Structural Analysis of Portfolio Risk Using. Beta Impulse Response Functions. April Abstract

Transcription

1 Structural Analysis of Portfolio Risk Using Beta Impulse Response Functions Christian M. Hafner Helmut Herwartz April 1998 Abstract We estimate the data generating process of daily excess returns of 20 major German stocks in a CAPM framework with time varying betas. Our sample spans a 23 year period from 1974 to An asymmetric dependence of volatility on lagged innovations is taken into account. We introduce beta impulse response functions to shed light on the structural implications of systematic risk associated with competing volatility models. The dependence of beta on news is characterized with respect to dierent sources (asset specic vs. market general news). The empirical results suggest that negative news emerging from the market involve a stronger impact on beta relative to positive news. Concerning rm specic news the opposite relation is found for the majority of the analysed data sets. Keywords: CAPM Multivariate GARCH{Models, asymmetry, impulse response analysis, Institut fur Statistik und Okonometrie and Sonderforschungsbereich 373, Wirtschaftswissenschaftliche Fakultat, Humboldt{Universitat zu Berlin, Spandauer Str. 1, D Berlin, Germany. hafner@wiwi.hu-berlin.de or helmut@wiwi.hu-berlin.de. Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. We thank two anonymous referees and Rolf Tschernig for helpful suggestions. 0

2 1 Introduction As a standard model in nance theory, the capital asset pricing model (CAPM) as invented by Sharpe (1964) and Lintner (1965) has been subject of many empirical tests and applications. One of the principal achievements was the distinction between systematic and unsystematic risk, where only the latter can be diversied away. Portfolio managers were thus enabled to consider as risk of one asset its beta, and the remaining unsystematic risk diminished in his broad portfolio. Thus, the beta of one asset was the standard measure of the asset's contribution to portfolio risk and, once calculated, this was not recalculated frequently. However, the success of the ARCH models (Engle (1982)) in describing stock market volatility suggested to allow for time-varying betas. This was rst investigated by Bollerslev, Engle and Wooldridge (1988, BEW hereafter), who established a link between bivariate GARCH models and the CAPM. BEW analyse quarterly data, but it should be noted that ARCH-eects have very often been shown to be stronger for shorter periods, weekly or daily data. In this paper, we will concentrate on the German stock market for a long time period (1974 to 1996) with daily observations. We thus deal with a huge data set (22 time series of 5748 observations each { a total of observations). On the one hand, this is a potential to detect and exploit typical structures in multivariate time series, on the other hand it challenges the computing power when complex multivariate models are applied. Our main topic is the following question: how sensitive are time-varying betas with respect to independent innovations to the system? These innovations can be rm-specic news or news that is relevant in general. By specifying models that allow for asymmetric volatility impacts, we investigate the issue whether good news (rm-specic or marketgeneral) has the same impact as bad news. Our ndings suggest that there is distinct asymmetry for the impact of shocks on the prediction of betas. Both positive and negative rm-specic innovations tend to increase systematic risk. The reverse tends to hold for market-general news, which for the majority of the analysed series had a decreasing eect on the betas. The overall variability of the betas is large on a daily frequency. This suggests that portfolio managers should revise their beta `estimates' daily and should adjust their beta forecasts given news on that day. This can be easily performed with the tool of impulse response functions, adapted to the analysis of time-varying betas in this paper. It is based on the development of volatility impulse reponse functions by Hafner and Herwartz (1997). The outline of the paper is as follows. Section 2 reviews the theory of the CAPM with time-varying betas. Section 3 deals with multivariate GARCH models and presents models allowing for asymmetry. In Section 4, beta impulse response functions (BIRF) are dened. Section 5 reports the results from estimating dierent multivariate GARCH 1

3 models and presents the resulting BIRF. Section 6 concludes. 2 The CAPM with Time{varying Betas Within the CAPM the return on an asset (r i ) corrected for a riskfree rate (r f ) is linearly related to the excess return on the so{called market portfolio (r m? r f ) covering all traded assets. The covariance between the asset's return and the return on the market portfolio ( im ) and the variance of market returns ( 2 m) govern this linear relationship: r i = r f + im (r m? r f ) m 2 = r f + i (r m? r f ): (1) The quantity i = im = 2 m has become popular as the asset's `beta' which is thought of as a measure for the undiversiable risk associated with a specic asset. Returns on assets having a beta larger than one are more aected by market movements relative to assets with a beta less than one and thus are relatively more risky. A model such as (1) is easily transformed into an econometric regression model explaining excess asset returns in period t, i.e. r i;t = + i r m;t + u t (2) In (2) r i;t and r m;t denote the excess returns, r i;t? r f;t and r m;t? r f;t respectively. The error term u t captures eects stemming from unsystematic risk associated with asset i. Note that the inclusion of a constant term is not implied by the economic model in (1). The static model in (1) is usually tested by checking whiteness of estimated residuals obtained from (2) and a formal test of the hypothesis H 0 : = 0. In its static form the CAPM implies time invariance of the involved variances and covariances and thus of `beta'. However, from an economic viewpoint one may imagine that these quantities vary through time. BEW and Hall, Miles and Taylor (1989) provide a time series framework allowing for time dependence of the relevant quantities. Let t?1 denote the set of information which is available up to time t? 1. Assuming E[r f;t j t?1] = r f;t?1 a time dependent version of (1) is obtained as (see Mills (1993), pp.192{194): E[r i;t j t?1] = r f;t?1 + i;t (E[r m;t j t?1]? r f;t?1 ); (3) with i;t = Cov(r i;t ; r m;t j t?1)=var(r m;t j t?1): (4) An econometric model corresponding to (3) is not immediately available since the expected excess return on the market portfolio is not observed. One means to overcome 2

