Minimax Portfolio Optimization Under Interval Uncertainty

In the 1950s, Markowitz proposed to combine different investment instruments to design a portfolio that either maximizes the expected return under constraints on volatility (risk) or minimizes the risk under given expected return. Markowitz’s formulas are still widely used in financial practice. However, these formulas assume that we know the exact values of expected return and variance for each instrument, and that we know the exact covariance of every two instruments. In practice, we only know these values with some uncertainty. Often, we only know the bounds of these values – i.e., in other words, we only know the intervals that contain these values. In this paper, we show how to select an optimal portfolio under such interval uncertainty. 1 Formulation of the Problem Variety of investments. There are different ways to invest money: we can deposit the money in a bank, we can buy stocks or bonds, we can buy securities, derivatives, and other financial instruments. Most investments come with risk: stocks or bounds can decrease their values, companies can go bankrupt, etc. Usually, the less risky investments – such as depositing money in a bank – are the least profitable, while the most profitable schemes – such as investing in promising start-ups – are the most risky ones. Every investor has a certain tolerance to risk, so he/she would like select