SKEWNESS AND THE RELATION BETWEEN RISK AND RETURN. Panayiotis Theodossiou and Christos S. Savva

Transcription

1 SKEWNESS AND THE RELATION BETWEEN RISK AND RETURN Panayiotis Theodossiou and Christos S. Savva Department of Commerce, Finance and Shipping, Cyprus University of Technology, Cyprus Forthcoming in Management Science Abstract The relationship between ris and return has been one of the most important and extensively investigated issues in the financial economics literature. The theoretical results predict a positive relation between the two. Nevertheless, the empirical findings so far have been contradictory. Evidence presented in this paper show that these contradictions are the result of negative sewness in the distribution of portfolio excess return and the fact that the estimation of intertemporal asset pricing models are based on symmetric loglielihood specifications. Keywords: Ris-return tradeoff; SGT distribution; GARCH-M JEL Classification: C8, C, G5

2 . Introduction The financial and economic literature on the relationship between ris and return is voluminous and the findings thus far have been inconclusive. Many well nown scholars have found a positive, others a negative and an equal number no relationship. For example, a significant positive ris-return relation for the US is reported in French et al. (987), Lundblad (007) and Lanne and Saionen (007), a significant negative relation in Glosten et al. (993), an insignificant one in Nelson (99), Campbell and Hentschel (99), Glosten et al. (993), Theodossiou and Lee (995) and Bansal and Lundblad (00) and mixed findings in Baillie and DeGennaro (990). In standard intertemporal capital asset pricing models, stochastic factors influence the investment opportunity set and through that the equilibrium ris premia of financial assets, e.g., Merton (973). These factors trigger fluctuations in the ris-return tradeoff and as such, they are a source of sewness and urtosis when returns are computed over discrete time intervals. Because investors hedge constantly against such fluctuations, higher moments are liely to be priced. This paper investigates the impact of sewness and urtosis on the ris-return relationship using an analytical framewor based on the popular sewed generalized t (SGT) distribution, e.g., Theodossiou (998). The SGT distribution is chosen because of its flexibility in modeling fat-tails, peaness and sewness, often observed in financial data. Furthermore, it includes several well nown symmetric distributions used in the finance literature, such as the generalized t (GT), generalized error (GED), student t (T) and normal (N), e.g., Bali and Theodossiou (008) and Hansen et al. (00).. Impact of Sewness on the Pricing of Ris SGT Framewor The Intertemporal relationship between ris and returns is investigated using the GARCH-in-mean process, which has been the standard in the literature, e.g., Engle et al. (987) and Glosten et al. (993). Contradictory findings are also reported in studies using other methodologies, such as Scruggs (998), Harrison and Zhang (999), Bali and Peng (006), Ludvigson and Ng (007), Pastór et al. (008) and Chan et al. (99). The SGT distribution has been used widely in finance for computing VaR measures, pricing options and estimating asset pricing models. It is also incorporated in econometric pacages such as GAUSS.

3 That is, a portfolio s excess returns are specified as: r c abr u, () t t t t where t = var(r t I t ) is the conditional variance of r t based on the information set I t available prior to the realization of r t, r t is past value of excess returns included in I t, a and b are typical regression coefficients and c, also nown as the GARCH-in-mean coefficient, lins σ t to μ t. For practical purposes and without loss of generality, a single lag value of r t is used. Under the SGT framewor, r t is modeled as where n n, u t t t t n sign( u ) t f r I B t t t t t t t n, () u r m r c a br (3) are deviations of returns r t from their conditional mode m t (in the case of the symmetric GT the mean and the mode are equal). The scaling parameter t is a time-varying dispersion measure related to σ t when it exists, and n are positive urtosis parameters controlling respectively the peaness around the mode and the tails of the distribution, λ is a sewness parameter with domain the open interval (, ), sign(u t ) is the sign function (i.e., sign(u t )= for u t < 0 and for u t > 0) and B(w, z) = Γ(w) Γ(z) / Γ(w+z) is the beta function. Values of < are associated with leptourtic (peaed) distributions relative to the normal distribution and smaller values of n with fat tailed distributions. The SGT includes the GT of McDonald and Newey (988) for λ = 0, the sewed t of Hansen (994) for =, the student t for λ = 0 and = and the Cauchy for λ = 0, = and n =. For n, it yields the sewed GED of Theodossiou (000), which includes the GED for λ = 0, the Laplace or double exponential for λ = 0 and =, the normal for λ = 0 and = and the uniform for λ = 0 and. When n >, the conditional mean and variance of r t, see eq. (A9) and (A) in the Appendix, are: and E r I m E u I m p (4) t t t t t t t t u I A A var, (5) t t t t

