Estimating time-varying risk prices with a multivariate GARCH model

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 1981 2004. 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, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan 1

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

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

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

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

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 2004. 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 2004. 2 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 1984. 6

price times shares outstanding) at the end of September of each year t (1981 2003). 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 2004. 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

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 2003. 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 2003. 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

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

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

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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 0.880 34.806 0.334 0.826 Highest 2.032 69.049 0.571 2.107 Size 2 1.245 30.193 0.216 1.183 BE/ME 2 1.838 45.395 0.147 1.006 Size 3 1.341 29.942 0.171 1.669 BE/ME 3 1.861 45.686 0.267 1.718 Size 4 1.396 33.943-0.014 1.533 BE/ME 4 1.746 48.948 0.608 1.510 Size 5 1.279 35.505 0.168 1.758 BE/ME 5 1.707 48.172 0.548 0.863 Size 6 1.373 37.241 0.108 1.755 BE/ME 6 1.778 42.823 0.144 0.609 Size 7 1.521 33.475 0.249 2.946 BE/ME 7 1.950 44.786 0.246 0.447 Size 8 1.448 39.028 0.100 1.765 BE/ME 8 1.550 44.244 0.442 1.332 Size 9 1.334 42.063 0.113 1.819 BE/ME 9 1.537 40.285-0.013 0.814 Size 10 1.587 43.404 0.068 1.272 BE/ME 10 1.881 45.487 0.350 1.349 Size 11 1.502 42.696 0.131 1.806 BE/ME 11 1.719 39.362 0.500 0.983 Size 12 1.786 47.146 0.334 1.435 BE/ME 12 1.207 35.977 0.098 1.588 Size 13 1.601 46.298 0.069 0.930 BE/ME 13 1.600 38.687 0.408 1.601 Size 14 1.596 45.948 0.133 0.826 BE/ME 14 1.308 36.352 0.219 0.980 Size 15 1.782 51.738 0.301 1.599 BE/ME 15 1.297 33.159-0.021 0.740 Size 16 1.812 55.766 0.279 1.116 BE/ME 16 1.288 34.835 0.406 2.994 Size 17 1.626 54.122 0.196 1.207 BE/ME 17 1.350 39.235 0.748 4.757 Size 18 1.904 57.538 0.402 1.410 BE/ME 18 1.153 38.323 0.605 3.220 Size 19 2.168 65.859 0.915 4.109 BE/ME 19 1.045 37.947 0.708 2.993 Size 20 1.719 58.811 0.522 2.349 BE/ME 20 1.001 33.143 0.314 1.585 Size 21 2.170 61.365 0.309 0.967 BE/ME 21 1.079 41.158 0.467 2.477 Size 22 2.092 66.998 0.302 1.645 BE/ME 22 1.117 41.062 0.344 1.337 Size 23 2.546 72.813 0.400 0.588 BE/ME 23 0.811 42.155 0.562 2.689 Size 24 2.775 80.312 0.596 1.847 BE/ME 24 0.565 43.652 0.182 1.171 Smallest 3.762 111.198 1.055 2.935 Lowest 0.545 47.542 0.105 2.233 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 2004. 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 (1981-2003). 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.

