A Log-Robust Optimization Approach to Portfolio Management

We present a robust optimization approach to portfolio management under uncertainty that builds upon insights gained from the well-known Lognormal model for stock prices, while addressing that model’s limitations, in particular, the issue of fat tails being underestimated in the Gaussian framework and the active debate on the correct distribution to use. Our approach, which we call Log-robust in the spirit of the Lognormal model, does not require any probabilistic assumption, and incorporates the randomness on the continuously compounded rates of return by using range forecasts and a budget of uncertainty, thus capturing the decision-maker’s degree of risk aversion through a single, intuitive parameter. Our objective is to maximize the worst-case portfolio value (over a set of allowable deviations of the uncertain parameters from their nominal values) at the end of the time horizon in a one-period setting; short sales are not allowed. We formulate the robust problem as a linear programming problem 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 applied directly to the stock returns. 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.