The intertemporal relation between expected returns and risk $

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1 Journal of Financial Economics 87 (2008) The intertemporal relation between expected returns and risk $ Turan G. Bali Baruch College, Zicklin School of Business, One Bernard Baruch Way, New York, NY, 10010, USA Received 12 August 2004; received in revised form 30 January 2007; accepted 2 March 2007 Available online 4 September 2007 Abstract This paper explores the time-series relation between expected returns and risk for a large cross section of industry and size/book-to-market portfolios. I use a bivariate generalized autoregressive conditional heteroskedasticity (GARCH) model to estimate a portfolio s conditional covariance with the market and then test whether the conditional covariance predicts time variation in the portfolio s expected return. Restricting the slope to be the same across assets, the risk-return coefficient is highly significant with a risk aversion coefficient (slope) between one and five. The results are robust to different portfolio formations, alternative GARCH specifications, additional state variables, and small sample biases. When conditional covariances are replaced by conditional betas, the risk premium on beta is estimated to be in the range of 3% to 5% per annum and is statistically significant. r 2007 Elsevier B.V. All rights reserved. JEL classification: G12; G13; C51 Keywords: ICAPM; Conditional CAPM; Conditional covariance; Risk aversion; Conditional beta; Market risk premium; Intertemporal hedging demand 1. Introduction In his seminal paper, Merton (1973) derives an intertemporal capital asset pricing model (ICAPM) that has formed the basis for much empirical research. The model predicts that an asset s expected return depends on its covariance with the market portfolio and with state variables that proxy for investment opportunities. Tests of the model have taken two different forms. The first type follows Merton (1980) and French, Schwert, and Stambaugh (1987) and explores the time-series relation between the market s conditional expected return and its conditional variance. The second type, of which tests of the Sharpe and Lintner capital asset pricing model (CAPM) are a special case, focuses on the cross-sectional relation between expected return and risk. $ I thank G. William Schwert (the editor) an anonymous referee, Gurdip Bakshi, Peter Carr, John Campbell, David Chapman, Ozgur Demirtas, Wayne Ferson, Edward Kane, John Merrick, Lin Peng, Pedro Santa-Clara, Robert Schwartz, Jun Wang, Chu Zhang, and seminar participants at Baruch College, Boston College, and Cornell University for helpful discussions. I also thank Kenneth R. French for making a large amount of historical data publicly available in his online data library. I owe a great debt to Liuren Wu for his extremely helpful comments and suggestions on earlier versions of this paper. Excellent research support was provided by Yi Tang. address: turan_bali@baruch.cuny.edu X/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi: /j.jfineco

2 102 T.G. Bali / Journal of Financial Economics 87 (2008) My study extends time-series tests of the ICAPM to many risky assets. The prediction of Merton (1980) that expected returns should be related to conditional risk applies not only to the market portfolio but also to individual stocks and portfolios: Expected returns for any asset should vary through time with the asset s conditional covariance with the market portfolio (assuming that hedging demands are not too large). To be internally consistent, the relation should be the same for all stocks and portfolios, i.e., the predictive slope on the conditional covariance for any asset should be the representative investor s relative risk aversion. I exploit this cross-sectional consistency condition and estimate the common time-series relation across a wide variety of stock portfolios formed based on industry, size, book-to-market, and beta. With monthly data from 1926 to 2002, I use a bivariate generalized autoregressive conditional heteroskedasticity (GARCH) model to estimate the conditional covariance between the excess returns on each portfolio and the market portfolio. Then, I estimate a system of time-series regressions of portfolios excess returns on their conditional covariances with the market, while constraining all regressions to have the same slope coefficient. According to Merton (1973), the common slope coefficient represents the average relative risk aversion of market investors. My estimation generates positive and highly significant relative risk aversion coefficients, with magnitudes between one and five. The identified positive risk return relation is robust to different portfolio formations, alternative GARCH specifications, additional state variables, and small sample biases. To compare with the literature, I also estimate the risk return relation using a single series on the market portfolio and, furthermore, on each of the industry portfolios. I find that the relative risk aversion estimates from these single-series regressions are often statistically insignificant, as found in many earlier studies. Furthermore, the estimates vary greatly from negative to positive as I estimate the relation using different series, illustrating the large sample variation in the single-series coefficient estimates. By expanding the study of the time-series relation to a large cross section, I not only achieve cross-sectional consistency among the portfolios, but also gain statistical power in identifying a significant slope coefficient. My expansion of the time-series relation to multiple stock portfolios while maintaining cross-sectional consistency also moves my paper closer to the conditional CAPM literature, which studies the cross-sectional implications of the ICAPM. When I restrict the intercepts and slopes to be the same across portfolios, my system of equations becomes closer to the cross-sectional regressions, but with intertemporal stability constraint on the relative risk aversion coefficient. Alternatively, if one replaces the conditional covariance with the conditional beta, she can regard the system of equations with common intercepts and slopes as crosssectional regressions with intertemporal stability constraint on the market risk premium. My estimation on the conditional beta formulation also generates highly significant and positive market risk premium estimates, in the range of 3 5% per annum. The literature often reports insignificant estimates from either time-series or cross-sectional regressions. I attribute these insignificant estimates to the low statistical power caused by focusing narrowly on the intertemporal risk return relation of a single market return series or by focusing on the unconditional measures or rolling window estimates of market risk. I contribute to the literature by expanding the analysis of the intertemporal relation to a large cross section of stock portfolios and by expanding the crosssectional analysis to each conditional time step based on the bivariate GARCH estimates. Also, restricting the intercepts and slopes to be the same across portfolios in the system of equations, I consider both the intertemporal and cross-sectional implications of the ICAPM. I achieve consistency across both dimensions and gain statistical power by using GARCH-based conditional risk measures and by pooling the data. When the investment opportunity is stochastic, investors adjust their investment to hedge against future shifts in the investment opportunity and achieve intertemporal consumption smoothing. Hence, covariations with state of the investment opportunity induce additional risk premiums on an asset. I identify a series of financial and macroeconomic factors and study whether their conditional covariances with portfolio excess returns induce additional risk premiums. Estimation shows that covariances with the size factor (SMB) of Fama and French (1993) and the Treasury bill rate generate significantly negative slope coefficients. Hence, increases in a portfolio s covariance with both factors predict a lower excess return on the portfolio. In the context of the Merton (1973) ICAPM, the negative slope estimates suggest that an increase in SMB or the riskfree rate predicts a decrease in optimal consumption and hence an unfavorable shift in the investment

3 opportunity. Meanwhile, covariances with default spread and aggregate dividend yield generate significantly positive slope coefficients. The paper is structured as follows. Section 2 discusses the studies that form the background of my work. Section 3 describes the data set and the estimation methodology. Section 4 presents the estimation results. Section 5 performs robustness analysis. Section 6 concludes. 2. Literature review T.G. Bali / Journal of Financial Economics 87 (2008) The Merton (1973) ICAPM predicts both a time-series and a cross-sectional relation between expected returns and risk. My paper focuses on the time-series aspect of the relation and is hence related to a large body of empirical literature that attempts to measure the intertemporal risk return relation on the stock market portfolio. The findings from the literature are inconclusive. Using different sample periods, different data sets, or different econometric methodologies often leads to different findings on the risk return trade-off. Many studies fail to identify a statistically significant intertemporal relation between risk and return of the market portfolio. French, Schwert, and Stambaugh (1987) find that the coefficient estimate is not significantly different from zero when they use past daily returns to estimate the monthly conditional variance. Goyal and Santa-Clara (2003) and Bali, Cakici, Yan, and Zhang (2005) obtain similar insignificant results using the same conditional variance estimator but over a longer sample period. Chan, Karolyi, and Stulz (1992) employ a bivariate GARCH-in-mean model to estimate the conditional variance. They also fail to obtain a significant coefficient estimate for the United States. Baillie and DeGennaro (1990) replace the normal distribution assumption in the GARCH-in-mean specification with a fat-tailed t-distribution. Their estimates remain insignificant. Campbell and Hentchel (1992) use the quadratic GARCH (QGARCH) model of Sentana (1995) to determine the existence of a risk-return trade-off within an asymmetric GARCH-in-mean framework. Their estimate is positive for one sample period and negative for another sample period, but neither is statistically significant. Glosten, Jagannathan, and Runkle (1993) use monthly data and find a negative but statistically insignificant relation from two asymmetric GARCH-in-mean models. Based on semi-nonparametric density estimation and Monte Carlo integration, Harrison and Zhang (1999) find a significantly positive risk and return relation at one-year horizon, but they do not find a significant relation at shorter holding periods such as one month. Several studies even find that the intertemporal relation between risk and return is negative. Examples include Campbell (1987), Breen, Glosten, and Jagannathan (1989), Turner, Startz, and Nelson (1989), Nelson (1991), Glosten, Jagannathan, and Runkle (1993), Whitelaw (1994), and Harvey (2001). A few studies do provide evidence supporting a positive risk return relation. French, Schwert, and Stambaugh (1987) identify a negative relation between the portfolio return and the unpredictable component of volatility. They interpret this finding as indirect evidence that the ex ante conditional volatility is positively related to the ex ante excess return. Furthermore, when they estimate a GARCH-in-mean model with daily data, they find a positive and statistically significant relation at the daily level. Using the symmetric GARCH model of Bollerslev (1986), Chou (1988) finds a significantly positive relation with weekly data. Bollerslev, Engle, and Wooldridge (1988) use a multivariate GARCH-in-mean process to model the conditional mean and the conditional covariance of returns on stocks, bonds, and bills with the excess market return. They find a small but nevertheless significant risk return relation. Scruggs (1998) includes the long-term government bond returns as a second factor of the bivariate EGARCH-in-mean model and finds the partial relation between the conditional mean and conditional variance to be positive and significant. Ghysels, Santa-Clara, and Valkanov (2005) introduce a new variance estimator that uses past daily squared returns, and they conclude that the monthly data are consistent with a positive relation between conditional expected excess return and conditional variance. Bali and Peng (2006) examine the intertemporal relation between risk and return using high-frequency data. Based on realized, GARCH, implied, and range-based volatility estimators, they find a positive and significant link between the conditional mean and conditional volatility of market returns at the daily level. By emphasizing the cross-sectional consistency of the intertemporal relation, my work is also related to the conditional CAPM literature, the focus of which is to examine whether allowing beta to be time-varying can better explain the cross-sectional behavior of excess returns. Important studies include Ang and Chen (2007),