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 0.109 ** -0.083 ** -0.094 ** 0.036 0.007 0.043-0.018-0.124 ** -0.089 ** 0.087 ** 0.309 ** 0.062 ** p -value 0.000 0.005 0.000 0.122 0.777 0.273 0.437 0.000 0.001 0.010 0.000 0.003 1983 Risk price 0.138 ** 0.089 ** 0.285 ** 0.236 ** 0.168 ** 0.102 * 0.304 ** 0.162 ** 0.023 0.064 * 0.093 ** 0.321 ** p -value 0.002 0.004 0.000 0.000 0.000 0.014 0.000 0.000 0.381 0.047 0.000 0.000 1984 Risk price 0.279 ** 0.052 0.247 ** -0.008-0.274 ** 0.123 ** -0.008 0.219 * 0.019 0.219 ** 0.146 ** 0.009 p -value 0.000 0.127 0.000 0.765 0.000 0.000 0.856 0.037 0.489 0.000 0.000 0.677 1985 Risk price 0.179 ** 0.135 * 0.081 ** 0.013 0.214 ** 0.149 ** -0.001 0.256 ** 0.182 ** 0.117 ** 0.139 ** 0.101 ** p -value 0.000 0.022 0.000 0.567 0.000 0.000 0.980 0.000 0.000 0.001 0.001 0.000 1986 Risk price 0.169 ** 0.405 ** 0.371 0.087 0.153 ** 0.333 ** 0.009 0.058-0.229 ** -0.026 0.401 ** -0.040 * p -value 0.000 0.000 0.000 0.055 0.001 0.000 0.789 0.358 0.008 0.122 0.000 0.025 1987 Risk price 0.144 ** 0.135 ** 0.069 ** 0.136 ** 0.611 ** 0.140 ** 0.297 ** 0.315 ** 0.086 ** -0.246 ** -0.058 ** -0.027 p -value 0.000 0.000 0.006 0.000 0.000 0.001 0.006 0.000 0.000 0.009 0.000 0.168 1988 Risk price 0.452 ** 0.194 ** 0.076 ** 0.245 ** 0.142 ** 0.114 ** -0.122 ** -0.084 ** -0.025-0.063 ** 0.325 ** 0.062 ** p -value 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.005 0.257 0.005 0.000 0.002 1989 Risk price 0.425 ** -0.032 0.165 ** 0.213 ** 0.174 ** -0.087 ** 0.273 ** 0.103 ** 0.357 ** 0.011 0.208 ** 0.178 ** p -value 0.000 0.090 0.000 0.000 0.000 0.000 0.000 0.005 0.000 0.483 0.000 0.000 1990 Risk price -0.028-0.121 ** -0.462 ** -0.099 ** 0.408 ** 0.040 * 0.018-0.510 ** -0.393 ** 0.261 ** -0.143 ** 0.003 p -value 0.503 0.001 0.000 0.009 0.000 0.017 0.327 0.000 0.000 0.000 0.000 0.612 1991 Risk price -0.075 ** 0.329 ** 0.037 ** 0.004-0.001-0.110 ** -0.007-0.228 ** 0.184 ** 0.088 ** -0.244 ** -0.008 p -value 0.000 0.000 0.000 0.330 0.782 0.000 0.328 0.000 0.000 0.000 0.000 0.068 1992 Risk price -0.087 ** -0.041 ** -0.241 ** -0.179 ** 0.187 ** -0.242 ** -0.113 ** 0.235 ** -0.061 ** -0.066 ** 0.065 ** 0.005 p -value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.738 1993 Risk price -0.053 ** 0.002 0.507 ** 0.349 ** 0.158 ** -0.131 ** 0.068 ** 0.050 ** -0.075 ** -0.168 ** -0.598 ** 0.101 ** p -value 0.000 