system based estimation and testing procedures are likely to perform better than the two pass type procedures adopted in 15

16 the literature. The results obtained with beta and industry portfolios are available from the authors upon request. Table 5 reports the estimates of the parameters and the associated t-values obtained in the estimation of the system of the cubic market model subject to the restriction of a three factor arbitrage equilibrium. As in Table 3 the coskewness coefficient is not statistically significant from zero in any of the portfolios. The evidence on the significance of cokurtosis is stronger when the estimation is performed subject to the APT restrictions. The cokurtosis coefficient is significant in more portfolios compared to the unrestricted case. Further, the coskewness coefficients in the restricted and the unrestricted models are negative. The estimates of the beta coefficients are slightly higher in the restricted model relative to the unrestricted case. No particular pattern is observed for the coskewness coefficients. The cokurtosis coefficients are slightly lower in the restricted case. Finally in Table 6 we report the QML based test of the restriction imposed by the arbitrage equilibrium for the three portfolio schemes. In all three cases the arbitrage restrictions are not rejected suggesting the appropriateness of higher order co-moments in the emerging market under consideration. Although not reported here the analysis also confirms that the arbitrage equilibrium is supported in system 3 as well. B. Fama-French model Table 7 presents the parameter estimate for the size portfolios in the three-factor Fama-French model. The intercepts are not significantly different from zero indicating that the Fama-French model may adequately explain the variation in returns. The big size portfolios tend to have larger coefficients for the systematic risk (beta) factor. All beta coefficients are significantly different from zero. The coefficients associated with the SMB factor indicates that the returns are nonlinear 16

17 with respect to this factor as the associated coefficients are positive for the small size portfolios while they are negative for the big size portfolios. This indicates a small firm premium effect. The HML factor is negatively related to portfolio returns in all cases and their association is significant in a majority of the size portfolios. The bottom row of Table 7 presents multivariate tests. The joint significance of the intercept is tested through an F-test as developed by Gibbons, Ross and Shanken (1989). The test indicates an overall adequacy of the Fama French factor model and no indication of any abnormal returns. The joint significance of the coefficients of the size and book-to-market factors is tested by a Wald test similar to that in Tables 3 and 5. Jointly the coefficients of the three Fama French factors are significantly different from zero across portfolios. Thus the multivariate tests appear to indicate a stronger joint relationship between portfolio return and the three factors. This is in contrast to the unrestricted coskewness cokurtosis model (Table 3) where one of the factors namely the coskewness is found significant in the multivariate tests. C. Comparing mispricing of alternative models Figures 1-3 plot the annualized estimated pricing errors 10 (the alpha coefficients) of the unrestricted higher co-moment and the Fama-French models for the three types of portfolios. Figure 1 clearly shows that the higher moment factor model results in greater mispricing for the smaller and the larger size portfolios compared to the Fama -French factor model. For portfolios of medium size both models perform well. These observations are contrary to the so called small-firm premium effect found in the developed markets which stipulates that small size firms earn higher average returns compared to the big size firms. However these observations should be treated with caution given that the mispricing estimates are statistically not different firm zero as evident in Tables (3) and (7). Figure 1 reveals that the relationship between average 10 Also known as abnormal returns in the market efficiency literature. 17

18 return and size appears to be quadratic. However, this association is not prominent in the Fama-French model. Based on the mispricing analysis the Fama-French model appears to be better than the high moment model. The evidence is stronger with the beta and industry portfolios as shown in Figures and 3. With industry portfolios the textile spinning and weaving, paper and board and miscellaneous firm portfolios result in greater mispricing than other industrial groups. D. Non-nested test of alternative APT factors models Here we discuss the results of the non-nested test described in section 3 for the two competing factor models for the size portfolios 11. The estimated coefficients α i and the corresponding t-statistics are reported in Table 8. The non-nested test corroborate the findings from the mispricing analysis in that for smaller size portfolios and a few larger portfolios the Fama-French model could not be rejected. The higher co-moment factor model is supported in 5 medium size portfolios. Overall the number of portfolios for which the Fama-French factor model is superior is twice the number for the higher co-moment model. Two portfolios do not favour either of the two models. The multivariate test results reported in the bottom row of Table 8 clearly supports the Fama-French alternative. E. Goodness of fit measures of alternative models Table 9 presents some goodness of fit measures of the two competing factor models. Once again the Fama-French alternative performs slightly better in explaining the variation in portfolio returns in all three types of portfolios. The average coefficient of 11 The beta and industry portfolio results in a similar conclusion. 18

