The document discusses insurance of natural catastrophes and when governments should intervene. It presents: 1) A model of private insurance with limited liability where insurers face ruin if claims exceed capital. Ruin probability increases with the share of the population claiming losses. 2) Government intervention could tax all policyholders to indemnify claims above insurer capital, reducing ruin probability. 3) A common shock model where a latent catastrophe variable determines correlated claims, giving the loss share distribution a spike at high losses. This increases ruin risk relative to independent claims.

1. 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
arthur.charpentier@univ-rennes1.fr
http ://freakonometrics.blog.free.fr/
Congrès Annuel de la SCSE, Sherbrooke, May 2011.
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2. 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)
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3. 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.
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4. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
GOVERNMENT
RE-INSURANCE COMPANY
CAISSE CENTRALE DE REASSURANCE
INSURANCE INSURANCE INSURANCE
COMPANY COMPANY COMPANY
4

5. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
GOVERNMENT
RE-INSURANCE COMPANY
CAISSE CENTRALE DE REASSURANCE
INSURANCE INSURANCE INSURANCE
COMPANY COMPANY COMPANY
5

6. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
GOVERNMENT
RE-INSURANCE COMPANY
CAISSE CENTRALE DE REASSURANCE
INSURANCE INSURANCE INSURANCE
COMPANY COMPANY COMPANY
6

7. 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)
no insurance insurance
i.e.
p · u(ω − l) + [1 − p] · u(ω − 0) ≤ u(ω − α)
no insurance insurance
i.e.
E[u(ω − X)] ≤ E[u(ω − α−l + I)]
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.
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8. 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.
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9. 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.
Ins. 1 Ins. 1 Ins. 3 Ins. 4 Ins. 5 Total
Premium 100 100 100 100 100 500
Loss - 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
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10. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
2.2 Possible government intervention
Ins. 1 Ins. 1 Ins. 3 Ins. 4 Ins. 5 Total
Premium 100 100 100 100 100 500
Loss - 1,000 - 1,000 - 2,000
Case 2 : possible government intervention
Tax -100 100 100 100 100 500
indemnity - 1,000 - 1,000 - 2,000
net -200 -200 -200 -200 -200 -1000
(note that it is a zero-sum game).
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11. 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
c+α
=⇒ ruin of the insurance company if X ≥ x = l
The indemnity function is

 l if X ≤ x
I(x) =
 c + α if X > x
n
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12. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
I(X)
I l
Positive profit Negative profit Ruin
[0 ; nα[ ]–cn ; 0[ –cn
c α
X
0 α α c 1
x
l l
Probability of no ruin: Probability of ruin:
F(x) 1–F(x)
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13. 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
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
1
V = [x · U (−α − l + I(x)) + (1 − x) · U (−α)]f (x)dx
0
that can be written
1
V = U (−α) − 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).
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14. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
3.2 Distorted risk perception by the insured
We’ve seen that
1
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
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).
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15. 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
1
V = [x · U (−α − T (x)) + (1 − x) · U (−α − T (x))]f (x)dx
0
i.e.
1 1
V = U (−α + T (x))f (x)dx = F (x) · U (−α) + U (−α − T (x))f (x)dx
0 x
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16. 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
n
= (pN )j (1 − pN )n−j (1 − p∗ ) + (pC )j (1 − pC )n−j p∗
j=0
j
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17. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Cumulative distribution function F
1.0
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqq
qq
qq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qq
qq
qq
1−p* q
q qq
qq
qq
qq
qq
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q
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qqqqqqqqqq
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qq
q
q
q
q
0.8
q
q
q
q
q
q
q
q
q
0.6
q
q
q
q
q
q
0.4
q
q
q
q
q
q
0.2 q
q
q
q
qq
q
q
q
qq
q
0.0
q
qqqqqqqqqqqqqqqqqqqqqpN
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq p pC
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
20
Probability density function f
15
10
5
pN p pC
0
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
17

18. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Cumulative distribution function F
1.0
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqq
qq
qq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qq
qq
qq
1−p* q
q qq
qq
qq
qq
qq
qq
q
qqq
qqqqqqqqq
qqqqqqqqqq
qqqqqqqqqq
qqqqqqqqqq
qqqqqqqqqq
qqqqqqqq
qq
q
q
q
q
0.8
q
q
q
q
q
q
q
q
q
0.6
q
q
q
q
q
q
0.4
q
q
q
q
q
q
0.2 q
q
q
q
qq
q
q
q
qq
q
0.0
q
qqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq pN p pC
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
20
Probability density function f
15
10
5
0
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
18

19. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Cumulative distribution function F
1.0
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqq
qq
qq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qq
qq
qq
1−p* q
q qq
qq
qq
qq
qq
qq
q
qqq
qqqqqqqqq
qqqqqqqqqq
qqqqqqqqqq
qqqqqqqqqq
qqqqqqqqqq
qqqqqqqq
qq
q
q
q
q
0.8
q
q
q
q
q
q
q
q
q
0.6
q
q
q
q
q
q
0.4
q
q
q
q
q
q
0.2 q
q
q
q
qq
q
q
q
qq
q
0.0
q
qqqqqqqqqqqqqqqqqqqqqpN
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq p pC
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
20
Probability density function f
15
10
5
0
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
19

20. 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
x
¯
Π(α, p, δ, c) = [nα − xnl] f (x)dx − [1 − F (¯)]cn
x (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.
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21. 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
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22. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Expected profit<0 Expected profit>0
0
−20
q
Expected utility
−40
q
−60
pU(−l)= −63.9
q q
0.00 0.05 0.10 0.15 0.20 0.25
Premium
22

23. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
4.2 Equilibriums in the non-EU framework
Assuming that the agents distort probabilities, they have to compare two
integrals,
1
V = U (−α) − Ak (x)f (x)dx
x
utility
Expected utility
U(–α)
Without government
intervention
(1–p)U(–α)+pU(c–l)
With government
intervention
U(c–l)
X
x 1
Probability density function
Slightly high correlation High correlation Very high correlation
X
x p p p 1
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24. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Expected profit<0 Expected profit>0
0
q
−20
q
Expected utility
−40
−60
pU(−l)= −63.9
q q
0.00 0.05 0.10 0.15 0.20 0.25
Premium
24

25. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
5 The 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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26. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
X1 ∼ F1 (x1 |p, δ1 ) = F1 (x1 ), (2)
X2 ∼ F2 (x2 |p, δ2 ) = F2 (x2 ), (3)
X0 ∼ F0 (x0 |F1 , F2 , θ) = F0 (x0 |p, δ1 , δ2 , θ) = F0 (x0 ), (4)
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 }.
26

27. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
1.0
Cumulative distribution function
1−p*
0.8
− Correlation beween: 0.01
0.6
− Region 1: within−correlation: 0.5
− Region 2: within−correlation: 0.5
0.4
0.2
0.0
pN1 p pC1
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
30
Probability density function
25
20
15
10
5
0
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
27

28. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
1.0
Cumulative distribution function
1−p*
0.8
− Correlation beween: 0.1
0.6
− Region 1: within−correlation: 0.5
− Region 2: within−correlation: 0.5
0.4
0.2
0.0
pN1 p pC1
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
30
Probability density function
25
20
15
10
5
0
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
28

29. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
1.0
Cumulative distribution function
1−p*
0.8
− Correlation beween: 0.01
0.6
− Region 1: within−correlation: 0.5
− Region 2: within−correlation: 0.7
0.4
0.2
0.0
pN1 p pC1
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
30
Probability density function
25
20
15
10
5
0
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
29

30. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
1.0
Cumulative distribution function
1−p*
0.8
− Correlation beween: 0.01
0.6
− Region 1: within−correlation: 0.5
− Region 2: within−correlation: 0.3
0.4
0.2
0.0
pN1 p pC1
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
30
Probability density function
25
20
15
10
5
0
0.0 0.2 0.4 0.6 0.8 1.0
Share of the population claiming a loss
30

31. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
Proposition4
When both regions decide to purchase insurance, the two-region models of natural
catastrophe insurance lead to the following comparative static derivatives :
∗∗
∂Vi,0 ∂αi
> 0, ∗∗ > 0, for i = 1, 2 and j = i.
∂αj ∂αj
31

32. Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ?
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
32