Informations Shocks, Systemic Risk and the Fama-French Model: Evidence from the US Stock Market

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1 Informations Shocks, Systemic Risk and the Fama-French Model: Evidence from the US Stock Market Leon Zolotoy March 9, 2011 Abstract We examine the impact of news on systemic equity risk as measured by the Fama- French three-factor model of the US industry portfolios. By using both semiparametric and parameric models we test for the presence of asymmetric e ect of information shocks on the variance-covariance matrix of monthly industry portfolio and Fama-French factor returns and factor loadings. We nd market factor betas to increase (decrease) following large negative (postive) market innovations. No evidence of asymmetric response to the news is found for the size (SMB) factor betas. The book-to-market betas, on the other hand, seems to decrease (increase) following large negative (positive) shocks to the book-to-market (HML) portfolio. Finally, we use dynamic estimates of the Fama- French betas to assess the risk pro le of the industry portfolios using the concepts of stochastic dominance. 1 Motivation and Background In this study we examine the impact of news on measures of undiversi able risk. Financial theory suggests that such a link should exist. Previous studies nd the volatility of equity returns to be higher in bear markets than in bull markets 1. One potential explanation links this nding to the "volatility-feedback" hypothesis (Bekaert and Wu, 2000; Wu, 2001). Holding dividends constant, if expected returns are increasing in volatility, then stock prices should fall when volatility rises. An alternative explanation attributes the asymmetry in equity return volatility to so-called "leverage-e ect" (Black, 1976). As the debt-to-equity (leverage) ratio rises, equity holders, who bear the residual risk of rm s cash ows, perceive their investment as being more risky. If linear factor model is an adequate representation of how the equity risk is priced then one would expect equity betas to be a ected by the news as well. There is also the extant body of research that links rm leverage directly to equity beta. A positive relation between equity beta and rm leverage was suggested by, among others, by Hamada (1972), Appleyard and Strong (1989), and Monkhouse (1997). Braun, Nelson and Sunier(1995), Engle and Cho (1999), Brooks and Henry (2002), Dean and Wa (2004) and Zolotoy (2010) explore time-variation and asymmetry in betas. Most of these studies nd equity betas to rise (fall) following the release of bad (good) news. 1 See Engle (2001) for a comprehensive review of these studies. 1

2 We seek to contribute to the existing literature in a number of ways. First, previous studies model systemic risk using the Capital Asset Pricing Model (CAPM) and, thus, focus on the dynamics of the market factor loadings. We propose a straightforward generalization of Hamada (1972) formula that links CAPM beta to rm leverage to multi-factor model and seek to extend previous results by examining the link between the news and systemic risk using a three-factor Fama-French model (1993). Previous studies (Fama and French, 1993; Fama and French, 1996) document the failure of CAPM to price size and book-to-market sorted portfolios. Vassalou (2003), Hahn and Lee (2006), and Petkova (2006), among others, suggest that these factors capture variations in investment opportunities. Overall, there seems to be a consensus that both the size (SMB) and book-to-market (HML) factors are important determinants of the undiversi able risk in addition to traditional market factor. Therefore, understanding the dynamics of size and book-to-market betas in important both in context of asset pricing and risk management. Second, in contrast to the previous studies which use parametric approach to model factor loadings we use both the semiparametric and parametric methods to study the dynamics of factor loadings. More speci cally, we examine the impact of news on the conditional variance-covariance matrix of equity and factor returns using a non-parametric approach following Stoker (1991), Long and Ulah (2005) and Linton (2008). The lagged equity returns are decomposed into systemic and sector-speci c components using multivariate asymmetric BEKK model (Baba et al., 1995; Brooks and Henry, 2002). This approach allows to examine separately potential sources of asymmetry via covariance and variance asymmetry channels (Kroner and Ng, 1996) while making our results more robust to model assumptions. Using monthly returns on the US industry portfolios we nd strong evidence of asymmetry in market factor loadings, where both the variance of market portfolio and the covariances of equity returns with market portfolio are increasing (decreasing) following the release of bad (good) news. Overall, large negative (positive) market shocks lead to an increase (decrease) in market factor loadings. On the other hand, we nd no evidence of either covariance or variance asymmetry in the size factor loadings, a nding which suggests that time-variation in the SMB equity betas is mainly driven by the magnitude and not the sign of news. In contrast to market factor loadings, we nd strong evidence of HML betas being positively related to the lagged innovations in HML factor, a nding which suggests that the asymmetry in HML loading is most likely to be attributed to the asymmetry in HML factor volatility. Finally, we propose and discuss some applications of the dynamic Fama-French three-factor model in context of portfolio selection and risk management. The remainder of the paper is as follows. In Section 2 we provide a brief description of our data. Section 3 outlines the methodology. Sections 4,5 and 6 present the empirical results. Finally, Section 7 provides a summary and concluding remarks. 2 Data Description and Preliminary Analysis Monthly US industry portfolio returns and the returns on Fama-French factors for the period of January 1951 to December 2009 were obtained from the Kenneth French website. To estimate excess returns we use the yield on a 1-month T-bill as a risk free rate. This data was obtained from CRSP. Selected descriptive statistics of the industry portfolio excess returns series are reported 2

