Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin

We consider a rm that is faced with an uncontrollable stochastic cash ow, or random risk process. There is one investment opportunity, a risky stock, and we study the optimal investment decision for such rms. There is a fundamental incompleteness in the market, in that the risk to the investor of going bankrupt can not be eliminated under any investment strategy, since the random risk process ensures that there is always a positive probability of ruin (bankruptcy). We therefore focus on obtaining investment strategies which are optimal in the sense of minimizing the risk of ruin. In particular, we solve for the strategy that maximizes the probability of achieving a given upper wealth level before hitting a given lower level. This policy also minimizes the probability of ruin. We prove that when there is no risk-free interest rate, this policy is equivalent to the policy that maximizes utility from terminal wealth, for a xed terminal time, when the rm has an exponential utility function. This validates a longstanding conjecture about the relation between minimizing probability of ruin and exponential utility. When there is a positive risk-free interest rate, the conjecture is shown to be false.