Expected Shortfall风险度量下的最优再保险(英文)

AbstractTo avoid the default due to a high claim amount， the insurer usually transfers part of its risk to the reinsurer by signing a reinsurance contract. To seek the optimal reinsurance strategy， recent studies focus on minimizing variance of the retained loss， value-at-risk or conditional tail expectation of the insurer’s total risk exposure. This note studies the reinsurance strategy under the expected shortfall criterion. We build the optimal increasing and convex ceded loss function， both cases without and with the constraint of premium are discussed.

Key wordsCeded loss； Premium principle； Quota-share strategy； Stop-loss strategy

CLC numberO 212Document codeA

1Introduction

To avoid paying for a high claim amount， the insurer transfers part of its risk to the reinsurer by signing a reinsurance contract. Let the random loss X≥0 due to an insurance claim be with distribution function F（x） = Pr{X≤x} and survival function S（x） = Pr{X > x}. Intuitively， as some loss is transferred to the reinsurer， the ceded loss r（X） should be nonnegative and never exceed the initial risk， i.e.， the ceded loss function satisfies 0≤r（x）≤x for all x≥0. Correspondingly， the retained loss of the insurer isˉr（X） = X？r（X）. The reinsurer undertakes a portion of loss for the insurer， in return， the insurer has to pay reinsurance premiumπr（X） to the reinsurer in correspondence to the ceded loss function r. Then， the insurer gets the total risk exposure Tr（X） =ˉr（X） +πr（X） = X？r（X） +πr（X）.

Naturally， it is of interest to seek the optimal reinsurance strategy in the context of certain criterion on loss and a specific premium principle. For example， under the criterion of minimizing the variance of the retained loss and the standard deviation premium principle， Gajek and Zagrodny[3]demonstrated that change loss reinsurance contract is optimal subject to the condition that the premium is not larger than a given amount， and for the premium principle based on the mean and variance of the ceded loss， Kaluszka[4]showed that change loss contract is the optimal one. Also， Gajek and Zagrodny[2]found out some optimal reinsurance strategies under several general risk measures. Recently， Cai and Tan[5]developed two new criteria： minimizing the value-at-risk （VaR） and minimizing the conditional tail expectation（CTE） of the insurer’s total risk exposure. For the optimal reinsurance strategies corresponding to these two new criteria， please refer to [1，6].

For ease of reference， let usfirst review some risk measures and related properties. For more details please refer to [7，8].

At a certain level p∈（0，1）， VaR tells the amount that will maximally be lost with probability p， but it does not give any information about the thickness of the upper tail of the distribution function， while CTE is the average loss in the worst 100pˉ% cases. By contrast， ES is the stop-loss premium with retention VaR[X；p]. Intuitively， from the perspective of the insurer， ES of the total risk exposure should be as small as possible. It is practical to consider the criterion of minimizing ES of the total risk exposure.

The following lemma lists some nice properties of these risk measures.

Lemma 1For any nonnegative random risk X and a confidence level p∈（0，1），

（i） VaR[X + c；p] = VaR[X；p] + c and TVaR[X + c；p] = TVaR[X；p] + c for any constant c；

This note aims to build the optimal increasing and convex ceded loss function under ES criterion. The rest of this note is organized as follows： Section 2 considers the criterion of minimizing ES of the total risk exposure. Setting aside the premium principle， we derive the general optimal ceded loss function. Specifically， in Section 3， the optimal ceded loss function is also built for the expected value premium principle subject to the condition that the premium is not larger than a given amount.

2Without the Constraint of Premium

Lemma 3The ceded loss function minimizing ES of the insurer’s total risk exposure in class H， if exists， is also optimal in class F.

By definition of ES， we have ES[Tr（X）；p]= E[（Tr（X）？VaR（Tr（X）；p））+]. According to Theorem 1， we could choose particular ceded loss functions such that ES of the total risk exposure equals to 0. And thus， at a given confidence level， by choosing particular ceded loss functions， the expected total risk exposure would not excess its VaR. This is of great help in allocating the economic capital for the insurance claim. Further， by （5）， we have： （i） for dn，1=···= dn，n= S？1（pˉ）， the optimal ceded loss function r？（x） =（x？S？1（pˉ））+， i.e.， stop-loss reinsurance strategy， and （ii） for dn，1=···= dn，n= 0， the optimal ceded loss function r？（x） = x， i.e.， quota-share reinsurance strategy.

References

[1] Cai J， Tan K S， Weng C et al. Optimal reinsurance under VaR and CTE risk measures. Insur Math Econom， 2008， 43： 185-196.

[2] Gajek L， Zagrodny D. Optimal reinsurance under general risk measures. Insur Math Econom， 2004， 34： 227-240.

[3] Gajek L， Zagrodny D. Insurer’s optimal reinsurance strategies. Insur Math Econom， 2000， 27： 105-112.

[4] Kaluszka M. Optimal reinsurance under mean-variance premium principles. Insur Math Econom， 2001， 28： 61-67.

[5] Cai J， Tan K S. Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures. Astin Bull， 2007， 37： 93-112.

[6] Tan K S， Weng C， Zhang Y. Optimality of general reinsurance contracts under CTE risk measure. Insur Math Econom， 2011， 49： 175-187.

[7] Denuit M， Dhaene J， Goovaerts M J et al. Actuarial theory for dependent risks. Chichester， Wiley Press， 2005.

[8] Kaas R， Goovaerts M J， Dhaene J et al. Modern actuarial risk theory， using R. Heidelberg， Springer Press， 2008.

[9] Tan K S， Weng C. Enhancing insurer value using reinsurance and value-at-risk criterion. Geneva Risk and Insurance Revi， 2012， 37： 109-140.