On the Expected Maximum Deficit and the Optimal Allocation of Reserves

Claude Lefevre and Pierre Zuyderhoff

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Abstract: Let $L$ be a c\`adl\`ag net-loss process and $M_t=\sup_{0\le s\le t}L_s$. We study the distorted expected maximum deficit $$ D_g^{(t)}(u)=\int_u^\infty g\!\left(P(M_t>v)\right) d v, $$ which combines the probability and severity of liquidity shortfalls over a fixed horizon. For concave distortions, the distorted expectation of the running maximum defines a coherent process risk measure. We then introduce two implicit capital requirements: a fixed-tolerance convex measure and a proportional-tolerance coherent measure. Their ordering properties and closed-form expressions are derived for the compound-Poisson model with exponential claims. We also study reserve allocation across multiple business lines. Minimizing the sum of line-specific deficits equalizes marginal distorted ruin probabilities and, under a common increasing distortion, yields an allocation invariant to the distortion. A worst-line deficit criterion produces a dependence-sensitive convex allocation problem, with explicit results for two independent exponential lines. Finally, we examine conditional updating: the distorted running-maximum evaluation satisfies a supermartingale inequality, while the implicit capital requirements retain an expected frozen-capital adequacy property. We also indicate how the static allocation criteria can be reapplied at discrete review dates using conditional future-increment laws and an externally given capital budget.

Date: 2026-05, Revised 2026-08
New Economics Papers: this item is included in nep-mac and nep-rmg
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