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. Purpose of the research The one-region model The two-region model
The 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. Purpose of the research The one-region model The two-region model
Two important issues
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.
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. Purpose of the research The one-region model The two-region model
Two 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. Purpose of the research The one-region model The two-region model
Theoretical 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. Purpose of the research The one-region model The two-region model
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 Public Choice Societies - World Meeting, March 9-11, 2012

7. Purpose of the research The one-region model The two-region model
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 Public Choice Societies - World Meeting, March 9-11, 2012

8. Purpose of the research The one-region model The two-region model
The 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∗
∂
company's 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. Purpose of the research The one-region model The two-region model
A 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. Purpose of the research The one-region model The two-region model
Unlimited 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. Purpose of the research The one-region model The two-region model
Theoretical 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 Public Choice Societies - World Meeting, March 9-11, 2012

12. Purpose of the research The one-region model The two-region model
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 Public Choice Societies - World Meeting, March 9-11, 2012

13. Purpose of the research The one-region model The two-region model
A 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. Purpose of the research The one-region model The two-region model
Conclusion
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