1 SCOR Paper 43 - Parametric divisibility of stochastic losses Oskar Laverny12

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SCOR Paper #43 - Parametric divisibility of stochastic losses

Oskar Laverny1,2*

Alessandro Ferriero2

Ecaterina Nisipasu2

Parametric divisibility

of stochastic losses

#43

June 2022

1Insitut Camille Jordan, UMR 5208,

Université Claude Bernard, Lyon 1, France

2SCOR SE, France

*Corresponding author: oskar.laverny@gmail.com

SCOR Paper #43 - Parametric divisibility of stochastic losses

Abstract

A probability distribution is n-divisible if its nth convolution root exists. While modeling the

dependence structure between several (re)insurance losses by an additive risk factor model,

the infinite divisibility, that is the n-divisibility for all , is a very desirable property.

Moreover, the capacity to compute the distribution of a piece (i.e., a convolution root) is also

desirable. Unfortunately, if many useful distributions are infinitely divisible, computing the

distributions of their pieces is usually a challenging task that requires heavy numerical

computations. However, in a few selected cases, particularly the Gamma case, the extraction

of the distribution of the pieces can be performed fully parametrically, that is with negligible

numerical cost and zero error. We show how this neat property of Gamma distributions can be

leveraged to approximate the pieces of other distributions, and we provide several illustrations

of the resulting algorithms.

SCOR Paper #43 - Parametric divisibility of stochastic losses

Contents

1. Introduction ....................................................................................... 4

2. Parametric divisibility ......................................................................... 5

3. Approximation schemes .................................................................. 10

3-a. The Gamma Approximation ...................................................... 10

3-b. Matching more moments with a wider class of distributions ...... 11

4. Illustrative examples ........................................................................ 12

4-a. Division of the Pareto distribution ............................................. 13

4-b. Approximate sampling from a risk factor model ........................ 14

5. Conclusion ...................................................................................... 16

References ............................................................................................. 17

SCOR Paper #43 - Parametric divisibility of stochastic losses

1. Introduction

Insurance and reinsurance risks are classically modeled via positive random variables

representing loss amounts, whose distributions are estimated from empirical data

and/or specific information about the underlying process producing these losses (such

as contract details or physical phenomena). Examples of the wide range of literature

on modeling methods and associated tools can be found in the Reference section [1–

5] at the end of this document. We consider here the internal modeling point of view,

where the distributions of several losses ,...,

are supposed to be known, but the dependence

structure between them must still be evaluated

and taken into account to assess the variability

and the extremal behavior of the total loss





 .

Setting such a dependency is an important but

complex matter. Important, since the

diversification effect between the ’s and their

potential tail dependencies might induce

drastically different behaviors for the total risk,

especially in the extremes. Complex, since the

quality and quantity of available information is

usually not very good.

When there is relevant data about the joint

behavior of the marginals, we rely on

parametric estimations of copulas. However,

when the quantity and/or quality of available

data is not sufficient to produce relevant

estimations of copulas, we must take the

viewpoints of experts into account. Experts’

viewpoints might not be in a format adapted to

deducing the parametrization of a given copula

model, since the interpretability of a Clayton’s

, for example, is hard to grasp. Let alone

choosing between several parametric families.

To handle and aggregate several of these

experts’ viewpoints, we consider the use of an

additive risk factor model. These models

manage the dependencies by using latent risk

factors, which are allowed to produce losses in

each of the marginals. More formally, consider

that there exist uniform random variables

,...,, all independent of each other, such that the random variables , . . .  are

each written as:

=,



 . (1)

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1SCORPaper#43-ParametricdivisibilityofstochasticlossesOskarLaverny1,2*AlessandroFerriero2EcaterinaNisipasu2Parametricdivisibilityofstochasticlosses#43June20221InsitutCamilleJordan,UMR5208,UniversitéClaudeBernard,Lyon1,France2SCORSE,France*Correspondingauthor:oskar.laverny@gmail.com2SCORPaper#43-Para...

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1 SCOR Paper 43 - Parametric divisibility of stochastic losses Oskar Laverny12

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