4 this diculty is to assume that the so{called market{price of risk is invariant through time, i.e. E[r m;t j t?1]? r f;t?1 = : (5) Var(r m;t j t?1) Dening " i;t = r i;t? E[r i;t j t?1] and " m;t = r m;t? E[r m;t j t?1] one obtains from substituting (5) and (4) into (3) and r i;t = r f;t?1 + Cov(r i;t ; r m;t j t?1) + " i;t r m;t = r f;t?1 + Var(r m;t j t?1) + " m;t from (5). A candidate model to estimate conditional variances and covariances of " t = (" i;t ; " m;t ) 0 t = E[" t " 0 t j t?1] = 2 i;t im;t im;t 2 m;t is the bivariate GARCH model which is briey outlined in Section 3. Stacking excess returns into a two dimensional vector r t = (r i;t? r f;t?1 ; r m;t? r f;t?1 ) 0 one obtains a multivariate GARCH in Mean (GARCH{M) model (see e.g. Engle, Lilien and Robins (1987)): r t = g + im;t 2 m;t 1 A 1 A + "t : (6) As already mentioned for the static model in (2) the inclusion of a vector of constants g = (g 1 ; g 2 ) 0 does not directly follow from the economic model in (3). Time varying betas for asset i are obtained from ( t ) as: i;t = im;t : m;t 2 The rst equation of (6) contains time dependent risk premia for asset i, g 1 + im;t. Note that = 0 implies a constant risk premium for the asset of interest. 3 Multivariate GARCH Models { Specication, Identication, and Estimation The generalization of univariate ARCH{type models of conditional heteroskedasticity (see e.g. Engle (1982), Bollerslev (1986)) to the multivariate case is more or less straightforward. An N-dimensional random vector " t = (" 1;t ; : : : ; " N;t ) 0 may be written as " t = 1=2 t t (7) 3

5 where t denotes an i:i:d: random vector with mean zero and covariance matrix equal to the identity matrix I N. If t is specied as GARCH, then (7) is a strong GARCH model in the sense of Drost and Nijman (1993). We employ this strong GARCH form, because we will later dene news to appear in the i:i:d: innovation vector t. Given information up to time t? 1, the elements of t are completely determined by their own history t?i ; i = 1; : : : ; p and lagged observations " t?i ; i = 1; : : : ; q. Let vech(:) denote the operator that stacks the lower fraction of an N N{matrix into an N(N + 1)=2 dimensional vector. To make the dynamics of t explicit, the multivariate GARCH(p; q) model may be given as follows: vech( t ) = c + qx A i vech(" t?i px " 0 ) + t?i i=1 i=1 G i vech( t?i ): (8) In (8), A i and G i denote parameter matrices each containing (N(N + 1)=2) 2 parameters. The vector c accounts for time invariant variance components and contains N(N + 1)=2 coecients. Oering a very general structure of multivariate interdependence the vec{ model may easily suer from overparametrization in econometric practice. As given in (8), the multivariate GARCH(1,1) model contains already 21 parameters for a bivariate series (N = 2) under study. To improve the econometric tractability of the model (8), BEW advocate the diagonal model in which the matrices A i and G i are assumed to be diagonal such that the (k; l){ element in t depends linearly on the respective elements of the matrices " t?i " 0 t?i and t?i. Within the framework of the diagonal model the price paid for parsimony is the a priori exclusion of potentially important cross dynamics for the dependence of t on t?1. Engle and Kroner (1995) discuss the following model named after Baba, Engle, Kraft and Kroner (1990): t = C 0 0 C 0 + KX qx k=1 i=1 A 0 ki " t?i" 0 t?i A ki + KX px k=1 i=1 G 0 ki t?i G ki : (9) In (9), C 0 is an upper triangular matrix and A ki and G ki are N N parameter matrices. Even in the simplest case of K = 1, the so{called BEKK representation given in (9) allows for direct dependence of the conditional variance of one variable on the history of other variables within the system. In addition, (9) always yields positive denite covariance matrices without imposing further parameter restrictions. Engle and Kroner (1995) discuss the specication given in (9) in detail. For the present analysis we take K = 1 and concentrate on the GARCH(1,1) model. In this case the assumption that the upper left elements of A 11 and G 11 are greater than zero is sucient for the model parameters to be identied. From empirical analysis the so{called leverage eect has become a stylized fact of the conditional variance of stock market returns (see e.g. Black (1976)). It states that 4

