ar X iv : 0 70 6 . 04 80 v 1 [ q - fi n . PM ] 4 J un 2 00 7 Maximizing the Growth Rate under Risk Constraints 1 December 3 , 2008

We investigate the ergodic problem of growth-rate maximization under a class of risk constraints in the context of incomplete, Itô-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.