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  • 1. 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
  • 2. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 3. Arthur CHARPENTIER, Insurance of Natural Catastrophes 1.1 The French cat nat mecanism Drought risk frequency, over the past 30 years. 3
  • 4. Arthur CHARPENTIER, Insurance of Natural Catastrophes INSURANCE COMPANY INSURANCE COMPANY INSURANCE COMPANY RE-INSURANCE COMPANY CAISSE CENTRALE DE REASSURANCE GOVERNMENT 4
  • 5. Arthur CHARPENTIER, Insurance of Natural Catastrophes INSURANCE COMPANY INSURANCE COMPANY RE-INSURANCE COMPANY CAISSE CENTRALE DE REASSURANCE GOVERNMENT INSURANCE COMPANY 5
  • 6. Arthur CHARPENTIER, Insurance of Natural Catastrophes INSURANCE COMPANY INSURANCE COMPANY RE-INSURANCE COMPANY CAISSE CENTRALE DE REASSURANCE GOVERNMENT INSURANCE COMPANY 6
  • 7. Arthur CHARPENTIER, Insurance of Natural Catastrophes 1.2 Agenda • demand for insurance, integrating possible ruin • model equilibriums on one homogeneous region • policy implication • an extension to two (correlated) regions 7
  • 8. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 9. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 10. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 11. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 12. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 13. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 14. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 15. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 16. Arthur CHARPENTIER, Insurance of Natural Catastrophes • 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
  • 17. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 18. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 19. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 20. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 21. Arthur CHARPENTIER, Insurance of Natural Catastrophes 3.4.1 Distribution of X, F(·) 21
  • 22. Arthur CHARPENTIER, Insurance of Natural Catastrophes 3.4.2 Impact of p on F(·) 22
  • 23. Arthur CHARPENTIER, Insurance of Natural Catastrophes 3.4.3 Impact of n on F(·) 23
  • 24. Arthur CHARPENTIER, Insurance of Natural Catastrophes 3.4.4 Impact of p on F(·) 24
  • 25. Arthur CHARPENTIER, Insurance of Natural Catastrophes 3.4.5 Impact of δ on F(·) 25
  • 26. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 27. Arthur CHARPENTIER, Insurance of Natural Catastrophes 3.6 Optimal premiums for alternative covers 27
  • 28. Arthur CHARPENTIER, Insurance of Natural Catastrophes 3.7 Impact of δ (increasing) on optimal premiums 28
  • 29. Arthur CHARPENTIER, Insurance of Natural Catastrophes Impact of δ (increasing) on optimal premiums 29
  • 30. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 31. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 32. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 33. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 34. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 35. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 36. Arthur CHARPENTIER, Insurance of Natural Catastrophes 4.2 Distribution of X = (N1 + N2)/(n1 + n2), F0(·) 36
  • 37. Arthur CHARPENTIER, Insurance of Natural Catastrophes 4.2.1 Impact of n1 on F0(·) (decrease) 37
  • 38. Arthur CHARPENTIER, Insurance of Natural Catastrophes 4.2.2 Impact of δ1 on F0(·) (decrease) 38
  • 39. Arthur CHARPENTIER, Insurance of Natural Catastrophes 4.2.3 Impact of p1 on F0(·) (decrease) 39
  • 40. Arthur CHARPENTIER, Insurance of Natural Catastrophes 4.2.4 Impact of θ on F0(·) (decrease) 40
  • 41. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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). 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 REGION 1 BUYS INSURANCE REGION 1 BUYS NO INSURANCE 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 REGION 1 BUYS INSURANCE REGION 1 BUYS NO INSURANCE 41
  • 42. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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). 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 REGION 2 BUYS NO INSURANCE REGION 2 BUYS INSURANCE 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 REGION 2 BUYS NO INSURANCE REGION 2 BUYS INSURANCE 42
  • 43. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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. 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 REGION 1 BUYS INSURANCE REGION 1 BUYS NO INSURANCE 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 REGION 1 BUYS INSURANCE REGION 1 BUYS NO INSURANCE 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 REGION 2 BUYS NO INSURANCE REGION 2 BUYS INSURANCE 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 REGION 2 BUYS NO INSURANCE REGION 2 BUYS INSURANCE 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 43
  • 44. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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. 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 1: insured, 2: insured 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 1: insured, 2: non−insured 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 1: non−insured, 2: insured 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 1: non−insured, 2: non−insured 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 44
  • 45. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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 1: insured, 2: insured 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 1: insured, 2: non−insured 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 1: non−insured, 2: insured 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 1: non−insured, 2: non−insured 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 1: insured, 2: insured 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 1: insured, 2: non−insured 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 1: non−insured, 2: insured 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 1: non−insured, 2: non−insured 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
  • 46. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 47. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 48. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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
  • 49. Arthur CHARPENTIER, Insurance of Natural Catastrophes 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