Pooling natural disaster risks in a community

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Introduction
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Model
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Results
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Conclusion
Pooling natural disaster risks in a community
Arnaud Gousseba¨ıle and Alexis Louaas
CREST and POLYTECHNIQUE
ACTINFO Chair
International conference on shocks and development
Dresden, October 6th and 7th, 2016
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
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Results
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Conclusion
Motivation
Figure : Natural disaster losses in the Caribbean countries (www.emdat.be)
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
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Results
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Conclusion
Motivation
How does the not-for-profit insurance facility of the Caribbean
countries (the CCRIF) manage the collective risk?
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
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Results
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Conclusion
Motivation
How does the not-for-profit insurance facility of the Caribbean
countries (the CCRIF) manage the collective risk?
The CCRIF partially reinsures the collective risk on reinsurance
markets.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
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Results
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Conclusion
Motivation
How does the not-for-profit insurance facility of the Caribbean
countries (the CCRIF) manage the collective risk?
The CCRIF partially reinsures the collective risk on reinsurance
markets.
The CCRIF supplies mutual insurance contracts to the countries
such that:
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
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Results
.
Conclusion
Motivation
How does the not-for-profit insurance facility of the Caribbean
countries (the CCRIF) manage the collective risk?
The CCRIF partially reinsures the collective risk on reinsurance
markets.
The CCRIF supplies mutual insurance contracts to the countries
such that:
an additional reserve is built and given back through dividend to
each country when the collective losses are not catastrophic,
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
. . .
Results
.
Conclusion
Motivation
How does the not-for-profit insurance facility of the Caribbean
countries (the CCRIF) manage the collective risk?
The CCRIF partially reinsures the collective risk on reinsurance
markets.
The CCRIF supplies mutual insurance contracts to the countries
such that:
an additional reserve is built and given back through dividend to
each country when the collective losses are not catastrophic,
the indemnity for a given country loss is lowered when the collective
losses are catastrophic.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
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Results
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Conclusion
Questions
In the context of a community with correlated risks and costly
reinsurance and reserve:
How should be designed the insurance contracts?
How much reinsurance should be purchased?
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
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Results
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Conclusion
Questions
In the context of a community with correlated risks and costly
reinsurance and reserve:
How should be designed the insurance contracts?
How much reinsurance should be purchased?
Literature review:
Insurance contracts with indemnity contingent on collective losses:
Doherty and Schlesinger (1990), Cummins and Mahul (2004),
Charpentier and Le Maux (2014).
Insurance contracts with dividend contingent on collective losses:
Borch (1962), Marshall (1974), Doherty and Dionne (1993), Doherty
and Schlesinger (2002).
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Questions
In the context of a community with correlated risks and costly
reinsurance and reserve:
How should be designed the insurance contracts?
How much reinsurance should be purchased?
Literature review:
Insurance contracts with indemnity contingent on collective losses:
Doherty and Schlesinger (1990), Cummins and Mahul (2004),
Charpentier and Le Maux (2014).
Insurance contracts with dividend contingent on collective losses:
Borch (1962), Marshall (1974), Doherty and Dionne (1993), Doherty
and Schlesinger (2002).
In the present paper: insurance contracts with both indemnity and
dividend contingent on collective losses.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
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Results
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Conclusion
The community
A community of N identical risk-averse agents, with VNM utility
function u(.), wealth w and risk of loss l.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
. . .
Results
.
Conclusion
The community
A community of N identical risk-averse agents, with VNM utility
function u(.), wealth w and risk of loss l.
Risks with two individual states and two collective states:
.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
. . .
Results
.
Conclusion
The community
A community of N identical risk-averse agents, with VNM utility
function u(.), wealth w and risk of loss l.
Risks with two individual states and two collective states:
..
p
.
catastrophic state:
.
affected agents =
.
fraction qc of N agents
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
. . .
Results
.
Conclusion
The community
A community of N identical risk-averse agents, with VNM utility
function u(.), wealth w and risk of loss l.
Risks with two individual states and two collective states:
..
p
.
catastrophic state:
.
affected agents =
.
fraction qc of N agents
.
1 − p
.
normal state:
.
affected agents =
.
fraction qn of N agents
. qn < qc
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
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Model
. . .
Results
.
Conclusion
The community
A community of N identical risk-averse agents, with VNM utility
function u(.), wealth w and risk of loss l.
Risks with two individual states and two collective states:
..
p
.
catastrophic state:
.
affected agents =
.
fraction qc of N agents
.
