Resurrecting the (C)CAPM: A Cross-Sectional Test When Risk Premia Are Time-Varying

Martin Lettau, Sydney Ludvigson. “Resurrecting the (C)CAPM: A Cross-Sectional Test When Risk Premia Are Time-Varying.” Journal of Political Economy Vol. 109, No. 6 (2001): 1238-1287. (link)

In this paper, Lettau and Ludvigson provide an asset pricing model that combines both the consumption CAPM and conditional CAPM in order to develop a factor model with reference to macroeconomic factors that drive risk premia and returns.

The authors review the effectiveness of the traditional CAPM, Fama French three factor model (FF3), and consumption CAPM (CCAPM) at explaining the cross-section of size and book-to-market related returns. Pricing errors for each model are shown in Figure 1 with a) CAPM, b) Fama French, c) CCAPM, d) the scaled consumption CAPM, ©CAPM, developed in this paper. For each two digit number, the first digit represents portfolios sorted on size (with 1 the smallest) while the second number represents portfolios sorted by book-to-market (with 1 the lowest book-to-market ratio).




The issue of the CAPM for explaining the cross-sectional returns of size and book-to-market are clear in the figure (1a) and have been empirically show in Fama and French 1992, 1993, among others. The consumption CAPM has not held up empirically but has theoretical weight which the authors utilize in their model. FF3 is successful at capturing much of the cross-sectional returns but Lettau and Ludvigson are interested in understanding the underlying economic, non-deversifiable risk which is proxied for by size and book-to-market.

Lettau and Ludvigson argue that the empirical evidence points to time-variation of risk premia which implies that investors’ expectations of future returns will affect the stochastic discount factor of the model. This approach contrasts with the traditional and consumption CAPM which assume constant risk premia.

To measure the expectations of future returns, the authors use the cointegration of c consumption, a log asset wealth, and y log labor income, referred to as cay. This measure is observable and Lettau and Ludvigson 2001 show that it serves as a reasonable proxy for the unobservable ratio of consumption to aggregate wealth, human and non-human. The factors in their model are scaled by the estimated value, $\widehat{cay}$, with stars representing measured values.

\[\widehat{cay_{t}}=c_{t}^*-0.31a_{t}^*-0.59y_{t}^*-0.60\]

Table 1 gives the results of Fama-MacBeth regressions on the 25 Fama French portfolios. The first row is the traditional CAPM and, as expected, the t-statistic for beta is insignificant and the $R^2$ is only 1%. The human capital CAPM, row 2, and FF3, row 3, explain increasing amounts of the cross-sectional variation in returns.

\[E[R_{i,t+1}]=E[R_{0,t}]+\beta_{zi}\lambda{z}+\beta_{vwi}\lambda_{vw}+\beta_{vwzi}\lambda_{vwz}\>\>\>\>\>\>\>(13)\]

\[E[R_{i,t+1}]=E[R_{0,t}]+\beta_{zi}\lambda_{z}+\beta_{vwi}\lambda_{vw}+\beta_{vwzi}\lambda_{vwz}+\beta_{\Delta yi}\lambda_{\Delta y}+\beta_{\Delta yzi}\lambda_{\Delta yz}\>\>\>\>\>(14)\]

Rows 4 and 5 of table 1 gives results for (13), the scaled CAPM, where $z_{t}=\widehat{cay}$, is the scaled factor and $f_{t+1}=R_{vw,t+1}$ is the fundamental factor.  Row 4 shows the time-varying intercept term, $\lambda_{z}$, is not significant and so row 5 gives the results without the intercept, showing it has little effect.   Rows 6 presents the regression (14), the scaled human capital CAPM, which succeeds in explaining 71% of the cross-sectional returns after correcting the t-statistic. Finally, row 7 presents regression (14) but without the time-varying intercept, $\lambda_{z}$, which has little effect on the results.

\[E[R_{i,t+1}]=E[R_{0,t}]+\beta_{zi}\lambda_{z}+\beta_{\Delta ci}\lambda_{\Delta c}+\beta_{\Delta czi}\lambda_{\Delta cz}\>\>\>\>\>\>(15)\]

Table 3 presents the results of the scaled multi factor consumption CAPM (15) developed in this paper.  Where $\Delta c$ represents the log difference of consumption, measured in Lettau and Ludvigson 2001. Row 1 presents the unconditional consumption CAPM for comparison and row 2 presents the results of (15). Row 3 is (15) but without the time-varying intercept as it is statistically insignificant. Note that $\lambda_{\Delta cz}$, the conditional scaled coefficient, is very significant and the $R^2$ statistic is nearly 70%, almost as high as the 80% cross-sectional returns explained by FF3. Furthermore, there is a dramatic increase in $R^2$ from the unconditional CCAPM in row 1 and the conditional CCAPM in row 2.

The authors demonstrate that their conditional version of the consumption CAPM performs nearly as well as the Fama French 3 factor model at explaining cross-sectional returns of portfolios sorted on size and book-to-market. Their intuition is that the risk associated with a value portfolio comes from the covariance with consumption growth that is time-variant rather than unconditional. During times of high risk aversion (high $\widehat{cay}$) this correlation of value with consumption growth will be high where as during times of low risk aversion it will be low. Figure 2 charts the conditional consumption betas for four pairs of size and book-to-market portfolios during good states ( $\widehat{cay}$ at least one standard deviation above its mean) and bad states (at least one standard deviation below). As expected, the high value portfolios have larger consumption betas during bad times than good times and higher consumption betas during bad times, than value stocks during bad times.  Since this high correlation with consumption growth comes when investors are most averse to it (times of high risk aversion) the value portfolios have greater time-variant risk and hence, greater return.

The scaled multi-factor model presented by Lettau and Ludvigson attempts to explain the cross-section of expected stock returns of portfolios sorted on size and book-to-market. The failure of the traditional CAPM to explain these returns has lead to research into the source of risk. The static consumption CAPM provides strong economic theory but ultimately falls short after empirical tests. The Fama French 3 factor model provides strong empirical results but does not offer an explanation of the fundamental macro economic sources of risk. The (C)CAPM model derived by the authors provides an economic framework in which to explain the cross-section of size and book-to-market while empirically explaining a large amount of the expected returns.