Bad, Good and Excellent: An ICAPM with bond risk premia JOB MARKET PAPER

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1 Bad, Good and Excellent: An ICAPM with bond risk premia JOB MARKET PAPER Paulo Maio* Abstract In this paper I derive an ICAPM model based on an augmented definition of market wealth by incorporating bonds, and by decomposing excess stock return news into bond premia news and the remainder, news in the "true" equity premia. This model which represents an extension of the Bad Beta Good Beta (BBGB) from Campbell and Vuolteenaho (2004), has 4 factors: Cash flow news, equity premia news, bond premia news and news on future interest rates. The betas associated with bond premia news are relatively stable across individual assets, in opposition to the equity premia betas. The risk prices estimates of cash flow news (bad beta) are higher relative to equity premia news (good beta) and this one has higher risk prices than bond premia news (excellent beta). The model outperforms the BBGB model in pricing both growth and value stocks, being able to price the value premium. An augmented model which incorporates scaled factors related with time-varying risk aversion, shows that risk aversion is negatively correlated with bond premia news. This model has very low pricing errors and is also able to price the value premium. Preliminary results show that the Fama-French HML and UMD factors are mostly insignificant in the presence of the ICAPM factors, and this suggests that at least partially both factors measure the same types of risks as the ICAPM with bond risk premia. Keywords: Asset pricing; Asset pricing models; Conditional pricing models; Consumption-based models; Equity premia; ICAPM; Bond risk premia; Linear multifactor models; Predictability of returns; Risk aversion; Stock and bond returns; Time-varying returns. JEL classification: G11; G12; G14 *Universidade Nova de Lisboa, Faculdade de Economia, Ph.D. program, Campus de Campolide, Lisboa. 1: pdm @fe.unl.pt. 2: paulo.maio@sapo.pt. Part of this paper was written when I was a visiting scholar at Anderson School of Management-University of California Los Angeles (UCLA), I thank my advisors Pedro Santa Clara and Joao Amaro de Matos for helpful discussions and suggestions. I have benefited from comments by Bernard Dumas, Daniel Ferreira and Quinglei Dai. I thank the financial support from Fundacao para a Ciencia e Tecnologia (Portuguese Government). All errors are mine. 1

2 I. Introduction Following the Merton (1973) ICAPM, state variables that predict market returns, should act as risk factors that price the cross-section of ex-post average returns. Despite this prediction - and the existence of a vast literature showing that the market equity premium is time-varying and predictable at several horizons by a set of state variables linked to short term interest rates, bond yields and financial ratios - there has been not many attempts to test the ICAPM, even in the presence of the CAPM failure to explain the cross section of average returns. Among the papers that implemented empirically testable versions of the original ICAPM, are Campbell (1993, 1996), and more recently Chen (2003), Brennan et al (2004) and Campbell and Vuolteenaho (2004) (CV hereafter). Using the same framework employed by Campbell (1993,1996), with an Epstein and Zin utility function, and employing the decomposition for unexpected market returns, CV derive an ICAPM with only 2 factors: covariance with discount-rate news (good factor) and covariance with cash flow news (bad factor). A decline in future cash flows, lead to a decline in current wealth, and investment opportunities are unchanged, thus this represents a permanent decrease in wealth. On the other hand, an increase in future discount rates leads to a decline in the current value of wealth, but future investment opportunities improve, since current wealth will be invested at higher future returns, thus we have a transitory decline in wealth. In this sense, CV argue that the risk price of cash flow news should be higher than the risk price of discount rate news, i.e., investors will demand a higher premium to hold assets that are correlated with cash flow news, than to hold assets that covary with discount rate news. In their model, this relation is valid for the case of a risk-averse investor, with relative-risk aversion coefficient greater than 1. CV have found that growth stocks have higher discount-rate betas and lower cash flow betas than value stocks, in their modern sub-sample. By testing the model, CV found that the first order condition of a long-term investor that holds the market portfolio is not violated, and the 2

3 difference in average returns between value stocks and growth stocks is explained by their different composition of cash flow and discount rate betas, and thus such an investor does not have incentives to overweight value stocks in his portfolio. In this paper, I extend CV paper in 3 critical points. First, in response to the Roll (1977) critique that the stock market index is an imperfect measure for total financial wealth, I use a measure of the market portfolio as a weighted average of a stock index and a long maturity government bond. Under this assumption, and using the same framework as in Campbell (1993, 1996) and CV, I derive a ICAPM model with 4 factors: Cash flow news, excess stock return news, excess bond return news, and nominal interest rate news. Second, I decompose expected excess stock returns into expected excess stock-bond returns and expected bond premia. Since the cash flows associated with stocks are uncertain, as opposed to bonds which have fixed cash flows, that are known beforehand, in order to compensate for the risk associated with cash-flows, stocks earn a risk premium over long-term bonds, originating higher expected returns, in average. Thus, we can reinterpret stock excess return as being composed by two components: Bond risk premia, used to discount the "certain" part of future stock cash flows, and the stock-bond risk premia used to discount the random or "risky" part of future cash flows, which represents the "true" equity risk premia. Using this decomposition for stock excess returns, news about future excess stock returns can be decomposed into news about future stock-bond premia and news about bond premia. A rise in both news components is associated with an improvement in investment opportunities, since current wealth will be reinvested at higher returns, but while future bond returns are known a priori, since they are used to discount certain cash-flows, future stock returns are uncertain given they are used to discount uncertain future cash-flows. Using the "bad beta good beta" terminology from CV, we can speak of a bad beta, a good beta and a very good beta, which is the covariance with bond premia news. Thus apart from the fact that the risk price of cash flow 3

