The Long-Run Risk of Stock Returns

Abstract

There is a large debate about whether stocks are less risky in the long-run than in the short run. Siegel (2008) in particular has argued that long-run variances are smaller than one-year variances on a per year basis. A recent article by Pastor and Stambaugh (2010) questions this evidence because of various uncertainties in the estimation of a typical data generating process (DGP, henceforth) for returns. As most research in this area however, the model of Pastor and Stambaugh assumes that there is short run predictability in stock returns, which generates mean reversion and thus affects the long-run variance, whereas the variability of stock return shocks is assumed to be constant over time. Our point of departure of the standard literature is simple: the standard DGP, such as the one used by Pastor and Stambaugh, is easily rejected by actual stock return data, where the evidence for time-varying volatility is many times stronger than the evidence for time-varying conditional means. We introduce a new dynamic model that does fit the intricacies of stock return dynamics, while still allowing for potential time-variation in the conditional mean. Our “Bad Environment-Good Environment” (BEGE for short) framework has two types of shocks: ”good environment” positively skewed shocks and ”bad environment” negatively skewed shocks. The relative importance of these shocks varies through time. By a judicious choice of the distribution to model these shocks, we can compute closed-form solutions for all moments of the stock return distribution at all horizons, even though the conditional variance, skewness and kurtosis of stock returns vary through time. The model is also easily estimable from the data and a normally distributed world is a special case of the BEGE framework. Thinking about the recent crisis period provides good motivation for a BEGE framework. It is hard to generate the type of day to day returns, or even month to month returns, we witnessed in 2008 within a Gaussian model. We view the shocks in October 2008 as likely coming from a negatively skewed distribution, which is characteristic for the “bad environment” that dominated the stock market in 2008 and the beginning of 2009 (and perhaps still does). However, even in the midst of the crisis, mini rallies did occur. In our framework, these are driven by a positively skewed shock distribution, which is more typical of a crisis-free environment but is still lurking in the background even in the midst of the crisis. We call the random variable generating such shocks the “good environment” variable. The BEGE framework has two implications for the long-run risk variance question. First, the ratio of long-run to short-term risks varies through time, as a function of the relative importance of the good and bad environment shocks and current volatility. For example, whatever the assumption on the conditional mean, it is difficult to imagine short-term risk not being higher than long-term stock market risk during most of 2008, and again now in August 2010. Second, the non-normalities imply that the variance is not a sufficient statistic to characterize risk, and we compute closed form solutions for higher order moments at all horizons as well.