Risk, Uncertainty, and Expected Returns—Internet Appendix A Variance Risk Premium and Empirical Measurement

The central empirical variable of this paper, as a proxy for economic uncertainty, is the market variance risk premium (VRP)—which is not directly observable but can be esti-mated from the difference between model-free option-implied variance and the conditional expectation of realized variance. A.1 Variance Risk Premium: Definition and Measurement In order to define the model-free implied variance, let Ct(T,K) denote the price of a European call option maturing at time T with strike price K, and B(t, T) denote the price of a time t zero-coupon bond maturing at time T. As shown by Carr and Madan (1998) and Britten-Jones and Neuberger (2000), among others, the market’s risk-neutral Q expectation of the return variance σ2t+1 conditional on the information set Ωt, or the implied variance IVt at time-t, can be expressed in a “model-free ” fashion as a portfolio of European calls