Mean-risk portfolio optimization problems with risk-adjusted measures

We consider the problem of optimizing a portfolio of finitely many assets whose returns are described by a joint discrete distribution. We formulate the mean-risk model, using as risk functions the semideviation and weighted deviation from quantile. Using representation theorems from convex analysis, we write the portfolio problem equivalently as a zero-sum matrix game, and provide convex optimization techniques for its solution. A new set of risk-adjusted probability measures is derived from the optimal saddle point solution of the game.;The risk-adjusted probability measures can be used to evaluate portfolio performance. An illustrative example is provided in which these measures are derived for a portfolio of 200 assets, and are used to evaluate a market portfolio and optimal risk-averse portfolio. The results suggest the mean-risk portfolio is more robust than a market portfolio.;We extend the above mean-risk model to the two-stage portfolio problem, where there are two investment periods and the option to rebalance in between. The resulting model is a two-stage stochastic programming problem, with mean-risk objectives in each stage. First and second stage risk-adjusted probability measures are derived in a similar fashion to the one investment period problem.;Using as risk functions semideviation and weighted deviation from quantile in both stages, we calculate the risk adjusted measures in a numerical example with 100 assets. These measures are used to evaluate a two-stage market portfolio and optimal risk-averse portfolio.;We extend the cutting-plane and multi-cut algorithms for solving linear two-stage stochastic problems to the two-stage mean-risk portfolio problem. The two-stage portfolio problem is also formulated as one large linear program. We provide an illustrative example, where a two-stage portfolio problem with risk functions semideviation and weighted deviation from quantile is solved, using these two methods and the simplex method. The performance of these three methods is compared for solving the portfolio problem. On large examples, the extended cutting-plane and multi-cut plane algorithms solve where the linear program fails.