Optimal reinsurance under variance related premium principles

Insurance: Mathematics and Economics 51 (2012) 310–321

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Optimal reinsurance under variance related premium principles Yichun Chi ∗ China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China

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Article history: Received January 2011 Received in revised form September 2011 Accepted 21 May 2012 Keywords: Conditional value at risk Value at risk Layer reinsurance Variance premium principle Standard deviation premium principle

abstract In this paper, we investigate the optimal form of reinsurance when the insurer seeks to minimize the value at risk(VaR) or the conditional value at risk(CVaR) of his/her total risk exposure. In order to exclude the moral hazard from a reinsurance treaty, both the ceded and retained loss functions are constrained to be increasing. Under the additional assumption that the reinsurance premium is calculated by a variance related principle, we show that the layer reinsurance is always optimal over both the VaR and CVaR criteria. Finally, the variance and standard deviation premium principles are applied to illustrate how to derive the optimal deductible and the upper limit of layer reinsurance. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Suppose that X denotes the amount of loss initially assumed by an insurer in a given time period. We assume X is a non-negative random variable defined on a probability space (Ω , F , P) with cumulative distribution function FX (x) = P(X ≤ x) for any x ∈ R and E[X ] < ∞. Under a reinsurance treaty, the loss X is partitioned into f (X ) and Rf (X ), where f (X ), satisfying 0 ≤ f (X ) ≤ X , captures the portion of the loss that is ceded to a reinsurer while Rf (X ) = X − f (X ) represents the residual loss retained by the insurer. Thus, f (x) and Rf (x) are referred to as the ceded and retained loss functions, respectively. Due to cover the loss f (X ) for the insurer, the reinsurer is compensated with a payment in the name of reinsurance premium denoted by π (f (X )). As a result, in the presence of reinsurance, the total risk exposure of the insurer is no longer X . It is now the sum of the retained loss and the incurred reinsurance premium. More precisely, using Tf (X ) to represent the total risk exposure of the insurer, we have Tf (X ) = Rf (X ) + π (f (X )). By choosing different optimality criteria for Tf (X ) or different reinsurance premium principles, a large number of optimal reinsurance models have been formulated from the perspective of the insurer and have been studied extensively over the past half-century. Just to name a few, Borch (1960) considers to minimize the variance of the total risk exposure with a fixed reinsurance premium, and shows that the stop-loss reinsurance

∗

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is optimal when the reinsurance premium is calculated by the expected value principle. Arrow (1963), who also assumes the expected value premium principle, demonstrates that the same stop-loss reinsurance contract maximizes the expected utility of the terminal wealth of a risk-averse insurer. Recently, motivated by the prevalent use of risk measures such as the value at risk(VaR) and the conditional value at risk(CVaR) among banks, insurance companies and other financial institutions for quantifying risk and setting capital requirements, Cai and Tan (2007) and Cai et al. (2008) have proposed two classes of optimal reinsurance models by minimizing the VaR and CVaR of the total risk exposure of an insurer, namely, VaR-minimization : VaRα (Tf ∗ (X )) = min VaRα (Tf (X )) f ∈C

(1.1)

and CVaR-minimization : CVaRα (Tf ∗ (X )) = min CVaRα (Tf (X )), (1.2) f ∈C

where C represents the set of admissible ceded loss functions and f ∗ is the desired feasible solution among C . For more general risk measure based optimal reinsurance problems, see e.g. Bernard and Tian (2009), Balbás et al. (2009), and references therein. More recently, the study of optimal reinsurance models (1.1) and (1.2) has attracted great attention. Cai and Tan (2007) assume the expected value reinsurance premium principle, and derive analytically the optimal retention for a stop-loss reinsurance. Their result is generalized by Cai et al. (2008) to explore the optimal reinsurance designs among the class of increasing convex ceded loss functions.1 Cheung (2010) revisits these two classes

1 Throughout this paper, the terms ‘‘increasing’’ and ‘‘decreasing’’ mean ‘‘nondecreasing’’ and ‘‘non-increasing’’, respectively.

