Long-Run Cash-Flow and Discount-Rate Risk in the Cross-Section of US Returns

Long-Run Cash-Flow and Discount-Rate Risks in the Cross-Section of US Returns Michail Koubourosy , Dimitrios Malliaropulosz , Ekaterini Panopouloux This version: May 2005 Abstract This paper decomposes the overall market (CAPM) risk into parts re‡ecting uncertainty related to the long-run dynamics of portfolio-speci…c and market cash ‡ows and discount rates. We decompose market betas into four sub-betas (associated with assets’and market’s cash ‡ows and discount rates) and we employ a discrete time version of the I-CAPM to derive a four-beta model. The model performs well in pricing average returns on single- and double-sorted portfolios according to size, book-to-market, dividend-price ratios and past risk, by producing high estimates for the explained cross-sectional variation in average returns and economically and statistically acceptable estimates for the coe¢ cient of relative risk aversion. JEL:G11, G12, G14 Keywords: CAPM, cash-‡ow risk, discount-rate risk, VAR-GARCH, BEKK, asset pricing We are grateful to Gikas Hardouvelis, Jack Meyer, Dimitrios Thomakos and the participants at the University of Piraeus-Athens Derivatives Stock Exchange Seminar for their helpful comments and suggestions. We acknowledge …nancial support from the Greek Ministry of Education and the European Union under “Pythagoras”grant. The usual disclaimer applies. yCorresponding author: University of Peloponnese, Department of Economics, Terma Karaiskaki, 22 100 Tripolis, Greece. Phone: (+30) 2710-230129, fax: (+30) 2710-230139, e-mail: [email protected]. zDepartment of Banking and Financial Management, University of Piraeus, and National Bank of Greece. xDepartment of Economics, National University of Ireland, Maynooth. 1 1 Introduction Since the original statement of the Sharpe-Lintner one-factor Capital Asset Pricing Model (CAPM), there is a considerable ongoing debate on whether its single risk measure, the market beta, can adequately describe the cross-section of average returns on individual stocks and portfolios sorted according to risk measures and …rm-speci…c characteristics. Numerous studies have shown that the single beta CAPM, at least in its unconditional form, performs poorly, since the cross-sectional variation in unconditional market betas cannot match the observed spread in average excess returns.1 In this paper, we decompose the market systematic risk (CAPM beta) of common stocks into four long-run risk components related to the covariance of unexpected changes in stock-speci…c cash-‡ows and discount rates with unexpected changes in market-wide cash-‡ows and discount rates. Further, we empirically test whether these sources of risk are priced using a discrete time version of the intertemporal asset pricing model of Merton (1973), recently developed by Campbell (1993, 1996). Our paper is related to the work of Campbell (1991), Campbell and Mei (1993), Campbell and Vuolteenaho (2005) and Campbell, Polk and Vuolteenaho (2003). In a novel paper, Campbell (1991) shows that unexpected stock returns can be decomposed into the discounted sum of revisions in expectations about future cash ‡ows and future discount rates. Campbell and Mei (1993) extend this analysis by studying the behavior of asset speci…c cash-‡ow and discount-rate components of portfolio betas but do not provide any evidence on whether these parts of systematic risk carry individual risk prices. More recently, Campbell and Vuolteenaho (2005) show that the market beta can be decomposed into a relatively “bad” cash-‡ow beta, re‡ecting news about the market’s future cash ‡ows, and a relatively “good”discount rate beta, re‡ecting news about the market’s future discount rates. They argue that the two components of return innovations have di¤erent implications for the rational investor. Since shocks to market cash ‡ows and market discount rates represent permanent and temporary shocks to overall wealth respectively, rational conservative investors are particularly averse to the former and require a premium which is a multiple of their attitude towards risk. As a result, discount rate betas are relatively “good”betas with low risk prices, whereas cash ‡ow betas are “bad”betas with high risk prices. Empirically, Campbell and Vuolteenaho …nd that small stocks and value stocks have considerably higher cash-‡ow (“bad”) betas than growth stocks and large stocks, and this can explain their higher average returns. However, they restrict their analysis by assuming that “good” and “bad” betas are independent of whether the innovation in individual returns is due to unexpected changes in future 1For a recent review on the CAPM see, among others, Fama and French (2004). 