4 news should be higher relative to both stock-bond premia and bond premia news, news on stock-bond excess returns should have a higher risk price than news on future bond excess returns, due to the uncertainty involved in reinvested wealth. I calculated betas associated with the 2 components of excess stock return news, and find that bond premia news has relatively stable betas across the book-to-market quintiles, as expected since the type of risk involved as to due with changes in long-term interest rates, which have no cash-flow risk involved. On the other hand, in the case of the equity-bond premia factor, growth stocks have significantly higher (magnitude) betas than value stocks. I derive the ICAPM associated with this decomposition for news in the equity premia, and find that the risk price for stock-bond excess returns is higher relative to the risk price associated with bond premia news, and in addition, the model significantly improve the pricing ability of both growth and value stocks, relative to the bad beta, good beta from CV. In third place, I derive and estimate a generalized ICAPM that allows for time-varying risk aversion, assuming that risk aversion is explained by both stock-bond premia news and bond premia news. In fact, the risk aversion of the representative investor, affects both his demand and valuation for both stocks and bonds, thus affecting expected stock and bond returns and the implied news components. The results show that a rise in bond risk premia is associated with lower risk aversion. A rise in current bond yields or future expected bond excess returns, usually takes place in periods of rising business conditions - and thus increasing earnings and cash flows - which benefit the prospects for stocks, and thus lowers risk aversion. Thus, bond premia news are not only a better "beta" relative to news in future excess stock-bond returns, due to representing a certain increase in reinvested wealth, but it also has a second order effect, associated with declining risk aversion. An alternative and complementar interpretation relies on the interaction between portfolio rebalances and risk aversion: A decrease in risk aversion is associated with an investment flow from bonds to all stocks in general, in 4

5 investors portfolios. This increases (decreases) the demand for stocks (bonds), leading to higher (lower) current prices and returns for stocks (bonds). Higher current stock market returns are associated with lower expected exycess returns, leading to the positive correlation between risk aversion and news on future equity risk premia. On the other hand, lower current bond returns lead to higher future bond excess returns, explaining the negative correlation between risk aversion and news on future bond premia. The ICAPM with time-varying risk aversion produces average pricing error of less than 2% annually, and the average pricing errors across the book-to-market are close to zero, which is mainly due to the presence of the scaled factors related to time-varying risk aversion. Hence, both the benchmark ICAPM with bond risk premia, and an augmented ICAPM incorporating time-varying risk aversion, are able to price the value premium. In addition when the less theoretically based factors HML and UMD from Fama and French are added, both HML and UMD are not priced, and thus the ICAPM takes into account, at least partially, the momentum observed in stock prices. II. Theoretical framework A. Measuring the market wealth Roll (1977) argues that we can not test the CAPM with a noisy proxy for the market portfolio, and that the stock market index is an imperfect measure of the market portfolio. To address this issue, I assume that the representative investor holds bonds in addition to stocks in his portfolio. Therefore, the market portfolio is a weighted average of a stock market index and a proxy for the bond market, where the weights are given by the respective market value. Data from the NYSE show that the total capitalization of the 3 Exchanges, NYSE, NASDAQ and AMEX at the end of the first semester, 2005, is around 16.5 trillion usd. In addition, data from the FRED II database, available from the St. Louis FED s website, indicate that the value 5

6 of public debt at the end of first quarter 2005 is around 8 trillion usd. Given the existence of corporate bonds in addition to government bonds, I assume that stocks represent 70% of the market portfolio, while bonds represent 30%. This will be the benchmark weights used in the paper. As an alternative, I consider the case where each class of assets has a weight of 50%. B. A 4 factor ICAPM The ICAPM model developed in this paper will make use of the decompositions for stock market excess returns and bond excess returns, derived in Appendices A and B. Following Campbell and Ammer (1993), the unexpected stock market excess return can be decomposed as, E E t r m, r f, r CF r H Y r 1 where r CF E E t j Δd j, r H j0 E E t j1 j r m,j r f,j and Y r E E t j1 j r f,j represent news about future cash flows, news about future excess stock returns and news about future nominal interest rates, respectively. This equation says that innovations in current stock excess returns are associated with either an increase in expected future cash-flows, a decrease in expected future excess returns or a decrease in expected future nominal interest rates. The current unexpected bond excess return can be decomposed as E E t r b, r f, r B Y r 2 Y where r B is the same as in equation 1 and r E E t j1 j r b,j r f,j represents news about future excess bond returns. This equation is similar to the bond return decomposition presented in Campbell and Ammer (1993), and it shows that a rise in current unexpected bond excess returns is the result of either a decrease in expected future bond excess returns or a decline in future nominal interest rates. 6