19 determination and the composite correlation coefficient are higher by about 5 per cent for the Fama-French model compared to the higher co-moment alternative Conclusion This paper extends the multivariate test of BAGU for arbitrage pricing with coskewness to incorporate the cokurtosis in the return generating process and provides empirical evidence for an emerging market. The empirical results support the three factor arbitrage pricing restrictions with the common factor representing the systematic risk, the coskewness and the cokurtosis. Comparing the relative importance of the higher comments it is shown that in a system of the cubic market model equation either unrestricted or carrying the arbitrage pricing restrictions the cokurtosis remains to be an important explanatory factor while the impact of coskewness is almost negligible. In the literature Fama French factors are advocated as another possible set of APT factors. The comparative studies of the alternative APT factors are extremely rare for emerging markets. This paper compares the relative merit of the Fama French and the higher co-moment model employing Pakistan s stock market data as a case study. We provide the empirical evidence from a non-nested test and compare the pricing error resulting from the two competing models. Some goodness of fit measures for individual portfolio and the joint multivariate tests are also performed. The empirical analysis prefer the Fama French factors to the higher co-moment factors although the explanatory power of the later model is only slightly less than the former model. This 1 The results are further supported when instead of the observed portfolio returns, empirically extracted factors obtained by Principal Component method are employed as the dependent variables. The squared composite correlation coefficients for this case are and for the higher comoment model and the Fama French model respectively. The empirically extracted factors have been used in classical studies of APT tests such as Roll and Ross (1980). 19

20 conclusion is apparently contradictory to the US study of Chung, Johnson and Schill (006) where the Fama French factors were no longer significant once the first ten comoments were employed. In this study to keep consistency in the number of factors we have employed only first three co-moments. The results however generally points to the fact that rules of the game of the risk return analysis may be different in emerging markets. References Aggarwal, R., Inclean, C., Leal, R., Volatility in emerging stock markets. The Journal of Financial and Quantitative Analysis 34, Barone-Adesi, G., Arbitrage equilibrium with skewed asset returns. Journal of Financial and Quantitative Analysis 0, Barone Adesi, G., Gagliardini, P., Urga, G., 004. Testing asset pricing models with coskewness. Journal of Business and Economic Statistics, Bawa, V., Lindenberg, E., Capital market equilibrium in a mean lower partial moment framework. Journal of Financial Economics 5, Bollerslev, T., Engle, R.F., Wooldridge, J.M., A capital asset pricing model with time-varying covariances. Journal of Political Economy 96, Brockett, P.L., Kahaney, Y., 199. Risk, return, skewness and preference. Management Science 38, Brooks, R.D. and Faff, R., A test of two-factor APT based on the quadratic market model: International evidence. Journal of Studies in Economics and Econometrics, Chen, N.-F., Some empirical tests of the theory of arbitrage pricing. The Journal of Finance 38,

