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Consider a risk portfolio with aggregate loss random variable $S=X_1+\dots+X_n$ defined as the sum of the $n$ individual losses $X_1, \dots, X_n$. Theexpected allocation, $E[X_i \times 1_{\{S = k\}}]$, for $i = 1, \dots, n$ and$k \in \mathbb{N}$, is a vital quantity for risk allocation and risk-sharing.For example, one uses this value to compute peer-to-peer contributions underthe conditional mean risk-sharing rule and capital allocated to a line ofbusiness under the Euler risk allocation paradigm. This paper introduces anordinary generating function for expected allocations, a power seriesrepresentation of the expected allocation of an individual risk given the totalrisks in the portfolio when all risks are discrete. First, we provide a simplerelationship between the ordinary generating function for expected allocationsand the probability generating function. Then, leveraging properties ofordinary generating functions, we reveal new theoretical results onclosed-formed solutions to risk allocation problems, especially when dealingwith Katz or compound Katz distributions. Then, we present an efficientalgorithm to recover the expected allocations using the fast Fourier transform,providing a new practical tool to compute expected allocations quickly. Thelatter approach is exceptionally efficient for a portfolio of independentrisks.

Generating function method for the efficient computation of expected allocations