7 Following Campbell (1993,1996) and Campbell and Vuolteenaho (2004) (CV thereafter), I use an Epstein and Zin utility function, 1 U t 1 C t E t U where 1 1 1, is the elasticity of intertemporal substitution, is the relative risk aversion parameter, is a time discount factor, and C t denotes consumption. This utility function has the advantage of allowing to separate and, contrary to the power utility function, where is the reciprocal of. In this context, the intertemporal budget constraint is given by W R p, W t C t 4 where W t represents total market wealth and R p, is the simple return on the "market" portfolio. The market portfolio return finances the stream of consumption, and is equal to a weighted average of the returns on a stock market index and a long-maturity bond which represents a proxy for the bond market, R p, R m, 1 R b, 5 where R m, denotes the simple return on the stock market portfolio and R b, denotes the simple return on a long-maturity bond. As shown in the Appendix C, the log market return can be approximated as r p, r m, 1 r b, 6 with r p, lnr p, representing the log market return and similarly r m, and r b, denote the log return on stocks and bonds, respectively. Given equation (6) the expected return on total wealth, is given by E t r p, E t r m, 1 E t r b, 7 Following Epstein and Zin (1989, 1991), the objective function 3 has an associated pricing equation in simple returns given by 7

8 1 E t C C t 1 1 R p, 1 R i, 8 with a corresponding stochastic discount factor (SDF) equal to M C C t 1 R p, 1 9 and a corresponding log SDF given by, m ln Δc 1 r p, 10 Summing and subtracting both t E t Δc and 1 t E t r p, yields, m ln E t Δc 1 E t r p, Δc E t Δc 1 r p, E t r p, E t m c E t c 1 r p, E t r p, 11 where the second equality makes use of the fact that Δc E t Δc c E t c. Substituting c E t c by its expression derived in the Appendix E, it follows m E t m r p, E t r p, 1 E E t j r p,j 1 r p, E t j1 E t m r p, E t r p, 1 E E t j r p,j 12 j1 where the last equality follows from substituting the expression for. By adding and subtracting the risk-free rate r f,, one has m E t m E E t r p, r f, E E t r f, 1 E E t j r p,j r f,j 1 E E t j1 j r f,j 13 j1 Using equation 7 and the fact that E E t r f, 0, itfollows m E t m E E t r m, r f, 1 E E t r b, r f, 1 E E t j r m,j r f,j j1 1 1 E E t j r b,j r f,j 1 E E t j1 j r f,j 14 j1 If we employ the decompositions for current unexpected stock and bond excess returns in equations 1 and 2 above, we have, m E t m r CF r H Making f r CF,r r Y 1 1 r B 1 r B Y 1 r E t m r CF r H 1 r B Y r H,r B r Y H 1 r 15,r Y and b,,1,1 and using Theorem 1 in Appendix 8

9 D, we have finally the pricing equation, Er i, r f, i 2 2 i,cf i,h 1 i,b i,y 16 where i,cf Covr i,,r CF, i,h Covr i,,r H, i,b Covr i,,r B, and i,y Covr i,,r Y denote the asset covariance with cash flow news, stock excess return news, bond excess return news and nominal interest rate news, respectively. In the ICAPM model represented in equation 16, there are 4 factors which help to price assets, and the covariances risk prices are theoretically constrained. The only free parameter to be estimated in the cross-section is the relative risk aversion parameter which affects the price of covariance with cash-flow news. For a risk-averse investor 1 the risk price associated with cash-flow news should be higher than (minus) the risk price of excess stock return news 1. In addition for 0.5, the risk price of covariance with excess stock return news should be higher (in magnitude) than the risk price of covariance with bond premia news. Equation 16 represents a generalization of the bad beta, good beta model (BBGB) from CV. If we allow 1, i.e., financial wealth is composed only by a stock index, then the BBGB model arises as a special case of 16, Er i, r f, i 2 2 i,cf i,h i,y 17 This is equivalent to the model in CV, with the only difference being the inclusion of i,y, which is ignored in their paper, since they assume that the log real risk-free rate is approximately constant, and therefore news about nominal interest rates are zero. Since most asset pricing models are estimated and evaluated in terms of factor betas risk prices, we can restate equation 16 in terms of single regression betas, originating the following model, Er i, r f, i CF i,cf 2 H i,h 1 2 B i,b 2 Y i,y 18 where 2 CF, H2, B2 and Y2 represent the variances of r CF, r H, r B and r Y, respectively. The risk prices for betas can be derived by CF, H, B, Y f b, where f is a diagonal 9

10 matrix with the factor variances on its main diagonal. In addition, the model in covariances 16, can be represented in an expected return-beta form with multiple-regression coefficients, as shown in Theorem 1 in Appendix D, Er i, r f, i 2 2 i CF i,cf H i,h B i,b Y i,y 19 where CF, H, B, Y Varf b denote the vector of factor risk prices, and i Varf 1 Covr i,,f represents the 4x1 vector of multiple-regression betas for asset i. The s represent the risk prices of multiple-regression beta risk for each of the factors. The risk prices depend on the SDF coefficients - as in the case of risk prices of covariances and single regression betas - but also on the variances and covariances between the risk prices, since we are working with multiple regression betas. Given f b, f Varf, standard errors for the factor risk price estimates can be calculated as, Var f Varb f 20 since f f, and given Varb Varb 0 1X X1 0 3X3 with b representing the SDF parameters to be estimated in the cross-section. Since some of the risk prices are fixed a priori by the model, and hence are not estimated, estimating the model with multiple regression betas, allow us to derive t-statistics for the individual factors, in order to evaluate whether they are priced. III. Asset pricing tests A. Data The test assets used in the asset pricing tests are the Fama-French 25 portfolios sorted on size and book-to-market (SBV25), and 38 industry sorted portfolios (IND38), all obtained from 10

