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Insurance of Natural Catastrophes When Should Government Intervene ? - - PowerPoint PPT Presentation

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Arthur CHARPENTIER, Insurance of natural catastrophes: when should governments intervene ? Insurance of Natural Catastrophes When Should Government Intervene ? Arthur Charpentier & Benot le Maux Universit Rennes 1 & cole

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✶✶♣t ✶✶♣t ◆♦t❡ ❊①❛♠♣❧❡ ❊①❛♠♣❧❡ ✶✶♣t Pr♦♦❢

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/

Séminaire Université Laval, Québec, Février 2011. 1

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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) 2

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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. 3

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

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

4

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

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

5

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

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

6

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

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 possible bankruptcy of the insurance company, but as an exogenous variable. Here, we want to make ruin endogenous. 7

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

Yi =    0 if agent i claims a loss 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

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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

9

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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). 10

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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

= ⇒ ruin of the insurance company if X ≥ x = c+α

l

The indemnity function is I(x) =    l if X ≤ x c + α n if X > x 11

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

Ruin Ruin Ruin Ruin – – – –cn cn cn cn

x α c l α l 1

Positive profit Positive profit Positive profit Positive profit [0 ; [0 ; [0 ; [0 ; n n n nα[ [ [ [ Negative profit Negative profit Negative profit Negative profit ] ] ] ]– – – –cn cn cn cn ; ; ; ; 0[ 0[ 0[ 0[

I I I I(X) (X) (X) (X) Probability of no ruin: F(x ) Probability of ruin: 1–F(x ) X X X X Il cα

12

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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 V = 1 [x · U(−α − l + I(x)) + (1 − x) · U(−α)]f(x)dx that can be written V = U(−α) − 1 x[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

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

3.2 Distorted risk perception by the insured

We’ve seen that V = U(−α) − 1 x[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 p[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

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

3.3 Government intervention (or mutual fund insurance)

The tax function is T(x) =    0 if X ≤ x Nl − (α + c)n n = Xl − α − c if X > x Then V = 1 [x · U(−α − T(x)) + (1 − x) · U(−α − T(x))]f(x)dx i.e. V = 1 U(−α + T(x))f(x)dx = F(x) · U(−α) + 1

x

U(−α − T(x))f(x)dx 15

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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(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) (1) =

k

  • j=0

n j (pN)j(1 − pN)n−j(1 − p∗) + (pC)j(1 − pC)n−jp∗ (2) 16

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

  • 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 Cumulative distribution function F

pN pC p 1−p*

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Share of the population claiming a loss Probability density function f

pN pC p

17

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

  • 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 Cumulative distribution function F

pN pC p 1−p*

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Share of the population claiming a loss Probability density function f

18

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

  • 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 Cumulative distribution function F

pN pC p 1−p*

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Share of the population claiming a loss Probability density function f

19

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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 Π(α, p, δ, c) = ¯

x

[nα − xnl] f(x)dx − [1 − F (¯ x)]cn (3) 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

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

Proposition2

From the expected utilities V , we obtain the following comparative static derivatives : ∂V ∂δ < 0 for ¯ x > x∗, ∂V ∂p < 0 for ¯ x > x∗, ∂V ∂c > 0 for ¯ x ∈ [0; 1], ∂V ∂α =? for ¯ x ∈ [0; 1].

Proposition3

From the equilibrium premium α∗, we obtain the following comparative static derivatives : ∂α∗ ∂δ < 0 for ¯ x > x∗, ∂α∗ ∂p =? for ¯ x > x∗, ∂α∗ ∂c > 0 for ¯ x ∈ [0; 1], 21

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

0.00 0.05 0.10 0.15 0.20 0.25 −60 −40 −20 Premium Expected utility

pU(−l)= −63.9

  • pU(−l)= −63.9

Expected profit<0 Expected profit>0

22

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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, 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 u Expected u Expected u Expected utility tility tility tility

1 x X

Probability Probability Probability Probability density density density density function function function function

1 p p

Slightly high correlation Very high correlation Without government intervention High correlation

p

23

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

0.00 0.05 0.10 0.15 0.20 0.25 −60 −40 −20 Premium Expected utility

pU(−l)= −63.9

  • pU(−l)= −63.9

Expected profit<0 Expected profit>0

24

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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. 25

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

X1 ∼ F1(x1|p, δ1) = F1(x1), (4) X2 ∼ F2(x2|p, δ2) = F2(x2), (5) X0 ∼ F0(x0|F1, F2, θ) = F0(x0|p, δ1, δ2, θ) = F0(x0), (6) 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

N/pi C, 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

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

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 Cumulative distribution function

pN1 pC1 p 1−p*

− Correlation beween: 0.01 − Region 1: within−correlation: 0.5 − Region 2: within−correlation: 0.5

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 25 30 Share of the population claiming a loss Probability density function

27

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

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 Cumulative distribution function

pN1 pC1 p 1−p*

− Correlation beween: 0.1 − Region 1: within−correlation: 0.5 − Region 2: within−correlation: 0.5

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 25 30 Share of the population claiming a loss Probability density function

28

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

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 Cumulative distribution function

pN1 pC1 p 1−p*

− Correlation beween: 0.01 − Region 1: within−correlation: 0.5 − Region 2: within−correlation: 0.7

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 25 30 Share of the population claiming a loss Probability density function

29

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

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 Cumulative distribution function

pN1 pC1 p 1−p*

− Correlation beween: 0.01 − Region 1: within−correlation: 0.5 − Region 2: within−correlation: 0.3

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 25 30 Share of the population claiming a loss Probability density function

30

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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 ∂αj > 0, ∂α∗∗

i

∂α∗∗

j

> 0, for i = 1, 2 and j = i. 31

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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).

