Slides hec-v3

Arthur CHARPENTIER, Insurance of Natural Catastrophes
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/
Séminaire HEC Montréal, November 2010.
1

1 Introduction and motivation
Insurance is “the contribution of the many
to the misfortune of the few”
The TELEMAQUE working group, 2005
Insurability requires independence (Cummins & Mahul (JRI, 2004) or C. (GP,
2008)).
2

1.1 The French cat nat mecanism
Drought risk frequency, over the past 30 years.
3

INSURANCE
COMPANY
INSURANCE
COMPANY
INSURANCE
COMPANY
RE-INSURANCE COMPANY
CAISSE CENTRALE DE REASSURANCE
GOVERNMENT
4

INSURANCE
COMPANY
INSURANCE
COMPANY
GOVERNMENT
INSURANCE
COMPANY
5

INSURANCE
COMPANY
INSURANCE
COMPANY
GOVERNMENT
INSURANCE
COMPANY
6

1.2 Agenda
• demand for insurance, integrating possible ruin
• model equilibriums on one homogeneous region
• policy implication
• an extension to two (correlated) regions
7

2 Demand for insurance
In Rothschild & Stiglitz (QJE, 1976), agent buy insurance if
E[u(ω − X)]
no insurance
≤ u(ω − α)
insurance
where L denote a random loss. Consider a binomial loss, then
pu(ω − l) + (1 − p)u(ω − 0) ≤ u(ω − α)
where agent can face loss l with probability p.
Doherty & Schlessinger (QJE, 1990) considered a model which integrates possible
bankruptcy of the insurance company, but as an exogenous variable. Here, we
want to make ruin endogenous.
8

Here we consider, where I denotes the indemnity
E[u(ω − X)]
no insurance
≤ u(ω − α − l + I)
insurance
2.1 Limited liability versus government intervention
Consider n = 10 insurance policies, possible loss $100 with probability 1/10.
Insurance company has capital u = 150.
policy 1 2 3 4 5 6 7 8 9 10
premium -11 -11 -11 -11 -11 -11 -11 -11 -11 -11
loss -100 -100 -100 -100
indemnity 65 65 65 65
net -46 -11 -46 -11 -11 -11 -46 -11 -11 -46
9

policy 1 2 3 4 5 6 7 8 9 10
premium -11 -11 -11 -11 -11 -11 -11 -11 -11 -11
loss -100 -100 -100 -100
indemnity 100 100 100 100
insurance 65 65 65 65
gvment 35 35 35 -35
net (b.t.) -11 -11 -11 -11 -11 -11 -11 -11 -11 -11
taxes -14 -14 -14 -14 -14 -14 -14 -14 -14 -14
net -25 -25 -25 -25 -25 -25 -25 -25 -25 -25
Here governement intervention might be understood as participating contracts or
mutual fund insurance.
10

3 The one region model
Consider an homogeneous region with n agents, and let N = Y1 + · · · + Yn denote
the (random) number of agents claiming a loss, where
Yi =



1 if agent i claims a loss
0 if not.
Let X = N/n denote the associated proportion, with distribution F(·).
Assume that all agents are homogeneous, i.e. have identical wealth ω and
identical vNM utility function u(·) (identical risk aversion). Here
P(Yi = 1) = p for all i = 1, · · · , n.
The insurance company has intial capital C = n · c, and ask for premium α.
11

The company
• has a positive proﬁt if N · l ≤ n · α,
• has a negative proﬁt if n · α ≤ N · l ≤ C + n · α,
• is bankrupted if C + n · α ≤ N · l.
The case of bankrucpcy, total capital (C + n · α) is splitted beween insured
claiming a loss. The indemnity function if then I(·)
I(X) =
n(c + α)
N
=
c + α
X
if X > x
where x is such that, over a proportion x claiming a loss the insurance company
is ruined, i.e.
x =
c + α
l
.
Let U(x) = u(ω + x), and U(0) = 0.
12

RuinRuinRuinRuin
––––cncncncn
xത ൌ
α ൅ c
l
α
l
10
Positive profitPositive profitPositive profitPositive profit
[0 ;[0 ;[0 ;[0 ; nnnnα[[[[
Negative profitNegative profitNegative profitNegative profit
]]]]––––cncncncn ;;;; 0[0[0[0[
IIII(X)(X)(X)(X)
Probability of no ruin:
F(xത)
Probability of ruin:
1–F(xത)
XXXX
Iൌl
c൅α
13