4 where and p A A A, (6) n n n A B, B, 3 3,, n n n A B B. (8) (7) It follows easily from eq. (4) that the parameter p = (μ t m t ) / σ t. This measure, nown as the Pearson s sewness, is a symmetric function of the sewness parameter λ and a highly non-linear function of the urtosis parameters and n. This is a ey measure for the issues investigated in this paper. Figure provides a graphical illustration of the parameter p for various values of λ, and n. It is clear from the figure that as λ increases in magnitude, p also increases in magnitude. Negative values of λ are associated with negative values of p and vice versa. For λ =0, p =0. Interestingly, larger values of and n are also associated with larger values of p. Clearly, p is a monotonic function of λ, and n. Intertemporal Pricing Model The substitution of mt c t a brt into the conditional mean excess return eq. (4) gives t ct pt a brt, (9) where c σ t is the pure ris premium, which is expected to be positive and p σ t is the sewness-urtosis premium. The latter, depending on the direction of sewness in the distribution of excess returns, can be negative, zero or positive. The regression in () can be written in the following equivalent form + r c p a br. (0) t t t t t t Unlie u t, the error term ε t = u t p σ t has a zero expected value. Note that the term (c + p) σ t = ξ σ t measures the combined impact of pure and sewness-urtosis ris on the mean of a portfolio s excess returns. This equation provides the foundation for exploring and explaining the contradictory findings in the literature regarding the ris-return relationship. In case of negative sewness, depending on the size of p, the value of ξ can be positive, zero or negative. 3

5 Conditional Variance The conditional variance of excess returns is specified as a function of the past regression errors, including their squared, absolute and standardized values and past conditional variances. The following four popular GARCH models are considered: GARCH of Bollerslev (986): v (a) t t t GJR-GARCH of Glosten, et al. (993): v N (b) t t t t QGARCH of Sentana (995): v (c) t t t t EGARCH of Nelson (99): ln v z g z ln (d) t t t t where N t = 0 for ε t > 0 and N t = for ε t < 0 and g(z t ) = z t E z t and z t = ε t / σ t. In eq. (b)-(d), the parameter δ captures asymmetric volatility. In eq. (b), δ is expected to be positive and in (c) and (d) negative. Such values would imply that volatility is higher in stoc maret downturns than in upturns. This ind of asymmetry is typically an indication of negative sewness in the distribution of excess returns. Estimation Parameter estimates for eq. () are obtained via numerical optimization of the sample log-lielihood T max L log f r, I t t, () t where f is a conditional probability density function for r t and θ = [ c, a, b, ν, δ, β, γ, λ,, n]'. For eq. (a), δ = 0. The t-values for the estimators are computed using robust standard errors. Moreover, the parameter p is endogenously determined using the MLE estimators for λ, and n along with eq. (6) (8). At this point, it is important to note that in the presence of sewness, the estimation of eq. () using a symmetric log-lielihood specification will result in a biased estimator for the price of pure ris or the GARCH-in-mean effect, measured by c. In fact, the resulting estimator will be ξ = c + p and not c. Moreover, the computed standardized errors will not possess a zero mean and a unit variance and the conditional variances will be misspecified. 4

6 3. Monte Carlo Simulations Random Samples For the simulations,000 samples per GARCH model are used. Each sample includes,05 randomly generated returns using similar parameters to those of the monthly models estimated in the next section. The returns are generated as follows:. A vector z= [z 0,z,..,z T ] of,053 standardized random errors is drawn from the SGT distribution below: z~ SGT 0,, 0.85,, n 0, with Pearson s sewness p = 0.43, standardized sewness SK = and standardized urtosis KU = The latter values are computed using eq. (6), (A5) and (A8), respectively.. The arithmetic mean and standard deviation of monthly excess returns are used as starting values for σ t μ t, ε t and r t. That is, σ 0 = 5.64, μ 0 = 0.50, z0and r Random monthly excess returns for each of the four GARCH specifications are generated using the recursive equations below: GARCH: t t t GJR-GARCH: N t t t t QGARCH: t t t t EGARCH: z z E z ln ln t t t t t z t t t r r, t t t where N t = for ε t < 0 and zero otherwise and t =,,,,05. Note that the parameter ξ in the return equation above is set to zero, i.e., c = p =