2 cash-‡ows or discount rates of the company. In a paper closest to ours, Campbell, Polk and Vuolteenaho (2003) decompose the overall market beta into four betas which re‡ect the covariance of unexpected changes in stock-speci…c cash-‡ows and discount rates with unexpected changes in market-wide cash-‡ows and discount rates. This decomposition of the market beta allows the authors to answer the question whether the high “bad”beta of small and value stocks and the high “good”beta of growth stocks and large stocks are attributable to their cash ‡ows or their discount rates. Campbell, Polk and Vuolteenaho estimate sample betas for growth and value portfolios and show that growth portfolios’cash ‡ows are particularly sensitive to temporary movements in aggregate stock prices (driven by market-wide shocks to discount rates) while value portfolios’cash-‡ows are highly correlated with temporary movements in market returns (driven by market-wide shocks to cash-‡ows). However, they do not test the asset pricing implications of this four factor model, leaving the question unanswered as to what economic forces determine the risk prices associated with these four sources of risk. Our four-beta model aims in investigating whether these four components of the overall market beta are priced according to a standard asset pricing model that identi…es changes in expectations about future cash ‡ows and future discount rates as the long-run risk factors that can explain the cross-section of mean returns. Using the discrete time version of Merton’s (1973) Intertemporal CAPM (I-CAPM) proposed by Campbell (1993, 1996), our structural four-beta model shows considerable in-sample success in pricing average returns on single- and double-sorted portfolios according to market capitalization, book-to-market, dividend-price ratios and risk. The model generates low and insigni…cant pricing errors, high estimates for the explained cross-sectional variation in average returns and statistically and economically acceptable estimates for the degree of relative risk aversion. We …nd that, as predicted by economic theory, permanent shocks to market returns are the main determinant of the overall risk premium, their covariances with both portfolio cash-‡ow and discount-rate dynamics earn equilibrium risk premia that are indistinguishable from zero, but the premia associated with asset-speci…c cash-‡ow news are greater than those linked to asset-speci…c discount-rate news. More importantly, we provide evidence that the coe¢ cient of proportionality between the two premia is equal to the constant coe¢ cient of relative risk aversion, as predicted by theory. The remainder of the paper is as follows: Section 2 provides the theoretical decomposition of total market risk into four parts: cash-‡ow and discount-rate portfolio risks associated with market’s cash-‡ow and discount-rate dynamics. Also, it develops the asset pricing framework that will be used for estimation. Section 3 describes the data set and the econometric model used to extract the news components of unexpected returns. 3 Section 4 presents the empirical results. Finally, section 5 concludes. 2 Decomposing Risk and Return 2.1 Cash-Flow and Discount-Rate Risk The starting point of our analysis is the decomposition of the unexpected return, developed by Campbell and Shiller (1988) and further expanded by Campbell (1991). We de…ne the one-period holding real gross return on asset ias ri;t+1 = log(Pi;t+1 +Di;t+1) log(Pi;t), where Pi;t+1 is the real stock price measured at the end of period t+ 1 (exdividend) and Di;t+1 is the real dividend payment during this period. Approximating this return with a …rst-order Taylor expansion around the, assumed constant, mean log dividend-price ratio, i=E[log(di;t pi;t)], we obtain: ri;t+1 ki+ipi;t+1 pi;t + (1 i)di;t+1;(1) with ki=log(i)(1 i) log[(1=i)1]; and i= 1=[1 + exp(i)] being …rm-speci…c constants. Campbell (1991), using this approximation of log returns, goes one step further and derives a decomposition of the unexpected return, ei;t+1 = ri;t+1 Et[ri;t+1], into revisions in expectations about future dividend growth rates (that is growth rates of future cash ‡ows) and revisions in expectations about future log returns (that is future discount rates): ei;t+1 =NC i;t+1 ND i;t+1;(2) with NC i;t+1 and ND i;t+1 de…ned as: NC i;t+1 = (Et+1 Et) 1 X j=0 j idi;t+1+j; and ND i;t+1 = (Et+1 Et) 1 X j=1 j iri;t+1+j; respectively. The above sums can be viewed as representing cash-‡ow