1. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Couverture du risque de catastrophe
naturelle dans un jeu non coop´ratif
e
Arthur Charpentier∗ & Benoˆ Le Maux
ıt
http ://blogperso.univ-rennes1.fr/arthur.charpentier/
∗ ´
CREM-Universit´ Rennes 1 & Ecole Polytechnique
e
Journ´es AST&Risk, Grenoble Aoˆ t 2009
e u
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Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Decision theory with uncertainty, classical models
Consider an agent with utility function u(·), facing possible loss l > 0 with
probability 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
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3. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Decision theory with uncertainty, with possible ruin
Assume that ruin is possible. Let N denote the number of insured claiming, then
there is ruin if
N C/n + α c+α
C + nα < N l or =X> = =x
n l l
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
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4. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
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
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5. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
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.
0.0 0.2 0.4 0.6 0.8 1.0
proportion claiming
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6. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Possible ruin, insurance company with limited liability
With insurance, the expected value for an agent with utility function u(·) was (in
the 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,
1
E(u(L)) = p · F (x) · U (−α) + p · F (x) · U (−α − l + I(x))f (x)dx + (1 − p)u(−α)
x
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7. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Possible ruin, insurance company with limited liability
Expected utility of a representative agent without ruin, or with possible ruin,
0.0 0.5 1.0 1.5 2.0
Premium
=⇒ a low premium is not (necessarily) interesting (high risk of bankrupt).
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8. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
A model with dependence among risks
Consider a common chock model, i.e. Θ is the indicator of occurrence of a
catastrophe. 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
1.0
0.8
q
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
proportion claiming
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9. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
A model with dependence among risks
k
j
P(N ≤ k) = pj (1 − pC )n−j p + pj (1 − pN )n−j (1 − p )
C N
j=0
n
where p = P(Θ = 1), pC = P(Zi = 1|Θ = 1) and pN = P(Zi = 1|Θ = 0).
1.0
q
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
proportion claiming
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10. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Getting a better understanding of the model
Let p = P(Zi = 1) (individual loss probability). Consider an increase of p
1.0
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qqqq
qq
q qq
qq qqq
qqqqqqqqqqqqqqq
qq
q
0.8
q
q
Probability (cumulated)
q
0.6
q
0.4
q
q
0.2
q
q
q
0.0
qq
0.0 0.2 0.4 0.6 0.8 1.0
Proportion of insured claiming
p1 ≤ p2 =⇒ Θ1 1 Θ2 =⇒ F1 1 F2
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Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Getting a better understanding of the model
Let δ = 1 − pN /pC , so that δ can be seen as a correlation within the region.
1.0
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqq
qq
q qq
qq qqq
qqqqqqqqqqqqqqq
qq
q
0.8
q
q
Probability (cumulated)
q
0.6
q
0.4
q
q
0.2
q
q
q
0.0
qq
0.0 0.2 0.4 0.6 0.8 1.0
Proportion of insured claiming
δ1 ≤ δ2 =⇒ Θ1 2 Θ2 =⇒ F1 2 F2
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Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Getting a better understanding of the model
The impact of the number of insurerd n
1.0
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqq
qq
qq
qq
qqqqqqqqqqqqqq qqqq
qq
q
0.8
q
q
Probability (cumulated)
q
0.6
q
0.4
q
q
0.2
q
q
q
0.0
qq
0.0 0.2 0.4 0.6 0.8 1.0
Proportion of insured claiming
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Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Impact of parameters on ruin probability
• increase of p, claim occurrence probability : increase of ruin probability for all
x ∈ [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]
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Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Impact of parameters on utility functions
... but the impact on utility is more complex, for instance, the impact of δ on the
utility is the following
0.0 0.2 0.4 0.6 0.8 1.0
Correlation
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Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
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, and
set N0 = N1 + N2 .
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16. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
A two region model Θ = (1, 1)
1.0
1.0
q
q
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
proportion claiming proportion claiming
1.0
q
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
proportion claiming
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17. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
A two region model Θ = (1, 0)
1.0
1.0
q
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
q
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
proportion claiming proportion claiming
1.0
q
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
proportion claiming
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Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
A two region model Θ = (0, 1)
1.0
1.0
q
0.8
0.8
0.6
0.6
q
0.4
0.4
0.2
0.2
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
proportion claiming proportion claiming
1.0
q
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
proportion claiming
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19. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
A two region model Θ = (0, 0)
1.0
1.0
0.8
0.8
0.6
0.6
q
0.4
0.4
0.2
0.2
q
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
proportion claiming proportion claiming
1.0
0.8
0.6
0.4
q
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
proportion claiming
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Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
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 δ0
Here, δi = 1 − pi /pi , where i = 1, 2 (Regions), while δ0 ∈ [0, 1] is such that
N C
P(Θ = (1, 1)) = δ0 × min{P(Θ = (1, ·)), P(Θ = (·, 1))} = δ0 × min{p1 , p2 }.
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21. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
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 insurance
Hence, here everything should be visualized in the the plan (α1 , α2 ). In each
state s, a representative agent in Region i maximumizes
1
Vi,s = pi ·Fs (xs )·U (−αi )+p·F s (xs )· U (−αi −l +I(x))fs (x)dx+(1−pi )u(−αi )
x
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22. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
A noncooperative game : who buys insurance ?
Hence, agents take decisions given the following payoff matrix for a given set of
premiums (α1 , α2 ),
Region 2
Insure Don’t
Region 1 Insure V1,(I,I) , V2,(I,I) V1,(I,N ) , p2 U (−l)
Don’t p1 U (−l), V2,(N,I) p1 U (−l), p2 U (−l)
Note that agents in the two regions can have different risk aversions (or risk
perception).
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Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Study of the two region model
The following graphs show the decision in Region 1, given that Region 2 buy
insurance (on the left) or not (on the right).
50
50
q q
q q
q q
q q
q q
q q
q q
q q
q q
q q
40
40
REGION 1 q
q REGION 1 q
q
q q
BUYS q
q BUYS q
q
q q
Premium in Region 2
Premium in Region 2
INSURANCE q
q INSURANCE q
q
q q
q q
30
30
q q
q q
q q
q q
q q
q q
q q
q q
q q
q q
20
20
q
q REGION 1 q
q REGION 1
q q
q
q BUYS NO q
q BUYS NO
q q
q
q INSURANCE q
q INSURANCE
q q
q q
10
10
q q
q q
q q
q q
q q
q q
q q
q q
q q
q q
q q
0
0
0 10 20 30 40 50 0 10 20 30 40 50
Premium in Region 1 Premium in Region 1
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24. ´
Arthur CHARPENTIER - Couverture du risque de catastrophe naturelle dans un jeu non cooperatif
Study of the two region model
The following graphs show the decision in Region 2, given that Region 1 buy
insurance (on the left) or not (on the right).
50
50
q q
q q
q q
q q
q q
q q
q q
q q
q q
q q
40
40
REGION 2 q
q REGION 2 q
q
q q
BUYS NO q
q BUYS NO q
q
q q
Premium in Region 2
Premium in Region 2
INSURANCE q
q INSURANCE q
q
q q
q q
30
30
q q
q q
q q
q q
q q
q q
q q
q q
q q
q q
20
20
q q
q q
q q
q q
q q
q
q REGION 2 q
q REGION 2
q q
q
q BUYS q
q BUYS
10
10
q q
q
q INSURANCE q
q INSURANCE
q q
q q
q q
q q
q q
q q
q q
q q
0
0
0 10 20 30 40 50 0 10 20 30 40 50
Premium in Region 1 Premium in Region 1
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