7 Simulation Results Eq. () for each GARCH model is estimated,000 times using the randomly generated samples of excess returns and the log-lielihood specifications of a) SGT, b) GT, c) GED, d) T and e) N. The first panel of Table reports the arithmetic mean and standard deviation of the estimated values of the pure price of ris (parameter c) and the sewness-urtosis price of ris (parameter p) for each of the four SGT-GARCH specifications in the,000 random samples and the percentage of times that estimates of c, p and ξ are statistically different from their theoretical values. The remaining panels report similar statistics for the (contaminated) pure price ris, c, of the four GARCH specifications under the symmetric log-lielihood specifications of GT, GED, T and N. Interestingly, in the case of the SGT distribution, the simulated values of the parameters c and p are extremely close to their respective theoretical values of 0.43 and 0.43 for all four GARCH models. The SGT simulation results uncover fully the hypothesized positive relationship between ris and return. This is, however, not the case with the estimated values of c based on the GT, GED, T and N log-lielihood specifications, where the estimated values of c are close to the theoretical value of ξ = c + p = 0. Generally, these simulation results indicate that, depending on the extent of negative sewness in the distribution of excess returns, the estimation of intertemporal pricing models using symmetric loglielihood specifications may yield conflicting findings regarding the ris-return relationship. 4. Empirical Findings Preliminary Statistics The data includes daily, weely and monthly value-weighted excess returns over the one-month Treasury bill rate of all CRSP firms incorporated in the US and listed on the NYSE, AMEX, or NASDAQ. It covers the period July, 96 to February 8, 04 and is obtained from French s website. 3 It is transformed into continuously compounded returns using the equation r 00 log y 00 where y t is a geometric return expressed as a percentage and log is the natural logarithm. t, t 3 6

8 Preliminary statistics for the daily, weely and monthly frequencies of CRSP excess returns are reported on Table. It is worth noting that all three frequencies exhibit negative sewness and standardized urtosis ranging between 6. and 7, which is at least twice that of the normal distribution, which is equal to 3. These findings along with the KS statistics indicate serious departures of the data from normality. 4 Main Results Table 3, 4 and 5 present the parameter estimates of the intertemporal excess return equation in () for all four GARCH models under the SGT log-lielihood specification based on daily, weely and monthly data. The results are quite similar for all GARCH models and frequencies. In all cases the estimated values for the pure price of ris coefficient, c, are positive and statistically significant. On the other hand, the estimated values for the sewness-urtosis price of ris, p, are negative and statistically significant. The combined impact of the two, measured by ξ = c + p, is close to zero and statistically insignificant in all cases. Interestingly, the asymmetric volatility coefficient, δ, is statistically significant and consistent in all three cases with the presence of asymmetric volatility. Despite the fact that the form of negative sewness implied by asymmetric volatility is factored out to some extent by the conditional variance equations, still the results indicate that the main force driving the mixed findings in the literature regarding the ris-return relationship is sewness in the return distribution coupled with fact that the estimation of the relevant pricing models is based on symmetric log-lielihood specifications. Replication of Previous Studies To investigate further the conflicting findings in the literature, the results of several important studies related to the US are replicated using the same data frequencies and periods and similar GARCH and log-lielihood specifications. The replications are performed for studies comparable to the framewor employed in this paper. 4 Data are Winsorized to plus/minus four standard deviations from their sample means. 7

9 The results regarding the estimated combined ris-sewness effect (parameter ξ = c + p), presented in panel A of Table 6, are qualitatively similar to those found in the original studies. There are, however, some minor differences which may be attributed to the omission of explanatory variables, which could not be replicated and the use of robust standard errors for the estimators, which may not have been used in previous studies. The results for the SGT specifications, presented on Panel B of Table 6, indicate a positive pure ris premium and a negative sewness-urtosis premium. In both cases, the relevant coefficients are statistically significant at the % level, while their combined impact, measured by ξ = c + p, is in line with that of panel A. Once more, the results point out to sewness as being the reason for the conflicting findings in the literature on the ris-return relation. 5 Other Ris-Return Specifications The investigation into the ris-return relationship is continued by augmenting eq. () with an extra term for the lag-value of the conditional standard deviation of returns, i.e. rt c t d t abrt ut. (3) The estimated parameters for the three data frequencies and the four GARCH models using an SGT loglielihood specification are presented on Table 7. Notice that the estimated values of d are quite mixed in terms of their signs and statistical significance. These results are liely to be triggered by the strong correlation between σ t and σ t, which ranges from 0.95 to Note, however, that the estimated values of the sewness-urtosis coefficient p and the combined impact ξ = c + d + p are almost identical to those on Tables 3, 4 and 5. The next step was to run the model with the lag value of σ t, only. The results, which are available upon request, are almost identical to those of the aforementioned tables. Eq. () is further augmented with the inclusion of the conditional variance, i.e. r c d abr u. (4) t t t t t 5 The replication was also performed on the entire dataset. The results, however, are qualitatively similar. 8

