Slides anr-jeu

Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif

Couverture du risque de catastrophenaturelle dans un jeu non cooperatif

Arthur Charpentier∗ & Benoıt Le Maux

http ://blogperso.univ-rennes1.fr/arthur.charpentier/

∗ CREM-Universite Rennes 1 & Ecole Polytechnique

Journees AST&Risk, Grenoble Aout 2009

1


Decision theory with uncertainty, classical models

Consider an agent with utility function u(·), facing possible loss l > 0 withprobability p.

• without insurance, E(u(X)) = p · u(−l) + (1− p) · u(0) = p · u(−l)• without insurance, E(u(X)) = u(−α)

where α denotes the insurance premium, with convention u(0) = 0. Thus, let

α? = −u−1 (pu(−l))

• if α < α? the agent buys insurance• if α > α? the agent does not buys insurance

2


Decision theory with uncertainty, with possible ruin

Assume that ruin is possible. Let N denote the number of insured claiming, thenthere is ruin if

C + nα < Nl orN

n= X >

C/n+ α

l=c+ α

l= x

In case of ruin two cases can be considered• ‘insurance company with limited liability’ : the total amount of asset is divided

among the insured claiming losses (prorata capita)• ‘mutual insurance company’ : there is no ruin since the insurance company will

ask for an additional premium to all the insured to pay that difference

3


insurance company Ltd. versus mutual insurance

Ins. 1 Ins. 1 Ins. 3 Ins. 4 Ins. 5 Total

Premium 100 100 100 200 200 700

Loss - 400 - 600 - 1000

Case 1 : insurance company with limited liability

payment - 320 - 480 - 800

Case 2 : mutual insurance company

payment 400 600 1000

-29 -29 -29 -57 -57 -200

final payment -29 371 -29 543 -57 800

4


Possible ruin, insurance company with limited liability

• if X = N/n ∈ [0, α/l] : positive profit, I(x) = l

• if X = N/n ∈ [α/l, x] : negative profit, I(x) = l

• if X = N/n ∈ [x, 1] : ruin, I(x) ∈ [0, l]

where x = (α+ c)/l.

proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

5



With insurance, the expected value for an agent with utility function u(·) was (inthe classical model), with random loss L

E(u(L)) = pu(−α− l + l) + (1− p)u(−α) = u(−α)

but if ruin is possible, it becomes

E(u(L)) = pE((u(−α− l + I(X))) + (1− p)u(−α)

Hence, if F and f are cdf and the pdf of X,

E(u(L)) = p ·F (x) ·U(−α) + p ·F (x) ·∫ 1

x

U(−α− l+ I(x))f(x)dx+ (1− p)u(−α)

6



Expected utility of a representative agent without ruin, or with possible ruin,

Premium

0.0 0.5 1.0 1.5 2.0

=⇒ a low premium is not (necessarily) interesting (high risk of bankrupt).

7


A model with dependence among risks

Consider a common chock model, i.e. Θ is the indicator of occurrence of acatastrophe. Given Θ, the Zi are independent,• if Θ = 0, Zi ∼ B(pN ), i.e. N = Z1 + · · ·+ Zn ∼ B(n, pN ), no catastrophe• if Θ = 1, Zi ∼ B(pC), i.e. N = Z1 + · · ·+ Zn ∼ B(n, pC), catastrophe

proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

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8


A model with dependence among risks

P(N ≤ k) =k∑

j=0

(j

n

)[pj

C(1− pC)n−jp? + pjN (1− pN )n−j(1− p?)

]where p? = P(Θ = 1), pC = P(Zi = 1|Θ = 1) and pN = P(Zi = 1|Θ = 0).

proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

0.0

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0.6

0.8

1.0

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9


Getting a better understanding of the model

Let p = P(Zi = 1) (individual loss probability). Consider an increase of p

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Proportion of insured claiming

Pro

babi

lity

(cum

ulat

ed)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p1 ≤ p2 =⇒ Θ1 �1 Θ2 =⇒ F1 �1 F2

10



Let δ = 1− pN/pC , so that δ can be seen as a correlation within the region.

