1 M ay 2 00 7 Mutual Fund Theorems when Minimizing the Probability of Lifetime Ruin

The contribution of this paper is two-fold. First, we show that mutual fund theorems hold when minimizing the probability of lifetime ruin (that is, wealth reaching zero before death), as they do when maximizing the utility of consumption (Merton, 1971). Bayraktar and Young (2007a) determine when the investment strategies are identical under the two problems of maximizing utility of consumption or minimizing the probability of lifetime ruin. They show that a necessary condition is that the utility exhibit hyperbolic risk aversion, a commonly used utility function in mathematical finance. Therefore, the present paper extends the work of Bayraktar and Young (2007a) in further tying together the areas of utility maximization and ruin probability minimization by showing that mutual fund theorems hold for the latter problem. See Bayraktar and Young (2007b) for motivation and additional references concerning minimizing the probability of lifetime ruin as a criterion for optimization. Second, we show that when consumption is random, then the optimal investment strategy can be expressed in terms of investing in two risky mutual funds–that is, a two-fund theorem holds with both funds risky. Then, once the problem is reduced to minimizing the probability of lifetime ruin in the presence of two risky assets, one can refer to Bayraktar and Young (2007c) who provide the solution to this problem. The two-fund result when consumption is random is implicit in the work of Duffie et al. (1997) in the context of utility maximization with random income; however, they do not state it explicitly. We do not obtain these mutual fund theorems in the most general setting; see Khanna and Kulldorff (1999) for more general mutual fund theorems when maximizing expected utility. For clarity of presentation, we apply the financial market of Björk (2004, Section 19.7), although we fully expect that mutual fund theorems hold when minimizing the probability of lifetime ruin in the financial markets of Khanna and Kulldorff (1999). The remainder of the paper is organized as follows: In Section 2, we describe the financial market in which the decision maker invests, namely, n risky assets either with or without a riskless asset; we define the probability of lifetime ruin; and we define what we mean by a relative portfolio vector and the corresponding mutual fund. In Section 3, we consider the case for which the