Actuarial Pricing Game A. Charpentier (Université de Rennes 1 & Chaire actinfo)

Aegina, Greece, May 2017 Dependence Modelling with Applications in Finance and Insurance Conference

@freakonometrics

freakonometrics

freakonometrics.hypotheses.org

1

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking “the contribution of the many to the misfortune of the few”

N1 X Finance: risk neutral valuation π = EQ S1 F0 = EQ0 S1 , where S1 = Yi i=1

Insurance: risk sharing (pooling) π = EP S1 or, with segmentation π(ω) = EP S1 Ω = ω for some (unobservable?) risk factor Ω imperfect information given some (observable) risk variables X = (X1 , · · · , Xk ) π(x) = EP S1 X = x @freakonometrics

freakonometrics

freakonometrics.hypotheses.org

2

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition In a competitive market, insurers can use different sets of variables and different models, with GLMs, Nt |X ∼ P(λX · t) and Y |X ∼ G(µX , ϕ) b b b T x) · zj = π bj (x) = E N1 X = x · E Y X = x = exp(α | {z }

T

b x) exp(β | {z }

Poisson P(λx ) Gamma G(µX ,ϕ)

(see Kaas et al. (2008)) or any other statistical model (see Hastie et al. (2009)) ( n ) X zj = π bj (x) where π bj ∈ argmin `(si , m(xi )) m∈Fj :ΠXj →R

i=1

With d competitors, each insured i has to choose among d premiums, πi = π b1 (xi ), · · · , π bd (xi ) ∈ Rd+ .

@freakonometrics

freakonometrics

freakonometrics.hypotheses.org

3

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

More and more price differentiation ? Consider π1 = E S1 and π2 (x) = E S1 |X = x X Observe that E π(X) = π(x) · P[x] = π1

• Insured with x ∈ X1 : choose Ins1 • Insured with x ∈ X2 : choose Ins2 X

π1 (x) · PIns1 [x] 6= E[S|X ∈ X1 ] = EIns1 [S]

x∈X1

X

0.15 0.10

x∈X2

0.05

x∈X1

π(x) · P[x]

0.00

π(x) · P[x] +

Claims Annual Frequency

=

X

0.20

x∈X

X

20

π2 (x) · PIns2 [x] = E[S|X ∈ X2 ] = EIns2 [S]

30

40

50

60

70

80

90

Age of the Driver

x∈X2

@freakonometrics

freakonometrics

freakonometrics.hypotheses.org

4

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition (episode 1, season 3) comonotonicity? ●

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Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

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Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

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Age in [30,50]

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A11

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Premium

500

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Insurance Ratemaking Competition (episode 1, season 1)

A1

@freakonometrics

A2

freakonometrics

freakonometrics.hypotheses.org

A10

10

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition (episode 1, season 1) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

400

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A5

A6

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● Age in [70,100] ● ● ● ● ●

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A1

@freakonometrics

A2

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A4

freakonometrics

A7

A8

freakonometrics.hypotheses.org

A9

A10

A11

A13

A14

11

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition (episode 1, season 1) ● ● ● ● ● ● ● ● ● ● ● ● ●

Category = 'Small'

400 300 200 100 0

Premium

500

600

700

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A1

@freakonometrics

A2

A3

A4

freakonometrics

A5

A6

A7

A8

freakonometrics.hypotheses.org

A9

A10

A11

A13

A14

12

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition (episode 1, season 1) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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Category = 'Medium'

400 300 200 100 0

Premium

500

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700

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A1

@freakonometrics

A2

A3

A4

freakonometrics

A5

A6

A7

A8

freakonometrics.hypotheses.org

A9

A10

A11

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13

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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● ● ● ● ● ● ● ● ●

A3

A4

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A6

A7

A8

A9

Category = 'Large'

● ● ● ● ● ●

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A10

A11

A13

A14

●

400 300 200 100 0

Premium

500

600

700

Insurance Ratemaking Competition (episode 1, season 1)

A1

@freakonometrics

A2

freakonometrics

A5

freakonometrics.hypotheses.org

14

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition We need a Decision Rule to select premium chosen by insured i cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc

@freakonometrics

Ins1

Ins2

Ins3

Ins4

Ins5

Ins6

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

freakonometrics.hypotheses.org

15

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition Basic ‘rational rule’ πi = min{b π1 (xi ), · · · , π bd (xi )} = π b1:d (xi ) cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc

@freakonometrics

Ins1

Ins2

Ins3

Ins4

Ins5

Ins6

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

freakonometrics.hypotheses.org

16

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition A more realistic rule πi ∈ {b π1:d (xi ), π b2:d (xi ), π b3:d (xi )} cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc

@freakonometrics

Ins1

Ins2

Ins3

Ins4

Ins5

Ins6

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

freakonometrics.hypotheses.org

17

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition A1

A10

A11 A12

A13

A2

A3

A A4

A5

A6

A7

A8

A9 Pearson residuals: 5.4

Female

4.0

Gender

2.0

0.0

Male

−2.0

−4.0 −5.5 p−value = < 2.22e−16

@freakonometrics

freakonometrics

freakonometrics.hypotheses.org

18

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition A1

A10

A11 A12

A13

A2

A3

A A4

A5

A6

A7

A8

A9

Large

Pearson residuals: 13

Category Medium

4 2 0 −2

Small

−4

−11 p−value = < 2.22e−16

@freakonometrics

freakonometrics

freakonometrics.hypotheses.org

19

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition A10

A11 A12

A13

A2

A3

A5

Housewife Employed

A1

A A4

A7

A8

A9 Pearson residuals: 15

Occupation Retired

4 2 0 −2 −4

UnemployedSelf−employed @freakonometrics

A6

−19 p−value = < 2.22e−16

freakonometrics

freakonometrics.hypotheses.org

20

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Insurance Ratemaking Competition A1