6 the conditional variance of a return series is more aected by bad news (negative lagged innovations) relative to good news (positive lagged innovations). To account for the potential of an asymmetric dependence of multivariate volatility on lagged innovations we adopt generalizations of (9) which may be regarded as multivariate extensions of the asymmetric univariate GARCH model introduced by Glosten, Jagannathan and Runkle (1993). Since negative news might occur in the system through each of the components of " t separately or simultaneously we investigate for the case N = 2 and K = 1 the following specications in addition to the symmetric model given in (9): M1: t = C 0 0 C 0 + A 0 11 " t?1" 0 t?1 A 11 + A 0 21 " t?1" 0 t?1 A 21I "1;t?1<0 + A 0 31 " t?1" 0 t?1 A 31I "2;t?1<0 + G 0 11 t?1g 11 (10) M2: t = C 0 0 C 0 + A 0 11 " t?1" 0 t?1 A 11 + A 0 21 " t?1" 0 t?1 A 21I "1;t?1<0 + G 0 11 t?1 G 11 (11) M3: t = C 0 0 C 0 + A 0 11 " t?1" 0 t?1 A 11 + A 0 31 " t?1" 0 t?1 A 31I "2;t?1<0 + G 0 11 t?1 G 11 (12) M4: t = C 0 0 C 0 + A 0 11 " t?1" 0 t?1 A 11 + G 0 11 t?1g 11 : (13) In (10) to (12), I () denotes an indicator function. The model denoted as M4 is just the symmetric model implied by K = p = q = 1 in (9) and is given above in order to provide a comprehensive list of the empirical models which were under study for the present analysis. Recall that one result of Engle and Kroner (1995) was that for each BEKK model there is a unique equivalent vec representation. Thus, when discussing the properties of M1 to M4, we can also consider the equivalent vec-specication by dening c = (C 0 C 0 ) 0 vec(i N ), A 1 = (A 11 A 11 ) 0, A 2 = (A 21 A 21 ) 0, A 3 = (A 31 A 31 ) 0, and G 1 = (G 11 G 11 ) 0. After eliminating from c, A, 1 A, 2 A, and 3 G 1 those rows and columns that are superuous due to the symmetry of the covariances, one obtains the matrices c, A 1, A 2, A 3, and G 1 as in (8). This transformation is notationally more convenient and is consistent with the next section where we discuss impulse response functions only for the vec-specication. The following proposition provides a result for the covariance stationarity of the model M1. The result applies immediately to M2, M3 and M4 by setting the corresponding matrices A 2 and/or A 3 to zero. Proposition 1 Assume that both components of the error t have a symmetric density around zero. Then the process " t is covariance stationary if and only if all eigenvalues of A 1 + A 2 =2 + A 3 =2 + G 1 (14) have moduli less than one. For the implied unconditional covariance matrix, one obtains Proof: see Appendix. vech() = E[vech(" t " 0 t)] = (I? A 1? A 2 =2? A 3 =2? G 1 )?1 c: (15) 5