10 The estimated parameters are presented on Table 8. Note that in almost every case the estimated values of d are positive and statistically insignificant. Nevertheless, the estimated values of p are in line with the previous results. Prices of Ris across Data Frequencies The stochastic behavior of pure and sewness-urtosis prices of ris across data frequencies requires further investigation. For this purpose, the pricing model of eq. (9) is rewritten as c p a b r, (5) tt, t t tt, t t tt, where Δt denotes the time length between two consecutive excess returns. For example, for daily, weely and monthly returns Δt = /5, 5/5 and /5, respectively, where 5 is for the trading days in each year. Clearly, all parameters in the above conditional mean excess return equation, including those for the conditional standard deviation, depend on the data frequency indicated by Δt. Eq. (5) is also written as where * tt, tt, t and *,, c p a b r t t t * t t * t t tt, tt, tt, tt tt t, (6) are annualized measures of the conditional mean and * * conditional standard deviation of excess returns for each data frequency. Moreover, t E t, t * * t t andt E t, t t t, where t and t are respectively the unconditional expected values of tt, and. tt, Under normality, or more generally when returns are Levy processes, E * * tt, and E tt * *,, regardless of Δt. In general, departures from these equalities occur when higher order moment dependencies (non-linearities), including sewness and urtosis, are present in the return series. It follows easily from eq. (6), under standard regularity conditions, that ct pt at t b t bt * * t t, (7) t 9

11 where and * t t t c c b t, * t t t p p b t * * * t c t p t are annualized measures for the pure and sewness-urtosis prices of ris. Figure presents the estimated values of the above annualized measures for the GJR model and data frequencies ranging from one to twenty-one trading days. For all data frequencies, the pure price of ris c * t is positive, the sewness-urtosis price of ris p * t negative and their combined value * t close to zero. Figure 3 presents their t-values, denoted by t c, t p and t ξ, respectively. All t c and t p values lie outside the interval ±.96, therefore, c and * t p * t are statistically significant. On the contrary, the t ξ values lie inside the interval indicating that is statistically insignificant across all data frequencies considered. Interestingly, as Δt gets larger, c and * t * t p * t decrease in magnitude and at the monthly frequency radiates around ±.6. The fact that these estimates are larger in higher data frequencies may be attributed to the presence of strong moment dependencies, which become weaer in lower data frequencies. For the remaining GARCH models, the results are remarably similar. For each data frequency, the time-series behavior of λ (main determinant of investigated using Hansen s (994) specification * p t, see eq. 6), is. (8) t 0 t t The estimates for λ range between 0.07 and and for λ between and These estimates are statistically insignificant, except for that of λ at the daily data frequency. Similar findings are documented for the remaining two determinants of results indicate that * p t is not time-varying within each data frequency. p * t, i.e., the urtosis parameters and n. These 0

12 The results in this sub-section are consistent with the hypothesis that because investors hedge constantly against fluctuations in the distribution of stoc returns, higher moments, including sewness and urtosis, are priced more in the short- than in the long-term. 5. Summary and Conclusions The theoretical analysis carried out using an analytical framewor based on the popular SGT distribution indicates that the conditional mean of a portfolio s excess returns is a function of a pure ris premium, which is expected to be positive and a sewness-urtosis premium which has the same sign as sewness. Depending on the extent of negative sewness, the combined size of the latter premia can be positive, zero or negative. Estimation of intertemporal pricing models using log-lielihood specifications based on symmetric probability distributions results in price of ris measures contaminated by that of the sewness-urtosis price of ris. This is in fact the main reason behind the contradictory findings in the finance literature regarding the ris-return relationship. The latter finding is confirmed via Monte Carlo simulations. Estimation of standard intertemporal pricing models based on a SGT log-lielihood specification using daily, weely and monthly CRSP excess returns over a long period, confirms the presence of a positive ris premium and a negative sewness-urtosis premium with zero combined impact. Replication of the results of previous studies using the same data frequencies and periods and model specifications yields mixed results when estimated using symmetric log-lielihood specification and similar results, as the ones presented previously, when estimated using a sewed log-lielihood specification.