Y. Chi / Insurance: Mathematics and Economics 51 (2012) 310–321

of optimal reinsurance models with the help of a geometric approach, and extends the results in Cai et al. (2008) by studying the VaR-minimization problem with Wang’s premium principle. Moreover, Chi and Tan (2011a), who also assume the expected value principle, are concerned with the robustness of the optimal form of reinsurance over the different optimality criteria and the constraints of the ceded loss functions. They find that under the VaR-minimization model, the optimal reinsurance policies among three admissible sets of ceded loss functions (with increasing degrees of generality) are varying. Specifically, in contrast to the optimality of stop-loss reinsurance in Cai et al. (2008), they show that the stop-loss reinsurance with an upper limit is optimal when both the ceded and retained loss functions are assumed to be increasing, and the truncated stop-loss reinsurance is optimal among the set of general increasing left-continuous retained loss functions. However, the stop-loss reinsurance is shown to be always optimal under the CVaR risk measure. In the aforementioned studies except Cheung (2010), they all assume that the reinsurance premium is calculated by the expected value principle. While Bühlmann (1970) declares that the expected value principle is used frequently in life insurance, there are many other premium principles the practicing actuaries and academics are interested. For example, Young (2004) lists as many as eleven well-known premium principles: net, expected value, exponential, proportional hazards, principle of equivalent utility, Wang’s, Swiss, Dutch, variance, standard deviation and Esscher premium principles. In the past few years, some attempts have been made to study the VaR and CVaR based optimal reinsurance models (1.1) and (1.2) with other reinsurance premium principles. For instance, Tan et al. (2009) consider as many as seventeen premium principles when investigating the optimal quota-share and stop-loss reinsurance policies. Chi and Tan (2011b) instead assume that the reinsurance premium principle follows three basic axioms: distribution invariance, risk loading and stop-loss ordering preserving. Their proposed class of premium principles is large enough to include eight of eleven principles in Young (2004), where the variance, standard deviation and Esscher principles are excluded as they fail to preserve stop-loss order. Under the additional assumption that both the insurer and reinsurer are obligated to pay more for greater losses, Chi and Tan (2011b) show that the layer reinsurance in the form of L(a,b] (x) , min {(x − a)+ , b − a} = (x − a)+ − (x − b)+ ,

0≤a≤b≤∞

(1.3)

where (x)+ = max{x, 0}, is robust in the sense that it is always optimal over both the VaR and CVaR criteria and the prescribed premium principles. Here, the parameters a and b − a are usually called the deductible and the upper limit of layer reinsurance, respectively. The variance and standard deviation premium principles, which are excluded from the analysis of Chi and Tan (2011b), play important roles in actuarial science. As pointed out by Bühlmann (1970), the standard deviation principle is most frequently used to determine the premium of property and casualty insurance, while the variance principle is preferred by academics. There have been some results on optimal reinsurance that use these two principles to calculate the reinsurance premium. Just to name a few, under mean–variance premium principles, Kaluszka (2001) investigates the optimal form of reinsurance by minimizing the variance of total risk exposure Tf (X ). Moreover, Kaluszka (2005) and Balbás et al. (2009) study the optimal reinsurance problems by assuming a class of convex principles of premium calculation, which contains the variance and standard deviation principles due to their convex and Gâteaux differentiable properties in Deprez and Gerber (1985). It should be emphasized that Kaluszka (2005) seeks to minimize the expectation of a convex function of the

311

retained loss or maximize the expected utility of the terminal wealth of a risk-averse insurer, and hence his optimality criteria do not encompass the VaR and CVaR risk measures. Furthermore, while CVaR is one of the risk measures analyzed in Balbás et al. (2009), it is not an easy task to derive optimal reinsurance by verifying the necessary and sufficient optimality conditions in their paper. In this paper, we extend the study of optimal reinsurance models (1.1) and (1.2) in Chi and Tan (2011b) by considering the variance related reinsurance premium principles. The variance related principles, which were first introduced by Guerra and Centeno (2010), are of the form