11 Prof. Kenneth French s website. Due to missing observations, the returns associated with five industries - Sanitary services (GARBG), Public Administration (GOVT), Steam supply (STEAM), Irrigation systems (WATER) and the residual class of industries (OTHER) - are excluded from the sample, leading to a total of 33 industry portfolios. The 1 month Treasury bill rate used to calculate excess returns, and data on the book-to-market ratios of small value and small growth stocks, are also obtained from Prof. French s website. Return data on the value-weighted market index and 10 year government bond is from CRSP, while monthly data on prices and earnings associated with the Standard & Poor s (S&P) Composite Index is obtained from Professor Robert Shiller s website. Macroeconomic and interest rate data, including the Federal funds rate, 10 year Treasury bond yield, 3 month Treasury bill rate, and the consumer price index are all obtained from the FRED II database, available from the St. Louis FED s website. B. Estimating the news components of stock and bond excess returns: a VAR approach Following Campbell and Ammer (1993), I rely on a first-order VAR in order to estimate the news components for unexpected stock and bond returns, r CF, r H B, r and r Y. The VAR 2 equation assumed to govern the behavior of a state vector X t, which includes the stock market excess return, bond excess return and other variables known in time t which help to forecast changes in expected stock and bond returns, is given by X AX t 22 I Follow Campbell and Ammer (1993) in estimating the revisions in expected excess bond returns and nominal interest rates, Y r B r E E t j1 E E t j1 n 1 j r f,j e3 I A 1 n 1I I A 1 n A n A j Y r b,j r f,j E E t r b, r f, r e2 Here is a discount coefficient linked to the average log consumption to wealth ratio 1 expc w, or average dividend yield, e2 and e3 are indicator vectors that take a value of 11

12 one in the cell corresponding to the position in the VAR of the bond excess return and interest rate, respectively, A is the VAR coefficient matrix, n is the maturity of the bond, and e3 I A 1 n 1I I A 1 n A n A is the function that relates the VAR shocks with interest rate news. Since the longest maturity bond available in the sample is 10 years, I B truncate the infinite sums in both r Y and r to n 120, as in Campbell and Ammer (1993). Equations 23 and 24 are approximations rather than exact relations, and include the parameter, contrary to Campbell and Ammer (1993). The difference in my results arises from using a coupon bond as opposed to zero-coupon bonds, as in Campbell and Ammer (1993). This decomposition threats the news in bond excess returns as the residual component of unexpected bond returns, which has the advantage of avoiding giving too much weight to interest rate news. The stock return news and cash flow news, are also estimated in a similar way to Campbell and Ammer (1993) and CV, r H E E t j r m,j r f,j e1 AI A 1 25 j1 r CF E E t j Δd j E E t r m, r f, r H Y r j0 e1 26 where e1 is the indicator vector that assigns a value of one in the cell corresponding to the position of the stock excess market return in the VAR, and e1 AI A 1 is the function that relates the VAR shocks with revisions in expected future stock market returns. Treating cash-flow news as the residual component of unexpected stock returns, has the advantage that one does not have to model directly the dynamics of dividends. Both stock and bond return decompositions are dynamic accounting identities that arise from the definition of stock and bond returns, and thus are not behavioral models for asset returns. In order to be consistent, with previous work (CV), I assume , which corresponds to an average consumption to wealth ratio of approximately 5% per year. 12

13 The state-vector associated with the first-order VAR is given by X t Δr f,t,ffprem t,term t,vs t,ey t,r bt,r mt, which follows the representations used in Campbell and Ammer (1993) and CV. The change in the 1 month Treasury bill rate, Δr f,t is an indicator of short-term interest rates and it is used to estimate the nominal interest rate news component. FFPREM represents the spread between the Federal Funds rate and the 3 month Treasury bill rate, and thus it is a measure of both monetary policy actions and short-term interest rates. Its inclusion in the VAR is justified by previous evidence that both monetary policy (Patelis (1997), Goto and Valkanov (2002)) and short-term interest rates (Ang and Bekaert (2003)) do forecast future expected equity market returns, at least for short term forecasting horizons. TERM refers to the term structure spread - measured here as the difference between the 10 year Treasury bond yield and the 1 month bill rate - which represents a proxy for the yield curve slope, and has been widely used in the predictability of returns literature, since Fama and French (1989) have found that TERM tracks the business cycle. EY denotes the log earnings yield (calculated as the log of the earnings to price ratio associated with the S&P Composite index), used instead of the dividend yield, in light of recent evidence that the forecasting power of the dividend yield has decreased since the 90 s, which might be related to a possible structural break in the firms dividend policy, causing more firms to paying less dividends (Fama and French (2001)). The value spread, VS, defined as the difference between the log book-to-market ratios of small value and small growth stocks, is used in the VAR system employed by CV, which argue that this spread being related with the value premium - an anomaly not priced by the CAPM - should help to predict market returns, if the ICAPM is true. In a dynamic context, if growth (value) stocks have lower (higher) expected returns than predicted by the CAPM, then it must be the case that the returns of growth (value) stocks forecast lower (higher) expected market returns, or shifts in the investment opportunity set. Thus, a decrease in the book-to-market ratio of growth stocks (equivalent to an increase in the current returns of growth stocks) forecasts lower stock 13

