How Should Investors Respond to Increases in Volatility?

They should reduce their equity position. We study the portfolio problem of a long-horizon investor that allocates between a risk-less and a risky asset in an environment where both volatility and expected returns are time-varying. We find that investors, regardless of their horizon, should substantially decrease risk exposure after an increase in volatility. Ignoring variation in volatility leads to large utility losses (on the order of 35% of lifetime utility). The utility benefits of volatility timing are larger than those coming from expected return timing (i.e., from return predictability) for all investment horizons we consider, particularly when parameter uncertainty is taken into account. We approximate the optimal volatility timing portfolio and find that a simple two fund strategy holds: all investors choose constant weights on a buy-and-hold portfolio and a volatility timing portfolio that scales the risky-asset exposure by the inverse of expected variance. We then show robustness to cases where the degree of mean-reversion in stock returns co-moves with volatility over time. ∗Yale School of Management and UCLA Anderson School of Management. We thank John Campbell, John Cochrane, William Goetzmann, Ben Hebert, Jon Ingersoll, Ravi Jagannathan, Serhiy Kosak, Hanno Lustig, Justin Murfin, Stefan Nagel, Lubos Pastor, Myron Scholes, Ivan Shaliastovich, Ken Singleton, Tuomo Vuoltenahoo, Lu Zhang and participants at Yale SOM, UCLA Anderson, Stanford GSB, and Arrowstreet Capital for comments. We especially thank Nick Barberis for many useful discussions. Stock market volatility is highly variable and easily forecastable, yet it is a conventional view among many practitioners and academics that investors should sit tight and not sell after increases in volatility which typically follow market downturns. Furthermore, it is often argued that long-term investors should view these high volatility periods as unique buying opportunities. In this paper, we investigate this conventional view. Specifically, we answer two questions: (1) how much volatility timing should investors do, if any, and (2) what are the utility benefits of volatility timing? Our approach is to study the portfolio problem of a long-lived investor that allocates her wealth between a risk-less and a risky asset in an environment where both volatility and expected returns are time-varying. We then provide comprehensive and quantitative answers to these questions and show how our answers depend on the investor’s horizon and their risk aversion. Importantly, our analysis also takes into account that investors’ face parameter uncertainty regarding the dynamics of volatility and expected returns. Our main finding is that investors should substantially decrease risk exposure after an increase in volatility and that ignoring variation in volatility leads to large utility losses. The benefits of volatility timing are on the order of 35% of lifetime utility for our preferred parameterization of an investor with risk aversion of 5 and a 20 year horizon. These benefits are significantly larger than those coming from expected return timing (i.e., from return predictability), particularly when parameter uncertainty is taken into account. We approximate the optimal volatility timing portfolio and find that its dependence on volatility is very simple: all investors, regardless of horizon, will choose fixed weights on a buy-and-hold portfolio that invests a constant amount in the risky-asset, and a volatility managed portfolio that scales the risky-asset exposure by the inverse of expected variance 1/σt . Further, we show that the weight on the volatility timing portfolio is independent of the investors’ horizon in our baseline results. In contrast, the weight on the buy-and-hold portfolio depends strongly on horizon and the amount of mean reversion investors’ perceive in stock returns, but doesn’t depend on the dynamics of volatility. Thus, despite an apparently complex numerical exercise, our solution turns out to be simple and intuitive. We begin our analysis by estimating a rich model for the dynamics of excess stock