1 − p
.
normal state:
.
affected agents =
.
fraction qn of N agents
. qn < qc.
qn
.
w − l
.
affected
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
The community
A community of N identical risk-averse agents, with VNM utility
function u(.), wealth w and risk of loss l.
Risks with two individual states and two collective states:
..
p
.
catastrophic state:
.
affected agents =
.
fraction qc of N agents
.
1 − p
.
normal state:
.
affected agents =
.
fraction qn of N agents
. qn < qc.
qn
.
w − l
.
affected
.
1 − qn
.
w
.
not affected
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
The community
A community of N identical risk-averse agents, with VNM utility
function u(.), wealth w and risk of loss l.
Risks with two individual states and two collective states:
..
p
.
catastrophic state:
.
affected agents =
.
fraction qc of N agents
.
1 − p
.
normal state:
.
affected agents =
.
fraction qn of N agents
. qn < qc.
qn
.
w − l
.
affected
.
1 − qn
.
w
.
not affected
.
qc
.
w − l
.
affected
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
The community
A community of N identical risk-averse agents, with VNM utility
function u(.), wealth w and risk of loss l.
Risks with two individual states and two collective states:
..
p
.
catastrophic state:
.
affected agents =
.
fraction qc of N agents
.
1 − p
.
normal state:
.
affected agents =
.
fraction qn of N agents
. qn < qc.
qn
.
w − l
.
affected
.
1 − qn
.
w
.
not affected
.
qc
.
w − l
.
affected
.
1 − qc
.
w
.
not affected
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
The community
A community of N identical risk-averse agents, with VNM utility
function u(.), wealth w and risk of loss l.
Risks with two individual states and two collective states:
..
p
.
catastrophic state:
.
affected agents =
.
fraction qc of N agents
.
1 − p
.
normal state:
.
affected agents =
.
fraction qn of N agents
. qn < qc.
qn
.
w − l
.
affected
.
1 − qn
.
w
.
not affected
.
qc
.
w − l
.
affected
.
1 − qc
.
w
.
not affected
Individual probability of being affected: q = (1 − p)qn + pqc .
Risk correlation between individuals: δ = p(1−p)
µ(1−µ)
(qc − qn)2
.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
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Results
.
Conclusion
Agent wealth profile
Insurance contract characteristics for one agent:
premium α,
indemnity τ in the normal state and τ − ϵ in the catastrophic state,
dividend π in the normal state.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
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Model
. . .
Results
.
Conclusion
Agent wealth profile
Insurance contract characteristics for one agent:
premium α,
indemnity τ in the normal state and τ − ϵ in the catastrophic state,
dividend π in the normal state.
Because the premium is usually required ex-ante, there is potentially
a reserve cost λl
(α) (increasing with α).
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Agent wealth profile
Insurance contract characteristics for one agent:
premium α,
indemnity τ in the normal state and τ − ϵ in the catastrophic state,
dividend π in the normal state.
Because the premium is usually required ex-ante, there is potentially
a reserve cost λl
(α) (increasing with α).
Agent wealth profile with insurance contract:
..
1 − p
.
p
.
1 − qn
.
w − α − λl
(α) + π
.
qn
.
w − α − λl
(α) − l + τ + π
.
1 − qc
.
w − α − λl
(α)
.
qc
.
w − α − λl
(α) − l + τ − ϵ
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
. . . .
Model
. . .
Results
.
Conclusion
Insurer profit profile
Reinsurance contract characteristics for the insurer:
premium αR
,
indemnity τR
in the catastrophic state.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Insurer profit profile
Reinsurance contract characteristics for the insurer:
premium αR
,
indemnity τR
in the catastrophic state.
The required premium is αR
= (1 + λR
)pτR
, in which λR
> 0 is the
reinsurance loading factor.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Insurer profit profile
Reinsurance contract characteristics for the insurer:
premium αR
,
indemnity τR
in the catastrophic state.
The required premium is αR
= (1 + λR
)pτR
, in which λR
> 0 is the
reinsurance loading factor.
Insurer profit profile with insurance and reinsurance contracts:
..
1 − p
.
Nα − Nqnτ − Nπ − (1 + λR
)pτR
.
p
.
