Optimal reinsurance with premium constraint under distortion risk measures

Insurance: Mathematics and Economics 59 (2014) 109–120

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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime

Optimal reinsurance with premium constraint under distortion risk measures Yanting Zheng a,∗ , Wei Cui b a

Department of Finance, Beijing Technology and Business University, Beijing, 100048, China

b

Institute, Shenzhen Stock Exchange, Shenzhen, 518028, China

article

info

Article history: Received March 2014 Received in revised form August 2014 Accepted 28 August 2014 Available online 6 September 2014 Keywords: VaR TVaR Distortion risk measure Stop loss reinsurance Truncated stop-loss reinsurance Expected value premium principle

abstract Recently distortion risk measure has been an interesting tool for the insurer to reflect its attitude toward risk when forming the optimal reinsurance strategy. Under the distortion risk measure, this paper discusses the reinsurance design with unbinding premium constraint and the ceded loss function in a general feasible region which requiring the retained loss function to be increasing and left-continuous. Explicit solution of the optimal reinsurance strategy is obtained by introducing a premium-adjustment function. Our result has the form of layer reinsurance with the mixture of normal reinsurance strategies in each layer. Finally, to illustrate the applicability of our results, we derive the optimal reinsurance solutions with premium constraint under two special distortion risk measures—VaR and TVaR. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Reinsurance strategy is an important risk management tool for insurers. By resorting to reinsurance strategy, insurers not only control their total risk but also extend their business. Both academicians and practitioners give significant interests in reinsurance designs. With different purposes, the insurers may design their reinsurance strategies by maximizing the expected utility of their wealth (see Sung et al., 2011), or minimizing the risk value or the risk-adjusted value of their liabilities (Chi and Weng, 2013). From the view of risk management, the optimal reinsurance strategy minimizing the risk value of the insurer’s liability is one type of popular reinsurance models (see Kaluszka, 2005; Promislow and Young, 2005; Cai and Tan, 2007; Zhu et al., 2013 and the references therein). Suppose the initial loss X , caused by insurance claims, satisfies X ≥ 0 and 0 < E [X ] < ∞. By reinsurance strategy, the direct insurer cedes part of its loss f (X ) to a reinsurer and pays the reinsurer µf (X ) as reinsurance premium. Then the direct insurer’s total liability Tf (X ) contains two part: one is the retained loss risk If (X ) = X − f (X ) and the other is the reinsurance premium µf (X ),

∗

Corresponding author. Tel.: +86 18612481350. E-mail addresses: [email protected], [email protected] (Y. Zheng), [email protected] (W. Cui). http://dx.doi.org/10.1016/j.insmatheco.2014.08.010 0167-6687/© 2014 Elsevier B.V. All rights reserved.

that is, Tf (X ) = If (X ) + µf (X ). The common risk measures include classical standard deviation (variance) (Borch, 1960; Gajek and Zagrodny, 2000 and Kaluszka, 2001), convex risk measure (Kaluszka, 2005), VaR and TVaR risk measures (see Cai and Tan, 2007; Cheung, 2010; Chi and Meng, 2014 and Chi and Weng, 2013). With the wide application of Basel II agreement, VaR and TVaR become two popular risk measures in reinsurance models (see Tan et al., 2011 and Lu et al., 2013). From a general point, VaR and TVaR are two special cases of the distortion risk measures which also include the Wang’s transform risk measure. By Dhaene et al. (2006), a distortion function is defined as a non-decreasing function g (x) : [0, 1] → [0, 1] satisfying that g (0) = 0 and g (1) = 1. The distortion risk measure associated with g is defined as follows. For a non-negative random variable X ,

ρg [X ] =

∞



g (SX (t ))dt ,

(1.1)

0

where SX (t ) = P (X > t ) is the survival distribution of X . Actually, the distortion risk measures are perspective risk measures because they allow an insurer to reflect its attitude toward risk by choosing the appropriate distortion function. Based on the empirical studies of the optimal (re)insurances, Sung et al. (2011) considered the behavioral optimal insurance strategy by maximizing the S-shaped utility of the insurer’s gains and losses with the existence of probability distortion. For generality, we will use the distortion risk measure to minimize the direct insurer’s total liability Tf (X ) in the

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paper. Then the reinsurance problem becomes min{ρg [Tf (X )]},

s.t. 0 ≤ f (x) ≤ x, for all x ≥ 0,

(1.2)

where g is a distortion function. In addition, different optimal reinsurance designs have been proposed with different assumptions on a reinsurance premium principle and a set of admissible ceded loss functions. For mathematical convenience, this paper focuses on the optimal reinsurance model (1.2) with the expected value premium principle which is widely used (see Cai et al., 2008). On the other hand, the normal types of f (x) include stop-loss reinsurance with reinsurance’s loss payment (X − d)+ , d > 0, quota-share reinsurance with (1 − α)X , 0 < α < 1, truncated stop-loss reinsurance with (X − d1 )+ 1{X ≤d2 } , d2 > d1 > 0, limited stop-loss reinsurance with min{(X − d1 )+ , d2 − d1 }, d2 > d1 > 0 and the mixture of the above forms. Several feasible classes of the cede loss function f (x) in the literature can be concluded as:

F := {f (x) : 0 ≤ f (x) ≤ x,

f (x) is an increasing and convex function};

 C :=

f (x) : 0 ≤ f (x) ≤ x, both If (x) and

f (x) x

 are increasing in x ;

H := {f (x) : 0 ≤ f (x) ≤ x,

both f (x) and If (x) are increasing functions};

L := {f (x) : 0 ≤ f (x) ≤ x,

If (x) is an increasing and left-continuous function}.

If f (x) ∈ F , then the larger the original risk X , the larger the ceded risk is transferred, and the optimal reinsurance strategy in F is always stop-loss function (see Cai et al., 2008 and Cheung, 2010); If f (x) ∈ C , the reinsurer pays an increasing proportion of indemnity to reflect the spirit of reinsurance of protecting the insurer, which is called the Vajda condition (see Chi and Weng, 2013). The mixture of quota-share and stop-loss reinsurance strategies in C may be the optimal reinsurance which is not convex; If f (x) ∈ H , both the ceded risk and the retained risk will increase as X increases, which is helpful to avoid moral hazard. The stop-loss with an upper limit function, which does not belong to F , could be the optimal reinsurance strategy in H (see Chi and Tan, 2011 and Cui et al., 2013); If f (x) ∈ L, the optimal reinsurance strategy can be the truncated stop-loss function, which is not included in both F and H . In fact, Chi and Tan (2011) and Chi and Weng (2013) verified the relationship F $ C $ H $ L. In our paper, we consider the reinsurance problem in the general set L. Finally, an insurer is concerned with not only risk management but also its profits in practice. Thus, it is more desirable to design the reinsurance by considering both risk exposure and expected profit. Although it is always mathematically complicated to optimize the reinsurance model with constraints, many literatures show great interest in this kind of problems. Particularly, Gajek and Zagrodny (2000) considered the variance minimization reinsurance model with the maximum premium budget and Gajek and Zagrodny (2004) extended the reinsurance model with premium constraint under general risk measures. Zhou et al. (2010) presented an optimal insurance policy when the insurer has a loss limit constraint. Tan et al. (2011) considered the optimal reinsurance problem with premium constraint under CTE1 risk measure, and analyzed the premium budget’s effects on the optimal

solutions for both binding and unbinding constraints. Balbás et al. (2009) considered the optimal reinsurance problem with general risk measures when the reinsurer’s premium is given by a convex function and is also constrained by premium budget. Sung et al. (2011) also discussed the behavioral optimal insurance with the premium limited by the insurer’s total wealth. In this paper, we follow Zheng et al. (2014) and explore optimal reinsurance treaties with the expected value premium constraint by minimizing the distortion risk value of an insurer’s liability in set L. We conclude our models from the following aspects:

• The distortion risk measures assumed in our paper allow an insurer to reflect its risk attitude by choosing proper distortion function, and also include some widely used measures, such as VaR, TVaR and Wang’s transform risk measures. • We consider the reinsurance model in the general set of ceded loss functions which includes some important ones, such as limited stop-loss reinsurance and truncated stop-loss reinsurance. • Our proposed reinsurance model is a constraint optimization model which takes into consideration the level of risk exposure in the presence of reinsurance while guaranteeing certain minimum level of expected profit. • The optimal reinsurance strategy obtained in this paper has the form of truncated layer reinsurance. On different layers, the insurer may adopt distinct reinsurance strategies, such as limited stop-loss reinsurance and the combination of quotashare and limited stop-loss reinsurance. Following Zheng et al. (2014), we introduce a premium adjustment function to reflect the premium constraint. On each layer, if the expected premium exceeds the budget in a low level, the insurer chooses to increase its retained loss; if the expected premium exceeds the budget in a medium level, the insurer should proportionally increase its retained loss; if the expected premium exceeds the budget in a high level, the insurer should not increase its retained loss for risk management. The paper is organized as follows. Section 2 reviews the main results of the optimal reinsurance problem without premium constraint under the distortion risk measure. Section 3 investigates the optimal reinsurance problem with premium constraint under the distortion risk measure, and give two special examples. Section 4 gives the proofs of the main results in Section 3. Conclusions are given in Section 5. 2. Preparation works In this section, we first recall the main results of the reinsurance model without premium constraint in Zheng et al. (2014). For any non-negative random variable X , the insurer cedes the loss f (X ) to the reinsurer. We assume the premium is calculated by the expected value premium principle, i.e.

