Generating function method for the efficient computation of expected allocations

Abstract

Consider a risk portfolio with aggregate loss random variable S = X 1 + • • • + X n defined as the sum of the n individual losses X 1 , . . . , X n . The expected allocation, E[X i × 1 {S=k} ], for i = 1, . . . , n and k ∈ 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.

A fixed point approach for computing actuarially fair Pareto optimal risk-sharing rules

“…When using the FFT algorithm for a discrete random variable, one important factor to consider is selecting a truncation point that is large enough such that f S (k max ) = 0 for k max the maximal value of S. A large value of k max may be required if S represents a large portfolio or if the individual risks have heavy tails. In order to apply the generating function approach (see Blier-Wong et al (2022) for more details of this approach) with the FFT algorithm (or other efficient convolution algorithms) to continuous random variables, it is necessary to discretize their continuous cumulative distribution functions with a step size δ ∈ R + . For a brief overview of the upper, lower, and mean preserving discretization methods and their applications using the FFT algorithm, we refer to Section 2 of Embrechts and Frei (2009).…”