Journal of Risk
ISSN:
1465-1211 (print)
1755-2842 (online)
Editor-in-chief: Farid AitSahlia
Need to know
- The author studies optimal reinsurance designs by minimizing the adjusted value of the liability of an insurer and the risk margin is determined by expectile
- The author considers a class of ceded loss functions that are subject to Vajda condition
- The premium principles are assumed to satisfy the properties of law invariance, risk loading and convex order preserving
- We show that the optimal ceded loss functions are of the form of three interconnected line segments
Abstract
In this paper, we revisit optimal reinsurance problems by minimizing the adjusted value of the liability of an insurer, which encompasses a risk margin. The risk margin is determined by expectile. To reflect the spirit of reinsurance of protecting the insurer, we assume that both the insurer’s retained loss and the proportion paid by a reinsurer are increasing in indemnity. The premium principles are assumed to satisfy the following three properties: law invariance, risk loading and convex order preservation. We show that the optimal ceded loss functions take the form of three interconnected line segments. Further, if the reinsurance premium is translation invariant or follows the expected value principle, simplified forms of the optimal reinsurance treaties are obtained. Finally, when the reinsurance premium is assumed to be the expected value principle or Wang’s premium principle, the explicit expression for the optimal reinsurance treaty is also given.
Introduction
1 Introduction
Reinsurance is a contract between an insurance company (insurer) and a reinsurance company (reinsurer); it is a popular risk management strategy for an insurer. In a reinsurance treaty, there exist three basic elements: a set of admissible ceded loss functions, a reinsurance premium principle and an optimal criterion. By changing one or more of the above three aspects, a number of optimal reinsurance treaties have been studied.
Regarding the variety of different classes of ceded loss functions considered for optimal reinsurance problems, if one does not take into account the constraints on the risk of the insurer or the reinsurer, there are two typical classes of ceded loss functions employed in the literature. The first is the class of ceded loss functions such that the retained and ceded loss functions are increasing, which was suggested by Huberman et al (1983) to preclude moral hazard. For example, Chi (2012) studied optimal reinsurance treaties by minimizing the adjusted value of the liability of an insurer, where value-at-risk (VaR) and conditional value-at-risk (CVaR) were used to determine the risk margin. Chi (2012) proved that layer reinsurance is often optimal. For more studies about optimal reinsurance treaties with VaR or CVaR, we refer to Liu et al (2016), Chi et al (2017), Cai et al (2017) and the references therein. Cai and Weng (2016) studied optimal reinsurance treaties by minimizing the adjusted value of the liability of an insurer, where the risk margin is determined by expectile. They proved that a two-layer reinsurance treaty is optimal. Optimal reinsurance treaties under distortion risk measures have also been studied in the literature, see, for example, Cui et al (2013), Assa (2015), Zhuang et al (2016), Cheung and Lo (2017), Lo (2017a,b) and Jiang et al (2018). However, in reality, when an insurer is facing an increasing total claim amount, the insurer will want to ask the reinsurer to pay not only an increasing amount of the liability but also an increasing proportion of the total claim amount. Therefore, the second is the class of ceded loss functions such that the retained loss functions are increasing and the ceded loss functions satisfy the Vajda condition, which was proposed by Vajda (1962) to reflect the spirit of reinsurance of protecting the insurer. A ceded loss function is called to satisfy the Vajda condition if the proportion of the ceded loss function is increasing, and such a ceded loss function is usually called a Vajda function. Hesselager (1990, 1993) studied some optimal reinsurance problems under the Vajda condition. Chi and Weng (2013) studied optimal reinsurance treaties under VaR or CVaR over the class of ceded loss functions such that the retained loss functions were increasing and the ceded loss functions satisfied the Vajda condition. They proved that the optimal ceded loss functions take the form of three interconnected line segments. Chen and Hu (2020) studied optimal reinsurance from the perspectives of both insurers and reinsurers under the VaR risk measure and the Vajda condition.
