On minimising a portfolio's shortfall probability

TL;DR

This paper derives a tight lower bound on the decay rate of the probability that a portfolio underperforms a benchmark over a long horizon, considering complex economic factors modeled by stochastic processes.

Contribution

It introduces a novel asymptotic bound on shortfall probability decay rates for portfolios influenced by nonlinear economic factors, with epsilon-optimal portfolios constructed.

Findings

Established a tight lower asymptotic bound on shortfall probability decay.

Demonstrated epsilon-optimal portfolios under general Ito process models.

Applicable to portfolios with securities driven by geometric Brownian motions and complex economic factors.

Abstract

We obtain a lower asymptotic bound on the decay rate of the probability of a portfolio's underperformance against a benchmark over a large time horizon. It is assumed that the prices of the securities are governed by geometric Brownian motions with the coefficients depending on an economic factor, possibly nonlinearly. The economic factor is modelled with a general Ito equation. The bound is shown to be tight. More specifically, epsilon-optimal portfolios are obtained under additional conditions.