µ[f (X )] = (1 + β)E [f (X )],

(2.1)

where β > 0 is the safety loading factor. Then the total risk Tf (X ) of the insurer becomes Tf (X ) = If (X ) + (1 + β)E [f (X )],

where If (X ) is the retained loss. Under the distortion risk measure ρg in Eq. (1.1), the reinsurance model without premium constraint is min ρg If (X ) + (1 + β)E [f (X )] .



f ∈L

1 CTE means tail conditional expectation which is equal to TVaR (tail Value-atRisk) and CVaR (conditional Value-at-Risk).

(2.2)



(2.3)

For any ceded function f ∈ L, denoting the general inverse function of the retain loss function If as If−1 (x) := inf{t : If (t ) > x}.

Y. Zheng, W. Cui / Insurance: Mathematics and Economics 59 (2014) 109–120

From the monotonicity and left-continuity of If , it is easy to show that for any non-negative random variable X ≥ 0,

{If (X ) > t } = {X > If−1 (t )}.

(2.4)

In order to obtain the optimal reinsurance solution in Eq. (2.3), we introduce several main properties of the distortion risk measures ρg in the following (see the detailed discussions in Denneberg, 1994). 1. Positive homogeneity: ρg (aX ) = aρg (X ), for all a ≥ 0; 2. Translation invariance: ρg (X + b) = ρg (X ) + b, for all b ∈ R; 3. Comonotonic additivity: If two non-negative random variables X and Y are comonotonic risks, then ρg (X +Y ) = ρg (X )+ρg (Y ); Combining the properties of distortion risk measures above with Eqs. (2.2) and (2.4), Zheng et al. (2014) proved there exists a measure νf such that

ρg [Tf (X )] = (1 + β)E [X ] −

 R¯ +

BX dνf ,

(2.5)

where BX (t ) = (1 + β)SX (t ) − g (SX (t ))

(2.6)

and the measure νf is defined by νf (B) = u({x : If−1 (x) ∈ B}) for any Borel set B ⊆ BR¯ + (see the detailed proof in Zheng et al., 2014). Note u represents the Lebesgue measure. Remark 2.1. For any [a, b) ∈ BR¯ + , since If (s) > t ⇔ s > If−1 (t ), we have νf ([a, b)) = u({x : a ≤ If (x) < b}) = u([If (a), If (b))) = If (b) − If (a) ≤ b. Then the class of νf generated by all the If (x), where f (x) = x − If (x) ∈ L, consists of all the measures defined on [0, ∞) such that νf ([a, b)) ≤ b for any 0 ≤ a < b < ∞. −1

max f ∈L

R¯ +

BX dνf .

Then the function max{BX (t ), BX (t −)}, t ≥ 0 also has finitely many local maximum values. For convenience, we denote the set of local maximum values of max{BX (t ), BX (t −)}, t ≥ 0 as loc (max {BX (t ), BX (t −)}, t ∈ (0, ∞)). The main idea in Zheng et al. (2014) is to construct the optimal ceded loss function by finding the positive local maximum values in loc (max{BX (t ), BX (t −)}, t ∈ (0, ∞)) in order. In particular we define the first local maximum value as M1 = max{BX (0), M1,0 }, where M1,0 = max{y : y ∈ loc (max{BX (t ), BX (t −)}, t ∈ (0, ∞))}. Then, corresponding to M1 , define its rightmost local maximum point as

 r1 =

sup{t : max{BX (t ), BX (t −)} = M1,0 }, 0,

if M1,0 = M1 otherwise

(2.7)

From Eq. (2.7), the properties of function BX (t ), t ≥ 0 and integral measure νf play important roles in obtaining the optimal results. In order to obtain the maximal solution, under Assumption A in the following, Zheng et al. (2014) constructed such the measure νf that it allocates relative maximal measure to each positive local maximum values of BX (t ). Assumption A. Assume that g (x), x ∈ [0, 1] is left continuous, and its domain has a finite partition [0, 1] = ∪ni=1 [αi , αi+1 ], such that for i = 1, 2, . . . , n, g (x) is either concave or convex on (αi , αi+1 ). Remark 2.2. Sung et al. (2011) made use of two continuously differentiable distortion functions to reflect the incentive of the insurer, and both the distortion functions are assumed to be concave on [0, 1]. By contrast, the conditions in Assumption A are general. As we will see in Examples 2.1 and 2.2, the distortion functions, associated with VaR and TVaR, all satisfy Assumption A. From Assumption A, the right-continuity of SX (t ) and the left-continuity of g (x), x ≥ 0 guarantee that the function BX (t ), t ≥ 0 is rightcontinuous and the function BX (t −), t ≥ 0 is left-continuous. Furthermore, Lemma 2.1 shows the local maximum values of BX (t ), t ≥ 0 and BX (t −), t ≥ 0 are finite (see the detailed proof in Zheng et al., 2014). Lemma 2.1. Under Assumption A, the two functions BX (t ), t ≥ 0 and BX (t −), t ≥ 0 have finitely many local maximum values.

(2.8)

and its last leftmost local maximum point as inf{0 ≤ t ≤ r1 : max{BX (s), BX (s−)} = M1 , ∀s ∈ [t , r1 ]}, if M1,0 = M1 0, otherwise.

 m1 =

Remark 2.3. If it is the case that m1 = 0 and BX (0−) > BX (0), then by the definitions of M1 and m1 , we have M1 = BX (0) which is not the local maximum values of max{BX (t ), BX (t −)}, t ≥ 0 actually. For easy explanation without ambiguity, we still call M1 in this case as the local maximum value. Orderly, for 2 ≤ i ≤ s and on the interval (ri−1 , ∞), defining the ith local maximum value Mi , the ith rightmost local maximum point ri and the ith leftmost local maximum point mi ≤ ri as Mi = max y : y ∈ loc (max{BX (t ), BX (t −)}, t ∈ (ri−1 , ∞)) ,





ri = sup{t : max{BX (t ), BX (t −)} = Mi } and mi = inf{t ≤ ri : max{BX (t ), BX (t −)} = Mi for each s ∈ [t , ri ]}.

By Eq. (2.5), the optimal problem (2.3) equals



111

(2.9)

Continuing the process above until we find all the positive local maximum values {M1 , M2 , . . . , Ms }, where s ∈ Z is finite by Lemma 2.1. Let ms = 0, ms+1 = ∞. Then we have 0 < Ms < · · · < M1

and

0 = m0 ≤ m1 < · · · < ms < ms+1 = ∞. Now, we obtain the partition {[mi , mi+1 )}si=0 of [0, ∞) by the properties of max{BX (t ), BX (t −)}. To refine the partition above, we introduce another function as

 B¯ X (t ) :=

BX (t −), if t = m1 , . . . , ms and BX (t −) > BX (t ) BX (t ), otherwise.

(2.10)

From the definition of B¯ X (t ), it is easy to find that (i) BX (t ) ≤ B¯ X (t ) ≤ max{BX (t ), BX (t −)}, (ii) B¯ X (t ) = BX (t ),

t ≥ 0;

for all t ̸= mi , i ≤ s;

(iii) B¯ X (mi ) = max{BX (mi ), BX (mi −)} = Mi ,

i = 1, . . . , s.

Combining with definitions of mi , i = 0, 1, . . . , s + 1, a sequence m∗i , i ≤ s can be defined. Let m∗0 = 0, and for 1 ≤ i ≤ s, m∗i = inf t : t ≥ mi and BX (t ) ≤ B¯ X (mi+1 ) .





(2.11)

Here let inf ∅ = ∞. By Eq. (2.11), we have m0 = m∗0 = 0 ≤ m1 ≤ m∗1 < m2 ≤ · · · < ms−1

≤ m∗s−1 < ms ≤ m∗s ≤ ms+1 = ∞.

(2.12)

Then each subinterval (mi , mi+1 ] is divided into two subintervals (mi , m∗i ) and [m∗i , mi+1 ]. For understanding the above partition, we give the VaR and TVaR’s partition points {mi , m∗i }si=1 in the next two examples.

