Slides hec-v3

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Arthur CHARPENTIER, Insurance of Natural Catastrophes

Insurance of Natural CatastrophesWhen Should Government Intervene ?

Arthur Charpentier & Benoît le Maux

Université Rennes 1 & École Polytechnique

[email protected]

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

Séminaire HEC Montréal, November 2010.

1

Arthur CHARPENTIER, Insurance of Natural Catastrophes 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 requires independence (Cummins & Mahul (JRI, 2004) or C. (GP,2008)).

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1.1 The French cat nat mecanism

Drought risk frequency, over the past 30 years.

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INSURANCE COMPANY

INSURANCE COMPANY

INSURANCE COMPANY

RE-INSURANCE COMPANYCAISSE CENTRALE DE REASSURANCE

GOVERNMENT

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INSURANCE COMPANY

INSURANCE COMPANY


GOVERNMENT

INSURANCE COMPANY

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INSURANCE COMPANY

INSURANCE COMPANY


GOVERNMENT

INSURANCE COMPANY

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1.2 Agenda

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

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Arthur CHARPENTIER, Insurance of Natural Catastrophes 2 DEMAND FOR INSURANCE

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 possiblebankruptcy of the insurance company, but as an exogenous variable. Here, wewant to make ruin endogenous.

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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

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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 ormutual fund insurance.

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Arthur CHARPENTIER, Insurance of Natural Catastrophes 3 THE ONE REGION MODEL

3 The one region model

Consider an homogeneous region with n agents, and let N = Y1 + · · ·+ Yn denotethe (random) number of agents claiming a loss, where

Yi =

1 if agent i claims a loss0 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 andidentical 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.

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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 insuredclaiming a loss. The indemnity function if then I(·)

I(X) = n(c+ α)N

= c+ α

Xif X > x

where x is such that, over a proportion x claiming a loss the insurance companyis ruined, i.e.

x = c+ α

l.

Let U(x) = u(ω + x), and U(0) = 0.

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RuinRuinRuinRuin

––––cncncncn

x α � c

l

α

l

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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

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3.1 Private insurance companies with limited liability

The objective function of the insured is V (α, p, δ, c) defined as

pF (x̄)U (−α) + p

∫ 1

x̄

U (−α− l + I(x)) f(x)dx+ (1− p)U(−α),

where F (x̄) = P(no ruin) which can be written as

V (α, p, δ, c) = U (−α)− p∫ 1

x̄

[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)

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3.2 (possible) Government intervention

T is a tax (or additional premium) collected by the government when X > x̄ :

T (X) = Nl − (α+ c)nn

= Xl − α− c ≥ 0

The expected utility of an agent buying insurance is

V (α, p, δ, c) = F (x̄)× U (−α) +∫ 1

x̄

U (−α− T (x)) f(x)dx,

which can be written as

V (α, p, δ, c) =

(−α)−∫ 1x̄

[U (−α)− U (−α− T (x))] f(x)dx

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• without governement intervention

V (α, p, δ, c) = U (−α)− p∫ 1

x̄

[U (−α)− U (−α− l + I(x))] f(x)dx

= U (−α)−∫ 1

x̄

A(x)f(x)dx

• with (possible) governement intervention

V (α, p, δ, c) = U (−α)−∫ 1

x̄

[U (−α)− U (−α− T (x))] f(x)dx

= U (−α)−∫ 1

x̄

B(x)f(x)dx

where x̄, I and T (i.e. A and B) depend on c and α, while f depends on p, δ (andn).

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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

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3.3 Some assumptions on F

∂F

∂p< 0 ∀x ∈ [0; 1], ∂F

∂δ< 0 ∀x > x∗,

∂2F

∂δ2 > 0 ∀x > x∗,∂2F

∂p2 > 0 ∀x > x∗,

for some x∗, where δ stands for the correlation between the individual risks.Proposition1The 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.Proposition1When the market has a monopolistic structure, a finite number of dependent risks implieslower equilibrium premiums than an infinite number of independent risks.

Let α? denote the equilibium premium.

