Estimating time-varying risk prices with a multivariate GARCH model

Transcription

1 Estimating time-varying risk prices with a multivariate GARCH model Chikashi TSUJI December 30, 2007 Abstract This paper examines the pricing of month-by-month time-varying risks on the Japanese stock market over the period Using a multivariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, we tested the conditional version of the Sharpe (1964) Lintner (1965) Mossin Capital Asset Pricing Model (CAPM) and Black s (1972) zero-beta CAPM. To focus strictly on the time-varying characteristics of the monthly risk prices, we employ twenty-five size-ranked and twenty-five BE/ME (book equity to market equity)-ranked portfolio returns. The method used in constructing the portfolios follows Fama and French (1993). The empirical results show that the price of market risk in the conditional version of the Sharpe Lintner Mossin CAPM is generally positive and significant. This provides evidence contrary to the findings of many international studies where the traditional CAPM is very often rejected. Keywords: Conditional CAPM; Jensen s alpha; Multivariate GARCH Model; Time-varying price of risk. JEL classification: G12; G15. Associate Professor, Department of Social Systems and Management, Graduate School of Systems and Information Engineering, University of Tsukuba, Tennodai, Tsukuba, Ibaraki , Japan 1

2 1 Introduction The time-varying characteristics of both covariance risks and the prices of risk are clearly crucial for asset pricing. In the Sharpe (1964) Lintner (1965) Mossin Capital Asset Pricing Model (CAPM), for example, as in most other one-period equilibrium models, the expected risk premium of an asset is equal to the price of risk times the amount of non-diversifiable asset risk, where the price of risk is related to individuals preferences and is positive and identical across assets. However, there is substantial empirical evidence that the amount of risk varies over time (see Bollerslev et al. (1988), Harvey (1989), Ng (1991), and Zhou (1994), amongst others). Thus, it is natural to extend the traditional unconditional CAPM to the conditional CAPM (see Jagannathan and Wang (1996), Guo (2006), and Lewellen and Nagel (2006), amongst others). In many earlier studies, including Harvey (1989), Campbell (1996), Hansson and Hördahl (1998), and Guo (2006), covariance risks are regarded as time-varying, but the prices of risk are evaluated at a certain value given the particular period specified by the authors. Thus, the dynamics of the risk prices and the degree of pricing of the risks from a time-series viewpoint appears to be unclear in existing work. From this specific viewpoint and motivation, the primary objective of this analysis is to reveal the dynamics of the time-varying prices of risk in traditional asset pricing models. Our interest lies particularly in clarifying the monthly statistical significance of the time-varying price of risk. To avoid blurring this motivation for our research, we focus on the conditional versions of just two traditional and representative asset-pricing models: the CAPM and Black s (1972) zero-beta CAPM. To conduct our investigation towards the above goal, we construct twenty-five portfolios formed by BE/ME (book equity to market equity) and twenty-five portfolios by size following Fama and French (1993). We then use a multivariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model to derive the time-varying covariance risks. By exploiting the covariances and performing cross-sectional regressions month-by-month in Japan, we inspect the monthly position of not only the time-varying risk as in other studies but also the time-varying risk prices. From a methodological point of view, research into asset pricing using multivariate 2