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Y. Zheng, W. Cui / Insurance: Mathematics and Economics 59 (2014) 109–120

Example 2.1 (Value at Risk). For a positive random variable X , we suppose its distribution is continuous and strictly increasing on (0, ∞) with a possible jump at 0. Given the confidence level 1 − α , the VaRα (X ) is defined as VaRα (X ) = inf{t ≤ 0 : SX (t ) ≤ α} = SX−1 (α), where SX−1 is the general inverse function of SX . Dhaene et al. (2006) proved the distortion function associated with VaRα (X ) is g (x) = 1{x>α} which satisfies Assumption A. Taking g (x) = 1{x>α} into Eq. (2.6), we have BX (t ) =

 (1 + β)SX (t ) − 1, (1 + β)SX (t ),

It is easy to find that BX (t ) is decreasing on both [0, VaRα (X )) and [VaRα (X ), ∞). From the discussions in previous section, we obtain the local maximum values Mi and partition points {mi , m∗i }, i = 1, 2, . . . , s of BX (t ) as follows. 1 (1) If BX (VaRα (X )) < BX (0), i.e. α < SX (0) − 1+β , we have s = 2, the first order local maximum M1 = BX (0) and the second order local maximum M2 = BX (VaRα (X )). Accordingly, their corresponding local maximum points are m1 = 0, m2 = VaRα (X ) and m3 = ∞. By Eq. (2.11), we also have





and m∗2 = ∞,

1 . where q = α + 1+β

Example 2.2 (Tail Value at Risk). For given positive random variable X and the confidence level 1 −α , the tail value-at-risk measure TVaRα (X ) is TVaRα (X ) =

α



1

VaRp (X )dp. 1−α

Corresponding to TVaR, the distortion function is g (x) = min{ αx , 1} which also satisfies Assumption A, and BX (t ) =

f # (x) =

s  (x − m∗i )+ 1(m∗i ,mi+1 ] (x) + (x − m∗s )+

 (1 + β)SX (t ) − 1, (1 + β)SX (t ) −

1

α

if t < VaRα (X ) SX (t ),

otherwise.

It is easy to find that BX (t ) is decreasing on [0, VaRα (X )), BX (VaRα (X )) ≤ BX (0) and BX (∞) = 0, then we have M1 = BX (0) and m1 = 0. In order to find other local maximums, we focus on the interval [VaRα (X ), ∞]. In the interval [VaRα (X ), ∞], we have BX (t ) = [(1 + β) − α1 ]SX (t ). So (1) If (1 + β) − α = 0, i.e. α = 1+β , then BX (t ) ≡ 0, t ∈ [VaRα (X ), ∞]. So M2 = 0, m2 = VaRα (X ) and   m∗1 = inf t ≥ m1 : (1 + β)SX (t ) − 1 ≤ M2 = VaRα (X ) = VaR 1 (X ). 1

(2.13)

i =0

If # (x) =

s 

min{x, m∗i }1(mi ,mi+1 ] (x).

1

1+β

1 , then BX (t ) is decreasing on (2) If (1 + β) − α1 > 0, i.e. α > 1+β [0, ∞]. Thus, we obtain s = 1, M1 = BX (0), m1 = 0, m2 = ∞ and m∗1 = ∞. 1 (3) If (1 + β) − α1 < 0, i.e. α < 1+β , then BX (t ) is increasing on [VaRα (X ), ∞]. So M2 = BX (∞) = 0, m2 = ∞ and m∗1 = VaR 1 (X ). 1+β

The following corollary shows the properties of function BX (t ) under the partitions in Eq. (2.12) (see its proof in Zheng et al., 2014).

(2.14)

i=1

By the relationship between νf # and the associated retained loss function If # , i.e. νf # [a, b) = If # (b) − If # (a), we have

νf # ([m∗i−1 , mi )) = 0,

νf # ({mi }) = mi − m∗i−1 ,  ∀B ⊆ (mi , m∗i ),

i = 1, . . . , s,

s

νf # (B) = u(B),

i=1

νf # ([m∗s , ∞)) = 0.

(2.15)

That means for any Borel set A ⊆ [0, ∞),



1 , then s = 1. (2) If BX (VaRα (X )) ≥ BX (0), i.e. α ≥ SX (0) − 1+β Thus M1 = BX (VaRα (X )) = (1 + β)α , m1 = VaRα (X ) and m∗1 = m2 = ∞.

1

Based on the partitions {mi , m∗i }si=0 of the interval (0, +∞) in Eq. (2.12), Zheng et al. (2014) constructed the ceded loss function as

and the associated retained loss function is

if t < VaRα (X ) otherwise.

m∗1 = inf t > m1 : BX (t ) ≤ BX (m2 ) = VaRq (X )

Corollary 2.1 (Zheng et al., 2014). For each 0 ≤ i ≤ s, BX (t ) > B¯ X (mi+1 ) for t ∈ (mi , m∗i ) and BX (t ) ≤ B¯ X (mi+1 ) for t ∈ [m∗i , mi+1 ], and the function BX (t ) is decreasing on (mi , m∗i ).



s  νf # (A) = u A ∩ (mi , m∗i ]



i=0 s   + (mi − m∗i−1 )1{mi ∈A} .

(2.16)

i =1

For i = 1, 2, . . . , s, by Corollary 2.1, we have BX (t ) < B¯ X (mi ), t ∈ [m∗i−1 , mi ) and B¯ X (mi ) is the local maximum value. From Eq. (2.15), νf # allocates no measure on the interval [m∗i−1 , mi ) and allocates the biggest measure mi − m∗i−1 on the local maximum point mi ; νf # becomes Lebesgue measure for other situations. Based on the structure of νf # , Zheng et al. (2014) proved that f # is the optimal solution of problem (2.3) under some mild conditions. We introduce their theorem without proof. Theorem 2.1 (Zheng et al., 2014). Suppose that g (x) satisfies Assumption A, then there exists an infimum under the distortion risk measure: inf ρg [Tf (X )] = (1 + β)E [X ] −

f ∈L

 R¯ +

B¯ X dνf # .

If BX (mi ) = B¯ X (mi ) for all i = 1, . . . , s, then f # in Eq. (2.13) is one optimal reinsurance solution of problem (2.3), and the distortion risk measure of the insurer is

ρg [Tf # (X )] =

s   i =1

+

mi m∗ i−1

s   i =1

[(1 + β)(SX (t ) − SX (mi )) + g (SX (mi ))]dt

m∗ i

g (SX (t ))dt ;

(2.17)

mi

Otherwise, there is no optimal solution for the reinsurance problem (2.3). 3. The optimal reinsurance strategy with premium constraint Following the ideas of the optimal reinsurance problem without premium constraint in Zheng et al. (2014), in this section, we

Y. Zheng, W. Cui / Insurance: Mathematics and Economics 59 (2014) 109–120

continue to consider the optimization problem with unbinding premium constraint min ρg [Tf (X )]



f ∈L

(3.1)

s.t. (1 + β)E [f (X )] ≤ π ,

where the constant π is the largest premium that the insurer would like to pay for the reinsurance. It is always a mathematical challenge to obtain the optimal solution of problem (3.1). Our strategy of solving (3.1) involves the following. We transform problem (3.1) into the optimal problem without constraint on the subset of L. Then we obtain the optimal solution by making proper adjustment on f # ∈ L in Section 2. More specifically, let D = π/(1 + β). We rewrite the premium restriction in Eq. (3.1) as E [f (X )] ≤ D which equals E [If (X )] =

 R¯ +

Defining LD = f : f ∈ L and R¯ SX dνf ≥ E [X ] − D , we trans+ form the optimization reinsurance problem (3.1) into the model without constraint, that is,



min ρg [Tf (X )].

f ∈LD



(3.2)

Note that LD ⊂ L. We will solve the optimal reinsurance problem (3.2) by reconsidering the optimal reinsurance results without premium constraint in Section 2. As we know,

ρg [Tf # (X )] = (1 + β)E [f # (X )] + ρg [If # (X )]

  B¯ X (mi+1 ) − BX (t ) , γi (t ) = SX (t ) − SX (mi+1 )  ∞, otherwise.

Λ(y) =

s   i =0

[m∗i ,mi+1 ]∩(γi (t )
if SX (t ) > SX (mi+1 )

(3.4)

(3.3)

where I1 represents the premium cap, I2 represents the possible part beyond the premium constraint and I3 is the distortion risk of the optimal retained loss function without premium constraint. If the premium of the ceded loss f # (X ) is within the constraint, i.e. E [f # (X )] ≤ D, then the insurer makes no adjustment; Otherwise, if E [f # (X )] > D, the insurer needs to decrease the ceded loss and increase the retained loss in order to satisfy the premium constraint, while the insurer also needs to control the total loss during the arrangement. For technical convenience, we assume that the distortion function g (x) also satisfies the following assumption except Assumption A.

(SX (t ) − SX (mi+1 ))dt ,

where (γi (t ) < y) controls the ratio of the change of the total loss to that of the retained loss. Then, Λ(y) represents the changes of the retained loss corresponding to the adjusting ratio y. It is easy to prove that Λ(y) is an increasing left-continuous function with lower bound Λ(0) = 0 and upper bound

Λ(∞) =

s   i =0

= (1 + β)D + (1 + β){E [f # (X )] − D} + ρg [If # (X )] =: I1 + I2 + I3 ,

In order to satisfy the reinsurance budget, the insurer needs to properly increase the retained loss function If # for the case E [f # (X )] > D. Note that the retained loss satisfies 0 ≤ If # (x) ≤ x. Combining with Eq. (2.14), we find If # (x) has reached its upper bound x in subintervals (mi , m∗i ] and can only be increased in subintervals [m∗i , mi+1 ). From the view of the integration mea sure νf # , both E [If # (X )] = R¯ + SX dνf # and ρg [Tf # (X )] = (1 + β)  E [X ] − R¯ + BX dνf # will be changed if the insurer regulates If # in [m∗i , mi+1 ). In order to control the adjustment range on subinterval [m∗i , mi+1 ), i = 0, 1, . . . , s, we introduce the adjusting function γi (t ), t ∈ [m∗i , mi+1 ) as follows. For each i = 0, . . . , s,

Based on the adjustment function γi (t ), t ∈ [m∗i , mi+1 ), we define

SX dνf ≥ E [X ] − D.