4 104 T.G. Bali / Journal of Financial Economics 87 (2008) Harvey (1989), Jagannathan and Wang (1996), Ferson and Harvey (1999), Lettau and Ludvigson (2001), and Lewellen and Nagel (2006). 3. Data and estimation The Merton (1973) ICAPM implies the following equilibrium relation between risk and return: m ¼ AS þ OB, (1) where m 2< n denotes the expected excess return on a vector of n risky assets, A reflects the average relative risk aversion of market investors, S 2< n denotes the covariance of the excess returns with the market portfolio, B 2< k measures the market s aggregate reaction to shifts in a k-dimensional state vector that governs the stochastic investment opportunity, and O 2< nk measures the covariance between excess returns on the n risky assets and the k state variables. For any risky asset or portfolio i, the relation becomes m i r ¼ As im þ o ix B, (2) where s im denotes the covariance between the returns on the risky asset i and the market portfolio m, and o ix denotes a (1 k) row of covariances between the return on risky asset i and the k state variables x. Many empirical studies focus on the time-series implication of the equilibrium relation in Eq. (2) and apply it narrowly to the market portfolio. Without the hedging demand component ðo ¼ 0Þ, this focus leads to the following risk-return relation: m m r ¼ As 2 m. (3) When considering stochastic investment opportunity, the literature often implicitly or explicitly projects the covariance vector o ix linearly to the state variables x to obtain the following relation: m m r ¼ As 2 m þ gx. (4) My work in this article differs from the above literature in two major ways. First, I estimate the intertemporal relation Eq. (2) not on the single series of the market portfolio, but simultaneously on many stock portfolios, and constrain all these portfolios to have the same cross-sectionally consistent proportionality coefficients A and B. Second, I directly estimate the conditional covariances s im and o ix using bivariate GARCH models. I do not make any linear projection assumptions on the state variables. In the Merton (1973) original setup, the two covariance matrices S and O are assumed to be constant. Nevertheless, the empirical literature has estimated the relation assuming time-varying covariances. I do the same in this paper. In principle, if the covariances are stochastic, they would represent additional sources of variation in the investment opportunity and induce extra intertemporal hedging demand terms Data I estimate the intertemporal relation in Eq. (2) using a wide variety of stock portfolios. Using portfolios instead of individual stocks reduces the workload to a manageable level and reduces the noise in individual stocks. Based on Kenneth French s online data library, I estimate the intertemporal relation using two broad sets of portfolios. One set is constructed according to different industry groups as in Fama and French (1997). The other is formed based on the quintiles of firm sizes and book-to-market ratios, as described in Fama and French (1993). For state variables, I use the firm-size (SMB) and book-to-market (HML) risk factors formed by Fama and French (1993). I also consider commonly used macroeconomic variables such as the relative Treasury bill rate, default spread, term spread, and dividend-price ratio Industry portfolios Fama and French (1997) assign each NYSE, AMEX, and Nasdaq stock to an industry portfolio at the end of June of each year based on its four-digit standard industrial classification (SIC) code at that time. Then, they compute both the value-weighted and the equal-weighted returns from July 1926 to December I use the value-weighted monthly returns following Fama and French (1997).

5 T.G. Bali / Journal of Financial Economics 87 (2008) French s online data library provides return data on several different industry partitions. The definitions of the industries and the corresponding SIC codes are available on his data library. I repeat my analysis using all these different partitions when possible. The stocks are partitioned into five, ten, 12, 17, 30, 38, and 48 industries. Data on five-, ten-, 12-, 17-, and 30-industry portfolios are available monthly from July 1926 to December 2002, yielding a total of 918 monthly observations for each series. I use these portfolios for the full sample analysis. Data on 38-industry portfolios have missing observations in six of the 38 industries. I drop this partition from my analysis. Data on 48-industry portfolios are fully available after July I include this partition when I perform subsample analysis from July 1963 to December 2002, with 474 observations for each series. Over time, French has revised the industry definitions. I perform a robustness check by repeating my analysis on some of the recently revised portfolios. The estimated risk return relations are similar across different portfolio definitions or revisions or both. The results reported in this paper are based on industry definitions and portfolio returns that I downloaded from the library in early Fama and French size/book-to-market portfolios Fama and French (1993) form 25 stock portfolios according to the quintiles of the stocks market capitalization (size, ME) and book-to-market equity ratios (BM). Monthly data from July 1926 to December 2002 on these 25 portfolios are available on French s online data library. To construct the portfolios, in June of each year, they rank all NYSE stocks in Center for Research in Security Prices (CRSP) based on the quintiles of ME. Then, they break NYSE, Amex, and Nasdaq stocks into five size groups based on the breakpoints of the NYSE stock quintiles. They also break all the NYSE, Amex, and Nasdaq stocks into five BM groups based on NYSE stock quintiles on BM ratios. The S5B5 portfolio (biggest size, highest BM) has missing observations from July 1930 to June I use two approaches to deal with these missing observations in my estimation. First, I replace the missing values with the average returns on the other four portfolios within the same size quintile (S5B1, S5B2, S5B3, and S5B4). Second, I replace the missing values with the average returns on the other four portfolios within the same BM quintile (S1B5, S2B5, S3B5, and S4B5). Both approaches yield similar results. I report the results based on the first approach Fama and French size and book-to-market risk factors Fama and French (1993) also form two common risk factors related to firm sizes (SMB) and book-tomarket equity ratios (HML). In a series of papers, Fama and French (1993, 1995, 1996) show the importance of these two factors. To construct these two factors, Fama and French first construct six portfolios according to the rankings on ME and BM ratios. In June of each year, they rank all NYSE stocks in CRSP based on ME. Then they use the median NYSE size to split NYSE, Amex, and Nasdaq stocks into two groups, small and big (S and B). They also break NYSE, Amex, and Nasdaq stocks into three BM groups based on the breakpoints for bottom 30% (Low), middle 40% (Medium), and top 30% (High) of the ranked values of BM for NYSE stocks. They construct the SMB factor as the difference between the return on the portfolio of small size stocks and the return on the portfolio of large size stocks, and the HML factor as the difference between the return on the portfolio of high BM stocks and the return on the portfolio of low BM stocks. In addition to SMB and HML, Fama and French (1993) use the excess market return as a proxy for the market factor in stock returns. The excess return on the market portfolio is the value-weighted return on all NYSE, Amex, and Nasdaq stocks (from CRSP) minus the one-month Treasury bill rate (from Ibbotson Associates). The three factors are also available on French s data library Macroeconomic variables Several studies find that macroeconomic variables associated with business cycle fluctuations can predict the stock market (see Fama and Schwert, 1977; Keim and Stambaugh, 1986; Chen, Roll, and Ross, 1986; Campbell and Shiller, 1988; Fama and French, 1988, 1989; Schwert, 1989, 1990; Fama, 1990; Campbell, 1991; Ferson and Harvey, 1991; Goyal and Santa-Clara, 2003; Ghysels, Santa-Clara, and Valkanov, 2005; Bali, Cakici, Yan, and Zhang, 2005). The commonly chosen variables include Treasury bill rates, default spreads, term spreads, and dividend-price ratios. I study how variations in these variables predict variations in the