7 The elements of the parameter matrices in (10) to (13) are estimated by numerical procedures. Within this study we used the BHHH{algorithm as described e.g. in Judge et al. (1988) to maximize the log{likelihood function assuming normally distributed innovations. For t N(0; I P N ), the contribution of an observation " t to the joint log{ likelihood of the sample (l = l t ) is obtained as: l t =? N 2 ln(2)? 1 2 ln j tj? 1 2 "0 t?1 t " t : (16) If the multivariate distribution of t is other than normal, (16) is the quasi{likelihood function, and estimation is performed by quasi maximum likelihood (QML). To our knowledge the consistency of QML for our general model has not been proven yet, but we conjecture such a result along the lines of Bollerslev and Wooldridge (1992) and Lumsdaine (1996). 4 Beta Impulse Response Functions As already mentioned we will only consider vec GARCH models since every BEKK model can be uniquely transformed into an equivalent vec specication. Consider the GARCH(1,1) model with two threshold components, vech( t ) = c + A 1 vech(" t?1 " 0 t?1) + A 2 vech(" t?1 " 0 t?1)i "1;t?1<0 with unconditional covariance matrix, and + A 3 vech(" t?1 " 0 t?1 )I " 2;t?1<0 + G 1 vech( t?1 ) (17) vech() = (I? A 1? A 2 =2? A 3 =2? G 1 )?1 c: All other restricted models (M2, M3, and M4) can be considered as special cases of (17). Assuming covariance stationarity we can dene an impulse response function for the CAPM beta of individual assets. At t = 0 some independent news is reected by 0, and it is not specied whether the news is `good' or `bad', as indicated by the sign of the individual components, and how important the news is, as indicated by the size. as Hafner and Herwartz (1997) dened the volatility impulse response function (VIRF) V t ( 0 ) = E[vech( t ) j 0 ; 0 ]; (18) where V t ( 0 ) is an N(N + 1)=2 dimensional vector. For example, if N = 2, the rst and third element of V t ( 0 ) represent the impulse response function of the conditional variances of the rst and second variable, respectively, and the second element of V t ( 0 ) is the impulse response function of the conditional covariance. For the condition of the volatility state at time t = 0 we will in the following consider the steady state, i.e. 0 =. 6

8 Of course, other states may be considered as well, but the type of inuence of 0 will not change in principal. An impulse response function for the time-varying betas of asset i, is naturally dened to be i;t = im;t 2 m;t B i;t ( 0 ) = V im;t( 0 ) V m;t ( 0 ) ; (19) where V im;t ( 0 ) represents the VIRF for the conditional covariance of the asset with the market portfolio, and V m;t ( 0 ) corresponds to the conditional variance of the market portfolio. Due to the law of iterated expectations, (19) can also be represented as B i;t ( 0 ) = E[ i;t j 0 ; 0 ], that is, the optimal forecast for beta at time t, given information at time 0. Starting at t = 1, it can be seen that V 1 ( 0 ) = c + A 1 vech( 1= =2 ) + A 2 vech( 1= =2 )I 1;0 <0 + A 3 vech( 1= =2 )I 2;0 <0 + G 1 vech(); (20) and, for t 2, V t ( 0 ) = c + (A 1 + A 2 =2 + A 3 =2 + G 1 )V t?1 ( 0 ): (21) This recursive formula can be implemented eciently. Note that lim t?!1 V t ( 0 ) = (I? A 1? A 2 =2? A 3 =2? G 1 )?1 c; so for non{persistent multivariate GARCH processes V t ( 0 ) naturally approaches the unconditional covariance matrix, and thus B t ( 0 ) approaches the unconditional beta. 5 Empirical Analysis 5.1 Dataset and Estimation Results For the empirical analysis we investigate daily prices for 20 German stocks. The sample period is from January 2, 1974 to December 30, 1996 which amounts to 5748 observations for each series under study. Stock price data were obtained from the Deutsche Kapitalmarktdatenbank, Karlsruhe and SFB 373 (Teilprojekt C1) at Humboldt University Berlin. As the risk free rate a money market rate for deposits with one month time to maturity was chosen. Daily rates were kindly provided by the Deutsche Bundesbank. To represent the evolution of the market portfolio we have chosen the so{called DAFOX series, as provided by the University of Karlsruhe. This index is computed for research purposes and is composed of all stocks traded at the Frankfurt stock exchange. All stock market series were adjusted for payments out of the stock and for changes of their nominal value. As an important intermediate result a univariate investigation of the DAFOX series shows that the market returns within our study are signicantly correlated and their 7

