Congrès SCSE Cat Nat

11pt

11pt

Note Example Example

11pt

Proof

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

Insurance of Natural CatastrophesWhen 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

http://freakonometrics.blog.free.fr/

Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ? 1 INTRODUCTION AND MOTIVATION

1 Introduction and motivation

Insurance is “the contribution of the manyto the misfortune of the few”.

The TELEMAQUE working group, 2005.

Insurability requieres independenceCummins & 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


INSURANCE COMPANY

INSURANCE COMPANY

INSURANCE COMPANY

RE-INSURANCE COMPANYCAISSE CENTRALE DE REASSURANCE

GOVERNMENT

4


INSURANCE COMPANY

INSURANCE COMPANY


GOVERNMENT

INSURANCE COMPANY

5


INSURANCE COMPANY

INSURANCE COMPANY


GOVERNMENT

INSURANCE COMPANY

6

Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ? 2 DEMAND FOR INSURANCE

2 Demand for insurance

An agent purchases insurance if

E[u(ω −X)]︸︷︷︸no insurance

≤ u(ω − α)︸︷︷︸insurance

i.e.p · u(ω − l) + [1− p] · u(ω − 0)︸︷︷︸

no insurance

≤ u(ω − α)︸︷︷︸insurance

i.e.E[u(ω −X)]︸︷︷︸

no insurance

≤ E[u(ω − α−l + I)]︸︷︷︸insurance

Doherty & Schlessinger (1990) considered a model which integrates possiblebankruptcy of the insurance company, but as an exogenous variable. Here, wewant to make ruin endogenous.

7


Yi =

0 if agent i claims a loss1 if not

Let N = Y1 + · · ·+Xn denote the number of insured claiming a loss, andX = 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 functionsu(·).

=⇒ 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 TotalPremium 100 100 100 100 100 500Loss - 1,000 - 1,000 - 2,000

Case 1 : insurance company with limited liabilityindemnity - 750 - 750 - 1,500loss - -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 TotalPremium 100 100 100 100 100 500Loss - 1,000 - 1,000 - 2,000

Case 2 : possible government interventionTax -100 100 100 100 100 500indemnity - 1,000 - 1,000 - 2,000net -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

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+αl

The indemnity function is

I(x) =

l if X ≤ xc+ α

nif X > x

11


RuinRuinRuinRuin

––––cncncncn

x 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(x)

Probability of ruin:

1–F(x)

XXXX

Il

c�α

12


Without ruin, the objective function of the insured is V (α, p, δ, c) defined asU(−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 thatV =

∫ 1

0[x · U(−α− l + I(x)) + (1− x) · U(−α)]f(x)dx

that can be written

V = U(−α)−∫ 1

0x[U(−α)− U(−α− l + I(x))]f(x)dx

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

V = U(−α)−∫ 1

0x[U(−α)− U(−α− l + I(x))]f(x)dx

since P(Yi = 1|X = x) = x (while P(Yi = 1) = p).

But in the model in the Working Paper (first version), we wrote

V = U(−α)−∫ 1

0p[U(−α)− U(−α− l + I(x))]f(x)dx

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

T (x) =

0 if X ≤ xNl − (α+ c)n

n= Xl − α− c if X > x

Then

V =∫ 1

0[x · U(−α− T (x)) + (1− x) · U(−α− T (x))]f(x)dx

i.e.

V =∫ 1

0U(−α+ T (x))f(x)dx = F (x) · U(−α) +

∫ 1

x

U(−α− T (x))f(x)dx

15

Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ? 4 THE COMMON SHOCK MODEL

4 The common shock model

Consider a possible natural castrophe, modeled as an heterogeneous latentvariable Θ, such that given Θ, the Yi’s are independent, and P(Yi = 1|Θ = Catastrophe) = pC

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∑j=0

(n

j

)[(pN )j(1− pN )n−j(1− p∗) + (pC)j(1− pC)n−jp∗]

16


●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Share of the population claiming a loss