π (X ) = E[X ] + g (v ar (X )),

(1.4)

where v ar (X ) is the variance of X and the loading function g : [0, ∞] → [0, ∞] is increasing with g (0) = 0. Especially, when g (√ x) = γ x for γ > 0, we recover the variance principle. If g (x) = θ x for θ > 0, it is the standard deviation principle. Further, in order to eliminate the moral hazard from a reinsurance treaty, we follow the way of Chi and Tan (2011b) to assume that both the insurer and reinsurer are obligated to pay more for greater losses. In other words, the set of admissible ceded loss functions in this paper is given by C , 0 ≤ f (x) ≤ x : both Rf (x) and f (x)



are increasing functions .



(1.5)

As a result, we obtain that the layer reinsurance is always optimal over both the VaR and CVaR risk measures and all the variance related premium principles. To further illustrate the applicability of our results, we use variance and standard deviation principles to show how to derive the optimal parameters of layer reinsurance. The contributions of this paper are threefold. First, we enrich the study of VaR and CVaR based optimal reinsurance models and extend the results in Chi and Tan (2011b) by considering variance related premium principles. Our result, together with that in Chi and Tan (2011b), indicates that the layer reinsurance is a very robust reinsurance treaty in the sense that it is always optimal over both the VaR and CVaR criteria and almost all the well-known premium principles listed in Young (2004). Second, it is necessary to point out that Tan et al. (2009) also study the optimal reinsurance models (1.1) and (1.2) with variance and standard deviation premium principles, and establish the sufficient and necessary conditions for the existence of nontrivial optimal reinsurance. However, they impose a very stringent assumption on the ceded loss functions, under which it is either a quota share or a stop loss. The stop-loss reinsurance with an upper limit, a well-known type of reinsurance in practice and academic research, is excluded from their analysis. In this paper, we relax such a severe constraint by requiring only that both the ceded and retained loss functions are increasing. Third, it should be emphasized that the optimal reinsurance problems (1.1) and (1.2) with variance related premium principles have also been studied by Guerra and Centeno (2012). However, there are some distinctive differences between their study and ours. One is that Guerra and Centeno (2012) assume the set of general admissible ceded loss functions {f : 0 ≤ f (x) ≤ x}, and we impose a more strict constraint on the ceded loss functions in order to exclude the moral hazard. Consequently, similar to the results of Chi and Tan (2011a), different optimal reinsurance policies are derived under the criterion of VaR risk measure: we show that the layer reinsurance is optimal, while they demonstrate the optimality of the truncated stop loss. Under the CVaR criterion, the layer reinsurance is shown to be optimal in both papers. The other is that the different approaches are applied to derive the optimal reinsurance. Guerra and Centeno (2012) use

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Y. Chi / Insurance: Mathematics and Economics 51 (2012) 310–321

the method of random treaties such that their proof appears to be rather technical. In contrast, we provide a relatively simple approach using the theory of stochastic orders and the increasing Lipschitz-continuous properties of the ceded loss functions in (1.5). In particular, for any admissible ceded loss function f , we can construct a layer reinsurance treaty such that it is better than f in the sense to minimize the total risk exposure of an insurer under the VaR or CVaR risk measure. The rest of this paper is organized as follows. Section 2 introduces some properties associated with the VaR and CVaR risk measures. Section 3 shows that the layer reinsurance is always optimal over both the VaR and CVaR criteria and the variance related premium principles. Consequently, the study of optimal reinsurance models (1.1) and (1.2) is simplified to solving one-parameter and two-parameter minimization problems, respectively. Section 4 is applied to solve these parameter-dependent optimization problems and derive the optimal layer reinsurance explicitly by assuming that the reinsurance premium is calculated by the variance or the standard deviation principle. Finally, some concluding remarks are provided in Section 5. 2. Some properties associated with VaR and CVaR