9 conditional variance shows a dierent impact of volatility to negative lagged innovations relative to positive ones which is signicant at the 10 % level. Due to signicant autocorrelation of at least one series involved in bivariate modelling we adopt a vector autoregressive model of order one, VAR(1), to investigate the dynamics of bivariate excess return series in the light of the CAPM given in Section 2. Using the econometric programme EVIEWS the following univariate threshold GARCH(1,1) model is obtained for the rst dierences of the logarithm of the DAFOX series (p{values in parentheses): r m;t = 3:52e-04 (:0008) ^ 2 t = 8:25e-07 (:0000) + 0:1378 r m;t?1 + ^e t ; ^e t j t?1 (0; ^ 2 t ); (:0000) + 0:0459 ^e 2 + 0:0591 t?1 ^e 2 t?1 I^e t?1<0 + 0:9182 ^ 2 t?1 (22) (:0379) (:0795) (:0000) The value of the log{likelihood function obtained for this model is Restricting the model in (22) to be symmetric in the variance equation one obtains a value of the log{likelihood function of and assuming the mean of r m;t to be uncorrelated with r m;t?1 one obtains a maximum of the log{likelihood function of Without relying too much on formal test procedures (see e.g. p-values in (22)), both types of restrictions obviously imply a considerable loss in accuracy of t. Further diagnostic checking of standardized residuals (^ t = ^e t =^ t ) obtained from (22) reveals that the GARCH(1,1){ model is sucient to cope with heteroskedasticity. ARCH LM(1) and LM(5) tests (see e.g. Engle (1982)) for ^ t yield p{values of 0.86 and 0.99 respectively. However, the normality hypothesis is clearly rejected for the sequence of standardized errors. Using the Jarque{Bera statistic (see e.g. Lutkepohl (1991)) the obtained p{value is The multivariate GARCH-M models outlined in (6) and (10) to (13) were estimated using numerical procedures written in GAUSS 3.2. It turned out that a few extreme outliers in the bivariate data sets had strong eects on the results of the optimization procedure. In order to reduce the dependence of our results on these outlying observations we trimmed the bivariate series. Let # i denote the empirical standard deviation of a return series in question. For a given return series each observation having an absolute value which was greater than 4# i was set to be equal to 2# i by preserving the sign of the respective observation. Depending on the data set it turned out that between 30 and 40 return observations were regarded as outliers. Relative to the sample size this amounts to 0.5% to 0.7% trimmed observations for each series under study. Without trimming the return series the obtained maximum values of the log{likelihood function were about 1000 points lower relative to a sample with trimmed observations and irrespective of the data set under study. In addition, it turned out that checking estimated residuals for autocorrelation became very dicult without trimming. Portmanteau type statistics became rather large for all data sets under study making a meaningful choice of a candidate VAR-order almost impossible. Note that the adopted trimming procedure is `weak' in the sense that observations which are large in absolute value remain large (2# i ) after 8

10 trimming. To account for conditional heteroskedasticity all multivariate models M1 to M4 given in Section 3 were employed alternatively. In addition, all CAPM{specications were estimated assuming constant risk{premia ( = 0) or time varying risk{premia as given in (6). Thus, we considered eight time series models per data set. Estimation results characterizing the relative performance of the empirical models are provided in Table 1. In order to keep the results readable we do not provide detailed estimates because of the large number of empirical models under study. Table 1 reports the log{likelihood values obtained for the most general model allowing for time dependent risk premia and for two complementary sources of asymmetry of volatility (M1 with unrestricted). The other entries indicate 2 times the dierence of the log{likelihood function of a specic model relative to the general model. Note that under normality these dierences provide the likelihood{ratio (LR) statistic to test simultaneously the zero restrictions implied by the `smaller' model. In addition, these results also provide implicitly test statistics for the restrictions implied by the specication M4 relative to M2 and M3. Asymptotically this statistic follows a 2 (q){distribution where q denotes the number of excess parameters in the general model relative to the model under the null{ hypothesis. Since the necessary assumptions for validity of the asymptotic distribution are possibly not met within the present application we do not interpret our results as formal tests. However, Table 1 provides informal insights into shortcomings of specic restrictions which are hardly met by the data. To indicate possible misspecication of the empirical model we also compute the multivariate Portmanteau{statistic (see e.g. Lutkepohl (1991)) which is obtained from empirical autocorrelations of the estimated error sequences ^" t of order 1 to 10 (Q(10)). Under the hypothesis that the underlying true autocorrelations are zero, Q(10) follows asymptotically a 2 {distribution with 36 degrees of freedom. However, with respect to the necessary assumptions to achieve validity of the asymptotic distribution the same argument as given above applies. The 95% (99%) critical value for the 2 (36) distribution is (58.56). It turned out that the obtained Q(10){statistics did not depend very much on the empirical model which was tted to a specic data set. To economize on space we do not provide Portmanteau statistics for each model under study but the observed range of the 8 Q(10){statistics obtained for each data set. These results are also given in Table 1. Compared with the critical values of the portmanteau statistic all empirical models appear to suer from signicant residual autocorrelation. However, a closer inspection of the estimated residuals reveals that the multivariate correlation measures are spoilt by only a few (say 10) outlying estimates. Therefore an augmentation of the empirical model with further lagged vector returns is regarded as inconvenient. On the other hand reducing the vector autoregressive order to the VAR(0) case would yield values of the 9