0.834 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1994 Risk price 0.330 ** 0.025 ** 0.014 0.080 ** 0.100 ** 0.085 ** -0.053 ** -0.021-0.180 ** 0.017-0.203 ** 0.108 ** p -value 0.000 0.000 0.296 0.000 0.000 0.000 0.000 0.052 0.000 0.199 0.000 0.000 1995 Risk price -0.142 ** -0.315 ** -0.119 ** 0.015-0.302 ** -0.128 ** 0.407 ** 0.280 ** -0.010 0.004 0.192 ** 0.261 ** p -value 0.000 0.000 0.000 0.052 0.000 0.000 0.000 0.000 0.221 0.638 0.000 0.000 1996 Risk price 0.194 ** -0.102 ** 0.192 ** 0.299 ** -0.025 * 0.044 ** -0.344 ** -0.045 ** 0.152 ** -0.156 ** -0.061 ** -0.329 ** p -value 0.000 0.000 0.000 0.000 0.045 0.006 0.000 0.004 0.000 0.000 0.001 0.000 1997 Risk price -0.128 ** 0.006-0.120 ** 0.113 ** 0.293 ** 0.074 ** -0.151 ** -0.350 ** -0.302 ** 0.085 * -0.341 ** -0.388 ** p -value 0.000 0.636 0.000 0.000 0.000 0.000 0.001 0.000 0.000 0.014 0.000 0.000 1998 Risk price 0.641 ** 0.128 ** -0.051 ** -0.109 ** 0.039 ** 0.111 ** 0.115 ** -0.381 ** -0.148 ** -0.026 * 0.415 ** -0.143 ** p -value 0.000 0.000 0.000 0.000 0.001 0.001 0.000 0.000 0.000 0.025 0.000 0.000 1999 Risk price 0.086 ** 0.035 * 0.438 ** 0.248 ** -0.014 0.357 ** 0.071 ** -0.009 0.014-0.074 ** -0.074-0.152 * p -value 0.000 0.017 0.000 0.000 0.602 0.000 0.000 0.485 0.502 0.000 0.140 0.014 2000 Risk price 0.263 ** 0.033 0.258 ** -0.114 ** 0.200 ** 0.713 ** -0.213 ** 0.153 ** -0.066 ** -0.273 ** 0.107 ** -0.141 ** p -value 0.000 0.356 0.000 0.009 0.000 0.000 0.001 0.000 0.000 0.000 0.000 0.000 2001 Risk price 0.065 ** 0.104 ** 0.316 ** 0.414 ** -0.078 ** 0.144 ** -0.251 ** -0.048-0.241 ** 0.199 ** -0.086 ** -0.153 ** p -value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.080 0.000 0.000 0.000 0.000 2002 Risk price -0.064 * 0.288 ** 0.153 ** 0.136 ** 0.279 ** -0.290 ** -0.055 ** -0.063 ** -0.023-0.270 ** 0.022-0.127 ** p -value 0.028 0.000 0.000 0.000 0.000 0.000 0.005 0.000 0.058 0.000 0.328 0.000 2003 Risk price 0.072 * 0.262 ** 0.059 * 0.295 ** 0.431 ** 0.528 ** 0.049 0.284 ** 0.136 ** 0.144 ** -0.220 ** 0.158 ** p -value 0.018 0.000 0.020 0.000 0.000 0.000 0.093 0.000 0.000 0.000 0.000 0.000 Monthly time-varying price of risk on twenty-five size-ranked portfolios are displayed for the sample period from January 1982 to December 2003. 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 (1981-2003), 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.