3 in Table 1. The estimated risk premia signi cantly varies between sectors, ranging between 44 (others) and 74 basis points (health) per month. All return series are negatively skewed, an observation potentially attributed to asymmetric dependence between individual stock returns reported in previous studies. 2. Some of the return series exhibit signi cant serial correlation as evident by Ljung-Box Q-statistics. The economic signi cance of this observation is, however, limited with the estimated rst-order serial correlation being around Squared returns are displaying a strong evidence of serial correlation, an observation consistent with "volatility clustering" phenomenon. Insert Table 1 approximately here Table 2 displays the estimated parameters of the unconditional Fama-French model. For each industry portfolio we report the estimated alpha, betas and the corresponding t statistics. Signi cant estimates of alphas for a number of sectors indicate the failure of Fama-French model to correctly price the risk, at least for the time-period of our sample. Also, it appears that the majority of sectors are also sensitive to the size and book-to-market risk, as evident by the statistically signi cant estimates of the corresponding betas. This observation is consistent with Lewellen (1999). 3 Analytical Framework Insert Table 2 approximately here Our notations are: let D t and E t be the values of total debt and equity as of period t:also, let r e;t+1 and r d;t+1 be the returns on equity and debt at t + 1, respectively. Also, let r a;t+1 be the return on assets at t + 1:Finally, assume that a rm holds a perpetual debt, so that the present value of tax shields is T D t where T is the marginal corporate tax rate and let r T;t+1 denote the return on dollar worth of tax shields. Then, the following relation should hold (see, for instance, Brealey and Myers, 2008) E t E t + D t r e;t+1 + D t r d;t+1 = E t + D t T D t r a;t+1 + T D t r T;t+1 (1) E t + D t E t + D t E t + D t Next, assume that stock returns are governed by the linear n-factor model with f = [f 1; ; f 2 ::::f n ] being 1n vector of factor realizations with conditional covariance matrix f;t+1. After applying a covariance operator and re-arranging terms the following relation should hold cov(r e ; f) 0 = cov(r a ; f) 0 + (1 T ) D t E t cov(r a ; f) 0 cov(r d ; f) 0 (2) Multiplying both sides of equation by f;t+1 gives the following result e;t+1 = a;t+1 + ( a;t+1 d;t+1 ) D t E t (1 T ) (3) 2 See Ang and Chen (2002), Campbell, Keodijk and Kofman (2002) and Bae, Karolyj and Stulz (2003) among others 3

