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Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Actuarial Pricing GameA. Charpentier(UQAM & Université de Rennes 1)
with R. Élie & J. Jakubowicz
Paris, 100% Actuaires, November 2015.
http://freakonometrics.hypotheses.org
@freakonometrics 1
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Pricing Game, Charpentier, Denuit & Élie (2015)
@freakonometrics 2
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Pricing Game, Charpentier, Denuit & Élie (2015)
@freakonometrics 3
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Pricing Game, with a Toy Dataset
@freakonometrics 4
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
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A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
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070
0
Gender = 'Male'
@freakonometrics 5
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
Gender = 'Female'
@freakonometrics 6
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
Pre
miu
m
●●●●●●●●●●●●●
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
Age in [17,25]
@freakonometrics 7
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
Pre
miu
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A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
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Age in [30,50]
@freakonometrics 8
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
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A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
Age in [70,100]
@freakonometrics 9
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
Pre
miu
m
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A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
Category = 'Small'
@freakonometrics 10
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
Pre
miu
m
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A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
Category = 'Medium'
@freakonometrics 11
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
Pre
miu
m
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A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
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Category = 'Large'
@freakonometrics 12
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
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A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
010
020
030
040
050
060
070
0
Bonus = −50
@freakonometrics 13
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Comparing Models
Source: unkown, so far...
@freakonometrics 14
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Comparing Models
Consider an ordered sample {y1, · · · , yn} of incomes,with y1 ≤ y2 ≤ · · · ≤ yn, then Lorenz curve is
{Fi, Li} with Fi = i
nand Li =
∑ij=1 yj∑nj=1 yj
We have observed losses yi and premiums π̂(xi). Con-sider an ordered sample by the model, see Frees, Meyers& Cummins (2014), π̂(x1) ≥ π̂(x2) ≥ · · · ≥ π̂(xn), thenplot
{Fi, Li} with Fi = i
nand Li =
∑ij=1 yj∑nj=1 yj
0 20 40 60 80 100
020
4060
8010
0
Proportion (%)
Inco
me
(%)
poorest ← → richest
0 20 40 60 80 100
020
4060
8010
0
Proportion (%)
Loss
es (
%)
more risky ← → less risky
@freakonometrics 15
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Models, Brief Summary
@freakonometrics 16
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Market Competition
Decision Rule: the insured selects the cheapeast premium, ccccccccccccccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc ccccccccccccccccccccccccc cccccccccccc ccccccccccccc
A B C D E F
787.93 706.97 1032.62 907.64 822.58 603.83
170.04 197.81 285.99 212.71 177.87 265.13
473.15 447.58 343.64 410.76 414.23 425.23
337.98 336.20 468.45 339.33 383.55 672.91
@freakonometrics 17
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Market Competition
Decision Rule: the insured selects randomly from the three cheapeast premiumcccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccccccccccccccc ccccccccccccc cccccccccccc ccccccccccccc
A B C D E F
787.93 706.97 1032.62 907.64 822.58 603.83
170.04 197.81 285.99 212.71 177.87 265.13
473.15 447.58 343.64 410.76 414.23 425.23
337.98 336.20 468.45 339.33 383.55 672.91
@freakonometrics 18
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Market Competition
Decision Rule: the insured were assigned randomly to some insurancecompany for year n− 1. For year n, they stay with their company if the premiumis one of the three cheapeast premium, if not, random choice among the four
A B C D E F
787.93 706.97 1032.62 907.64 822.58 603.83
170.04 197.81 285.99 212.71 177.87 265.13
473.15 447.58 343.64 410.76 414.23 425.23
337.98 336.20 468.45 339.33 383.55 672.91
@freakonometrics 19
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Financial Results, Market Share, Method 2
●
●
●●
●
●
●
●
●
●
●
●●●
●●
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
1000
2000
3000
4000
5000
6000
Num
ber
of C
ontr
acts
@freakonometrics 20
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Financial Results, Market Share, Method 3
●
●●
●
●
●
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
2000
3000
4000
5000
Num
ber
of C
ontr
acts
@freakonometrics 21
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Financial Results, Loss Ratio, Method 2
Market Loss Ratio ∼ 154%.
●
●
●
●
●
●●
●●
●
●
●
●
●
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
100
150
200
250
Loss
Rat
io
@freakonometrics 22
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
●
−2 −1 0 1 2 3
−1.