6 106 T.G. Bali / Journal of Financial Economics 87 (2008) investment opportunity and how incorporating covariances with these variables affects the intertemporal riskreturn relation. I obtain monthly data on the three-month Treasury bill and ten-year Treasury bond yields from CRSP. The three-month Treasury bill data are available from July 1926 to December I follow standard practice in using the detrended relative rate (RREL), defined as the difference between the three-month T-bill rate and its 12-month backward moving average. I construct the term spread (TERM) as the difference between the yields on the ten-year Treasury bond and the three-month Treasury bill. The ten-year Treasury data are available from May 1941 to December I obtain monthly BAA- and AAA-rated corporate bond yields during my sample period from the Federal Reserve statistical release, and I construct the default spread as the yield difference between the two rating groups. I also download monthly aggregate dividend price ratio data over my sample period from Robert Shiller s website, The log dividend-price ratio (DP) is defined as the log difference between last 12-month dividends and the current level of the Standard & Poor s (S&P) 500 index Estimating conditional covariances I estimate the conditional covariance between excess returns on asset i and the market portfolio m based on the following bivariate GARCH(1,1) specification: R i;tþ1 ¼ a i 0 þ ai 1 R i;t þ i;tþ1, (5) R m;tþ1 ¼ a m 0 þ am 1 R m;t þ m;tþ1, (6) E t ½ 2 i;tþ1 Šs2 i;tþ1 ¼ gi 0 þ gi 1 2 i;t þ gi 2 s2 i;t, (7) E t ½ 2 m;tþ1 Šs2 m;tþ1 ¼ gm 0 þ gm 1 2 m;t þ gm 2 s2 m;t, (8) E t ½ i;tþ1 m;tþ1 Šs im;tþ1 ¼ g im 0 þ gim 1 i;t m;t þ g im 2 s im;t, (9) where R i;tþ1 and R m;tþ1 denote the time (t+1) excess return on asset i and the market portfolio m over a riskfree rate, respectively, and E t ½:Š denotes the expectation operator conditional on time t information. The GARCH specifications do not arise directly from the ICAPM model, but they provide a parsimonious approximation of the form of heteroskedasticity typically encountered with financial time-series data. Similar conditional covariance specifications are used in Baillie and Bollerslev (1992), Bollerslev (1990), Bollerslev, Engle, and Wooldridge (1988), Bollerslev and Wooldridge (1992), Ding and Engle (2001), Engle and Kroner (1995), Engle and Mezrich (1996), Engle, Ng, and Wooldridge (1990), andkroner and Ng (1998). When considering stochastic investment opportunities governed by a set of state variables, I estimate the conditional covariance between each portfolio i and each state variable x, o ix, using an analogous bivariate GARCH specification: R i;tþ1 ¼ a i 0 þ ai 1 R i;t þ i;tþ1, (10) x tþ1 ¼ a x 0 þ ax 1 x t þ x;tþ1, (11) E t ½ 2 i;tþ1 Šs2 i;tþ1 ¼ gi 0 þ gi 1 2 i;t þ gi 2 s2 i;t, (12) E t ½ 2 x;tþ1 Šs2 x;tþ1 ¼ gx 0 þ gx 1 2 x;t þ gx 2 s2 x;t, (13) E t ½ i;tþ1 x;tþ1 Šo ix;tþ1 ¼ g ix 0 þ gix 1 i;t x;t þ g ix 2 o ix;t. (14) I estimate the conditional covariances of each portfolio with the market portfolio and with each state variable using the maximum likelihood method. Using i;t and V t to denote the bivariate demeaned excess

7 T.G. Bali / Journal of Financial Economics 87 (2008) return vector and the conditional covariance matrix forecasts, " t ¼ R i;t a i 0 ai 1 R # " i;t 1 R i;t a i 0 R m;t a m 0 am 1 R or ai 1 R # i;t 1 m;t 1 x t a x 0 ax 1 x, (15) t 1 " # V t ¼ s2 i;t s im;t s im;t s 2 m;t or " # s 2 i;t o ix;t o ix;t s 2 x;t, (16) I can write the log-likelihood function as LðYÞ ¼ 1 2 X N t¼1 lnð2pþþln jv t jþ T t V 1 t t, (17) where Y denotes the vector of parameters in the specifications Eqs. (5) (9) or Eqs. (10) (14) and N denotes the number of monthly observations for each series. Table 1 Maximum likelihood estimates of the conditional covariance dynamics Entries report the maximum likelihood estimates of the parameters that govern the dynamics of the conditional covariance between the excess returns on the market portfolio and the excess returns on each of the 30 industry portfolios (Panel A) and the 25 size/book-tomarket (BM) portfolios (Panel B): s im;tþ1 ¼ g im 0 þ gim 1 i;t m;t þ g im 2 s im;t. In the notation for the size/bm portfolios, the firm size increases from S1 to S5, and the BM ratio increases from B1 to B5. The t-statistics of the parameter estimates are in parentheses. The estimation is based on monthly returns from July 1926 to December Panel A. Covariance between excess returns on the market portfolio and on each of the 30 industry portfolios Industry a g im 0 g im 1 g im 2 Food (5.43) (9.81) (59.07) Beer (4.54) (7.66) (61.13) Smoke (4.07) (9.13) (64.06) Games (6.02) (10.08) (64.84) Books (8.37) (10.88) (40.63) Hshld (9.29) (10.95) (26.55) Clths (9.28) (9.75) (35.27) Hlth (9.88) (9.13) (29.63) Chems (7.80) (7.67) (50.87) Txtls (6.64) (10.04) (67.91) Cnstr (8.89) (7.91) (62.74) Steel (7.18) (5.48) (40.05) FabPr (10.04) (7.23) (34.70)

8 108 T.G. Bali / Journal of Financial Economics 87 (2008) Table 1 (continued ) Panel A. Covariance between excess returns on the market portfolio and on each of the 30 industry portfolios Industry a g im 0 g im 1 g im 2 ElcEq (5.22) (10.12) (55.24) Autos (7.31) (10.39) (49.99) Carry (8.24) (9.25) (69.99) Mines (5.19) (7.52) (51.59) Coal (3.58) (6.57) (77.16) Oil (5.00) (8.18) (72.75) Util (5.10) (12.11) (111.92) Telcm (4.71) (9.17) (66.29) Servs (5.66) (12.18) (111.52) BusEq (6.56) (9.98) (63.55) Paper (4.69) (10.30) (66.38) Trans (5.07) (8.45) (44.74) Whlsl (6.61) (9.17) (77.50) Rtail (18.48) (7.94) (22.55) Meals (4.69) (8.65) (57.18) Fin (6.89) (8.68) (53.34) Other (4.35) (10.57) (93.71) Panel B. Covariance between excess returns on the market portfolio and on each of the 25 size/bm portfolios Size/BM g im 0 g im 1 g im 2 S1B (5.41) (9.06) (59.29) S1B (6.41) (10.34) (55.87) S1B (7.76) (8.50) (58.16) S1B (6.31) (8.59) (64.43) S1B (5.55) (6.71) (56.22) S2B (6.30) (9.99) (44.47) S2B (5.45) (7.65) (48.16) S2B (7.14) (10.01) (80.37) S2B (6.32) (7.88) (55.43)

