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Purpose of the research                        The one-region model                        The two-region model           ...
Purpose of the research                        The one-region model                        The two-region modelThe problem...
Purpose of the research                        The one-region model                        The two-region modelTwo importa...
Purpose of the research                        The one-region model                        The two-region modelTwo models ...
Purpose of the research                        The one-region model                        The two-region modelTheoretical...
Purpose of the research                               The one-region model                        The two-region modelSupp...
Purpose of the research                           The one-region model                               The two-region modelD...
Purpose of the research                        The one-region model                        The two-region modelThe role of...
Purpose of the research                        The one-region model                        The two-region modelA regulated...
Purpose of the research                        The one-region model                        The two-region modelUnlimited g...
Purpose of the research                                 The one-region model                                     The two-r...
Purpose of the research                                   The one-region model                                           T...
Purpose of the research                        The one-region model                        The two-region modelA governmen...
Purpose of the research                        The one-region model                        The two-region modelConclusion ...
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Miami

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  1. 1. Purpose of the research The one-region model The two-region model 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 Public Choice Societies - World Meeting, March 9-11, 2012 Benoît Le Maux, Arthur Charpentier Public Choice Societies - World Meeting, March 9-11, 2012
  2. 2. Purpose of the research The one-region model The two-region modelThe problem Both the frequency and strength of natural catastrophes such as hurricanes, oods and droughts have increased during the past few years (Intergovernmental Panel on Climate Change, 2007). This trend could endanger the viability of the insurance and reinsurance industry. 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). The present paper aims to investigate these failures by developing a model of natural catastrophe insurance market and evaluating the ways the government can intervene. Benoît Le Maux, Arthur Charpentier Public Choice Societies - World Meeting, March 9-11, 2012
  3. 3. Purpose of the research The one-region model The two-region modelTwo important issues 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. No formal study has ever been undertaken to compare these two possible alternatives. Large insurance companies can pool the risks with independent risks from other regions (Cummins, 2006; Charpentier,2008). As the largest entity in a given jurisdiction, the government would be the most eective agency for spreading risks and losses (Priest, 1996). Are taxpayers from less risky regions willing to show solidarity with taxpayers from riskier regions ? This is especially relevant if we want to build an insurance program that is politically viable in the long run. Benoît Le Maux, Arthur Charpentier Public Choice Societies - World Meeting, March 9-11, 2012
  4. 4. Purpose of the research The one-region model The two-region modelTwo models One-region model Natural risks are correlated, i.e., may imply a considerable number of claims at the same time. An insurer may have a non-zero probability of insolvency depending on 1 the distribution of the risks (Kunreuther, 2001), 2 the premium rate (Tapiero et al., 1986), 3 the amount of capital in the company (Charpentier, 2008). Two-region model 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 Public Choice Societies - World Meeting, March 9-11, 2012
  5. 5. Purpose of the research The one-region model The two-region modelTheoretical framework 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 Public Choice Societies - World Meeting, March 9-11, 2012
  6. 6. Purpose of the research The one-region model The two-region modelSupply 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 Public Choice Societies - World Meeting, March 9-11, 2012
  7. 7. Purpose of the research The one-region model The two-region modelDemand 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 Public Choice Societies - World Meeting, March 9-11, 2012
  8. 8. Purpose of the research The one-region model The two-region modelThe role of capital requirements ∂Π ∂c 0: An increase in the capital will increase the exposition of the shareholders to industry failure. c 0: The WTP for an insurance contract is a positive function of the ∂α∗ ∂ companys capital. The company can consequently increase the price of a contract. The decision to increase the capital requirements depends on the demand sensitivity to insurers capital resources. Other possibilities: capital market instruments such as CAT bonds or CAT options, creation of tax-deferred catastrophe reserves (Kousky, 2011). Benoît Le Maux, Arthur Charpentier Public Choice Societies - World Meeting, March 9-11, 2012
  9. 9. Purpose of the research The one-region model The two-region modelA regulated premium Private insurers advocate high levels of premium when faced with natural disasters. ∂Π ∂α 0: The higher the premium, the higher the expected prots. However, the WTP for a catastrophe coverage is a negative function of δ , because correlated risks imply a higher default risk. Given this controversial impact, a regulated price cannot be of any use, unless the idea is to solve the market ineciencies due to imperfect competition and imperfect information (see, e.g., Epple and Schäfer, 1996; Jaee and Russell, 1997). Benoît Le Maux, Arthur Charpentier Public Choice Societies - World Meeting, March 9-11, 2012
  10. 10. Purpose of the research The one-region model The two-region modelUnlimited guarantee from the government Coverage does not exist for too risky areas (e.g., ood insurance in the US). Such problems are usually solved by the creation of government programs. Because an unlimited guarantee insurance allow to spread the risks equally among the policyholders, the WTP for insurance is higher with government intervention. Consequence: the insurer can put forward higher premiums, which will reduce the insolvency probability, lead to higher expected prots, and could guarantee the existence of a catastrophe coverage. Question: does this result hold in a two-region economy with heterogenous risks? Benoît Le Maux, Arthur Charpentier Public Choice Societies - World Meeting, March 9-11, 2012
  11. 11. Purpose of the research The one-region model The two-region modelTheoretical 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 Public Choice Societies - World Meeting, March 9-11, 2012
  12. 12. Purpose of the research The one-region model The two-region modelSet 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 Public Choice Societies - World Meeting, March 9-11, 2012
  13. 13. Purpose of the research The one-region model The two-region modelA government program should be priced appropriately 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 Public Choice Societies - World Meeting, March 9-11, 2012
  14. 14. Purpose of the research The one-region model The two-region modelConclusion Compared to a purely private market, the chance of failure of a government program should be reduced. 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. Benoît Le Maux, Arthur Charpentier Public Choice Societies - World Meeting, March 9-11, 2012
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