14 market returns, or equivalently, a increase in the value spread forecasts lower stock market returns. Finally, the sample used in estimating the VAR is Descriptive statistics for the VAR state variables are presented in table I. The first-order autocorrelation coefficients show that both VS and EY are very persistent variables, whereas, to a lower degree both FFPREM and TERM exhibit some short term momentum. Furthermore, the VAR state variables are not significantly correlated, with the most relevant correlation arising between VS and EY (-0.607). The VAR coefficient estimates and associated Newey-West t-statistics (calculated with 5 lags) are presented at Table II, Panel A. The bottom row of Panel A shows that FFPREM, EY and r b have short-term forecasting power over the stock market return: FFPREM predicts negative market excess returns 1 month ahead, consistent with previous evidence (Patelis (1997)), and both EY and r b predict positive market returns, also consistent with previous evidence, with all three regressors being statistically significant at the 1% level, which is remarkable in the case of EY, given previous evidence that the forecasting power of financial ratios is greater for long-horizon returns (beyond 1 year). On the other hand VS forecasts positive stock returns, but the effect is not statistically significant. In the equation for bond returns, TERM predicts positive bond returns, in line with previous evidence, whereas VS has marginal positive predictive power (10% level) over bond returns. In addition, bond returns exhibit short-term momentum, and stock returns strongly forecast negative bond returns. The adjusted R 2 for the stock and bond forecasting regressions are 3.8% and 4.3%, respectively, in line with the values for monthly predictive regressions existent in the literature. Regarding the other equations in the VAR, Δr f is positively forecasted by TERM and negatively forecasted by r b, FFPREM is mainly explained by its lagged value, but Δr f and r b also have some forecasting power on it. TERM and VS are close to AR(1) processes, although EY helps to predict (negatively) VS. In the equation for EY, the lagged values for EY and FFPREM 14

15 forecast a rise in EY, whereas lagged VS is negatively correlated with EY. In addition, both bond and stock market returns forecast negatively EY, which in the latter case might be related to the mean reversion observed in stock prices. The results for the estimated "news" components associated with bond and stock excess returns are presented in Table II, Panels B and C respectively, which are similar to the Table 3 presented in CV. News about bond excess returns contribute the most for the variance of unexpected bond returns (0.853), whereas news about future nominal interest rates have a very small contribution to the overall bond variance (0.04), although the covariance between the 2 components is positive, and has a weight of about 11%. These results confirm previous evidence (Campbell and Ammer (1993), Cochrane and Piazzesi (2004)), that the "Expectation theory" of the term structure which states that bond risk premia should be constant trough time, is not validated by the data. In respect to the stock return decomposition, excess return news represents more than 100% of the total stock market return variance. This is possible because the covariance between excess return news and cash flow news has a negative contribution (-0.318). The variance of cash-flow news has a lower contribution than excess return news for the overall variance (0.331), a result that goes in line with previous evidence (Campbell (1991), Campbell and Ammer (1993), CV) which emphasizes the fact that excess return news is the main determinant of unexpected equity market return s volatility. On the other hand, news about future nominal interest rates have a negligible contribution, representing less than 1% of the stock market variance, also in line with previous evidence (Campbell and Ammer (1993)). B By analyzing the correlations of shocks in the individual VAR state variables with both r and r Y, the most relevant results are that TERM and Δr f,t are negatively and positively correlated with r Y, respectively, which is a sign that short-term interest rates exhibit some 15

16 Y persistency. Shocks on bond excess returns are weakly negatively correlated with r strongly negatively correlated with r B, indicating that bond prices exhibit long term mean reversion. and From the correlations between individual shocks and both r H and r CF, we can verify that the innovations on VS are weakly negatively correlated with both cash-flow news and excess return news, in line with the results obtained in CV. Innovations on EY are strongly positively correlated with r H, confirming that EY forecasts positive stock market returns, in part due to the mean reversion in stock prices. Finally, innovations in stock market returns are strongly negatively correlated with stock excess return news, reflecting the existence of long term mean reversion in stock prices, also in line with the results produced in CV. C. Estimating factor betas In table III, I present single regression betas for the news components associated with stock and bond returns, in the case of the 25 size/book-to-market portfolios. The cash-flow betas are positive, while the betas associated with both stock excess return and bond return news are negative. Thus, an increase in future cash-flows and a decrease in future excess stock market and bond returns lead to higher individual stock returns today, as expected. Comparing the two betas related with discount rate news, stock premia news have higher (magnitude) betas than bond premia news, i.e., the individual assets are more sensible to rises in future excess stock returns than rises in future bond premia. On the other hand, the betas associated with nominal interest rate news are positive, i.e., rises in future nominal interest rates are associated with higher average returns for individual assets. In Panels B and C, I present the average betas across book-to-market and size quintiles. The average betas within the book-to-market quintiles indicate that value stocks have higher cash-flow betas than growth stocks (0.518 for BV5 versus for BV1), although there is no 16

