Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Insurance of Natural Catastrophes When Should Government Intervene ? Arthur Charpentier & Benoît le Maux Université Rennes 1 & École Polytechnique
[email protected]
http ://freakonometrics.blog.free.fr/
Congrès Annuel de la SCSE, Sherbrooke, May 2011.
1
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
1
Introduction and motivation
Insurance is “the contribution of the many to the misfortune of the few”.
The TELEMAQUE working group, 2005.
Insurability requieres independence Cummins & Mahul (JRI, 2004) or C. (GP, 2008)
2
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
1.1
The French cat nat mecanism
=⇒ natural catastrophes means no independence
Drought risk frequency, over 30 years, in France. 3
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
GOVERNMENT
RE-INSURANCE COMPANY CAISSE CENTRALE DE REASSURANCE
INSURANCE COMPANY
INSURANCE COMPANY
INSURANCE COMPANY
4
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
GOVERNMENT
RE-INSURANCE COMPANY CAISSE CENTRALE DE REASSURANCE
INSURANCE COMPANY
INSURANCE COMPANY
INSURANCE COMPANY
5
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
GOVERNMENT
RE-INSURANCE COMPANY CAISSE CENTRALE DE REASSURANCE
INSURANCE COMPANY
INSURANCE COMPANY
INSURANCE COMPANY
6
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
2
Demand for insurance
An agent purchases insurance if E[u(ω − X)] ≤ u(ω − α) {z } | {z } | no insurance
insurance
i.e. p · u(ω − l) + [1 − p] · u(ω − 0) ≤ u(ω − α) | {z } | {z } no insurance
insurance
i.e. E[u(ω − X)] ≤ E[u(ω − α−l + I)] | {z } | {z } no insurance
insurance
Doherty & Schlessinger (1990) considered a model which integrates possible bankruptcy of the insurance company, but as an exogenous variable. Here, we want to make ruin endogenous. 7
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
0 if agent i claims a loss Yi = 1 if not Let N = Y1 + · · · + Xn denote the number of insured claiming a loss, and X = N/n denote the proportions of insured claiming a loss, F (x) = P(X ≤ x). P(Yi = 1) = p for all i = 1, 2, · · · , n Assume that agents have identical wealth ω and identical vNM utility functions u(·). =⇒ exchangeable risks Further, insurance company has capital C = n · c, and ask for premium α.
8
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
2.1
Private insurance companies with limited liability
Consider n = 5 insurance policies, possible loss $1, 000 with probability 10%. Company has capital C = 1, 000.
Premium Loss
Ins. 1
Ins. 1
Ins. 3
Ins. 4
Ins. 5
Total
100
100
100
100
100
500
-
1,000
-
1,000
-
2,000
Case 1 : insurance company with limited liability indemnity
-
750
-
750
-
1,500
loss
-
-250
-
-250
-
-500
net
-100
-350
-100
-350
-100
-1000
9
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
2.2
Possible government intervention
Premium Loss
Ins. 1
Ins. 1
Ins. 3
Ins. 4
Ins. 5
Total
100
100
100
100
100
500
-
1,000
-
1,000
-
2,000
Case 2 : possible government intervention Tax indemnity net
-100
100
100
100
100
500
-
1,000
-
1,000
-
2,000
-200
-200
-200
-200
-200
-1000
(note that it is a zero-sum game).
10
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
3
A one region model with homogeneous agents
Let U (x) = u(ω + x) and U (0) = 0.
3.1
Private insurance companies with limited liability
• the company has a positive profit if N · l ≤ n · α • the company has a negative profit if n · α ≤ N · l ≤ C + n · α • the company is bankrupted if C + n · α ≤ N · l =⇒ ruin of the insurance company if X ≥ x =
c+α l
The indemnity function is l if X ≤ x I(x) = c + α if X > x n
11
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
I(X)
I l
Negative profit ]–cn ; 0[
Positive profit [0 ; nα[
Ruin –cn
c α 0
X α l Probability of no ruin: F(x )
α c x
l
1
Probability of ruin: 1–F(x )
12
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Without ruin, the objective function of the insured is V (α, p, δ, c) defined as U (−α). With possible ruin, it is Z E[E(U (−α − loss)|X)]) = E(U (−α − loss)|X = x)f (x)dx where E(U (−α − loss)|X = x) is equal to P(claim a loss|X = x) · U (α − loss(x)) + P(no loss|X = x) · U (−α) i.e. E(U (−α − loss)|X = x) = x · U (−α − l + I(x)) + (1 − x) · U (−α) so that Z
1
[x · U (−α − l + I(x)) + (1 − x) · U (−α)]f (x)dx
V = 0
that can be written Z V = U (−α) −
1
x[U (−α) − U (−α − l + I(x))]f (x)dx 0
And an agent will purchase insurance if and only if V > p · U (−l). 13
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
3.2
Distorted risk perception by the insured
We’ve seen that 1
Z V = U (−α) −
x[U (−α) − U (−α − l + I(x))]f (x)dx 0
since P(Yi = 1|X = x) = x (while P(Yi = 1) = p). But in the model in the Working Paper (first version), we wrote Z 1 V = U (−α) − p[U (−α) − U (−α − l + I(x))]f (x)dx 0
i.e. the agent see x through the payoff function, not the occurence probability (which remains exogeneous).