and discount-rate “news”for the investor, since any upward or downward revision in her expectations at 4 time t+ 1 must be consistent with the arrival of new valuable information at time t+ 1. Moreover, as Campbell and Shiller (1988a, 1993), and Campbell (1991) argue, equation (2) must be considered as a consistent model of expectations, since a positive (negative) unexpected return today must be only associated with an upward (downward) revision in expectations about future cash-‡ows, a downward (upward) revision in expectations about future returns, or a combination of the two. That is, although equation (2) does not restrict the generating mechanism of expectations or the asset pricing model that derives equilibrium expected returns, it restricts the way through which changing expectations due to “good”or “bad”news a¤ect unexpected returns on any asset if investors’ expectations are to be consistent with the observed asset prices. As, among others, Campbell and Vuolteenaho (2005) and Campbell, Polk and Vuolteenaho (2003) argue, the two components of unexpected returns can be viewed as permanent and transitory shocks to the value of the underlying asset. A positive unexpected return caused by an upward revision in cash-‡ow expectations represents a permanent positive e¤ect on the value of the asset since it is never reversed subsequently, whereas a positive unexpected return generated from a downward revision in expectations about future returns can be viewed as a temporary shock to the asset price, since the capital gain today is at a cost of lower future investment opportunities. In the case where the underlying asset is the total wealth portfolio held by investors, these e¤ects can be viewed as permanent and temporary movements in total wealth. We now turn to link the sources of time variation in asset returns with the associated sources in the total wealth portfolio. Following Campbell and Shiller (1993), we de…ne the “market” or CAPM beta as the ratio of the conditional covariance of asset’s and market’s unexpected returns divided by the conditional variance of market unexpected returns: im;t =Covt(ei;t+1; em;t+1) Vart(em;t+1);(3) where Vart(:)and Covt(:)are the conditional, at time t; variance and covariance operators, respectively. Given that the current innovation in returns on both the asset iand the market portfolio can be written as the sum of cash-‡ow and (the negative of) discountrate news (equation (2)), we obtain the following decomposition of the conditional market sensitivity im;t which can be now written as the sum of four conditional “beta-like measures”of systematic risk: im;t =Covt(NC i;t+1 ND i;t+1; NC m;t+1 ND m;t+1) Vart(em;t+1) =i;CC;t +i;CD;t +i;DC;t +i;DD;t;(4) 5 where the individual components of total market risk, i;CC;t; i;CD;t; i;DC;t;and i;DD;t; are de…ned as: i;CC;t =Covt(NC i;t+1; NC m;t+1) Vart(em;t+1); i;CD;t =Covt(NC i;t+1;ND m;t+1) Vart(em;t+1); and i;DC;t =Covt(ND i;t+1; NC m;t+1) Vart(em;t+1); i;DD;t =Covt(ND i;t+1;ND m;t+1) Vart(em;t+1)(5) These “beta-like”ratios in (5) are not the traditional conditional sensitivities used in APT models. These models identify betas to be the univariate slope coe¢ cient of a regression of unexpected returns on the unexpected component (or return) of the risk factor (or factor mimicking portfolio). Rather, the “beta-like” measures of systematic risk in (5) represent the part of total market (CAPM) risk attributed to portfolio and market shocks to time-varying economic fundamentals and shocks to time-varying returns. 2.2 Pricing cash-‡ow and discount rate risk The approach of decomposing and pricing the sources of systematic risk is not new to the …nance literature. Campbell and Mei (1993) decompose the unexpected component of assets’returns into cash-‡ow and discount-rate news and examine their covariation with market total unexpected return. However, they do not consider shocks to market portfolio returns in their calculations and thus they work with (i;CC;t +i;CD;t)and (i;DC;t +i;DD;t)as representing aggregate risk quantities. Campbell and Vuolteenaho (2005) examine the opposite story and, while they do not split the full unexpected return on the asset, they decompose the market return innovation into permanent and transitory shocks. Further, using a discrete time variant of Merton’s (1973) I-CAPM (see Campbell (1993, 1996)) they show that their two-beta model performs well in describing the cross section of average returns on size-value and risk-loading sorted portfolios.2In a recent paper, Campbell, Polk and Vuolteenaho (2003) decompose the market beta in a similar way to ours and