4 This expression is a multivariate generalization of Hamada (1972) formula for levered CAPM beta. It shows that Hamada (1972) result can be extended to any linear factor model and, in particular, to the Fama-French three-factor model which is the focus of this paper. Equations (2) and (3) suggest two possible channels through which stock betas can be a ected by past performance. First is the covariance asymmetry channel (Kroner and Ng, 1996), through the link between stock return covariance with factors and rm s debt-toequity ratio (leverage). Welch (2004) nds that a substantial portion of variation in leverage is due to changes in the market value of equity. Combined with eq. (3) this nding links equity betas to past stock performance through the negative link between the value of equity and nancial leverage. Second, asymmetric response of equity betas to past stock performance can occur due to asymmetric response of conditional factor covariance matrix, f;t+1 ; to past factor realizations and, possibly, to sector-speci c shocks. A vast body of previous studies report conditional volatility of stock returns to be negatively correlated with past returns (see Engle and Ng (1993), Glosten, Jagannathan and Runkle (1993) and Wu(2001), among others). Thus, a second source of asymmetric behavior of equity betas with respect to past stock performance can be own and cross-variance asymmetry channels. We proceed as follows. First, for each industry portfolio and Fama-French factors we estimate the asymmetric BEKK model (Baba et al.,1995; Brooks and Henry, 2002). H t+1 = C 0 C + A 0 H t A + B 0 t 0 tb + D 0 t 0 td (4) where A; B; C and D are the 44 coe cient matrices. Since a four dimensional asymmetric BEKK model requires a relatively large number of parameters to be estimated we impose a diagonality restriction on A: Note that cross-variance and covariance spillovers are still allowed since no restrictions were imposed on the B and D coe cient matrices. Further, let t be a 4 1 vector of industry portfolio and factor innovations. Also, we de ne i;t =min( i;t ; 0) for i =IND,MKT,SMB,HML. The industry (IND) and factor (MKT,SMB,HML) portfolio innovations are estimated as for the factor portfolios and j;t+1 = f j;t+1 j ; j = MKT; SMB; HML (5) ind;t+1 = r ind;t+1 ind 0 ind;t+1 (6) where is a 31 vector of estimated factor risk premia and ind;t+1 is de ned as ind;t+1 = H 1 f;t+1 H f;ind;t+1 (7) To complete model speci cation we assume that innovations to industry and factor portfolios follow a multivariate Student-t distribution. We use the estimated betas to decompose industry returns into systemic (factor innovations) and industry speci c components. Cho and Engle (1999) and Brooks and Henry (2002) emphasize the importance of dichotomizing returns into systemic and idiosyncratic components. We estimate industry speci c component as r ind;t+1 ind 0 ind;t+1 f t+1: 4

5 Next, we estimate conditional covariances between the industry and factor portfolios using the Nadaraya-Watson kernel estimate (Long and Ulah,2005) ^ cov(r ind; f js t ) = TP r ind; f K h (s s t ) =2 TP K h (s s t ) =2 (8) where K h () is a kernel function and s is the variable the model is conditioned on. We condition our model separately on systemic (factor) innovations and industry speci c components. Similarly, conditional factor variance can be estimated as ^ var(f js t ) = TP f 2 K h (s s t ) =2 TP K h (s s t ) =2 (9) To evaluate the impact of lagged factor and industry innovations on the factor variances and covariances with industry portfolio returns we use the average derivative approach of Stoker (1991). This approach allows to test for the presence of both the covariance and variance asymmetry channels. This is particularly important since if the systemic shocks a ect both covariances of industry returns with risk factors and the variance of factor portfolios, the impact of past stock/portfolio performance on betas could be hard to detect (Bekaert and Wu, 2000). Also, parametric rate of convergence makes this estimator particularly useful for the purpose of our analysis which makes our analysis robust to di erent model speci cations on the one hand, while allowing for a reasonable degree of estimation accuracy, on the other. 4 Empirical Findings-Semiparametric Analysis We start with examining the impact of factor and industry speci c shocks on the industry portfolio covariances with factors. The estimation results are reported in Table 3. For each industry portfolio we report the estimated average derivative of its covariance with each of the three Fama-French factors, conditioned separately on the industry speci c (reported under "idiosync." heading) and the corresponding factor (reported under "factor" heading) innovations. 3 Insert Table 3 approximately here The estimated derivatives of the covariances with MKT factor with respect to lagged MKT factor innovations are negative and statistically signi cant. This observation suggests that the covariance of industry portfolio returns with factors tends to increase (decrease) 3 Following Lewellen (1999) we run the regression of SMB and HML factors on MKT factor and use the regression residuals as the orthogonalized size and book-to-market factors. Bandwidths used to construct non-parametric estimates were selected using cross-validation method (Pagan and Ulah, 1994). 5