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Pro

babi

lity

(cum

ulat

ed)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

δ1 ≤ δ2 =⇒ Θ1 �2 Θ2 =⇒ F1 �2 F2

11



The impact of the number of insurerd n

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Pro

babi

lity

(cum

ulat

ed)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

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Impact of parameters on ruin probability

• increase of p, claim occurrence probability : increase of ruin probability for allx ∈ [0, 1]

• increase of δ, within correlation : there exists x0 ∈ (0, 1) such that◦ increase of ruin probability for all x ∈ [x0, 1]◦ decrease of ruin probability for all x ∈ [0, x0]

• increase of n, number of insured : there exists x0 ∈ (0, 1) such that◦ decrease of ruin probability for all x ∈ [pC , 1]◦ increase of ruin probability for all x ∈ [x0, pC ]◦ decrease of ruin probability for all x ∈ [pN , x0]◦ increase of ruin probability for all x ∈ [0, pN ]

• increase of c, economic capital : decrease of ruin probability for all x ∈ [0, 1]• increase of α, individual premium : decrease of ruin probability for all x ∈ [0, 1]

13


Impact of parameters on utility functions

... but the impact on utility is more complex, for instance, the impact of δ on theutility is the following

Correlation

0.0 0.2 0.4 0.6 0.8 1.0

14


A two region model

Consider here a two-region chock model such that• Θ = (0, 0), no catastrophe in the two regions,• Θ = (1, 0), catastrophe in region 1 but not in region 2,• Θ = (0, 1), catastrophe in region 2 but not in region 1,• Θ = (1, 1), catastrophe in the two regions.Let N1 and N2 denote the number of claims in the two regions, respectively, andset N0 = N1 +N2.

15


A two region model Θ = (1, 1)

proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

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proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

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proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

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proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

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proportion claiming

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proportion claiming

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proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

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proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

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proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

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proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

0.0

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1.0

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proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

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proportion claiming

0.0 0.2 0.4 0.6 0.8 1.0

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19


On correlations in this two region model

Note that there are two kinds of correlation in this model,• a within region correlation, with coefficients δ1 and δ2

• a between region correlation, with coefficient δ0Here, δi = 1− pi

N/piC , where i = 1, 2 (Regions), while δ0 ∈ [0, 1] is such that

P(Θ = (1, 1)) = δ0 ×min{P(Θ = (1, ·)),P(Θ = (·, 1))} = δ0 ×min{p?1, p

?2}.

20


A noncooperative game : who buys insurance ?

In that case, 4 cases should be considered, depending on the region profiles• state s = (I, I), agents buy insurance in the two regions• state s = (I,N), only agents in region 1 buy insurance• state s = (N, I), only agents in region 2 buy insurance• state s = (N,N), no one buys insuranceHence, here everything should be visualized in the the plan (α1, α2). In eachstate s, a representative agent in Region i maximumizes

Vi,s = pi ·Fs(xs) ·U(−αi)+p ·F s(xs) ·∫ 1

x

U(−αi− l+I(x))fs(x)dx+(1−pi)u(−αi)

21


A noncooperative game : who buys insurance ?

Hence, agents take decisions given the following payoff matrix for a given set ofpremiums (α1, α2),

Region 2

Insure Don’t

Region 1 Insure V1,(I,I), V2,(I,I) V1,(I,N), p2U(−l)

Don’t p1U(−l), V2,(N,I) p1U(−l), p2U(−l)

Note that agents in the two regions can have different risk aversions (or riskperception).

22


Study of the two region model

The following graphs show the decision in Region 1, given that Region 2 buyinsurance (on the left) or not (on the right).

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Study of the two region model

The following graphs show the decision in Region 2, given that Region 1 buyinsurance (on the left) or not (on the right).

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Nash equilibrium for a two region model

Definition 1. In a Nash equilibrium which each player is assumed to know theequilibrium strategies of the other players, and no player has anything to gain bychanging only his or her own strategy unilaterally.

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Nash equilibrium for a two region model

Definition 2. In a Nash equilibrium which each player is assumed to know theequilibrium strategies of the other players, and no player has anything to gain bychanging only his or her own strategy unilaterally.

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1: insured, 2: insured

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Possible Nash equilibriums

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Sensilibity to parameters

Impact of catastrophe probability (in region 2), while individual claim occurrenceprobability remains unchanged.

increase of catastrophe probability p?2 decrease of catastrophe probability p?

2

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Impact of correlation between the two regions, and impact of the economiccapital of the insurance company.

increase of between (region) correlation δ0 increase of economic capital c

31



Impact when the populations are different in the two regions,

increase of number of unhabitants n1 increase of number of unhabitants n2

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increase of within correlations δ1 and δ2 increase of within correlation δ1

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A short word about Pareto optimality

Definition 3. An allocation is defined as Pareto optimal when no further Paretoimprovements can be made, i.e. a change from one allocation to another that canmake at least one individual better off without making any other individual worseoff.

34

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