17 monotonic relation between book-to-market and cash flow betas, with the betas decreasing from BV1 to BV3, and then rising from BV3 to BV5. On the other hand, growth stocks have higher (absolute) stock return news betas, relative to value stocks ( for BV1 versus for BV5), and there is a similar pattern concerning the bond premia news beta ( versus ), with the relation between betas and book-to-market being close to monotonic in both cases. The findings for cash-flow and stock market premia betas are close to the results obtained by CV, which have found in their modern sub-sample that value stocks have higher (lower) cash flow (discount rate) betas relative to growth stocks. An important finding in these results, is that there is much more dispersion across the quintiles for the stock news betas than the bond news betas (absolute difference between BV1 and BV5 of for stock premia beta versus for the bond news beta), thus the betas associated with bond news are more stable across assets. This result shows that the risk premium associated with news in future bond excess returns is approximately the same across the individual stocks, while the risk premium associated with news in stock market excess returns presents sharp differences within the cross-section. This goes in line with the proposition that the type of risks measured by bond premia news - changes in discount rates used to discount certain cash-flows in the future - should be similar across individual stocks. The fact that growth stocks have marginally higher bond betas than value stocks has to due with the higher duration risk of growth stocks, i.e., since they discount more distant cash flows into the future, they are more sensible to rises in future long term interest rates. In what concerns the interest rate betas, growth stocks are also more sensible to upwards revisions in future nominal interest rates (1.910 for BV1 versus for BV5). The average betas associated with size quintiles, indicate that small stocks have lower cash-flow betas than large stocks (0.374 for S1 versus for S5). In what concerns the stock news beta, small stocks have higher absolute betas than big caps ( for S1 versus 17

18 for S5), while the opposite relation holds for bond news betas ( for S1 versus for S5). In addition small stocks are more sensible to the risk associated with future nominal interest rates (1.219 for S1 versus for S5). Therefore, as it is the case of growth relative to value stocks, small caps are not unambiguously riskier than big caps, since small stocks have both lower cash-flow and (absolute) bond premia betas. D. Estimating the factor risk premia The natural econometric framework to estimate and test the asset pricing models presented in the previous section, is first-stage GMM using as weighting matrix the identity matrix, where the N sample moments correspond to the pricing errors for each of the N test assets at hand, g T b 1 T T ri, r f, i 2 2 i,cf i,h 1 i,b i,y 0 i 1,...,N The standard errors for the parameter estimates and moments are presented in the Appendix, and the asymptotic test that the pricing errors are jointly zero, with g T b, is given by var 1 ~ 2 N K, with K being the number of parameters estimated by the system, and N K denoting the number of overidentifying conditions. Following Cochrane (1996), and given the fact that var is singular in most of the cases, I perform a eigenvalue decomposition of the moments variance-covariance matrix, var QQ, where Q is a matrix containing the eigenvectors of var on its columns, and is a diagonal matrix of eigenvalues, and then I invert only the non-zero eigenvalues of. In Table IV, I present the estimation and evaluation results for the ICAPM model of equations 16 and 18 above. Following Lo and Mackinlay (1990), who argue against testing asset-pricing models by using returns on portfolios sorted on some characteristic associated with returns themselves, I use the returns on 38 industry portfolios (IND38), and the combination of size/book-to-market and industry portfolios (SBV25IND38), as additional 18

19 groups of test assets. I also present the risk prices for multiple regression betas in addition to single regression betas. In terms of single regression betas, the estimates for both H, B and Y are the same across the 3 classes of portfolios, since they are constrained a priori by the model. H have higher magnitudes than B, as a result of the higher variance of stock excess return news relative to bond premia news. CF is much higher than the negative of H across the 3 classes of test assets, confirming that cash-flow news has a higher risk price than stock excess return news, as predicting by the model. On the other hand, news on nominal interest rates has a very low price of risk, as a consequence of the low variance of interest rate news. The risk aversion parameter is higher for SBV25 than IND38 (28.861versus ), assuming an intermediate value (23.800) for the combine portfolios (SBV25IND38). In terms of statistical significance, is highly significant for SBV25 (1% level), being significant at the 5% level for the other two classes of portfolios. The ICAPM model is not rejected by the asymptotic 2 test, for all 3 classes of test assets. The average pricing error (RMSE) is slightly higher for SBV25 than the industry portfolios (0.259 versus 0.221). Looking at the risk prices associated with multiple regression betas, B is strongly significant in the 3 classes of portfolios, and although B has lower magnitudes than H, its t-statistics are much higher. In Panel B, I present the results for the alternative weight for stocks in the market portfolio, 0.5. The results are similar to Panel A, although assumes higher values, thus originating higher values for CF. Given the increased weight of bonds in the market portfolio, B H has higher (lower) magnitudes in comparison with the respective estimates in Panel A. Nevertheless, H still has higher risk prices than B, given the higher variance of stock excess return news relative to bond premia news. In what concerns the risk prices of the multiple regression betas, B continues to be strongly significant. 19

