Compositional risk capital allocations

Risk capital allocation refers to the problem of disaggregating the total capital requirement of a complex organization, such as a financial or insurance company, into additive contributions from its various units. The capital shares corresponding to each unit can be viewed as quantitative descriptions of the components of a whole, subject to a fixed sum constraint (full allocation). This interpretation suggests interesting connections between capital allocation principles and Compositional Data (CoDa) analysis. Prioritizing the compositional perspective, we propose a new optimality criterion for proportional risk capital allocations in insurance contexts. Our criterion requires that capital shares assigned to individual units should be "sufficiently close" to the corresponding loss proportions in a metric that is compatible with the Aitchison distance on the simplex. We solve this optimization problem under risk scenarios aligned with managerial concerns at the corporate level and we study the behavior of the resulting compositional allocations in alternative situations, reflecting different distributional assumptions and dependencies among risks. The outcomes of our numerical studies, including an empirical application to a public database of cyber-related losses, suggest that compositional risk capital allocations can offer a valuable alternative to traditional methods, particularly in scenarios involving heavy-tailed risks or requiring flexible allocation rules based on proportional rather then absolute contributions from the various risk sources.