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4 190 R. Petkova, L. Zhang / Journal of Financial Economics 78 (2005) We use two approaches to estimate conditional betas. First, we regress value and growth portfolio excess returns on the market excess return using a 60-month rolling window. Using 24-, 36-, or 48-month rolling windows yields similar results (not reported). We call the resulting beta the rolling beta. Second, we use conditional market regressions: r itþ1 ¼ a i þðb i0 þ b i1 DIV t þ b i2 DEF t þ b i3 TERM t þ b i4 TB t Þ r mtþ1 þ itþ1 (3) and ^b it ¼ ^b i0 þ ^b i1 DIV t þ ^b i2 DEF t þ ^b i3 TERM t þ ^b i4 TB t, (4) where r itþ1 is portfolio i s excess return. We call ^b it from Eq. (4) the fitted beta Beta-premium regressions Albeit informal, the sorting procedure is perhaps the simplest way to study timevarying risk. We supplement this informal procedure with a more formal test motivated by the conditional CAPM. In a world with time-varying risk, the conditional CAPM serves as a natural benchmark model for asset pricing tests Econometric framework Define conditional beta as b it Cov t ½r itþ1 ; r mtþ1 Š=Var t ½r mtþ1 Š and let g t denote the expected market risk premium. Both b it and g t are conditional on the information set at time t. The conditional CAPM states that E t ½r itþ1 Š¼g t b it. To measure the effects of time-varying risk on average returns, we follow Jagannathan and Wang (1996) and take unconditional expectations on both sides of the conditional CAPM to obtain E½r itþ1 Š¼ḡ b i þ Cov½g t ; b it Š¼ḡ b i þ Var½g t Šj i, (5) where ḡ E½g t Š is the average market excess return, b i E½b it Š is the average beta, and j i is the beta-premium sensitivity, defined as j i Cov½b it ; g t Š=Var½g t Š. From Eq. (5), the average value-minus-growth return depends on the dispersion in average beta and on the dispersion in beta-premium sensitivity between value and growth. The beta-premium sensitivity in Eq. (5) is unique to the conditional CAPM. As pointed out by Jagannathan and Wang (1996), this sensitivity measures the instability of an asset s beta over the business cycle. Intuitively, stocks with positive beta-premium sensitivities have high risk precisely during recessionary periods when investors dislike risk or when the price of risk is high. Therefore, these stocks earn higher average returns than stocks with low or even negative beta-premium sensitivities. In the conditional CAPM, the effect of time-varying beta on average returns is captured entirely by the beta-premium sensitivity. And the part of the conditional beta correlated with the unexpected market excess return, apart from its impact on the average beta, has no direct effect on average returns. However, this result is specific to the conditional CAPM and is not a general property of efficient markets.

5 R. Petkova, L. Zhang / Journal of Financial Economics 78 (2005) The reason is that the assumptions under which the conditional CAPM can be derived are restrictive. Examples include investors with quadratic utility but no labor income and exponential utility with normally distributed returns (see, e.g., Cochrane, 2001, Chapter 9). If value stocks expose investors to a greater downside risk, then the beta-premium sensitivities of these stocks will be positive. To test this hypothesis, we regress the conditional betas of value portfolios on the expected market risk premium to estimate j i : ^b it ¼ c i þ j i^g t þ Z it. (6) And the one-sided null hypothesis then is j i 40. We also test whether growth portfolios have negative beta-premium sensitivities and whether value-minus-growth strategies have positive beta-premium sensitivities Estimation Several sources of measurement errors in regression Eq. (6) can affect our inferences. The estimated expected market risk premium, ^g t, is a generated regressor that is only a proxy for the true premium. Similarly, ^b it is only a proxy for the true conditional beta. With fitted betas, the inferences on j i based on the sequential regression Eqs. (1), (3), and (6) are likely to be biased. The