  • Premium in Region 1

Premium in Region 2 10 20 30 40 50 10 20 30 40 50

REGION 1 BUYS INSURANCE REGION 1 BUYS NO INSURANCE

  • Premium in Region 1

Premium in Region 2 10 20 30 40 50 10 20 30 40 50

REGION 1 BUYS INSURANCE REGION 1 BUYS NO INSURANCE

32

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

Study of the two region model

The following graphs show the decision in Region 2, given that Region 1 buy insurance (on the left) or not (on the right).

  • Premium in Region 1

Premium in Region 2 10 20 30 40 50 10 20 30 40 50

REGION 2 BUYS NO INSURANCE REGION 2 BUYS INSURANCE

  • Premium in Region 1

Premium in Region 2 10 20 30 40 50 10 20 30 40 50

REGION 2 BUYS NO INSURANCE REGION 2 BUYS INSURANCE

33

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

Definition1

In a Nash equilibrium which each player is assumed to know the equilibrium strategies

  • f the other players, and no player has anything to gain by changing only his or her own

strategy unilaterally.

  • Premium in Region 1

Premium in Region 2 10 20 30 40 50 10 20 30 40 50

REGION 1 BUYS INSURANCE REGION 1 BUYS NO INSURANCE

  • Premium in Region 1

Premium in Region 2 10 20 30 40 50 10 20 30 40 50

REGION 1 BUYS INSURANCE REGION 1 BUYS NO INSURANCE

  • Premium in Region 1

Premium in Region 2 10 20 30 40 50 10 20 30 40 50

REGION 2 BUYS NO INSURANCE REGION 2 BUYS INSURANCE

  • Premium in Region 1

Premium in Region 2 10 20 30 40 50 10 20 30 40 50

REGION 2 BUYS NO INSURANCE REGION 2 BUYS INSURANCE

  • Premium in Region 1

Premium in Region 2 10 20 30 40 50 10 20 30 40 50

34

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

Definition2

In a Nash equilibrium which each player is assumed to know the equilibrium strategies

  • f the other players, and no player has anything to gain by changing only his or her own

strategy unilaterally.

  • 1: insured, 2: insured

Premium in Region 1 Premium in Region 2 10 20 30 40 50 10 20 30 40 50

  • 1: insured, 2: non−insured

Premium in Region 1 Premium in Region 2 10 20 30 40 50 10 20 30 40 50

  • 1: non−insured, 2: insured

Premium in Region 1 Premium in Region 2 10 20 30 40 50 10 20 30 40 50

  • 1: non−insured, 2: non−insured

Premium in Region 1 Premium in Region 2 10 20 30 40 50 10 20 30 40 50

  • Premium in Region 1

Premium in Region 2 10 20 30 40 50 10 20 30 40 50

35

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

Possible Nash equilibriums

  • 1: insured, 2: insured

Premium in Region 1 Premium in Region 2 10 20 30 40 50 10 20 30 40 50

  • 1: insured, 2: non−insured

Premium in Region 1 Premium in Region 2 10 20 30 40 50 10 20 30 40 50

  • 1: non−insured, 2: insured

Premium in Region 1 Premium in Region 2 10 20 30 40 50 10 20 30 40 50

  • 1: non−insured, 2: non−insured

Premium in Region 1 Premium in Region 2 10 20 30 40 50 10 20 30 40 50

  • Premium in Region 1

Premium in Region 2 10 20 30 40 50 10 20 30 40 50

  • 1: insured, 2: insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: insured, 2: non−insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: non−insured, 2: insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: non−insured, 2: non−insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • −10

10 20 30 40 −10 10 20 30 40 (−10:40) (−10:40)

36

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

Possible Nash equilibriums

  • 1: insured, 2: insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: insured, 2: non−insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: non−insured, 2: insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: non−insured, 2: non−insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • −10

10 20 30 40 −10 10 20 30 40 (−10:40) (−10:40)

  • 1: insured, 2: insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: insured, 2: non−insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: non−insured, 2: insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: non−insured, 2: non−insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • −10

10 20 30 40 −10 10 20 30 40 (−10:40) (−10:40)

37

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

Possible Nash equilibriums

  • 1: insured, 2: insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: insured, 2: non−insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: non−insured, 2: insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: non−insured, 2: non−insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • −10

10 20 30 40 −10 10 20 30 40 (−10:40) (−10:40)

  • 1: insured, 2: insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: insured, 2: non−insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: non−insured, 2: insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • 1: non−insured, 2: non−insured

Premium in Region 1 Premium in Region 2 −10 10 20 30 40 −10 10 20 30 40

  • −10

10 20 30 40 −10 10 20 30 40 (−10:40) (−10:40)

38

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

When the risks between two regions are not sufficiently independent, the pooling

  • f 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. 39

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

  • α

α

  • α
  • α

α

  • α
  • a Starting situation: Q=P

b Decreasing between-correlation c Increasing between-correlation d Increasing within-correlation in Region 1

Q P

  • α

α

  • α
  • α

α

  • α
  • P
  • α

α α

  • α
  • P

Q

  • α

α

  • α
  • α

α

  • α
  • Q

P α

α

  • Q
  • 40