Recently, the expectile, as first proposed by Newey and Powell (1987), has been of interest in statistics and finance. Ziegel (2014) (see also Bellini et al 2014) has pointed out that VaR is elicitable but not coherent and CVaR is coherent but not elicitable, and that the expectile is the only elicitable, law-invariant and coherent risk measure. However, in order to score the estimation of risks, according to Gneiting (2011), elicitability is a natural property that must be satisfied by a risk measure. From this point of view, the expectile can be considered as a natural candidate beyond VaR and CVaR. For instance, Bellini and Di Bernardino (2017) studied risk management with expectiles. Emmer et al (2015) compared these risk measures from a practical point of view. Maume-Deschamps et al (2017) and Herrmann et al (2018) proposed two kinds of multivariate expectiles. Hu and Zheng (2020) studied the application of expectiles in a capital asset pricing model. Cai and Weng (2016) studied expectiles’ applications in reinsurance as well as optimal reinsurance with expectiles over the class of ceded loss functions such that the retained and ceded loss functions were increasing. Among a class of reinsurance premium principles satisfying the law invariance, risk loading and convex order preservation properties, they proved that a two-layer reinsurance treaty is optimal. To account for the spirit of reinsurance of protecting the insurer, as suggested by Vajda (1962), we must answer an important question: what happens to the optimal reinsurance treaty with the expectile if the class of ceded loss functions is replaced by the smaller class of Vajda functions, which means that the retained loss functions are increasing and the ceded loss functions are Vajda functions? It will turn out that the optimal reinsurance treaty obtained in the case of Vajda functions is quite different from that of Cai and Weng (2016).
In this paper, motivated by Chi and Weng (2013) and Cai and Weng (2016), we study optimal reinsurance treaties that minimize the risk-adjusted value of an insurer’s liability over the class of ceded loss functions such that the retained loss functions are increasing and the ceded loss functions satisfy the Vajda condition. By employing the expectile as a risk measure to calculate the capital at risk included in the risk-adjusted value of an insurer’s liability, we show that the optimal ceded loss functions always take the form of three interconnected line segments for a wide class of reinsurance premium principles that satisfy three properties: law invariance, risk loading and convex order preservation. Moreover, further simplified forms of optimal reinsurance treaties are obtained for the expected value principle or the reinsurance premium principles that satisfy the translation invariant property.
The rest of the paper is organized as follows. In Section 2, we will introduce some preliminaries, including a definition of the expectile and the formulation of our optimal reinsurance treaties. In Section 3, we will study the optimal reinsurance design that minimizes the expectile of the insurer over the class of ceded loss functions such that the retained loss functions are increasing and the ceded loss functions satisfy the Vajda condition. The optimal reinsurance treaties will be provided, but their proofs will be postponed to Section 5. Some examples will be presented in Section 4 to demonstrate how the parameters in the optimal ceded loss functions can be determined explicitly.
2 Preliminaries
In this section, we will briefly introduce some preliminaries. The claim faced by an insurer is characterized by a nonnegative random variable on some probability space with finite expectation . We denote by , , the distribution function of , and by the survival function of . We denote by the class of nonnegative random variables with finite expectation.
In a classical reinsurance design, the insurer would cede part of the loss , say , to a reinsurer and retain part of the loss , say . In a reinsurance contract, the functions and are referred to as the ceded loss function and the retained loss function, respectively. When an insurer cedes part of the loss to a reinsurer, the insurer needs to pay a reinsurance premium to the reinsurer according to a premium principle . In the presence of reinsurance, the liability of an insurer, denoted by , is
According to Risk Margin Working Group (2009), the risk-adjusted value of the insurer’s liability is calculated as
(2.1) |
where is a constant and is a risk measure used to quantify the gap between the total risk exposure and its actuarial reserve . For more works on evaluating the insurer’s liability using the risk margin, we refer to Swiss Federal Office of Private Insurance (2006), Risk Margin Working Group (2009), Wüthrich et al (2010), Chi (2012), Asmit et al (2013), Chi and Weng (2013), Cai and Weng (2016), Cheung and Lo (2017), Chi et al (2017) and the references therein.
To preclude the moral hazard and, further, to reflect the spirit of reinsurance of protecting the insurer, the set of admissible ceded loss functions is assumed to be the class of ceded loss functions such that the insurer’s retained loss and the proportion paid by a reinsurer are increasing in indemnity, which was first suggested by Vajda (1962). Namely, we search for the optimal reinsurance treaties among the following set of ceded loss functions:
(2.2) |
Note that, as proved by Chi and Weng (2013), , where
(2.3) |
was suggested by Huberman et al (1983).
Meanwhile, we consider a wide class of reinsurance premium principles satisfying the following three properties.
- (i)
Law invariance: depends only on the cumulative distribution function of .
- (ii)
Risk loading: for any .
- (iii)
Convex order preservation: for any with , ie,
provided that the expectations exist, where for .
In this paper, we denote by the set of all the premium principles of satisfying the above three properties. As pointed out by Chi (2012) and Chi and Weng (2013), the proposed class of premium principles includes all the premium principles listed in Young (2004) except the Esscher principle.