9 T.G. Bali / Journal of Financial Economics 87 (2008) Table 1 (continued ) Panel B. Covariance between excess returns on the market portfolio and on each of the 25 size/bm portfolios Size/BM g im 0 g im 1 g im 2 S2B (5.17) (7.94) (73.40) S3B (6.92) (13.65) (59.16) S3B (8.88) (7.42) (50.75) S3B (7.07) (8.83) (56.47) S3B (7.11) (10.79) (62.29) S3B (5.99) (9.96) (50.27) S4B (5.36) (12.76) (64.89) S4B (5.79) (10.19) (59.89) S4B (6.51) (7.58) (53.25) S4B (5.79) (8.35) (58.08) S4B (5.91) (13.99) (75.31) S5B (6.57) (9.58) (66.38) S5B (6.55) (9.23) (46.22) S5B (6.01) (8.76) (55.56) S5B (4.99) (7.98) (44.88) S5B (4.47) (8.73) (49.95) a Food: Food Products; Beer: Beer & Liquor; Smoke: Tobacco Products; Games: Recreation; Books: Printing and Publishing; Hshld: Consumer Goods; Clths: Apparel; Hlth: Healthcare, Medical Equipment, Pharmaceutical Products; Chems: Chemicals; Txtls: Textiles; Cnstr: Construction and Construction Materials; Steel: Steel Works; FabPr: Fabricated Products and Machinery; ElcEq: Electrical Equipment; Autos: Automobiles and Trucks; Carry: Aircraft, ships, and railroad equipment; Mines: Precious Metals, Non-Metallic, and Industrial Metal Mining; Coal: Coal; Util: Utilities; Telcm: Communication; Servs: Personal and Business Services; BusEq: Business Equipment; Paper: Business Supplies and Shipping Containers; Trans: Transportation; Whlsl: Wholesale; Rtail: Retail; Meals: Restaraunts, Hotels, Motels; Fin: Banking, Insurance, Real Estate, Trading; Other: Everything Else. Table 1 presents the parameter estimates for the conditional covariance between the excess portfolio returns and the excess market returns for the 30 industry and 25 size/bm portfolios. The persistence of the conditional covariance dynamics on each series is measured by the sum of g im 1 and gim 2. The estimated value of gim 1 þ gim 2 is in the range of for the 30 industry portfolios and for the 25 size/bm portfolios. Hence, I observe more cross-sectional variations in the persistence of the conditional covariances among different industry portfolios than across the size/bm partitions. Table 2 reports the sample mean, standard deviation, minimum, maximum, and autocorrelation statistics of the estimated conditional covariance series for the whole sample period of July 1926 to December For comparison, I also report the unconditional covariances between the excess portfolio return with the excess market return. For both the 30 industry and 25 size/bm portfolios, the time-series averages of the conditional covariances are similar to the corresponding unconditional covariance estimates. The standard deviations of the conditional covariances are large compared with their means. Hence, it is important to allow the

10 110 T.G. Bali / Journal of Financial Economics 87 (2008) Table 2 Summary statistics for the conditional covariance and conditional beta estimates for Entries report the unconditional covariances (Uncond) between the excess monthly returns on market portfolio and 30 industry (Panel A) and 25 size/book-to-market (BM) (Panel B) portfolios, and the sample average (Average), standard deviation (Std), minimum (Min), maximum (Max), and first- and 12th-order monthly autocorrelations (Auto) of the corresponding conditional covariance estimates. For comparison, the last three columns of the table report the corresponding unconditional beta and the sample average and standard deviation of the conditional beta, b i;t ¼ s im;t =s 2 m;t. The sample period is from July 1926 to December Panel A. 30 industry portfolios (Un)conditional covariances (Un)conditional betas Industry a Uncond Average Std Min Max Auto(1) Auto(12) Uncond Average Std Food Beer Smoke Games Books Hshld Clths Hlth Chems Txtls Cnstr Steel FabPr ElcEq Autos Carry Mines Coal Oil Util Telcm Servs BusEq Paper Trans Whlsl Rtail Meals Fin Other Panel B. 25 size/bm portfolios (Un)conditional covariances (Un)conditional betas Size/BM Uncond Average Std Min Max Auto(1) Auto(12) Uncond Average Std S1B S1B S1B S1B S1B S2B S2B S2B S2B S2B S3B S3B

11 T.G. Bali / Journal of Financial Economics 87 (2008) Table 2 (continued ) Panel B. 25 size/bm portfolios (Un)conditional covariances (Un)conditional betas Size/BM Uncond Average Std Min Max Auto(1) Auto(12) Uncond Average Std S3B S3B S3B S4B S4B S4B S4B S4B S5B S5B S5B S5B S5B a Food: Food Products; Beer: Beer & Liquor; Smoke: Tobacco Products; Games: Recreation; Books: Printing and Publishing; Hshld: Consumer Goods; Clths: Apparel; Hlth: Healthcare, Medical Equipment, Pharmaceutical Products; Chems: Chemicals; Txtls: Textiles; Cnstr: Construction and Construction Materials; Steel: Steel Works; FabPr: Fabricated Products and Machinery; ElcEq: Electrical Equipment; Autos: Automobiles and Trucks; Carry: Aircraft, ships, and railroad equipment; Mines: Precious Metals, Non-Metallic, and Industrial Metal Mining; Coal: Coal; Util: Utilities; Telcm: Communication; Servs: Personal and Business Services; BusEq: Business Equipment; Paper: Business Supplies and Shipping Containers; Trans: Transportation; Whlsl: Wholesale; Rtail: Retail; Meals: Restaraunts, Hotels, Motels; Fin: Banking, Insurance, Real Estate, Trading; Other: Everything Else. conditional covariances to vary over time when estimating the ICAPM relations. I measure the cross correlations between the different conditional covariance and market portfolio conditional variance estimates, and I find that the conditional covariance series show highly positive cross correlations with one another and with the conditional variance of the market portfolio. The full-sample correlation estimates range from 0.85 to Given the conditional covariance estimates for each portfolio ðs im;t Þ and the conditional variance estimates for the market portfolio ðs 2 m;t Þ, I also compute the conditional betas for each portfolio i as b i;t ¼ s im;t =s 2 m;t.in the last three columns of Table 2, I report the sample average and standard deviation of the conditional beta estimates, as well as the unconditional beta estimates for each portfolio, b i ¼ CovðR i ; R m Þ=VarðR m Þ. The sample averages of the conditional betas are close to the magnitudes of the corresponding unconditional betas. Table 3 presents the same descriptive statistics for the unconditional and conditional covariance and beta estimates over the subsample period of July 1963 to December Similar to my findings in Table 2, the time-series averages of the conditional covariances and betas are similar to their corresponding unconditional estimates for both the 30 industry and 25 size/bm portfolios. The major difference between the whole sample and subsample estimates is that the volatilities of the conditional covariances and betas are much lower for the Compustat era because the subsample period ( ) does not include the extremely volatile Great Depression era. As shown in Table 3, the standard deviations of conditional betas are in the range of for 30 industry portfolios and for 25 size/bm portfolios. These estimates are similar to the findings of Lewellen and Nagel (2006) but larger than the rolling window estimates of Fama and French (2006). I cannot compare my estimates with those of Ang and Chen (2007) because they do not report the unconditional standard deviation of their stochastic betas. For the sample period of , Lewellen and Nagel (2006) find the standard deviation of monthly betas to be roughly 0.30 for a small-minus-big portfolio, 0.25 for a value-minus-growth portfolio, and 0.60 for a winner-minus-loser portfolio. For the sample period of , the standard deviation of 60-month rolling window beta estimates of Fama and French (2006) is roughly 0.26 for a small-minus-big portfolio and 0.24 for a value-minus-growth portfolio. Although it is not comparable with the unconditional standard deviation of my conditional betas, Ang and Chen (2007) find the conditional standard deviation of their stochastic betas in the range of for the value and growth portfolios.

