Catnat

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.

1


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


1.1 The French cat nat mecanism
=⇒ natural catastrophes means no independence

Drought risk frequency, over 30 years, in France.

3


GOVERNMENT

RE-INSURANCE COMPANY
CAISSE CENTRALE DE REASSURANCE

INSURANCE INSURANCE INSURANCE
COMPANY COMPANY COMPANY

4


GOVERNMENT



5


GOVERNMENT



6


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)
i.e.
E[u(ω − X)] ≤ E[u(ω − α−l + I)]

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



 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


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

9


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).

10


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

11


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)

12


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).

13


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).

14


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

15


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

16


Cumulative distribution function F

1.0
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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 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


17



1.0
qqq
qq
qq
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 pN p pC

0.0 0.2 0.4 0.6 0.8 1.0

20

15
10
5
0

0.0 0.2 0.4 0.6 0.8 1.0


18



1.0
qqq
qq
qq
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 p pC

0.0 0.2 0.4 0.6 0.8 1.0

20

15
10
5
0

0.0 0.2 0.4 0.6 0.8 1.0


19


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.

20


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


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


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

23


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


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 .

25


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


1.0
Cumulative distribution function
1−p*

0.8
− Correlation beween: 0.01

0.6
− Region 1: 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

30

25
20
15
10
5
0

0.0 0.2 0.4 0.6 0.8 1.0


27


1.0
1−p*

0.8

0.6

0.4
0.2
0.0

pN1 p pC1

0.0 0.2 0.4 0.6 0.8 1.0

30

25
20
15
10
5
0

0.0 0.2 0.4 0.6 0.8 1.0


28


1.0
1−p*

0.8

0.6

0.4
0.2
0.0

pN1 p pC1

0.0 0.2 0.4 0.6 0.8 1.0

30

25
20
15
10
5
0

0.0 0.2 0.4 0.6 0.8 1.0


29


1.0
1−p*

0.8

0.6

0.4
0.2
0.0

pN1 p pC1

0.0 0.2 0.4 0.6 0.8 1.0

30

25
20
15
10
5
0

0.0 0.2 0.4 0.6 0.8 1.0


30


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


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

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