In the present paper, we use the expectile to determine the risk margin of (2.1). It is defined as follows.
Definition 2.1.
The expectile of a random variable with at a given confidence level is defined as
(2.4) |
where and .
Note that is strictly increasing in and . In insurance and finance, the economic meaning of a risk measure of a loss random variable is usually considered as a premium or regulatory capital, which is often required to be larger than the expected loss . Hence, in insurance and finance, we are usually interested in , since for , for , and for . Bellini et al (2014) proved that is a coherent risk measure when . Ziegel (2014) proved that the expectile is the only elicitable, law-invariant and coherent risk measure. Hence, from the point of view that diversification can reduce risk, and in order to score the estimation of risks, the expectile can be considered a natural candidate beyond the VaR and CVaR. For more studies on properties of the expectile, we refer to Emmer et al (2015), Bellini and Di Bernardino (2017), Maume-Deschamps et al (2017), Herrmann et al (2018) and the references therein.
3 Optimal reinsurance design
In this section, we shall investigate the solution to the optimal reinsurance problem (2.5).
We begin with some notation. For and , we define
(3.1) |
Note that . Moreover, can be rewritten as
(3.2) |
Graphs of the functions with , , and with , , are given in Figure 1 and Figure 2, respectively, by the solid curve.
Note that and , hence, stop-loss reinsurance and quota share reinsurance are special cases of . can be considered as a combination of quota share reinsurance and stop-loss reinsurance.
Denote by
the expectile of the insurer’s total risk exposure and the retained loss , respectively. For any , define
(3.3) |
Then, , which was proved by Lemma 3.2 of Cai and Weng (2016) since . Based on , we further define
(3.4) |
with a constant satisfying
(3.5) |
The following lemma shows the well definedness and some properties of , defined by (3.4), which can be considered as analogous to that of Lemma 3.2 in Cai and Weng (2016).
Lemma 3.1.
Lemma 3.2.
For any , there exists a ceded loss function satisfying
(3.6) | ||||
(3.7) | ||||
(3.8) |
and
(3.9) |
where
(3.10) |
Now we are in a position to state the main result of the present paper, which provides the optimal reinsurance treaty for problem (2.5).
Theorem 3.3.
For any premium principle ,
(3.11) |
Remark 3.4.
Comparing Theorem 3.3 of this paper with Theorem 3.4 of Chi and Weng (2013), the optimal reinsurance form under with the expectile and the CVaR all take the form of three interconnected line segments, ie, they have the same shape but the conditions satisfied by the parameters are different. More precisely, Theorem 3.3 shows that the optimal ceded loss functions are with and satisfying some conditions, ie, they take the form of three interconnected line segments. In other words, the optimal reinsurance form for (2.5) is a combination of quota share reinsurance and stop-loss reinsurance. We believe that this kind of combination could be practical in insurance companies.
Theorem 3.3 shows that the study of infinite-dimensional optimal reinsurance model (2.5) can be simplified to solve an optimal problem of three variables in (3.11). The following theorem shows that, with an additional mild condition on the premium principle , the dimension of this problem can be further reduced.
Theorem 3.5.
If the premium principle is translation invariant, ie,
or if is the expected value principle, ie,
where is a loading factor, then
(3.12) |
where
(3.13) |
Remark 3.6.
Comparing Theorem 3.5 of this paper with Corollary 3.5 of Chi and Weng (2013), when the premium principle is translation invariant or is the expected value principle, the optimal ceded loss functions under with the expectile and the CVaR are ; the difference is the conditions satisfied by and .
4 Examples
In this section, we will use the results obtained in the previous section to derive explicit expressions for optimal reinsurance treaties, assuming that the reinsurance premium principle is calculated by the expected value premium principle and Wang’s premium principle.
Under the conditions in Theorem 3.5, the optimal ceded loss function is of the following form:
(4.1) |
with . In this section, we will show that parameters and in the optimal ceded loss function can be determined explicitly under the expected value premium principle or Wang’s premium principle.
Denote
(4.2) |
then, for any and , we have .
Moreover, denote
(4.3) |
where . Then, under the conditions in Theorem 3.5, a ceded loss function is the optimal solution to (2.1) if and only if the parameter is the solution of the following problem:
(4.4) |
where .
Given , is strictly decreasing in and thus has at most one solution for , and the solution is unique if it exists. This means that the implicit function , constrained with and , has the domain of
The next lemma will show that the set is nonempty, which implies that the function is well defined, and it will give some properties of the function . Let
Lemma 4.1 of Cai and Weng (2016) has shown that admits a unique solution over .
