On the Expected Maximum Deficit and the Optimal Allocation of Reserves

Abstract

Let $L=(L_s)_{0\le s\le t}$ be a cumulative net-loss process and let $M_t=\sup_{0\le s\le t}L_s$. For a candidate reserve $u$ and a distortion function $g$, define $D_g^{(t)}(u)=\int_u^\infty g(P(M_t>v))d v$. This function measures the tail-weighted residual severity of the largest cumulative loss over the horizon. We derive three monetary risk measures: its value at zero reserve and two measures based on fixed and proportional deficit tolerances. Under a concave distortion, the zero-reserve and proportional-tolerance measures are coherent, while the fixed-tolerance measure is convex. Given a reserve budget distributed among several units, we analyze two allocation criteria: the sum of the units' distorted deficits, which depends only on their marginal distributions, and a distorted expectation of the largest residual unit deficit, which uses their joint distribution. We characterize the optimal allocations and corresponding near-optimal sets. A two-unit exponential benchmark shows that the optimum can be unique and unequal. We then apply both criteria to the Building, Contents and loss-of-Profits components of the public Danish large-fire data. Under the baseline predictive model, both yield interior allocations and localized 0.5\%-optimal regions, while sensitivity analyses illustrate model uncertainty. We also provide Monte Carlo estimators, conditional counterparts and a review-date allocation formulation.