13 Acnowledgements The authors are grateful to George Constantinides, Yannis Yatracos, Denise R. Osborn, Andrew Harvey, Neophytos Lambertides, Netarios Aslanidis, and Konstantinos Tolias, as well as seminar/conference participants at the MFS Conference 03, Sofie/INET Conference - Cambridge 04, University of Aarhus - CREATES seminar series, 4 th International Conference of the Financial Engineering and Baning Society and Conference in honour of Denise Osborn - The University of Manchester. The authors are especially grateful for the insightful comments of one referee, an associate editor, and department editor Jerome Detemple.

14 References Baillie RT., De Gennaro RP (990) Stoc returns and volatility. Journal of Financial and Quantitative Analysis 5: Bali TG, Peng L (006) Is there a ris-return trade-off? Evidence from high-frequency data. Journal of Applied Econometrics : Bali TG, Theodossiou P (008) Ris measurement performance of alternative distribution functions. The Journal of Ris and Insurance 75(): Bansal R, Lundblad C (00) Maret efficiency, asset returns, and the size of the ris premium in global equity marets. Journal of Econometrics 09: Bollerslev T (986) Generalized autoregressive conditional heterosedasticity. Journal of Econometrics 3: Campbell JY, Hentschel L (99) No news is good news: An asymmetric model of changing volatility in stoc returns. Journal of Financial Economics 3: Chan KC, Karolyi GA, Stulz RM (99) Global financial marets and the ris premium on US equity. Journal of Financial Economics 3: Engle RF, Lilien DM, Robins RP (987) Estimating time varying ris premia in the term structure: The ARCH-M model. Econometrica 55: French KR, Schwert GW, Stambaugh RF (987) Expected stoc returns and volatility. Journal of Financial Economics 9: 3-9. Glosten LR, Jagannathan R., Runle DE (993) On the relation between the expected value of the volatility of the nominal excess return on stocs. Journal of Finance 48: Hansen, BE (994) Autoregressive Conditional Density Estimation. International Economic Review 35: Hansen JV, McDonald JB, Theodossiou P, Larsen BJ (00) Partially adaptive econometric methods for regression and classification. Computational Economics 36: Harrison P, Zhang HH (999) An investigation of the ris and return relation at long horizons. The Review of Economics and Statistics 8(3): Lanne M, Saionen P (007) Modeling conditional sewness in stoc returns. The European Journal of Finance 3(8):

15 Ludvigson SC, Ng S (007) The empirical ris-return relation: A factor analysis approach. Journal of Financial Economics 83: 7-. Lundblad C (007) The ris-return trade-off in the long run: Journal of Financial Economics 85: McDonald JB, Newey WK (988) Partially adaptive estimation of regression models via the generalized t distribution. Econometric Theory 4: Merton RC (973) An intertemporal capital asset pricing model. Econometrica 4: Nelson D (99) Conditional heterosedasticity in asset returns: A new approach. Econometrica 45: Pastór L, Sinha M, Swaminathan B (008) Estimating the intertemporal ris-return tradeoff using the implied cost of capital. Journal of Finance 63(6): Rapach D, Wohar ME (009) Multi-period portfolio choice and the intertemporal hedging demands for stocs and bonds: International evidence. Journal of International Money and Finance 8: Sentana E (995) Quadratic ARCH models. Review of Economic Studies 6: Scruggs TJ (998) Resolving the puzzling intertemporal relation between the maret ris premium and conditional maret variance: A two-factor approach. Journal of Finance 5: Theodossiou P (000) Sewed generalized error distribution of financial assets and option pricing. SSRN woring paper, Theodossiou P (998) Financial data and the sewed generalized T distribution. Management Science 44(): Theodossiou P, Lee U (995) Relation between volatility and expected returns across international stoc marets. Journal of Business Finance and Accounting ():