Nα − Nqc (τ − ϵ) − (1 + λR
)pτR
+ τR
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Optimal insurance and reinsurance contracts
The optimal insurance and reinsurance contracts are such that:
max
α,τ,ϵ,π,τR
E(u(˜w))
s.t. τ ≥ 0, τ − ϵ ≥ 0, π ≥ 0, τR
≥ 0,
Nα − Nqnτ − (1 + λR
)pτR
− Nπ ≥ 0,
Nα − Nqc (τ − ϵ) − (1 + λR
)pτR
+ τR
≥ 0.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Reinsurance coverage and insurance premium
The insurer budget constraints are binding.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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Introduction
. . . .
Model
. . .
Results
.
Conclusion
Reinsurance coverage and insurance premium
The insurer budget constraints are binding.
Purchased reinsurance coverage:
τR
= Nqc (τ − ϵ)
catastrophic state
− (Nqnτ + Nπ)
normal state
≥ 0
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Reinsurance coverage and insurance premium
The insurer budget constraints are binding.
Purchased reinsurance coverage:
τR
= Nqc (τ − ϵ)
catastrophic state
− (Nqnτ + Nπ)
normal state
≥ 0
Required insurance premium:
α =
(
1 +
p(qc − qn)
q
λR
0<...<λR
)
qτ+
(
−1 −λR
<0
)
pqc ϵ+
(
1 −
p
1 − p
λR
<0
)
(1−p)π
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Without reserve cost (λl′
(α) = 0)
.
Result 1a: optimal contracts
..
......
The optimal contracts are such that:
(i) τ = l, ϵ = 0 and π > 0
full indemnity plus dividend
;
(ii) τR
> 0 iff λR
< λR ∗
.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Without reserve cost (λl′
(α) = 0)
.
Result 1a: optimal contracts
..
......
The optimal contracts are such that:
(i) τ = l, ϵ = 0 and π > 0
full indemnity plus dividend
;
(ii) τR
> 0 iff λR
< λR ∗
.
.
Result 1b: comparative statics
..
......
With a CARA utility function and λR
< λR ∗
, we have:
(i) dτR
dλR < 0, dα
dλR > 0 (reins. cost ↗ ⇒ ins. premium ↗);
(ii) dτR
dδ > 0, dα
dδ > 0 (correlation ↗ ⇒ ins. premium ↗).
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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.
. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
With reserve cost (λl′
(α) > 0)
.
Result 2a: optimal contracts
..
......
With λR
< λR ∗
, the optimal contracts are such that:
(i) if λl ′
(α) < λl ∗
: τ = l, ϵ > 0 and π > 0
contingent indemnity plus dividend
; τR
> 0;
(ii) if λl ′
(α) > λl ∗
: τ < l, ϵ > 0 and π = 0
contingent indemnity
; τR
> 0.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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.....
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....
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.....
.
....
.
....
.
. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
With reserve cost (λl′
(α) > 0)
.
Result 2a: optimal contracts
..
......
With λR
< λR ∗
, the optimal contracts are such that:
(i) if λl ′
(α) < λl ∗
: τ = l, ϵ > 0 and π > 0
contingent indemnity plus dividend
; τR
> 0;
(ii) if λl ′
(α) > λl ∗
: τ < l, ϵ > 0 and π = 0
contingent indemnity
; τR
> 0.
.
Result 2b: comparative statics
..
......
With a CARA utility function and λl ′
(α) = λl
< λl ∗
, we have:
(i) dα
dλl < 0, dτR
dλl > 0 (reserve cost ↗ ⇒ reins. purchase ↗);
(ii) dτR
dδ > 0, dα
dδ ? (correlation ↗ ⇒ ins. premium ambiguous).
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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.....
.
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.
....
.
. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Conclusion
We develop a simple model with correlated individual risks within a
community.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

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.....
.
....
.
.....
.
....
.
....
.
. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Conclusion
We develop a simple model with correlated individual risks within a
community.
We take into account that reinsurance and reserve are costly.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

36. .....
.
....
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....
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.
.....
.
....
.
....
.
. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Conclusion
We develop a simple model with correlated individual risks within a
community.
We take into account that reinsurance and reserve are costly.
We show that the optimal insurance contract can have both
indemnity and dividend contingent on collective losses.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community

37. .....
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.....
.
....
.
.....
.
....
.
....
.
. . .
Introduction
. . . .
Model
. . .
Results
.
Conclusion
Conclusion
We develop a simple model with correlated individual risks within a
community.
We take into account that reinsurance and reserve are costly.
We show that the optimal insurance contract can have both
indemnity and dividend contingent on collective losses.
We exhibit the example of the Caribbean countries which implement
this type of contract for natural disaster risks.
Arnaud Gousseba¨ıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair
Pooling natural disaster risks in a community