3.1 Private insurance companies with limited liability
The objective function of the insured is V (α, p, δ, c) deﬁned as
pF(¯x)U (−α) + p
1
¯x
U (−α − l + I(x)) f(x)dx + (1 − p)U(−α),
where F(¯x) = P(no ruin) which can be written as
V (α, p, δ, c) = U (−α) − p
1
¯x
[U (−α) − U (−α − l + I(x))] f(x)dx
Hence, the agent will buy insurance if and only if :
V (α, p, δ, c) ≥ pU(−l)
14

3.2 (possible) Government intervention
T is a tax (or additional premium) collected by the government when X > ¯x :
T(X) =
Nl − (α + c)n
n
= Xl − α − c ≥ 0
The expected utility of an agent buying insurance is
V (α, p, δ, c) = F(¯x) × U (−α) +
1
¯x
U (−α − T(x)) f(x)dx,
which can be written as
V (α, p, δ, c) =
(−α) −
1
¯x
[U (−α) − U (−α − T(x))] f(x)dx
15

• without governement intervention
V (α, p, δ, c) = U (−α) − p
1
¯x
[U (−α) − U (−α − l + I(x))] f(x)dx
= U (−α) −
1
¯x
A(x)f(x)dx
• with (possible) governement intervention
V (α, p, δ, c) = U (−α) −
1
¯x
[U (−α) − U (−α − T(x))] f(x)dx
= U (−α) −
1
¯x
B(x)f(x)dx
where ¯x, I and T (i.e. A and B) depend on c and α, while f depends on p, δ (and
n).
16

With government
intervention
x
(1–p)U(–α)+pU(c–l)
U(–α)
U(c–l)
X
Expected uExpected uExpected uExpected utilitytilitytilitytility
1
x
X
ProbabilityProbabilityProbabilityProbability densitydensitydensitydensity functionfunctionfunctionfunction
1pେ pେ
Slightly high correlation Very high correlation
Without government
intervention
High correlation
pେ
17

3.3 Some assumptions on F
∂F
∂p
< 0 ∀x ∈ [0; 1],
∂F
∂δ
< 0 ∀x > x∗
,
∂2
F
∂δ2
> 0 ∀x > x∗
,
∂2
F
∂p2
> 0 ∀x > x∗
,
for some x∗
, where δ stands for the correlation between the individual risks.
Proposition1
The expected profit of the insurance company is an increasing function of the premium.
As a result, maximizing the expected profit is equivalent to minimizing ruin probability.
In our two models, an insured agent will reach a utility below U(α). As a result,
the premium an agent is willing to pay is lower when ruin is possible.
Proposition1
When the market has a monopolistic structure, a finite number of dependent risks implies
lower equilibrium premiums than an infinite number of independent risks.
Let α denote the equilibium premium.
18

Proposition2
The two models of natural catastrophe insurance lead to the following comparative static
derivatives :
∂V
∂δ
< 0 and
∂α∗
∂δ
< 0, if ¯x > x∗
,
∂V
∂p
< 0 and
∂α∗
∂p
=?, if ¯x > x∗
,
∂V
∂c
> 0 and
∂α∗
∂c
> 0, if ¯x ∈ [0; 1],
∂Π
∂c
< 0 and
∂Π
∂α∗
> 0, if ¯x ∈ [0; 1].
19

3.4 A common shock model
Consider a natural catastrophe (based on some heterogeneous latent factor Θ)
such that given Θ probabilities to claim a loss are independent and identically
distributed. E.g.
P(Yi = 1|Θ = catastrophe ) = pC and P(Yi = 1|Θ = no catastrophe ) = pN .
Let p = P(Θ = catastrophe ). The distribution of X is
F(x) = P(N ≤ k) where k = nx
= P(N ≤ k|no cat) × P(no cat) + P(N ≤ k|cat) × P(cat)
=
k
j=0
n
j
(pN )j
(1 − pN )n−j
(1 − p∗
) + (pC)j
(1 − pC)n−j
p∗
Deﬁne
pC =
1
1 − δ
· pN
so that pC and pN are function of p, p and δ, where δ is a correlation coeﬃcient.
20

