pooling natural disaster risks in a community

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.....................................................................

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......

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.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Pooling natural disaster risks in a community

Arnaud Goussebaıle and Alexis Louaas

CREST and POLYTECHNIQUEACTINFO Chair

International conference on shocks and developmentDresden, October 6th and 7th, 2016

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

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.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Motivation

Figure : Natural disaster losses in the Caribbean countries (www.emdat.be)

Arnaud Goussebaıle 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

Motivation

I How does the not-for-profit insurance facility of the Caribbeancountries (the CCRIF) manage the collective risk?

I The CCRIF partially reinsures the collective risk on reinsurancemarkets.

I The CCRIF supplies mutual insurance contracts to the countriessuch that:

I an additional reserve is built and given back through dividend toeach country when the collective losses are not catastrophic,

I the indemnity for a given country loss is lowered when the collectivelosses are catastrophic.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

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......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Motivation

I How does the not-for-profit insurance facility of the Caribbeancountries (the CCRIF) manage the collective risk?

I The CCRIF partially reinsures the collective risk on reinsurancemarkets.

I The CCRIF supplies mutual insurance contracts to the countriessuch that:

I an additional reserve is built and given back through dividend toeach country when the collective losses are not catastrophic,

I the indemnity for a given country loss is lowered when the collectivelosses are catastrophic.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Motivation

I How does the not-for-profit insurance facility of the Caribbeancountries (the CCRIF) manage the collective risk?

I The CCRIF partially reinsures the collective risk on reinsurancemarkets.

I The CCRIF supplies mutual insurance contracts to the countriessuch that:

I an additional reserve is built and given back through dividend toeach country when the collective losses are not catastrophic,

I the indemnity for a given country loss is lowered when the collectivelosses are catastrophic.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Motivation

I How does the not-for-profit insurance facility of the Caribbeancountries (the CCRIF) manage the collective risk?

I The CCRIF partially reinsures the collective risk on reinsurancemarkets.

I The CCRIF supplies mutual insurance contracts to the countriessuch that:

I an additional reserve is built and given back through dividend toeach country when the collective losses are not catastrophic,

I the indemnity for a given country loss is lowered when the collectivelosses are catastrophic.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Motivation

I How does the not-for-profit insurance facility of the Caribbeancountries (the CCRIF) manage the collective risk?

I The CCRIF partially reinsures the collective risk on reinsurancemarkets.

I The CCRIF supplies mutual insurance contracts to the countriessuch that:

I an additional reserve is built and given back through dividend toeach country when the collective losses are not catastrophic,

I the indemnity for a given country loss is lowered when the collectivelosses are catastrophic.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Questions

I In the context of a community with correlated risks and costlyreinsurance and reserve:

I How should be designed the insurance contracts?

I How much reinsurance should be purchased?

I Literature review:

I Insurance contracts with indemnity contingent on collective losses:Doherty and Schlesinger (1990), Cummins and Mahul (2004),Charpentier and Le Maux (2014).

I Insurance contracts with dividend contingent on collective losses:Borch (1962), Marshall (1974), Doherty and Dionne (1993), Dohertyand Schlesinger (2002).

I In the present paper: insurance contracts with both indemnity anddividend contingent on collective losses.

Arnaud Goussebaıle 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

Questions

I In the context of a community with correlated risks and costlyreinsurance and reserve:

I How should be designed the insurance contracts?

I How much reinsurance should be purchased?

I Literature review:

I Insurance contracts with indemnity contingent on collective losses:Doherty and Schlesinger (1990), Cummins and Mahul (2004),Charpentier and Le Maux (2014).

I Insurance contracts with dividend contingent on collective losses:Borch (1962), Marshall (1974), Doherty and Dionne (1993), Dohertyand Schlesinger (2002).

I In the present paper: insurance contracts with both indemnity anddividend contingent on collective losses.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Questions

I In the context of a community with correlated risks and costlyreinsurance and reserve:

I How should be designed the insurance contracts?

I How much reinsurance should be purchased?

I Literature review:

I Insurance contracts with indemnity contingent on collective losses:Doherty and Schlesinger (1990), Cummins and Mahul (2004),Charpentier and Le Maux (2014).

I Insurance contracts with dividend contingent on collective losses:Borch (1962), Marshall (1974), Doherty and Dionne (1993), Dohertyand Schlesinger (2002).

