# The Conditional CAPM and the Cross-section of Expected Returns

Jagannathan, Ravi, and Zhenyu Wang, “The Conditional CAPM and the Cross-section of Expected Returns,” The Journal of Finance, Vol 51, No 1 (1996), 3-53.

Purpose:  Use a modifed “conditional” CAPM, which includes the return on human capital as part of the return on the market portfolio and which allows betas to vary across time periods, to explain the cross-section of stock returns.

Results:  The Conditional CAPM (CCAPM) explains the cross-section of a large portfolio of stock returns very well.  The CCAPM explains 30% of the cross-sectional variation, while the static CAPM (assuming constant betas) explains only 1%.  When the return on human capital is included in the market return, the CCAPM explains 50%, and size and book-to-market factors have very little explanatory power.

Theory:  The static CAPM of Sharpe, Lintner, and Black is given by the equation $E(R_i) = \gamma_0 + \gamma_1 \beta_i$, where $\beta_i = \frac{\text{cov}(R_i, \: R_m)}{\text{var}(R_m)}$ is the regression coefficient of stock return i on the market return.  Letting the variables on the right-hand side vary depending on investors’ information set $I_{t-1}$.  The Conditional CAPM is

$E(R_{it}|I{t-1}) = \gamma_{0t-1} + \gamma_{1t-1} \beta_{it-1}$ where $\beta_{it-1} = \frac{\text{cov}(R_{it}, \: R_{mt} | I_{t-1})}{\text{var}(R_{mt} | I_{t-1})}$.

Taking the unconditional expectation of both sides,

$E(R_{it}) = \gamma_0 + \gamma_1 \overline{\beta}_i + \text{cov}(\gamma_{1t-1}, \: \beta_{it-1})$.

Now define the “beta-premium sensitivity” as $\vartheta_i = \frac{\text{cov}(\beta_{it-1}, \: \gamma_{1t-1})}{\text{var}(\gamma_{1t-1})}$, and substitute to get the CCAPM form

$E(R_{it}) = \gamma_0 + \gamma_1 \overline{\beta}_i + \text{var}(\gamma_{1t-1}) \vartheta_i$.

The PL-Model:

• $\overline{\beta}_i \text{ and } \vartheta_i$ are not observable, so define two unconditional betas:
• $\beta_i \equiv \frac{\text{cov}(R_{it}, \: R_{mt})}{\text{var}(R_{mt})}$.
• This “market beta” is the standard beta from the CAPM.
• Decompose the market beta into one beta for the stock market, from a regression of stock returns on market returns, and another beta for the return to human capital, from a regression of stock returns on the growth rate of per capita labor income.
• $\beta_i^{\gamma} \equiv \frac{\text{cov}(R_{it}, \: \gamma_{1t-1})}{\text{var}(\gamma_{1t-1})}$.
• This “premium beta” is not a linear function of the market beta.
• Use the yield spread between BAA- and AAA-rated bonds to proxy for $\gamma_{1t-1}$ in calculating the premium beta.
• The “PL-model” (Premium-Labor):  $E(R_{it}) = c_0 + c_{vw} {\beta_i}^{vw} + c_{prem} {\beta_i}^{prem} + c_{labor} {\beta_i}^{labor}$.
• Also add size to the right-hand side to verify whether there is any residual size effect (the coefficient of this term should be zero if the PL-model holds).
• Use the Generalized Method of Moments to test the PL-model.

Empirical Tests:

• Starting in 1963, create 100 size and beta portfolios at the end of each June as in Fama and French (1992)
• Use all NYSE and AMEX stocks in CRSP.
• Sort into size deciles (based on the NYSE/AMEX universe)
• Subdivide each size decile into beta sub-deciles based on pre-ranking betas
• Find pre-ranking betas for each stock over the previous 60 months, and require at least 24 months of data
• Regress stock returns on CRSP value-weighted returns
• Calculate each portfolio’s equal-weighted return for each of the next 12 months after portfolio formation
• Find portfolio value-weighted betas by regressing monthly portfolio returns on CRSP value-weighted index returns over the entire sample period.
• Portfolio premium betas come from month-by-month regressions of portfolio return on the yield spread between AAA- and BAA-bonds.
• Get bond yields from Table 1.35 in the Federal Reserve Bulletin.
• Portfolio labor betas are found by regressing portfolio return on the growth in (the two-period moving average of) labor income, defined by
• ${R_t}^{labor} = [L_{t-1} + L_{t-2}]/ [L_{t-2} + L_{t-3}]$.
• Personal income growth data come from Table 2.2 in the National Income and Product Account of the U.S.A., from the Bureau of Economic Analysis.
• Portfolio size is the equal-weighted market value, defined as the natural log of (number of shares x share price) from CRSP.
• Perform time-series regressions of portfolio returns on size and on the variables in the PL-model.

The GMM Test:

• It is possible to define a stochastic discount factor (SDF) $d_t{\delta} = \delta_0 + \delta_{vw} {R_t}^{vw} + \delta_{prem} {R_{t-1}}^{prem} + \delta_{labor} {R_t}^{labor}$ such that $E(R_{it}d_t) = 1$.
• $E[w_t(\delta)]$ is the vector of pricing errors in the model.
• Choose $\delta$ to minimize the quadratic form $E[w_t(\delta)]'[A]E[w_t(\delta)]$.
• ${E[w_t(\delta)]} = D_T \delta - 1_N$.
• $D_T = \frac{1}{T} {\sum_{t=1}}^T R_t Y_t'$.
• $R_t$ is the Nx1 vector of portfolio returns in month t.
• $Y_t$ is the (up to) 4×1 vector of parameters $(1, {R_t}^{vw}, {R_{t-1}}^{prem}, {R_t}^{labor})$.
• The Hansen-Singleton (1982) “optimal” weighting matrix $A = [\text{var}(w_t(\delta))]^{-1}$ is model-specific, so it cannot be used to compare models.
• Instead, choose the NxN weighting matrix ${G_T}^{-1} = \left[ \frac{1}{T} {\sum_{t=1}}^T R_t R_t' \right]^{-1}$, which does not vary across models.
• $\delta_T = {({D_T}' {G_T}^{-1} {D_T})}^{-1} {D_T}' {G_T}^{-1} 1_N$
• The square-root of the minimized quadratic form, or Hansen-Jagannathan (HJ) distance, is the distance between the model’s SDF and the set of SDFs that correctly price the sample assets.  It is also the largest pricing error among all the portfolios being tested.