3.4.1 Distribution of X, F(·)
21

3.4.2 Impact of p on F(·)
22

3.4.3 Impact of n on F(·)
23

3.4.4 Impact of p on F(·)
24

3.4.5 Impact of δ on F(·)
25

3.5 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 xc ∈ (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]
26

3.6 Optimal premiums for alternative covers
27

3.7 Impact of δ (increasing) on optimal premiums
28

Impact of δ (increasing) on optimal premiums
29

Result1
Without government intervention, the expected utility function first increases with the
premium and then decreases.
Result2
A premium less than the pure premium can lead to a positive expected profit.
In Rothschild and Stiglitz (QJE, 1976), the expected profit for one individual
contract is Π = (1 − p)α + p(α − l), which is positive only if α > pl
Here we considered insurance companies with limited liability (i.e. loss has a
lower bound). This can be related to ∂Π/∂c < 0.
Remark1
Here we consider that insurance companies and agents have perfect (and identical)
information. For instance, insured know perfectly correlation implied by natural events
(see discussion on demand sensitiviy to financial ratings).
30

In insurance, the bigger, the stronger : natural monopoly.
Cf. Epple and Schäfer (1996), von Ungern-Sternberg (1996) and Felder (1996)
local state monopolies in Switzerland and Germany provide more efficient
housing and fire insurance than do private companies in competitive markets.
Policy implication1
The government should encourage the emergence of a monopoly and, as a result,
discipline the industry through regulated premiums. This can be done by (1) instituting a
strong limit on the companies exposure (minimum capital per head for instance) or (2)
implementing a minimum insurance requirement such as a minimum financial strength
rating.
31

3.8 With or without government intervention ?
the expected utility with government intervention is higher than without
intervention if and only if :
1
¯x
U (B) f(x)dx ≥ (1 − p) (1 − F(¯x)) U(−α) + p
1
¯x
U (A) f(x)dx (1)
where B = α − T(x) = c − xl and A = −α − l + I(x) = −α − l + c+α
x .
Let denote ∆ the diﬀerence between the expected utilities :
∆ =
1
¯x
[U (B) − (1 − p)U(−α) − pU (A)] f(x)dx. (2)
Policy implication2
Government intervention of last resort is not needed when the correlation between the
individual risks is high.
32

With government
intervention
x
(1–p)U(–α)+pU(c–l)
U(–α)
U(c–l)
X
Expected uExpected uExpected uExpected utilitytilitytilitytility
1
x
X
ProbabilityProbabilityProbabilityProbability densitydensitydensitydensity functionfunctionfunctionfunction
1pେ pେ
Slightly high correlation Very high correlation
Without government
intervention
High correlation
pେ
33

4 The two region model
Consider two regions, with n1 and n2 agents, respectively.
• either no one buy insurance,
• either only people in region 1 buy insurance,
• either only people in region 2 buy insurance,
• or everyone buy insurance.
This can be seen as a standard non-cooperative game between agents in region 1,
and agents in region 2.
Region 2
Insure Don’t
Region 1 Insure V1,0(α1, α2, F0, c), V2,0(α1, α2, F0, c) V1,1(α1, F1, c), p2U(−l)
Don’t p1U(−l), V2,2(α2, F2, c) p1U(−l), p2U(−l)
34

4.1 The two region common shock model
Consider a four state latent variable Θ = (Θ1, Θ2),
• Θ = (cat, cat) with probability pCC = θ,
max{p1 + p2 − 1, 0} ≤ θ ≤ min{p1, p2},
• Θ = (cat, no cat) with probability pCN = p1 − θ,
• Θ = (no cat, cat) with probability pNC = p2 − θ,
• Θ = (no cat, no cat) with probability pNN = 1 − p1 − p2 + θ.
The (nonconditional) distribution F0 is a mixture of four distributions,
F0(x) = pCCFCC(x) + pCN FCN (x) + pNCFNC(x) + pNN FNN (x),
with
Fij(·) ∼ B(n1, p1
i ) B(n2, p2
j ), where i, j ∈ {N, C}
where is the convolution operator.
If δ was considered as a within correlation coeﬃcient, θ will be between
correlation coeﬃcient
35