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 0.097 ** -0.238 ** -0.156 ** 0.140 ** -0.013-0.078 ** -0.091 ** -0.029-0.056 ** 0.182 ** 0.337 ** 0.115 ** p -value 0.000 0.000 0.000 0.000 0.550 0.000 0.001 0.119 0.001 0.000 0.000 0.000 1983 Risk price -0.053 0.034 0.277 ** 0.131 ** 0.070 * 0.138 ** 0.130 ** 0.111 ** 0.091 ** -0.003 0.031 0.379 ** p -value 0.071 0.159 0.000 0.000 0.020 0.000 0.000 0.000 0.001 0.889 0.140 0.000 1984 Risk price 0.150 ** -0.073 ** 0.317 ** -0.030-0.305 ** 0.087 ** -0.094 ** 0.279 ** 0.030 0.128 ** 0.005 0.125 ** p -value 0.000 0.005 0.000 0.172 0.000 0.000 0.000 0.000 0.113 0.000 0.856 0.009 1985 Risk price 0.064 0.033 0.073-0.022 0.202 ** 0.091 ** -0.119 0.193 ** 0.226 ** 0.017-0.026 0.144 ** p -value 0.115 0.108 0.061 0.447 0.003 0.004 0.077 0.000 0.000 0.789 0.360 0.000 1986 Risk price 0.028 0.288 ** 0.653 ** 0.042 0.110 ** 0.184 ** 0.236 ** 0.209 ** -0.019-0.121 ** 0.278 ** 0.067 ** p -value 0.178 0.000 0.000 0.114 0.000 0.000 0.004 0.003 0.698 0.000 0.000 0.000 1987 Risk price 0.114 ** 0.107 ** 0.092 * 0.191 ** 0.317 ** 0.040 0.052 * 0.240 ** 0.121 * -0.413 ** -0.060 ** -0.086 ** p -value 0.007 0.008 0.031 0.001 0.000 0.293 0.017 0.000 0.011 0.000 0.000 0.009 1988 Risk price 0.323 ** 0.202 ** 0.107 ** 0.159 ** -0.040 * 0.167 ** 0.149 ** -0.200 ** 0.054-0.007 0.251 ** 0.128 * p -value 0.000 0.000 0.000 0.000 0.035 0.000 0.004 0.000 0.102 0.850 0.000 0.019 1989 Risk price 0.261 ** 0.013 0.118 ** 0.121 ** 0.129 ** -0.170 ** 0.310-0.046 0.203 ** -0.014 0.170 ** 0.084 ** p -value 0.000 0.701 0.000 0.000 0.000 0.000 0.000 0.051 0.000 0.511 0.000 0.003 1990 Risk price -0.153 ** -0.208 ** -0.432 ** 0.016 0.307 ** -0.053 * -0.076 ** -0.427 ** -0.417 ** 0.254 ** -0.144 ** 0.037 ** p -value 0.000 0.000 0.000 0.238 0.000 0.037 0.000 0.000 0.000 0.000 0.000 0.000 1991 Risk price -0.044 ** 0.356 ** 0.002-0.022 ** -0.023 * -0.203 ** 0.030 ** -0.261 ** 0.227 ** 0.054 ** -0.319-0.007 p -value 0.000 0.000 0.629 0.001 0.032 0.000 0.009 0.000 0.000 0.000 0.000 0.280 1992 Risk price -0.094 ** -0.097 ** -0.273 ** -0.122 ** 0.142 ** -0.316 ** -0.092 ** 0.330 ** -0.113 ** -0.068 ** 0.093 ** -0.041 ** p -value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 1993 Risk price -0.018-0.034 * 0.563 ** 0.331 ** 0.010-0.136 ** 0.117 ** 0.066 ** -0.078 ** -0.044 ** -0.561 ** 0.107 ** p -value 0.077 0.032 0.000 0.000 0.311 0.000 0.000 0.000 0.000 0.010 0.000 0.000 1994 Risk price 0.355 ** 0.024 * -0.071 ** 0.058 ** 0.133 ** 0.023-0.073 ** 0.034 * -0.141 ** 0.025 ** -0.171 ** 0.082 ** p -value 0.000 0.016 0.000 0.000 0.000 0.091 0.000 0.013 0.000 0.010 0.000 0.000 1995 Risk price -0.273 ** -0.266 ** -0.047 ** 0.016-0.255 ** -0.096 ** 0.411 ** 0.268 ** 0.027 * -0.015 0.148 ** 0.244 ** p -value 0.000 0.000 0.001 0.071 0.000 0.000 0.000 0.000 0.025 0.139 0.000 0.000 1996 Risk price 0.140 ** -0.092 ** 0.182 ** 0.229 ** -0.054 ** 0.104 ** -0.301 ** -0.085 ** 0.213 ** -0.168 ** 0.035-0.176 ** p -value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.117 0.000 1997 Risk price -0.118 ** 0.030 * -0.022 0.254 ** 0.172 ** 0.108 ** 0.045-0.341 ** -0.094 * -0.049-0.166 ** -0.268 ** p -value 0.000 0.043 0.197 0.000 0.000 0.000 0.192 0.000 0.016 0.223 0.001 0.000 1998 Risk price 0.442 ** 0.088 ** -0.015-0.107 ** 0.068 ** 0.076 ** 0.179 ** -0.471 ** -0.109 ** -0.037 * 0.405 ** -0.108 ** p -value 0.000 0.000 0.405 0.000 0.001 0.000 0.000 0.000 0.000 0.029 0.000 0.000 1999 Risk price 0.110 ** 0.009 0.524 ** 0.211 ** -0.032 * 0.368 ** 0.107 ** -0.049 * 0.050-0.062-0.007 0.087 p -value 0.000 0.546 0.000 0.000 0.020 0.000 0.000 0.031 0.105 0.166 0.936 0.462 2000 Risk price 0.070 0.030 0.222 ** -0.069 0.125 0.460 ** -0.211 ** 0.134 ** -0.063 * -0.182 ** 0.126 ** -0.096 * p -value 0.227 0.549 0.000 0.067 0.055 0.000 0.000 0.000 0.023 0.000 0.000 0.011 2001 Risk price 0.050 0.039 0.264 ** 0.473 ** -0.068 ** 0.123 ** -0.243 ** -0.123 ** -0.214 ** 0.137 ** -0.066 * -0.074 ** p -value 0.076 0.399 0.000 0.000 0.000 0.001 0.000 0.004 0.000 0.000 0.049 0.001 2002 Risk price -0.138 ** 0.252 0.164 ** 0.115 ** 0.260 ** -0.284 ** -0.108 ** -0.060 ** -0.015-0.201 ** 0.086 ** -0.125 ** p -value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.533 0.000 0.003 0.000 2003 Risk price -0.010 0.153 ** -0.025 0.176 ** 0.307 ** 0.438 ** 0.087 ** 0.274 ** 0.108 ** 0.161 ** -0.189 ** 0.210 ** p -value 0.709 0.000 0.244 0.000 0.000 0.000 0.006 0.000 0.001 0.000 0.000 0.000 Monthly time-varying price of risk on twenty-five BE/ME-ranked portfolios ratios are displayed for the period from January 1982 to December 2003. 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.