3 GARCH models also seems limited. 1 Therefore, clarifying the monthly statistical significance of time-varying risk prices in Japan using the multivariate GARCH model is the main contribution of this paper. The remainder of this paper is organized as follows. Section 2 describes the data employed. Section 3 presents the models and Section 4 documents the methodology used. The empirical results and their interpretation are supplied in Section 5. Section 6 presents our conclusions. 2 Model As mentioned, we focus on two traditional conditional models in this paper. The first is the conditional CAPM. The model for period t is an equilibrium relation for the conditional expected return of an asset in excess of the risk-free rate when agents use the information available at the end of period t 1: E [(r i,t r f,t ) Ω t 1 ] = E [(r m,t r f,t ) Ω t 1 ] Cov [r i,t,r m,t Ω t 1 ] (1) Var[r m,t Ω t 1 ] = β i,t E [(r m,t r f,t ) Ω t 1 ]. β i,t Cov [r m,t,r i,t Ω t 1 ], (2) Var[r m,t Ω t 1 ] where r i,t and r m,t are the one-period returns on an asset and the market portfolio, respectively, r f,t is the one-period risk-free rate, and Ω t 1 is the information available to academic researchers or practitioners at time t 1. From a cross-sectional perspective, the model first has the implication 1 Risk as the variation in asset returns has generally been modelled using the Autoregressive Conditional Heteroskedasticity (ARCH) model of Engle (1982) and its successors, the Generalized ARCH (GARCH) model (Bollerslev (1986)), or EGARCH model (Nelson (1991)). In most early studies, a univariate analysis is employed in asset pricing by using the conditional variance estimated via ARCH, GARCH, or the EGARCH models (French et al. (1987), Poon and Taylor (1992), and Hansson and Hördahl (1997)). Bollerslev et al. (1988) perform one of the first multivariate analyses in a test of CAPM. Other studies that analysed asset pricing models using the multivariate GARCH model include Engle et al. (1995), Koutmos and Booth (1995), Braun et al. (1995), Kroner and Ng (1998), and Leon et al. (2007). In the above studies, risk prices were either evaluated as a certain value in a particular period specified by the authors (Engle et al. (1995) and Leon et al. (2007)), or the focus was not on risk prices at all (Kroner and Ng (1998), Koutmos and Booth (1995), and Braun et al. (1995)). 3

4 that the conditional expected excess returns vary with the different conditional beta values or different conditional covariances. From a time-series perspective, the model has the implication that the conditional expected excess returns change over time with three time-varying components: the market risk premium, the market conditional variance, and the conditional covariance between an asset s return and the market s return. The conditional CAPM is a static one-period model that holds period by period. It is also a generalization of the one-period CAPM developed by Sharp Lintner Mossin in the sense that agents have common conditional expectations of the first two moments of future returns. Unlike Harvey (1989), Engel et al. (1995), and Hansson and Hördahl (1998), among others, we do not assume that the market price of risk is stable over time but rather assume that it is time varying: δ t E [(r m,t r f,t ) Ω t 1 ]. (3) Var[r m,t Ω t 1 ] Thus, again differently from other studies, we have the following conditional version of CAPM for asingleasseti: E [(r i,t r f,t ) Ω t 1 ]=δ t Cov [r i,t,r m,t Ω t 1 ]. (4) In this formulation, the estimation of the time-varying covariances, Cov[r i,t,r m,t Ω t 1 ], is necessary for evaluating this model and, by inspecting the statistical significance of δ t using these covariances, we can judge whether the covariance risk is priced. Thus, this model is the first focus for the empirical work in this analysis. In model (4), δ t is interpreted as the Arrow Pratt measure of aggregate relative risk aversion and should be positive if agents are risk averse. Model (4) could also be considered as a statistical implementation of the intertemporal CAPM. Our second focus is the following model (5), which could be interpreted as the conditional version of Black s (1972) zero-beta CAPM: E [(r i,t r f,t ) Ω t 1 ]=γ + δ t Cov [r i,t,r m,t Ω t 1 ], (5) where γ is a common intercept for all portfolios and, again differently from earlier analyses, we assume a time-varying risk price, δ t. 4

5 3 Methodology As an excellent survey by Bauwens et al. (2006) argues, the multivariate GARCH model is crucially important in the context of asset pricing since the model is useful for calculating the time-varying covariances or factor loadings. To evaluate the time-varying risk prices, δ t and λ t above, we first estimate the time-varying covariances, Cov[r i,t,r m,t Ω t 1 ] by the multivariate BEKK GARCH model. The BEKK version of the multivariate GARCH model was introduced by Engle and Kroner (1995). This particular BEKK model ensures that the H matrix is always positive definite, and is specified by H t = W + B 0 H t 1 B + A 0 Ξ t 1 Ξ 0 t 1A, (6) where W, A, and B are 2 2 matrices of parameters, and W is assumed to be symmetric and positive definite. The positive definiteness of the covariance matrix is ensured because of the quadratic nature of the terms on the right-hand side of equation (6). For the purpose of clarity, in the case of two assets, we define the matrices as below, H t = h 11,t h 12,t, W = w 11 w 12, A = a 11 a 12, h 12,t h 22,t w 12 w 22 a 21 a 22 B = b 11 b 12, Ξ t = u 1,t ; b 21 b 22 u 2,t the model is then written in full as: h 11,t = w 11 + a 2 11u 2 1,t 1 + a 2 21u 2 2,t 1 +2a 11 a 21 u 1,t 1 u 2,t 1 +b 2 11h 11,t 1 + b 2 21h 22,t 1 +2b 11 b 21 h 12,t 1, (7) h 22,t = w 22 + a 2 12u 2 1,t 1 + a 2 22u 2 2,t 1 +2a 12 a 22 u 1,t 1 u 2,t 1 +b 2 12h 11,t 1 + b 2 22h 22,t 1 +2b 12 b 22 h 12,t 1, (8) h 12,t = w 12 + a 11 a 12 u 2 1,t 1 + a 21 a 22 u 2 2,t 1 +(a 12 a 21 + a 11 a 22 )u 1,t 1 u 2,t 1 +b 11 b 12 h 11,t 1 + b 21 b 22 h 22,t 1 +(b 11 b 22 + b 12 b 21 )h 12,t 1. (9) In regard to the model estimation, the parameters of the multivariate GARCH models of any 5

