Time optimal portfolio selection – a survey of basic ideas
Portfolio selection is a core subject in finance, which aims to find the optimal combination of risky financial assets for an investor with given preferences. Classic portfolio selection models, following the Nobel price winning work of Markowitz, require typically that the investor can specify his investment horizon. As a consequence, the duration of the investment is fixed, and risk and return are characterized via the probability distribution of value of the investment portfolio at the given point in time. Time optimal portfolio selection takes an alternative approach, as many investors in real life have difficulty to specify an investment horizon, but can specify their goal in monetary term, like to reach 200 000 Euro from an investment of 100000 Euro. In such a case, the essential question is how long it takes the investment to reach the set goal value. As a consequence, risk is no measured based on the the probability distribution of value of the investment, but on the probability distribution of the time needed to reach the goal. For risky investments, this time is a random variable, and the resulting probability distribution a so called first passage time distribution. The talk surveys the concept as developed by the author, presents some key results, and hints at some interesting mathematical questions related to improvements of the existing models and potential applications of stochastic optimal control techniques in this context.