segmentation et mutualisation en assurance

1. Arthur Charpentier, UQ`aM Economics Seminar 2019
‘Segmentation’ & ‘Mutualisation’
Arthur Charpentier (Universit´e du Qu´ebec `a Montr´eal)
UQ`aM, Economics Seminar, March 2019, Montr´eal
@freakonometrics freakonometrics freakonometrics.hypotheses.org 1

2. Arthur Charpentier, UQ`aM Economics Seminar 2019
Insurance, “segmentation” & “mutualisation”
“Insurance is the contribution of the many to the misfortune of the few ”
Insurance: risk sharing (pooling)
π = EP S1
segmentation / price differentiation
π(ω) = EP S1 Ω = ω
for some (unobservable) risk factor Ω
imperfect information
given some (observable) risk variables x
π(x) = EP S1 X = x = EPx
S1
why a “spirale de la segmentation” ?
@freakonometrics freakonometrics freakonometrics.hypotheses.org 2

3. Arthur Charpentier, UQ`aM Economics Seminar 2019
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).
@freakonometrics freakonometrics freakonometrics.hypotheses.org 3

4. Arthur Charpentier, UQ`aM Economics Seminar 2019
Risk Transfert with Segmentation and Perfect Information
Assume that information Ω is observable,
Insured Insurer
Loss E[S|Ω] S − E[S|Ω]
Average Loss E[S] 0
Variance Var E[S|Ω] Var S − E[S|Ω]
Observe that Var S − E[S|Ω] = E Var[S|Ω] , so that
Var[S] = E Var[S|Ω]
→ insurer
+ Var E[S|Ω]
→ insured
.
@freakonometrics freakonometrics freakonometrics.hypotheses.org 4

5. Arthur Charpentier, UQ`aM Economics Seminar 2019
Risk Transfert with Segmentation and Imperfect Information
Assume that X ⊂ Ω is observable
Insured Insurer
Loss E[S|X] S − E[S|X]
Average Loss E[S] 0
Variance Var E[S|X] E Var[S|X]
Now
E Var[S|X] = E E Var[S|Ω] X + E Var E[S|Ω] X
= E Var[S|Ω]
perfect pricing
+ E Var E[S|Ω] X
misfit
.
@freakonometrics freakonometrics freakonometrics.hypotheses.org 5

6. Arthur Charpentier, UQ`aM Economics Seminar 2019
Actuarial Pricing Model
Premium is E S|X = x = E
N
i=1
Yi X = x = E [N|X = x] · E [Y |X = x]
Statistical and modeling issues to approximate based on some training datasets,
with claims frequency {ni, xi} and individual losses {yi, xi}.
Use GLM to approximate E [N|X = x] and E [Y |X = x]
Recall that E E S|X = E S
How can we claim that model π(x) = E S|X = x is “good” ?
@freakonometrics freakonometrics freakonometrics.hypotheses.org 6

7. Arthur Charpentier, UQ`aM Economics Seminar 2019
How can we visualize the goodness of a model ?
In the context of classification, use ROC curve
sensitivity = TNR =
TN
TN + FP
= P[m(X) > s
Y =1
|Y = 1] vs. specificity = TPR =
TP
TP + FN
Specificity
Sensitivity
1.0 0.8 0.6 0.4 0.2 0.0
0.00.20.40.60.81.0
q
q
0.285 (0.860, 0.927)
0.500 (0.907, 0.860)
logit
bagging
random forest
boosting
Specificity
Sensitivity
1.0 0.8 0.6 0.4 0.2 0.0
0.00.20.40.60.81.0
logit
bagging
random forest
boosting
Source : Charpentier, Flachaire & Ly (2018)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 7

8. Arthur Charpentier, UQ`aM Economics Seminar 2019
How can we visualize the goodness of a model ?
Source : https://www.progressive.com/jobs/analyst-program/
@freakonometrics freakonometrics freakonometrics.hypotheses.org 8