16 Appendix Mean, Variance, Sewness and Kurtosis of the SGT The j th non-centered moment, for 0 j n, is M j j u f du j j u Cu 0 n n j u Cu 0 n where u = r m,, n > 0 and < λ <. The substitution of du n du, (A) and into M j gives u n t t, (A) j j j j j j udu n t t dt (A3) j j n j j j j j M j C n t t dt, j j j j j j n j C n B,, (A4) j n j j n j B, t t dt. where For f to be a proper p.d.f., M 0 =, and The substitution of (A5) into (A4), gives 0 n n C 0.5 B, 0. (A5) where and j M A. (A6) j j j j j A j G j (A7) j n j n j n Gj B, B,, for j =,,,< n. (A8)

17 Expected Value of u For j = and n >, where Variance For j = and n >, where The variance of u is M Eu A, (A9) A G G. M Eu A, (A0) A 3 3 G 3 G. var u M M A A, (A) where A Substitution of A 0. A A into (A9) gives M Eu A A A p. (A) Third Centered Moment and Sewness For j = 3 and n > 3, 3 3 M 3 Eu A3, (A3) where A The third centered moment of u is The standardized sewness is 4 4 G G A3 3AA A E u Eu Eu Eu Eu Eu E u SK Fourth Centered Moment and Kurtosis For j = 4 and n > 4, 3 3 Eu A3 3AAA 3 3 A A. (A4). (A5) where M Eu A, (A6)

18 A G 4 4 The fourth centered moment of u is The standardized urtosis is 0 5 G E u Eu Eu Eu Eu Eu Eu Eu A 4A A 6A A 3 A. (A7) E u KU 4 4 Eu A4 4A3A 6AA 3A 4 A A. (A8) 7

19 Table. Monte Carlo Simulations of the Intertemporal Asset Pricing Model Distribution Statistics GARCH-M GJR-GARCH-M QGARCH-M EGARCH-M A. SGT - Sewed Generalized t c Mean Std Reject rate, % p Mean Std Reject rate, % ξ = c + p Mean Std Reject rate, % B. GT - Generalized t c Mean Std Reject rate, % C. GED - Generalized Error Distribution c Mean Std Reject rate, % D. T - Student t c Mean Std Reject rate, % E. N - Normal c Mean Std Reject rate, % Notes: Simulations are based on,000 random samples with theoretical values for c = 0.43 and p = Mean and Std are the simple arithmetic mean and standard deviation of the estimates in the random samples. Reject rate is the percent of estimates which are statistically different from their theoretical values.

20 Table. Preliminary Statistics for CRSP Excess Returns Statistics Daily Weely Monthly Mean Std Sewness Kurtosis KS OBS 3,73 4,574,05 Notes: Data covers the period July, 96 - February 8, 04. Excess returns are continuously compounded. Sewness is equal to m 3 / m 3/ and urtosis m 4 / m, where m j is the sample estimate for the jth moment around the mean. KS is the Kolmogorov-Smirnov statistic.

21 Table 3. Excess Return Pricing Model Estimation Using Daily Data Parameters GARCH-M GJR-GARCH-M QGARCH-M EGARCH-M A. Conditional mean a * (0.0) (0.0) (0.0) (0.0) b (0.007) (0.007) (0.007) (0.007) c (0.0) (0.0) (0.0) (0.0) B. Sewness price of ris p [ ] [ ] [ ] [ ] ξ = c + p 0.0# 0.00# 0.00# -0.0# (0.08) (0.07) (0.08) (0.07) C. Conditional variance v (0.00) (0.00) (0.00) (0.00) β (0.005) (0.003) (0.005) (0.008) δ (0.009) (0.006) (0.005) γ (0.006) (0.006) (0.006) (0.00) D. Unconditional mean and standard deviation μ σ E. Sewness and urtosis parameters λ (0.008) (0.008) (0.008) (0.008) n F. Other statistics L (θ) -7, ,90.7-6, ,86.4 SK KU OBS 3,73 3,73 3,73 3,73 Notes: Estimation of eq. under the SGT log-lielihood specification for each GARCH model. All coefficients are significant at the % level, unless otherwise noted. *, ** Statistically significant at the 5% and 0%, respectively. # statistically insignificant. Parentheses include the standard errors and bracets the confidence intervals for the estimators Confidence Intervals for p are based on Rapach and Wohar (009) bootstrapping procedure L(θ) is the log-lielihood value. SK and KU are standardized sewness and urtosis, computed using eq. (A4) and (A8) Computation of unconditional variance of excess returns GARCH & QGARCH GJR-GARCH EGARCH σ =ν /( -β-γ) σ =ν /( - 0.5δ -β-γ) σ exp(ν /(-γ )) Computation of unconditional mean return μ =(α +ξ σ)/(-b ) c is the pure price of ris and p the sewness-urtosis price of ris.