6 of the above specifications can be estimated by maximizing the log-likelihood function: l(θ) = TN 2 log 2π 1 2 TX (log H t + Ξ 0 t H 1 t Ξ t ), (10) t=1 where θ denotes all of the unknown parameters to be estimated, N is the number of assets, T is the number of observations, and H t and Ξ t are as defined earlier. The maximum-likelihood estimate for θ is asymptotically normal, and thus traditional procedures for statistical inference are applicable. After deriving the time-varying covariances, Cov[r i,t,r m,t Ω t 1 ], from the multivariate GARCH model, we perform regressions (4) and (5) using cross-sections in each month. Then, the timevarying prices of risk δ t and λ t can be evaluated month by month. 4 Data The data analysed in this article are from the sample period from October 1981 to July The individual data series are outlined below, and the notations of the data are risk-free percentage rate, R f,t, the market portfolio percentage return, R m,t,andr i,t is the returns of twenty-five portfolios constructed using stocks listed on the Tokyo Stock Exchange s (TSE) 1st Section. First, R f is the gensaki rate from the Japan Securities Dealers Association from October 1981 to May 1984 and the one-month median rate on negotiable-time certificates of deposit (CD) from the Bank of Japan from June 1984 to July The market return R m is the value-weighted return of all stocks in the 1st Section of the TSE as provided by the Japan Securities Research Institute. In regards to the twenty-five portfolio returns, we constructed twenty-five size-ranked portfolio returns and twenty-five BE/ME (book equity to market equity)-ranked portfolio returns, following the manner of Fama and French (1993). To construct the size-ranked portfolios, all TSE 1st Section stocks are allocated to one of twenty-five groups based on their market equity (ME, stock 2 Before June 1984, one-month CD rates are not available. Thus, following Hamao (1988), we specified the gensaki rate as the risk-free rate before June

7 price times shares outstanding) at the end of September of each year t ( ). Valueweighted monthly returns on the portfolios are then calculated from the following October to the next September. 3 When constructing the BE/ME portfolios, the BE/ME ratio used to form the portfolios in September of year t is the book value of common equity for fiscal year t 1, divided by the market value of equity at the end of March in calendar year t. We do not use negative BE firms when forming the BE/ME portfolios. The value-weighted monthly returns on the BE/ME portfolios are then calculated from October to the following September as for the size-ranked portfolios. Further, only firms with ordinary common equity are included in our analysis. This means that REITs (Real Estate Investment Trusts) and beneficial interest units are excluded. 4 5 Empirical Results 5.1 The pricing degree of the time-varying risk prices This section provides our empirical results and their interpretation. First, we present sample statistics of the value-weighted returns of the twenty-five size-ranked and BE/ME-ranked portfolios over the period from October 1981 to July The mean returns of the size-ranked portfolios show a rather clear pattern of a monotonic increase from the biggest-size portfolio to the smallestsize portfolio. The mean returns of BE/ME-ranked portfolios also show evidence of a monotonic increase from the lowest BE/ME portfolio to the highest BE/ME portfolio, although the pattern is not as strong as that found in the size-ranked portfolios. Thus, we recognize both a size effect and a BE/ME effect; however, as shown in Table 1, we can see that the former is stronger than 3 We rebalanced the portfolios every September following Fama and French s (1993) suggestion: We calculate returns beginning in July of year t to be sure that book equity for year t 1isknown(FamaandFrench1993,p. 9). In Japan, the fiscal year that most companies close is not at the end of December as in the United States, but at the end of March; that is, the end of the fiscal year in Japan is generally three months after the United States. Thus, we calculate returns not from July but from October of year t to September of year t +1, after rebalancing portfolios in every September of year t, tobesurethatbookequityforthemostrecentfiscal year is known in the Japanese market. 4 The BE/ME-ranked portfolios were formed following the manner and intention of Fama and French (1993) with the size-ranked portfolios. 7