reason is that both ^b it and ^g t are estimated using the same instrumental variables, and their measurement errors are likely to be correlated. To account for these measurement errors, we estimate ^bit, ^g t, and ^j i simultaneously by the generalized method of moments (GMM). Define Z t ½i DIV t DEF t TERM t TB t Š to be the vector of instrumental variables including a constant term, where i is a vector of ones. And define b i ½b i0 b i1 b i2 b i3 b i4 Š 0 and d ½d 0 d 1 d 2 d 3 d 4 Š 0 to be vectors of regression coefficients. The set of moment conditions is naturally the set of orthogonality conditions associated with Eqs. (1), (3), and (6): E½ðr mtþ1 Z t dþz 0 tš¼0, (7) E½½r itþ1 a i ðz t r mtþ1 Þb i Šði Z t r mtþ1 Þ 0 Š¼0, (8) and E½ðZ t b i c i j i Z t dþði Z t dþ 0 Š¼0. (9) For each portfolio i, there are in total 13 moment conditions and 13 parameters, so the system is exactly identified. Because the rolling betas are not estimated using instruments, their measurement errors are less likely to correlate with those in the expected market risk premium. And because the errors in beta only enter the left-hand side of Eq. (6), their effect can be absorbed by the disturbance term, Z it (see, e.g., Green, 1997, p. 436). In this case, we adjust only for the sampling variation in ^g t via GMM by using the moment conditions E½ðr mtþ1 Z t dþz 0 tš¼0 (10)

6 192 R. Petkova, L. Zhang / Journal of Financial Economics 78 (2005) and E½ð^b it c i j i Z t dþði Z t dþ 0 Š¼0. (11) Discussion Our tests so far focus only on the null hypothesis that value-minus-growth strategies have positive beta-premium sensitivities. This is a weak restriction implied by the conditional CAPM. Failing to reject this hypothesis, while suggesting that value exposes investors to a greater downside risk than growth, does not imply that the conditional CAPM can explain the value anomaly. To test this stronger restriction, we test whether the intercepts of value-minus-growth portfolios equal zero in conditional market regressions. 3. Empirical results Section 3.1 describes our data. The heart of this paper is Section 3.2, which studies the time-varying risk of value and growth. Section 3.3 tests the conditional CAPM Data Our conditioning variables are defined and obtained as follows. The dividend yield is the sum of dividends accruing to the Center for Research in Securities Prices (CRSP) value-weighted market portfolio over the previous 12 months divided by the level of the market index. The default premium is the yield spread between Moody s Baa and Aaa corporate bonds, and the term premium is the yield spread between the ten-year and the one-year Treasury bond. The default yields are from the monthly database of the Federal Reserve Bank of St. Louis, and the government bond yields are from the Ibbotson database. Finally, the short-term interest rate is the onemonth Treasury bill rate from CRSP. We use two value-minus-growth strategies. HML is the value portfolio (H) minus the growth portfolio (L) in a two-by-three sort on size and book-to-market. 2 And the small-stock value premium, denoted HMLs, is the small-value (Hs) minus the smallgrowth (Ls) portfolio in a five-by-five sort on size and book-to-market. HML has been a standard measure of the value premium since Fama and French (1993). HMLs is economically interesting because the value anomaly is strongest in the smallest quintile. All the monthly portfolio data from January 1927 to December 2001 are from Kenneth R. French s website. 3 2 Compared with the results from using HML, using the difference between decile ten and decile one in a one-way sort on book-to-market yields quantitatively similar but somewhat weaker results (not reported). 3