6 when the lagged factor innovations are negative (positive), a nding which is consistent with changes in leverage. Turning to SMB and HML factors, most of the estimates lack statistical signi cance. Turning to the estimated derivatives with respect to lagged industry speci c shocks, we nd no statistical evidence of covariance asymmetry. The sign and statistical signi cance of estimated derivatives, apparently, vary randomly across sectors. Overall, we nd strong evidence of the covariance asymmetry for the MKT factor linked to the lagged market portfolio returns, a nding which suggests that covariance asymmetry has a systemic rather than idiosyncratic nature. Next, we test for the presence of the own-and cross-variance asymmetry channels, by examining the link between factor variance and lagged returns. We start with estimating average factor variance derivatives with respect to lagged industry speci c shocks. We test for the presence of both the symmetric and asymmetric spillovers, by separately estimating the derivatives with respect to the raw and squared industry innovations. Insert table 4 approximately here The results are displayed in Table 4. For each industry portfolio and for each factor we report estimated average derivatives with respect to raw and squared industry speci c return components under the "innov" and "innov sq.", respectively along with the corresponding t-statistics. Overall, the evidence of asymmetric volatility spillovers from industry sectors to factor volatility is very limited. The sign of the estimated derivative seems to vary randomly across sectors and factors. Turning to the estimated derivatives with respect to lagged squared industry innovations a positive sign is consistent with volatility spillovers from industry to factor portfolios. For the MKT and SMB factors, however, the estimates lack statistical signi cance with the Utils sector being the only exception. On the other hand, we nd a signi cant evidence of symmetric volatility spillovers from industry to the HML factor. Petkova (2006) and Lee (2005) nd the HML portfolio to be positively correlated with the changes in term structure. Thus, it is possible that volatility spillovers from industries to the HML factor re ect the impact of industry related news on the forecasted future interest rates. 5 Empirical Findings-Parametric Analysis Table 5 displays the estimates of the asymmetric BEKK model described in Section 3. The estimated alphas are statistically signi cant for the Busec, Durbl, Hlth, Shops and Other sectors. However, it seems that allowing factor loadings to vary over time still leads to some limited reduction in the magnitude of pricing errors, with the mean absolute pricing error decreasing from 17 basis points for the unconditional Fama-French model to 14.5 basis points for the BEKK model. Also, for most of the industry sectors we nd a signi cant reduction in the residual portfolio variance compared to static estimates of the Fama-French betas. 4 4 The results are available upon request. Insert Table 5 approximately here 6

7 Next, we examine the estimates of the covariance matrix. Large values of the estimates of the A matrix suggest that both the conditional factor variances and covariances of industry portfolio with factors are highly persistent, a nding consistent with previous studies. Turning to the estimates of the B matrix the diagonal elements are both statistically and economically signi cant, a nding which indicates a well documented "volatility clustering" phenomenon. Also, we nd some evidence of the cross-variance and covariance spillovers with some of the o -diagonal elements being signi cantly di erent from zero. Finally, we inspect the estimates of the D matrix. First, there is strong evidence of both the industry, MKT and HML factor variance asymmetric response to their past innovations. Interestingly enough, no such evidence is found for the SMB factor. Also, we nd a signi cant evidence of the cross-variance asymmetry for the industry and MKT factors with the estimates of d 12 and d 21 being statistically signi cant. This nding is consistent with the results of semiparametric analysis reported in the previous section. We turn now to the analysis of Fama-French dynamic factor loadings. To provide some preliminary insight we plot the estimated MKT, SMB and HML betas in Figures 1 to 3. The time variation in factor loadings is evident. Interestingly enough, the time variation in SMB and HML betas appears to be more pronounced compared to MKT betas for most of the industry portfolios. We formally test the equality between the coe cients of variation of the MKT, SMB and HML betas and nd the di erences to be statistically signi cant as well. 5 Large volatility of both the SMB and HML factor loadings as opposed to the MKT betas is potentially attributed to the di erent sources of asymmetry a ecting factor betas. The results of semiparametric analysis reported in Section 3 along with the estimates of BEKK model suggest that, in contrast to SMB and HML betas, MKT beta is a ected by both the covariance and variance asymmetry with respect to MKT factor innovations which mitigate the uctuations in MKT factor loadings. Insert Figures 1 to 3 approximately here To assess statistical signi cance of the relation between the news and factor loadings we estimate the following regression model (Brooks and Henry, 2002) j;t+1 j;t = 1 ( j;t 2 ) + 3 I j;t + 4 j;t + 5 I ind;t + 6 ind;t (10) + 7 I j;t ind;t + 8 I ind;t j;t + u j;t+1 for j = MKT; SMB; HML: Here, I j (I ind ) is an indicator function which takes value of one if lagged factor (industry) innovation is negative. Similarly, j;t ( ind;t ) is an interaction term of I j (I ind ) with j ( ind ) to control for the magnitude of lagged factor (industry) related news. The coe cients 7 and 8 control for potential interactions between factor and industry innovations. Finally, 1 is a mean-reversion parameter which measures the speed of adjustment of factor loading to it s long-run value, 2 : The model estimates are reported in Tables 6,7 and 8 for the MKT, SMB and HML factor loadings, respectively. Starting with the estimates of the MKT beta equation we nd strong evidence of the asymmetric response of MKT factor loading to the lagged innovations in MKT factor. More speci cally, positive and signi cant estimates of 4 suggest that MKT 5 The results are available upon request. 7

