Optimal Reciprocal Reinsurance under GlueVaR Distortion Risk Measures ()
1. Introduction
Reinsurance is an effective risk management tool for the insurer to transfer part of its risk to the reinsurer. Let X be the original loss,if the insurer cedes a part of loss
(f is called the ceded loss function,or indemnification function) to the reinsurer and pays reinsurance premium
,then the insurer’s total liability
contains two parts: one is the retained loss risk
and the other is the reinsurance premium
,that is
(1.1)
The reinsurer’s total liability
also contains two parts: one is the ceded loss risk
and the other is the received reinsurance premium
,that is
(1.2)
For any
,we define total risks
in the presence of an insurer and a reinsurer as
(1.3)
Due to the development and application of risk measures in finance and insurance,many workers formulate the optimal reinsurance problem with Value at Risk (VaR) and Tail Value at Risk (TVaR). [1] proposed two optimization criterion that minimize total loss of the insurer by the Value at Risk (VaR) and the Conditional Tail Expectation (CTE). [2] showed that quota-share and stop-loss reinsurance are optimal when they studied a class of increasing convex ceded loss functions by VaR and CTE under the expected value principle. Many works extended the fundamental results,for example,[3] - [15] . [16] extended the conclusion of [15] to the general convex risk measure that satisfied regular invariance. Recently,there has a surge of interest in more generally distortion risk measures. [17] discussed the general model of the distortion risk measure and assumed that the distortion function is piecewise convex or concave. [18] studied the general model with distortion risk measures under general reinsurance premium principles. [19] expended the model of [18] under the cost-benefit framework. [20] studied the optimal reinsurance model of [18] without the premium constraint by a marginal indemnification function (MIF) formula. [21] studied the optimal reinsurance with premium constraint by combining the MIF formula and the Lagrangian dual method. [22] and [23] studied the optimal reinsurance with constraints under the distortion risk measure.
VaR has been adopted as the standard tool for assessing the risks and calculating the capital requirements in finance and insurance,however,it has two drawbacks in financial industry. One is that the capital requirements can be underestimated and the underestimated may be aggravated when heavy tail losses are incorrectly modeled by mild tail distribution. The second one is that the VaR may fail the subadditivity. Though TVaR has no these two disadvantages of VaR,it has not been widely accepted by practitioners in finance and insurance. In order to overcome this weakness,[24] proposed a new family of risk measures,namely GlueVaR distortion risk measures. We take different definitions of VaR from [24] ,therefore,a new definition of GlueVaR has been given in this paper.
Optimal reinsurance from an insurer’s viewpoint or from a reinsurer’s viewpoint has been studied for a long time in the literatures. However,as two parties of a reinsurance contract,there has a conflict of interests between an insurer and a reinsurer. The optimal reinsurance policy from one party’s perspective may not be optimal for another party. Therefore,we consider a reciprocal reinsurance. Motivated by [21] and [24] ,we want to study the optimal reciprocal reinsurance strategy under GlueVaR distortion risk measures with MIF formula.
The rest of this paper is organized as follows. In Section 2,we give some notations and proposal a reciprocal reinsurance model. In Section 3,we derive the sufficient conditions that guarantee the existence of a reinsurance contract. In Section 4,we obtain the specific expression of optimal reinsurance. Section 5 concludes this paper.
2. The Model
2.1. Preliminaries and Notations
Definition 2.1. (Distortion risk measure or distorted expectation) A distortion function is a non-decreasing function
such that
and
. The distortion risk measure or distorted expectation of the random variable X associated with distortion function g,notation
,is defined as
(2.1)
The most well-known examples of distortion risk measures are the VaR and TVaR,if we define the distortion functions,respectively,as follows
(2.2)
and
(2.3)
then the distorted expectation
can be equivalently expressed as
(2.4)
and
(2.5)
Definition 2.2. (GlueVaR distortion risk measure) Given the confidence levels
and
,when the distortion function for GlueVaR is specified to the following function
(2.6)
with
,
,
,and
,then the corresponding distortion risk measure
is the GlueVaR distortion risk measure,which is denoted by
.
Remark 2.1. If the following notation is used,
(2.7)
then the distortion function
in (2.6) may be rewritten as
(2.8)
where
,
and
are the distortion functions corresponding to the
,
and
,respectively. Therefore,GlueVaR is a risk measure that can be expressed as a linear combination of three risk measures as follows,
(2.9)
where
for
,and
.
