Optimal strategies in high risk investments

Abstract

A decision-maker observes sequentially a given permutation of $n$ uniquely rankable options. He has to invest capital into these opportunities at the moment when they appear. At each step only relative ranks are known. At the end the true rank of the option, at which the investment has been made, is known. [Bruss and Ferguson] have considered such problems under the assumption that an investment on the very best opportunity yields a lucrative, possibly time-dependent, rate of return. Uninvested capital keeps its risk-free value. Wrong investments lose their value. In this paper we partially extend results by [Bruss and Ferguson]. We confine our study to linear utility but a wider range of payoffs is taken into account. Two cases are considered. The first-type payoff gives a positive rate of return if the investment is made on the best or the second best option. The second-type payoff pays when the investment is at the second best option. We motivate these payoff choices. A few examples are explicitly solved.