14
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
3.3
Government intervention (or mutual fund insurance)
The tax function is 0 if X ≤ x T (x) = N l − (α + c)n = Xl − α − c if X > x n Then Z
1
[x · U (−α − T (x)) + (1 − x) · U (−α − T (x))]f (x)dx
V = 0
i.e. Z
1
Z
1
U (−α + T (x))f (x)dx = F (x) · U (−α) +
V = 0
U (−α − T (x))f (x)dx x
15
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
4
The common shock model
Consider a possible natural castrophe, modeled as an heterogeneous latent variable Θ, such that given Θ, the Yi ’s are independent, and P(Y = 1|Θ = Catastrophe) = p i C P(Yi = 1|Θ = No Catastrophe) = pN Let p? = P(Cat). Then the distribution of X is F (x)
= =
P(N ≤ [nx]) = P(N ≤ k|No Cat) × P(No Cat) + P(N ≤ k|Cat) × P(Cat) k X n j n−j ∗ j n−j ∗ (pN ) (1 − pN ) (1 − p ) + (pC ) (1 − pC ) p j j=0
16
1.0 0.8 0.6 0.4 0.2 0.0
Cumulative distribution function F
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● pN p pC
1−p*
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
20 15 10 5
pN
0
Probability density function f
Share of the population claiming a loss
0.0
0.2
p
pC
0.4
0.6
Share of the population claiming a loss
17
1.0 0.8 0.6 0.4 0.2 0.0
Cumulative distribution function F
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● p ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● pN pC
1−p*
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
20 15 10 5 0
Probability density function f
Share of the population claiming a loss
0.0
0.2
0.4
0.6
Share of the population claiming a loss
18
1.0 0.8 0.6 0.4 0.2 0.0
Cumulative distribution function F
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● pN p pC
1−p*
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
20 15 10 5 0
Probability density function f
Share of the population claiming a loss
0.0
0.2
0.4
0.6
Share of the population claiming a loss
19
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
4.1
Equilibriums in the EU framework
The expected profit of the insurance company is Z x¯ Π(α, p, δ, c) = [nα − xnl] f (x)dx − [1 − F (¯ x)]cn
(1)
0
Note that a premium less than the pure premium can lead to a positive expected profit. In Rothschild & Stiglitz (QJE, 1976) a positive profit was obtained if and only if α > p · l. Here companies have limited liabilities.
Proposition1 If agents are risk adverse, for a given premium , their expected utility is always higher with government intervention. Démonstration. Risk adverse agents look for mean preserving spread lotteries. 20
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Proposition2 From the expected utilities V , we obtain the following comparative static derivatives : ∂V ∂V ∂V ∂V < 0 for x ¯ > x∗ , < 0 for x ¯ > x∗ , > 0 for x ¯ ∈ [0; 1], =? ∂δ ∂p ∂c ∂α for x ¯ ∈ [0; 1].
Proposition3 From the equilibrium premium α∗ , we obtain the following comparative static derivatives : ∂α∗ < 0 for x ¯ > x∗ , ∂δ ∂α∗ =? for x ¯ > x∗ , ∂p ∂α∗ > 0 for x ¯ ∈ [0; 1], ∂c
21
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Expected profit>0
−20
0
Expected profit 0, ∂αj
∂αi∗∗ > 0, for i = 1, 2 and j 6= i. ∂αj∗∗
31
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Study of the two region model
●
0
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
50 40
●
10
REGION 1 BUYS
REGION 1 BUYS NO
INSURANCE
30
●
●
●
●
●
●
INSURANCE
10
●
●
●
●
●
●
●
●
●
●
●
0
10
20
30
INSURANCE
0
Premium in Region 2
BUYS
●
●
●
20
REGION 1
●
●
Premium in Region 2
40
50
The following graphs show the decision in Region 1, given that Region 2 buy insurance (on the left) or not (on the right).