7 R. Petkova, L. Zhang / Journal of Financial Economics 78 (2005) Table 1 Descriptive statistics for two value-minus-growth strategies across different samples: the full sample from 1927 to 2001; the post-depression sample from 1935 to 2001; the postwar sample from 1946 to 2001; the post-compustat sample after 1963; and the pre-compustat sample before 1963 We report average returns, m, unconditional alphas, and unconditional market betas, as well as their standard errors adjusted for heteroskedasticity and autocorrelations of up to six lags. Average returns, alphas, and their standard errors are in monthly percent. HMLs is defined as the small-value minus the small-growth portfolio in the Fama and French 25 size and book-to-market portfolios. The last column reports the correlation between HML and HMLs in different samples. Sample HML HMLs Correlation m a b m a b (ste) (ste) (ste) (ste) (ste) (ste) January December 2001 (0.12) (0.13) (0.08) (0.27) (0.19) (0.10) January December 2001 (0.11) (0.12) (0.05) (0.18) (0.18) (0.06) January December 2001 (0.11) (0.12) (0.04) (0.16) (0.17) (0.05) January December 2001 (0.14) (0.15) (0.04) (0.20) (0.21) (0.06) January December 1962 (0.20) (0.18) (0.07) (0.51) (0.33) (0.15) The value premium can be defined as the average returns or the unconditional alphas of value-minus-growth portfolios. Table 1 reports both results. Consistent with Davis et al. (2000), the value premium exists in the long run, especially among small stocks. From the first row of Table 1, the average returns of HML and HMLs in the full sample are 0.39% and 0.89%, and their unconditional alphas are 0.30% and 1.05% per month, respectively. From the third row, excluding the great depression of the 1930s increases somewhat the unconditional alphas, but not necessarily the average returns. For example, the average return of HMLs goes down slightly from 0.89% to 0.88%, even though its unconditional alpha goes up slightly from 1.05% to 1.07%. The unconditional alpha of HML rises from 0.30% to 0.47%, and its average return rises from 0.39% to 0.44%. Using the postwar sample from January 1946 to December 2001 further increases the alpha of HML but decreases the alpha of the small-stock value strategy. And the sample gives lower average returns for both value strategies. In the post-compustat sample from January 1963 to December 2001, the value strategies have slightly higher alphas than, but similar average returns as those from the longer samples. Only HMLs has a significant unconditional alpha in the pre-compustat sample from January 1927 to December The average returns of HML and HMLs are not significant. Neither is the alpha of HML. We therefore do not focus on this subsample in subsequent analysis on time-varying risk. The correlation between

9 R. Petkova, L. Zhang / Journal of Financial Economics 78 (2005) Table 2 Average conditional betas in different states of the world for portfolios HML and HMLs The rolling betas are estimated using 60-month rolling-window regressions, and the fitted betas are estimated from conditional market regressions. We use four conditioning variables, a constant term, dividend yield, term premium, default premium, and one-month Treasury bill rate. We define four states by sorting on the expected market risk premium, which is estimated as a linear function of the same instruments. State peak corresponds to the 10% lowest observations for the premium; state expansion corresponds to below average premium; state recession corresponds to above average premium, excluding the 10% highest observations; and state trough corresponds to the 10% highest observations for the premium. We report the standard errors of the difference between the HML and the HMLs betas in trough and peak, denoted ste(diff). We also report the estimates of beta-premium sensitivity, denoted j i, and their standard errors, denoted ste(j i ), for portfolios H, L, HML, Hs (small-value), Ls (small-growth), and HMLs. All standard errors are adjusted for heteroskedasticity and autocorrelations of up to 60 lags. Panel A. The full sample (January 1927 December 2001) Rolling beta Fitted beta Peak Expansion Recession Trough ste(diff) Peak Expansion Recession Trough ste(diff) HML HMLs j i ste(j i ) j i ste(j i ) j i ste(j i ) j i ste(j i ) H Hs H Hs L Ls L Ls HML HMLs HML HMLs Panel B. The post-depression sample (January 1935 December 2001) Rolling beta Fitted beta Peak Expansion Recession Trough ste(diff) Peak Expansion Recession Trough ste(diff) HML HMLs j i ste(j i ) j i ste(j i ) j i ste(j i ) j i ste(j i ) H Hs H Hs L Ls L Ls HML HMLs HML HMLs Panel C. The post-compustat sample (January 1963 December 2001) Rolling beta Fitted beta Peak Expansion Recession Trough ste(diff) Peak Expansion Recession Trough ste(diff) HML HMLs j i ste(j i ) j i ste(j i ) j i ste(j i ) j i ste(j i ) H Hs H Hs L Ls L Ls HML HMLs HML HMLs

