Secretarial Investment Models

F. Thomas Bruss and Thomas S. Ferguson


How should we invest capital into a sequence of investment opportunities, if, for reasons of external competition, our interest focuses on trying to invest in the very best opportunity? We introduce new models to answer such questions. Our objective is to formulate them in a way that makes results high-risk specific in order to present true alternatives to other models. At the same time we try to keep them applicable in quite some generality, also for different utility functions. Viewing high risk situations we assume that an investment on the very best opportunity yields a lucrative, possibly time-dependent, rate of return, that uninvested capital keeps its risk-free value, whereas ``wrong'' investments lose their value. Several models are presented, mainly for the so-called rank-based case. Optimal strategies and values are found, also for different utility functions, and several examples are explicitly solved. We also include results for the so-called full-information case, where, in addition, the quality distribution of investment opportunities is supposed to be known. In addition we present tractable models for an unknown number of opportunities in terms of Pascal arrival processes. Effort is made throghout the paper to justify assumptions in the view of applicability.