The skewness risk premium in equilibrium and stock return predictability

Abstract

In this study, we investigate the skewness risk premium in the financial market under a general equilibrium setting. Extending the long-run risks (LRR) model proposed by Bansal and Yaron (J Financ 59:1481–1509, 2004) by introducing a stochastic jump intensity for jumps in the LRR factor and the variance of consumption growth rate, we provide an explicit representation for the skewness risk premium, as well as the volatility risk premium, in equilibrium. On the basis of the representation for the skewness risk premium, we propose a possible reason for the empirical facts of time-varying and negative risk-neutral skewness. Moreover, we also provide an equity risk premium representation of a linear factor pricing model with the variance and skewness risk premiums. The empirical results imply that the skewness risk premium, as well as the variance risk premium, has superior predictive power for future aggregate stock market index returns, which are consistent with the theoretical implication derived by our model. Compared with the variance risk premium, the results show that the skewness risk premium plays an independent and essential role for predicting the market index returns.

Appendices

Appendix 1: Proof of Proposition 2

From (14),

$$\begin{aligned} \begin{aligned} \displaystyle \mathrm{Var}_{t+1}^\mathbb {P} (r_{m,t+2})&= B_r^t G_{t+1} G_{t+1}^t B_r + \sum _i B_r^2(i) \mathrm{Var}_{t+1}^\mathbb {P} (J_{i,t+2}) \\&= B_r^t G_{t+1} G_{t+1}^t B_r + {B_r^2}^t \Psi _{t+1}^{(2)} (0), \end{aligned} \end{aligned}$$

(28)

where

$$\begin{aligned} \begin{aligned} \displaystyle B_r&= \kappa _{1,m} A_m + e_d (\because (15)) \\&\equiv \begin{pmatrix} B_r(1)&B_r(2)&B_r(3)&B_r(4)&B_r(5)&B_r(6) \end{pmatrix} ^t \in \mathbb {R}^6, \\ B_r^2&\equiv \begin{pmatrix} B_r^2(1)&B_r^2(2)&B_r^2(3)&B_r^2(4)&B_r^2(5)&B_r^2(6) \end{pmatrix} ^t \in \mathbb {R}^6, \\ \Psi _{t+1}^{(2)} (0)&\equiv \begin{pmatrix} 0&\Psi _{t+1,x}^{(2)} (0)&\Psi _{t+1,\sigma ^2}^{(2)} (0)&0&0&0 \end{pmatrix} ^t \in \mathbb {R}^6, \nonumber \end{aligned} \end{aligned}$$

and $\Psi _{t+1,x}^{(2)} (0)$ and $\Psi _{t+1,\sigma ^2}^{(2)} (0)$ are respectively the second derivative of the cgf (cumulant-generating function) for $J_{x,t+1}$ and $J_{\sigma ^2, t+1}$ evaluated at 0, that is,

$$\begin{aligned} \begin{aligned} \displaystyle \Psi _{t+1,x}^{(2)} (0)&\equiv \frac{\partial ^2}{\partial u^2} \Psi _{t+1,x} (u) \mid _{u=0} = \frac{\partial ^2}{\partial u^2} \lambda _{x,t+1} (\psi _x (u) -1) \mid _{u=0}, \\ \Psi _{t+1,\sigma ^2}^{(2)} (0)&\equiv \frac{\partial ^2}{\partial u^2} \Psi _{t+1,\sigma ^2} (u) \mid _{u=0} = \frac{\partial ^2}{\partial u^2} \lambda _{\sigma ^2,t+1} (\psi _{\sigma ^2} (u) -1) \mid _{u=0}. \nonumber \end{aligned} \end{aligned}$$

Thus the expression of (28) is rearranged to the following representation,

$$\begin{aligned} \begin{aligned} \displaystyle \mathrm{Var}_{t+1}^\mathbb {P} (r_{m,t+2})&= B_r^t G_{t+1} G_{t+1}^t B_r + {B_r^2}^t \Psi _{t+1}^{(2)} (0) \\&= B_r^t (H_{\sigma ^2} \sigma _{t+1}^2 + H_q q_{t+1}) B_r + {B_r^2}^t \mathrm{diag} \Big ( \psi ^{(2)} (0) \Big ) \Pi _{t+1}, \nonumber \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \displaystyle H_{\sigma ^2}&\equiv \begin{pmatrix} \varphi _\eta ^2 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \varphi _e^2 &{} \quad 0 &{} \quad 0&{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \varphi _\zeta ^2 \end{pmatrix}, H_q \equiv \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \varphi _\xi ^2 &{} \rho \varphi _\xi \varphi _u &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \rho \varphi _\xi \varphi _u &{} \varphi _u^2 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\ \mathrm{diag} \Big ( \psi ^{(2)} (0) \Big )&\equiv \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \psi _x^{(2)} (0) &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \psi _{\sigma ^2}^{(2)} (0) &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \Pi _{t+1} \equiv \begin{pmatrix} 0 \\ \lambda _{x, t+1} \\ \lambda _{\sigma ^2, t+1} \\ 0 \\ 0 \\ 0 \end{pmatrix}. \nonumber \end{aligned} \end{aligned}$$

