In this paper we present a robust optimization approach to portfolio management under uncertainty that (i) builds upon the well-established Lognormal model for stock prices while addressing its limitations, and (ii) incorporates the imperfect knowledge on the true distribution of the continuously compounded rates of return, i.e., the increments of the logarithm of the stock prices, in an intuitive manner. Real-life data suggests that a Gaussian distribution for those parameters, although convenient from a mathematical standpoint, does not necessarily provide a good fit for the historical stock prices, because the tails of the distribution tend to be fatter than what the Lognormal model indicates. In contrast, our approach, which we call Log-robust in the spirit of the Lognormal model, presents the advantage of not requiring any distributional assumption. Our objective is to maximize, in a static framework, the worst-case portfolio value at the end of the time period; short sales are not allowed. We formulate the robust problem as a linear programming problem, which can be solved efficiently, and derive theoretical insights into the worst-case uncertainty and the optimal allocation. We then compare in numerical experiments the Log-robust approach with the traditional robust approach, where range forecasts are used for the stock returns themselves. Our results indicate that the Log-robust approach significantly outperforms the benchmark with respect to 95% or 99% Value-at-Risk. This is because the traditional robust approach leads to portfolios that are far less diversified.
Technical Report, Lehigh University, Department of Industrial and Systems Engineering, 2007.