113

[m∗i ,mi+1 ]

(SX (t ) − SX (mi+1 ))dt

= E [f # (X )] > E [f # (X )] − D. For the case E [f # (X )] > D, let

  ρ = sup y > 0 : Λ(y) ≤ E [f # (X )] − D . It is easy to see 0 < ρ < +∞, which means it is possible for the insurer to increase the retained loss as well as controlling the total loss. Proposition 3.2 discusses the properties of function Λ(y), y > 0 and the proof of Proposition 3.2 will be given in Section 4. Proposition 3.2. Suppose that for the non-negative random variable X , the function g (x), x ∈ [0, 1] satisfies Assumption B. Then for each i = 0, . . . , s, the function γi (t ) is an increasing function on [m∗i , mi+1 ). Moreover, for the case E [f # (X )] > D, if there exists y > 0 such that Λ(y) = E [f # (X )] − D, then

Assumption B. For each i = 0, . . . , s, g (x) is concave on the interval (SX (mi+1 ), SX (m∗i )].

Λ(ρ) = E [f # (X )] − D.

Comparing with Assumption A, Assumption B depends not only on the distortion function g (x) but also on the distribution of random variable X . Moreover, Assumption A focuses on the domain of the function g (x), x ∈ [0, 1], while Assumption B only concerns the property of g (x) on some subsets of [0, 1]. The following proposition shows that some common distortion risk measures satisfy Assumption B.

Λ(ρ) < E [f # (X )] − D ≤ Λ(ρ+).

Proposition 3.1. For any non-negative random variable X , its associated VaR and concave distortion risk measures satisfy Assumption B. Proof. The concave distortion risk measure means that its associated distortion function g (x) is concave on [0, 1]. Thus, it is easy to see that the concave distortion risk measures satisfy Assumption B. For VaR with confidence level 1 − α , the distortion function g (x) = 1{x>α} is concave on either [0, α) or [α, 1]. According to the discussions in Example 2.1, we can directly check that each (SX (mi+1 ), SX (m∗i )] belongs to either [0, α) or [α, 1]. 

(3.5)

Otherwise, we have (3.6)

For i = 0, 1, . . . , s, the increasing property of γi (t ) on the interval [m∗i , mi+1 ) will guarantee that there exists unique mDi ≥ m∗i such that

[m∗i , mDi ) ⊆ [m∗i , mi+1 ) ∩ (γi (t ) < ρ) ⊆ [m∗i , mDi ]. mDi

Similarly, there exists some ¯

≥

mDi

(3.7)

such that

¯ Di ) ⊆ [m∗i , mi+1 ] ∩ (γi (t ) ≤ ρ) ⊆ [m∗i , m ¯ Di ]. [m∗i , m

(3.8)

Thus, we obtain

Λ(ρ) =

s   i =0

Λ(ρ+) =

[m∗i ,mD ] i

(SX (t ) − SX (mi+1 ))dt ,

s   i =0

¯D [m∗i ,m ] i

(3.9)

(SX (t ) − SX (mi+1 ))dt .

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Y. Zheng, W. Cui / Insurance: Mathematics and Economics 59 (2014) 109–120

Fig. 1. Comparisons of f ## (t ) and If ## (t ) with f # (t ) and If # (t ) on the interval (mi , mi+1 ].

¯ Di are unique, and Proposition 3.3. For each i = 0, . . . , s, mDi and m ¯ Di ). γi (t ) = ρ for all t ∈ (mDi , m Proof. For each i, if both points we can obtain

mDi,1

<

mDi,2

satisfy Eq. (3.7), then

[m∗i , mDi,2 ) ⊆ [m∗i , mi+1 ) ∩ (γi (t ) < ρ) ⊆ [m∗i , mDi,1 ], which leads to that [m∗i , mDi,2 ) ⊆ [m∗i , mDi,1 ], contradictory to mDi,1 <

mDi,2 . Thus the uniqueness of mDi is proved. Similarly we can prove

¯ Di . Finally, from the definitions of mDi and m ¯ Di , the uniqueness of m D D ¯ i ).  we can assure that γi (t ) = ρ for each t ∈ (mi , m 3.1. Main results of the optimal reinsurance strategy with premium constraint Based the adjustment function γi (t ), t ∈ [m∗i , mi+1 ) and the associated function Λ(y), in this section, we will construct the adjusted ceded loss function f ## for the case E [f # (X )] > D, and show it is the optimal solution of problem (3.2) under mild conditions. Λ(ρ+)−(E [f # (X )]−D) . Then from Proposition 3.2, we have Let λ = Λ(ρ+)−Λ(ρ)

¯ Di , we construct a 0 ≤ λ ≤ 1. Based on definitions of mDi and m ceded loss function as follows. f ## (x) =

s   ¯ Di )+ 1{m¯ D ≤x≤mi+1 } (x − m i

i=0



¯ Di − mDi }1{x≤mi+1 } . + λ min{(x − mDi )+ , m

(3.10)

To understand the adjustment process, Fig. 1 shows the comparisons of f ## (t ) and If ## (t ) with f # (t ) and If # (t ) on the interval (mi , mi+1 ] respectively. By the relationship νf ## ([a, b)) = If ## (b)− If ## (a), for any Borel set A ⊆ [0, ∞), we have s s       ¯ Di ] νf ## (A) = u A ∩ (mi , mDi ] + (1 − λ)u A ∩ (mDi , m i =0

+

s−1 

i=0

¯ Di − mDi ) . 1{mi+1 ∈A} mi+1 − mDi − (1 − λ)(m 



Lemma 3.1. For the measure νf ## defined in Eq. (3.11), we have

R¯ +

That is, f ## ∈ LD and its premium reaches the premium cap π . Now, we give the main result of the optimal problem (3.1) in Theorem 3.1, and its proof will be shown in Section 4. Theorem 3.1. Suppose that Assumption B holds.

(1) In the case E [f # (X )] ≤ D. The optimization problem (3.1) has an infimum inf ρg [Tf (X )] = (1 + β)E (X ) −

(3.12)



f ∈LD

R¯ +

B¯ X dνf # ,

and if B¯ X (mi ) = BX (mi ) for each i = 1, . . . , s, then f # is one solution of the optimization problem (3.1); Otherwise, the optimization problem (3.1) has no solutions;

(2) In the case E [f # (X )] > D. The optimization problem (3.1) exists an infimum inf ρg [Tf (X )] = (1 + β)E (X ) −



f ∈LD

R¯ +

B¯ X dνf ## .

Furthermore, if B¯ X (mi ) = BX (mi ) for each i = 1, . . . , s, then f ## is the solution for problem (3.1); In the case E [f # (X )] > D, Theorem 3.1 shows if B¯ X (mi ) = BX (mi ) for each i = 1, . . . , s, then f ## is the solution for problem (3.1). The following proposition gives another situation for f ## still being the solution for problem (3.1), even though B¯ X (mi ) ̸= BX (mi ) for some i = 1, . . . , s. Proposition 3.4. For any i ∈ {0, 1, . . . , s} such that B¯ X (mi ) > BX (mi ), if γi (t ) < ρ, ∀t ∈ [m∗i , mi+1 ], then f ## is one solution of the optimal reinsurance problem (3.1). Proof. It is obvious to note that if γi (t ) < ρ, ∀t ∈ [m∗i , mi+1 ], we ¯ Di = mi+1 . According to the definition of νf ## , we have have mDi = m νf ## ({mi }) = 0. Thus,

ρg [Tf ## (X )] − inf ρg [Tf (X )] =

From Eq. (3.10), the following lemma shows the properties of νf ## and we give its proof in Section 4.

SX dνf ## = E [X ] − D.

+

µ(f ## (X )) = (1 + β)E [f ## (X )]   = (1 + β) E [X ] − E [If ## (X )] = π .

(3.11)

i=0



We leave the proof of Lemma 3.1 in Section 4. From Lemma 3.1  and E [If ## (X )] = R¯ SX dνf ## , we have

f ∈LD

=

 R¯ +

B¯ X dνf ## −

s  (B¯ X (mi ) − BX (mi ))νf ## ({mi }) = 0.

 R¯ +

BX dνf ## (3.13)

i=1

That is, the infimum of the optimization problem (3.1) can be attained by f ## . So results of the optimal reinsurance problem (3.1) exist. 