13 R. Petkova, L. Zhang / Journal of Financial Economics 78 (2005) up-market beta, b iu, and a down-market beta, b id, can then be defined from r itþ1 ¼ a i þ b iu r mtþ1 d þ b id r mtþ1 ð1 dþþ itþ1. (12) Upon finding that the long-term loser or value portfolio has a lower down-market beta and a higher up-market beta than the long-term winner or growth portfolio, DeBondt and Thaler and Chopra, Lakonishok, and Ritter conclude that value does not carry a greater downside risk than growth. Panel A in Table 4 replicates the DeBondt and Thaler (1987) regressions using our full sample from January 1927 to December Using their sample from January 1927 to December 1982 yields similar results (not reported). The results largely confirm theirs. First, once betas are allowed to vary with the market, the alphas are no longer significant, a result first obtained by Chan (1988). Second, as emphasized by DeBondt and Thaler, the insignificant alphas are not impressive because value stocks have higher up-market betas but lower down-market betas than growth stocks. Panel B of Table 4 reports the same regressions as Eq. (12), but conditional on low and high expected market risk premium. The dummy variable now takes the value of one if the estimated ^g t is lower than its sample average ḡ (good times) and zero otherwise (bad times). The panel shows that the HML beta is 0:26 in good times and 0.34 in bad times. Both are estimated with relatively small standard errors. And Table 4 Market regressions conditional on up and down realized market excess returns and on low and high expected market risk premium Panel A reports market regressions conditional on up and down realized market excess returns: r itþ1 ¼ a i þ b iu r mtþ1 d þ b id r mtþ1 ð1 dþþ itþ1, where r mtþ1 denotes the realized market excess return. d equals one if the market excess return is positive and zero otherwise, i.e., d ¼ 1 frmtþ1 40g. Panel B reports market regressions conditional on low and high expected market risk premium. The dummy variable is redefined as one if the estimated expected market risk premium is higher than its average level and zero otherwise, i.e., d ¼ 1 f^gt 4ḡg. The sample is from January 1927 to December Each panel contains regressions for both HML and the small-stock value premium, HMLs, defined as the small-value minus the small-growth portfolios in the Fama and French 25 size and book-to-market portfolios. The numbers in parentheses are standard errors adjusted for heteroskedasticity and autocorrelations of up to six lags. a r mtþ1 d r mtþ1 ð1 dþ Adjusted R 2 Panel A. d ¼ 1 frmtþ1 40g HML (0.33) (0.13) (0.07) HMLs (0.67) (0.25) (0.12) Panel B. d ¼ 1 fgt 4ḡg HML (0.12) (0.05) (0.08) HMLs (0.19) (0.07) (0.15)

15 R. Petkova, L. Zhang / Journal of Financial Economics 78 (2005) than to expand capital. In bad times, firms want to scale down, especially value firms that are less productive than growth firms (see, e.g., Fama and French, 1995). Because scaling down is more difficult, value firms are more adversely affected by economic downturns. In good times, growth firms face less flexibility because they tend to invest more. Expanding is less urgent for value firms because their previously unproductive assets have become more productive. In sum, costly reversibility causes value betas to be countercyclical and growth betas to be procyclical. Although time-varying risk goes in the right direction, the magnitude of the value premium remains positive and mostly significant after we control for time-varying risk using conditional market regressions. Therefore, it is necessary to consider other possible drivers of the value