9. Arthur Charpentier, UQ`aM Economics Seminar 2019
Constructing the (pseudo)-Lorenz curve
Sort the n risks according to the model π(x1) ≥ π(x2) ≥ · · · ≥ π(xn)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 9

10. Arthur Charpentier, UQ`aM Economics Seminar 2019
Constructing the (pseudo)-Lorenz curve
On the x-axis, xi = i/n, on the y-axis, yi =
i
j=1 yj/
n
j=1 yj
@freakonometrics freakonometrics freakonometrics.hypotheses.org 10

11. Arthur Charpentier, UQ`aM Economics Seminar 2019
Constructing the (pseudo)-Lorenz curve
Connect points (xi, yi)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 11

12. Arthur Charpentier, UQ`aM Economics Seminar 2019
Constructing the (pseudo)-Lorenz curve
see Frees, Meyers & Cummins (2014).
@freakonometrics freakonometrics freakonometrics.hypotheses.org 12

13. Arthur Charpentier, UQ`aM Economics Seminar 2019
Practice of (pseudo)-Lorenz curves
What if π and π are not perfectly correctly correlated...?
@freakonometrics freakonometrics freakonometrics.hypotheses.org 13

14. Arthur Charpentier, UQ`aM Economics Seminar 2019
Practice of (pseudo)-Lorenz curves
What if m and m are not perfectly correctly correlated...?
@freakonometrics freakonometrics freakonometrics.hypotheses.org 14

15. Arthur Charpentier, UQ`aM Economics Seminar 2019
Practice of (pseudo)-Lorenz curves
What if m and m are not perfectly correctly correlated...?
@freakonometrics freakonometrics freakonometrics.hypotheses.org 15

16. Arthur Charpentier, UQ`aM Economics Seminar 2019
Practice of (pseudo)-Lorenz curves
What if m and m are not perfectly correctly correlated...?
@freakonometrics freakonometrics freakonometrics.hypotheses.org 16

17. Arthur Charpentier, UQ`aM Economics Seminar 2019
What is the “average” model ?
What is this “average pricing” ?
@freakonometrics freakonometrics freakonometrics.hypotheses.org 17

18. Arthur Charpentier, UQ`aM Economics Seminar 2019
Can it be worst than the “average” model ?
Is it a lower bond ? Is it possible to be below that curve ?
@freakonometrics freakonometrics freakonometrics.hypotheses.org 18

19. Arthur Charpentier, UQ`aM Economics Seminar 2019
What is in the upper corner ?
What is the upper bond ? Ex-post pricing...
@freakonometrics freakonometrics freakonometrics.hypotheses.org 19

20. Arthur Charpentier, UQ`aM Economics Seminar 2019
How to understand this (pseudo)-Lorenz curve ?
Is there a continuity between mutualization and hyper-segmentation ?
@freakonometrics freakonometrics freakonometrics.hypotheses.org 20

21. Arthur Charpentier, UQ`aM Economics Seminar 2019
Insurance, Risk Pooling and Solidarity
Consider flood risk, in France
One can look at the “Lorenz curve”
@freakonometrics freakonometrics freakonometrics.hypotheses.org 21

22. Arthur Charpentier, UQ`aM Economics Seminar 2019
Price Differentiation, a Toy Example
Claims frequency N ∈ {0, 1} (average cost = 1,000)
X1
Young Experienced Senior Total
X2
Town
12%
(500)
9%
(2,000)
9%
(500)
9.5%
(3,000)
Outside
8%
(500)
6.67%
(1,000)
4%
(500)
6.33%
(2,000)
Total
10%
(1,000)
8.22%
(3,000)
6.5%
(1,000)
8.23%
(5,000)
from Charpentier, Denuit & ´Elie (2015)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 22

