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On Merton's Optimal Portfolio Problem under Sporadic Bankruptcy

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  • Yaacov Kopeliovich
  • Michael Pokojovy

Abstract

Consider a stock market following a geometric Brownian motion and a riskless asset continuously compounded at a constant rate. Assuming the stock can go bankrupt, i.e., lose all of its value, at some exogenous random time (independent of the stock price) modeled as the first arrival time of a homogeneous Poisson process, we study the Merton's optimal portfolio problem consisting of maximizing the expected logarithmic utility of the total wealth at a preselected finite maturity time. First, we present a heuristic derivation based on a new type of Hamilton-Jacobi-Bellman equation. Then, we formally reduce the problem to a classical controlled Markovian diffusion with a new type of terminal and running costs. A new version of Merton's ratio is rigorously derived using Bellman's dynamic programming principle and validated with a suitable type of verification theorem. A real-world example comparing the latter ratio to the classical Merton's ratio is given.

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  • Yaacov Kopeliovich & Michael Pokojovy, 2024. "On Merton's Optimal Portfolio Problem under Sporadic Bankruptcy," Papers 2403.15923, arXiv.org.
    • Handle: RePEc:arx:papers:2403.15923
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    References listed on IDEAS

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