Y. Zheng, W. Cui / Insurance: Mathematics and Economics 59 (2014) 109–120

Remark 3.1. In Theorem 3.1, we prove that there is an infimum for the optimal reinsurance problem with premium constraint, and show one special situation that the infimum can be reached in Proposition 3.4. It is still an open question to show when the infimum of the optimal reinsurance problem with premium constraint cannot be reached. 3.2. Two special examples Continuing to consider Examples 2.1 and 2.2 in Section 2, we discuss the optimal reinsurance problem with unbinding premium constraint under two special distortion risk measures, VaR and TVaR, respectively. Example 3.1 (VaR). Based on the discussions in Example 2.1, we consider the optimal reinsurance problem under VaR risk measure. We first obtain the optimal reinsurance solution without premium constraint, and then obtain the optimal reinsurance strategy with premium constraint by the adjustment function γ (t ). From the reviews in Section 2, 1 , the optimal ceded loss function without (1) If α < SX (0) − 1+β premium constraint is



B¯ X (VaRα (X )) − B¯ X (t )

SX (t ) − SX (VaRα (X )) 1 = − (1 + β). SX (t ) − α

=

(1 + β)α − (1 + β)SX (t ) + 1 SX (t ) − α

f # (x) = (x − VaR

1 1+β

(X ))+ 1{x=VaR

1 (X )} 1+β

= 0,

which means there is no reinsurance in this case. Thus, the optimal ceded loss function with premium constraint is also f ## (x) = f # (x) = 0. (2) If (1 + β) − α1 > 0, the optimal ceded loss function without premium constraint is f # (x) = 0, which also means there is no reinsurance in this case. Then f ## (x) = f # (x) = 0. (3) If (1 + β) − α1 < 0, the optimal ceded loss function without premium constraint is f # (x) = (x − VaR

1 1+β

(X ))+ .

If E [f # (X )] ≤ D, the optimal ceded loss function f # (x) satisfies premium constraint and the optimal ceded loss function with premium constraint f ## (x) = f # (x); If E [f # (X )] > D, then the insurer needs to make adjustments on [m∗1 , m2 ) = [VaR 1 (X ), ∞).

γ1 (t ) =

1 1+β

(X ), ∞), the adjustment function is

B¯ X (∞) − B¯ X (t )

SX (t ) − SX (∞)  1  − (1 + β), if VaR 1 (X ) ≤ t < VaRα (X )  1+β (3.16) = SX (t )   1 − (1 + β), otherwise. α γ1 (t ) is also an increasing function on (VaR 1 (X ), ∞). Then from 1+β

Theorem 3.1, the optimal reinsurance strategy with premium constraint is f ## (x) = (x − d)+ ,

Obviously, γ1 (t ) is an increasing and continuous function on [VaRq (X ), VaRα (X )). That means mD1 = mD1 . From Theorem 3.1, the optimal ceded loss function with premium constraint can be expressed as f ## (x) = (x − b)+ I{x≤VaRα (X )} ,

1 (1) If (1 + β) − α1 = 0, i.e. α = 1+β , then the optimal ceded loss function without premium constraint is

For t ∈ [VaR

For the case E [f # (X )] ≤ D, the optimal ceded loss function f # (x) satisfies premium constraint and the optimal ceded loss function with premium constraint f ## (x) = f # (x); For the case E [f # (X )] > D, the insurer needs to make adjustments on each subinterval [m∗i , mi+1 ), i = 0, 1, 2. Based on the discussions in Section 3, it is necessary to only focus on subinterval [m∗1 , m2 ) = [VaRq (X ), VaRα (X )) in this example. That is, for t ∈ [VaRq (X ), VaRα (X )), the adjustment function is

γ1 (t ) =

Example 3.2 (TVaR). Based on the results in Example 2.2, we obtain that

1+β

f # (x) = x − VaRq (X ) + 1{x≤VaRα (X )} .



115

(3.14)

where the constant b ∈ [VaRq (X ), VaRα (X )] is determined by E [f ## (X )] = D.

where d ∈ [VaR

1 1+β

(X ), ∞) is determined by E [f ## (X )] = D.

Remark 3.2. The parameter α in VaR and TVaR is always less than 5%, thus (1 +β)− α1 < 0 is often satisfied. By the above discussions, with the expected value premium principle constraint, the stoploss reinsurance is also optimal under the TVaR-based reinsurance model. The result is consistent with Proposition 4.2 in Chi and Weng (2013) and Proposition 4.1 in Chi and Meng (2014).

1 , the optimal ceded loss function without (2) If α ≥ SX (0) − 1+β premium constraint is

4. Proofs

f # (x) = xI{x≤VaRα (X )} .

Proof of Proposition 3.2

For the case E [f # (X )] ≤ D, the optimal ceded loss function f # (x) satisfies premium constraint and the optimal ceded loss function with premium constraint f ## (x) = f # (x); For the case E [f # (X )] > D, the insurer makes adjustments on subinterval [0, VaRα (X )), that is, for 0 ≤ t < VaRα (X ), the adjustment function

Proof. First, we will prove that under Assumption B, the adjustment function γi (t ) is an increasing function on [m∗i , mi+1 ). From the property of increasing function, it is necessary to prove γi (t1 ) ≤ γi (t2 ), ∀m∗i ≤ t1 < t2 < mi+1 . Based on the fact that SX (t ) is decreasing on [0, ∞), for any m∗i ≤ t1 < t2 < mi+1 , we have SX (m∗i ) ≥ SX (t1 ) ≥ SX (t2 ) ≥ SX (mi+1 ). If either SX (t2 ) = SX (mi+1 ) or SX (t1 ) = SX (t2 ) holds, it is easy to see γi (t1 ) ≤ γi (t2 ). Next, we consider the case SX (m∗i ) ≥ SX (t1 ) > SX (t2 ) > SX (mi+1 ). From definitions of BX (t ) and γi (t ) in Eqs. (2.6) and (3.4), we obtain

γ0 (t ) =

1 SX (t ) − α

− (1 + β),

and γ0 (t ) is also an increasing and continuous function on [0, VaRα (X )). Thus mD0 = mD0 . From Theorem 3.1, the optimal ceded loss function with premium constraint is f ## (x) = (x − b)+ I{x≤VaRα (X )} , where b ∈ [0, VaRα (X )) is determined by E [f ## (X )] = D.

γi (t ) = (3.15)

=

B¯ X (mi+1 ) − (1 + β)SX (t ) + g (SX (t )) SX (t ) − SX (mi+1 )

BX (mi+1 ) − BX (mi+1 ) + BX (mi+1 ) − (1 + β)SX (t ) + g (SX (t )) SX (t ) − SX (mi+1 )

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Y. Zheng, W. Cui / Insurance: Mathematics and Economics 59 (2014) 109–120

g (SX (t )) − g (SX (mi+1 ))

=

SX (t ) − SX (mi+1 )

+

B¯ X (mi+1 ) − BX (mi+1 ) SX (t ) − SX (mi+1 )

Definition 4.1. For any function f ∈ LD , we define the measure νf˜ satisfying:

− (1 + β)

(1) νf˜ ([d1 , d2 ]) = min{νf ([d1 , d2 ]), d2 − d1 }, for any m∗i−1 ≤ d1 < d2 < mi , where 1 ≤ i ≤ s + 1;

,

(2) νf˜ (B) = u(B), for any Borel set B ∈ (mi , m∗i ), 1 ≤ i ≤ s;

thus we have

(3) νf˜ ({mi }) = mi − νf˜ ([0, mi )), 1 ≤ i ≤ s.

γi (t1 ) − γi (t2 ) g (SX (t1 )) − g (SX (mi+1 )) g (SX (t2 )) − g (SX (mi+1 )) = − SX (t1 ) − SX (mi+1 ) SX (t2 ) − SX (mi+1 ) B¯ X (mi+1 ) − BX (mi+1 ) B¯ X (mi+1 ) − BX (mi+1 ) − + SX (t1 ) − SX (mi+1 ) SX (t2 ) − SX (mi+1 ) g (SX (t1 )) − g (SX (mi+1 ))

≤

SX (t1 ) − SX (mi+1 )

−

g (SX (t2 )) − g (SX (mi+1 )) SX (t2 ) − SX (mi+1 )

Based on the relationship between νf˜ and f˜ , i.e. f˜ (x) = x −

νf˜ ([0, x)), it is easy to see that f˜ ∈ L. The main reason to introduce function f˜ is to consider the optimal reinsurance problem with premium constraint in the subset L˜ D of LD , where L˜ D =: {f˜ ∈ L : f˜ (x) = x − νf˜ ([0, x))}. The following lemma explains this reason .

(4.1)

By the concavity of g (x) on interval (SX (mi+1 ), SX (m∗i )], we have g (SX (t2 )) ≥

SX (t2 ) − SX (mi+1 ) SX (t1 ) − SX (mi+1 )

+

SX (t1 ) − SX (t2 )

Lemma 4.1. For any function f ∈ LD , its associated function f˜ from Definition 4.1 belongs to LD . On the other hand, under the distortion risk measure ρg and the expected value premium principle with safety loading factor β , the optimal reinsurance strategy associated with f˜ is better than the one associated with f , that is,

g (SX (t1 ))

SX (t1 ) − SX (mi+1 )

exactly.

g (SX (mi+1 )),

 R¯ +

B¯ X dνf˜ ≥

 R¯ +

B¯ X dνf .

(4.4)

which leads to g (SX (t1 )) − g (SX (mi+1 )) SX (t1 ) − SX (mi+1 )

−

g (SX (t2 )) − g (SX (mi+1 )) SX (t2 ) − SX (mi+1 )

Proof. Given f ∈ LD .

