Business-cycle pattern of asset returns: a general equilibrium explanation

Abstract

I develop an analytical general-equilibrium model to explain economic sources of business-cycle pattern of aggregate stock market returns. With concave production functions and capital accumulation, a technology shock has a pro-cyclical direct effect and a counter-cyclical indirect effect on expected returns. The indirect effect, reflecting the “feedback” effect of consumers’ behavior on asset returns, dominates the direct effect and causes counter-cyclical variations of expected returns. I show that the conditional mean, volatility, and Sharpe ratios of asset returns all vary counter-cyclically and they are persistent and predictable, and that stock market behavior has forecasting power for real economic activity.

Appendix: Proofs

Proof of Corollary 1

Using Proposition 1, the net asset return $r_{t+1}\equiv \frac{ P_{t+1}+d_{t+1}}{P_{t}}-1=$$\frac{k_{t+2}+d_{t+1}}{k_{t+1}}-1 = \alpha z_{t+1}k_{t+1}^{\alpha -1}h_{t+1}^{1-\alpha }-\delta \equiv r_{t+1}^{I}$. $\square$

Proof of Proposition 2

• (1) If $\rho <1,$ then Eq. (2) implies that $\ln z_{t+1}|\ln z_{t}\sim N\left( \rho \ln z_{t},\sigma _{\varepsilon }^{2}\right)$. Then use the formula on the first two moments of a lognormal distribution to obtain the first two equations. The remaining two equations are obtained using the first two equations and Eq. (2).

• (2) $\frac{\partial \mu _{z,t}}{\partial \rho }=\mu _{z,t}\ln z_{t} >0 \ \left( <0\right)$ and $\frac{\partial \sigma _{z,t}}{\partial \rho }=\sigma _{z,t}\ln z_{t}>0$$\left( <0\right)$ if $z_{t}>1\ \left( <1\right) .$

• (3) $\frac{\partial \mu _{z,t}}{\partial z_{t}}=\mu _{z,t} \frac{\rho }{z_{t}}\geqslant 0$ and $\frac{\partial \sigma _{z,t}}{\partial z_{t}} =\sigma _{z,t}\frac{\rho }{z_{t}}\geqslant 0$.

$\square$

Proof of Proposition 3

The proof is trivial since $b=\alpha \phi ^{\alpha -1}$ is a decreasing function of $\phi$ as $\alpha <1$ (the law of diminishing returns to capital). If $\alpha =1,$$b=1$ is a constant. $\square$

Proof of Lemma 1

The first part is trivial given Eqs. (32) and (31).

For the second part, realize that $\frac{\partial v_{t}}{\partial y_{t}}=-\frac{2\mu _{z,t}^{2}\left( 2Ay_{t}-\gamma \sigma _{z,t}^{2}\right) }{\left( 2Ay_{t}+\gamma \sigma _{z,t}^{2}\right) ^{3}}<0$$\left( >0\right)$ if $2Ay_{t}-\gamma \sigma _{z,t}^{2}>0$$\left( <0\right) \!.$

For the last part, note that $\phi _{t}=\frac{1}{2}v_{t}y_{t},$ so $\frac{\partial \phi _{t}}{\partial y_{t}}=\frac{1}{2} \left( \frac{\partial v_{t}}{\partial y_{t}}y_{t}+v_{t}\right) \! =\!\frac{2\gamma y_{t}\mu _{z,t}^{2}\sigma _{z,t}^{2}}{\left( 2Ay_{t}+\gamma \sigma _{z,t}^{2}\right) ^{3}}>0$. $\square$

Proof of Proposition 6

Note that

$\frac{\partial b_{t}}{\partial \gamma }=24\cdot 2^{\frac{2}{3} }A^{2}y_{t}^{2}\sigma _{z,t}^{2} Q_{t}^{-\frac{2}{3}}/\sqrt{\gamma ^{2} \sigma _{z,t}^{4}+48Ay_{t}^{2}\mu _{z,t}}>0$, and

$\frac{\partial b_{t}}{\partial y_{t}}=-24\cdot 2^{\frac{2}{3} }A^{2}y_{t}\gamma \sigma _{z,t}^{2}Q_{t}^{-\frac{2}{3}}/\sqrt{\gamma ^{2} \sigma _{z,t}^{4}+48Ay_{t}^{2}\mu _{z,t}}<0$.

Because $\mu _{r,t}^{e}=b_{t}\mu _{z,t}-A$, $\sigma _{r,t} =b_{t}\sigma _{z,t}$, and $SR_{r,t}=\frac{b_{t}\mu _{z,t}-A}{b_{t}\sigma _{z,t}}$, I have $\frac{\partial \mu _{r,t}^{e}}{\partial X}=\frac{\partial b_{t}}{\partial X} \mu _{z,t}$, $\frac{\partial \sigma _{r,t}}{\partial X} =\frac{\partial b_{t} }{\partial X}\sigma _{z,t}$, and

$\frac{\partial SR_{r,t}}{\partial X}=\frac{\partial b_{t}}{\partial X}\frac{ A}{b_{t}^{2}\sigma _{z,t}}$, for $X=\gamma$ and $y_{t}$, respectively. $\square$