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Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions Equilibria for Insurance Covers of Natural Catastrophes on Heterogeneous Regions Arthur Charpentier (Université de Rennes 1, Chaire ACTINFO ) & Benoît le Maux, Arnaud Goussebaïle, Alexis Louaas International Conference on Applied Business and Economics ICABE, Paris, June 2016 http://freakonometrics.hypotheses.org 1 @freakonometrics

Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions Major (Winter) Storms in France Proportion of insurance policy that did claim a loss after storms, for a large insurance company in France ( ∼ 1.2 million household policies) 2 @freakonometrics

Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions Demand for Insurance An agent purchases insurance if E [ u ( ω − X )] ≤ u ( ω − α ) � �� � � �� � no insurance insurance i.e. p · u ( ω − l ) + [1 − p ] · u ( ω − 0) ≤ u ( ω − α ) � �� � � �� � no insurance insurance i.e. E [ u ( ω − X )] ≤ E [ u ( ω − α − l + I )] � �� � � �� � no insurance 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. 3 @freakonometrics

Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions Notations  0 if agent i claims a loss  Y i = 1 if not  Let N = Y 1 + · · · + Y n 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 ( Y i = 1) = p for all i = 1 , 2 , · · · , n Assume that agents have identical wealth ω and identical utility functions u ( · ). Further, insurance company has capital C = n · c , and ask for premium α . 4 @freakonometrics

Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions 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 5 @freakonometrics

Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions 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). 6 @freakonometrics

Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions A one region model with homogeneous agents Let U ( x ) = u ( ω + x ) and U (0) = 0. 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  ℓ if x ≤ x  I ( x ) = c + α if x > x  n 7 @freakonometrics

Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions I I I I(X) (X) (X) (X) I�l Negative profit Negative profit Negative profit Negative profit Ruin Ruin Ruin Ruin Positive profit Positive profit Positive profit Positive profit ]– ] ] ] –cn – – cn cn cn ; ; ; ; 0[ 0[ 0[ 0[ –cn – – – cn cn cn n α [ [0 ; [0 ; [0 ; [0 ; n n n [ [ [ c� α X X X X α � � α � c 0 1 x l l Probability of no ruin: Probability of ruin: F(x �) 1–F(x �) 8 @freakonometrics

Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions The objective function of the insured is V defined as � E [ E ( U ( − α − loss) | X )]) = E ( U ( − α − loss) | X = x ) dF ( x ) 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 � 1 V = [ x · U ( − α − l + I ( x )) + (1 − x ) · U ( − α )] dF ( x ) 0 that can be written � 1 V = U ( − α ) − x [ U ( − α ) − U ( − α − ℓ + I ( x ))] f ( x ) dx 0 An agent will purchase insurance if and only if V > p · U ( − l ). 9 @freakonometrics

Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions with government intervention (or mutual fund insurance), the tax function is  0 if x ≤ x  T ( x ) = Nℓ − ( α + c ) n = Xℓ − α − c if x > x  n Then � 1 V = [ x · U ( − α − T ( x )) + (1 − x ) · U ( − α − T ( x ))] dF ( x ) 0 i.e. � 1 � 1 V = U ( − α + T ( x )) dF ( x ) = F ( x ) · U ( − α ) + U ( − α − T ( x )) dF ( x ) x 0 10 @freakonometrics

Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions A common shock model for natural catastrophes risks Consider a possible natural castrophe, modeled as an heterogeneous latent variable Θ, such that given Θ, the Y i ’s are independent, and  P ( Y i = 1 | Θ = Catastrophe) = p C  P ( Y i = 1 | Θ = No Catastrophe) = p N  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 � n � � � ( p N ) j (1 − p N ) n − j (1 − p ∗ ) + ( p C ) j (1 − p C ) n − j p ∗ � = j j =0 11 @freakonometrics

Arthur Charpentier, Equilibira for Insurance Covers of Natural Catastrophes on Heterogeneous Regions Cumulative distribution function F 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1−p* ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.8 ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● ● 0.4 ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● p ● ● ● pN pC ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 Share of the population claiming a loss 20 Probability density function f 15 10 5 p pN pC 0 0.0 0.2 0.4 0.6 0.8 1.0 Share of the population claiming a loss 12 @freakonometrics