Portfolio optimization under l(infinity) 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(infinity) function as a risk aversion measure. This differs from previous studies where either an l(2) function or an l(1) 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(infinity) 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 assests to be invested in accordance with the ratio of the difference in their return rates to their risks.