Robust Portfolio Selection with and without Relative Entropy

We analyze two robust portfolio selection models, where a mean-variance investor considers possible deviations from a reference distribution of asset returns, adopting a maxmin criterion. The two models differ in the metric used to measure the distance between the reference distribution of asset returns and the alternative probability distributions. In the first model, where relative entropy is used as a measure of distance between distributions, an observational equivalence result obtains, whereby introducing robustness is equivalent to increasing risk aversion and, therefore, the percentage composition of the optimal portfolio of risky assets is equal to that of the optimal portfolio held by an investor without concerns for robustness. In the second model, introducing an alternative measure of distance between distributions, we show that observational equivalence ceases to hold and the proportions between risky assets are altered. We exploit the natural game-theoretic interpretation of the maxmin setting to illustrate the differences between the two models.