12 112 T.G. Bali / Journal of Financial Economics 87 (2008) Table 3 Summary statistics for the conditional covariance and conditional beta estimates for Entries report the unconditional covariances (Uncond) between the excess monthly returns on market portfolio and 30 industry (Panel A) and 25 size/book-to-market (BM) (Panel B) portfolios, and the sample average (Average), standard deviation (Std), minimum (Min), maximum (Max), and first- and 12th-order monthly autocorrelations (Auto) of the corresponding conditional covariance estimates. For comparison, the last three columns of the table report the corresponding unconditional beta and the sample average and standard deviation of the conditional beta, b i;t ¼ s im;t =s 2 m;t. The sample period is from July 1963 to December Panel A. 30 industry portfolios Industry a (Un)conditional covariances (Un)conditional betas Uncond Average Std Min Max Auto(1) Auto(12) Uncond Average Std Food Beer Smoke Games Books Hshld Clths Hlth Chems Txtls Cnstr Steel FabPr ElcEq Autos Carry Mines Coal Oil Util Telcm Servs BusEq Paper Trans Whlsl Rtail Meals Fin Other Panel B. 25 size/bm portfolios (Un)conditional covariances (Un)conditional betas Size/BM Uncond Average Std Min Max Auto(1) Auto(12) Uncond Average Std S1B S1B S1B S1B S1B S2B S2B S2B S2B S2B S3B S3B

13 T.G. Bali / Journal of Financial Economics 87 (2008) Table 3 (continued ) Panel B. 25 size/bm portfolios (Un)conditional covariances (Un)conditional betas Size/BM Uncond Average Std Min Max Auto(1) Auto(12) Uncond Average Std S3B S3B S3B S4B S4B S4B S4B S4B S5B S5B S5B S5B S5B a Food: Food Products; Beer: Beer & Liquor; Smoke: Tobacco Products; Games: Recreation; Books: Printing and Publishing; Hshld: Consumer Goods; Clths: Apparel; Hlth: Healthcare, Medical Equipment, Pharmaceutical Products; Chems: Chemicals; Txtls: Textiles; Cnstr: Construction and Construction Materials; Steel: Steel Works; FabPr: Fabricated Products and Machinery; ElcEq: Electrical Equipment; Autos: Automobiles and Trucks; Carry: Aircraft, ships, and railroad equipment; Mines: Precious Metals, Non-Metallic, and Industrial Metal Mining; Coal: Coal; Util: Utilities; Telcm: Communication; Servs: Personal and Business Services; BusEq: Business Equipment; Paper: Business Supplies and Shipping Containers; Trans: Transportation; Whlsl: Wholesale; Rtail: Retail; Meals: Restaraunts, Hotels, Motels; Fin: Banking, Insurance, Real Estate, Trading; Other: Everything Else. The affinity in magnitudes between the sample averages of the conditional estimates and their unconditional counterparts show that my bivariate GARCH specifications generate reasonable conditional variance and covariance estimates. As a further sensibility check, I compute the value-weighted averages of the conditional covariances of all the portfolios within each portfolio partition, using the average firm size of each portfolio reported in Kenneth French s online data library to determine the weight. Then, I compare the properties of the weighted average conditional covariances with the conditional variance of the market portfolio. The top two panels of Fig. 1 plot the time series of the value-weighted covariances and the market variances for the 30 industry and 25 size/bm portfolios. In each panel, the solid line denotes the weighted averages of the conditional covariances, and the dashed line is the market portfolio conditional variance benchmark. The weighted-average covariances are all in the same range as the market portfolio variances. The two series in each panel move closely together. The bottom two panels of Fig. 1 plot the time series of the value-weighted conditional betas (solid lines) and compare them with the market beta (one, dashed lines) benchmark for the 30 industry and 25 size/bm portfolios. The value-weighted averages fluctuate around one within a narrow band. The deviations come from temporal mismatches in portfolio weights. The composition of the market portfolio is updated monthly based on market capitalization. In contrast, the compositions of the industry and size/bm portfolios are updated annually. Furthermore, when I average the conditional covariances and betas of these portfolios, I use firm size information reported in French s data library, which is also updated annually. The weights mismatches from these two layers generate deviations between the weighted-average conditional betas and the market portfolio beta. As a further robustness check, I use the GARCH conditional betas in a conditional market-model regression of Shanken (1990): R i;t ¼ a 0;i þ a 1;i b i;t þðb 0;i þ b 1;i b i;t ÞR m;t þ e i;t ; i ¼ 1; 2;...; n, (18) where R i;t is the return on portfolio i at month t, R m;t is the return on market portfolio at month t, and b i;t is the time-varying conditional beta of portfolio i at month t. If the GARCH conditional betas are unbiased then the coefficients b 0;i and b 1;i should be zero and one, respectively. Pagan and Schwert (1990) use a similar test to

14 114 T.G. Bali / Journal of Financial Economics 87 (2008) Fig. 1. Value-weighted conditional covariances and conditional betas. In each panel, the solid line denotes the value-weighted average of the conditional covariance (top panels) and conditional beta (bottom panels) estimates across the different portfolios within the 30 industry portfolio partition (left panels) and the 25 size/book-to-market (BM) portfolio partition (right panels). The dashed line in top panels denotes the conditional variance of the market portfolio and the dashed line in bottom panels denotes the beta of market portfolio. compare the relative performance of alternative conditional volatility models in terms of their ability to produce unbiased forecasts of future monthly volatility. I use the Wald test to determine whether the joint hypothesis of b 0;i ¼ 0, b 1;i ¼ 1 can be rejected for the 30 industry and 25 size/bm portfolios. As shown in Appendix A, the Wald statistics distributed as Chi-square with two degrees of freedom are smaller than the critical values, indicating that the conditional betas are unbiased for both the whole sample and subsample periods. I also compute the cross-sectional average of b 0;i and b 1;i as well as the cross-sectional average of their standard errors for 55 portfolios pooled from the 30 industry and 25 size/bm portfolios. For the sample period of July 1963 December 2002, the cross-sectional mean of b 0;i is about with the average standard error of and the cross-sectional mean of b 1;i is about with the average standard error of Estimating the risk-return relations Given the conditional covariances, we estimate the intertemporal relation from the following system of equations, R i;tþ1 ¼ C i þ As im;tþ1 þ Bo ix;tþ1 þ e i;tþ1 ; i ¼ 1; 2;...; n, (19) where n denotes the number of portfolios and also the number of equations in the estimation. For example, in the five-industry partition case, I simultaneously estimate n ¼ 5 equations. As I make the industry partition finer from five to ten, 12, 17, 30, and 48, I am estimating increasingly larger systems of equations. I constrain the slope coefficients (A, B) to be the same across all portfolios for cross-sectional consistency. I allow the intercept C i to differ across different portfolios. Under the null hypothesis of ICAPM, the intercepts should be zero. I use deviations of the intercept estimates from zero as a test against the validity and sufficiency of the ICAPM specification. I estimate the system of equations using a weighted least square method that allows me to place constraints on coefficients across equations. I compute the t-statistics of the parameter estimates accounting for heteroskedasticity and autocorrelation as well as contemporaneous cross-correlations in the errors from different equations. The estimation methodology can be regarded as an extension of the seemingly unrelated regression (SUR) method, the details of which are in Appendix B.

15 T.G. Bali / Journal of Financial Economics 87 (2008) The risk-return relation I first report the estimation results on the intertemporal risk-return relation assuming zero intertemporal hedging demand. Then, I discuss the cross-sectional implications from the perspective of the conditional CAPM. I conclude this section by discussing additional risk premiums induced by conditional covariances with various financial and macroeconomic variables The intertemporal risk return relation Table 4 reports the common slope estimates and the t-statistics from the following system of equations: R i;tþ1 ¼ C i þ As im;tþ1 þ e i;tþ1 ; i ¼ 1; 2;...; n, (20) under different portfolio partitions and for both the full sample period from July 1926 to December 2002 (Panel A) and a more recent subsample from July 1963 to December 2002 (Panel B). The estimates are relatively stable across different portfolio partitions, between 1.22 and 1.78 during the full sample period and between 2.14 and 4.16 during the subsample period. The t-statistics show that all estimates are highly significant. The consistent estimates and high t-statistics across different portfolio partitions and sample periods suggest that the identified positive risk return relation is not only significant, but also robust. Based on the relative risk aversion interpretation, the magnitudes of the estimates are also economically sensible. For comparison, I also do what the traditional literature does and estimate the risk-return relation over the full sample period using the single series of the market portfolio: R m;tþ1 ¼ 0:0038 þ 0:8034 ð1:4162þ ð0:8777þ s2 m;tþ1 þ e m;tþ1, (21) where I report the t-statistics in parentheses below the coefficient estimates. The slope estimate is lower than what I obtain in Table 4. More important, the t-statistic for the slope estimate is small. The slope estimate is no longer significantly different from zero. To carry this exercise one step further, I also estimate the Table 4 The intertemporal risk-return relation with cross-sectional consistency Entries report the common slope estimates and the t-statistics (in parentheses) from the following system of equations: R i;tþ1 ¼ C i þ As im;tþ1 þ e i;tþ1 ; i ¼ 1; 2;...; n, where n denotes the number of portfolio partitions and the number of regression equations in the estimation. Each row reports the estimates based on one portfolio partition. The t-statistics adjust for heteroskedasticity and autocorrelation for each series and crosscorrelations among the portfolios. BM ¼ book-to-market. Sample Period July 1926 December 2002 July 1963 December Industry (2.4901) (2.1767) 10 Industry (2.5128) (2.4274) 12 Industry (2.5731) (2.6354) 17 Industry (2.5366) (2.5573) 30 Industry (2.5207) (2.4961) 48 Industry (2.5806) 25 Size/BM (2.5487) (2.5164)