8 betas tend to increase following negative market-wide news. This observation is consistent with the results of semiparametric analysis reported in a previous section which indicate the presence of covariance asymmetry with respect to MKT lagged innovations. On the other hand, the sign and signi cance level of the estimates of 5 and 6 which measure the impact of industry-speci c news on factor betas seem to vary randomly across the industry portfolios. Market betas appear to be highly persistent with the estimates of the meanreversion parameter, 1 ; varying between (Manuf) and (Money), suggesting that it takes between 8 and 25 months for the shocks to market beta to dissipate. Insert Tables 6,7, and 8 approximately here Turning to the estimates of SMB and HML betas we nd no evidence of asymmetric response of the SMB factor loadings to either SMB or industry innovations. This ndings is consistent with the results of the semiparametric analysis which nds no evidence of the covariance or cross-variance asymmetry, on the one hand, and the results of parametric analysis, which nds no evidence of the own variance asymmetry for the SMB factor, on the other. This nding suggests that the observed time-variation in SMB beta is most likely to be driven by the magnitude rather than the sign of news. On the other hand, we nd strong evidence of HML factor loadings decreasing following negative shocks to HML factor. This result along with the ndings of the semiparametric analysis suggests that the asymmetry in HML loading with respect to HML innovations is most likely to be caused by the own variance asymmetry of the HML factor rather than via covariance asymmetry channel. 6 Dynamic Fama-French Model: Applications in Risk Management We examine the risk pro le of industry portfolios using the concepts of stochastic dominance where the least risky sector will dominate. To do so, for each sector portfolio return series we estimate the residuals using the estimated dynamic factor loadings from the BEKK model. Next we compare their empirical distributions using the concept of the rst-order stochastic dominance criterion. Let F X () and G Y () be the cumulative distribution functions of the industry portfolio residuals for the industry sectors X and Y, respectively. If F X (z) G Y (z) for all values of z then Y dominates X in a rst-order sense (see, for instance, Bawa, 1982; Barrett and Donald, 2003 among others). Empirical distributions are displayed in Figure 4. A visual inspection of the distribution plots provides some initial insights into risk pro le of industry portfolios. It seems that among industries Busec and Hlth sectors are the least risky ones. However, since the empirical distribution of all industry sectors intersect with each other it is hard to infer whether there exists a sector that dominates all others in a rst-order sense based on a visual inspection. Insert Figure 4 approximately here 8

9 We formally test the rst-order stochastic dominance hypothesis using the Barret- Donald (2003) S-statistic S = N 1=2 sup( G ^ Y (z) 2 z ^ F x (z)) (11) The null and alternative hypotheses are de ned as H 0 : G y (z) F x (z) for all z 2 [0; z] H 1 : G y (z) > F x (z) for some z 2 [0; z] where [0; z] is the support of F x (z) and G y (z): The critical values are , 1.224, and for 1%,5%, and 10% levels of signi cance, respectively (Barret and Donald, 2003). The null hypothesis of Y dominating X is rejected at 1 level of con dence if S exceeds the corresponding critical value. The test results are reported in Table 9. The S-statistics are displayed in the 1212 matrix where S i;j corresponds to the null hypothesis that sector i dominates sector j in a rst-order sense. Overall, our results suggest that Hlth appears to be the least risky sector for which the null hypothesis of the rst-order stochastic dominance cannot be rejected with respect to eight out of eleven remaining sectors. More speci cally, Hlth sector does not seem to dominate Manuf, Nodur and Other industries. Insert Table 9 approximately here Finally, we examine the systemic risk pro le of industry sectors. In terms of methodology the approach we propose is similar to the Gonzales-Rivera (1996). She suggests using stochastic dominance criteria to compare the distribution of CAPM betas across di erent stocks. However, this approach is valid when there is a single risk factor. Since in case of Fama-French model we have multiple sources of risk, instead of examining individual factor betas we look at the weighted average of factor loadings, where the weight of factor j at time h t + 1 is calculated as j;t+1 h MKT;t+1 +h smb;t+1 +h hml;t+1 for j = MKT; SMB; HML: The underlying intuition is that each factor beta is weighed by the relative riskiness of the corresponding factor. The empirical distributions are displayed in Figure 5 and the corresponding S-statistics are reported in Table 10. Note that in this case S i;j corresponds to the null hypothesis that sector i is dominated by sector j in a rst-order sense. Overall, Durbl sector appears to be the most risky sector in terms of systemic risk pro le. The null hypothesis cannot be rejected for all but Other sector. On the other hand, Telcm seems to be the least risky sector which appears to dominate all but Hlth sectors. It is followed by Hlth and Utils industries which both seem to dominate nine out of eleven sectors. Insert Figure 5 approximately here Insert Table 10 approximately here 9