Cum

ulat

ive

dist

ribut

ion

func

tion

F

pN pCp

1−p*

0.0 0.2 0.4 0.6 0.8 1.0

05

1015

20


Pro

babi

lity

dens

ity fu

nctio

n f

pN pCp

17


●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0


Cum

ulat

ive

dist

ribut

ion

func

tion

F

pN pCp

1−p*

0.0 0.2 0.4 0.6 0.8 1.0

05

1015

20


Pro

babi

lity

dens

ity fu

nctio

n f

18


●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0


Cum

ulat

ive

dist

ribut

ion

func

tion

F

pN pCp

1−p*

0.0 0.2 0.4 0.6 0.8 1.0

05

1015

20


Pro

babi

lity

dens

ity fu

nctio

n f

19


4.1 Equilibriums in the EU framework

The expected profit of the insurance company is

Π(α, p, δ, c) =∫ x̄

0[nα− xnl] f(x)dx− [1− F (x̄)]cn (1)

Note that a premium less than the pure premium can lead to a positive expectedprofit.

In Rothschild & Stiglitz (QJE, 1976) a positive profit was obtained if and only ifα > p · l. Here companies have limited liabilities.Proposition1If agents are risk adverse, for a given premium , their expected utility is always higherwith government intervention.

Démonstration. Risk adverse agents look for mean preserving spreadlotteries.

20


Proposition2From the expected utilities V , we obtain the following comparative static derivatives :

∂V

∂δ< 0 for x̄ > x∗,

∂V

∂p< 0 for x̄ > x∗,

∂V

∂c> 0 for x̄ ∈ [0; 1], ∂V

∂α=?

for x̄ ∈ [0; 1].Proposition3From the equilibrium premium α∗, we obtain the following comparative staticderivatives :

∂α∗

∂δ< 0 for x̄ > x∗,

∂α∗

∂p=? for x̄ > x∗,

∂α∗

∂c> 0 for x̄ ∈ [0; 1],

21


0.00 0.05 0.10 0.15 0.20 0.25

−60

−40

−20

0

Premium

Exp

ecte

d ut

ility

pU(−l)= −63.9

●●

●

●● ●pU(−l)= −63.9

Expected profit<0 Expected profit>0

22


4.2 Equilibriums in the non-EU framework

Assuming that the agents distort probabilities, they have to compare twointegrals,

V = U(−α)−∫ 1

x

Ak(x)f(x)dx

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

Probability Probability Probability Probability densitydensitydensitydensity functionfunctionfunctionfunction

1 p p

Slightly high correlation Very high correlation

Without government

intervention

High correlation

p

23


0.00 0.05 0.10 0.15 0.20 0.25

−60

−40

−20

0

Premium

Exp

ecte

d ut

ility

pU(−l)= −63.9

●●

●

●●●pU(−l)= −63.9

Expected profit<0 Expected profit>0

24

Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ? 5 THE TWO REGION MODEL

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, andset 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 δ0Here, δi = 1− piN/piC , where i = 1, 2 (Regions), while δ0 ∈ [0, 1] is such that

P(Θ = (1, 1)) = δ0 ×min{P(Θ = (1, ·)),P(Θ = (·, 1))} = δ0 ×min{p?1, p?2}.

26


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0


Cum

ulat

ive

dist

ribut

ion

func

tion

pN1 pC1p

1−p*

− Correlation beween: 0.01

− Region 1: within−correlation: 0.5


0.0 0.2 0.4 0.6 0.8 1.0

05

1015

2025

30


Pro

babi

lity

dens

ity fu

nctio

n

27


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0


Cum

ulat

ive

dist

ribut

ion

func

tion

pN1 pC1p

1−p*




0.0 0.2 0.4 0.6 0.8 1.0

05

1015

2025

30


Pro

babi

lity

dens

ity fu

nctio

n

28


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0


Cum

ulat

ive

dist

ribut

ion

func

tion

pN1 pC1p

1−p*




0.0 0.2 0.4 0.6 0.8 1.0

05

1015

2025

30


Pro

babi

lity

dens

ity fu

nctio

n

29


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0


Cum

ulat

ive

dist

ribut

ion

func

tion

pN1 pC1p

1−p*




0.0 0.2 0.4 0.6 0.8 1.0

05

1015

2025

30


Pro

babi

lity

dens

ity fu

nctio

n

30


Proposition4When both regions decide to purchase insurance, the two-region models of naturalcatastrophe insurance lead to the following comparative static derivatives :

