slides pricing game

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


Pricing Game, Charpentier, Denuit & Élie (2015)

@freakonometrics 2


Pricing Game, Charpentier, Denuit & Élie (2015)

@freakonometrics 3


Pricing Game, with a Toy Dataset

@freakonometrics 4


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

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Gender = 'Male'

@freakonometrics 5



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

Gender = 'Female'

@freakonometrics 6



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



A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14

010

020

030

040

050

060

070

0

Pre

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A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14

010

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0

Age in [30,50]

@freakonometrics 8



A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14

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0

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A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14

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Age in [70,100]

@freakonometrics 9



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

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@freakonometrics 10



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Category = 'Medium'

@freakonometrics 11



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

Category = 'Large'

@freakonometrics 12



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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0

Bonus = −50

@freakonometrics 13


Comparing Models

Source: unkown, so far...

@freakonometrics 14


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

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2438129

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2438129


The Models, Brief Summary

@freakonometrics 16


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


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


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


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


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


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


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



−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



−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



−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



−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



−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



−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



−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


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



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


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



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



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


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


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


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

slides pricing game

Education