Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
@freakonometrics
1
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Actuarial Pricing Game A. Charpentier (UQAM & Université de Rennes 1) with R. Élie & J. Jakubowicz
Paris, 100% Actuaires, November 2015. http://freakonometrics.hypotheses.org
@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, Charpentier, Denuit & Élie (2015)
@freakonometrics
4
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Pricing Game, with a Toy Dataset
@freakonometrics
5
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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● ● ● ● ● ● ● ● ● ● ●
Gender = 'Male'
400 300 0
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200
Premium
500
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A1
@freakonometrics
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
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A14
6
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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A6
A7
A8
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A9
A10
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A11
A13
300
400
Gender = 'Female'
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0
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The Premiums, Brief Summary
A1
@freakonometrics
A2
A3
A4
A5
A14
7
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary ● ●
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100
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Premium
500
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Age in [17,25]
A1
@freakonometrics
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
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8
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
A3
A4
A5
A6
A7
A8
A9
Age in [30,50]
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
A11
A13
A14
400 300 0
100
200
Premium
500
600
700
The Premiums, Brief Summary
A1
@freakonometrics
A2
A10
9
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary
400 300
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●
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● Age in [70,100] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
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500
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
A1
@freakonometrics
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A13
A14
10
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Category = 'Small'
400 300 0
100
200
Premium
500
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700
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A1
@freakonometrics
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A13
A14
11
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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●
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Category = 'Medium'
400 300 0
100
200
Premium
500
600
700
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A1
@freakonometrics
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A13
A14
12
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
A3
A4
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
A6
A7
A8
A9
Category = 'Large'
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
A10
A11
A13
A14
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400 300 0
100
200
Premium
500
600
700
The Premiums, Brief Summary
A1
@freakonometrics
A2
A5
13
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Premiums, Brief Summary
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ●
Bonus = −50 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
300
400
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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●
●
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●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
A13
A14
●
200
Premium
500
600
700
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
100
●
A1
@freakonometrics
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
14
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Comparing Models
Source: unkown, so far...
@freakonometrics
15
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
100
Comparing Models
poorest ←
60 40 20
Consider an ordered sample {y1 , · · · , yn } of incomes, with y1 ≤ y2 ≤ · · · ≤ yn , then Lorenz curve is
Income (%)
80
→ richest
Pi
0
i j=1 yj {Fi , Li } with Fi = and Li = Pn n j=1 yj
0
20
40
60
80
100
80 60 40 20
i j=1 yj {Fi , Li } with Fi = and Li = Pn n j=1 yj
more risky ←
0
Pi
Losses (%)
We have observed losses yi and premiums π b(xi ). Consider an ordered sample by the model, see Frees, Meyers & Cummins (2014), π b(x1 ) ≥ π b(x2 ) ≥ · · · ≥ π b(xn ), then plot
100
Proportion (%)
0
20
→ less risky 40
60
80
100
Proportion (%)
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16
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
The Models, Brief Summary
@freakonometrics
17
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Market Competition Decision Rule: the insured selects the cheapeast premium, cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc
@freakonometrics
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
18
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Market Competition Decision Rule: the insured selects randomly from the three cheapeast premium cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc
@freakonometrics
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
19
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Market Competition Decision Rule: the insured were assigned randomly to some insurance company for year n − 1. For year n, they stay with their company if the premium is one of the three cheapeast premium, if not, random choice among the four
@freakonometrics
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
20
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Financial Results, Market Share, Method 2
6000
●
●
5000 4000 3000
●
● ● ●
2000
Number of Contracts
●
1000
● ●
● ●
A1
@freakonometrics
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A13
A14
21
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Financial Results, Market Share, Method 3
4000 3000
●
● ●
●
2000
Number of Contracts
5000
●
A1
@freakonometrics
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A13
A14
22
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Financial Results, Loss Ratio, Method 2 Market Loss Ratio ∼ 154%.