10 7 Summary and Conclusions In this paper we examine the impact of news on the time-varying equity betas of the Fama- French three-factor model. Employing monthly returns on the US industry portfolios we test for the presence of asymmetry in Fama-French factor loadings using both semiparametric and parametric methods. Following Kroner and Ng (1996) we separately test for the presence of covariance and variance asymmetry in Fama-French betas with respect to systemic and industry-speci c information shocks. Overall, we nd MKT, SMB and HML betas to display di erent asymmetry patterns with respect to new information. Market factor betas seem to increase (decrease) following large negative (positive) shocks to the market portfolio. This nding is consistent with the results of Brooks and Henry (2002) for the UK stock market. On the other hand, we nd no evidence of asymmetric response of SMB beta to news, suggesting that the time variation in size factor loadings is mostly driven by the magnitude and not the sign of information shocks. Finally, we nd the HML factor loadings to be decreasing (increasing) following negative (positive) shocks to the HML portfolio. This nding suggests that the asymmetric response of HML factor loading to systemic shocks is most likely to be attributed to the asymmetric response of factor volatility to the lagged innovations. The evidence of factor betas being a ected by the industry-speci c shocks is limited. We also examine some applications of the dynamic Fama-French model in context of portfolio selection and risk management. In particular, we examine the risk pro les of the US industry portfolios using the concepts of stochastic dominance. While none of the sectors appears to uniformly dominate all the rest, in terms of risk-adjusted performance Health sector to be the least risky one, while in terms of systemic risk pro le it is Telcm sector that seems to dominate most of the other industry sectors. 10

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19 Table 5 Busec Chems Durbl Energy Hlth Manuf Money Nodur Other Shops Telcm Utils a11 0:926 a22 0:92 a33 0:938 a44 0:88 0:204 MKT 0:712 (0:13) smb 0:151 hml 0:248 9:228 (1:36) 0:916 0:893 0:929 0:848 0:052 0:7 (0:13) 0:17 0:255 9:883 (1:49) 0:942 0:897 0:934 0:859 0:203 (0:11) 0:645 (0:13) 0:196 0:294 7:97 0:956 0:94 0:928 0:868 0:103 (0:12) 0:633 (0:13) 0:165 0:226 8:82 (1:25) 0:945 0:936 0:933 0:907 0:368 0:705 (0:12) 0:132 0:222 8:57 (1:19) 0:942 0:92 0:933 0:871 0:093 0:695 (0:13) 0:162 0:21 10:22 (1:64) 0:885 0:906 0:928 0:898 0:032 0:639 (0:13) 0:183 0:254 0:937 0:928 0:929 0:867 0:053 0:624 (0:12) 0:181 0:275 0:942 0:934 0:936 0:914 0:218 0:664 (0:13) 0:144 0:24 0:941 0:937 0:931 0:892 0:142 0:626 0:159 0:232 0:939 0:927 0:934 0:902 0:115 0:675 (0:13) 0:139 0:226 (1:03) (1:46) (1:39) (1:16) (1:29) (1:55) Log-lik :664 9:265 8:462 9:698 (1:44) 9:032 0:947 0:924 0:915 0:893 0:159 0:712 (0:12) 0:175 0:255 10:02 This Table displays the estimates of the asymmetric BEKK model with Student-t innovations as described in Section 3. Corresponding standard errors appear in parentheses. ** (*) denotes signi cance at 5 (10) % level. 19

