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- 1. Purpose of the research The one-region model The two-region model Conclusion Natural Catastrophe Insurance How Should Government Intervene? Benoît LE MAUX Université de Rennes 1 CREM-CNRS Condorcet Center Arthur CHARPENTIER Université de Rennes 1 CREM-CNRS Ecole Polytechnique The 9th PEARL Conference, June 7-8, 2012 Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 2. Purpose of the research The one-region model The two-region model ConclusionThe problem Between 1969 and 1998, 36 US insurers became insolvent primarily as a result of catastrophe losses. Of these companies, 20 became insolvent between 1989 and 1993, the same time period as Hurricane Hugo (Matthews, 1999). Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 3. Purpose of the research The one-region model The two-region model ConclusionTwo important questions 1. Purely private market vs Government program Purely private market: only policyholders at risk have to deal with their insurers insolvency. Government program: policyholders participate to a collective sharing practice based on solidarity from the taxpayers. Question 1: Which one is the best? 2. Viability of a government program Are taxpayers from less risky regions willing to show solidarity with taxpayers from riskier regions ? Example: Michigan wants the end of the US Flood Insurance Program because it has to pay for the costs that some other states are incurring. Question 2: Under which conditions is an insurance program viable in the long run? Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 4. Purpose of the research The one-region model The two-region model ConclusionHow can we address these two issues? 1. Purely private market vs Government program In contrast with the usual literature, we need a model where the insurer may have a non-zero probability of insolvency depending on the distribution of the risks (Kunreuther, 2001), the premium rate (Tapiero et al., 1986), the amount of capital in the company (Charpentier, 2008). 2. Viability of a government program The participation of a region can strongly inuence the solvency of a public program, as well as the indemnities received and the amount of additionnal taxes. We extend our theoretical framework by focusing on a simultaneous non-cooperative game combining two regions with heterogeneous natural risks. Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 5. Purpose of the research The one-region model The two-region model ConclusionMain assumptions Population: n Natural events: cause a loss l to N individuals. Share of population claiming a loss: X = N . n Distribution of X : depends on the probability p for each individual to claim a loss and the correlation δ between the individual risks. x F = F (x |p , δ) = F (x ) = f (t )dt ∈ [0; 1] 0 δ : determines the total number of people that will be claiming a loss at the same time. p : represents the odds for each individual to be one of the victims. The inhabitants will decide simultaneously whether or not to pay full insurance coverage. Premium=α ; Capital per policy of the insurance company=c Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 6. Purpose of the research The one-region model The two-region model ConclusionSupply of insurance Probability of insolvency The insurer becomes insolvent when it is not possible to pay the full coverage l to the victims anymore, i.e., when the total losses (Nl ) become higher than the total revenue (nα) and the total economic capital (nc ). α+c P (Nl nα + nc ) = P X = 1 − F (¯) x l where x = (α + c )/l denotes the largest possible event without default. ¯ Expected prot of the company x ¯ Π(c , α, p , δ) = [nα − xnl ] f (x )dx − [1 − F (¯)]cn. x 0 Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 7. Purpose of the research The one-region model The two-region model ConclusionDemand for insurance Scenario with limited liability (i.e. no government intervention) 1 1 V (c , α, p , δ) = xU (−α − l + I (x ))f (x )dx + (1 − x )U (−α)f (x )dx , 0 0 with I (X ) = c +α = reduced indemnity in case of insolvency. X Scenario with unlimited guarantee from the government 1 V (c , α, p , δ) = U (−α − T (x ))f (x )dx . 0 T (X ) = Xl − α − c = tax to compensate the default of payment. An agent will buy insurance if V (c , α, p , δ) ≥ pU (−l ) + (1 − p )U (0). Let denote α∗ the WTP for insurance. Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 8. Purpose of the research The one-region model The two-region model ConclusionMain result of the model I The controversial impact of capital requirements The capital c has a negative impact on the expected prot because it increases the exposition of the shareholders to industry failure. On the other hand, the WTP for an insurance contract is a positive function of the companys capital because it reduces the insolvency probability. A better possibility: capital market instruments such as CAT bonds or CAT options, creation of tax-deferred catastrophe reserves (Kousky, 2011). Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 9. Purpose of the research The one-region model The two-region model ConclusionMain result of the model II The controversial impact of a regulated premium The higher δ , the higher the insolvency probability. Private insurers advocate high levels of premium when faced with natural disasters. However, the WTP for a catastrophe coverage is a negative function of δ , because correlated risks imply a higher default risk. A regulated price cannot be of any use, unless the idea is to solve the market ineciencies due to imperfect competition and imperfect information. Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 10. Purpose of the research The one-region model The two-region model ConclusionMain result of the model III A free market is not necessarily the ecient solution An insurance with unlimited guarantee from the government proves to be a mean preserving spread of a limited liability insurance. Government programs allow to spread the risks equally among the policyholders and, therefore, are less risky and more attractive in terms of expected utility. Consequence: the insurer can put forward higher premiums, which will reduce the insolvency probability!!! Question: does this result still hold in a two-region economy with heterogenous risks? Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 11. Purpose of the research The one-region model The two-region model ConclusionThe extended framework Settings Two populations: n1 and n2 living in two dierent jurisdictions Natural events: cause a loss l to Ni inhabitants in Region i , i = 1, 2. Share of people claiming a loss in the total population: X0 = N1 +n22 n 1 +N The distribution of X0 depends on a new parameter : θ, the between-correlation: X0 ∼= F0 (x0 |p , δ1 , δ2 , θ) = F0 (x0 ), Insure Dont Insure V1 (c , α1 , α2 , p, δ1 , δ2 , θ), V2 (c , α1 , α2 , p, δ1 , δ2 , θ) V1 (c , α1 , p, δ1 ), pU (−l ) Dont pU (−l ), V2 (c , α2 , p, δ2 ) pU (−l ), pU (−l ) Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 12. Purpose of the research The one-region model The two-region model ConclusionSet of Nash Equilibria כ ככ כ ככ હ αଵ αଵ ሺαଶ ሻ હ αଵ αଵ ሺαଶ ሻ α ככሺαଵ ሻ ଶ α ככሺαଵ ሻ ଶ Q P αכ αכ ଶ ଶ Q P 0 હ 0 હ ሺaሻ Starting situation: Q=P ሺbሻ Decreasing between-correlation ככ כ כ ככ હ αଵ ሺαଶ ሻ αଵ હ αଵ αଵ ሺαଶ ሻ P αכ αכ ଶ ଶ P α ככሺαଵ ሻ α ככሺαଵ ሻ ଶ ଶ Q Q 0 હ હ ሺcሻ Increasing between-correlation ሺdሻ Increasing within-correlation in Region 1 Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 13. Purpose of the research The one-region model The two-region model ConclusionMain result of the model IV Region 1 Region 2 Region 3 Regions 1+2 Regions 1+3 Loss per inhabitant in Year 1 5 65 35 35 20 Loss per inhabitant in Year 2 95 35 65 65 80 Average of annual losses (p) 50 50 50 50 50 Variance of annual losses (δ) 2025 225 225 225 900 Pearson correlation coecient (θ) -1 +1 a The number of inhabitants is the same in each region. The rates of a government program should be computed based not only on the level of risks (p ), i.e., on the expected losses (a basic actuarial principle), but also on how the risks are correlated within and between the regions (δ and θ), i.e., on the variance of the losses (which has never been applied to our knowledge). In particular, government ocials must be prepared to announce rates lower than usual to attract low-correlation regions. Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 14. Purpose of the research The one-region model The two-region model ConclusionConclusion Risk-averse policyholders will accept to pay higher rates for an unlimited guarantee insurance, thus reducing the probability of insolvency. To limit the protests of the less correlated areas, these rates should be computed based on how the risks are correlated within and between the jurisdictions involved. Future research There are several problems of related interest which were not examined in the present paper: The inuence of risk mitigation The role of bounded rationality in insurance decisions. We have tested the robustness of the model with respect to statistical modeling. It could be nice to test the model on a real database. Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 15. Purpose of the research The one-region model The two-region model ConclusionSimulations Results ________________________ Expected profit0 Expected profit0 THEORETICAL MODEL 0 Loss: 1 Capital: 0.05 ________________________ PROBABILISTIC MODEL n: 1000 p*: 0.05 p: 0.1 −20 Correlation: 0.9 pC: 0.69 pN: 0.069 ________________________ q WILLINGNESS TO PAY Expected utility Limited liability: 0.206 q Unlimited guarantee: 0.216 −40 −60 pU(−l)= −63.9 q q 0.00 0.05 0.10 0.15 0.20 0.25 Premium Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012
- 16. Purpose of the research The one-region model The two-region model ConclusionThe US Flood Insurance Program Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012

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