Rencontres Mutualistes

Arthur Charpentier, Rencontres Mutualistes - November 2018
Segmentation et mutualisation,
les deux faces d’une même pièce ?
A. Charpentier (Université du Québec à Montréal)
Rencontres Mutualistes, Beaune, November 2018.
@freakonometrics freakonometrics freakonometrics.hypotheses.org 1

A. Charpentier (Université du Québec à Montréal)
Professor Mathematics Department, Université du Québec à Montréal
previously Econ. Dept, Université de Rennes & ENSAE Paristech
actuary in Hong Kong, IT & Stats FFA
director Data Science for Actuaries Program, Institute of Actuaries
PhD in Statistics (KU Leuven), Fellow of the Institute of Actuaries
MSc in Financial Mathematics (Paris Dauphine) & ENSAE
Research Chair :
ACTINFO (valorisation et nouveaux usages actuariels de l’information)
Editor of the freakonometrics.hypotheses.org’s blog
Editor of Computational Actuarial Science, CRC
Author of Mathématiques de l’Assurance Non-Vie (2 vol.), Economica

Insurance, “segmentation” & “mutualisation”
Insurance is the contribution of the many
to the misfortune of the few
• what is actuarial pricing ?
• why a “spirale de la segmentation” ?
• how to compare actuarial models ?
• ﬁeld experiment : pricing games

Actuarial Pricing in a Nutshell
Consider some portfolio with n insured

Consider some portfolio with n insured

The n insured have heterogeneous unobservable risk factors ω1, · · · , ωn

Nevertheless, they have characteristics x1, · · · , xn (that can approximate ωi’s)

e.g. x1 can be the color of the car, x2 can the gender, etc.
Those characteristics are observable, a priori.

Actuaries can use a regression tree to create homogeneous rate classes (or GLMs)
Here classes are based on x1 (only)

Actuaries can use a regression tree to create homogeneous rate classes (or GLMs)
But classes can be based on x1 and x2 – see Bailey (1963)

Risk Transfert without Segmentation
Insured Insurer
Loss E[S] S − E[S]
Average Loss E[S] 0
Variance 0 Var[S]
All the risk - Var[S] - is kept by the insurance company.
Remark: all those interpretation are discussed in Denuit & Charpentier (2004).

How can we visualize the goodness of a model ?
Source : https://www.progressive.com/jobs/analyst-program/

Constructing the (pseudo)-Lorenz curve
Sort the n risks according to the model m(x1) ≥ m(x2) ≥ · · · m(xn)

On the x-axis, xi = i/n, on the y-axis, yi =
i
j=1 yj/
n
j=1 yj

Connect points (xi, yi)

see Frees, Meyers & Cummins (2014).

Practice of (pseudo)-Lorenz curves
What if m and m are not perfectly correctly correlated...?

What is the “average” model ?
What is this “average pricing” ?

Can it be worst than the “average” model ?
Is it a lower bond ? Is it possible to be below that curve ?

What is in the upper corner ?
What is the upper bond ? Ex-post pricing...

How to understand this (pseudo)-Lorenz curve ?
Is there a continuity between mutualization and hyper-segmentation ?

Insurance, Risk Pooling and Solidarity
Consider ﬂood risk
One can look at the “Lorenz curve”

Field experiment: the actuarial pricing games
Actuarial pricing is data based, and model based
To understand how model inﬂuence pricing
we ran some actuarial pricing games

Insurance Ratemaking Before Competition
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p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23
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