8 the latter in Japan. Next, applying equations (4) and (5) cross-sectionally month by month, we obtain the monthly time-varying prices of risk from the two conditional models on the size-ranked and BE/ME-ranked portfolios. Because each regression comprises a cross-section, White s (1980) heteroskedasticityconsistent covariance matrix is used to calculate the p-values. Table 2 displays the monthly timevarying prices of risk from the conditional CAPM of the portfolios formed on the basis of size for the period from January 1982 to December From Table 2, we understand that, in general, the monthly time-varying prices of risk from the conditional CAPM are statistically significant, and the number of the significant risk prices with theoretically consistent positive signs comprise 135 of the 264 cases in total. Table 3 also displays the monthly time-varying prices of risk from the conditional CAPM for the portfolios formed on the basis of BE/ME for the same sample period as in Table 2. The trends found are very similar to those of the size-ranked portfolios. In general, the monthly time-varying prices of risk from the conditional CAPM are also statistically significant for the BE/ME-ranked portfolios, and the number of the significant risk prices with positive signs represent 123 of the 264 cases in total. Table 4 presents the monthly time-varying prices of risk and alphas from the conditional zerobeta CAPM for the twenty-fivesize-rankedportfolios for the periodfromjanuary 1982 to December From Table 4, we can see that, in general, the monthly time-varying prices of risk from the conditional zero-beta CAPM are not statistically significant; in only 42 of 264 cases are the risk prices significant and with positive signs. In contrast, Table 4 also shows that the monthly timevarying alphas from the conditional zero-beta CAPM for the size-ranked portfolios are statistically significant and with positive signs in 69 of 264 cases, and there are 153 cases of positive alphas, regardless of their statistical significance, in the 264 total cases. Table 5 also displays the monthly time-varying prices of risk and alphas from the conditional zero-beta CAPM for the portfolios formed on the basis of BE/ME. The trends found are very similar to those for the size-ranked portfolios. In general, the monthly time-varying prices of risk from the conditional zero-beta CAPM are not statistically significant in the BE/ME-ranked portfolios; in just 30 of 264 cases are the risk prices significantly positive. In contrast, Table 5 also shows that the monthly time-varying 8

9 alphas from the conditional zero-beta CAPM on the BE/ME-ranked portfolios are statistically significant with positive signs in 41 of 264 cases, and this increases to 146 cases if all positive alphas are included regardless of statistical significance. Hence, the application of the conditional version of the zero-beta CAPM using the multivariate GARCH model suggests that positive alphas exist in Japan, although not all of them are always statistically significant. 6 Conclusions This paper has originally and minutely investigated the degree of pricing of the month-by-month time varying risks on the Japanese stock market by using a multivariate GARCH model. The significant facts and implications derived in this analysis are as follows. First, we demonstrated that conditional covariance risks in CAPM, as derived by a multivariate GARCH model, are generally positively priced in Japan. The clarification of the situation in regard to the time-varying risk prices, rather than the time-varying covariance risks analysed in many other studies, is our focus and primary contribution in this article. Second, from the viewpoint of the conditional version of the zero-beta CAPM, positive alphas generally exist in Japan. In particular, the alphas obtained from the size-ranked portfolios are higher than the alphas from the BE/ME-ranked portfolios. In this paper, we have focused on two asset pricing models: the conditional CAPM and conditional zero-beta CAPM. Therefore, risk factors other than the covariances between the return on an asset and the market return and other types of asset pricing models are beyond the scope of this article. However, as a result of our research, we have revealed that an investigation of other risk factors and other models by especially focusing on the time-varying characteristics of risk prices would be interesting. This is best left to future work. Also, as pointed out earlier, studies using multivariate GARCH models in the field of asset pricing are limited. Hence, international research using this model in the field of asset pricing would be a valuable contribution to the entire finance literature. 9

