Portfolio optimization under a minimax rule revisited

In this paper, we revisit the bi-criteria portfolio optimization model where the short selling is permitted, and a trade-off is sought between the expected return rate of a portfolio and the maximum of the uncertainty measured by a general deviation measure for all the investments comprising a portfolio. We solve this bi-criteria model by first converting it into a collection of weighted sum piecewise linear convex programs, and then analysing their optimality conditions. We not only provide explicit analytical formulas for all the efficient portfolios, but also explore as a whole the set of all the efficient portfolios and its structure such as dimensionality and distribution. We generalize the classical Two-fund Theorem by providing some collections of finitely many efficient portfolios to generate or estimate the set of all the efficient portfolios. We also notice that our efficient portfolios are almost the risk parity ones in the sense that the risks are allocated equally across the investments. Moreover, we illustrate the reliability of our model by carrying out Monte Carlo simulations to test the performance of some efficient portfolios versus inefficient ones.