16 116 T.G. Bali / Journal of Financial Economics 87 (2008) risk-return relation based on a single portfolio series: R i;tþ1 ¼ C i þ A i s im;tþ1 þ e i;tþ1. (22) I repeat this exercise for all the industry portfolios and the market portfolio, altogether 75 series. The 75 series are the collection of five-, ten-, 12-, 17-, 30-industry portfolios and the market portfolio. The estimation differs from Eq. (20) by relaxing the cross-sectional consistency requirement. The single-series estimation can generate different relative risk aversion coefficients from different portfolio series and hence does not guarantee consistency across different portfolios. Fig. 2 plots the histogram of the 75 estimates in the left panel and the corresponding t-statistics in the right panel. From the left panel, I observe that the slope estimates are widely dispersed, from 0.24 to The right panel shows that the t-statistics also vary dramatically from 0.28 to Of the 75 estimates, only 31 have t-statistics greater than 1.96 in absolute magnitude. Therefore, by narrowly focusing on one series instead of making full use of the cross sections, the estimation loses its statistical power. The estimates show both large cross-sectional sample variation and low statistical significance. The R-squared estimates for each return series in Eq. (20) are between 1% and 5%. Since the equations are forecasting equations on portfolio returns, the low percentages are consistent with market efficiency and are in 15 Frequency Estimates for A i Frequency t-statistics for A i Fig. 2. Risk return coefficient estimates based on single portfolio series. The two panels plot the histograms of the risk return coefficient estimates (left panel) and t-statistics (right panel) based on each of the 75 different industry and market portfolio series, which are collections of the five-, ten-, 12-, 17-, and 30-industry portfolio series and the market portfolio for the sample period of July 1926 to December 2002.

17 T.G. Bali / Journal of Financial Economics 87 (2008) the same range as those from the single-series forecasting regressions without cross-sectional constraints. The low explained percentages also suggest that the fundamental risk return relation is inherently difficult to identify. By using a large cross section of return series and constraining them to share the same slope coefficient, I not only achieve cross-sectional consistency in pricing all portfolios according to the same risk return relation, but also gain statistical power in identifying significant, robust, and economically sensible risk return coefficients The cross-sectional implications of the ICAPM By expanding my analysis of the time-series relation to a large cross section of stock portfolios while maintaining cross-sectional consistency, I also move closer to the conditional CAPM literature, which focuses on the cross-sectional relations between excess returns and market beta. Ignoring intertemporal hedging demands, the ICAPM predicts that the expected excess returns vary both cross sectionally and over time with their conditional covariances with the market. My results in Section 4.1 confirm the time-series prediction of the ICAPM as I identify a positive and highly significant intertemporal risk return trade-off. However, the existing literature finds little evidence supporting the cross-sectional implication. Many studies find that little cross-sectional relation exists between unconditional betas and expected excess returns (e.g., Fama and French, 1992; Jagannathan and Wang, 1996). In this subsection, I reconcile with the literature and highlight the fundamental connection between the time-series and the crosssectional risk return relations implied by the ICAPM. Generically, I can write the risk return relation in the following two forms in terms of covariance and beta, respectively: Covariance : R i;tþ1 ¼ C i þ As im;tþ1 þ e i;tþ1 ; Beta : R i;tþ1 ¼ f 0;t þ f m;t b im;tþ1 þ e i;tþ1 ; (23) both of which can be interpreted as either time-series or cross-sectional relations. According to Merton (1973), the slope estimate from the covariance regression represents the average relative risk aversion of market investors, whereas the slope estimate from the beta regression represents the market risk premium. Section 4.1 focuses on the covariance equation and treats it as a time-series relation with time-constant coefficients. I estimate the time-series relation while constraining the slope coefficients to be the same across all portfolios, but I allow the intercept to vary across different portfolios. In contrast, the unconditional and conditional CAPM literatures often focus on the beta equation and treat it as a cross-sectional relation across different stocks or portfolios. Unconditional CAPM assumes that beta is constant over time and hence the same crosssectional relation holds across all time periods. Conditional CAPM allows beta to be time-varying and contends that the cross-sectional relation holds period by period. Fama and MacBeth (1973) estimate the cross-sectional relation period by period and then report the summary statistics of time series on the intercept and slope estimates. For comparison, I start my analysis by repeating a similar exercise using the whole-sample estimates of unconditional covariance and beta. First, I estimate the unconditional covariance and unconditional beta of each portfolio using the entire return data and then run the cross-sectional regressions for each month from July 1926 to December In this exercise, the conditional covariances and betas (s im;tþ1 and b im;tþ1 ) in Eq. (23) are replaced by their unconditional counterparts. In Panel A of Table 5, I report the time-series averages and the Newey and West (1987) t-statistics of the intercept and slope estimates based on the 30 industry and 25 size/bm portfolios. I also repeat the estimation using the one hundred size/beta portfolios formed as in Jagannathan and Wang (1996). The results are similar to those presented in Table 5. To save space, henceforth I report results only based on the 30 industry portfolios and the 25 size/bm portfolios. The results are similar to those reported in the existing literature: The average slope estimate is not significantly different from zero, and the average intercept is large and highly significant. The results are similar whether the cross-sectional regressions are on beta or covariance. These findings have prompted some to claim that unconditional beta does not matter. At an earlier stage of the study, I also estimate the unconditional covariance and unconditional beta of each portfolio using 60 months

18 118 T.G. Bali / Journal of Financial Economics 87 (2008) Table 5 The cross-sectional implications of the intertemporal capital asset pricing model (ICAPM) Sample period: July 1926 December Industry portfolios 25 Size/BM portfolios Intercept Slope Intercept Slope Panel A. Fama and MacBeth regressions with unconditional covariance and beta estimates Covariance (5.7439) ( ) (2.5128) (0.9411) Beta (5.7439) ( ) (2.5128) (0.9411) Panel B. Fama and MacBeth regressions with conditional covariance and beta estimates Covariance (3.1682) (2.2977) (3.2744) (2.3432) Beta (3.1682) (2.1706) (3.2744) (2.2054) Panel C. Cross-sectional relations with intertemporal stability Covariance (2.9671) (2.4702) (3.0105) (2.5809) Beta (3.1370) (2.2804) (3.1982) (2.3965) Sample period: July 1963 December 2002 Panel A. Fama and MacBeth regressions with unconditional covariance and beta estimates Covariance (3.4344) ( ) (4.4060) ( ) Beta (3.4344) ( ) (4.4060) ( ) Panel B. Fama and MacBeth regressions with conditional covariance and beta estimates Covariance ( ) (2.3051) ( ) (2.3682) Beta ( ) (2.2140) ( ) (2.2405) Panel C. Cross-sectional relations with intertemporal stability Covariance ( ) (2.4880) ( ) (2.5333) Beta ( ) (2.3117) ( ) (2.3779) of past-return data, as in Fama and French (1992). Then, I run the cross-sectional regressions for each month from July 1931 to December In this exercise, the conditional covariances and betas are replaced by their unconditional counterparts based on a five-year rolling window. The qualitative results are similar to those obtained from the whole-sample unconditional estimates of market risk. The average slope coefficient is not significantly different from zero, whereas the average intercept is highly significant. Similar results are also obtained for the subsample period of July 1963 December Some of the earlier studies on conditional CAPM, e.g., Jagannathan and Wang (1996), focus on an unconditional implication of the conditional relation by regressing excess returns on the unconditional beta and an unconditional covariance term that accounts for the covariation between the conditional beta and the conditional market risk premium. They find significant slope estimates on the additional covariance term, but the slope estimate on the unconditional beta remains insignificant. However, a more direct test of the conditional CAPM is to estimate the conditional relation directly, instead of estimating an unconditional implication of the conditional relation. Given my GARCH estimates on the