I In the present paper: insurance contracts with both indemnity anddividend contingent on collective losses.

Arnaud Goussebaıle 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

The community

I A community of N identical risk-averse agents, with VNM utilityfunction u(.), wealth w and risk of loss l .

I 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

I Individual probability of being affected: q = (1− p)qn + pqc .I Risk correlation between individuals: δ = p(1−p)

µ(1−µ)(qc − qn)

2.

Arnaud Goussebaıle 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

The community

I A community of N identical risk-averse agents, with VNM utilityfunction u(.), wealth w and risk of loss l .

I 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

I Individual probability of being affected: q = (1− p)qn + pqc .I Risk correlation between individuals: δ = p(1−p)

µ(1−µ)(qc − qn)

2.

Arnaud Goussebaıle 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

The community

I A community of N identical risk-averse agents, with VNM utilityfunction u(.), wealth w and risk of loss l .

I 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

I Individual probability of being affected: q = (1− p)qn + pqc .I Risk correlation between individuals: δ = p(1−p)

µ(1−µ)(qc − qn)

2.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

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.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community

I A community of N identical risk-averse agents, with VNM utilityfunction u(.), wealth w and risk of loss l .

I 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

I Individual probability of being affected: q = (1− p)qn + pqc .I Risk correlation between individuals: δ = p(1−p)

µ(1−µ)(qc − qn)

2.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community

I A community of N identical risk-averse agents, with VNM utilityfunction u(.), wealth w and risk of loss l .

I 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

I Individual probability of being affected: q = (1− p)qn + pqc .I Risk correlation between individuals: δ = p(1−p)

µ(1−µ)(qc − qn)

2.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community

I A community of N identical risk-averse agents, with VNM utilityfunction u(.), wealth w and risk of loss l .

I 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

I Individual probability of being affected: q = (1− p)qn + pqc .I Risk correlation between individuals: δ = p(1−p)

µ(1−µ)(qc − qn)

2.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community

I A community of N identical risk-averse agents, with VNM utilityfunction u(.), wealth w and risk of loss l .

I 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

I Individual probability of being affected: q = (1− p)qn + pqc .I Risk correlation between individuals: δ = p(1−p)

µ(1−µ)(qc − qn)

2.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community

I A community of N identical risk-averse agents, with VNM utilityfunction u(.), wealth w and risk of loss l .

I 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

I Individual probability of being affected: q = (1− p)qn + pqc .I Risk correlation between individuals: δ = p(1−p)

µ(1−µ)(qc − qn)

2.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community

I A community of N identical risk-averse agents, with VNM utilityfunction u(.), wealth w and risk of loss l .

I 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

I Individual probability of being affected: q = (1− p)qn + pqc .I Risk correlation between individuals: δ = p(1−p)

µ(1−µ)(qc − qn)

2.

Arnaud Goussebaıle 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

I Insurance contract characteristics for one agent:I premium α,I indemnity τ in the normal state and τ − ϵ in the catastrophic state,I dividend π in the normal state.

I Because the premium is usually required ex-ante, there is potentiallya reserve cost λl(α) (increasing with α).

I 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 Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Agent wealth profile

I Insurance contract characteristics for one agent:I premium α,I indemnity τ in the normal state and τ − ϵ in the catastrophic state,I dividend π in the normal state.

I Because the premium is usually required ex-ante, there is potentiallya reserve cost λl(α) (increasing with α).

I 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 Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Agent wealth profile

I Insurance contract characteristics for one agent:I premium α,I indemnity τ in the normal state and τ − ϵ in the catastrophic state,I dividend π in the normal state.

I Because the premium is usually required ex-ante, there is potentiallya reserve cost λl(α) (increasing with α).

I 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 Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Insurer profit profile

I Reinsurance contract characteristics for the insurer:

I premium αR ,I indemnity τR in the catastrophic state.

I The required premium is αR = (1 + λR)pτR , in which λR > 0 is thereinsurance loading factor.

I 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 Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Insurer profit profile

I Reinsurance contract characteristics for the insurer:

I premium αR ,I indemnity τR in the catastrophic state.

I The required premium is αR = (1 + λR)pτR , in which λR > 0 is thereinsurance loading factor.

I 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 Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Insurer profit profile

I Reinsurance contract characteristics for the insurer:

I premium αR ,I indemnity τR in the catastrophic state.

I The required premium is αR = (1 + λR)pτR , in which λR > 0 is thereinsurance loading factor.