21 Chung, Y.P., Johnson, H., Schill, M., 006. Asset pricing when returns are nonnormal: Fama-French factors versus higher-order systematic co-moments. The Journal of Business 79, Dittmar, R., 00. Nonlinear pricing kernals, kurtosis preference, and evidence from cross section of equity returns. Journal of Finance 57, Faff, R.W., 199. A multivariate test of an equilibrium APT with time varying risk and risk premia in the Australian equity market. Australian Journal of Management 17, Fama, E., French, K.R., 199. The Cross-Section of Expected Stock Returns. Journal of Finance 48, 6-3. Fama, E. and French, K.R., Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33, Fang, H., Lai, T.Y., Cokurtosis and capital asset pricing. The Financial Review 3, Gallant, A.R., Nonlinear statistical models, John Wiley New York. Gibbons, M.R., 198. Multivariate tests of financial models: A new approach. Journal of Financial Economics 10, Gibbons, M.R., Ross, S.A. and Shanken, J., Testing the efficiency of a given portfolio. Econometrica 57, Glahn, H., Some relationships derived from canonical correlation theory. Econometrica 37, Holmes, K. and Faff, R., 004. Stability, asymmetry and seasonality of fund performance: An analysis of Australian multi-sector managed funds. Journal of Business Finance and Accounting 31, Homaifar, G., Graddy, D.B., Equity yields in models considering higher moments of the return distribution. Applied Economics 0,

22 Hung, D.C., Shackleton, M., Xu, X., 004. CAPM, higher co-moment and factor models of UK stock returns. Journal of Business Finance and Accounting 31, Hwang, S., Satchell, S.E., Modelling emerging market risk premia using higher moments. International Journal of Finance and Economics 4, Iqbal, J. and R.D., B., 006. Multivariate tests of asset pricing in emerging markers. Paper presented at Econometric Society Australasian Meetings 006, Alice Springs Australia. Iqbal, J., Brooks, R.D., 007. Alternative beta risk estimators and asset pricing tests in emerging markets: the case of Pakistan. Journal of Multinational Financial Management 17, Iqbal, J., Brooks, R.D., 008. A test of CAPM on Karachi Stock Exchange. International Journal of Business 13, forthcoming. Jobson, J. and Korkie, B., 198. Potential performance and tests of portfolio efficiency. Journal of Financial Economics 10, Jones, C.S., 001. Extracting factors from heteroskedastic asset returns. Journal of Financial Economics 6, Khawaja, A.I. and Mian, A., 005. Unchecked intermediaries: price manipulation in an emerging stock market. Journal of Financial Economics, 78: Kraus, A., Litzenberger, R., Skewness preference and the valuation of risk assets. Journal of Finance 31, Mittelhammer, R.C., Judge, G.C., Miller, D.J., 000. Econometric Foundations, Cambridge University Press. Roll, R. and Ross, S.A., An empirical investigation of the arbitrage pricing theory. The Journal of Finance 35,

23 Ross, S.A., Arbitrage theory of capital asset pricing. Journal of Economic Theory 13, Scott, R., Horvath, P., On the direction of preference for moments of higher order than the variance. Journal of Finance 35, Sears, R.S., Wei, K.C., Asset pricing, higher moments and the market risk premium: A note. The Journal of Finance 40, Sears, R.S., Wei, K.C., The structure of skewness preferences in asset pricing models with higher moments: An empirical test. The Financial Review 3, Srivastava, V.K., Giles, D.E., Seemingly unrelated regression equations models, Marcell Dekker Inc. Tan, K.-J., Risk return and the three moment capital asset pricing model: Another look. Journal of Banking and Finance 15,

24 Table 1: Descriptive statistics of size portfolio returns Portfolio Mean SD Skewness Kurtosis Jarque- P-value Bera (JB) Market Portfolio

25 Table : Goodness of fit measures of the alternative unrestricted system of SUR equations System 1 System System 3 Panel A: Size Portfolio Average R System R Panel B: Beta Portfolio Average R System R Panel C: Industry Portfolio Average R System R = α + β + γ + ε Notes: System 1 (covariance coskewness): rt rmt qmt t System (covariance cokurtosis): rt = α + β rmt + δ cmt + ε t System 3 (covariance coskewness cokurtosis): rt = α + β rmt + γ qmt + δ cmt + ε t The system R-square is Glahn s (1969) squared composite correlation coefficient computed as tr( Yˆ' AYˆ) R = where Y = [ r1 r... rn ] and Y ˆ = [ ˆ1 ˆ... ˆN ] tr( YAY ) r r r are each T N matrices of excess returns and OLS fitted values of the excess returns respectively and A = [ IT jj' ] where j is a T 1 vector of ones. 1 n 5