● ●
●
●
● ●
150
● ●
100
Loss Ratio
200
250
●
●
●
A1
@freakonometrics
●
●
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A13
A14
23
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
1.5
MCA factor map
1.0
A3 Retired
0.5
Small Unemployed A2 Employed
A1
0.0
E A5 CD A7 B● A6 Female Male A8 A13 AA9 A11 A10 A4 Medium F A12
(50,70]
(25,35]
−0.5
Dim 2 (7.25%)
(70,101]
(17,25]
Large
Housewife −1.0
(35,50]
−2
−1
Self−employed
0
1
2
3
Dim 1 (11.02%)
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24
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies A1
A10
A11 A12
A13
A2
A3
A A4
A5
A6
A7
A8
A9
Female
Pearson residuals: 5.4 4.0
Gender
2.0
0.0
Male
−2.0
−4.0 −5.5 p−value = < 2.22e−16
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25
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies A1
A10
A11 A12
A13
A2
A3
A A4
A5
A6
A7
A8
A9
Category Medium
Large
Pearson residuals: 13
4 2 0 −2
Small
−4
−11 p−value = < 2.22e−16
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26
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
UnemployedSelf−employed
Occupation Retired
Housewife Employed
A1
@freakonometrics
A10
A11 A12
A13
A2
A3
A A4
A5
A6
A7
A8
A9 Pearson residuals: 15
4 2 0 −2 −4
−19 p−value = < 2.22e−16
27
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies A1
A10
A11 A12
A13
A2
A3
A A4
A5
A6
A7
A8
A9
(70,101]
(50,70]
AgeF (35,50] (25,35]
(17,25]
Pearson residuals: 48
@freakonometrics
4 0 −4
−20 p−value = < 2.22e−16
28
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies A10
A11 A12
A13
A2
A3
A A4
A5
A6
A7
A8
A9 Pearson residuals: 31
(200,300]
DensityF (125,200] (70,125]
(40,70]
(10,40]
A1
@freakonometrics
4 2 0 −2 −4
−26 p−value = < 2.22e−16
29
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
(100,200]
BonusF (30,100] (0,30] (−10,0](−30,−10] (−49,−30] [−50,−49]
A1
@freakonometrics
A10
A11 A12
A13
A2
A3
A A4
A5
A6
A7
A8
A9 Pearson residuals: 29
4 2 0 −2 −4
−25 p−value = < 2.22e−16
30
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Insurers and Insurance Companies
PoldurF (5,10] (1,5]
[0,1]
A1
A10
A11 A12
A13
A2
A3
A A4
A5
A6
A7
A8
A9 Pearson residuals: 8.7
4.0 2.0 0.0 −2.0
(10,16]
−4.0
@freakonometrics
−7.5 p−value = < 2.22e−16
31
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
8 6 4 0
2
Market Share (in %)
10
Market Competition, A2
A1 A2 A3 A4 A5 A6 A7 A8 A9
A11
A13
A1 A2 A3 A4 A5 A6 A7 A8 A9
A11
A13
150 100 0
50
Loss Ratio (in %)
200
No segmentation, unique price
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32
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
8 6 4 2 0
A11
A13
A1 A2 A3 A4 A5 A6 A7 A8 A9
A11
A13
100
150
200
A1 A2 A3 A4 A5 A6 A7 A8 A9
0
50
Sent by an actuary working for a mutuelle
Loss Ratio (in %)
Three models (GLM) frequency bodily injury - frequency material damage - loss material Age in 4 categories, 30−, 30 − 45, 45 − 60 and 60+, an interaction with occupation. Manual Smoothing of parameters Done with SAS and Excel.
Market Share (in %)
10
Market Competition, A1
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33
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
8 6 4 2 0
A1 A2 A3 A4 A5 A6 A7 A8 A9
A11
A13
A1 A2 A3 A4 A5 A6 A7 A8 A9
A11
A13
150 100 0
50
Sent by an actuary working for a mutuelle
Loss Ratio (in %)
200
GLM for frequency and standard cost (large claimes were removed, above 15k) Interaction Age and Gender All variables but density and group2. Done with a software developped by Actuaris.
Market Share (in %)
10
Market Competition, A8-A9
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34
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
8 6 4 2 0
A11
A13
A1 A2 A3 A4 A5 A6 A7 A8 A9
A11
A13
100
150
200
A1 A2 A3 A4 A5 A6 A7 A8 A9
0
50
Sent by an actuary working for a private insurance company (sudent at the Data Science for Actuaries program)
Loss Ratio (in %)
Use of all variables, but Subgroup2, Use of two XGBoost models (bodily injury and material), Correction for negative premiums Use of Python
Market Share (in %)
10
Market Competition, A11
@freakonometrics
35
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
8 6 4 2 0
A11
A13
A1 A2 A3 A4 A5 A6 A7 A8 A9
A11
A13
100
150
200
A1 A2 A3 A4 A5 A6 A7 A8 A9
0
50
Sent by an actuary working for a private insurance company, in Europe, but not in France.
Loss Ratio (in %)
Use of two xgboost models (bodily injury and material), Correction for negative premiums Use of R
Market Share (in %)
10
Market Competition, A12
@freakonometrics
36
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Distorting the Market: the Price Aggregator as a Market Player
25 20 15 0
5
10
Part de marché (%)
15 10 0
5
Part de marché (%)
20
25
Partnership between A4 and the price aggregator: if A4 is either the 4th or 5th , it returns the 3rd price (−ε)
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A13
A14
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A13
A14
Happens for 33% of the prices. Market share 8.8% up to 22.5%
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37
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Distorting the Market: the Price Aggregator as a Market Player
2.5 2.0 1.5 0.0
0.5
1.0
Part de marché (%)
1.5 1.0 0.0
0.5
Part de marché (%)
2.0
2.5
Increase of loss ratio of A4 from 128% to 143%
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A13
A14
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A13
A14
Much higher volatility on other companies...
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38
Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015
Take-Home Message • different but consistent models, from different actuaries, different backgrounds, different softwares, different languages, different models • hard to predict model’s behavior in a competitive market from standard tools (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
39
