The document discusses natural catastrophe insurance and how governments should intervene. It examines a one-region and two-region model. In the one-region model, capital requirements have a negative impact on insurer profit but reduce insolvency risk, while regulated premiums increase insolvency risk but reduce demand. A free market is not efficient as government insurance spreads risk better. The two-region model finds different Nash equilibria depending on the correlation between regions' risks, showing rates should consider within- and between-region correlations to limit protests from less correlated areas. Simulations support the theoretical models.

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 Conclusion
The 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 Conclusion
Two important questions
1. Purely private market vs Government program
Purely private market: only policyholders at risk have to deal with their
insurer's 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 Conclusion
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 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 Conclusion
Main 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 Conclusion
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 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 Conclusion
Demand 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 Conclusion
Main 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 company's 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 Conclusion
Main 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 Conclusion
Main 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 Conclusion
The 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 Don't
Insure 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 )
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 Conclusion
Set 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 Conclusion
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 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 Conclusion
Conclusion
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 Conclusion
Simulations 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 Conclusion
The US Flood Insurance Program
Benoît Le Maux, Arthur Charpentier The 9th PEARL Conference, June 7-8, 2012