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 0.792-1.177 3.047-3.488-1.805-3.336-1.905-2.957-8.002 ** 0.592 4.619 0.866 p -value 0.856 0.800 0.313 0.162 0.454 0.533 0.378 0.067 0.000 0.837 0.063 0.578 Risk price 0.075-0.030-0.226 0.189 0.091 0.203 0.073 0.022 0.308 ** 0.058 0.072 0.023 p -value 0.709 0.887 0.113 0.098 0.409 0.468 0.511 0.787 0.002 0.718 0.560 0.767 1983 Intercept 7.179 * 7.017 3.516 5.943 * 2.130 2.893 10.403 5.197 1.382 5.596 * 0.522 3.817 p -value 0.015 0.125 0.070 0.025 0.549 0.453 0.130 0.046 0.434 0.025 0.615 0.053 Risk price -0.223-0.265 0.102-0.046 0.065-0.039-0.221-0.102-0.047-0.237 0.066 0.114 p -value 0.136 0.211 0.314 0.732 0.730 0.858 0.504 0.391 0.590 0.083 0.266 0.336 1984 Intercept 9.055 ** -1.460 1.450 0.891-4.390 ** 3.117 * 13.396 7.936 ** 5.189 ** 7.082 ** -0.673 4.456 ** p -value 0.008 0.608 0.609 0.315 0.003 0.026 0.273 0.000 0.001 0.000 0.771 0.002 Risk price -0.130 0.116 0.177-0.046-0.111 0.020-0.497 0.029 ** -0.137 ** -0.047 0.168 * -0.177 ** p -value 0.308 0.434 0.233 0.397 0.071 0.670 0.233 0.003 0.001 0.098 0.027 0.006 1985 Intercept 4.586 2.042 1.411 4.150 1.778-1.871-4.507 6.238 ** -0.228 5.334 ** 5.973 * -0.070 p -value 0.070 0.758 0.089 0.205 0.365 0.541 0.148 0.002 0.910 0.001 0.017 0.950 Risk price -0.021 0.038 0.018-0.181 0.130 0.237 0.216-0.042 0.193-0.141-0.167 0.105 p -value 0.833 0.915 0.557 0.233 0.169 0.127 0.219 0.617 0.096 0.079 0.243 0.109 1986 Intercept 6.174 ** 6.751 ** 0.772 9.613 ** 7.203 ** 10.621 ** -5.680 * -6.157 * -13.805 ** -0.146 11.175 ** -3.508 ** p -value 0.002 0.007 0.784 0.000 0.000 0.000 0.011 0.024 0.000 0.874 0.000 0.008 Risk price -0.159 0.037 0.336-0.128 ** -0.031-0.073 0.209 * 0.326 0.331 ** -0.021-0.083 0.074 p -value 0.167 0.747 0.060 0.000 0.187 0.350 0.025 0.051 0.001 0.462 0.356 0.081 1987 Intercept 1.768 3.285 * 2.024 0.419 18.666 ** 10.037 ** 7.115 ** 7.734 ** 0.171-18.729 ** 3.404 3.336 p -value 0.561 0.028 0.086 0.760 0.000 0.000 0.008 0.001 0.886 0.000 0.212 0.230 Risk price 0.067-0.011 0.017 0.119 * -0.203-0.163 ** -0.050 0.065 0.080 0.479 * -0.159 * -0.137 p -value 0.605 0.857 0.433 0.045 0.118 0.002 0.780 0.510 0.058 0.016 0.049 0.174 1988 Intercept 15.240 8.694 ** 3.284-0.650-3.972-4.216-2.297-4.790-3.738-4.440 8.437 ** 4.144 p -value 0.149 0.000 0.144 0.841 0.438 0.104 0.613 0.225 0.141 0.092 0.000 0.079 Risk price -0.098-0.027-0.012 0.265 * 0.270 0.273 ** -0.035 0.128 0.128 0.132-0.053-0.109 p -value 0.787 0.372 0.832 0.017 0.099 0.005 0.842 0.491 0.250 0.205 0.550 0.234 1989 Intercept 20.678 ** -0.557 9.608 * 8.379 * 6.331 * 0.716 3.015 9.917 ** 20.006 ** 3.706 6.294 ** 6.798 ** p -value 0.000 0.893 0.022 0.015 0.023 0.697 0.058 0.001 0.000 0.187 0.004 0.008 Risk price -0.484 * -0.010-0.245-0.150-0.113-0.120 0.136-0.312 * -0.583 ** -0.150-0.089-0.132 p -value 0.013 0.950 0.176 0.279 0.359 0.167 0.097 0.015 0.000 0.200 0.354 0.248 1990 Intercept 2.999 6.041 * -3.581 * -15.789 ** 17.636 ** -8.709 ** 4.192-12.954 ** -14.713 ** 18.342 ** -16.046 ** 4.529 ** p -value 0.578 0.030 0.042 0.000 0.000 0.002 0.145 0.000 0.000 0.000 0.000 0.010 Risk price -0.171-0.402 ** -0.308 ** 0.304 ** -0.143 0.251 ** -0.110-0.085-0.111 0.055 * 0.005-0.041 ** p -value 0.533 0.006 0.000 0.000 0.055 0.000 0.235 0.297 0.114 0.022 0.668 0.008 1991 Intercept -4.422 * 18.216 ** -2.441-1.429-0.445-6.786 ** 0.368-7.478 ** 8.298 ** 0.840-10.603 ** -0.865 p -value 0.024 0.000 0.284 0.210 0.587 0.000 0.813 0.000 0.000 0.644 0.000 0.347 Risk price -0.024 0.086 * 0.062 * 0.022 0.005 0.005-0.013-0.077 ** 0.032 0.071-0.011 0.009 p -value 0.245 0.027 0.014 0.116 0.700 0.754 0.627 0.008 0.307 0.057 0.689 0.618 1992 Intercept -2.786 * -2.155-6.636 ** 0.897 1.575-7.950 ** 35.233 ** 11.260 ** -1.819-4.565 ** 1.936-1.326 p -value 0.029 0.084 0.000 0.749 0.507 0.000 0.000 0.000 0.622 0.000 0.224 0.608 Risk price -0.023 0.010-0.067 ** -0.199 ** 0.152 ** -0.055-2.293 ** -0.020-0.024 0.036 0.017 0.042 p -value 0.435 0.705 0.007 0.004 0.005 0.078 0.006 0.766 0.749 0.072 0.690 0.585