11 Q(10){statistics that are about three times higher relative to the VAR(1) case. Taking this argument into account and remembering the univariate properties for the DAFOX return series we adopt the VAR(1){model. With respect to the empirical issue of time varying risk premia we obtain for a few data sets weak evidence in favour of signicance of ^. The values of the log{likelihood function of empirical models specied with and without are always very close. Assuming the relevant LR{test statistic to follow a 2 (1)-distribution some tests for = 0 are signicant e.g. at the 10% signicance level. However, apart from looking at formal test{statistics one observes that for a number of data sets the estimated values of are relatively close to each other for dierent specications of the variance equation. Complementary to a comparison of the implied values of the log{likelihood functions one might interpret `similar' estimates for as evidence in favour of time varying risk{premia. Along these lines we found evidence supporting time varying risk{premia for 10 data sets under study. Since estimates for are essential for the interpretation of the CAPM (time dependent vs. time invariant risk premia) we also provide the obtained estimates for this parameter in Table 1. A few estimation results are obtained irrespective of the time series under study. A symmetric specication of the multivariate GARCH{model is clearly outperformed by the three remaining devices for specifying conditional volatility. Relative to the most general model the smallest LR{statistic is 43.4 (RWE) and for 10 data sets the respective statistic exceeds 100. Taking a log{likelihood dierence of 10 as an informal criterion to discriminate between variance specications one favours a restricted asymmetric model relative to the general model M1 in 9 cases under study. For ve (four) of these data sets asymmetry is best treated using innovations in asset returns (market returns), i.e. model specications M2 (M3) outperform M3 (M2). The most general variance model yields a value of the log{likelihood function which is about 17 points higher relative to the second best model in case of the THYSSEN series. Together with the MAN data set (16 points) these cases indicate the strongest improvement of the log{likelihood function obtained from the general model. For the preferred models, Table 2 reports the implied unconditional correlation, which is dened as = 12 = p The smallest correlation is found for PREUSSAG ( = 0:547), the largest for SIEMENS ( = 0:843). Also, the implied unconditional betas () as well as the two largest eigenvalues of the matrix (14) are reported. For all assets the obtained eigenvalues imply covariance stationarity, since all of them are smaller than one. Also we nd high persistence in the sense that the largest eigenvalues are very close to one. Across the assets, the unconditional beta varies strongly with a minimum for LINDE ( = 0:912) and a maximum for VOLKSWAGEN ( = 1:493). The estimated conditional betas also vary strongly over time, as can be seen in Figure 5. Obviously, the variability 10

12 of the betas for HOECHST is much higher than for SIEMENS, which can be interpreted as a higher second order risk. 5.2 BIRF for the estimated models Based on the estimation results reported in the previous section we calculated the BIRF as dened in Section 4 for each data set. To keep the presentation comprehensible, we selected four assets that can be considered as representative, and present the BIRF plots for these assets. Since we preferred dierent models for dierent assets, we selected BAYER (M1), SIEMENS (M2) and HOECHST (M3). Additionally, we plot the BIRF for all models for MANNESMANN. There are two possible ways of plotting B t ( 0 ): (1) to x t and plot B t ( 0 ) versus the two components of 0, and (2), to x one component of 0 to a constant (zero) and plot B t ( 0 ) versus the other component and time. In Figure 1 the BIRF are given for BAYER, where the upper two plots were obtained by the rst method by setting t = 5 (one week, left plot) and t = 20 (four weeks, right plot). The lower two plots are the BIRF for the second method, where the left plot visualizes the impact of an independent innovation to the asset, i;0, xing the market's innovation to zero. Analogously, the lower right plot visualizes the impact of an innovation to the market, setting the asset's innovation to zero. For the time horizon, a maximum period of 50 trading days (ten weeks) was chosen. This pattern is kept for Figure 2 (SIEMENS) and Figure 3 (HOECHST). First, a general result was that the BIRF tend to be convex in the direction of the asset's innovation, keeping the market's innovation xed, and concave the other way around. The only exception was DAIMLER, where the BIRF was convex in both dimensions. The concavity with respect to the market innovation can be explained by the following. The larger the innovation to the market, the larger also the volatility prediction for the market. Thus, the contribution of a single asset to market risk, as measured by its beta, decreases. On the other hand, an asset-specic innovation creates specic uncertainty, and its own variance and covariance with the market portfolio tend to increase. As a consequence, its contribution to market risk will have a higher price. This general fact is well represented by almost all return series analysed. Next, we will investigate the dierence between the impact of `good news' and `bad news' on the systematic risk. Consider rst BAYER (Figure 1). Obviously, a positive rm-specic innovation has a higher impact on the beta as a negative one of the same size. This observation held for the majority of the analysed series, with some exceptions being COMMERZBANK, DRESDNER BANK and MANNESMANN. The asymmetry is even more distinct for SIEMENS (Figure 2). It should be noted that the leverage eect predicts a higher impact of negative shocks on volatility, so the signs are reverted. Apparently rm-specic news may have a dierent type of impact on volatility and on systematic 11