study the properties of sub-betas of value and growth stocks but they do not investigate the asset pricing implications of this decomposition. That is, they do not estimate the individual risk premia associated with the asset-speci…c return shocks driven by either changing expectations about future fundamentals and/or returns. Filling this gap is the main purpose of the present study. In order to derive testable restrictions on the premia associated with the cash-‡ow and discount rate risks in (4) we need a risk story. For this purpose, we employ the 2These betas correspond to i;C;t =i;CC;t +i;DC;t and i;D;t =i;CD;t +i;DD;t in our decomposition in (5): 6 recursive utility framework provided by Epstein and Zin (1989, 1991) and Weil (1989). The lifetime utility function of the investor is given by the recursive utility function Ut; de…ned over current real consumption and future expected utility of real consumption: Ut[Ct; Et(Ut+1)] = h(1 )C 1  t+EtU1 t+1 1 i 1;(6) where Ctis current real consumption at time t,0<  < 1is the subjective discount factor,  > 0is the constant coe¢ cient of relative risk aversion (CRRA), is a parameter de…ned as = (1 )=(1 1);and  > 0is the elasticity of intertemporal substitution (EIS) between current and expected future consumption. Equation (6) has the advantage of breaking the tight link between CRRA and EIS given by power utility (=1), thus, disconnecting investors’risk attitude across states of nature and across time.3The consumer is assumed to …nance all her consumption plan entirely from her total real wealth Wt;given the following dynamic budget constraint: Wt+1 = (1 + Rm;t+1)(WtCt);(7) where Rm;t+1 is the net real return on total wealth (or the market portfolio, m). Epstein and Zin (1989) solve for the optimal portfolio and consumption policies and show that the following set of conditional moment restrictions hold for each asset i: Et"Ct+1 Ct  (1 + Rm;t+1)1(1 + Ri;t+1)#= 1 (8) The above set of non-linear moment restrictions can be linearized using the assumption of joint conditional log-normality of asset returns and consumption in the spirit of Hansen and Singleton (1983). Campbell (1993, 1996) goes one step further and, using these strong assumptions along with the dynamic budget constraint in (7), derives the following crosssectional linear restrictions on assets’risk premia:4 Et[Ri;t+1]Rf;t+1 =Covt(ei;t+1; em;t+1)+(1) Covt(ei;t+1; ND m;t+1);(9) 3For a discussion of the properties of this speci…cation see Campbell (2003) and the references therein. 4Campbell (1993, 1996) discusses how to handle heteroscedasticity of returns. Equation (8) approximately holds even if returns are heteroskedastic if one assumes that the elasticity of intertemporal substitution is equal to unity. Since our aim is to test the unconditional version of our the model, we employ this assumption (see also Guo (2003)). 7 which using equation (2) for any individual asset as well as the market portfolio, m; gives: Et[Ri;t+1]Rf;t+1 =Covt(NC i;t+1; NC m;t+1) + Covt(NC i;t+1;ND m;t+1) +Covt(ND m;t+1; NC m;t+1) + Covt(ND i;t+1;ND m;t+1)(10) The left part of equations (9) and (10) represent the risk premium in simple returns which are equal to Et[ri;t+1]rf;t+1 +1 2Vart(ei;t+1); an expression resulting from the log linearization of the …st-order condition in (8). The covariance-risk representation of the equity premium in (10) can have a “beta-like-premium”representation (see, for example, Cochrane (2001)). Multiplying and dividing each conditional covariance term in (10) by the conditional variance of market’s unexpected returns, Vart(em;t+1), we obtain the following representation for the risk premium on any risky asset i: Et[Ri;t+1]Rf;t+1 =0;t +CC;ti;CC;t +CD;ti;CD;t +DC;ti;DC;t +DD;ti;DD;t;(11) where 0;t represents the conditional Jensen’s alpha, the rest of the s represent timevarying prices of beta risks, de…ned as CC;t =DC;t =Vart(em;t+1)and CD;t = DD;t =Vart(em;t+1);respectively, and the betas are de…ned similarly to (5). Equation (11) states that the required risk premium on asset iis jointly determined by the covariances of asset’s shocks to cash ‡ows and discount rates with the corresponding components of the total market innovation. Similarly to Campbell and Vuolteenaho (2005), a conservative risk-averse investor with  > 1demands a higher risk price for risks associated with market cash ‡ow uncertainty (i;CC;t and i;DC;t) rather than for risks linked to shocks to market returns (i;CD;t and i;DD;t), since any positive (negative) shock to market discount rates is at a bene…t (cost) of worse future investment opportunities, whereas the investor