19 T.G. Bali / Journal of Financial Economics 87 (2008) conditional variance and covariances, I can estimate the conditional cross-sectional relations directly as in Eq. (23) month by month and then analyze the sample properties of the estimates following Fama and MacBeth (1973). Panel B of Table 5 reports the time-series averages and Newey and West (1987) t-statistics of the monthly estimates on the intercept and the slope of the conditional relations from July 1926 to December By testing the conditional CAPM conditionally, I obtain positive and significant slope estimate on both the conditional covariance and the conditional beta. The coefficient estimates on the conditional covariance are 1.32 (with a t statistic of 2.30) for the 30 industry portfolios and 1.21 (with a t-statistic of 2.34) for the 25 size/bm portfolios, similar to my time-series relation estimates in Section 4.1. The coefficient estimates on the conditional betas are 0.41% with a t-statistic of 2.17 for the 30 industry and 0.37% with a t-statistic of 2.21 for 25 size/bm portfolios. The corresponding annualized market risk premiums are 4.92% and 4.44%, respectively, both of which are in economically sensible ranges. Therefore, my estimation suggests that conditional beta matters when estimated conditionally using the bivariate-garch model. Another notable point in data for sample period July 1926 December 2002 in Table 5 is that the intercepts drop considerably (but remain significantly positive in the full sample) in moving from Panel A to Panel B. For the 25 size/bm portfolios in the full sample, the point estimates of the risk premiums in Panels A and B are similar and the greater statistical significance in Panel B comes primarily from lower standard errors because there is generally a larger spread in betas in the conditional regressions. I can go one step further and link the conditional CAPM result tightly to my time-series estimation in Section 4.1. By keeping the intercept and slope in Eq. (23) constant across both time and cross-section, I can interpret the estimates either as intertemporal relations with cross-sectional consistency or as cross-sectional relations with intertemporal stability constraint on the relative risk aversion (for the covariance regression) or the market risk premium (for the beta regression). Table 5 reports the results from the pooled regression in Panel C. Similar to what I found in Section 4.1, the slope coefficient on the conditional covariance regression is about 1.4 and highly significant. The slope estimates on the conditional beta regression are also highly significant. The monthly estimates of 0.46% and 0.40% correspond to annualized market risk premium of 5.52% and 4.8%, respectively. I have so far discussed the cross-sectional implications of the ICAPM for the whole-sample period of July 1926 December Table 5 also presents results for the subsample period of July 1963 December Similar to what I found from the whole sample, the slope coefficients on the conditional covariance and conditional beta regressions are statistically significant, whereas the cross-sectional relation between expected returns and unconditional measures of market risk is flat. Overall, my results from both samples indicate that conditional beta and conditional covariance do matter. The expected excess return on a risky portfolio increases with its conditional covariance with the market portfolio, or equivalently with its conditional beta. The estimated relative risk reversion coefficient and the market risk premium are not only statistically significant, but also in economically sensible ranges. The often insignificant estimates in the literature can be attributed to the low statistical power caused by focusing narrowly on either the intertemporal risk return relation of a single market return series or the unconditional measures of covariance and beta. I contribute to the literature by expanding the analysis of the intertemporal relation to a large cross section of stock portfolios and by expanding the cross-sectional analysis to each conditional time step based on bivariate-garch estimates. Furthermore, by simultaneously considering the intertemporal and cross-sectional implications of the ICAPM, I not only achieve consistency across both time and cross section, but I also gain statistical power by using the GARCH conditional risk measures and by pooling data along the two dimensions Risk premiums induced by conditional covariation with SMB and HML When the investment opportunity is stochastic, investors adjust their investment to hedge against future shifts in the investment opportunity and achieve intertemporal consumption smoothing. Hence, covariations with state of the investment opportunity induce additional risk premiums on an asset. In this subsection, I take the size (SMB) and book-to-market (HML) factors of Fama and French (1993) to describe the state of the investment opportunity, and I investigate whether covariations with these two factors induce additional risk premiums on a stock portfolio. I measure the conditional covariance of each portfolio with these two factors

20 120 T.G. Bali / Journal of Financial Economics 87 (2008) and estimate the following system of equations: R i;tþ1 ¼ C i þ As im;tþ1 þ B s o is;tþ1 þ B h o ih;tþ1 þ e i;tþ1 ; i ¼ 1; 2;...; n, (24) where o is;tþ1 and o ih;tþ1 measure the time-t expected conditional covariance between the time-(t +1) excess return on portfolio i and the two risk factors SMB and HML, respectively. I compute the sample mean, standard deviation, minimum, and maximum values of o is;tþ1 and o ih;tþ1 for the full sample period of July 1926 December The sample mean of o is;tþ1 is in the range of , and the standard deviations range from to The volatilities of o is;tþ1 are large compared with their means. For most of the 25 size/bm portfolios, both the mean and the standard deviation of o ih;tþ1 are small, especially as compared with the values of o is;tþ1. From the estimates on B s and B h, one can learn how investors react to the covariations with the two state variables. If upward shocks in the two state variables predict favorable shifts in the investment opportunity, we would expect risk averse investors to invest less in an asset when the asset s correlation with the state variables increases. As a consequence, I would expect the excess return on the asset to increase with increased correlation with the state variables and hence a positive estimate for the slope coefficient B. However, negative estimates for the coefficient B would indicate that upward shocks in the state variable predict unfavorable shifts in the investment opportunity. Increased correlation with the state variable would induce a positive intertemporal hedging demand and a negative risk premium. Table 6 reports the slope estimates and t-statistics of different restricted and unrestricted versions of the system of equations in Eq. (24) on the 30 industry and 25 size/bm portfolios. For both portfolio partitions and in all specifications that include conditional covariances with the SMB factor, the coefficient estimates Bs on the SMB factor are strongly negative. Thus, an increase in the covariance of an asset return with the SMB factor predicts a decrease in the asset s expected excess return for the next period. The negative estimates suggest that upward movements in the SMB factor predict unfavorable shifts in the investment opportunity. Campbell (1993, 1996) argues that an unfavorable shift in the investment opportunity can come in the form of heightened market volatility. Based on my negative coefficient estimate, I conjecture that an increase in the SMB factor can potentially predict an increase in future market volatility. To test this conjecture, I compute the correlation between SMB and the changes in the next month s realized variance on the market portfolio excess return, (s 2 m;tþ1 s2 m;t ). I compute the realized variance according to the following formula: s 2 m;t ¼ XD t d¼1 R 2 md þ 2 XD t d¼2 R m;d R m;d 1, (25) Table 6 Risk premiums induced by conditional covariation with firm-size (SMB) and book-to-market (HML) factors Entries report the slope estimates and t-statistics (in parentheses) from the system of equations, R i;tþ1 ¼ C i þ As im;tþ1 þ B s o is;tþ1 þ B h o ih;tþ1 þ e i;tþ1 ; i ¼ 1; 2;...; n, where o is;t and o ih;t measure the time-t expected conditional covariance between the excess return on portfolio i and the two risk factors SMB and HML, respectively. Estimation is based on monthly data from July 1926 to December BM ¼ book-to-market. Portfolios A B s B h 30 Industry (3.3150) ( ) 30 Industry (2.4836) (0.7392) 30 Industry (3.2713) ( ) (1.0004) 25 Size/BM (3.1529) ( ) 25 Size/BM (2.4588) (0.9284) 25 Size/BM (3.3219) ( ) ( )

21 T.G. Bali / Journal of Financial Economics 87 (2008) where D t is the number of trading days in month t and R m;d is the value-weighted market portfolio return on day d. The second term on the right-hand side adjusts for the autocorrelation in daily returns, as in French, Schwert, and Stambaugh (1987) and Goyal and Santa-Clara (2003). The daily data on the CRSP valueweighted index are available from July 2, 1962 to December 31, I estimate the market portfolio s monthly realized variance over this sample period and measure its correlation with the SMB factor. Consistent with my conjecture, the correlation coefficient between SMB t and monthly changes in realized variance ðs 2 m;tþ1 s2 m;tþ is positive and at about 13%. I experiment with alternative realized variance measures and obtain similarly positive correlation estimates. Moskowitz (2003) also reports evidence that the SMB factor predicts future market volatility. Therefore, the negative risk premium on the SMB factor might be partially induced by its positive prediction of future market volatility. By contrast, the coefficient estimates on the HML factor are not statistically significant, indicating that investors do not react consistently to movements in the HML factor. While not reported, similar results are obtained on the two risk factors from other portfolio partitions. From the cross-sectional perspective of the conditional CAPM, I can also estimate the risk-return relation following Fama and MacBeth (1973) by first estimating the cross-sectional relation at each month using the conditional covariances: R i;tþ1 ¼ C t þ A t s im;tþ1 þ B s;t o is;tþ1 þ B h;t o ih;tþ1 þ e i;tþ1, (26) and then analyzing the time-series properties of the cross-sectional slope estimates. Table 7 reports the timeseries averages and the Newey and West (1987) t-statistics of the slope coefficients from different versions of the cross-sectional regressions in Eq. (26). The results are similar to those in Table 6. The coefficient estimates on the covariance with the SMB factor are significantly negative, while the estimates on the covariance with the HML factor are insignificant. Taken together, the results indicate that the relations between the expected excess returns and their conditional covariances with the market, SMB, and HML factors are similar whether considering the relation intertemporally or cross-sectionally Risk premiums induced by conditional covariation with macroeconomic variables Researchers often choose certain macroeconomic variables to control for shifts in the investment opportunity. The commonly chosen variables include the short-term Treasury bill rates, default spreads on Table 7 Fama and MacBeth regressions using conditional covariances with firm-size (SMB) and book-to-market (HML) factors Entries report the time-series averages and the Newey and West t-statistics (in parentheses) of the slope coefficient estimates from the following cross-sectional regressions at each month t: R i;tþ1 ¼ C t þ A t s im;tþ1 þ B s;t o is;tþ1 þ B h;t o ih;tþ1 þ e i;tþ1, where o is;t and o ih;t measure the time-t expected conditional covariance between the excess return on portfolio i and the two risk factors SMB and HML, respectively. Estimation is based on monthly data from July 1926 to December BM ¼ book-to-market. Portfolios Ā B s B h 30 Industry (2.7953) ( ) 30 Industry (2.3914) (0.9221) 30 Industry (2.6586) ( ) (0.8434) 25 Size/BM (2.6126) ( ) 25 Size/BM (2.3530) (0.5865) 25 Size/BM (2.4026) ( ) (0.4172)

