  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.
