Maximizing The Growth Rate Under Risk Constraints

The authors investigate the ergodic problem of growth-rate maximization under a class of risk constraints in the context of incomplete, Ito-process models of financial markets with random ergodic coefficients. Including Value-at-Risk (VaR), Tail-Value-at-Risk (TVaR), and 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.