1993 Intercept -0.437-1.009 13.706 ** 11.281 ** -3.552-5.580 ** 6.163 ** 1.543-4.247 ** 5.615 * -8.019 2.331 p -value 0.826 0.485 0.000 0.008 0.731 0.000 0.001 0.085 0.030 0.028 0.071 0.090 Risk price -0.040 0.035 0.028 0.081 0.229 0.002-0.084 0.006 0.057-0.351 ** -0.317 * 0.061 * p -value 0.515 0.479 0.716 0.408 0.262 0.953 0.073 0.829 0.329 0.000 0.044 0.013 1994 Intercept 13.204 ** 3.237-2.772 2.367 4.422 ** -2.725-0.212 2.536 0.632-3.648-1.209-1.471 p -value 0.000 0.097 0.288 0.237 0.000 0.328 0.841 0.094 0.684 0.163 0.578 0.400 Risk price 0.053-0.030 0.072 0.025-0.017 0.164-0.046-0.106 * -0.203 ** 0.144-0.157 0.160 * p -value 0.160 0.330 0.179 0.581 0.452 0.051 0.223 0.047 0.002 0.137 0.062 0.018 1995 Intercept -9.247 * -12.406 ** 0.632 2.350 0.540 5.052 4.618 6.401 * 4.472 * 0.222-0.570-1.496 p -value 0.034 0.000 0.910 0.056 0.851 0.144 0.150 0.032 0.024 0.803 0.833 0.579 Risk price 0.223 0.141 * -0.138-0.062-0.322 ** -0.280 ** 0.262 * 0.121-0.120 * -0.002 0.210 * 0.311 ** p -value 0.178 0.035 0.394 0.139 0.003 0.009 0.014 0.106 0.012 0.936 0.027 0.003 1996 Intercept -1.782 0.134 1.868-4.380-4.166 * 6.569 ** -2.573-8.672 ** 1.985-7.726 * 7.549 3.613 p -value 0.566 0.935 0.339 0.281 0.031 0.000 0.183 0.000 0.778 0.015 0.261 0.369 Risk price 0.252 * -0.106 0.126 0.461 ** 0.125-0.211 ** -0.237 ** 0.231 ** 0.082 0.137-0.331-0.474 ** p -value 0.020 0.061 0.105 0.007 0.082 0.002 0.006 0.000 0.754 0.277 0.157 0.004 1997 Intercept -4.743 * 1.384 7.522 * 19.141 ** 7.257-1.664-11.342-10.964 ** -7.320-6.577-13.795 ** -11.507 ** p -value 0.033 0.634 0.032 0.000 0.250 0.744 0.060 0.000 0.445 0.274 0.006 0.000 Risk price 0.022-0.039-0.389 ** -0.602 ** 0.002 0.140 0.347 0.111-0.066 0.293 0.189-0.040 p -value 0.745 0.687 0.003 0.001 0.993 0.508 0.187 0.172 0.825 0.126 0.319 0.610 1998 Intercept -15.992-0.048 1.230-1.581 2.501 ** -2.248 7.115 ** -7.474 ** 2.801-4.707 ** 3.038 5.543 * p -value 0.181 0.976 0.358 0.155 0.008 0.480 0.000 0.000 0.100 0.001 0.238 0.012 Risk price 1.041 ** 0.129 ** -0.075 ** -0.074 ** -0.020 0.172-0.096 ** -0.135 ** -0.210 ** 0.082 * 0.335 ** -0.260 ** p -value 0.004 0.000 0.003 0.006 0.287 0.122 0.008 0.003 0.000 0.011 0.000 0.000 1999 Intercept 3.888 ** -3.212 6.978 1.547-0.487 2.137-2.952 0.716 2.706 0.660 3.526-5.758 p -value 0.005 0.055 0.251 0.675 0.797 0.229 0.117 0.689 0.303 0.721 0.449 0.053 Risk price 0.001 0.116 * 0.239 0.217 ** -0.003 0.307 ** 0.130 ** -0.027-0.058-0.095-0.200 0.078 p -value 0.971 0.021 0.245 0.004 0.933 0.000 0.005 0.571 0.429 0.069 0.213 0.437 2000 Intercept 7.268 ** 3.991 8.328 ** -2.564 2.274 17.345 ** -8.005 ** 8.546 ** 1.161-5.547-1.412-1.123 p -value 0.000 0.074 0.000 0.434 0.411 0.000 0.000 0.000 0.598 0.172 0.561 0.689 Risk price -0.045-0.122-0.102-0.007 0.102-0.274 * 0.014-0.109 ** -0.107-0.076 0.149 * -0.101 p -value 0.498 0.096 0.094 0.966 0.343 0.047 0.148 0.008 0.186 0.599 0.044 0.300 2001 Intercept -6.069 ** -18.627 ** 10.804 12.711 ** 1.655-3.310-2.283-16.209 ** -7.541 * 6.055 * -1.014 12.669 ** p -value 0.008 0.008 0.194 0.000 0.540 0.162 0.261 0.001 0.014 0.028 0.833 0.009 Risk price 0.267 ** 0.800 ** -0.138-0.029-0.131 0.264 ** -0.162 0.473 ** 0.011 0.025-0.054-0.584 ** p -value 0.001 0.002 0.673 0.263 0.166 0.003 0.051 0.002 0.913 0.733 0.720 0.002 2002 Intercept -17.019 ** 3.836 8.550 2.180 2.269-8.779 ** -1.843 1.245 3.918-6.004 * 25.977 ** -1.030 p -value 0.000 0.264 0.082 0.206 0.638 0.001 0.706 0.586 0.542 0.026 0.000 0.665 Risk price 0.520 ** 0.155-0.141 0.055 0.188 0.059 0.003-0.106-0.162-0.038-0.801 ** -0.088 p -value 0.001 0.190 0.381 0.401 0.320 0.518 0.983 0.157 0.475 0.703 0.000 0.361 2003 Intercept -10.891 * 37.130 ** -2.389 11.901-3.124-4.357 6.475 ** 5.114 ** 0.321 4.049 * -1.555 6.039 ** p -value 0.037 0.002 0.600 0.229 0.693 0.505 0.000 0.002 0.836 0.023 0.183 0.000 Risk price 0.459 * -1.234 ** 0.161-0.243 0.569 0.684 * -0.117 ** 0.114 * 0.126 * -0.004-0.160 ** -0.056 p -value 0.021 0.008 0.433 0.612 0.127 0.023 0.000 0.031 0.014 0.939 0.000 0.191 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 2003. 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 (1981-2003), 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.