≤ 0.

(4.2)

Combining Eqs. (4.1) and (4.2), we have γi (t1 ) ≤ γi (t2 ). Then, we conclude that γi (t ) is an increasing function on [m∗i , mi+1 ). Second, we will prove (3.5) and (3.6). By the definition of ρ and the left-continuity of Λ(y), we have Λ(ρ) ≤ E [f # (X )] − D and Λ(ρ + ϵ) > E [f # (X )] − D for ϵ > 0. Thus if there exists y > 0 such that Λ(y) = E [f # (X )] − D, we can conclude that (3.5) holds; Otherwise, we can get (3.6).  Before we give the main proof of Theorem 3.1, we introduce the following corollary in Zheng et al. (2014) without giving proof.

(1) This part will prove Eq. (4.4). Based on the partitions of [0, ∞) in Section 2, we have

 R¯ +

B¯ X dνf =

h(mh,1 ) ≥ h(mh,2 ) ≥ · · · ≥ h(mh,sh ) ≥ h(mh,sh+1 ) ≥ 0. Further, for each 0 ≤ i ≤ sh , there exists m∗h,i satisfying m∗h,0 = 0 and mh,i ≤ m∗h,i ≤ mh,i+1 , 1 ≤ i ≤ sh , such that

t ∈

f ∈L

=

[0,d] sh 

 h(mh,i )(mh,i −

m∗h,i−1 )

 +

i=1

+ h(mh,sh+1 )(mh,sh+1 − m∗h,sh ).

s −1  

[m∗i ,mi+1 ]

(mi ,m∗i )

B¯ X (t )dνf +

B¯ X (t )dνf

 [m∗s ,∞)

B¯ X (t )dνf (4.5)

and

 R¯ +

B¯ X dνf˜ =

 [0,m1 ]

+

B¯ X (t )dνf˜ +

i=1

s −1   i =1

s  

[m∗i ,mi+1 ]

(mi ,m∗i )

B¯ X (t )dνf˜ +

B¯ X (t )dt

 [m∗s ,∞)

B¯ X (t )dνf˜ (4.6)

(1a) Comparing the first part I1 in Eq. (4.5) with the first part I˜1 in Eq. (4.6). Based on the definitions of νf˜ and νf , we have νf˜ ([0, m1 )) = min{νf ([0, m1 )), m1 } = νf ([0, m1 )) and νf ([0, m1 )) ≤ m1 , that is,



B¯ X (t )dνf =

 [0,m1 )

B¯ X (t )dνf˜ .

Combining with the condition (3) in Definition 4.1, we obtain

I1 − I˜1 =

m∗ h, i

 [0,m1 ]

B¯ X (t )dνf −

 [0,m1 ]

B¯ X (t )dνf˜

  = B¯ X (m1 ) νf ({m1 }) − νf˜ ({m1 })   = −B¯ X (m1 ) m1 − (νf ({m1 }) + νf ([0, m1 )))

h(t )dνf (t )

max

i=1

=: I1 + I2 + I3 + I4

[0,m1 )

(mh,i , m∗h,i ), [m∗h,i , mh,i+1 ]

and h(t ) is decreasing on interval [mh,i , m∗h,i ). Then we have



s  

=: I˜1 + I˜2 + I˜3 + I˜4 .

and

h(t ) ≤ h(mh,i+1 ),

B¯ X (t )dνf +

i =1

0 = mh,0 ≤ mh,1 < mh,2 < · · · ≤ mh,sh+1 = d

t ∈

[0,m1 ]

+

Corollary 4.1 (Zheng et al., 2014). For given function h(t ) defined on interval s[0, d], 0 < d < ∞, suppose there exists a partition [0, d) = i=h 0 [mh,i , mh,i+1 ) satisfying

h(t ) ≥ h(mh,i+1 ),



 h(t )dt

= −B¯ X (m1 )(m1 − νf ([0, m1 ])).

mh,i

(4.3)

For simplicity, let 0 × ∞ = 0, ∞ − ∞ = ∞ and [∞, ∞) = ∅.

(4.7)

(1b) Comparing the second part I2 in Eq. (4.5) with the second part I˜2 in Eq. (4.6). Focusing on the interval (mk , m∗k ), k ∈ {1, 2, . . . , s}, if m∗k < ∞, then we introduce the following function JX (t ) = B¯ X (t )1{mk ≤t
k

t ∈ [0, m∗k ].

Y. Zheng, W. Cui / Insurance: Mathematics and Economics 59 (2014) 109–120

Meanwhile, we construct the measure νζ satisfying νζ ([0, mk ]) = νζ ({mk }) = νf ([0, mk ]), νζ (B) = νf (B), ∀B ∈ B(mk ,m∗k ) and νζ ({m∗k }) = m∗k − νf ([0, m∗k )). From the fact that ζ (x) = x − νζ ([0, x)), we have ζ ∈ L.

On the other hand, from the definition of νf˜ , we have

νf˜ ([m∗k , mk+1 ]) = νf˜ ([m∗k , mk+1 )) + νf˜ ({mk+1 }) = νf˜ ([m∗k , mk+1 )) + mk+1 − νf˜ ([0, mk+1 ))

On one hand, from Corollary 4.1, we have

 max l∈L

[0,m∗k ]

JX (t )dνl (t ) = B¯ X (mk )mk +

= B¯ X (mk )mk +

= mk+1 − νf˜ ([0, m∗k ))

 (mk ,m∗k ]

 (mk ,m∗k )

= mk+1 − νf˜ ([0, mk )) − νf˜ ({mk }) − νf˜ ((mk , m∗k ))

B¯ X (t )dt B¯ X (t )dt .

= mk+1 − m∗k . (4.8)

[0,m∗k ]

That is,

νf ([m∗k , mk+1 ]) − νf˜ ([m∗k , mk+1 ])     = m∗k − νf ([0, m∗k )) − mk+1 − νf ([0, mk+1 ]) .

On the other hand, by definitions of JX (t ) and νζ , we have





+ (mk ,m∗k )

B¯ X (t )dνf + B¯ X (m∗k −)(m∗k − νf ([0, m∗k ))).

[m∗k ,mk+1 ]

(4.9)

Thus, based on the fact that [0,m∗ ] JX dνζ ≤ maxl∈L [0,m∗ ] JX (t )dνl (t ), k k and combining Eq. (4.8) with Eq. (4.9), we obtain

(mk ,m∗k )

B¯ X (t )dνf −

 (mk ,m∗k )

B¯ X (t )dt

≤ B¯ X (mk )(mk − νf ([0, mk ])) − B¯ X (m∗k −)(m∗k − νf ([0, m∗k ))).

(4.10)



B¯ X (t )dνf −

(ms ,∞)

 (ms ,∞)

B¯ X (t )dt (4.11)

[m∗k ,mk+1 ]

B¯ X (t )dνf˜

(4.15)

(1d) Comparing the fourth part I4 in Eq. (4.5) and the fourth part I˜4 in Eq. (4.6). Focusing on the interval [m∗s , ∞). If m∗s = ∞, then [m∗s , ms+1 ) = ∅; If m∗s < ∞, based on B¯ X (t ) ≤ 0, t ∈ [m∗s , ∞) and νf (B) ≥ νf˜ (B), B ∈ [m∗s , ∞), we have

[m∗s ,∞)

B¯ X (t )dνf (t ) ≤

 [m∗s ,∞)

B¯ X (t )dνf˜ (t ).

(4.16)

Based on Eqs. (4.5) and (4.6), applying Eqs. (4.7), (4.10), (4.12), (4.15) and (4.16), we obtain



≤ B¯ X (ms )(ms − νf ([0, ms ])).



− B¯ X (mk+1 )(mk+1 − νf ([0, mk+1 ])).

 If m∗k = ∞, then we have k = s and B¯ X (m∗k −) = 0 from definitions of m∗k and B¯ X . Similarly with Eq. (4.10), we obtain

B¯ X (t )dνf −

≤ B¯ X (mk+1 )(νf ([m∗k , mk+1 ]) − νf˜ ([m∗k , mk+1 ])) ≤ B¯ X (m∗k −)(m∗k − νf ([0, m∗k )))







R¯ +

B¯ X dνf −

 R¯ +

B¯ X dνf˜ ≤ −B¯ X (m1 )(m1 − νf ([0, m1 ])) s  

As defined 0 × ∞ = 0, we know B¯ X (m∗s −)(m∗s − νf ([0, m∗s ))) = 0. For uniform with Eq. (4.10), we rewrite Eq. (4.11) as

+



 − B¯ X (m∗i −)(m∗i − νf ([0, m∗i )))

B¯ X (t )dνf −

(ms ,∞)

 (ms ,∞)

B¯ X (t )dt

+

[m∗k ,mk+1 ]

 = [m∗k ,mk+1 )

[m∗k ,mk+1 ]

B¯ X (t )dνf −

B¯ X (m∗i −)(m∗i − νf ([0, m∗i ]))

B¯ X (t )dνf˜

 [m∗k ,mk+1 )

− B¯ X (mi+1 )(mi+1 − νf ([0, mi+1 ])) = −B¯ X (m∗s −)(m∗s − νf ([0, m∗s ))) ≤ 0.