22 Table 4. Excess Return Pricing Model Estimation Using Weely Data Parameters GARCH-M GJR-GARCH-M QGARCH-M EGARCH-M A. Conditional mean a 0.3# 0.68* 0.49** 0.79* (0.086) (0.085) (0.088) (0.084) b -0.00# 0.08# 0.08# 0.07# (0.06) (0.06) (0.06) (0.06) c (0.060) (0.059) (0.06) (0.058) B. Sewness price of ris p [ ] [ ] [ ] [ ] ξ = c + p 0.03# # # -0.08# (0.048) (0.047) (0.050) (0.045) C. Conditional variance v (0.09) (0.07) (0.043) (0.009) β (0.03) (0.009) (0.06) (0.00) δ (0.04) (0.045) (0.0) γ (0.05) (0.09) (0.0) (0.006) D. Unconditional mean and standard deviation σ μ E. Sewness and urtosis parameters λ (0.0) (0.0) (0.0) (0.0) n F. Other statistics L (θ) -9, ,508-9,50.4-9,49.5 SK KU OBS 4,574 4,574 4,574 4,574 Notes: See Table 3

23 Table 5. Excess Return Pricing Model Estimation Using Monthly Data Parameters GARCH-M GJR-GARCH-M QGARCH-M EGARCH-M A. Conditional mean a 0.545# 0.70# 0.68# 0.605# (0.488) (0.59) (0.53) (0.464) b -0.00# 0.04# 0.06# 0.06# (0.030) (0.034) (0.033) (0.034) c (0.5) (0.34) (0.37) (0.9) B. Sewness price of ris p [ ] [ ] [ ] [ ] ξ = c + p 0.036# -0.0# # # (0.) (0.5) (0.3) (0.094) C. Conditional variance v *.500* 0.69 (0.74) (0.5) (0.65) (0.048) β ** (0.00) (0.06)** (0.03) (0.046) δ - 0.3* * (0.063) (0.97) (0.06) γ (0.05) (0.039) (0.04) (0.06) D. Unconditional mean and standard deviation σ μ E. Sewness and urtosis parameters λ (0.04) (0.043) (0.043) (0.048) n F. Other statistics L (θ) -3,06.7-3, , ,053.8 SK KU OBS,05,05,05,05 Notes: See Table 3.

24 Table 6. Replication of the Results of Previous Studies Model and Author(s) Pdf Dataset, span and frequency A. Original Specification B. SGT Specification Ris-return relation found ξ = c + p ξ = c + p c p. GARCH-M French et. al. (987) N CRSP /98 - /984 Monthly Signif. Posit # 0.08# 0.407* (0.9) (0.43) (0.60) [ ] Closten et. al. (993) N CRSP 4/95 - /989 Monthly Insign. Posit # 0.05# 0.447* (0.6) (0.8) (0.83) [ ] Baillie and DeGennaro (990) N CRSP /970 - /987 Daily Signif. Posit. 0.08** 0.079# 0.3* (0.06) (0.059) (0.065) [ ] Baillie and DeGennaro (990) T CRSP /970 - /987 Daily Insign. Posit # 0.079# 0.3* (0.058) (0.059) (0.065) [ ] Theodossiou and Lee (995) N S&P500 /976 - /99 Weely Insign. Posit. 0.65# 0.6# # (.564) (0.383) (0.40) [ ] Bansal and Lundblad (00) N DataStream /973 - /998 Monthly Insign. Posit..4#.04#.5# (0.753) (.83) (.97) [ ] Lanne and Saionen (007) T S&P500 /946 - /00 Monthly Signif. Posit * 0.86* 0.73* (0.43) (0.30) (0.34) [ ] Lundblad (007) N CRSP /836 - /003 Monthly Signif. Posit. 0.97* 0.9* (0.03) (0.095) (0.0) [ ]. GJR GARCH-M Closten et. al. (993) N CRSP 4/95 - /989 Monthly Signif. Neg. -0.9* -0.8* 0.084# -0.0 (0.08) (0.89) (0.0) [ ] Lundblad (007) N CRSP /836 - /003 Monthly Signif. Posit. 0.44** 0.4** (0.08) (0.08) (0.) [ ] 3. QGARCH-M Campbell and Hentschel (99) N CRSP /96 - /988 Daily Insign. Posit # -0.00# (0.05) (0.00) (0.05) [ ] Lundblad (007) N CRSP /836 - /003 Monthly Signif. Posit. 0.38* 0.3* (0.06) (0.063) (0.08) [ ] 4. EGARCH-M Nelson (99) GED CRSP 7/96 - /987 Daily Insign. Neg # -0.0# 0.084* (0.04) (0.035) (0.043) [ ] Lundblad (007) N CRSP /836 - /003 Monthly Signif. Posit. 0.58** 0.40** (0.096) (0.84) (0.33) [ ] Notes: Data for the years are compiled by Schwert. All coefficients are significant at the % level, unless otherwise noted. *, ** Statistically significant at the 5% and 0%, respectively. # statistically insignificant.