Lemma 4.1.
Suppose that the nonnegative random variable with has a continuous distribution function on with for any . Then, the following results hold.
- (1)
The function defined on is a strictly decreasing function with domain .
- (2)
, , and the derivative
(4.8)
In terms of the function , the optimal ceded loss function is given by , with solved via
(4.9) |
In the rest of this section, we will solve (4.9) for the expected value premium principle and Wang’s premium principle.
Example 4.2 (Expected value premium principle).
For the expected value premium principle, by (4.5), we have
for a loading factor . Thus, from (4.3), (4.5), (4.6) and (4.7), the objective function in (4.9) is reduced to
Hence, by (4.8), we obtain
(4.10) |
Let .
- (1)
Assume ; then it follows from (4.10) that for any . Thus, the objective function in (4.9) is decreasing in , and then its minimum is attained at . From Lemma 4.1, we know that . Hence, the optimal ceded loss function is , ie, the optimal strategy for the insurer is not to seek any reinsurance in this case.
- (2)
Assume ; let . Then, is decreasing on and increasing on . Thus, attains its minimum at and the optimal ceded loss function is . That is,
- (i)
if (note that ), we have the optimal ceded loss function is , ie, the optimal strategy for the insurer is not to seek any reinsurance in this case;
- (ii)
if , then the optimal ceded loss function is .
- (i)
Remark 4.3.
Under the expected value premium principle, Cai and Weng (2016, Example 4.1) show that a solution to the expectile-based optimal reinsurance model (2.5) with the admissible set is
where is the unique solution of
In contrast, an optimal ceded loss function among the set according to the above example is
where is the unique solution of
Obviously, .
Example 4.4 (Wang’s premium principle).
Assume the reinsurance premium is calculated by Wang’s premium principle, ie,
where the distortion function is increasing and concave with and . It is not hard to check that satisfies for any . Then, the optimal ceded loss function to (2.5) is given by
(4.11) |
where
(4.12) |
is a function of satisfying
and is a solution of
Proof.
Note that Wang’s premium principle is translation invariant:
Hence, from (4.3), (4.5), (4.6) and (4.7), the objective function in (4.9) is reduced to
This means that the minimization problem (2.5) can be reduced to
It is easy to see that defined in (4.12) is a solution to the above minimization problem. ∎
5 Proofs of main results
In this section, we will provide all the proofs of the results stated in Section 3 and the proof of Lemma 4.1.
Proof of Lemma 3.1.
- (1)
and (2) are obvious by using Cai and Weng (2016, Lemma 3.2(a,b)) and the fact that .
- (3)
- (4)
Note that if and only if . Thus,
(5.4) which implies the desired result.
- (5)
Using (5.4), we have
- (6)
- (7)
Using (5) and (6) in Lemma 3.1, we obtain that
(5.5) If , we may assume that , and then
which contradicts the assumption of . Hence, .
- (8)
We will prove this property in three separate cases: , and .
Case 1. Say . Then, the condition (3.5) is equal to
which, together with the fact that if and only if , yields that P-a.s. in the event that Thus, P-a.s. in the event that . Further, P-a.s., which, together with the law invariance property of , yields that .
Case 2. Say . In this case, using (2) in Lemma 3.1, from the condition (3.5), we have
(5.6) Since is increasing in , P-a.s in the event that , which, together with (5.6), yields that P-a.s. Hence, .
Case 3. If . Let
(5.7) We will show that , and then up-crosses at .11 1 A function is said to up-cross a function if there exists an such that for , and for .
First, we show that . In fact, if , then by the definition of we have for any . Thus, for any . Hence,
which contradicts the condition (3.5). Hence, .
Say . From the continuity of , we have . Further, , which, together with (2) in Lemma 3.1, yields that ; this contradicts the definition of . Hence, .
Next, we show that up-crosses at :
(5.8) where (5.8) follows from the increasing property of in . In fact, for any , , and by the continuity of , we have . Thus, for any .
Hence, up-crosses at . From (5) in Lemma 3.1, we have . Therefore, using Ohlin’s lemma, we obtain , which, together with the convex order preservation of , yields that . The proof of Lemma 3.1 is completed.22 2 For a random variable and two increasing functions and with , Ohlin’s lemma states that if up-crosses , then . See Ohlin (1969) for more details.
∎
Proof of Lemma 3.2.
