On the relation between the market risk premium and market volatility

Abstract

The Capital Asset Pricing Model (CAPM) suggests that the market risk premium should be positively related to the market systematic risk as measured by the market variance. However, the empirical evidence is conflicting. While some studies find a significantly positive relation, others find an insignificant or a significantly negative relation. This article attempts to resolve the market risk and return relation puzzle by recognizing that the market volatility is stochastic and should be treated as an important source of systematic risk – volatility risk. Investors demand a risk premium for bearing the market volatility risk in addition to the market systematic risk. As a result, the market risk premium consists of two components, both related to the market volatility. After taking into account the volatility risk premium, we find strong evidence of a significantly positive relation between the market risk premium and the market systematic risk. We also find that the volatility risk premium is negative and significant, which distorts the positive market risk and return relation.

Keywords:

• market risk premium
• market risk-return tradeoff
• volatility risk premium
• stochastic volatility
• EMM

JEL Classification::

• G12
• C32

Acknowledgements

I am grateful to Kerry Back, Phil Dybvig and Guofu Zhou for valuable suggestions and especially to George Tauchen for his help with the EMM programme. I wish to thank John Scruggs, Tao Li and seminar participants at Washington University in St. Louis for helpful comments.

Notes

¹ French et al. (1987), Harvey (1989), Turner et al. (1989), Campbell and Hentschel (1992), Lundblad (2007), Campello et al. (2008), Guo and Neely (2008) and Pástor et al. (2008) report a significantly positive relation between the market premium and market volatility; Fama and Schwert (1977), Campbell (1987), Pagan and Hong (1989), Nelson (1991), Glosten et al. (1993), Whitelaw (2000) and Chauvet and Potter (2001) report a significantly negative relation; Baillie and DeGennaro (1990) and Chan et al. (1992) find no significant relations at all.

² The volatility risk premium is nonzero only if the market volatility is correlated with the market return. It is essentially an empirical issue whether or not the volatility risk premium is zero. This article shows that the risk premium is indeed nonzero.

³ This assumptions holds when the changes in the coefficients are slower than the changes in the market pervasive risks. While it may be true that the aggregate relative risk aversion changes slowly, the changes in J_SQ/J_Q and thus λ₂ seem hard to determine. We will look at subperiod results to see if the estimates differ substantially over time.

⁴ It is conventional and appropriate to use continuously compounded returns to study the time series properties of stock returns, and to use simple returns for portfolio analysis. But in our specific formulation, it is appropriate to use simple returns; otherwise an additional term needs to be added into the return equation.

⁵ See the Appendix for more information on how to estimate continuous-time and discrete-time models.

⁶ The correlation coefficient ρ is given by