is never compensated later for every positive (negative) shock to dividends. Hence, the beta prices of market cash-‡ow risk, CC and DC, are a multiple of the beta risk prices of market discount-rate risk, CD and DD, respectively. We are interested in studying average returns for a long sample of U.S. stock market and macroeconomic data in order to get comparable results to the literature of the unconditional CAPM and, more importantly, to the empirical …ndings of the two-beta model of Campbell and Vuolteenaho (2005). Using the methodology described in the next section, we proceed with an unconditional version of (11): E[Ri;t+1]Rf;t+1 =0+CCi;CC +CDi;CD +DCi;DC +DDi;DD (12) 8 3 Data and Empirical Methodology We study monthly US asset and macroeconomic data from December 1928 to December 2001 (877 monthly observations). Our data consist of di¤erent sets of common stock portfolios sorted on various …rm-speci…c characteristics and risk measures, and a set of economy-wide variables that serve as instruments. Following common practice, these variables have been selected under the assumption that they forecast future returns. The test assets include monthly excess returns on (a) 25 size-BE/ME sorted portfolios from CRSP, corresponding to the Davis, Fama and French (2001) data …le, (b) 20 risksorted portfolios provided by Campbell and Vuolteenaho (2005),5and (c) a set of 10 book- to-market, 10 dividend-price ratio and 10 size sorted portfolios (30 in total). The valueweighted CRSP portfolio serves as the market portfolio of all traded wealth.6Although our model in (12) is written in real log returns, we assume that for the monthly test interval we employ, in‡ation rates are almost fully forecastable, and thus we proxy real log returns with nominal log returns. Variables that have been successful in predicting the future state of the economy and asset returns are used to generate cash-‡ows and discount rate news through the VAR speci…cation in (13). Following common practice, we use the following variables: (a) the log excess market return rmrf, de…ned as the di¤erence between the log return on the value-weighted CRSP stock index portfolio and the log return on the risk-free rate, constructed by CRSP from T-bills with approximately 3 month maturity, (b) the log price-earnings ratio, pe; taken from Shiller (2000) and de…ned as the log of the S&P 500 index, scaled by the 10-year moving average of aggregate earnings of companies in the S&P 500 index, (c) the term yield spread, TY; constructed by Global Financial Data and de…ned as the yield di¤erential between ten-year taxable bonds and short-term taxable notes, and (d) the small-stock value spread, V S; de…ned as the di¤erence between the log (BE/ME)of the small high-BE/ME portfolio and the log (BE/ME)of the small low-BE/ME portfolio.7 Measuring cash ‡ow news and discount rate news, as the main sources of risk, is central in our methodology. We follow Campbell (1991) and estimate the cash-‡ow-news and discount-rate-news series using a …rst-order vector autoregressive (VAR) model. We 5Campbell and Vuolteenaho (2005) sort common stocks into 20 portfolios according to their past loadings on the market return and innovations on the VAR variables. The purpose of their strategy is to generate portfolios with large spread in these loadings and thus overcome Daniel and Titman’s(1997) observation that sorting only on …rm characteristics could generate a spurious link between premia and risk measures. 6The returns on book-to-market, size, and dividend-price sorted portfolios are available at Kenneth’s French web site http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. 7The returns on the 20 risk sorted portfolios and the state variabes rmrf; T Y and V S are kindly provided by Tuomo Vuolteenaho and correspond to those used in Cambell and Vuolteenaho (2005). 9 do. The two-factor model performs quite well and generates insigni…cant pricing error (b 0=0:003;bt=0:229) and statistically signi…cant premia with the premium associated with market cash ‡ow risk being considerably higher than the premium associated with market’s discount rate risk (b C= 0:077 and b D= 0:013 with bt= 5:296 and 14:233 respectively). A high adj.-R2of 42:4% shows that much of the cross-sectional variation in average returns is explained. These results are in line with Campbell and Vuolteenaho (2005). The four-factor I-CAPM model performs even better. When the highly insignificant constant 0(bt=0:121) is removed, the model in its restricted version yields a highly statistically signi…cant and economically reasonable estimate for the RRA coe¢ - cient (^= 5:755 with bt= 7:878) and a higher adj.