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A Laplace-based perspective on conditional mean risk sharing

Christopher Blier-Wong

Abstract

The conditional mean risk-sharing (CMRS) rule is an important tool for distributing aggregate losses across individual risks, but its implementation in continuous multivariate models typically requires complicated multidimensional integrals. We develop a framework to compute CMRS allocations from the joint Laplace--Stieltjes transform of the risk vector. The LSTs of the allocation measures $ν_i(B)=\mathbb{E}[X_i\boldsymbol{1}_{\{S\in B\}}]$ are expressed as partial derivatives of the joint LST evaluated on the diagonal $t_1=\cdots=t_n$. When densities exist, this yields one-dimensional Laplace inversions for $f_S$ and $ξ_i$, and hence $h_i(s)=ξ_i(s)/f_S(s)$ on the absolutely continuous part, providing closed-form or semi-analytic solutions for a broad class of distributions. We also develop numerical inversion methods for cases where analytic inversion is unavailable. We introduce an exponential tilting procedure to stabilize numerical inversion in low-probability aggregate events. We provide several examples to illustrate the approach, including in some high-dimensional settings where existing approaches are infeasible.

A Laplace-based perspective on conditional mean risk sharing

Abstract

The conditional mean risk-sharing (CMRS) rule is an important tool for distributing aggregate losses across individual risks, but its implementation in continuous multivariate models typically requires complicated multidimensional integrals. We develop a framework to compute CMRS allocations from the joint Laplace--Stieltjes transform of the risk vector. The LSTs of the allocation measures are expressed as partial derivatives of the joint LST evaluated on the diagonal . When densities exist, this yields one-dimensional Laplace inversions for and , and hence on the absolutely continuous part, providing closed-form or semi-analytic solutions for a broad class of distributions. We also develop numerical inversion methods for cases where analytic inversion is unavailable. We introduce an exponential tilting procedure to stabilize numerical inversion in low-probability aggregate events. We provide several examples to illustrate the approach, including in some high-dimensional settings where existing approaches are infeasible.
Paper Structure (24 sections, 9 theorems, 95 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 24 sections, 9 theorems, 95 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Transform-based CMRS representation
  3. Conditional mean risk sharing and allocation measures
  4. A Laplace representation of CMRS
  5. Dealing with atoms in numerical inversion
  6. Allocation proportions and transform-level diagnostics
  7. Closed-form CMRS from the LST
  8. Frailty models and Archimedean dependence
  9. Exponential dispersion family and Tweedie margins under frailty
  10. Mixed exponentials under frailty
  11. Matrix-exponential margins
  12. Independent compound Katz distributions
  13. Numerical Laplace inversion for CMRS
  14. The Gaver--Stehfest inversion formula
  15. Complex-plane Euler inversion
  16. ...and 9 more sections

Key Result

Theorem 1

Let $X_1,\dots,X_n$ be nonnegative random variables, $S=\sum_{j=1}^n X_j$, $\mu_S$ the law of $S$, and $\nu_i$, $\mathcal{L}_{\bm X}$ as defined in eq:alloc-measures-intro--eq:joint-lst. Then, the following hold:

Figures (3)

  • Figure 1: Common-shock compound Poisson: closed-form benchmark, GS, and Euler inversion with $\theta=0.2$ and $\theta=0$. Note that we show the curves as faded once the allocation-sum diagnostic breaks down, which occurs at different values of $s$ across methods.
  • Figure 2: Scaling of full-grid mean runtime (seconds) under the same runtime basis as Table \ref{['tab:compound-poisson-runtime']}.
  • Figure 3: Independent lognormals, $s\in[0.5,25]$: GS (untilted) versus Euler inversion with $\theta=0.8$ and $\theta=0$. Panels show the three component allocations $h_i(s)$ and the allocation-sum diagnostic $\sum_i h_i(s)$ against the identity $y=s$.

Theorems & Definitions (23)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Remark 1
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Example 1
  • Example 2
  • ...and 13 more