Maximizing the Growth Rate under Risk Constraints

TL;DR

This paper studies how to maximize long-term growth rates in financial markets while adhering to various risk constraints, providing explicit optimal strategies that adapt to market conditions.

Contribution

It introduces a method to explicitly determine optimal growth strategies under risk constraints in incomplete markets with ergodic coefficients.

Findings

Optimal policies exist within an admissible class.

Strategies are explicit and involve uniform scaling of unconstrained portfolios.

The optimal policy locally resembles a CRRA-investor with market-dependent risk aversion.

Abstract

We investigate the ergodic problem of growth-rate maximization under a class of risk constraints in the context of incomplete, It\^{o}-process models of financial markets with random ergodic coefficients. Including {\em value-at-risk} (VaR), {\em tail-value-at-risk} (TVaR), and {\em limited expected loss} (LEL), these constraints can be both wealth-dependent(relative) and wealth-independent (absolute). The optimal policy is shown to exist in an appropriate admissibility class, and can be obtained explicitly by uniform, state-dependent scaling down of the unconstrained (Merton) optimal portfolio. This implies that the risk-constrained wealth-growth optimizer locally behaves like a CRRA-investor, with the relative risk-aversion coefficient depending on the current values of the market coefficients.