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 0.966 1.849 6.816-9.258 ** -5.625-4.359-2.085 0.545-1.488 15.112 * 8.900-0.878 p -value 0.778 0.808 0.117 0.009 0.116 0.211 0.708 0.871 0.392 0.049 0.002 0.672 Risk price 0.054-0.323-0.426 * 0.511 ** 0.225 0.119 0.004-0.054 0.013-0.535-0.061 ** 0.151 p -value 0.743 0.371 0.024 0.001 0.157 0.443 0.988 0.729 0.872 0.137 0.594 0.090 1983 Intercept 5.078 4.283 12.388 * 1.539 1.418 8.196 * 6.923 1.295-0.570-8.783 ** -3.614 9.047 * p -value 0.150 0.219 0.039 0.638 0.811 0.019 0.176 0.764 0.907 0.003 0.227 0.015 Risk price -0.276-0.157-0.308 0.066 0.004-0.249-0.202 0.049 0.119 0.429 ** 0.208-0.074 p -value 0.096 0.337 0.274 0.622 0.988 0.118 0.400 0.821 0.622 0.002 0.151 0.690 1984 Intercept 3.197-3.277 6.819-1.029-9.229 ** 4.259 ** -4.379 ** 2.017-6.682 1.697-3.975 8.336 p -value 0.452 0.137 0.070 0.493 0.007 0.004 0.002 0.530 0.085 0.730 0.475 0.126 Risk price 0.015 0.066 0.013 0.002 0.013-0.033 0.048 0.207 0.252 0.063 0.162-0.223 p -value 0.931 0.508 0.939 0.973 0.919 0.427 0.287 0.076 0.062 0.745 0.489 0.321 1985 Intercept 15.939 ** -1.875 7.434 0.917-18.471-2.420-7.680 0.965-1.505 14.124 ** 1.011 0.971 p -value 0.002 0.207 0.052 0.910 0.065 0.330 0.194 0.482 0.694 0.006 0.657 0.741 Risk price -0.608 ** 0.113 * -0.268-0.065 1.052 * 0.191 0.236 0.156 ** 0.282-0.537 ** -0.066 0.100 p -value 0.002 0.043 0.130 0.866 0.026 0.088 0.331 0.004 0.082 0.007 0.488 0.503 1986 Intercept 1.121-9.162 * 4.440 9.278 ** 2.171 3.485-21.044 ** 6.573-4.006-2.658 4.102 1.563 p -value 0.633 0.019 0.248 0.001 0.289 0.179 0.001 0.084 0.220 0.151 0.080 0.357 Risk price -0.024 0.724 ** 0.468 * -0.135 ** 0.047 0.077 0.967 ** 0.039 0.088-0.047 0.126 0.015 p -value 0.832 0.000 0.012 0.003 0.420 0.304 0.000 0.735 0.293 0.286 0.190 0.778 1987 Intercept -0.242 2.757 0.575 11.913 ** 13.017 ** 5.575 0.844 3.603 5.762-10.547 ** -7.380 ** -11.750 ** p -value 0.947 0.110 0.796 0.000 0.000 0.163 0.552 0.181 0.146 0.000 0.001 0.005 Risk price 0.124 0.020 0.073-0.309 * -0.121-0.150 0.023 0.099-0.091-0.014 0.120 * 0.290 * p -value 0.383 0.770 0.300 0.013 0.203 0.281 0.678 0.432 0.586 0.869 0.030 0.025 1988 Intercept 5.322 7.935 ** 6.828 * 5.260 * 0.415 12.368 0.389-1.517-3.588-16.117 0.474-1.237 p -value 0.087 0.000 0.028 0.041 0.898 0.089 0.927 0.520 0.483 0.059 0.901 0.921 Risk price 0.148-0.017-0.087-0.012-0.055-0.278 0.135-0.143 0.185 0.642 0.231 0.179 p -value 0.155 0.680 0.282 0.876 0.618 0.240 0.380 0.117 0.357 0.070 0.157 0.731 1989 Intercept 4.293-5.533 6.399-2.539-12.459-1.045 5.967 * 1.115 5.656-11.721 * 19.378 ** 14.402 * p -value 0.245 0.468 0.493 0.383 0.052 0.661 0.018 0.688 0.151 0.029 0.000 0.015 Risk price 0.084 0.232-0.160 0.232 0.713 * -0.121 0.047-0.090-0.035 0.484 * -0.683 ** -0.536 * p -value 0.593 0.444 0.694 0.075 0.026 0.296 0.626 0.449 0.832 0.032 0.001 0.027 1990 Intercept -8.559 2.919-0.263-3.535 * 4.985 ** 11.770 ** 4.995 * -11.546 ** -25.777 ** 19.562 ** -11.407 ** 3.409 p -value 0.136 0.293 0.947 0.030 0.004 0.003 0.034 0.000 0.000 0.006 0.000 0.107 Risk price 0.226-0.329 ** -0.422 ** 0.094 * 0.157 ** -0.358 ** -0.228 ** -0.066 0.125 0.010-0.027-0.003 p -value 0.373 0.008 0.009 0.015 0.005 0.001 0.001 0.363 0.118 0.898 0.149 0.902 1991 Intercept -4.282 ** 14.877 ** -0.067-2.370-2.924 ** -5.185 ** 3.358 * -9.061 ** 2.192-1.480-9.039 ** -1.521 p -value 0.008 0.000 0.966 0.244 0.003 0.000 0.012 0.000 0.346 0.459 0.000 0.319 Risk price 0.024 0.074 0.003 0.025 0.044 * -0.061 * -0.057 0.015 0.169 * 0.095-0.027 0.033 p -value 0.340 0.103 0.880 0.508 0.046 0.029 0.089 0.651 0.011 0.150 0.620 0.387 1992 Intercept -3.030-3.732 * -3.566 6.764-4.113-4.604-3.675 * -2.264-5.572 ** -0.250 3.459 ** 0.734 p -value 0.101 0.022 0.096 0.112 0.057 0.000 0.013 0.567 0.000 0.882 0.007 0.713 Risk price 0.002 0.026-0.151 * -0.318 * 0.265 ** -0.165 0.004 0.398 ** 0.017-0.061-0.009-0.065 p -value 0.965 0.576 0.042 0.011 0.000 0.000 0.918 0.003 0.515 0.142 0.814 0.357