le maux pearl-2012

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


The problem

Between 1969 and 1998, 36 US insurers became insolvent primarily as a resultof catastrophe losses. Of these companies, 20 became insolvent between 1989and 1993, the same time period as Hurricane Hugo (Matthews, 1999).



Two important questions

1. Purely private market vs Government program

Purely private market: only policyholders at risk have to deal with theirinsurer's insolvency.

Government program: policyholders participate to a collective sharingpractice 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 Programbecause 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?



How 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 mayhave 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 in�uence the solvency of a public

program, as well as the indemnities received and the amount of additionnaltaxes.

We extend our theoretical framework by focusing on a simultaneous

non-cooperative game combining two regions with heterogeneous naturalrisks.



Main assumptions

Population: n

Natural events: cause a loss l to N individuals.

Share of population claiming a loss: X = Nn.

Distribution of X : depends on the probability p for each individual toclaim a loss and the correlation δ between the individual risks.

F = F (x |p, δ) = F (x) =

∫ x

0

f (t)dt ∈ [0; 1]

δ: determines the total number of people that will be claiming a loss atthe 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 fullinsurance coverage.

Premium=α ; Capital per policy of the insurance company=c



Supply 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 thanthe total revenue (nα) and the total economic capital (nc).

P (Nl > nα + nc) = P(X >

α + c

l

)= 1− F (x̄)

where x̄ = (α + c)/l denotes the largest possible event without default.

Expected pro�t of the company

Π(c, α, p, δ) =

∫ x̄

0

[nα− xnl ] f (x)dx − [1− F (x̄)]cn.



Demand for insurance

Scenario with limited liability (i.e. no government intervention)

V (c, α, p, δ) =

∫ 1

0

xU(−α− l + I (x))f (x)dx +

∫ 1

0

(1− x)U(−α)f (x)dx ,

with I (X ) = c+αX

= reduced indemnity in case of insolvency.

Scenario with unlimited guarantee from the government

V (c, α, p, δ) =

∫ 1

0

U(−α− T (x))f (x)dx .

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). Letdenote α∗ the WTP for insurance.



Main result of the model I

The controversial impact of capital requirements

The capital c has a negative impact on the expected pro�t because itincreases the exposition of the shareholders to industry failure.

On the other hand, the WTP for an insurance contract is a positive

function of the company's capital because it reduces the insolvencyprobability.

A better possibility: capital market instruments such as CAT bonds orCAT options, creation of tax-deferred catastrophe reserves (Kousky,2011).



Main result of the model II

The controversial impact of a regulated premium

The higher δ, the higher the insolvency probability. Private insurersadvocate high levels of premium when faced with natural disasters.

However, the WTP for a catastrophe coverage is a negative function ofδ, because correlated risks imply a higher default risk.

A regulated price cannot be of any use, unless the idea is to solve themarket ine�ciencies due to imperfect competition and imperfectinformation.



Main result of the model III

A free market is not necessarily the e�cient solution

An insurance with unlimited guarantee from the government proves to bea 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 ofexpected utility.

Consequence: the insurer can put forward higher premiums, which willreduce the insolvency probability!!!

Question: does this result still hold in a two-region economy withheterogenous risks?



The extended framework

Settings

Two populations: n1 and n2 living in two di�erent 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+N2

n1+n2

The distribution of X0 depends on a new parameter : θ, thebetween-correlation:

X0 ∼= F0(x0|p, δ1, δ2, θ) = F0(x0),

Insure Don'tInsure V1(c, α1, α2, p, δ1, δ2, θ),V2(c, α1, α2, p, δ1, δ2, θ) V1(c, α1, p, δ1), pU(−l)Don't pU(−l),V2(c, α2, p, δ2) pU(−l), pU(−l)



Set of Nash Equilibria

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α��α�

α��

0000�a Starting situation: Q=P �b Decreasing between-correlation

�c Increasing between-correlation

�d Increasing within-correlation in Region 1

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P

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0000

P

Q

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Main 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 20Loss per inhabitant in Year 2 95 35 65 65 80

Average of annual losses (p) 50 50 50 50 50Variance of annual losses (δ) 2025 225 225 225 900Pearson correlation coe�cient (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 thelevel of risks (p), i.e., on the expected losses (a basic actuarial principle), butalso 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 ourknowledge).

In particular, government o�cials must be prepared to announce rates lowerthan usual to attract low-correlation regions.



Conclusion

Risk-averse policyholders will accept to pay higher rates for an unlimitedguarantee insurance, thus reducing the probability of insolvency.

To limit the protests of the less correlated areas, these rates should becomputed based on how the risks are correlated within and between thejurisdictions involved.

Future research

There are several problems of related interest which were not examined in thepresent paper:

The in�uence of risk mitigation

The role of bounded rationality in insurance decisions.

We have tested the robustness of the model with respect to statisticalmodeling. It could be nice to test the model on a real database.



Simulations Results

0.00 0.05 0.10 0.15 0.20 0.25

−60

−40

−20

0

Premium

Exp

ecte

d ut

ility

________________________THEORETICAL MODELLoss: 1Capital: 0.05________________________PROBABILISTIC MODELn: 1000p*: 0.05p: 0.1Correlation: 0.9pC: 0.69pN: 0.069________________________WILLINGNESS TO PAYLimited liability: 0.206Unlimited guarantee: 0.216●●

●

●● ●pU(−l)= −63.9

Expected profit<0 Expected profit>0



The US Flood Insurance Program


le maux pearl-2012

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