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.
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Notes
The volatility (variance) risk premium is defined by the difference between two expected values of the volatility (variance) under the risk-neutral and physical probability measures, respectively.
The long-run risks model, which is a stylized self-contained general equilibrium model incorporating the effects of time-varying economic uncertainty, is pioneered by Bansal and Yaron (2004).
We can find that the correlation value between \(VIX_{t}^2\) and \(I{\textit{Skew}}_{t}\) in the same period is also nearly zero (see Table 2).
Because the datails of the four parameters, \(A_{x,m}\), \(A_{\sigma _m}\), \(A_{q,m}\), and \(A_{\lambda ,m}\), are insignificant and do not affect the discussion explored in the following at all. For simplicity, we express the parameters, \(A_{x,m}\), \(A_{\sigma _m}\), \(A_{q,m}\), and \(A_{\lambda ,m}\), as they are and do not show explicit representations of those parameters in this paper.
The empirical evidence of the negative correlation between the one-step-ahead market return, \(r_{m,t+1}\), and the one-step-ahead change of risk-neutral skewness, \(\bigtriangleup {\textit{Skew}}_{t+1}^{\mathbb {Q}} \equiv {\textit{Skew}}_{t+1}^{\mathbb {Q}} - {\textit{Skew}}_{t}^{\mathbb {Q}}\), also essentially shown by Neuberger (2012). He shows that the period when the S&P500 index volatility was very low by historic standards (from the year of 2003 to 2007) was also one of relatively low skewness, whereas skewness was actually rather high in the volatility spike of the year of 2008. Under the leverage effect of Black (1976), that evidence suggests the same essentially as that of the negative correlation between the one-step-ahead market return and the one-step-ahead change of risk-neutral skewness.
According to the description of the CBOE’s SKEW index, we have the proxy for the risk-neutral expected skewness, \(\mathbb {E}_t^{\mathbb {Q}} [{\textit{Skew}}_{t+1}^\mathbb {Q} (r_{m,t+2})]\), as \(\frac{1}{10} (100- {\textit{Skew}} \quad index)\).
In our model setting, the time-t skewness of the one-step-ahead market return, \(r_{m,t+1}\), can be expressed by (21). Thus, we can find that the first-order autocorrelation of the skewness under the physical measure \(\mathbb {P}\) (or the risk-neutral measure \(\mathbb {Q}\)) is determined by that of the jump intensity \(\lambda _t\). From (6), we can also find that the first-order autocorrelation of the jump intensity \(\lambda _t\) is \(\rho _\lambda \). Previous studies such as those by Chan and Maheu (2002) and Christoffersen et al. (2012) also model the dynamics of the jump intensity of financial asset returns as a first-order autoregressive model in a similar fashion, and they find that the \(\rho _\lambda \) parameters in the jump intensity models, which are estimated with historical data of stock indices, are more than 0.9 and show the strong persistence in the conditional jump intensity. So, on the basis of these previous studies, we could assume that the objective expected future skewness, \(\mathbb {E}_t^{\mathbb {P}} [{\textit{Skew}}_{t+1}^{\mathbb {P}} (r_{m,t+2})] \), will be close to the value of the current empirical skewness so that the same qualitative implications hold true for the skewness difference obtained by replacing \(\mathbb {E}_t^{\mathbb {P}} [Skew_{t+1}^{\mathbb {P}} (r_{m,t+2})] \) in the definition (22) with the current empirical skewness.
We thank a referee so much for suggesting the idea on this point.
In this case, under the assumption of i.i.d distribution for daily returns, \({\textit{Skew}}_t^D\) is the historical 22 days actual skewness of daily return data, which is devided by \(\sqrt{22}\).
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For their helpful comments on this article, I especially wish to thank Hidetoshi Nakagawa, Kazuhiko Ohashi, Toshiki Honda, Nobuhiro Nakamura, Fumio Hayashi, Tatsuyoshi Okimoto, Ryozo Miura, and the participants at the Research Center for Mathematical Economics workshop in 2012 and the CSFI-CREST Seminar in 2013. I assume full responsibility for all errors.
Appendices
Appendix 1: Proof of Proposition 2
From (14),
where
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,
Thus the expression of (28) is rearranged to the following representation,
where
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,
where \(\Lambda _q \equiv (1-\theta ) \kappa _1 A_q\), \(\Lambda _\lambda \equiv (1-\theta ) \kappa _1 A_\lambda \) (see (13)), and
Substituting the following facts,
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 \(\lambda _t\).
where
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Sasaki, H. The skewness risk premium in equilibrium and stock return predictability. Ann Finance 12, 95–133 (2016). https://doi.org/10.1007/s10436-016-0275-7
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DOI: https://doi.org/10.1007/s10436-016-0275-7
Keywords
- Long-run risks model
- Epstein–Zin preferences
- Variance risk premium
- Skewness risk premium
- Stock return predictability
- Stochastic volatility
- Volatility of volatility
- Jump intensity