20
30
Premium in Region 1
40
50
●
0
●
●
●
●
●
●
●
●
●
●
●
10
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
REGION 1 BUYS NO INSURANCE
20
30
40
50
Premium in Region 1
32
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Study of the two region model
●
0
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
50 40
●
10
BUYS NO
REGION 2 BUYS INSURANCE
20
30
Premium in Region 1
REGION 2 INSURANCE
30
●
●
●
●
●
●
10
●
●
●
●
●
●
●
●
●
●
●
0
10
20
30
INSURANCE
0
Premium in Region 2
BUYS NO
●
●
●
20
REGION 2
●
●
Premium in Region 2
40
50
The following graphs show the decision in Region 2, given that Region 1 buy insurance (on the left) or not (on the right).
40
50
●
0
●
●
●
●
●
●
●
●
●
●
●
10
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
REGION 2 BUYS INSURANCE
20
30
40
50
Premium in Region 1
33
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Definition1
40
0
10
10
20
30
Premium in Region 1
40
50
50 40 30
Premium in Region 2
20 10 0
50 40 30
Premium in Region 2
20 10 0
● ● ● ● ● ● ● ● ● ● ● REGION 2 ● ● ● BUYS NO ● ● ● INSURANCE ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● REGION 2 ● ● ● BUYS ● ● ● INSURANCE ● ● ● ● ● ● ● ● ●
0
20
30
40
50
Premium in Region 1
● ● ● ● ● ● ● ● ● ● ● REGION 2 ● ● ● BUYS NO ● ● ● INSURANCE ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● REGION 2 ● ● ● BUYS ● ● ● INSURANCE ● ● ● ● ● ● ● ● ●
0
10
20
30
Premium in Region 1
40
50
50 40
● ● ● ● ● ● ● ● ●
30
50 40 30 20
Premium in Region 2
50
● ● ● ●
● ● ● ● ● ● ● ● ● ●
20
30
● ● ● ●
● ● ● ● ● ● ● ● ● ●
10
20
Premium in Region 1
● ● ●
● ● ● ● ● ● ● ● ● ●
0
10
● ● ● ● ● ● ● ● ● ● ● REGION 1 ● ● ● BUYS ● ● ● INSURANCE ● ● ● ● ● ● ● ● ● ● ● ● ● ● REGION 1 ● ● ● BUYS NO ● ● ● INSURANCE ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Premium in Region 2
0
10
● ● ● ● ● ● ● ● ● ● ● REGION 1 ● ● ● BUYS ● ● ● INSURANCE ● ● ● ● ● ● ● ● ● ● ● ● ● ● REGION 1 ● ● ● BUYS NO ● ● ● INSURANCE ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
30 20 0
10
Premium in Region 2
40
50
In a Nash equilibrium which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally.
●
0
10
20
30
40
50
Premium in Region 1
34
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Definition2 In a Nash equilibrium which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally.
40
0
10
20
30
Premium in Region 1
40
50
50 40 30
Premium in Region 2
20 10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
10
30
40
50
1: non−insured, 2: non−insured
0
30 20 10 0
Premium in Region 2
40
50
1: non−insured, 2: insured
0
20
Premium in Region 1
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
10
20
30
Premium in Region 1
40
50
50 40
● ● ● ● ● ● ● ● ●
30
40 30 20
Premium in Region 2
50
● ● ● ●
● ● ● ● ● ● ● ● ● ●
20
30
● ● ● ●
● ● ● ● ● ● ● ● ● ●
10
20
Premium in Region 1
● ● ●
● ● ● ● ● ● ● ● ● ●
0
10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Premium in Region 2
0
10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
50
1: insured, 2: non−insured
0
30 20 0
10
Premium in Region 2
40
50
1: insured, 2: insured
●
0
10
20
30
40
50
Premium in Region 1
35
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Possible Nash equilibriums
10
30
40
0
50 40 30 20
Premium in Region 2
10
50
0
10
20
30
40
30
40
50 40 30
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
50
10
20
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
0
Premium in Region 1
10
20
30
−10
0
10
20
Premium in Region 1
30
● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
1: non−insured, 2: non−insured
40
●
0
10
20
Premium in Region 1
30
40
● ● ● ● ● ● ● ● ● ●
0
● ● ● ● ● ● ● ● ●
−10
40 30 20
Premium in Region 2
10 0 −10
40 30 20
Premium in Region 2
10 0 −10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
50
●
10
1: non−insured, 2: insured
40
●
40
Premium in Region 1
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
30
20
20 10
Premium in Region 2
30
40
1: insured, 2: non−insured
20
● ● ● ● ●
Premium in Region 1
1: insured, 2: insured
10
50
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Premium in Region 1
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
40
Premium in Region 1
0
40 30 20 10
Premium in Region 2
0 −10
20
0
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
10
30
1: non−insured, 2: non−insured
−10
30 20 0
10
Premium in Region 2
40
50
1: non−insured, 2: insured
0
