Shortfall Minimizing Portfolios

Many institutional and private investors seek for a long run excess return relative to a reference strategy (e.g. money market, bond index, etc.) which they want to attain under a minimal shortfall probability. In this article it is shown that even in the long run in order to attain a substantial excess return a high shortfall probability has to be accepted. In the model the prices of the assets follow geometric Brownian motions. Two types of a shortfall are distinguished. A shortfall of type I occurs, if at some point of time the investment goal is missed by a given percentage. There is a shortfall of type II, if the investment goal is missed at the end of the planning horizon. To begin with, only constant portfolio weights are admitted. For both types it can be shown that minimizing the shortfall probability under a given excess return is equivalent to the Merton problem. Under realistic parameter values moderate shortfall probabilities are only compatible with very low excess returns. Finally, it is shown, that "Constant Proportion Portfolio Insurance" does not lead to a reduction of the shortfall probability.