I 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 Goussebaıle 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

Optimal insurance and reinsurance contracts

I The optimal insurance and reinsurance contracts are such that:

maxα,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 Goussebaıle 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

Reinsurance coverage and insurance premium

I The insurer budget constraints are binding.

I Purchased reinsurance coverage:

τR = Nqc(τ − ϵ)︸ ︷︷ ︸catastrophic state

− (Nqnτ + Nπ)︸ ︷︷ ︸normal state

≥ 0

I Required insurance premium:

α =

(1+

p(qc − qn)

qλR︸ ︷︷ ︸

0<...<λR

)qτ+

(−1−λR︸︷︷︸

<0

)pqcϵ+

(1− p

1− pλR︸ ︷︷ ︸

<0

)(1−p)π

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Reinsurance coverage and insurance premium

I The insurer budget constraints are binding.

I Purchased reinsurance coverage:

τR = Nqc(τ − ϵ)︸ ︷︷ ︸catastrophic state

− (Nqnτ + Nπ)︸ ︷︷ ︸normal state

≥ 0

I Required insurance premium:

α =

(1+

p(qc − qn)

qλR︸ ︷︷ ︸

0<...<λR

)qτ+

(−1−λR︸︷︷︸

<0

)pqcϵ+

(1− p

1− pλR︸ ︷︷ ︸

<0

)(1−p)π

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Reinsurance coverage and insurance premium

I The insurer budget constraints are binding.

I Purchased reinsurance coverage:

τR = Nqc(τ − ϵ)︸ ︷︷ ︸catastrophic state

− (Nqnτ + Nπ)︸ ︷︷ ︸normal state

≥ 0

I Required insurance premium:

α =

(1+

p(qc − qn)

qλR︸ ︷︷ ︸

0<...<λR

)qτ+

(−1−λR︸︷︷︸

<0

)pqcϵ+

(1− p

1− pλR︸ ︷︷ ︸

<0

)(1−p)π

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .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, dαdλR > 0 (reins. cost ↗ ⇒ ins. premium ↗);

(ii) dτR

dδ > 0, dαdδ > 0 (correlation ↗ ⇒ ins. premium ↗).

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .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, dαdλR > 0 (reins. cost ↗ ⇒ ins. premium ↗);

(ii) dτR

dδ > 0, dαdδ > 0 (correlation ↗ ⇒ ins. premium ↗).

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .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) dαdλl < 0, dτR

dλl > 0 (reserve cost ↗ ⇒ reins. purchase ↗);

(ii) dτR

dδ > 0, dαdδ ? (correlation ↗ ⇒ ins. premium ambiguous).

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .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) dαdλl < 0, dτR

dλl > 0 (reserve cost ↗ ⇒ reins. purchase ↗);

(ii) dτR

dδ > 0, dαdδ ? (correlation ↗ ⇒ ins. premium ambiguous).

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Conclusion

I We develop a simple model with correlated individual risks within acommunity.

I We take into account that reinsurance and reserve are costly.

I We show that the optimal insurance contract can have bothindemnity and dividend contingent on collective losses.

I We exhibit the example of the Caribbean countries which implementthis type of contract for natural disaster risks.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Conclusion

I We develop a simple model with correlated individual risks within acommunity.

I We take into account that reinsurance and reserve are costly.

I We show that the optimal insurance contract can have bothindemnity and dividend contingent on collective losses.

I We exhibit the example of the Caribbean countries which implementthis type of contract for natural disaster risks.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Conclusion

I We develop a simple model with correlated individual risks within acommunity.

I We take into account that reinsurance and reserve are costly.

I We show that the optimal insurance contract can have bothindemnity and dividend contingent on collective losses.

I We exhibit the example of the Caribbean countries which implementthis type of contract for natural disaster risks.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community

..........

.....

.....................................................................

.....

......

.....

.....

.

. . .Introduction

. . . .Model

. . .Results

.Conclusion

Conclusion

I We develop a simple model with correlated individual risks within acommunity.

I We take into account that reinsurance and reserve are costly.

I We show that the optimal insurance contract can have bothindemnity and dividend contingent on collective losses.

I We exhibit the example of the Caribbean countries which implementthis type of contract for natural disaster risks.

Arnaud Goussebaıle with Alexis Louaas Crest, Polytechnique, Actinfo Chair

Pooling natural disaster risks in a community