20 E. Constant stock-bond premia Both stocks and bonds share common characteristics, since in both cases the current value is the discounted sum of a long stream of future cash flows. Nevertheless, stocks exhibit two key differences relative to long maturity bonds: First, there is an infinite stream of future cash-flows, thus stocks have higher duration risk than bonds, which have fixed maturities. Thus, given a common discount rate, stocks are more sensible to changes in future discount rates. Secondly, and most important, the cash flows associated with stocks are uncertain, as opposed to bonds which have fixed cash flows, that are known beforehand. To compensate for the risk associated with cash-flows, stocks earn a risk premium over long-term bonds, originating higher returns in average. Thus, we can reinterpret stock excess return as being composed by two components: Bond risk premia, used to discount the "certain" part of future stock cash flows, and the stock-bond risk premia used to discount the "random" part of future equity cash flows, which represents the "true" equity risk premia. If we assume a constant stock-bond risk premia k, then expected stock market returns can be represented as, E t r m, E t r b, k 27 and this originates that stock and bond excess return news are coincident, r H B r 28 since E E t j1 k 0. Substituting 28 in equation 15 above, the log SDF is given by m E t m r CF r B Y r 29 Making f r follows, CF,r B,r Y and b,1,1 and using Theorem 1 in Appendix D, it Er i, r f, i 2 2 i,cf i,b i,y 30 20

21 In this model, news in stock-bond excess returns don t appear in the pricing equation, since the stock-bond risk premium is constant, and the covariance with bond premium news receives a risk price of 1 compared with 1 in the ICAPM presented in equation 16. The results for the ICAPM model in equation 30, are reported in table V. Both CF and B present slightly higher values relative to table IV, as a result of eliminating the stock-bond premia factor. The estimates of are also higher relative to table IV, producing higher values for CF. In addition, the asymptotic 2 test and RMSE present equivalent values to Table IV. These results seem to suggest that by incorporating a constant stock-bond premia, the global fitting of the ICAPM model does not change significantly relative to the ICAPM of the previous sub-section. F. Time-varying stock-bond premia: Benchmark ICAPM The assumption of a constant stock-bond risk premia presented in the previous sub-section, is not realistic given the extensive evidence that market equity premium is time-varying, and thus both components of stock excess returns should be time-varying. With a time-varying stock-bond risk premia, k, equation 27 generalizes to E t r m, E t r b, k 31 As shown in the Appendix F, this originates that excess stock return news can be split into stock-bond premia news, and bond excess return news, r H r K B r 32 where r K E E t j1 j k j denotes news about future stock-bond premia, which represent the true equity premium news. These two components have a distinct fundamental interpretation: Bond premia news represent revisions in future discount rates used to discount certain future cash-flows, whereas stock-bond premia news represent revisions in future 21

22 expected discount rates used to discount uncertain future cash flows. Substituting 32 in equation 15 above, the log SDF is given by m E t m r CF r K r B Y r 33 Making f r CF,r D, one has, K,r B Er i, r f, i 2,r Y and b,,1,1 and using Theorem 1 in the Appendix 2 i,cf i,k i,b i,y 34 where i,k Covr i,,r K, represents covariance with stock-bond premia news. The model in 34 will be the benchmark ICAPM analyzed in this paper, and the difference to the ICAPM in equation 16, is that now the covariance with stock excess return news is replaced by the covariance with stock-bond premia news, i,k, which has the same risk price given by. Thus, for a conservative investor 1, the risk price associated with cash-flow news should be higher than (minus) the risk price of "true" equity premia news. The second difference to model 16 is that the covariance with bond premium news receives a risk price of 1 compared with 1 previously. In consequence, since 1, the risk price of covariance with stock-bond premia news is lower (in magnitude) than the risk price of covariance with bond premia news. The model 34 can be restated in terms of betas, Er i, r f, i CF i,cf 2 K i,k 2 B i,b 2 Y i,y 35 K In terms of betas risk prices, it will be the variances of r whether i,k has a higher (magnitude) risk price than i,b or not. B and r that will determine, In order to identify the "true" equity premia news component, I use the same VAR model estimated above, with r K given by r K E E t j1 j r m,j r b,j e1 e2 AI A 1 36 where e1 e2 AI A 1 is the function that relates the VAR shocks with revisions in expected future stock-bond excess returns. Cash-flow news continues to be the residual 22

23 component of stock returns, r CF E E t j Δd j E E t r m, r f, r K r B Y r j0 e1 e2 37 The results for the estimated "news" components in 36 and 37 are presented in Table VI. We can see, that the correlation between r CF K and r is higher than the correlation between r CF and r H in Table II (0.426 versus 0.274), and the correlation between r K B and r is very low (0.114). r CF is also weakly correlated with both r Y and r B. The fact that r CF is more correlated with r K than previously, and r K is uncorrelated with r B, arises from decomposing r H into two different fundamental variables, bond premia news and news in the "true" stock premia, r K.In terms of variance decomposition, the variance of cash flow news is higher than in Table II (0.706 versus 0.331) but the contribution of the covariance between excess stock-bond return news and cash flow news is also more negative ( versus before), as a result of the higher covariance between cash flow news and stock-bond premia news, as compared to the covariance between cash flow and excess stock return news in Table II. The variance of bond premia news has some explanatory power over the stock return variance (0.201), whereas, covariance between news in stock-bond premia and bond premia have a marginal contribution (0.109). Finnally, the covariance between cash flow and bond premia news has a negative contribution of to the total stock return variance. The correlations between VAR innovations and news components are similar to those obtained in Table II, with the single difference, that shocks in bond returns are weakly negatively correlated with cash flow news: A rise in current bond prices and returns is associated with lower future bond returns, and is also associated with expectations of poor business conditions and decreasing future earnings and cash flows. The single regression beta estimates associated with both r CF K and r for the 25 size/book-to-market portfolios are presented in Table VII. The cash-flow betas are lower than in Table III, but growth stocks continue to have lower cash-flow betas than value stocks 23

