Consider a risk portfolio with aggregate loss random variable $S=X_1+\dots +X_n$ defined as the sum of the $n$ individual losses $X_1, \dots, X_n$. The expected 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 under the conditional mean risk-sharing rule and capital allocated to a line of business under the Euler risk allocation paradigm. This paper introduces an ordinary generating function for expected allocations, a power series representation of the expected allocation of an individual risk given the total risks in the portfolio when all risks are discrete. First, we provide a simple relationship between the ordinary generating function for expected allocations and the probability generating function. Then, leveraging properties of ordinary generating functions, we reveal new theoretical results on closed-formed solutions to risk allocation problems, especially when dealing with Katz or compound Katz distributions. Then, we present an efficient algorithm to recover the expected allocations using the fast Fourier transform, providing a new practical tool to compute expected allocations quickly. The latter approach is exceptionally efficient for a portfolio of independent risks.

翻译：考虑一个风险组合，其总损失随机变量 $S=X_1+\dots +X_n$ 定义为 $n$ 个个体损失 $X_1, \dots, X_n$ 之和。期望分配量 $E[X_i \times 1_{\{S = k\}}]$（其中 $i = 1, \dots, n$，$k \in \mathbb{N}$）是风险分配与风险分担中的关键量。例如，该值可用于计算条件均值风险分担规则下的点对点贡献，以及欧拉风险分配范式下分配给各业务线的资本。本文针对所有风险均为离散的情形，引入了一种关于期望分配量的普通生成函数，即个体风险在给定组合总风险下的期望分配量的幂级数表示。首先，我们给出了期望分配量的普通生成函数与概率生成函数之间的简单关系。进而，利用普通生成函数的性质，我们揭示了风险分配问题闭式解的新理论结果，尤其适用于Katz分布或复合Katz分布的情形。最后，我们提出了一种基于快速傅里叶变换的高效算法来恢复期望分配量，为快速计算期望分配量提供了新的实用工具。该算法对于独立风险组合尤为高效。