anomaly, both APT- or ICAPM-related risk (see, e.g., Fama and French, 1993, 1996; Liew and Vassalou, 2000; Petkova, 2005) and overreaction-related mispricing (see, e.g., Lakonishok et al., 1994). References Ang, A., Chen, J., CAPM over the long-run: Unpublished working paper, Columbia University, New York. Barberis, N., Thaler, R.H., A survey of behavioral finance. In: Constantinides, G., Harris, M., Stulz, R. (Eds.), Handbook of the Economics of Finance. North-Holland, Amsterdam, pp Bernanke, B., Nonmonetary effects of the financial crisis in the propagation of the great depression. American Economic Review 73 (3), Campbell, J.Y., Stock returns and the term structure. Journal of Financial Economics 18, Campbell, J.Y., Cochrane, J.H., By force of habit: a consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107 (2), Chan, K.C., On the contrarian investment strategy. Journal of Business 61 (2), Chopra, N., Lakonishok, J., Ritter, J.R., Measuring abnormal performance: do stocks overreact. Journal of Financial Economics 31, Cochrane, J.H., Asset Pricing. Princeton University Press, Princeton, NJ. Constantinides, G.M., Duffie, D., Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (2), Davis, J.L., Fama, E.F., French, K.R., Characteristics, covariances, and average returns: 1929 to Journal of Finance 55 (1), DeBondt, W.F.M., Thaler, R.H., Further evidence on investor overreaction and stock market seasonality. Journal of Finance 42 (3), Fama, E.F., Stock returns, real activity, inflation, and money. American Economic Review 71, Fama, E.F., Market efficiency, long-term returns, and behavioral finance. Journal of Financial Economics 49, Fama, E.F., French, K.R., Dividend yields and expected stock returns. Journal of Financial Economics 22, Fama, E.F., French, K.R., Business conditions and expected returns on stocks and bonds. Journal of Financial Economics 25, Fama, E.F., French, K.R., The cross-section of expected stock returns. Journal of Finance 47 (2), Fama, E.F., French, K.R., Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33, Fama, E.F., French, K.R., Size and book-to-market factors in earnings and returns. Journal of Finance 50,

16 202 R. Petkova, L. Zhang / Journal of Financial Economics 78 (2005) Fama, E.F., French, K.R., Multifactor explanations of asset pricing anomalies. Journal of Finance 51, Fama, E.F., Schwert, G.W., Asset returns and inflation. Journal of Financial Economics 5, Ferson, W., Harvey, C.R., Conditioning variables and the cross-section of stock returns. Journal of Finance 54, Green, W.H., Econometric Analysis, third ed. Prentice-Hall, Englewood Cliffs, NJ. Jagannathan, R., Wang, Z., The conditional CAPM and the cross-section of expected returns. Journal of Finance 51, Keim, D.B., Stambaugh, R.F., Predicting returns in the stock and bond markets. Journal of Financial Economics 17, Lakonishok, J., Shleifer, A., Vishny, R.W., Contrarian investment, extrapolation, and risk. Journal of Finance 49 (5), Liew, J., Vassalou, M., Can book-to-market, size, and momentum be risk factors that predict economic growth?journal of Financial Economics 57, Lettau, M., Ludvigson, S., Resurrecting the (C)CAPM: a cross-sectional test when risk premia are time-varying. Journal of Political Economy 109 (6), Lewellen, J., Nagel, S., The conditional CAPM does not explain asset-pricing anomalies. Unpublished working paper. Sloan School of Management, Massachusetts Institute of Technology, Cambridge Massachusetts. Petkova, R., Do the Fama-French factors proxy for innovations in predictive variables?journal of Finance, forthcoming. Rosenberg, B., Reid, K., Lanstein, R., Persuasive evidence of market inefficiency. Journal of Portfolio Management 11, Shanken, J., Intertemporal asset pricing: an empirical investigation. Journal of Econometrics 45, Stock, J.H., Watson, M.W., Business cycle fluctuations in U.S. macroeconomic time series. In: Taylor, J.B., Woodford, M. (Eds.), Handbook of Macroeconomics, vol. 1, pp Zhang, L., The value premium. Journal of Finance 60 (1),

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