-R2of almost 55%. Also, the model yields the predicted di¤erence between the level of risk prices for the components of market cash-‡ow and discount-rate risk: the premia associated with market cash-‡ows (CC and DC) are 5 to 6 times higher than those associated with market discount rates ( CD and DD). However, we can not establish a clear statistical relationship of equality between the two pairs since the equality hypothesis CC =DC cannot be rejected at the low 2% level of signi…cance and the equality hypothesis CD =DD is rejected even for lower levels of signi…cance. Panel B of Table 6 reports our model estimates for three sets of 10 portfolios sorted according to BE/ME, D/P and market value. This experiment with single-sorted portfolios may provide us with better empirical evidence on the observed pattern of mean excess returns on value and size portfolios and the cross-sectional variation in fundamental and discount-rate risks. Our model again improves the ability of the disappointing static CAPM and the well performing two-beta I-CAPM to capture the spread in mean asset premia. The proportion of cross-sectional variability explained increases from 46:4% (twobeta model) to an impressive 83:1%;while the pricing error is still highly insigni…cant (bt=0:566). Most importantly, and even when the insigni…cant constant is included in the regression, all the slope coe¢ cients (except CD (bt= 1:69)) are signi…cant, indicating that the approach of decomposing cash ‡ow and discount rate market risks yields interesting insights for the determination of average risk premia. Once 0is removed, all four risk prices are highly signi…cant and the high estimated values for CC and DC (0:019 and 0:023;respectively) provide further support on the results presented by Campbell and Vuolteenaho (2005) and Campbell, Polk and Vuolteenaho (2003). They argued that value and small stocks have considerably higher cash-‡ow betas than large and growth stocks and this can explain their high average returns. We extend their results by showing that the sign and magnitude of our estimated beta-risk prices of the decomposed cash-‡ow market risk are in line with a rational asset pricing model for a long-lived conservative investor. This investor requires a higher premium per unit of market cash-‡ow risk than 16 for market discount-rate risk. Further, the factor of proportionality that is restricted to be equal to the coe¢ cient of relative risk aversion is both economically and statistically signi…cant (b= 5:304,bt= 9:713). For this group of portfolios, although we again cannot reject the equality hypothesis for the market discount rate premia CD and DD, we can safely accept it for the cash ‡ow premia (2= 0:846; p = 0:357). Overall, for our four-beta speci…cation in (22) the spread in size and value portfolios seems to be not puzzling. We also test the empirical validity of our four-factor model using the 25 BE/ME portfolios as well as 20 risk portfolios sorted on market betas and betas associated with innovations to the state variables.10 The approach of sorting stocks according to past risk rather than …rm-speci…c characteristics can gauge the impact of data snooping on empirical …ndings that reveal relationships between characteristic-sorted portfolio trading strategies and average returns. Panel C of Table 6 shows that the static CAPM still performs badly and generates a very low adj.-R2of 0:9%, statistically signi…cant pricing errors (b 0= 0:014 with bt= 1:665) and an insigni…cant and economically rejected point estimate for the market premium (b m=0:007;bt=0:769). Thus, market beta, as a single aggregate risk measure, fails to capture the cross-sectional spread in returns. The four-beta model captures a large part of the cross-sectional average return variation and, compared to the two-beta speci…cation, impressively increases the percentage of explained cross-sectional return variability from 44% to almost 61%. It produces even more insigni…cant pricing errors and results in a signi…cant CRRA value (^= 5:788;bt= 11:130). It is interesting that the observed pattern in cash-‡ow and discount-rate prices of risk is in line with our previous tests: still, risk prices associated with the two components of market cash-‡ow risk (b CC = 0:062;b DC = 0:072) are much higher than the risk prices associated with market discount-rate risk (b CD = 0:008;b DD = 0:013). For experimental reasons we include the 5 sets of portfolios (25 size-BE/ME, 20 risk, 10 BE/ME, 10 D/P and 10 size sorted) in one cross-sectional regression. Panel D in Table 6 illustrates the