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Proposition2The two models of natural catastrophe insurance lead to the following comparative staticderivatives :

∂V

∂δ< 0 and ∂α∗

∂δ< 0, if x̄ > x∗,

∂V

∂p< 0 and ∂α∗

∂p=?, if x̄ > x∗,

∂V

∂c> 0 and ∂α∗

∂c> 0, if x̄ ∈ [0; 1],

∂Π∂c

< 0 and ∂Π∂α∗ > 0, if x̄ ∈ [0; 1].

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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 identicallydistributed. 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−jp∗]

DefinepC = 1

1− δ · pN

so that pC and pN are function of p, p? and δ, where δ is a correlation coefficient.

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3.4.1 Distribution of X, F (·)

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3.4.2 Impact of p on F (·)

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3.4.3 Impact of n on F (·)

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3.4.4 Impact of p? on F (·)

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3.4.5 Impact of δ on F (·)

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3.5 Impact of parameters on ruin probability

• increase of p, claim occurrence probability : increase of ruin probability for allx ∈ [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]

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3.6 Optimal premiums for alternative covers

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3.7 Impact of δ (increasing) on optimal premiums

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Impact of δ (increasing) on optimal premiums

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Result1Without government intervention, the expected utility function first increases with thepremium and then decreases.

Result2A premium less than the pure premium can lead to a positive expected profit.

In Rothschild and Stiglitz (QJE, 1976), the expected profit for one individualcontract 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 alower bound). This can be related to ∂Π/∂c < 0.Remark1Here 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).

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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 efficienthousing and fire insurance than do private companies in competitive markets.Policy implication1The 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 astrong limit on the companies exposure (minimum capital per head for instance) or (2)implementing a minimum insurance requirement such as a minimum financial strengthrating.

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3.8 With or without government intervention ?

the expected utility with government intervention is higher than withoutintervention if and only if :∫ 1

x̄

U (B) f(x)dx ≥ (1− p) (1− F (x̄))U(−α) + p

∫ 1

x̄

U (A) f(x)dx (1)

where B = α− T (x) = c− xl and A = −α− l + I(x) = −α− l + c+αx .

Let denote ∆ the difference between the expected utilities :

∆ =∫ 1

x̄

[U (B)− (1− p)U(−α)− pU (A)] f(x)dx. (2)

Policy implication2Government intervention of last resort is not needed when the correlation between theindividual risks is high.

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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

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Arthur CHARPENTIER, Insurance of Natural Catastrophes 4 THE TWO REGION MODEL

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 2Insure 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)

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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) + pCNFCN (x) + pNCFNC(x) + pNNFNN (x),

withFij(·) ∼ B(n1, p

1i ) ? B(n2, p

2j ), where i, j ∈ {N,C}

where ? is the convolution operator.If δ was considered as a within correlation coefficient, θ will be betweencorrelation coefficient

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

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

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

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4.2.3 Impact of p?1 on F0(·) (decrease)

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4.2.4 Impact of θ on F0(·) (decrease)

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4.3 Decisions as a function of (α1, α2)

The 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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Decisions as a function of (α1, α2)

The 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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In a Nash equilibrium which each player is assumed to know the equilibriumstrategies of the other players, and no player has anything to gain by changingonly his or her own strategy unilaterally.

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45


4.4 Impact of correlation on Nash equilibriums

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46


Proposition3The two models of natural catastrophe insurance lead to the following comparative staticderivatives (for i = 1, 2 and i 6= j) :

∂Vi,0∂δi

< 0 and ∂α∗∗i

∂δi< 0, for x̄ > x∗,

∂Vi,0∂δj

< 0 and ∂α∗∗i

∂δj< 0, for x̄ > x∗,

∂Vi,0∂θ

< 0 and ∂α∗∗i

∂θ< 0, for x̄ > x∗,

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.

47


Policy implication3When the risks between two regions are not sufficiently independent, the pooling of therisks can lead to a Pareto improvement only if the regions have identicalwithin-correlations, ceteris paribus. If the within-correlations are not equal, then the lesscorrelated region needs the premium to decrease to accept the pooling of the risks.

Remark2This two region model can be used to model one region with heterogenous agents(utilities U1(·) and U2(·).

48

Arthur CHARPENTIER, Insurance of Natural Catastrophes 5 CONCLUSION(S)

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 aretoo correlated.

In the two region model, when regions are not sufficently independent, pooling ofrisks can be Pareto optimal, even if it is not an equilibrium.

49

Economy & Finance

Slides hec-v3