Optimal capital growth with convex shortfall penalties

The optimal capital growth strategy or Kelly strategy, has many desirable properties such as maximizing the asympotic long run growth of capital. However, it has considerable short run risk since the utility is logarithmic, with essentially zero Arrow-Pratt risk aversion. It is common to control risk with a Value-at-Risk constraint defined on the end of horizon wealth. A more effective approach is to impose a VaR constraint at each time on the wealth path. In this paper we provide a method to obtain the maximum growth while staying above an ex-ante discrete time wealth path with high probability, where shortfalls below the path are penalized with a convex function of the shortfall. The effect of the path VaR condition and shortfall penalties is less growth than the Kelly strategy, but the downside risk is under control. The asset price dynamics are defined by a model with Markov transitions between several market regimes and geometric Brownian motion for prices within regime. The stochastic investment model is reformulated as a deterministic program which allows the calculation of the optimal constrained growth wagers at discrete points in time.