Example 2.1. Assume that initial risk X follows an exponential distribution with parameter 0.001,then
,
. When
,
and
,the values of VaR,TVaR and GlueVaR at different confidence levels are calculated in Table 1.
Given
and
,the values in Table 1 indicate that GlueVaR is more conservative than VaR. Note that
,which means that GlueVaR may overcome the VaR’s shortage of underestimating risks. On the other hand,GlueVaR is not,unlike TVaR,overly conservative. It seems clear that GlueVaR,a new risk measure based on distortion functions,can be valuable in the scope of finance and insurance.
Definition 2.3. (Marginal indemnification function) (See [ [20] ,Definition 2]) For any indemnification function
,the associated marginal indemnification is a function
such that
(2.10)
2.2. Model Set-Up
Based on the notations of the preceding subsection,we will introduce a reciprocal reinsurance model to study the optimal strategy which considers the interests of both an insurer and a reinsurer.
Problem 1 (Optimization model of a reciprocal reinsurance)
(2.11)
where
= {
:
and
are non-decreasing and
,
}.
Our objective is to find the optimal ceded loss function
and to characterize the corresponding
.
Table 1. VaR,TVaR and GlueVaR of initial risk X.
3. Existence of Optimal Reinsurance Strategy
Lemma 3.1 For any ceded loss functions
,
can be expressed as
(3.1)
where
for
,and
.
Proof. As proved in Lemma 2.1 of Zhuang et al. (2016),for any distortion function g,
Obviously,
may be rewritten as (3.1).
Lemma 3.2 For any
and ceded loss function
,total risks
can be expressed as
(3.2)
where
.
Proof. From definitions of
and
,
can be rewritten as
(3.3)
By the comonotonic additivity of the distortion risk measures,total risks
under the GlueVaR distortion risk measures can be expressed as
(3.4)
Based on the fact that
(3.5)
with the expressions (3.1),(3.4) and (3.5),we get
Lemma 3.3 Let
be the optimal marginal indemnification function,then it satisfies
(3.6)
Suppose that
for
. Then
solves (3.6) if and only if
solves (2.11).
Proof. This follows from the same arguments used in the proof to Proposition 2.1 of Zhuang et al. (2016).
Theorem 3.1 For
,
solves 3.6 if and only if it satisfies the followings.
1). If
,then
(3.7)
2). If
,then
(3.8)
3). If
,then
(3.9)
Proof. Note that minimizing
is equivalent to minimizing
of (3.2). In the next,we will prove the results from three cases.
1). For the cases
,
.
a) If
,then the minimum
is attained at
.
b) If
,then
for any
.
c) If
,then the minimum
is attained at
.
2). For the cases
,
for any
.
3). For the cases
,
.
a) If
,then the minimum
is attained at
.
b) If
,then
for any
.
c) If
,then the minimum
is attained at
.
4. Explicit Solutions
In Section 3,we have derived the optimal marginal indemnification function
. It seems very concise but we can not obtain the optimal reinsurance strategy
directly. In this section,we want to derive the optimal reinsurance contract
bases on optimal marginal indemnification function
.
Let
and denote
,we have
(4.1)
where
(4.2)
(4.3)
(4.4)
With the expression (4.1)-(4.4),
may be reexpressed as
(4.5)
which has two positive zeros,
where
(4.6)
(4.7)
(4.8)
Theorem 4.1 For any ceded loss function
,if
,then
Proof. From (2.10) and (3.8),we can derive above results easily.
Theorem 4.2 For
,and any ceded loss function
,optimal reinsurance contracts
to Problem 1 are given as follows:
1). If
and
,then
.
2). If
and
,then
3). If
,then
4). If
,then
Proof. Analyse the optimal reinsurance contract with (3.7) for the case
. From (4.5)-(4.8),clearly
and
. Note that
,but
,which means that
is discontinuous at the point
. Therefore,we consider the followings.