10 7 Acknowledgements The author acknowledge the generous financial assistance of the Japan Society for the Promotion of Science and the Zengin Foundation for Studies on Economics and Finance. I would also like to thank Jason McQueen, Visiting Professor at Tokyo University, for some helpful suggestions on the empirical analysis in this paper. 10

11 References [1] Bauwens, L., Laurent, S. and Rombouts, J.V.K. (2006) Multivariate GARCH models: A survey, Journal of Applied Econometrics 21, [2] Black, F. (1972) Capital market equilibrium with restricted borrowing, Journal of Business 45, [3] Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics 31, [4] Bollerslev, T., Engle, R.F. and Wooldridge, J.M. (1988) A capital asset pricing model with time-varying covariances, Journal of Political Economy 96, [5] Braun, P.A., Nelson, D.B. and Sunier, A.M. (1995) Good news, bud news, volatility, and betas, Journal of Finance 50, [6] Campbell, J (1996) Understanding risk and return Journal of Political Economy 104, [7] Engel, C., Frankel, J.A., Froot, K.A. and Rodrigues, A.P. (1995) Tests of conditional mean variance efficiency on the U.S. stock market, Journal of Empirical Finance 2, [8] Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50, [9] Engle, R.F. and Kroner, K.F. (1995) Multivariate simultaneous generalized ARCH, Econometric Theory 11, [10] Fama, E. and French, K. (1993) Common risk factors in the returns on stocks and bonds, Journal of Financial Economics 33, [11] French, K. R., Schwert, G.W. and Stambaugh, R.F. (1987) Expected stock returns and volatility, Journal of Financial Economics 19, [12] Guo, H. (2006) Time-varying risk premia and the cross section of stock returns, Journal of Banking & Finance 30, [13] Hamao, Y. (1988) An empirical examination of the arbitrage pricing theory, Japan and the World Economy, 1, [14] Hansson, B. and Hördahl, P. (1997) Estimating the price of risk in a conditional asset pricing model: Evidence from the Swedish stock market, Scandinavian Journal of Economics 99, [15] Hansson, B. and Hördahl, P. (1998) Testing the conditional CAPM using multivariate GARCH-M, Applied Financial Economics 8, [16] Harvey, C.R. (1989) Time-varying conditional covariances in tests of asset pricing models, Journal of Financial Economics 24, [17] Jagannathan, R. and Wang, Z. (1996) The Conditional CAPM and the Cross-Section of Expected Returns, Journal of Finance 51, [18] Koutmos, G. and Booth, G.G. (1995) Asymmetric volatility transmissions in international stock markets, Journal of International Money and Finance 14, [19] Kroner, K.F. and Ng, V.K. (1998) Modeling Asymmetric Comovement of Assets Returns, Review of Financial Studies 11, [20] Lewellen, J. and Nagel, S. (2006) The conditional CAPM does not explain asset-pricing anomalies, Journal of Financial Economics 82, [21] Leon, A., Nave, J.M. and Rubio, G. (2007) The relationship between risk and expected return in Europe, Journal of Banking & Finance 31,

12 [22] Lintner, J. (1965) The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics 47, [23] Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach, Econometrica 59, [24] Ng, L. (1991) Tests of the CAPM with time-varying covariances: A multivariate GARCH approach, Journal of Finance 46, [25] Poon, S.H. and Taylor, S.J. (1992) Stock returns and volatility: An empirical study of the U.K. stock market, Journal of Banking & Finance 16, [26] Sharpe, W. (1964) Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance 19, [27] White, H. (1980) A Heteroskedasticity-Consistent Covariance Matrix Estimator and Direct Test for Heteroskedasticity, Econometrica 48, [28] Zhou, G. (1994) Analytical GMM tests: asset pricing with time-varying risk premiums, Review of Financial Studies 7,