Appendix 2: Proof of Proposition 3

From the definition of the variance risk premium (18) and the expressions of the conditional variance of the market return $r_{m,t+2}$ at time $t+1$ under each of the probability measures, we can derive the following expression,

$$\begin{aligned} \displaystyle vp_t= & {} - B_r^t \Bigl [ \Lambda _{\sigma ^2} H_{\sigma ^2} + \varphi _\xi (\varphi _\xi \Lambda _q + \rho \varphi _u \Lambda _\lambda ) H_{q} \Bigr ] B_r q_t \nonumber \\&+ B_r^t H_{\sigma ^2} B_r \Bigl [ \mathbb {E}_t^\mathbb {Q} [J_{\sigma ^2,t+1}^\mathbb {Q}] - \mathbb {E}_t^\mathbb {P} [J_{\sigma ^2,t+1}^\mathbb {P}] \Bigr ] \nonumber \\&+ {B_r^2}^t \Bigl [ \mathrm {diag} (\psi ^{(2)} (-\Lambda ) ) \mathbb {E}_t^\mathbb {Q} [\Pi _{t+1}] - \mathrm {diag} (\psi ^{(2)} (0 ) ) \mathbb {E}_t^\mathbb {P} [\Pi _{t+1}] \Bigr ], \end{aligned}$$

(29)

where $\Lambda _q \equiv (1-\theta ) \kappa _1 A_q$, $\Lambda _\lambda \equiv (1-\theta ) \kappa _1 A_\lambda$ (see (13)), and

$$\begin{aligned} \begin{aligned} \displaystyle \mathrm{diag} \Big ( \psi ^{(2)} (-\Lambda ) \Big )&\equiv \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \psi _x^{(2)} (- \Lambda _x) &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \psi _{\sigma ^2}^{(2)} (- \Lambda _{\sigma ^2}) &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \\ \mathbb {E}_t^\mathbb {Q} [\Pi _{t+1}]&\equiv \begin{pmatrix} 0&\mathbb {E}_t^\mathbb {Q} [\lambda _{x,t+1}]&\mathbb {E}_t^\mathbb {Q} [\lambda _{\sigma ^2, t+1}]&0&0&0 \end{pmatrix} ^t, \\ \mathbb {E}_t^\mathbb {P} [\Pi _{t+1}]&\equiv \begin{pmatrix} 0&\mathbb {E}_t^\mathbb {P} [\lambda _{x,t+1}]&\mathbb {E}_t^\mathbb {P} [\lambda _{\sigma ^2, t+1}]&0&0&0 \end{pmatrix} ^t, \nonumber \end{aligned} \end{aligned}$$

Substituting the following facts,

$$\begin{aligned} \begin{aligned} \displaystyle \mathbb {E}_t^\mathbb {Q} [J_{\sigma ^2,t+1}^\mathbb {Q}]&= \lambda _{\sigma ^2, t} \psi _{\sigma ^2}^{(1)} (- \Lambda _{\sigma ^2}), \\ \mathbb {E}_t^\mathbb {P} [J_{\sigma ^2,t+1}^\mathbb {P}]&= \lambda _{\sigma ^2, t} \psi _{\sigma ^2}^{(1)} (0), \nonumber \end{aligned} \end{aligned}$$

into (29) and considering (6) and (16), we can obtain the representation (20). $\square$

Appendix 3: The risk-free rate

The explicit expression of the risk-free rate can be obtained by substituting $r_{f,t}$ into $r_{j,t+1}$ in (3). We finally provide the following proposition on the risk-free rate $r_{f,t}$.

Proposition 6

(The Risk-Free Rate) The risk free rate is expressed as follows with the state variables of $\sigma _t^2$, $q_t$, and $\lambda _t$.

$$\begin{aligned} \displaystyle r_{f,t} = \beta _{rf,c} + \beta _{rf,x} x_t + \beta _{rf,\sigma } \sigma _t^2 + \beta _{rf,q} q_t + \beta _{rf,\lambda } \lambda _t, \\ \end{aligned}$$

where

$$\begin{aligned} \beta _{rf,c}\equiv & {} - \theta \log \delta + \gamma \mu _g - (\theta -1)(\kappa _0 - A_0)\\&- (\theta -1) \kappa _1 (A_0 + A_\sigma \mu _\sigma + A_q \mu _q + A_\lambda \mu _\lambda ), \\ \beta _{rf,x}\equiv & {} \gamma - (\theta -1) A_x (\kappa _1 \rho _x -1), \\ \beta _{rf,\sigma }\equiv & {} (1-\theta ) A_\sigma (\kappa _1 \rho _\sigma -1) - \frac{1}{2} \Bigl [ \gamma ^2 \varphi _\eta ^2 + (\theta -1)^2 \kappa _1^2 A_x^2 \varphi _e^2 \Bigr ], \\ \beta _{rf,q}\equiv & {} (1-\theta ) A_q (\kappa _1 \rho _q -1) - \frac{1}{2} (\theta -1)^2 \kappa _1^2\\&\Bigl [ A_\sigma ^2 + A_q^2 \varphi _\xi ^2 + 2 A_q A_\lambda \varphi _\xi \varphi _u \rho + A_\lambda ^2 \varphi _u^2\Bigr ], \\ \beta _{rf,\lambda }\equiv & {} (1-\theta ) A_\lambda (\kappa _1 \rho _\lambda -1) \\&- l_x \Bigl [ \exp \left( \frac{1}{2} (\theta -1)^2 \kappa _1^2 A_x^2 \right) - 1 \Bigr ] - l_\sigma \Bigl [ \exp \left( \frac{1}{2} (\theta -1)^2 \kappa _1^2 A_\sigma ^2 \right) - 1 \Bigr ]. \nonumber \end{aligned}$$