13 risk. Of course, this type of asymmetry can not be found for HOECHST (Figure 3), since for this series we selected the asset-symmetric model (M3). Now consider a market-relevant innovation, with rm-specic news set to zero. For BAYER (Figure 1) and HOECHST (Figure 3) we detect a similar type of asymmetry: good news to the market tends to decrease systematic risk of the individual assets more than bad news of the same size. This was also a typical result for all market-asymmetric models. It may be explained by a higher conditional covariance for negative shocks to the market than for positive ones. Again, note that for the market-symmetric model applied to SIEMENS (Figure 2) this asymmetry cannot be found. Next, we compare the BIRF implied by the estimation of four dierent models to MANNESMANN, see Figure 4. Recall from Table 1 that for this data set the double asymmetric model M1 had a much higher likelihood than the restricted models. It can be seen that the principal shape of the BIRF is similar, but that the restriction to symmetry has to be paid with less reliable predictions especially at the boundaries. For example, the symmetric model M4 predicts a drop of the beta to 1.2 over the next week when there is a positive rm-specic innovation of two standard deviations and a negative marketrelevant news of the same size. The double-asymmetric model M1, on the other hand, predicts only a small drop to about Finally, we look at the persistence of shocks to the betas. Recall from Table 2 that the two assets with largest eigenvalues are RWE and COMMERZBANK. The BIRF of these assets (not shown) converge extremely slowly to their unconditional betas. As for the combination of two independent shocks, there is still some moderate variability in the betas for most of the assets after 20 days (upper right plots of Figure 1 to 3). For HOECHST and BAYER, market-relevant news appears to have some inuence even after 50 days whereas rm-specic news has disappeared. The inverse is true for SIEMENS. The general conclusion is that portfolio management should consider the sensitivity of time-varying betas with respect to rm-specic and market-general news. In particular, good and bad news appear to have dierent impact not only on volatility, but also on the betas for many assets. This should be taken into account by using appropriate models such as the threshold GARCH model. 6 Conclusions We employ bivariate GARCH in Mean time series models in order to explain excess returns of 20 major German stocks in the light of the CAPM. Using daily data we allow for autocorrelation via a VAR(1) framework and to cope with asymmetry of conditional variances we introduce, estimate and evaluate competing threshold model specications. Asymmetry is found to be `signicant' in the sense that a symmetric bivariate volatility model involves considerable loss in accuracy of t. We provide only weak evidence in 12

14 favour of time depending (i.e. daily varying) risk premia for one half of the data sets under study. For the remaining series the hypothesis of constant risk premia is supported. To facilitate structural analysis of the estimated volatility specications we introduce beta impulse response functions. The estimated BIRF are used to indicate the dependence of one asset's beta on independent news appearing at some xed point in time. Dependence on such news is characterized through time and with respect to its possible sources, i.e. general market vs. asset specic news. The implications of competing specications of asymmetry are discussed with respect to some indicative examples. Within the CAPM framework, beta indicates asset specic systematic risk. We nd that for almost all series under study beta is aected dierently by positive and negative news. For news generated by the market beta is higher in the sequel of bad news relative to good ones. With respect to news emerging from the asset we nd the inverse relationship, i.e. beta is more aected by good news. As mentioned, the interpretation of the estimated BIRF depends on the volatility model under study. Taking general and asset specic innovations as complementary sources of asymmetry the most general volatility model is found to yield superior volatility estimates for 11 data sets. Our results are of interest for time series econometricians as well as for portfolio management in practice. Appendix Proof of Proposition 1: Let t = vech(" t " 0 t). Due to the symmetry of the distribution of 1;t and 2;t, we have E t?1 [ t I "1;t <0] = E t?1 [ t I "2;t <0] = 1 2 E t?1[ t ]; where E t [] is short for E[ j t]. The proof proceeds along the lines of the proof of Proposition 2.7 of Engle and Kroner (1995). Succesively plugging in conditional expectations, one arrives at E t? [ t ] = (I + Z + : : : + Z?2 )c + Z?1 1X i=1 G i?1 f( t?i? +1 ) with Z = A 1 + A 2 =2 + A 3 =2 + G and f( t ) = c + A 1 t + A 2 t I "1;t <0 + A 3 t I "2;t <0. The matrix Z converges to the zero matrix for! 1 if and only if all the eigenvalues of Z have modulus less than one, in which case E t? [ t ] converges in probability (as! 1) to (I? Z)?1 c. Q.E.D. 13