For any (3.4), let
where is defined as (3.4) and satisfies (3.5) (Lemma 3.1 has proved the existence of satisfying (3.5), ie, the well definedness of ). Note that implies that , giving us
Further,
(5.9) |
where the last inequality above comes from the fact that if and only if :
When , using the increasing property of in , we have . Hence,
(5.10) |
Hence, from (5.9) and (5.10), and using the continuity of in , there exists a constant such that
(5.11) |
From (5) in Lemma 3.1, it follows that . Hence, we further have , ie, we obtain (3.6).
Then, (3.8) follows from (3.6), (3.7) and the definition of the expectile. Using (3.6)–(3.8), it is easy to check the above function .
To show (3.9), note that
Since , we have . Hence,
We will show that up-crosses . Let
where .
Case 1. Say . In this case, obviously, , and .
- (1)
When , we have, from the definition of , , ie, for any .
- (2)
When , , since is increasing in .
Hence, up-crosses at .
Case 2. Say . In this case, for any , it follows that .
- (1)
When , .
- (2)
When , obviously, .
Hence, up-crosses at .
Proof of Theorem 3.3.
On the one hand, obviously, ; thus,
(5.12) |
On the other hand, for any , we define , as given by Lemma 3.2, satisfying (3.6)–(3.9). We will further show that
(5.13) |
For or , define and . Then, in order to show (5.13), we only need to show
Note that , and for or . Hence,
(5.14) |
Proof of Theorem 3.5.
For any , define
where
We first show that there exists a constant
such that
(5.16) |
where is given by
and
In fact, when , we have
and
Hence,
which implies that
When , however, we have
Thus,
ie,
Since is continuous and decreasing in , there exists a constant such that
which further implies (5.16).
Further, it is easy to verify that up-crosses ; hence, using Ohlin’s lemma, we obtain
which, together with the convex order preservation property of , yields that
(5.17) |
Next, we will show
(5.18) |
where for or .
In order to show (5.18), we need only prove that
Note that
(5.19) |
where the last equality follows from (5.16).
On the other hand,
(5.20) |
Thus, we have
(5.21) |
where the second equality follows from (5.19), the third equality follows from (5.20) and the penultimate equality follows from the fact that satisfies
Hence, , ie,
(5.22) |
We first assume that is translation invariant. Note that (5.16) is equal to
(5.23) |
Hence,
(5.24) |
where the second inequality follows from (5.17) and (5.23), and the penultimate equality follows from the translation invariance of . Therefore,
When is the expected value principle, ie, for any loss random variable with , with a loading factor . Then,
and
Proof of Lemma 4.1.
- (1)
Similar to Cai and Weng (2016, Lemma 4.1), we can show that is a closed set. At the same time, is bounded, since
for any and . Thus, the set is bounded by from the above.
Second, we show that and . In fact,
Hence, if and only if , ie, . Hence, , .
When , from the definition of , we know that
(5.26) Hence,
Hence, and Further, taking the derivative with respect to on both sides of the equation , ie,
we obtain
ie,
(5.27) Hence, is strictly decreasing. Further, if there exists such that , then , which contradicts the condition . Thus, .
Finally, we show that , ie, for any , there exists such that .
Given , define function
Then
where the second inequality follows from the fact that is decreasing in and the last inequality follows from (5.26):
where the second inequality follows from the fact that is increasing in . Hence, there exists a constant such that since is continuous in . This yields that the domain of function is .
- (2)
This follows from the proof of (1). The proof of Lemma 4.1 is completed.
∎
6 Conclusions
In this paper, we study optimal reinsurance with expectile under the Vajda condition. Among a general class of premium principles, we prove that the optimal ceded loss functions take the form of three interconnected line segments, which can be considered as a combination of quota share reinsurance and stop-loss reinsurance. Simplified forms of the optimal reinsurance treaties can be obtained if the reinsurance premium is translation invariant or follows the expected value principle. Finally, the explicit expression for the optimal reinsurance treaty is also given when the reinsurance premium is assumed to be the expected value principle or Wang’s premium principle. Further research could discuss the application of a multivariate expectile to optimal reinsurance contracts.
Declaration of interest
The author reports no conflicts of interest. The author alone is responsible for the content and writing of this paper.
Acknowledgements
The author is very grateful to the editor-in-chief, the editors and the anonymous referees for their constructive comments and suggestions, which led to the present, greatly improved version of the manuscript. The author was supported by the National Natural Science Foundation of China (No. 11901184), the Natural Science Foundation of Hunan Province (No. 2020JJ5025) and the Fundamental Research Funds for the Central Universities (No. 531107051210).
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