∂Vi,0∂αj

> 0, ∂α∗∗i

∂α∗∗j

> 0, for i = 1, 2 and j 6= i.

31


Study of the two region modelThe following graphs show the decision in Region 1, given that Region 2 buyinsurance (on the left) or not (on the right).

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Study of the two region modelThe following graphs show the decision in Region 2, given that Region 1 buyinsurance (on the left) or not (on the right).

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Definition1In a Nash equilibrium which each player is assumed to know the equilibrium strategiesof the other players, and no player has anything to gain by changing only his or her ownstrategy unilaterally.

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Definition2In a Nash equilibrium which each player is assumed to know the equilibrium strategiesof the other players, and no player has anything to gain by changing only his or her ownstrategy unilaterally.

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●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●

1: non−insured, 2: insured

Premium in Region 1

Pre

miu

m in

Reg

ion

2

0 10 20 30 40 50

010

2030

4050

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●

1: non−insured, 2: non−insured

Premium in Region 1

Pre

miu

m in

Reg

ion

2

0 10 20 30 40 50

010

2030

4050

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

Premium in Region 1

Pre

miu

m in

Reg

ion

2

0 10 20 30 40 50

010

2030

4050

35


Possible Nash equilibriums

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●


Premium in Region 1

Prem

ium in

Reg

ion 2

0 10 20 30 40 50

010

2030

4050

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●


Premium in Region 1

Prem

ium in

Reg

ion 2

0 10 20 30 40 50

010

2030

4050

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●


Premium in Region 1

Prem

ium in

Reg

ion 2

0 10 20 30 40 50

010

2030

4050

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●


Premium in Region 1

Prem

ium in

Reg

ion 2

0 10 20 30 40 50

010

2030

4050

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

Premium in Region 1

Prem

ium in

Reg

ion 2

0 10 20 30 40 50

010

2030

4050

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●


Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●


Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●


Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●


Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

−10 0 10 20 30 40

−10

010

2030

40

(−10:40)

(−10

:40)

36



●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

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●●

●●

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●●

●●

●●

●●

●●

●●

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●●

●●

●●

●●

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●●

●●

●●

●●

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●

●

●

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●

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●

●

●

●

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●

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−10 0 10 20 30 40

−10

010

2030

40

(−10:40)

(−10

:40)

●●

●●

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●●

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●●

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●●

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●●

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●

●

●

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●

●

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●

●

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●

●

●

●

●

●

●

−10 0 10 20 30 40

−10

010

2030

40

(−10:40)

(−10

:40)

37



●●

●●

●●

●●

●●

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●●

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●

●

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●

●

●

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●

●

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●

●

●

●

●

●

●

●

●

−10 0 10 20 30 40

−10

010

2030

40

(−10:40)

(−10

:40)

●●

●●

●●

●●

●●

●●

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●●

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●●

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

●●

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Premium in Region 1

Prem

ium in

Reg

ion 2

−10 0 10 20 30 40

−10

010

2030

40

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Premium in Region 1

Prem

ium in

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−10 0 10 20 30 40

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−10 0 10 20 30 40

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010

2030

40

(−10:40)

(−10

:40)

38


When the risks between two regions are not sufficiently independent, the poolingof the risks can lead to a Pareto improvement only if the regions have identicalwithin-correlations, ceteris paribus. If the within-correlations are not equal, thenthe less correlated region needs the premium to decrease to accept the pooling ofthe risks.

39


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40

Documents

Congrès SCSE Cat Nat