1). When
,there has three cases about
,which are
,
and
.
a) If
,then
.
exists since
and. Note that in,in as Figure 1. With the expression (3.7),we have that for,for as Figure 2,thus.
b) If,then. Similar to 1),.
c) When,has three cases,and. Since is discontinuous at the point,we have to consider the cases of.
i) If,then. Therefore,exists. in,in. Furthermore,for,for,so.
ii) If,then exists. If,then exists since. Note that in and,in and. Furthermore,for,for,so .
iii) If,then exists. When,in,in. Furthermore,for,for,so.
2). When,from (4.6) and (4.7),we obtain that and. Next,we consider the cases of.
a) When,we can derive that in,in,and in and. Furthermore,for,for,and for. Therefore,.
b) When,in and in. Furthermore,for,and for . Therefore,.
3). When,note that and. There has three cases for.
a) When,in and in other cases. Furthermore,for,for. Therefore,.
b) If,then in. Therefore,when,.
Theorem 4.3 For,and any ceded loss function,optimal reinsurance contracts to Problem 1 are given as follows:
1). If and,then.
2). If and,then
3). If,then
4). If,then
Proof. Analyse the optimal reinsurance contract with (3.9) for the case.
1). When,there has three cases about.
a) If,then. Since and,then exists. Therefore,in and in as Figure 1. With the expression (3.9),we have that for and for as Figure 3,so.
b) If,then. Similar to 1),.
c) When,has three cases,and. Since is discontinuous at the point,we have to consider the cases of.
i) If,then,exists since. Note that in and in. Furthermore,for,for,so.
ii) If and,then and exists. Clearly in and,in and. Furthermore,for,for ,so .
iii) If and,then exists. Clearly,in and in. Furthermore,for,for,so.
2). When,from (4.6) and (4.7),we obtain that and. Next,we consider the cases of.
a) When,we can derive that in,in,and in and. Furthermore,for,for,and when. Therefore,.
b) When,in and in. Furthermore,for,and for . Therefore,.
3). When,note that and. There has three cases for.
a) When,in and in other cases. Furthermore,for,for. Therefore,.
b) If,then in. Therefore,when,.
Example 4.1. Similar to Example 2.1,we assume the risk is measured by the GlueVaR risk measures under the expectation premium principle,for,,,and
,optimal reinsurance contracts are given as follows.
From the reinsurer’s point of view,as Case 1 in Table 2,the optimal reinsurance strategy can be in form of limited quota-share,,which means that if initial loss X less than 405.47,the case that an insurer ceded all loss to a reinsurer is optimal,and if initial loss X more than 405.47,the case that an insurer ceded 405.47 to a reinsurer is optimal.
From the insurer’s point of view,as Case 6 in Table 2,the optimal reinsurance strategy
,which means that an insurer should retain all loss to achieve itself optimality.
Table 2. Optimal ceded loss function.
From the perspectives of an insurer and a reinsurer,as Cases 2 - 5. Note that Cases 2 and 5 include the parameter,which means that reinsurance contracts can be different forms when the loss risk has been minimized. Case 3 means that the stop-loss after quota-share reinsurance (which is to say a stop-loss will be applied after a quota-share reinsurance) is optimal. Case 4 means that stop-loss reinsurance is optimal.
5. Conclusion
This article has studied the optimal reciprocal reinsurance with the GlueVaR distortion risk measures under the expected value premium principle. The GlueVaR distortion risk measure is a linear combination of two TVaR and one VaR with different confidence levels,which adds the difficulty than the case of only one VaR or the case of only one TVaR when we derive the optimal reinsurance contract. In this paper,we have expressed GlueVaR as a linear combination of three distortion risk measures with different distortion functions. Therefore,we can use MIF formula to deal with the complex optimization problems easily. The results indicate that depending on the risk measures’s level of confidence (and),the safety loading () for the reinsurance premium,weight () of an insurer in the reciprocal reinsurance model and the proportions (and) of the three risk measures in the definition of GlueVaR,the optimal reinsurance can be in the forms of quota-share,stop-loss,change-loss,or their combination,for example,stop-loss after quota-share. This paper has not considered the practical constraints,such as risk constraints or reinsurance premium constraints,which can be studied at a later time.
Acknowledgements
The author would like to thank the anonymous referees for helpful comments and suggestions,which have led to significant improvements of the present paper.
Author Contributions
These authors contributed equally to this work.
Funding
The research was supported by the National Natural Science Foundation of China (No. 11171179,11571198).