20
Premium in Region 1
20
0
● ●
10
40 30 20
Premium in Region 2
10
50
● ● ●
0
40
● ● ● ● ●
40
30
● ● ● ● ●
30
20
Premium in Region 1
● ● ● ● ● ●
Premium in Region 2
10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
(−10:40)
0
50
1: insured, 2: non−insured
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
30 20 0
10
Premium in Region 2
40
50
1: insured, 2: insured
● ●
−10
0
10
20
30
40
(−10:40)
36
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Possible Nash equilibriums
0
10
20
30
40
● ● ● ● ● ● ● ● ●
30
● ● ● ● ● ● ● ● ● ● ●
20
30 20 10
Premium in Region 2
0
40
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
0
Premium in Region 1
10
20
30
●
40
Premium in Region 1
● ● ● ● ● ● ●
1: non−insured, 2: non−insured
●
30
0
10
20
30
30
40
0
● ●
−10
40 30 20
Premium in Region 2
10
● ●
−10
40
0
10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
0
Premium in Region 1
10
20
30
−10
0
10
20
Premium in Region 1
30
● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
1: non−insured, 2: non−insured
40
●
0
10
20
Premium in Region 1
30
40
● ● ● ● ● ● ● ● ● ●
0
● ● ● ● ● ● ● ● ●
−10
40 30 20
Premium in Region 2
10 0 −10
40 30 20
Premium in Region 2
10 0 −10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
40
●
10
1: non−insured, 2: insured
30
●
40
Premium in Region 1
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
20
20
10
20
30
40
1: insured, 2: non−insured
20
● ● ● ● ●
(−10:40)
1: insured, 2: insured
10
● ● ● ● ●
(−10:40)
0
−10
Premium in Region 1
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
0
40
● ● ● ● ● ●
40
20
●
30
10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Premium in Region 1
Premium in Region 2
0
0
20 10 −10
0
Premium in Region 2
30
40
−10
−10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
20 10 −10
0
Premium in Region 2
30
40
10
1: non−insured, 2: insured
●
(−10:40)
−10
40
1: insured, 2: non−insured
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
20 10 −10
0
Premium in Region 2
30
40
1: insured, 2: insured
● ●
−10
0
10
20
30
40
(−10:40)
37
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Possible Nash equilibriums
0
10
20
30
40
● ● ● ● ● ● ● ● ●
30
● ● ● ● ● ● ● ● ● ● ●
20
30 20 10
Premium in Region 2
0
40
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
0
Premium in Region 1
10
20
30
●
40
Premium in Region 1
● ● ● ● ● ● ●
1: non−insured, 2: non−insured
●
30
0
10
20
30
30
40
0
● ●
−10
40 30 20
Premium in Region 2
10
● ●
−10
40
0
10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
0
Premium in Region 1
10
20
30
−10
0
10
20
Premium in Region 1
30
● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
1: non−insured, 2: non−insured
40
●
0
10
20
Premium in Region 1
30
40
● ● ● ● ● ● ● ● ● ●
0
● ● ● ● ● ● ● ● ●
−10
40 30 20
Premium in Region 2
10 0 −10
40 30 20
Premium in Region 2
10 0 −10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
40
●
10
1: non−insured, 2: insured
30
●
40
Premium in Region 1
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
20
20
10
20
30
40
1: insured, 2: non−insured
20
● ● ● ● ●
(−10:40)
1: insured, 2: insured
10
● ● ● ● ●
(−10:40)
0
−10
Premium in Region 1
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
0
40
● ● ● ● ● ●
40
20
●
30
10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Premium in Region 1
Premium in Region 2
0
0
20 10 −10
0
Premium in Region 2
30
40
−10
−10
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
20 10 −10
0
Premium in Region 2
30
40
10
1: non−insured, 2: insured
●
(−10:40)
−10
40
1: insured, 2: non−insured
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10
20 10 −10
0
Premium in Region 2
30
40
1: insured, 2: insured
● ●
−10
0
10
20
30
40
(−10:40)
38
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
When the risks between two regions are not sufficiently independent, the pooling of the risks can lead to a Pareto improvement only if the regions have identical within-correlations, ceteris paribus. If the within-correlations are not equal, then the less correlated region needs the premium to decrease to accept the pooling of the risks.
39
Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
α
α α
P Q
0
a Starting situation: Q=P
α α
Q 0
c Increasing between-correlation
Q P
0
b Decreasing between-correlation
α
α α
α α
α α α
α α α
P
α
α
α
α α
α α α
P Q
d Increasing within-correlation in Region 1
40
