Fréchet inequalities via convex optimization

Quantifying the risk carried by an aggregate position Sd := ∑d i=1Xi comprising many risk factors Xi is fundamental to both insurance and financial risk management. Fréchet inequalities quantify the worst-case risk carried by the aggregate position given distributional information concerning its composing factors but without assuming independence. This marginal factor modeling of the aggregate position in terms of its risk factors Xi leaves, however, the distribution of Sd ambiguous. The resulting distributional ambiguity can be traced back directly to the lack of dependence structure imposed between the risk factors. Fréchet inequalities implicitly uphold already the work of George Boole on probabilistic logic but were explicitly derived by Maurice Fréchet only in 1935. These inequalities can be considered rules about how to bound calculations involving probabilities without assuming independence or, indeed, without making any dependence assumptions whatsoever. It is exactly this robustness to obscure risk dependencies which have stimulated their renewed interest in the post crisis economy. We will approach these Fréchet inequalities using a modern convex optimization lens. Our novel perspective provides data friendly computational tools applicable to practical Fréchet problems. Two important data friendly classes of Fréchet problems are presented, both of which are shown to admit exact tractable convex optimization reformulations. The efficacy of our approach is illustrated using a small insurance management problem.