23. Arthur Charpentier, UQ`aM Economics Seminar 2019
Price Differentiation, a Toy Example
Y-T
(500)
Y-O
(500)
E-T
(2,000)
E-O
(1,000)
S-T
(500)
S-O
(500)
none 82.3 82.3 82.3 82.3 82.3 82.3
X1 × X2 120 80 90 66.7 90 40
market 82.3 80 82.3 66.7 82.3 40
none 82.3 82.3 82.3 82.3 82.3 82.3
X1 100 100 82.2 82.2 65 65
X2 95 63.3 95 63.3 95 63.3
X1 × X2 120 80 90 66.7 90 40
market 82.3 63.3 82.2 63.3 65 40
@freakonometrics freakonometrics freakonometrics.hypotheses.org 23

24. Arthur Charpentier, UQ`aM Economics Seminar 2019
Price Differentiation, a Toy Example
premium losses loss 99.5% Market
ratio quantile Share
none 247 285 115.4% (±8.9%) 66.1%
X1 × X2 126.67 126.67 100.0% (±10.4%) 33.9%
market 373.67 411.67 110.2% (±5.1%)
none 41.17 60 145.7% (±34.6%) 189% 11.6%
X1 196.94 225 114.2% (±11.8%) 140% 55.8%
X2 95 106.67 112.3% (±15.1%) 134% 26.9%
X1 × X2 20 20 100.0% (±41.9%) 160% 5.7%
market 353.10 411.67 116.6% (±5.3%) 130%
@freakonometrics freakonometrics freakonometrics.hypotheses.org 24

25. Arthur Charpentier, UQ`aM Economics Seminar 2019
From Econometric to ‘Machine Learning’ Techniques
In a competitive market, insurers can use different sets of variables and different
models, e.g. GLMs, Nt|X ∼ P(λX · t) and Y |X ∼ G(µX, ϕ)
πj(x) = E N1 X = x · E Y X = x = exp(αT
x)
Poisson P(λx)
· exp(β
T
x)
Gamma G(µX ,ϕ)
that can be extended to GAMs,
πj(x) = exp
d
k=1
sk(xk)
Poisson P(λx)
· exp
d
k=1
tk(xk)
Gamma G(µX ,ϕ)
or some Tweedie model on St (compound Poisson, see Tweedie (1984))
conditional on X
@freakonometrics freakonometrics freakonometrics.hypotheses.org 25

26. Arthur Charpentier, UQ`aM Economics Seminar 2019
From Econometric to ‘Machine Learning’ Techniques
(see Charpentier & Denuit (2005) or Kaas et al. (2008)) or any other statistical
model
πj(x) where πj ∈ argmin
m∈Fj :Xj →R
n
i=1
(si, m(xi))
For some loss function : R2
→ R+ (usually an L2 based loss, (s, y) = (s − y)2
since argmin{E[ (S, m)], m ∈ R} is E(S), interpreted as the pure premium).
For instance, consider regression trees, forests, neural networks, or boosting
based techniques to approximate π(x), and various techniques for variable
selection, such as LASSO (see Hastie et al. (2009) or Charpentier et al. (2017) for
a description and a discussion).
With d competitors, each insured i has to choose among d premiums,
πi = π1(xi), · · · , πd(xi) ∈ Rd
+.
@freakonometrics freakonometrics freakonometrics.hypotheses.org 26

27. Arthur Charpentier, UQ`aM Economics Seminar 2019
Machine Learning & Credit
Before discussing the use of those models in insurance, note that the same issues
exist in credit, see Hardt, Price & Srebro (2017).
“the shift from traditional to machine learning lending models may have
important distributional effects for consumers [... ] machine learning would offer
lower rates to racial groups who already were at an advantage under the
traditional model, but it would also benefit disadvantaged groups by enabling them
to obtain a mortgage in the first place ” Fuster, Goldsmith-Pinkham, Ramadorai &
Walther (2017)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 27

28. Arthur Charpentier, UQ`aM Economics Seminar 2019
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
@freakonometrics freakonometrics freakonometrics.hypotheses.org 28

29. Arthur Charpentier, UQ`aM Economics Seminar 2019
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
02004006008001000
@freakonometrics freakonometrics freakonometrics.hypotheses.org 29