4.2 Distribution of X = (N1 + N2)/(n1 + n2), F0(·)
36

4.2.1 Impact of n1 on F0(·) (decrease)
37

4.2.2 Impact of δ1 on F0(·) (decrease)
38

4.2.3 Impact of p1 on F0(·) (decrease)
39

4.2.4 Impact of θ on F0(·) (decrease)
40

4.3 Decisions as a function of (α1, α2)
The following graphs show the decision in Region 1, given that Region 2 buy
insurance (on the left) or not (on the right).
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BUYS NO
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Premium in Region 1
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REGION 1
BUYS
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REGION 1
BUYS NO
INSURANCE
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Decisions as a function of (α1, α2)
The following graphs show the decision in Region 2, given that Region 1 buy
insurance (on the left) or not (on the right).
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Deﬁnition1
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.
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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.
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1: insured, 2: insured
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1: insured, 2: non−insured
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1: non−insured, 2: insured
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1: non−insured, 2: non−insured
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q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
Premium in Region 1
PremiuminRegion2
0 10 20 30 40 50
01020304050
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
Premium in Region 1
PremiuminRegion2
0 10 20 30 40 50
01020304050
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
Premium in Region 1
PremiuminRegion2
−10 0 10 20 30 40
−10010203040
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
Premium in Region 1
PremiuminRegion2
−10 0 10 20 30 40
−10010203040
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
Premium in Region 1
PremiuminRegion2
−10 0 10 20 30 40
−10010203040
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
Premium in Region 1
PremiuminRegion2
−10 0 10 20 30 40
−10010203040
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
−10 0 10 20 30 40
−10010203040
(−10:40)
(−10:40)
45

4.4 Impact of correlation on Nash equilibriums
હ૛ αଵ
‫ככ‬
ሺαଶ
‫ככ‬
ሻαଵ
‫כ‬
હ૚
αଶ
‫ככ‬
ሺαଵ
‫ככ‬
ሻ
αଶ
‫כ‬
0000
ሺaሻ Starting situation: Q=P ሺbሻ Decreasing between-correlation
ሺcሻ Increasing between-correlation ሺdሻ Increasing within-correlation in Region 1
Q
P
હ૛ αଵ
‫ככ‬
ሺαଶ
‫ככ‬
ሻαଵ
‫כ‬
αଶ
‫ככ‬
ሺαଵ
‫ככ‬
ሻ
αଶ
‫כ‬
P
હ૛ αଵ
‫ככ‬
ሺαଶ
‫ככ‬
ሻ αଵ
‫כ‬
હ૚
αଶ
‫כ‬
0000
P
Q
હ૛ αଵ
‫ככ‬
ሺαଶ
‫ככ‬
ሻαଵ
‫כ‬
હ૚
αଶ
‫ככ‬
ሺαଵ
‫ככ‬
ሻ
αଶ
‫כ‬
0000
Q
P
αଶ
‫ככ‬
ሺαଵ
‫ככ‬
ሻ
Q
હ૚
46

Proposition3
The two models of natural catastrophe insurance lead to the following comparative static
derivatives (for i = 1, 2 and i = j) :
∂Vi,0
∂δi
< 0 and
∂α∗∗
i
∂δi
< 0, for ¯x > x∗
,
∂Vi,0
∂δj
< 0 and
∂α∗∗
i
∂δj
< 0, for ¯x > x∗
,
∂Vi,0
∂θ
< 0 and
∂α∗∗
i
∂θ
< 0, for ¯x > x∗
,
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
∂αj
> 0,
∂α∗∗
i
∂α∗∗
j
> 0, for i = 1, 2 and j = i.
47

Policy implication3
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.
Remark2
This two region model can be used to model one region with heterogenous agents
(utilities U1(·) and U2(·).
48

5 Conclusion(s)
Correlations is a crucial element in the analysis, both within correlations (δi’s)
and between correlation (θ).
It helps to understand empirical evidences provided in Switzerland and Germany.
In the one region model, surprisingly, governement is not needed when risks are
too correlated.
In the two region model, when regions are not suﬃcently independent, pooling of
risks can be Pareto optimal, even if it is not an equilibrium.
49

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