Beta Risk in the Cross-section of Stocks and Options

In order to study beta risk, we develop a multivariate stochastic volatility model in which individual equity and market returns co-vary stochastically. In the model, the stochastic covariance matrix of an individual equity return with the market follows a bivariate Wishart process and the associated beta risk is priced. We estimate the model on returns and options jointly for a large cross-section of stocks. When analyzing the models empirical performance, we nd that it outperforms the standard rolling regression approach that is prevalent in empirical asset pricing. JEL Classi cation: G10; G12; G13. Keywords: Factor models; stochastic beta; option-implied beta; Wishart processes. For helpful comments we thank Daniel Andrei, Alexander David, Christian Dorion, Jan Ericsson, Pascal François, Alexandre Jeanneret, and seminar participants at the McGill-HEC Workshop. The authors gratefully acknowledge the nancial support from the Montreal Institute of Structured Finance and Derivatives (IFSID). Correspondence to: Mathieu Fournier, HEC Montréal, 3000 Chemin de la Côte-Sainte-Catherine, Montréal, QC H3T 2A7. E-mail: mathieu.fournier@hec.ca. 1 Introduction An extensive literature documents pronounced time-variation in equity betas.1 When modelling equity return dynamic, not accounting for the dynamics in market exposures might be relatively harmless during normal market conditions. However, missing sudden variation in beta during market downturns can be costly. Equity stock and option portfoliosoptimal allocation, hedging, and performance evaluation require appropriate measures of the joint dynamics of the equity and market index returns. Capturing the dependence of equity on the market index is also of paramount importance for cross-sectional asset pricing. Yet, little is known about the risks associated with time-varying betas. We study the dynamics of individual equity beta with the market index. To capture time variation in equity risks, we build on the work of Gouriéroux and Sufana (2003), Gouriéroux (2006), and Buraschi, Porchia, and Trojani (2010). We develop a stochastic volatility-covolatility model in which individual equity and market returns covary dynamically. In the model, the variancecovariance matrix of index and equity returns follows a bivariate Wishart process which can be viewed as a multivariate extension of Heston (1993). Our structural model extends the static Capital Asset Pricing Model (CAPM) by allowing beta to vary over time. This makes the model appropriate to study the cross-sectional implications of equity risk dynamics.2 The model provides several new insights about beta risk. First, it shows that part of equity premium can be attributed to a premium for time-varying betas. This prediction is consistent with the empirical ndings in Ang, Chen, and Xing (2006) and Petkova and Zhang (2005). Moreover, the model predicts that stock betas and rm-speci c innovations should comove negatively in the presence of a negative leverage e¤ect. Finally, the model suggests that a substantial part of comovements in betas is driven by market return and variance risks. Several studies argue that better capturing changes in equity beta helps decrease CAPM pricing anomalies (see, among others, Jagannathan and Wang, 1996; and Petkova and Zhang, 2003). When estimating the CAPM conditionally, the common practice is to adopt an ad-hoc recursive estimation method to capture uctuations in betas using ordinary least square regressions (OLS). An inherent problem with this methodology is that all observations in a given window impact equally the estimation of beta. The longer the estimation window the more backward looking the OLS beta becomes. This is in stark contrast with our model which, in addition to account for the dynamics 1See Jagannathan and Wang, (1996), Lettau and Ludvigson (2001), Lustig and Van Nieuwerburgh (2005), and Ang, Chen, and Xing (2006) for evidence on time-varying betas. 2See for example Da Fonseca, Grasselli, and Tebaldi (2007), Da Fonseca and Grasselli (2011), and Gruber, Tebaldi, and Trojani (2015) for other studies that use Wishart processes for modelling purposes. Note that none of these papers study the joint dynamic of market index and individual equity returns.