26 Table 3: Parameter estimates for the unrestricted SUR cubic market model for the size portfolio Portfolio αˆ βˆ γˆ δˆ (-0.70) (-1.04) (-0.68) (-0.67) (-0.46) (-0.86) (0.81) (0.49) (0.07) (0.0) (0.9) (-0.7) (0.40) (-0.0) (-0.73) (-0.59) (-1.8) F/Wald [0.715] (5.8) (1.85) (3.93) (3.01) (7.85) (7.60) (5.96) (6.15) (9.37) (9.59) (8.11) (8.55) (8.30) (10.93) (14.05) (16.49) (14.58) [0.000] (-1.9) (0.91) (-0.83) (-1.05) (-0.14) (-0.57) (-1.69) (-0.09) (-0.88) (-1.14) (-1.47) (-1.58) (-0.39) (-1.4) (0.90) (-0.95) (0.60) [0.418] (-.60) (-0.3) (-0.91) (-1.48) (-.65) (-3.35) (-3.08) (-1.95) (-4.97) (-3.19) (-3.73) (-3.75) (-.57) (-4.03) (-3.0) (-.90) (-1.7) [0.000] Notes: the t-statistics of the parameter estimates are reported in the parenthesis and the p-values of the Wald test are given in the square bracket. The test for intercept is the F test proposed by Gibbons, Ross and Shanken (1989) which is robust to non-normality in small samples. The test used for the remaining coefficients are the Wald tests. 6

27 Table 4: The residual variance covariance and correlation matrix for the unrestricted system of cubic market model for size portfolios Portfolio

28 Table 5: Parameter estimates for the restricted SUR cubic market model for the size portfolio Portfolio βˆ γˆ i i δˆ i (5.40) (-1.80) (-.84) (.00) (0.50) (-0.58) (4.05) (-1.31) (-1.1) (3.1) (-1.54) (-1.70) (7.95) (-0.34) (-.81) (7.75) (-1.0) (-3.61) (5.88) (-1.4) (-.94) (6.10) (0.1) (-1.85) (9.36) (-0.79) (-4.96) (9.65) (-1.1) (-3.4) (8.11) (-1.41) (-3.71) (8.63) (-1.8) (-3.88) (8.6) (-0.16) (-.50) (10.98) (-1.30) (-4.09) (14.18) (0.69) (-3.0) (16.67) (14.86) (-1.3) (-0.5) (-3.31) (-.0) Notes: the t-statistics of the parameter estimates are reported in the parenthesis 8

29 Table 6: QML test statistic and the Chi Square P-values Portfolio QML test statistic P-value Size Beta Industry

30 Table 7: Parameter estimates for the SUR system of Fama-French model for the size portfolio Portfolio ˆ β ˆ 0 β ˆ 1 β ˆ β (-1.14) (5.994) (4.081) (-0.870) (-0.00) (5.478) (7.366) (-.65) (-0.551) (5.654) (4.740) (-4.853) (-1.3) (5.778) (5.816) (0.469) (0.188) (8.178) (3.684) (-3.499) (-0.79) (6.73) (3.496) (-5.896) (0.16) (5.351) (.038) (0.16) (1.078) (6.107) (.6) (-.79) (0.86) (5.8) (0.953) (-.671) (-0.153) (6.983) (-0.935) (-.7) (0.455) (4.119) (-1.159) (-6.474) (-0.8) (5.445) (-0.888) (-1.698) (0.465) 0.41 (6.84) (-0.98) (-0.664) (0.157) (6.7) (-.161) (-6.175) (0.83) (11.47) (-1.818) (-3.33) (-0.95) (14.013) (-5.11) (-3.456) (-1.39) (1.768) (-5.684) (-5.81) F / Wald [0.88] [0.000] [0.000] [0.000] Note: the t-statistics of the parameter estimates are reported in the parenthesis and the p-values of the Wald and F tests are given in the square bracket. The test for intercept is the F test proposed by Gibbons, Ross and Shanken (1989) which is robust to non-normality in small samples. The test used for the remaining coefficients are the Wald tests. 30