Doubling an investment

We study the issue of optimal long-term portfolio management in continuous time multi-asset financial markets. Rather than following the abstract notion of ‘utility’ and its implied paradigm of ‘maximization of expected utility’ we suggest a different approach: The investor sets a goal—such as reaching a desired fortune level, or doubling the initial investment—and then operates to minimize the expected time-to-goal, i.e., achieving the goal as quick as possible. We assume the ‘standard model’ of multi-asset financial markets where assets are governed by correlated Geometric Brownian motion dynamics, and study optimality under the criteria of ‘minimization of the expected time-to-goal’. We explicitly compute: (i) the optimal holding strategies; (ii) the dynamics and behavior of the optimal investment portfolios; and, (iii) the statistics—mean, variance, and Laplace transform—of the time-to-goal (under the optimal investment strategy). Also, an investment paradox arising in this context—in which some portfolios have exponential mean growth but have a positive probability of never doubling their initial value—is discussed and explained.