Portfolio optimization under l∞ risk measure

In this paper, a new model for portfolio selection is introduced to address the situation where a risk averse investor wants to minimize the maximum individual risk among assets to be invested. The model uses an l_∞ function as a risk aversion measure. This differs from previous studies where either an l₂ function or an l₁ function is suggested, which may not model the concern of very cautious investors properly. We formulate our problem as a bi-criteria piecewise linear program, where one criterion is to minimize the l_∞ risk function while the other is to maximize the total expected return. This bi-criteria optimization problem is converted into an equivalent scalarized problem with a single combined criterion. An interesting finding is that an optimal solution to the scalarized optimization problem can be derived analytically. The solution exhibits a simple structure, which selects successively assets to be invested in accordance with the ratio of the difference in their return rates to their risks