24 (difference of , versus in Table III). The stock-bond news factor have lower (magnitude) betas when compared to the stock excess return news betas in Table III, but still growth stocks are more sensible to upwards revisions in future stock-bond risk premia, relative to value stocks (difference of 0.344). In terms of size quintiles, large stocks have higher cash-flow betas than small stocks but have lower (absolute) stock-bond premia news betas, similarly to Table III. The single regression betas associated with the 38 industry portfolios, which are presented in Table VIII, show that there is wide dispersion in betas across industries, similarly to the size/book-to-market portfolios. In general, the betas of stock-bond premia news have higher magnitudes than the betas for bond risk premia news, with the exception of Telephone and Telegraph communication (PHONE), Tobacco products (SMOKE) and Electric, Gas and Water supply (UTILS) industries. The estimation results of the asset pricing model in equations 34 and 35, are displayed in Table IX. The risk prices for both K and B are higher in magnitude relative to the corresponding values for H and B in the model of subsection III. D above. The estimates for are higher than in table IV, especially for SBV25 portfolios, leading to higher risk prices for cash-flow news, which is significant at the 1% level for SBV25, and 5% level in the case of both IND38 and SBV25IND38. Thus, By decomposing stock excess return news into its 2 components, we have an increase in the explanatory power of all 3 factors, CF, K and B.In this model, the risk price of stock-bond premia news is higher (in magnitude) than the risk price associated with bond news, given that the former factor has higher variance. In terms of multiple regression betas, although B has lower magnitudes than K, it presents similar statistical significance. In Panel B, the estimates for are higher, thus producing higher estimates for CF, and in result of the lower weight given to stocks, the estimates for K decrease in magnitude relative to Panel A, similar to Table IV. While this model does not improve the global pricing ability for the SBV25 portfolios, compared to the ICAPM in equation 24

25 16, it does improve the pricing ability of value and growth stocks, as will be analyzed in section V: the key feature, is that the bond premia factor, which has lower risk price than the stock-bond premia factor, has larger relative (to stock-bond premia) betas than in model 16, and this enables to price both value and growth stocks. G. An unrestricted factor model The ICAPM estimated in the last sub-section, has only one freely estimated parameter in the cross-section, which is the risk aversion coefficient,, and therefore cash flow news is the only factor with a risk price estimated from the cross-section of stock returns. The risk prices of the other 3 factors, r K B Y, r and r are constrained a priori by the model. Since most asset pricing models have freely estimated risk prices in the cross-section, even in the cases of theoretically based models as it is the case of the CAPM, I estimate a4factormodel, which represents an unrestricted version of the ICAPM in equation 34. Writing the log SDF as m E t m b CF r CF b K r K b B r B Y b Y r 38 making f r CF,r K,r B D, one has the following 4 factor pricing model, Er i, r f, i 2,r Y, b b CF,b K,b B,b Y and using Theorem 1 in Appendix 2 b CF i,cf b K i,k b B i,b b Y i,y 39 with the covariance risk prices being estimated in the cross section. The estimation results of model 39 are reported in Table X. In terms of beta risk prices estimates, cash-flow news have a lower price than (the negative of) stock-bond premia news, and bond premia news is not statistically significant. Thus the factor model does not confirm some of the restrictions imposed by the theoretical ICAPM. On the other hand, the model presents very low pricing errors, with the asymptotic 2 test having a p-value of 1 for the SBV25 portfolios, while the square root of the average pricing error is only 0.106% for SBV25 and 0.159% for the industry 25

26 portfolios. H. Assessing individual pricing errors All ICAPM models estimated above were not rejected using the asymptotic 2 test of joint nullity of the pricing errors. As emphasized before (Cochrane (1996, 2001), Hodrick and Zhang (2001)), inference using this test can be misleading due to the singularity of var, and the inherent problems in inverting it. As a consequence I have opted for a generalized inverse as described above. Nevertheless, it could be that the low test values, are not so much the result of individual low pricing errors - what we want - but rather the economic uninteresting result of low values for var 1. To address this issue, it is helpful to pursue an analysis of the individual pricing errors. Figure 1.A. presents a picture of the pricing errors for SBV25, associated with the BBGB model in equation 17, and the ICAPM models in equations 16, 30, 34, and 39 above. I will denote these models as ICAPM I, II, III and IV respectively. The graph shows that the unrestricted ICAPM (ICAPM IV) has the lowest pricing errors, and the errors associated with the other models are in general similar, confirming the values for RMSE presented before. Nevertheless, the benchmark model (ICAPM III) has lower pricing errors associated with the extreme book-to-market portfolios within each size quintile, an issue that will be further analyzed in section V. In addition, I compare the ICAPM individual pricing errors with those associated with the CAPM. Following Campbell (1993), I show in Appendix I, that in the framework of section II and for the case of a investor with log utility 1, the CAPM arises as a special case of the ICAPM in equations 16 and 34, Er i, r f, i 2 2 i,m 1 i,b 40 where i,m Covr i,,e E t r m, and i,b Covr i,,e E t r b, denote the covariances with stock market return and bond return, respectively. Equation 40 represents 26