0−
0.5
0.0
0.5
1.0
1.5
MCA factor map
Dim 1 (11.02%)
Dim
2 (
7.25
%)
Female MaleA
BCD
E
F
Large
Medium
Small
Employed
Housewife
Retired
Self−employed
Unemployed
(17,25]
(25,35]
(35,50]
(50,70]
(70,101]
A1
A10A11
A12
A13
A2
A3
A4
A5A6A7A8
A9
@freakonometrics 23
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
−5.5
−4.0
−2.0
0.0
2.0
4.0
5.4
Pearsonresiduals:
p−value =< 2.22e−16
A
Gen
der
Mal
eF
emal
e
A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9
@freakonometrics 24
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
−11
−4
−2
0
2
4
13
Pearsonresiduals:
p−value =< 2.22e−16
A
Cat
egor
yS
mal
lM
ediu
mLa
rge
A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9
@freakonometrics 25
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
−19
−4 −2 0 2 4
15
Pearsonresiduals:
p−value =< 2.22e−16
A
Occ
upat
ion
Une
mpl
oyed
Sel
f−em
ploy
edR
etire
dH
ouse
wife
Em
ploy
ed
A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9
@freakonometrics 26
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
−20
−4 0 4
48
Pearsonresiduals:
p−value =< 2.22e−16
A
Age
F(7
0,10
1](5
0,70
](3
5,50
](2
5,35
](1
7,25
]
A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9
@freakonometrics 27
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
−26
−4 −2 0 2 4
31
Pearsonresiduals:
p−value =< 2.22e−16
A
Den
sity
F(2
00,3
00]
(125
,200
](7
0,12
5](4
0,70
](1
0,40
]
A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9
@freakonometrics 28
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
−25
−4 −2 0 2 4
29
Pearsonresiduals:
p−value =< 2.22e−16
A
Bon
usF
(100
,200
](3
0,10
0](0
,30]
(−10
,0](−
30,−
10]
(−49
,−30
][−
50,−
49]
A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9
@freakonometrics 29
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
−7.5
−4.0
−2.0
0.0
2.0
4.0
8.7
Pearsonresiduals:
p−value =< 2.22e−16
A
Pol
durF
(10,
16]
(5,1
0](1
,5]
[0,1
]
A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9
@freakonometrics 30
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Market Competition, A2
No segmentation, unique priceA1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13
Mar
ket S
hare
(in
%)
02
46
810
A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13
Loss
Rat
io (
in %
)
050
100
150
200
@freakonometrics 31
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Market Competition, A1
Three models (GLM) frequency bodily injury - fre-quency material damage - loss materialAge in 4 categories, 30−, 30− 45, 45− 60 and 60+, aninteraction with occupation.Manual Smoothing of parametersDone with SAS and Excel.
Sent by an actuary working for a mutuelle
A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13
Mar
ket S
hare
(in
%)
02
46
810
A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13
Loss
Rat
io (
in %
)
050
100
150
200
@freakonometrics 32
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Market Competition, A8-A9
GLM for frequency and standard cost (large claimeswere removed, above 15k)Interaction Age and GenderAll variables but density and group2.Done with a software developped by Actuaris.
Sent by an actuary working for a mutuelle
A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13
Mar
ket S
hare
(in
%)
02
46
810
A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13
Loss
Rat
io (
in %
)
050
100
150
200
@freakonometrics 33
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Market Competition, A11
Use of all variables, but Subgroup2,Use of two XGBoost models (bodily injury and material),Correction for negative premiumsUse of Python
Sent by an actuary working for a private insurance com-pany (sudent at the Data Science for Actuaries pro-gram)
A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13
Mar
ket S
hare
(in
%)
02
46
810
A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13
Loss
Rat
io (
in %
)
050
100
150
200
@freakonometrics 34
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Market Competition, A12
Use of two xgboost models (bodily injury and material),Correction for negative premiumsUse of R
Sent by an actuary working for a private insurance com-pany, in Europe, but not in France.
A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13
Mar
ket S
hare
(in
%)
02
46
810
A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13
Loss
Rat
io (
in %
)
050
100
150
200
@freakonometrics 35
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Distorting the Market: the Price Aggregator as a Market Player
Partnership between A4 and the price aggregator: if A4 is either the 4th or 5th, itreturns the 3rd price (−ε)
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
Par
t de
mar
ché
(%)
05
1015
2025
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
Par
t de
mar
ché
(%)
05
1015
2025
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
05
1015
2025
Happens for 33% of the prices. Market share 8.8% up to 22.5%
@freakonometrics 36
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Distorting the Market: the Price Aggregator as a Market Player
Increase of loss ratio of A4 from 128% to 143%
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
Par
t de
mar
ché
(%)
0.0
0.5
1.0
1.5
2.0
2.5
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14
Par
t de
mar
ché
(%)
0.0
0.5
1.0
1.5
2.0
2.5
Much higher volatility on other companies...
@freakonometrics 37
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Take-Home Message
• different but consistent models, from different actuaries, differentbackgrounds, different softwares, different languages, different models
• hard to predict model’s behavior in a competitive market from standardtools (lift curves)
• the choice of the insurance company has no big impact on market results,but has a big impact on how to use information on competitors
@freakonometrics 38