The Role of Volatility Shocks and Rare Events in Long-Run Risk Models

We study a long-run risk model with a stochastic consumption growth rate, a stochastic volatility, a stochastic jump intensity, and a stochastic mean reversion level for the latter two processes. First, using a square-root specification instead of the Ornstein-Uhlenbeck process suggested by Drechsler and Yaron (2010) for the long-run mean reversion level of uncertainty has far-reaching economic consequences: the equity risk premium is increasing not only with short-run but also with long-run uncertainty, and the predictive power of the current price-dividend ratio for future excess returns increases and comes closer to empirically observed values. Second, we distinguish between two sources of a time-varying uncertainty of cash-flows, stochastic diffusive variance and stochastic jump intensity. We find that for most effects caused by time-varying uncertainty, time-variation in the jump intensity is much more important than time-variation in diffusive volatility risk. Third, the empirically observed low correlation between changes in the level and changes in the slope of the implied volatility smile for the S&P 500 index can only be matched with a model where jump intensity and conditional variance are locally uncorrelated