20 Table 5 (contd.) Busec Chems Durbl Energy Hlth Manuf Money Nodur Other Shops Telcm Utils b11 0:168 b12 0:057 b13 0:017 b14 0:026 b21 0:137 b22 0:321 b23 0:013 b24 0:041 b31 0:039 b32 0:07 b33 0:218 b34 0:035 b41 0:119 b42 0:065 b43 0:033 b44 0:339 0:012 (0:11) 0:025 0:039 0:017 0:221 0:202 0:006 0:053 0:008 0:11 0:272 0:12 0:008 0:007 0:047 0:232 0:201 0:033 0:006 0:01 0:169 0:029 0:019 0:067 0:109 0:242 0:042 0:015 0:065 0:016 0:337 0:207 0:019 0:021 0:039 0:071 0:186 0:015 0:035 0:126 0:096 0:245 0:064 0:015 0:038 0:018 0:334 0:187 0:008 0:019 0:038 0:013 0:217 0:024 0:061 0:032 0:057 0:233 0:043 0:058 0:093 0:013 0:268 0:274 (0:11) 0:079 0:107 0:065 (0:11) 0:138 (0:12) 0:023 0:13 0:036 0:085 0:227 0:038 0:047 0:063 0:045 0:298 0:254 0:051 0:01 0:077 0:02 0:32 0:022 0:074 0:086 0:076 0:231 0:076 0:068 0:058 0:056 0:257 0:279 0:026 0:034 0:041 0:072 0:198 0:063 0:046 0:031 0:261 0:011 0:077 0:087 0:033 0:341 0:301 0:112 0:01 0:018 0:098 0:069 0:055 0:019 0:062 0:059 0:218-0:012 0:039 0:037 0:055 0:307 0:232 0:019 0:007 0:026 0:007 0:205 0:021 0:005 0:021 0:021 0:232 0:01 0:099 0:054 0:088 0:276 0:257 0:048 0:035 0:242 0:058 0:005 0:01 0:026 0:232 0:071 0:033 0:081 0:021 0:249 0:112 0:122 0:011 0:096 0:033 0:218 0:051 0:048 0:051 0:033 0:236 0:042 0:029 0:043 0:063 0:287 Corresponding standard errors appear in parentheses. ** (*) denote signi cances at 5 (10) % level 20

21 Table 5 (contd.) Busec Chems Durbl Energy Hlth Manuf Money Nodur Other Shops Telcm Utils d11 0:251 (0:26) d12 0:178 (0:18) d13 0:063 d14 0:103 d21 0:192 (0:21) d22 0:105 (0:16) d23 0:097 d24 0:223 (0:16) d31 0:381 (0:29) d32 0:165 (0:15) d33 0:03 d34 0:06 (0:12) d41 0:049 (0:16) d42 0:091 d43 0:258 d44 0:062 0:238 0:13 0:012 0:085 0:014 0:407 0:016 0:269 0:081 (0:11) 0:213 0:08 0:012 0:069 0:176 0:022 0:297 0:175 0:088 0:014 0:051 0:309 0:374 0:013 0:115 0:219 0:029 0:082 0:093 0:15 0:255 0:052 0:109 0:183 0:008 0:018 0:278 0:287 0:036 0:133 0:012 0:168 0:063 0:254 0:189 0:083 0:289 0:031 0:079 0:061 0:049 0:052 0:057 0:074 0:097 0:128 0:26 0:092 0:089 0:038 0:055 0:156 0:242 0:023 0:723 (0:17) 0:655 (0:14) 0:121 0:189 0:97 (0:13) 0:855 (0:13) 0:141 0:356 0:143 (0:13) 0:197 0:113 0:056 (0:13) 0:108 0:159 0:283 0:141 0:359 (0:14) 0:273 0:012 0:008 0:364 (0:15) 0:18 (0:13) 0:065 0:144 0:252 (0:11) 0:019 0:075 0:038 0:131 0:071 0:257 0:026 0:094 0:159 0:033 0:22 0:132 0:074 0:01 0:178 0:041 0:036 0:095 0:159 0:081 0:127 0:146 0:049 0:313 (0:15) 0:11 (0:14) 0:069 0:546 (0:15) 0:363 (0:14) 0:045 0:139 0:139 0:267 0:041 0:178 0:126 0:154 0:215 0:147 0:416 0:403 0:005 0:098 0:375 0:334 0:059 0:02 0:163 (0:18) 0:216 0:031 0:254 0:125 0:06 0:138 0:231 0:122 0:211 0:109 0:097 0:046 0:067 (0:11) 0:013 0:205 0:201 0:026 0:017 0:064 0:123 0:214 0:024 0:212 0:286 0:149 0:104 0:007 0:068 0:12 0:173 0:209 0:089 0:066 0:181 0:334 0:146 0:077 Corresponding standard errors appear in parentheses. ** (*) denote signi cance at 5 (10)% level. 21