(4.12)

(1c) Comparing the third part I3 in Eq. (4.5) and the third part I˜3 in Eq. (4.6). Focusing on interval [m∗k , mk+1 ], k ≤ s − 1, from Definition 4.1, for any Borel set B ⊆ [m∗k , mk+1 ), we have νf˜ (B) ≤ νf (B). So νf˜ − νf is still a measure on the interval [m∗k , mk+1 ). Then



s −1   i =1

− B¯ X (m∗s −)(m∗s − νf ([0, m∗s ))).

B¯ X (t )dνf −

B¯ X (mi )(mi − νf ([0, mi ]))

i =1

≤ B¯ X (ms )(ms − νf ([0, ms ]))



(4.14)

Combining with B¯ X (m∗k −) ≥ B¯ X (mk+1 ) and Eq. (4.13), we have

JX dνζ = B¯ X (mk )νf ([0, mk ])



117

B¯ X (t )dνf˜



Now we finish the proof of Eq. (4.4). (2) We will prove f˜ ∈ LD in thispart. As we have showed that f˜ ∈ L, it is sufficient to prove that R¯ SX dνf˜ ≥ E [X ] − D. From the +

fact f ∈ LD , i.e. R¯ SX dνf ≥ E [X ] − D, it is equal to proving the + following equation







  + B¯ X (mk+1 ) νf ({mk+1 }) − νf˜ ({mk+1 })  B¯ X (t )d(νf − νf˜ ) =

R¯ +

 R¯ +

SX dνf .

(4.17)

Similarly by the case (1) above, we have

[m∗k ,mk+1 )

  + B¯ X (mk+1 ) νf ({mk+1 }) − νf˜ ({mk+1 })   ≤ B¯ X (mk+1 ) νf ([m∗k , mk+1 )) − νf˜ ([m∗k , mk+1 ))   + B¯ X (mk+1 ) νf ({mk+1 }) − νf˜ ({mk+1 })   = B¯ X (mk+1 ) νf ([m∗k , mk+1 ]) − νf˜ ([m∗k , mk+1 ]) .

SX dνf˜ ≥

 R¯ +

SX dνf =

 [0,m1 ]

+

s   i=1

s−1   i=1

(4.13)

SX (t )dνf +

[m∗i ,mi+1 ]

(mi ,m∗i )

SX (t )dνf +

=: S1 + S2 + S3 + S4

SX (t )dνf

 [m∗s ,∞)

SX (t )dνf (4.18)

118

Y. Zheng, W. Cui / Insurance: Mathematics and Economics 59 (2014) 109–120

Proof of Lemma 3.1

and

 R¯ +

SX dνf˜ =

Proof. According to Eq. (3.11) and SX (ms+1 ) = SX (∞) = 0, we have

s

 [0,m1 ]

+

SX (t )dνf˜ +

 (mi ,m∗i )

i=1

s−1  

[m∗i ,mi+1 ]

i=1

SX (t )dνf˜ +

SX (t )dt

 [m∗s ,∞)

 SX (t )dνf˜

=: S˜ 1 + S˜ 2 + S˜ 3 + S˜ 4 .

[0,m1 ]

SX (t )dνf −

(4.19)

[0,m1 ]

(mk ,m∗k )

i =0

(4.20)

(mk ,m∗k )

SX (t )dt

 [m∗k ,mk+1 ]

d→∞

[m∗s ,d)

(4.22)

 [m∗s ,∞)

SX (t )dνf (t ) −

 [m∗s ,d)

 R¯ +

R¯ +

¯ ∗i ] (mi ,m

B¯ X (t )dνf (t ) ≤

B¯ X (t )dνf˜ (t ) ≤

B¯ X (t )dνf˜ (t ) =

+ (mi+1 ,m∗i+1 )

B¯ X (t )dνf ## (t ).

(4.26)

 R¯ +

B¯ X (t )dνf ## (t ).

(4.27)

 s−1  [m∗i ,mi+1 ]

B¯ X (t )dt

B¯ X (t )dνf˜

 + [m∗s ,∞)

B¯ X (t )dνf˜ .

(4.28)

Meanwhile, (4.23)

R¯ +

B¯ X (t )dνf ## (t ) =

 s  i =0

+ (1 − λ)

+ 

R¯ +



SX dνf .

That is, we have f˜ ∈ LD .



i=0





R¯ +

¯D (mi ,m ] i

Proof. From Lemma 4.1, it is sufficient to prove that for f˜ in Definition 4.1, the following equation is correct.

SX (t )(dνf (t ) − dνf˜ (t ))

Combining Eqs. (4.20)–(4.23), we obtain SX dνf˜ ≥



¯ Di ) SX (t )dt + SX (mi+1 )(mi+1 − m

+ E [f (X )] − D = E [If # (X )] + E [f # (X )] − D = E [X ] − D. 



 SX (t )dνf˜ (t )

≤ SX (m∗s )(m∗s − νf ([0, m∗s ))).

R¯ +

s  

For the measure νf˜ in Definition 4.1,

SX (t )dνf˜ (t )

d→∞



(4.25)

#

R¯ +

≤ lim SX (m∗s )[νf ([m∗s , d)) − νf˜ ([m∗s , d))]



(4.24)

i

i =0

i =0

∗

SX (t )dνf (t ) −

= lim

i

SX dνf ## =



(2d) Comparing the fourth part S4 in Eq. (4.18) with the fourth part S˜ 4 in Eq. (4.19). If m∗s < ∞, for any d > m∗s , we have νf ([m∗s , d)) − νf˜ ([m∗s , d)) ≤ m∗s − νf ([0, m∗s )), thus



) .

Lemma 4.2. For any function f ∈ LD , we have

− SX (mk+1 )(mk+1 − νf ([0, mk+1 ])).

[m∗s ,d)

mDi

− Λ(ρ+) + E [f # (X )] − D s    SX (t )dt + SX (mi+1 )(mi+1 − m∗i ) =

≤ SX (mk −)(mk − νf ([0, mk ))) ∗

d→∞

mDi

The following lemma also plays an important role in the proof of Theorem 3.1.

∗

− SX (m∗k )(mk+1 − νf ([0, mk+1 ]))

= lim

SX (t )dt

s

i=0

(4.21)

SX (t )dνf˜

∗



¯D (mD ,m ] i i

Λ(ρ+) − (E [f # (X )] − D) .  ( S ( t ) − S ( m )) dt X i+1 ¯ D] X (mD ,m

λ=

R¯ +

≤ SX (m∗k −)(m∗k − νf ([0, m∗k )))

[m∗s ,∞)



By Eq. (3.9), we have



≤ SX (mk )(νf ([mk , mk+1 ]) − νf˜ ([mk , mk+1 ]))



¯D (mi ,m ] i

i=0

SX (t )dt − λ

Taking Eqs. (4.25) and (3.9) into Eq. (4.24), we have



SX (t )dνf −

∗

  

=

(2c) Comparing the third part S3 in Eq. (4.18) with the third part S˜ 3 in Eq. (4.19). As SX (t ) is non-increasing on interval [m∗k , mk+1 ]. By Eq. (4.14), we have

∗



+ SX (mi+1 )(mi+1 − ¯ ) + λSX (mi+1 )( ¯ −

− SX (m∗k −)(m∗k − νf ([0, m∗k ))).

[m∗k ,mk+1 ]

¯ Di − mDi )) SX (mi+1 )(mi+1 − mDi − (1 − λ)(m

s

SX (t )dνf˜

≤ SX (mk )(mk − νf ([0, mk ]))



SX (t )dt

¯D (mD ,m ] i i

s −1  

mDi

(2b) Comparing the second part S2 in Eq. (4.18) with the second part S˜ 2 in Eq. (4.19). It is easy to see that SX (t ), t ∈ [0, m∗k ] for each k = 1, 2, . . . , s satisfies the conditions in Corollary 4.1. Then from similar discussions with the case (1b), we obtain SX (t )dνf −

SX (t )dt + (1 − λ)









= −SX (m1 )(m1 − νf ([0, m1 ])).



(mi ,mD ] i

i=0

+

(2a) Comparing the first part S1 in Eq. (4.18) with the first part S˜ 1 in Eq. (4.19). Based on the similar discussions of the case (1a), we have



SX dνf ## =

R¯ +

 s 

s−1  

(mi ,mD ] i

B¯ X (t )dt



 ¯D (mD ,m ] i i

B¯ X (t )dt

¯ Di − mDi )) . B¯ X (mi+1 )(mi+1 − mDi − (1 − λ)(m

i=0



(4.29)

Y. Zheng, W. Cui / Insurance: Mathematics and Economics 59 (2014) 109–120

∗ ¯ Recalling that R¯ + B¯ X (t )dνf # (t ) = i=0 [BX (mi+1 )(mi+1 − mi ) +  ¯ (m ,m∗ ] BX (t )dt ], we can rewrite Eqs. (4.28) and (4.29) as

s−1



i+1



i+1

B¯ X (t )dνf˜ (t ) =

R¯ +

¯ Di , mi+1 ). So the second part in the right of γi (t ) > ρ , t ∈ (m Eq. (4.32) equals zero, and the third part of Eq. (4.32) is less than s  (ρ − γ ( t ))( SX (t ) − SX (mi+1 ))dνf˜ . ∗ D i i=0 [m ,m ] i

 R¯ +

B¯ X (t )dνf # (t )

+

[m∗i ,mi+1 ]

i=0



[B¯ X (t ) − B¯ X (mi+1 )]dνf˜ (t ),

(4.30)

R¯ +

B¯ X (t )dνf ## (t ) − s  

≥

and



B¯ X (t )dνf ## (t ) =

R¯ +



s



+

(1 − λ)

R¯ +

B¯ X (t )dνf # (t )

−

¯D (mD ,m ] i i

[B¯ X (t ) − B¯ X (mi+1 )]dt

=

 + [m∗i ,mD ] i

[m∗i ,mD ] i

i=0

 [B¯ X (t ) − B¯ X (mi+1 )]dt .