results. Similarly to the previous empirical …ndings, the market overall beta, im;explains none of the cross-sectional variation in average returns. The two-beta model again results insigni…cant pricing errors and signi…cant risk prices for the aggregate market cash-‡ow and discount-rate risk with the latter being much higher (0:068 and 0:012 respectively) as predicted by economic theory. However, our four-beta speci…cation in (22) increases the ability of the two-beta model by more than 20% in terms of explanatory power. The point estimate of the CRRA is signi…cant, economically acceptable and similar to the one generated from the previous samples (b= 5:594;bt= 14:640). However, 10For recent studies that relates loadings of unexpected returns on innovations of state variables and the global size and book-to-market premia of the Fama-French (1996) three factor model see Petkova (2002) and the references therein. 17 as in all previous tests, we cannot accept the hypothesis that the risk prices concerning markets cash-‡ows and market discount-rate risks are equal. 5 Conclusions This paper builds on the decomposition of the overall market, or CAPM, risk into parts re‡ecting time variation related to the dynamics of portfolio-speci…c and aggregate market cash ‡ows and discount rates. Extending the methodology of Campbell (1991), Campbell and Mei (1993), Campbell and Vuolteenaho (2005) and Campbell, Polk and Vuolteenaho (2003) we decompose market betas into four sub-betas, two associated with market cash- ‡ows and two with market discount-rates. The approach used attempts to …ll the gap between the time-series predictability of returns and the cross-sectional variation in average returns due to di¤erences in risk. Using a VAR-GARCH(1,1) approach and a discrete time version of Merton’s I-CAPM, we ask whether these parts of overall risk related to innovations to state variables that are related to changes in expectations about future dividends and future returns are rationally priced. Our four-beta model performs well in pricing average returns on single- and doublesorted portfolios according to market capitalization, book-to-market, dividend-price ratios and past risk, by producing insigni…cant pricing errors, high estimates for the explained cross-sectional variation (which in many cases exceeds 80%) in average monthly returns and both economically and statistically acceptable estimates for the coe¢ cient of relative risk aversion (values range between 5 and 6). We …nd that the risks associated with permanent shocks to market returns, as these are described by the two market cash- ‡ow betas, earn higher unconditional risk prices compared to the risk prices associated with market discount-rate risks, but all four components are required in order to improve the ability of the static CAPM to capture the cross-sectional variation of mean premia. 18 References [1] Bollerslev Tim (1986), Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, 307-328. [2] McCurdy Thomas and Morgan (1998), Testing the martingale hypothesis in Deutchmark futures with models specifying the form of heteroskedasticity, Journal of Applied Econometrics 3, 187-202. 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[34] Petkova, Ralitsa (2002), Do do the Fama-French factors proxy for innovations in predictive variables?, unpublished paper, University of Rochester. [35] Roze¤ (1984), Dividend Yields are Equity Risk Premiums, Journal of Portfolio Management 10, 68-75 [36] Scholes , Myron and Joseph Williams (1977), Estimating betas from nonsynchronous data, Journal of Financial Economics 5, 309-327. [37] Sharpe, William (1964), Capital asset prices: a theory of market equilibrium under conditions of risk, Journal of Finance 19, 425-442 [38] Shiller, Robert (2000), Irrational Exuberance, Princeton University Press, Princeton, N.J. 21 Table 1. VAR estimates for market portfolio Each column of the table corresponds to a di¤erent equation of the VAR model. The …rst …ve rows report coe¢ cients on the three state variables plus a constant and lagged values of the excess market return. OLS standard errors are reported in parentheses below coe¢ cients. The table also reports the R2and Fstatistics for each equation and a Lagrange Multiplier test for heteroscedasticity of the VAR residuals up to four lags. rm;t+1 TYt+1 pt+1 et+1 V St+1 constant 0:062 (0:020) 0:046 (0:097) 0:019 (0:013) 0:014 (0:017) rm;t 0:093 (0:034) 0:033 (0:165) 0:519 (0:022) 0:008 (0:029) TYt0:006 (0:003) 0:880 (0:016) 0:002 (0:002) 0:002 (0:003) ptet0:015 (0:005) 0:036 (0:026) 0:994 (0:004) 0:000 (0:005) V St0:012 (0:006) 0:082 (0:028) 0:003 (0:004) 0:991 (0:005) R22:6% 82:4% 99:1% 98:4% F-stat. 5:713 1020:446 22898:7 13400:13 LM Test for Heteroscedasticity (ARCH Test: lag = 4) ^urm^uTY ^upe^uV S F-stat. 25:071 24:652 13:264 22:841 p-value [0:000] [0:000] [0:000] [0:000] 22 Table 2: Estimates of univariate GARCH(1,1) models of market portfolio VAR innovations The table reports estimates of GARCH(1,1) models for the conditional variance of VAR innovations for the market portfolio, equation (17). Column 2 refers to innovations of market excess returns and columns 3-5 to innovations of the yield-spread (TY ), the log price-earnings ratio (pe) and the value spread (V S), respectively. The last rows report the results of a Lagrange Multiplier test for heteroscedasticity of the standardized residuals, zjt =ujt=phjt, up to four lags. Robust standard errors are reported in parentheses below coe¢ cients. Probability values are given in brackets. Parameter hrm;t hTY;t hpe;t hV S;t kj 5:75e05 (1:98e05) 9:39e05 (4:11e05) 3:08e05 (9:76e06) 0:000 (1:76e05) 2 j 0:107 (0:017) 0:135 (0:019) 0:082 (0:017) 0:048 (0:007) 2 j 0:877 (0:017) 0:805 (0:014) 0:892 (0:022) 0:905 (0:014) LM Test for Heteroscedasticity (ARCH Test: lag = 4) ^zrm^zTY ^zpe^zV S F-stat. 0:331 0:890 0:586 0:223 p-value [0:857] [0:469] [0:673] [0:926] 23 Table 3: Market portfolio cash-‡ow and discount-rate news The table reports the estimated covariance matrix (upper-left) and the correlation matrix with st. dev. (upper-right) of estimated market portfolio cash-‡ow and discount rate news using equations (13) to (16), the correlations of innovations of state variables with market news (lower-left) and the mapping functions de…ned in (16). Covariance matrix of news News corr/st.d. NC mND mNC mND m NC m0:385 0:486 NC m0:621 0:621 ND m0:486 1:590 ND m0:621 1:262 Correlations of innovations with news Functions Innovations/News NC mND mNC mND m rmrf0:162 0:874 rmrfshock 0:599 0:401 TY 0:039 0:011 TY shock 0:010 0:010 pe0:692 0:953 peshock 0:889 0:889 V S 0:377 0:221 V S shock 0:263 0:263 24 Table 4. Cash-‡ow and discount-rate betas for 25 book-to-market portfolios The table reports sample estimates of cash-‡ow and discount rate betas for the 25 size-BE/ME portfolios de…ned in (4) and calculated according to (18). Estimates for the “bad" and “good" betas are de…ned as ^ i;C =^ i;CC +^ i;DC and ^ i;D =^ i;CD +^ i;DD and the estimates for the full market beta are de…ned as the sum ^ i;m =^ i;CC +^ i;DC + ^ i;CD +^ i;DD:“Di¤." refers to the di¤erence between the extreme cell. Panel A. Cash-‡ow cash-‡ow betas (^ i;CC) Growth 2 3 4 Value Di¤. Small 0.753 0.732 0.476 0.409 0.641 -0.112 2 0.408 0.335 0.398 0.497 0.690 0.282 3 0.370 0.348 0.335 0.453 0.582 0.212 4 0.235 0.316 0.422 0.395 0.694 0.460 Large 0.139 0.242 0.368 0.418 0.471 0.332 Di¤. 0.614 0.490 0.108 -0.009 0.169 Panel B. Cash-‡ow discount-rate betas (^ i;CD) Growth 2 3 4 Value Di¤. small -1.711 -1.397 -1.273 -1.107 -1.640 0.071 2 -1.141 -0.936 -1.003 -1.258 -1.549 -0.409 3 -1.205 -0.787 -0.673 -0.931 -1.218 -0.013 4 -0.475 -0.694 -0.817 -0.774 -1.435 -0.960 Large -0.069 -0.200 -0.401 -0.683 -1.347 -1.277 Di¤. -1.642 -1.197 -0.873 -0.424 -0.293 Panel C. Discount-rate cash-‡ow betas (^ i;DC) Growth 2 3 4 Value Di¤. Small -0.818 -0.802 -0.527 -0.461 -0.681 0.137 2 -0.505 -0.413 -0.459 -0.541 -0.727 -0.222 3 -0.473 -0.421 -0.379 -0.489 -0.607 -0.134 4 -0.335 -0.382 -0.461 -0.423 -0.719 -0.384 Large -0.252 -0.325 -0.412 -0.438 -0.493 -0.242 Di¤. -0.566 -0.477 -0.115 -0.023 -0.187 Panel D. Discount-rate discount-rate betas (^ i;DD) Growth 2 3 4 Value Di¤. Small 2.696 2.409 2.271 2.075 2.592 -0.104 2 2.192 1.959 1.972 2.194 2.492 0.300 3 2.215 1.809 1.632 1.876 2.113 -0.102 4 1.503 1.679 1.789 1.717 2.373 0.871 Large 1.106 1.231 1.315 1.562 2.232 1.126 Di¤. 1.589 1.178 0.957 0.513 0.360 25 Panel D. 75 portfolios: 25 Size-BE/ME, 20 risk, 10 BE/ME, 10 D/P and 10 size sorted portfolios Two-factor Two-factor Four-factor Four-factor CAPM I-CAPM I-CAPMI-CAPM I-CAPM 00:012 (0:005) [2:339] 0:002 (0:004) [0:553] 0:001 (0:003) [0:413] m0:004 (0:005) [0:809] C0:073 (0:011) [6:750] 0:068 (0:007) [9:544] D0:015 (0:005) [3:103] 0:012 (0:000) [24:981] CC 0:052 (0:009) [5:262] 0:049 (0:007) [6:824] CD 0:009 (0:004) [2:272] 0:007 (0:001) [7:812] DC 0:060 (0:010) [5:601] 0:058 (0:008) [6:943] DD 0:013 (0:004) [3:294] 0:011 (0:001) [11:383] adj.R20:4% 44:4% 44:9% 64:8% 65:3% F-test (all zero) (p-value) 30:543 (0:000) 35:193 (0:000) 2-test (p-value) 0=D= 0 618:399 (0:000) 0=CD = 0 60:483 (0:000) CC =DC 9:889 (0:002) 2-test (p-value) CD =DD 29:874 (0:000) CRRA () (t-stat) 5:594 [14:640] 0:012 (0:000) [24:981] 32