A10

A11 A12

A13

A2

A3

A A4

A5

A6

A7

A8

A9

AgeF (35,50] (25,35]

(17,25]

Pearson residuals: 48

(70,101]

(50,70]

4 0 −4

@freakonometrics

−20 p−value = < 2.22e−16

freakonometrics

freakonometrics.hypotheses.org

21

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

A Game with Rules... but no Goal Two datasets : a training one, and a pricing one (without the losses in the later) Step 1 : provide premiums to all contracts in the pricing dataset Step 2 : allocate insured among players Season 1 13 players Season 2 14 players Step 3 [season 2] : provide additional information (premiums of competitors) Season 3 23 players (3 markets, 8+8+7) Step 3-6 [season 3] : dynamics, 4 years

@freakonometrics

freakonometrics

freakonometrics.hypotheses.org

22

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Actuarial Pricing Game (season 3)

@freakonometrics

freakonometrics

freakonometrics.hypotheses.org

23

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Actuarial Pricing Game (episode 1, season 3) MARKET 1 250

MARKET 1

200

8000

●

● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

100 50

2000

● ● ● ● ● ● ● ● ● ●

150

Loss Ratio (S/P, %)

4000

●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ●

Ins7

Ins10

Ins13

Ins16

Ins17

Ins21

●

0

0

Premiums (€)

6000

●

Ins3

@freakonometrics

freakonometrics

Ins3

freakonometrics.hypotheses.org

Ins7

Ins10

Ins13

Ins16

Ins17

Ins21

24

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Actuarial Pricing Game (episode 1, season 3) MARKET 2 250

1e+05

MARKET 2

200

8e+04

●

Loss Ratio (S/P, %)

Premiums (€)

● ●

100

4e+04

●

150

6e+04

●

50

2e+04

●

● ● ● ● ● ●

● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ●

Ins1

Ins4

●

● ●

● ●

● ●

● ●

Ins11

Ins14

Ins15

Ins19

Ins20

0

0e+00

●

Ins5

@freakonometrics

freakonometrics

Ins1

freakonometrics.hypotheses.org

Ins4

Ins5

Ins11

Ins14

Ins15

Ins19

Ins20

25

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Actuarial Pricing Game (episode 1, season 3) MARKET 3 250

MARKET 3

200 150

Loss Ratio (S/P, %)

100

100000 50000

●

50

●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●

●

●

● ●

● ● ●

● ● ● ● ● ● ● ●

●

Ins2

Ins6

Ins8

Ins9

Ins12

Ins18

Ins22

Ins23

0

0

Premiums (€)

150000

●

@freakonometrics

freakonometrics

Ins2

freakonometrics.hypotheses.org

Ins6

Ins8

Ins9

Ins12

Ins18

Ins22

Ins23

26

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Ins4

●

●

400

●

400

Ins1

400

400

Actuarial Pricing Game (episodes 1-3, season 3) Ins5

Ins11 ●

●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

300

● ●

● ● ● ● ● ● ● ● ● ● ● ●

300

● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ●

● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

200

200

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ●

200

● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

300

● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

200

300

● ● ● ●

●

●

losses premium

●

●

losses

●

premium

● ●

● ●

●

losses

●

premium

100

100

●

100

●

● ●

100

● ●

● ● ●

●

●

●

losses premium

2

Ins14

3

0 1

2

Ins15

3

Ins19

1

2

400

1

400

3

400

2

400

1

0

0

0

●

●

3

Ins20 ● ●

●

● ●

●

● ● ●

●

losses

●

premium

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ●

● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

300

300

●

● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●

● ●

●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●

● ●

losses premium

● ●

● ●

●

● ●

losses premium

2

3

@freakonometrics

1

freakonometrics

2

3

● ●

losses premium

0

1

0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0

● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ●

●

● ●

●

●

● ●

● ●

● ● ●

200

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ●

● ● ● ● ● ● ●

200

● ●

●

100

100

●

● ●

●

100

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

300

● ●

●

200

● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

100

200

300

● ● ●

●

1

freakonometrics.hypotheses.org

2

3

1

2

3

27

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

140

180

Actuarial Pricing Game (episodes 1-3, season 3)

160

● Ins12

120

● Ins23

● Ins22

Loss Ratio (%)

● Ins4

140

Loss Ratio (%)

● Ins5

● Ins2

● Ins18

100

● Ins9

● Ins1

120

● Ins6 ● Ins15

● Ins20

● Ins8

● Ins11

100

80

● Ins19 ● Ins14

0

5

10

15

20

25

30

0

Market Share (%)

@freakonometrics

freakonometrics

5

10

15

20

25

30

Market Share (%)

freakonometrics.hypotheses.org

28

Arthur Charpentier, Aegina 2017 (Actuarial Pricing Game)

Actuarial Pricing Game (season 3)

@freakonometrics

freakonometrics

freakonometrics.hypotheses.org

29