Pooling natural disaster risks in a community

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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


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. . .Introduction

. . . .Model

. . .Results

.Conclusion

Motivation

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



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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.



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. . .Introduction

. . . .Model

. . .Results

.Conclusion

Motivation








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. . .Introduction

. . . .Model

. . .Results

.Conclusion

Motivation








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. . .Introduction

. . . .Model

. . .Results

.Conclusion

Motivation








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. . .Introduction

. . . .Model

. . .Results

.Conclusion

Motivation








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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.



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. . .Introduction

. . . .Model

. . .Results

.Conclusion

Questions










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. . .Introduction

. . . .Model

. . .Results

.Conclusion

Questions










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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.



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. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community



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.

p

.

catastrophic state:

.

affected agents =

.


.

1− p

.

normal state:

.

affected agents =


. qn < qc.

qn

.w − l

.affected

.

1− qn

.

w

.

not affected

.

qc

.

w − l

.

affected

.

1− qc

.

w

.

not affected



2.



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. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community



..

p

.

catastrophic state:

.

affected agents =

.


.

1− p

.

normal state:

.

affected agents =


. qn < qc.

qn

.w − l

.affected

.

1− qn

.

w

.

not affected

.

qc

.

w − l

.

affected

.

1− qc

.

w

.

not affected



2.



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. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community



..

p

.

catastrophic state:

.

affected agents =

.


.

1− p

.

normal state:

.

affected agents =


. qn < qc

.

qn

.w − l

.affected

.

1− qn

.

w

.

not affected

.

qc

.

w − l

.

affected

.

1− qc

.

w

.

not affected



2.



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. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community



..

p

.

catastrophic state:

.

affected agents =

.


.

1− p

.

normal state:

.

affected agents =


. qn < qc.

qn

.w − l

.affected

.

1− qn

.

w

.

not affected

.

qc

.

w − l

.

affected

.

1− qc

.

w

.

not affected



2.



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. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community



..

p

.

catastrophic state:

.

affected agents =

.


.

1− p

.

normal state:

.

affected agents =


. qn < qc.

qn

.w − l

.affected

.

1− qn

.

w

.

not affected

.

qc

.

w − l

.

affected

.

1− qc

.

w

.

not affected



2.



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. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community



..

p

.

catastrophic state:

.

affected agents =

.


.

1− p

.

normal state:

.

affected agents =


. qn < qc.

qn

.w − l

.affected

.

1− qn

.

w

.

not affected

.

qc

.

w − l

.

affected

.

1− qc

.

w

.

not affected



2.



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. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community



..

p

.

catastrophic state:

.

affected agents =

.


.

1− p

.

normal state:

.

affected agents =


. qn < qc.

qn

.w − l

.affected

.

1− qn

.

w

.

not affected

.

qc

.

w − l

.

affected

.

1− qc

.

w

.

not affected



2.



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. . .Introduction

. . . .Model

. . .Results

.Conclusion

The community



..

p

.

catastrophic state:

.

affected agents =

.


.

1− p

.

normal state:

.

affected agents =


. qn < qc.

qn

.w − l

.affected

.

1− qn

.

w

.

not affected

.

qc

.

w − l

.

affected

.

1− qc

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w

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not affected



2.



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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:

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1− p

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p

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1− qn

.

w − α− λl(α) + π

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qn

.w − α− λl(α)− l + τ + π

.

1− qc

.

w − α− λl(α)

.

qc

.

w − α− λl(α)− l + τ − ϵ



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. . .Introduction

. . . .Model

. . .Results

.Conclusion





.

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1− p

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p

.

1− qn

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w − α− λl(α) + π

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qn

.w − α− λl(α)− l + τ + π

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1− qc

.

w − α− λl(α)

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qc

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w − α− λl(α)− l + τ − ϵ



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. . .Introduction

. . . .Model

. . .Results

.Conclusion





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1− p

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p

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1− qn

.

w − α− λl(α) + π

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qn

.w − α− λl(α)− l + τ + π

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1− qc

.

w − α− λl(α)

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qc

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w − α− λl(α)− l + τ − ϵ



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. . .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



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. . .Introduction

. . . .Model

. . .Results

.Conclusion






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.1− p

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p

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. . .Introduction

. . . .Model

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.Conclusion






..1− p

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p

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. . .Introduction

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.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.



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. . .Introduction

. . . .Model

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.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



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. . .Introduction

. . . .Model

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.Conclusion






≥ 0


α =

(1+

p(qc − qn)

qλR︸︷︷︸

0<...<λR

)qτ+

(−1−λR︸︷︷︸

<0

)pqcϵ+

(1− p

1− pλR︸︷︷︸

<0

)(1−p)π



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. . .Introduction

. . . .Model

. . .Results

.Conclusion






≥ 0


α =

(1+

p(qc − qn)

qλR︸︷︷︸

0<...<λR

)qτ+

(−1−λR︸︷︷︸

<0

)pqcϵ+

(1− p

1− pλR︸︷︷︸

<0

)(1−p)π



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. . .Introduction

. . . .Model

. . .Results

.Conclusion

Without reserve cost (λl ′(α) = 0)

.Result 1a: optimal contracts..

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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..

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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 ↗).



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. . .Introduction

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.Conclusion

Without reserve cost (λl ′(α) = 0)


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The optimal contracts are such that:

(i) τ = l , ϵ = 0 and π > 0︸︷︷︸full indemnity plus dividend

;

(ii) τR > 0 iff λR < λR∗.


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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 ↗).



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. . .Introduction

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.Conclusion

With reserve cost (λl ′(α) > 0)


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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.


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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).



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. . .Introduction

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.Conclusion

With reserve cost (λl ′(α) > 0)


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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.


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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).



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. . .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.



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Conclusion







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Conclusion







Economy & Finance

Pooling natural disaster risks in a community