13 Table 1 Sample statistics of the value-weighted returns on twenty-five portfolios formed on the basis of size and BE/ME: October 1981 to July 2004 Size-ranked portfolios BE/ME-ranked portfolios Portfolio Mean Variance Skewness Kurtosis Portfolio Mean Variance Skewness Kurtosis Biggest Highest Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Size BE/ME Smallest Lowest The sample statistics of the value-weighted returns of twenty-five portfolios formed on the basis of size or BE/ME (book equity to market equity) ratios are displayed. The sample period is from October 1981 to July The size- and BE/ME-ranked portfolios are constructed following Fama and French (1993). In constructing the size-ranked portfolios, TSE (Tokyo Stock Exchange) 1st Section stocks were allocated to twenty-five groups based on their market equity (ME, stock price times shares outstanding) at the end of September of each year t ( ). Value-weighted monthly returns on the portfolios were then calculated from October to the following September. When constructing the BE/ME portfolios, the BE/ME ratio used to form portfolios in September of year t is the book common equity for the fiscal year t 1, divided by the market equity at the end of March in calendar year t. Negative BE firms when not used in forming the BE/ME portfolios. The value-weighted monthly returns on the portfolios are then calculated from October to the following September as for the size-ranked portfolios. Only firms with ordinary common equity are included. REITs (Real Estate Investment Trusts) and units of beneficial interest are excluded.

14 Table 2 Monthly time-varying price of risk on twenty-five portfolios formed on the basis of size: the case of the conditional CAPM in Japan from January 1982 to December 2003 January February March April May June July August September October November December 1982 Risk price ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** * ** ** * ** ** p -value Risk price ** ** ** ** * ** ** p -value Risk price ** * ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** ** * p -value Risk price ** ** ** ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** * ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** * ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** * ** ** p -value Risk price ** ** ** ** ** ** ** ** ** * ** ** p -value Risk price ** * ** ** ** ** ** * p -value Risk price ** ** ** ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** ** ** ** p -value Risk price * ** ** ** ** ** ** ** ** ** p -value Risk price * ** * ** ** ** ** ** ** ** ** p -value Monthly time-varying price of risk on twenty-five size-ranked portfolios are displayed for the sample period from January 1982 to December The risk prices of the conditional CAPM are calculated using conditional time-varying covariances from a multivariate GARCH model. The portfolios are formed following the procedures in Fama and French (1993); that is, at the end of September of each year t ( ), TSE (Tokyo Stock Exchange) 1st Section stocks are allocated to one of twenty-five groups based on their September market equity (ME, stock price times shares outstanding). Value-weighted monthly returns on the portfolios are then calculated from October to the following September. Only firms with ordinary common equity are included. REITs (Real Estate Investment Trusts) and units of beneficial interest are excluded. p-values are calculated using White's (1980) heteroskedasticity consistent covariance matrix. ** and * denote statistical significance at the 1% and 5% level, respectively.

15 Table 3 Monthly time-varying price of risk for twenty-five portfolios formed on the basis of BE/ME: the case of the conditional CAPM in Japan from January 1982 to December 2003 January February March April May June July August September October November December 1982 Risk price ** ** ** ** ** ** ** ** ** ** p -value Risk price ** ** * ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** ** p -value Risk price ** ** * ** ** * ** * ** ** ** p -value Risk price ** ** ** ** * ** ** ** ** * p -value Risk price ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** * ** ** ** ** ** ** p -value Risk price ** ** ** * ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** ** ** ** ** p -value Risk price * ** ** ** ** ** ** ** ** ** p -value Risk price ** * ** ** ** ** * ** ** ** ** p -value Risk price ** ** ** ** ** ** ** * ** ** p -value Risk price ** ** ** ** ** ** ** ** ** ** ** p -value Risk price ** * ** ** ** ** * ** ** p -value Risk price ** ** ** ** ** ** ** ** * ** ** p -value Risk price ** ** ** * ** ** * p -value Risk price ** ** ** ** * ** ** * p -value Risk price ** ** ** ** ** ** ** ** * ** p -value Risk price ** ** ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** ** ** ** ** ** ** p -value Monthly time-varying price of risk on twenty-five BE/ME-ranked portfolios ratios are displayed for the period from January 1982 to December The risk prices of the conditional CAPM are calculated using the conditional time-varying covariances derived from the multivariate GARCH model. The portfolios are formed following the procedures in Fama and French (1993). That is, the BE/ME ratios used to form portfolios in September of year t is the book common equity for the fiscal year t 1, divided by the market equity at the end of March in calendar year t. We do not use negative BE firms when forming the BE/ME portfolios. Value-weighted monthly returns on the portfolios are then calculated from October to the following September. Only firms with ordinary common equity are included. REITs (Real Estate Investment Trusts) and units of beneficial interest are excluded. p-values are calculated by using White's (1980) heteroskedasticity consistent covariance matrix. ** and * denote statistical significance at the 1% and 5% level, respectively.

