Optimal guaranteed return portfolios and the casino effect

In this paper we address the problem of determining optimal portfolios that may include options in a framework of return maximization with risk constraints relative to a benchmark, as well as in terms of absolute returns. The model we propose allows for deterministic constraints as well as probabilistic constraints. We derive properties of optimal and feasible portfolios and present a linear programming model to solve the problem. The optimal portfolios have payoff functions that reflect a gambling policy. We show that optimal solutions to a large class of portfolio models that maximize expected return subject to downside risk constraints are driven by this casino effect and present tractable conditions under which it occurs in our model. We propose to control the casino effect by using chance constraints. Using results from financial theory, we formulate an LP model that maximizes expected return subject to worst-case return constraints and chance constraints on achieving prespecified levels of return. The results are illustrated with real-life data on the S&P 500 index.