The Conditional CAPM and the Cross-section of Expected Returns

The Conditional CAPM and the Cross-section of Expected Returns

Ravi Jagannathan and Zhenyu Wang. “The Conditional CAPM and the Cross-section of Expected Returns.” The Journal of Finance 51 (1996): 3-53.

Most empirical studies of static CAPM have assumed that betas remain constant over time and that the value-weighted return on a portfolio of all stocks well proxies for the return on aggregate wealth. This paper attributes the failure of static CAPM in explaining the real world data to these two assumptions of static CAPM. Throughout the paper, the authors discuss in what manner the assumptions in static CAPM may deviate from investor behaviors in the real world, theoretically propose an approach to address the identified potential sources of deviation, and then empirically test the new model, “Conditional CAPM,” to see whether their revised version of CAPM fits well with the real world data.

In contrast with static CAPM, conditional CAPM posits that betas are likely to vary over time. The underlying intuition is that, in the real world, investors live multiple periods, and therefore the relation between betas and expected returns should depend on the nature of information available at any given point in time. For example, the relative cash flow risk of a firm is likely to vary over the business cycle. In this sense, ‘conditioning on’ the available information at a given point in time, there should be a linear relationship between expected returns and betas.

Another point of differentiation of their model is that it incorporates a measure of return on human capital into the proxy for the return on market portfolio, in addition to the return on value-weighted portfolio of all stocks, the commonly used proxy for in existing studies. The rationale for doing so is in line with Mayers (1972), who points out that human capital forms a substantial part of the total capital in the economy, hence arguing that the return on all stocks alone may not be sufficient to measure the return on aggregate wealth.

The model to generate testable implications is as follows;

The first equation says that CAPM hold in a conditional sense (i.e. given information at a point in time, the conditional expected return is linearly related with the conditional beta). The second expression is derived by taking the unconditional expectation of the first equation. Note that the second equation says that, in unconditional sense, we should take the covariance between conditional risk premium and conditional betas into account, which is likely to be correlated and therefore non-zero. 

The following expression denotes the decomposition of conditional beta;

                                  where

This decomposition of conditional beta leads to three orthogonal parts; the constant expected beta, the beta-premium sensitivity, and the zero-mean residual beta, respectively.

Finally, the unconditional implication derived from the conditional CAPM model boils down to the expressions below;

Note that the third beta in the second expression, which is related with the third term in the first expression, represents the beta-instability risk, which arises when we consider the potential covariance between conditional risk premium and conditional betas.

The regression models for empirical test derived from the theoretical model discussed above is as below;

where the proxies used for the calculation of the betas regarding the conditional market risk premium and the return on human capital are the spread between the commercial paper rate and Treasury bill rate and the growth rate in per capita labor income, respectively.

The results are presented the table below;

The results indicate that the conditional CAPM with human capital (Panel C) outperforms the static CAPM without human capital (Panel A); the r-squared jumps from only 1.35% to 55.21%. Also, both C_prem and C_labor are statistically significant, and size does not explain what is left unexplained in this model anymore. However, even though we can see that the conditional CAPM specification does substantially better than the static CAPM in explaining the cross-section of average stock returns, it appears the model is still not sufficiently reflecting the aspects of reality. First, the slope of C_vw is negative, even though it is statistically not different from zero. Second, the intercept is still large and statistically significant.

The last two figures graphically illustrate the improvement of conditional CAPM in explaining returns relative to static CAPM;