16 Table 4 Monthly time-varying prices of risk and alphas on twenty-five portfolios formed on the basis of size: the case of the conditional zero-beta CAPM in Japan from January 1982 to December 2003 January February March April May June July August September October November December 1982 Intercept ** p -value Risk price ** p -value Intercept * * * p -value Risk price p -value Intercept ** ** * ** ** ** ** p -value Risk price ** ** * ** p -value Intercept ** ** * p -value Risk price p -value Intercept ** ** ** ** ** * * ** ** ** p -value Risk price ** * ** p -value Intercept * ** ** ** ** ** p -value Risk price * ** * * p -value Intercept ** ** p -value Risk price * ** p -value Intercept ** * * * ** ** ** ** p -value Risk price * * ** p -value Intercept * * ** ** ** ** ** ** ** ** p -value Risk price ** ** ** ** * ** p -value Intercept * ** ** ** ** ** p -value Risk price * * ** p -value Intercept * ** ** ** ** ** p -value Risk price ** ** ** ** p -value

17 1993 Intercept ** ** ** ** ** * p -value Risk price ** * * p -value Intercept ** ** p -value Risk price * ** * p -value Intercept * ** * * p -value Risk price * ** ** * * * ** p -value Intercept * ** ** * p -value Risk price * ** ** ** ** ** p -value Intercept * * ** ** ** ** p -value Risk price ** ** p -value Intercept ** ** ** ** * p -value Risk price ** ** ** ** ** ** ** * ** ** p -value Intercept ** p -value Risk price * ** ** ** p -value Intercept ** ** ** ** ** p -value Risk price * ** * p -value Intercept ** ** ** ** * * ** p -value Risk price ** ** ** ** ** p -value Intercept ** ** * ** p -value Risk price ** ** p -value Intercept * ** ** ** * ** p -value Risk price * ** * ** * * ** p -value Monthly time-varying prices of risk and the alphas of twenty-five portfolios formed on the basis of size are displayed for the period from January 1982 to December The risk prices and the alphas of the conditional zero-beta CAPM are calculated using the conditional time-varying covariances derived from the multivariate GARCH model. The portfolios are formed following the procedures in Fama and French (1993). That is, at the end of September of each year t ( ), TSE (Tokyo Stock Exchange) 1st Section stocks are allocated to twenty-five groups based on their September market equity (ME, stock price times shares outstanding). Value-weighted monthly returns on the portfolios are then calculated from October to the following September. Only firms with ordinary common equity are included. REITs (Real Estate Investment Trusts) and units of beneficial interest are excluded. p-values are calculated by using White's (1980) heteroskedasticity consistent covariance matrix. ** and * denote statistical significance at the 1% and 5% level, respectively.

18 Table 5 Monthly time-varying prices of risk and alphas of twenty-five portfolios formed on the basis of BE/ME: the case of the conditional zero-beta CAPM in Japan from January 1982 to December 2003 January February March April May June July August September October November December 1982 Intercept ** * p -value Risk price * ** ** p -value Intercept * * ** * p -value Risk price ** p -value Intercept ** ** ** p -value Risk price p -value Intercept ** ** p -value Risk price ** * * ** ** p -value Intercept * ** ** p -value Risk price ** * ** ** p -value Intercept ** ** ** ** ** p -value Risk price * * * p -value Intercept ** * * p -value Risk price p -value Intercept * * ** * p -value Risk price * * ** * p -value Intercept * ** ** * ** ** ** ** p -value Risk price ** ** * ** ** ** p -value Intercept ** ** ** ** * ** ** p -value Risk price * * * p -value Intercept * * ** ** p -value Risk price * * ** ** p -value