22 Table 6 Busec Chems Durbl Energy Hlth Manuf Money Nodur Other Shops Telcm Utils 1 0:098 (0:016) 2 1: :002 (0:006) 4 0:024 5 (0:006) 6 0: :011 0:12 (0:016) 1:056 0:011 0:008 0:006 0:125 1:267 0:013 0:012 0:027 0:007 0:089 1:002 0:005 0:006 0:001 0:002 0:0001 0:059 0:975 0:011 0:013 0:011 0:0005 0:007 0:038 1:189 0:006 0:007 0:002 (0:0025) 0:126 1:081 0:014 0:024 0:015 0:006 0:055 0:795 0:01 0:058 0:012 0:001 0:078 1:063 0:031 0:005 4 (0:0019) 0:043 0:985 0:013 0:013 0:015 0:0054 0:068 0:809 0:014 0:019 0:002 0:093 0:655 0:007 0:007 0: :019 0:002 0:01 0:005 Adj. R :008 0:011 0:01 0:008 In this Table we report the estimates of the regression model, described in Section 5, eq.(10). The dependent variable is market factor beta from the asymmetric BEKK model. Newey-West standard errors are reported in parentheses. ** (*) denotes signi cance at 5 (10)% level. 22

23 Table 7 Busec Chems Durbl Energy Hlth Manuf Money Nodur Other Shops Telcm Utils 1 0:11 (0:015) 2 0: : :014 (0:006) 5 0:017 (0:007) 6 7 0: :101 0:313 0:021 0:007 0:0008 0:005 0:053 0:164 (0:11) 0:01 (0:008) (0:006) 0:005 (0:008) 0:001 0:0018 (0:0013) 0:0008 0:063 0:079 (0:13) 0:012 (0:008) 0:002 0: :002 0:069 (0:014) 0:279 (0:11) 0:006 0:022 0:021 (0:008) 0:0007 0:001 0:011 0:078 (0:015) 0:163 0:015 0:006 0:0007 (0:0008) 0:0006 (0:0006) Adj. R :072 (0:014) 0:163 0:008 () 0:006 () 0:006 0:005 0:001 0:067 0:481 () 0:012 0:013 (0:008) 0:005 (0:0025) 0:041 0:283 0:013 0:012 0:001 0:002 0:0006 0:072 0:475 (0:007) 0:012 (0:006) 0:011 0:014 (0:006) 0:001 0:108 0:258 0:01 (0:006) 0:015 (0:006) 0:014 0:016 0:008 0:062 0:235 0:023 (0:007) (0:007) 0:002 0:001 0:001 In this Table we report the estimates of the regression model, described in Section 5, eq.(10). The dependent variable is size factor beta from the asymmetric BEKK model. Newey-West standard errors are reported in parentheses. ** (*) denotes signi cance at 5 (10)% level. 23

24 Table 8 Busec Chems Durbl Energy Hlth Manuf Money Nodur Other Shops Telcm Utils 1 0: : : : : : :002 0:124 0:197 (0:008) 0:024 0:043 () 0:011 0:002 0:128 0:282 0:012 0:008 0:015 0:006 0:005 0:005 0:098 0:773 0:005 (0:007) 0:016 0:018 0:015 0:087 0:139 0:017 (0:007) 0:007 0:039 () 0:002 0:01 0:002 0:097 0:25 0:015 0:015 (0:007) 0:012 0:072 0:558 (0:11) 0:011 (0:008) 0:008 0:019 () 0:002 0:079 0:102 0:017 (0:008) 0:008 (0:008) 0:006 0:055 () 0:505 (0:008) (0:006) 0:103 0:207 0:012 (0:007) 0:029 0:017 (0:007) 0:0019 Adj. R (0:0016) 0:008 0:006 (0:0016) 0:007 0:089 0:326 0:019 (0:007) 0:011 () 0:007 0:017 0:0002 0:042 0:451 (0:16) 0:018 () 0:005 0:005 () 0:002 (0:0023) In this Table we report the estimates of the regression model, described in Section 5, eq.(10). The dependent variable is book-tomarket factor beta from the asymmetric BEKK model. Newey-West standard errors are reported in parentheses. ** (*) denotes signi cance at 5 (10)% level. 24

25 25

26 26

27 Figure 1: Dynamic Fama-French factor loadings estimated using asymmetric BEKK model, as described in Section 3. 27

28 Figure 2: Dynamic Fama-French factor loadings (contd.) 28

29 Figure 3: Dynamic Fama-French factor loadings (contd.) 29

30 Figure 4: This Figure displays estimated empirical distributions of the hedged industry portfolio returns. The return on hedged industry portfolio is calculated as the raw return minus the return on hedge portfolio, estimated using dynamic Fama-French betas from asymmetric BEKK model. 30

31 Figure 5: This Figure displays estimated empirical distributions of the systemic risk pro les of industry portfolio returns. Systemic risk pro le is estimated as the weighted average of the dynamic Fama-French betas from asymmetric BEKK model, as described in Section 6. 31