15 References Baba, Y., R.F. Engle, D.F. Kraft, K.F. Kroner (1990), Multivariate simultaneous generalized ARCH, mimeo, Department of Economics, University of California, San Diego. Black, F. (1976), Studies in stock price volatility changes, Proceedings of the 1976 Meeting of the Business and Economics Statistics Section, American Statistical Association, 177{181. Bollerslev, T. (1986), Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, 307{327. Bollerslev, T., R.F. Engle, J.M. Wooldridge (1988), A capital asset pricing model with time varying covariances, Journal of Political Economy 96, 116{131. Bollerslev, T., J.M. Wooldridge (1992), Quasi{maximum likelihood estimation and inference in dynamic models with time{varying covariances, Econometric Reviews 11, 143{172. Drost, F., T. Nijman (1993), Temporal aggregation of GARCH processes, Econometrica 50, Engle, R. (1982), Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. ination, Econometrica 50, 987{1008. Engle, R.F., K.F. Kroner (1995), Multivariate simultaneous generalized ARCH, Econometric Theory 11, 122{150. Engle, R.F., D.M. Lilien, R.P. Robins (1987), Estimating time varying risk premia in the term structure: The ARCH-M model, Econometrica 55, 391{407. Glosten, L.R., R. Jagannathan, D.E. Runkle (1993), On the relation between the expected value and the volatility of the nominal excess return on stocks, Journal of Finance 48, 1779{1801. Hafner, C.M., H. Herwartz (1997), Structural analysis of multivariate GARCH models using impulse response functions, mimeo, Humboldt{Universitat zu Berlin, Germany. Hall, S.G., D.K. Miles, M.P. Taylor (1989), Modelling asset prices with time{varying betas, Manchester School 57, 340{356. Judge, G.G., R.C. Hill, W.E. Griths, H. Lutkepohl, T.C. Lee (1988), Introduction to the Theory and Practice of Econometrics, Wiley, New York. 14

16 Lintner, J. (1965), The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economis and Statistics 47, 13{37. Lutkepohl, H. (1991), Introduction to Multiple Time Series Analysis, Springer Verlag, Berlin. Lumsdaine, R.L. (1996), Consistency and asymptotic normality of the quasi maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models, Econometrica 64, 575{596. Mills, T.C. (1993), The Econometric Modelling of Financial Time Series, Cambridge University Press, Cambridge. Sharpe, W.F. (1964),Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance 19, 425{

17 1 M1 2 M2 1 M2 2 M3 1 M3 2 M4 1 M4 2 Q(10) M1 ^ 2 lo 2lo ^ 2 lo 2lo ^ 2 lo 2lo ^ 2 lo min - max lo ALLIANZ BASF BAYER BMW COMMERZBANK BENZ DAIMLER DEGUSSA BANK DEUTSCHE BANK DRESDNER HOECHST KARSTADT LINDE MAN MANNESMANN PREUSSAG RWE SCHERING SIEMENS THYSSEN VOLKSWAGEN 1: Estimation and diagnostic results for alternative variance specications M1 to M4. Subscripts 1 and 2 indicate the inclusion and Table of from the econometric model respectively. "lo" is short for log{likelihood and 2lo is the LR{statistic relative to the most exclusion 16 general model. Q(10) is the multivariate Portmanteau statistic. Preferred Models are indicated with an asterisk.

18 eig. eig. ALLIANZ KARSTADT BASF LINDE BAYER MAN BMW MANNESMANN COMMERZBANK PREUSSAG DAIMLER BENZ RWE DEGUSSA SCHERING DEUTSCHE BANK SIEMENS DRESDNER BANK THYSSEN HOECHST VOLKSWAGEN Table 2: Estimates of the implied unconditional correlation (), unconditional beta () and the two largest eigenvalues of the `preferred' GARCH{specication (eig.) 17

19 i;0 m;0 i;0 m;0 i;0 t m;0 t Figure 1: BIRF estimates for BAYER (upper panel: B 5 ( 0 ) (left) and B 20 ( 0 ) (right) with respect to asset specic ( i;0 ) and general market innovations ( m;0 ); lower panel: B t ( 0 ) versus time). 18

20 i;0 m;0 i;0 m;0 i;0 t m;0 t Figure 2: BIRF estimates for SIEMENS (upper panel: B 5 ( 0 ) (left) and B 20 ( 0 ) (right) with respect to asset specic ( i;0 ) and general market innovations ( m;0 ); lower panel: B t ( 0 ) versus time). 19

21 i;0 m;0 i;0 m;0 i;0 t m;0 t Figure 3: BIRF estimates for HOECHST (upper panel: B 5 ( 0 ) (left) and B 20 ( 0 ) (right) with respect to asset specic ( i;0 ) and general market innovations ( m;0 ); lower panel: B t ( 0 ) versus time). 20

22 i;0 m;0 i;0 m;0 m;0 m;0 i;0 i;0 Figure 4: BIRF estimates for MANNESMANN (B 5 ( 0 ) obtained for volatility models M1 (upper left), M2 (upper right), M3 (lower left), M4 (lower right) with respect to asset specic ( i;0 ) and general market innovations ( m;0 )). 21

23 y y y y BAYER x SIEMENS x HOECHST x MANNESMANN x Figure 5: The estimated betas for BAYER, SIEMENS, HOECHST, and MANNESMANN (from top to bottom) for the preferred models. The time axis covers the estimation period 1974 to