25 Table 7. Augmented Excess Return Pricing Model with σ t- Parameters GARCH-M GJR-GARCH-M QGARCH-M EGARCH-M A. Daily c # (0.30) (0.6) (0.33) (0.5) d # (0.7) (0.3) (0.37) (0.3) p [ ] [ ] [ ] [ ] c * = c + d ξ = c + d + p [ ] [ ] [ ] [ ] corr(σ t,σ t - ) B. Weely c 0.358** 0.77# 0.88# 0.49# (0.06) (0.04) (0.80) (0.6) d 0.040# 0.80# 0.07# 0.03* (0.94) (0.9) (0.7) (0.045) p [ ] [ ] [ ] [ ] c * = c + d ξ = c + d + p [ ] [ ] [ ] [ ] corr(σ t,σ t - ) C. Monthly c 0.5# 0.54# 0.356# 0.75# (0.75) (0.80) (0.334) (0.93) d 0.359# 0.67# 0.070# 0.80# (0.79) (0.93) (0.436) (0.38) p [ ] [ ] [ ] [ ] c * = c + d ξ = c + d + p [ ] [ ] [ ] [ ] corr(σ t,σ t - ) Notes: See Table 3 Regression is based on the augmented model r t =cσ t +dσ t- +α+br t- +u t Computation of unconditional mean return μ =(α +(c +d +p )σ)/(-b ) c * = c + d is for pure price of ris and p for sewness-urtosis price of ris ξ = c + d + p is the combined pure-sewness-urtosis price of ris

26 Table 8. Augmented Excess Return Pricing Model with σ t GARCH-M GJR-GARCH-M QGARCH-M EGARCH-M A. Daily c * 0.078# 0.00# (0.058) (0.048) (0.059) (0.04) d -0.08# 0.0# 0.035# (0.03) (0.05) (0.03) (0.09) p [ ] [ ] [ ] [ ] ξ = c + d σ + p corr(σ t,σ t ) [ ] [ ] [ ] [ ] B. Weely c 0.35* 0.6# 0.7# 0.05# (0.73) (0.90) (0.89) (0.6) d 0.00# 0.040# 0.05# 0.065* (0.038) (0.039) (0.039) (0.03) p [ ] [ ] [ ] [ ] ξ = c + d σ + p [ ] [ ] [ ] [ ] corr(σ t,σ t ) C. Monthly c 0.4# -0.3# -0.7# -0.9# (0.54) (.558) (.358) (0.70) d 0.007# 0.048# 0.055# 0.06# (0.055) (0.45) (0.5) (0.068) p [ ] [ ] [ ] [ ] ξ = c + d σ + p [ ] [ ] [ ] [ ] corr(σ t,σ t ) Notes: See Table 3 Regression is based on the augmented model r t =cσ t +dσ t +α+br t- +u t Computation of unconditional mean return μ =(α +(c +p +dσ )σ)/(-b )

27 Figure. Sewness-Kurtosis Price of Ris p SK price of ris n =30 n =4 n = Kurtosis parameter, Sewness parameter,

28 Figure. Pure and Sewness-Kurtosis Price of Ris: GJR Model c* * p* Data Frequency in Trading Days

29 Figure 3. T-values for Pure and Sewness-Kurtosis Price of Ris: GJR Model t c 0 t - -4 t p Data Frequency in Trading Days