Market Crashes , Correlated Illiquidity , and “ Flight to Quality ” ∗

Models with i.i.d. returns and perfect liquidity (e.g., Merton (1971)) suggest that after a stock price crash, investors should buy more stock. However, many investors sell stock to buy safe assets after a market crash (a “flight-to-quality”), even when market liquidity has significantly worsened. In this paper, we propose a tractable and flexible portfolio selection model where market crashes can trigger changes in the investment opportunity set (e.g., lower market liquidity, higher volatility, or greater probability for further crashes). In contrast to the standard theory, our model implies that “flight-to-quality” after a crash may well be optimal, even when the market becomes illiquid and stock prices are low. In addition, we find that large price movements might help reduce rebalancing costs. We also develop a simple iterative numerical solution procedure that can be applied to a more general class of problems. Journal of Economic Literature Classification Numbers: D11, D91, G11, C61. The recent financial crisis highlights several potentially important fundamental elements for optimal portfolio selection. First, event risks such as a market crash may be significant; second, market liquidity may dry up after a crash; third, the probability of another crash may increase after a crash; and fourth, other investment opportunity set parameters (e.g., market volatility) may also change after a crash. However, the optimal trading strategy in the presence of market crashes that can trigger changes in the investment opportunity set has not been studied in the existing literature. In this paper, we develop a flexible portfolio selection workhorse model for a small investor that incorporates correlated market crashes and changes in the investment opportunity set. For example, both liquidity and volatility may change after a crash and crashes themselves may be correlated. This model captures the essence of all the above-mentioned important features, but still remains tractable. More specifically, we consider the optimal trading strategy of a constant relative risk averse (CRRA) investor who derives utility from terminal wealth and can trade a riskless asset and a risky stock continuously. Stock price crashes in a liquid regime can trigger switching into an illiquid regime where other parameters such as crash intensity, expected return, and volatility can also change. Similarly, large upward price jumps in the illiquid regime can trigger regime switching into the liquid regime. Because of the possibility of price jumps, the coupled Hamilton-Jacobi-Bellman (HJB) equations become integro-differential equations, which makes our problem much more difficult to solve even numerically than that of Jang, Koo, Liu, and Loewenstein (2007), who did not consider event risks. Remarkably, we are able to We take these changes after a market crash as exogenously given. There is a large literature on why liquidity and other parameters may change after a crash. See for example, Geanakoplos (2003) and Diamond and Rajan (2009). Our model can also be consistent with a model where investors learn from crashes and update their beliefs about the investment opportunity set after a crash.