22 122 T.G. Bali / Journal of Financial Economics 87 (2008) corporate bonds, term spreads on interest rates, and aggregate log dividend price ratios. To analyze how these macroeconomic variables vary with the investment opportunity and whether covariations with them induce additional risk premiums, I first estimate the conditional covariance of these variables with excess returns on each portfolio and then analyze how the portfolio excess returns respond to their conditional covariance with these macroeconomic variables. In estimating the conditional covariances, I use the detrended relative Treasury bill rates, the monthly changes in default spreads, the term spreads, and the log dividendprice ratios, as described in Section Table 8 reports the estimation results for the 30 industry and the 25 size/bm portfolios. For each variable, I first estimate the intertemporal relation based on the longest common sample of the available data. Then, I re-estimate the relation using a more recent subsample that starts in July The estimates in Table 8 reveal several important results. First, incorporating any of these macroeconomic variables does not dramatically alter the relative risk aversion estimates. In all cases, the relative risk aversion estimates are positive and highly significant. Second, the coefficient on the covariance with the relative Treasury bill rate is significantly negative, indicating that a short rate increase predicts a downward shift in the optimal consumption. This negative estimate is consistent with Merton s conjecture in the original Merton (1973) paper, where he uses the example of stochastic interest rate to illustrate the role of intertemporal hedging demand. It is also consistent with the findings of several empirical studies (e.g., Campbell, 1987) that the Treasury bill rate predicts positively future market volatility. However, the coefficient estimates on Table 8 Risk premiums induced by conditional covariation with macroeconomic variables Entries report the slope estimates and t-statistics (in parentheses) from the following system of equations using the 30 industry and 25 size/book-to-market (BM) portfolios, R i;tþ1 ¼ C i þ As im;tþ1 þ B x o ix;tþ1 þ e i;tþ1 ; i ¼ 1; 2;...; n, where s im;tþ1 measures the conditional covariance between the excess return on each portfolio ðr i;tþ1 Þ and the market portfolio ðr m;tþ1 Þ, and o ix measures the conditional covariance of the portfolio return with a macroeconomic variable X t, which includes the relative Treasury bill rate (RREL t ), monthly changes in the default spread (DEF t ), monthly changes in the term spread (TERM t ), and monthly changes in the log dividend-price ratio (DIV t ). Estimation is based on both the full sample starting July 1926 (except for TERM t, which starts in May 1941) and a subsample starting July July 1926 December 2002 July 1963 December 2002 X t A B x A B x Panel A. 30 industry portfolios RREL t (2.6174) ( ) (2.5611) ( ) DEF t (2.7906) (2.4611) (2.5202) (1.9010) TERM t (2.5123) (1.2660) (2.6000) (0.9203) DIV t (2.9899) (2.6946) (2.8010) (2.3894) Panel B. 25 size/bm portfolios RREL t (2.5975) ( ) (2.5300) ( ) DEF t (2.6712) (2.3180) (2.4073) (2.0219) TERM t (2.4904) (1.6101) (2.6115) (1.2983) DIV t (2.9365) (2.4586) (2.7826) (2.2000)

23 T.G. Bali / Journal of Financial Economics 87 (2008) covariances with the default spread and dividend-price ratio are significantly positive, indicating that upward movements in these variables predict favorable shifts in the investment opportunity. As a robustness check, I also directly incorporate the lagged macroeconomic variables and the lagged excess returns to the system of equations. Table 9 presents the parameter estimates and t-statistics for the 30 industry and 25 size/bm portfolios for the common sample period from July 1963 to December The relative risk aversion estimates range from 3.05 to 3.72 for the 30 industry and from 3.84 to 4.11 for the 25 size/bm portfolios. They all remain highly significant for both portfolio partitions. In regressions with one lagged variable, I obtain significantly negative coefficient estimate on the relative T-bill rate and significantly positive coefficient estimates on the default spread, the term spread, the dividend yield, and the lagged market and portfolio returns. When I include all the lagged variables at the same time, the statistical significance of the default spread, term spread, lagged market return disappears, but the relative T-bill rate, the dividend yield, and the lagged portfolio return remain significant predictors of the one-month-ahead excess portfolio returns. Table 9 Directly incorporating lagged macroeconomic variables Entries report the slope estimates and t-statistics (in parentheses) from the following system of equations: R i;tþ1 ¼ C i þ As im;tþ1 þ BX t þ e i;tþ1 ; i ¼ 1; 2;...; n, where s im;tþ1 measures the conditional covariance between the excess return on each portfolio ðr i;tþ1 Þ and the market portfolio ðr m;tþ1 Þ, and X t denotes lagged macroeconomic variables, which includes the relative Treasury bill rate (RREL t ), default spread (DEF t ), term spread (TERM t ), log dividend-price ratio (DIV t ), lagged excess return on the market portfolio ðr m;t Þ,and lagged excess return on the corresponding industry or size/book-to-market (BM) portfolio ðr i;t Þ. Data are monthly from July 1963 to December s im;tþ1 RREL t DEF t TERM t DIV t R m;t R i;t Panel A. 30 industry portfolios (2.4492) ( ) (2.4784) (2.0258) (2.4512) (2.2611) (2.3951) (2.3180) (2.4524) (2.5620) (2.4658) (2.6572) (2.3209) ( ) (0.3561) ( ) (2.0550) ( ) (2.6044) Panel B. 25 size/bm portfolios (2.4780) ( ) (2.4296) (2.0402) (2.5341) (2.3204) (2.4880) (2.3402) (2.5341) (2.7919) (2.5102) (2.7480) (2.4710) ( ) (0.2621) ( ) (2.3984) ( ) (2.5251)

24 124 T.G. Bali / Journal of Financial Economics 87 (2008) Abnormal returns In estimating the system of time-series relations, I allow the intercept to be different for different portfolios. These intercepts capture the abnormal returns on each portfolio that cannot be explained by the conditional covariances with the market portfolio and other factors. Fig. 3 displays the histograms of the abnormal return estimates (left panel) and t-statistics (right panel) for the full sample period from 1926 to 2002 on various industry portfolios. The histogram contains the estimates on 74 industry portfolios pooled from the five-, ten-, 12-, 17-, and 30-industry partitions. Hence, some estimates refer to the same industry portfolios, or subsamples of them, in different portfolio partitions. To save space, I do not report results on the specifications including covariance with SMB, HML, and macroeconomic factors. Out of the 74 portfolios, only seven generate statistically significant estimates for the abnormal returns. Thus, the model explains fairly well the excess returns on different industry portfolios. When I collect the abnormal return estimates on the more recent subsample from 1963 to 2002 on all the 122 industry portfolios from the five-, ten-, 12-, 17-, 30-, and 48-industry partitions, we find that only 21 out of 122 abnormal return estimates are significantly different from zero. Incorporating conditional covariances with other state variables Frequency Estimates for monthly abnormal returns, in percent Frequency t-statistics for abnormal return estimates Fig. 3. Histograms of the estimates and t-statistics of abnormal returns on industry portfolios for the sample period of July 1926 December The left panel plots the histogram of the abnormal return estimates on different industry portfolio series. The right panel plots the histogram of the t-statistics of these estimates.