(4.31)

[m∗i ,mD ] i

s  

 R¯ +

B¯ X (t )dνf˜ (t )

(ρ − γi (t ))(SX (t ) − SX (mi+1 ))dt

s   i =0



i =0

[m∗i ,mD ] i

i =0



i

Summing up,

s



119

(ρ − γi (t ))(SX (t ) − SX (mi+1 ))dνf˜

(ρ − γi (t ))(SX (t ) − SX (mi+1 ))(dt − dνf˜ )

≥ 0. Thus, for any f ∈ LD , Eq. (4.26) is always correct.



Then

 R¯ +

B¯ X (t )dνf ## (t ) − s  

=

[m∗i ,mD ] i

i=0

+ (1 − λ) −

R¯ +

Proof of Theorem 3.1: (1) For the case that E [f # (X )] ≤ D, the optimal reinsurance problem with premium constraint (3.1) is the same with Problem (2.3). Thus we have the conclusion in Theorem 3.1 referring to Theorem 2.1. (2) For the case that E [f # (X )] > D, we prove that there exists an infimum for the optimal reinsurance Problem (3.1), that is,

B¯ X (t )dνf˜ (t )

[B¯ X (t ) − B¯ X (mi+1 )]dt

 ¯D (mD ,m ] i i

s  

[m∗i ,mi+1 ]

i =0



[B¯ X (t ) − B¯ X (mi+1 )]dt

 [B¯ X (t ) − B¯ X (mi+1 )]dνf˜ (t )

inf ρg [Tf (X )] = (1 + β)E (X ) −

f ∈LD

=: III . Based on Lemma 3.1 and the fact that R¯ SX (t )dνf˜ ≥ E [X ] − D, we + have



 0 ≥ R¯ +

SX (t )dνf ## (t ) −

 R¯ +

SX (t )dνf˜ (t )

=

 n→∞

[m∗i ,mD ] i

i=0

+ (1 − λ) −

[SX (t ) − SX (mi+1 )]dt

 ¯D (mD ,m ] i i

s  

[m∗i ,mi+1 ]

i=0

R¯ +

BX dvfn =

 R¯ +

B¯ X dvf ## .

(4.33)

¯ Dk ) such that limn→∞ BX (mk − δk(n) ) = BX (mk −) = ∈ (0, mk − m (n) B¯ X (mk ). On the other hand, for easy explanation, we define δi ≡ 0 for other mi , i ̸= k satisfying BX (mi ) = B¯ X (mi ). Thus, for all mi , i = (n) 1, 2, . . . , s, we always have limn→∞ BX (mi − δi ) = B¯ X (mi ). (n) Based on δi constructed above, we define the following ceded

[SX (t ) − SX (mi+1 )]dt

 [SX (t ) − SX (mi+1 )]dνf˜ (t )

loss function

From Eq. (3.4), we have B¯ X (mi+1 ) − B¯ X (t ) = γi (t )(SX (t ) − SX (mi+1 )), t ∈ [m∗i , mi+1 ) and then III + IV × ρ

[m∗i ,mD ] i

+(1 − λ)  − [m∗i ,mi+1 ]

fnd (x) =

s  

λ(x − mDi )1{x∈[mD ,mD )} i

i =0

s   i =0

B¯ X dvf ## .

In fact, if there exists mk > 0, 1 ≤ k ≤ s such that BX (mk ) < (n) (n) B¯ X (mk ) and B¯ X (mk ) = BX (mk −), we can find a series δk ↓ 0, δk

=: IV .

=

R¯ +

In order to obtain the above result, we need to prove two aspects: For the first aspect, we prove that there exists a sequence of functions {fn ∈ LD } such that lim

s





+ [x − mDi + λ(mDi − mDi )]1{x∈[mD ,mi+1 −δn )} i i  + (x − mi+1 − δi(n) )1{x∈(mi+1 −δin ,mi+1 ]} .

(ρ − γi (t ))(SX (t ) − SX (mi+1 ))dt

 ¯D (mD ,m ] i i

(ρ − γi (t ))(SX (t ) − SX (mi+1 ))dt

 (ρ − γi (t ))(SX (t ) − SX (mi+1 ))dνf˜ ≤ III .

i

(4.34)

It is easy to compute that fnd ∈ LD . Then, according to the relationship between fnd and vf d , we have n

(4.32)

Accordingly, from the definition of ρ and for i = 0, . . . , s, there ¯ Di ] and exists γi (t ) < ρ , t ∈ [m∗i , mDi ), γi (t ) = ρ , t ∈ [mDi , m

¯ Di−1 − λmDi−1 − δi(n) , vfnd ({mi − δi(n) }) = mi − (1 − λ)m vfnd ({mi }) = δi(n)

120

Y. Zheng, W. Cui / Insurance: Mathematics and Economics 59 (2014) 109–120

(n) and vf d (A) = vf## (A), A ∈ R¯ + /{mi , mi − δi ; i = 1, 2, . . . , s}. n

Thus, if n → ∞, combining with the fact that B¯ X (t ) = BX (t ), ∀t ∈ R¯ + /{mi , i = 0, 1, 2, . . . , s}, we have

 R¯ +

B¯ X dvf## −

=

 R¯ +

s  

(mi ,mD ] i

i=0

BX dvf d n

B¯ X dµ +

 ¯D (mD ,m ] i i

(1 − λ)B¯ X dµ

¯ Di−1 − λmDi−1 ] + B¯ X (mi+1 )[mi − (1 − λ)m −

s  

B X dµ +

(mi ,mD ] i

i=0

 ¯D (mD ,m ] i i



(1 − λ)BX dµ

¯ Di−1 − λmDi−1 − δi(+n)1 ] + BX (mi+1 − δi(+n)1 )[mi − (1 − λ)m  +BX (mi+1 )δi(n) =

s  

¯ Di−1 (B¯ X (mi+1 ) − BX (mi+1 − δi(+n)1 ))(mi − (1 − λ)m

i=0

−λ

−

δi(+n)1 )

+

δi(+n)1 (B¯ X (mi+1 )

ρg [Tf (X )] = (1 + β)E (X ) −



≥ (1 + β)E (X ) −



− BX (mi+1 ))

R¯ +

R¯ +

BX (t )dvf B¯ X (t )dvf ## ,

that is, BX (t )dvf ≤

 R¯ +

B¯ X (t )dvf ## .

In fact, combining BX (t ) ≤ B¯ X (t ) with the result in Lemma 4.2, we have

 R¯ +

BX (t )dvf ≤

 R¯ +

B¯ X (t )dvf ≤

 R¯ +

B¯ X (t )dvf ## .

Thus, we have proved that inf ρg (Tf (X )) = (1 + β)E (X ) −

f ∈LD

 R¯ +

B¯ X (t )dvf ## .

If BX (mi ) = B¯ X (mi ) for all i = 1, 2, . . . , s, i.e. BX (t ) = B¯ X (t ), ∀t ≥ 0, then from Lemma 4.2, we have

 max f ∈LD

R¯ +

BX dvf =

 R¯ +

BX dvf ## .

Thus, inf ρg (Tf (X )) = min {ρg [Tf (X )]} = (1 + β)E (X ) −

f ∈LD

Acknowledgments We are grateful to Jingping Yang and the anonymous referees for their valuable comments and suggestions. Zheng’s research was supported by the Young Program of National Natural Science Foundation of China (Grant No. 11201012).



Thus, Eq. (4.33) is correct; For the second aspect, we will prove that for any function f ∈ LD ,

R¯ +

In this paper, we discuss the optimal reinsurance strategy with unbinding premium constraint under distortion risk measure in a general set of ceded loss functions. Comparing with the optimal reinsurance solution without premium constraint, the insurer must make a balance between the loss risks and the insurance profits by adjusting the ceded loss function and the retained loss function in proper range. By introducing an adjustment function γ (t ), we constructed the optimal reinsurance solution with premium constraint from the optimal one without premium constraint in Zheng et al. (2014). Finally, we show the optimal reinsurance problems with premium constraint under two special distortion risk measures—VaR and TVaR.

References mDi−1

→ 0.



5. Conclusions

f ∈LD

 R¯ +

B¯ X dvf ## ,

which means f ## is a solution of the optimal reinsurance problem (3.1).

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