3. Optimal reinsurance under VaR and CVaR criteria In this section, we study the optimal reinsurance problems (1.1) and (1.2) under the assumptions that the reinsurance premium is calculated by a variance related premium principle (1.4) and the set of admissible ceded loss functions is defined in (1.5). Define

  w1 (d) , E L(d,VaRα (X )] (X ) = w2 (d) , E

L(2d,VaRα (X )]



VaRα (X )



(X ) = 2 

SX (t )dt

and

d VaRα (X )



(3.1)

(t − d)SX (t )dt

d

for any 0 ≤ d ≤ VaRα (X ), where L(a,b] (x) is given in (1.3). Theorem 3.1. When the reinsurance premium is calculated by a variance related premium principle (1.4), the layer reinsurance is optimal under the VaR-based reinsurance model (1.1) in the sense that for any admissible ceded loss function f defined in (1.5), we can construct a layer reinsurance treaty f˜ (x) = L(d,VaRα (X )] (x) for some 0 ≤ d ≤ VaRα (X ) such that VaRα (Tf˜ (X )) ≤ VaRα (Tf (X )).

We begin with the definitions of the VaR and CVaR risk measures. Definition 2.1. VaR of a non-negative random variable X with E[X ] < ∞ at a confidence level 1 − α where 0 < α < 1 is defined as

(3.2)

Thus, we have min VaRα (Tf (X )) = f ∈C

min

0≤d≤VaRα (X )

Q (d),

(3.3)

where

(2.1)

Q (d) , d + w1 (d) + g w2 (d) − w12 (d) , 0 ≤ d ≤ VaRα (X ). (3.4)

Based upon the definition of VaR, CVaR of X at a confidence level 1 − α is defined as

Proof. The proof of this theorem is lengthy and is provided in Appendix A. 

VaRα (X ) , inf{x ≥ 0 : P(X > x) ≤ α}.

CVaRα (X ) =

1

α

α



VaRs (X )ds.



(2.2)

0

It follows from the definition of VaRα (X ) that VaRα (X ) ≤ x ⇔ SX (x) ≤ α,

(2.3)

where SX (x) = 1 − FX (x). Therefore, VaRα (X ) = 0 for α ≥ SX (0). For this reason, we assume in this paper that the parameter α satisfies 0 < α < SX (0) to avoid the trivial cases. Another important property associated with VaRα (X ) is that for any increasing continuous function ρ(x), we have (see Theorem 1 in Dhaene et al. (2002)) VaRα (ρ(X )) = ρ(VaRα (X )).

(2.4)

Denote the essential infimum and the essential supremum of X by ess inf X , sup{x ∈ R : SX (x) = 1}

By the above theorem, we know that the layer reinsurance is optimal under the VaR-based reinsurance model (1.1) with variance related premium principles, where the optimal deductible of layer reinsurance depends on the specific form of the loading function g. By imposing a constraint on the loading function, we derive the optimal layer reinsurance analytically in the following corollary. Corollary 3.2. When the reinsurance premium is calculated by a variance related principle (1.4) with g ′′ (x) ≥ 0 for any x > 0, the ceded loss function f˜ ∗ that solves the optimal reinsurance model (1.1) is given by f˜ ∗ (x) = L(d˜ ∗ ,VaRα (X )] (x),

(3.5)

where

and

ess sup X , sup{x ∈ R : FX (x) < 1},



(2.5)

d˜ ∗ , min ess inf X ≤ d ≤ VaRα (X ) :



g ′ w2 (d) − w12 (d) w1 (d) ≤ 1/2 .



respectively, then we have 0 ≤ ess inf X ≤ VaRα (X ) ≤ ess sup X since 1 − SX (VaRα (X )) ≥ 1 − α > 0 according to (2.3). A main advantage of CVaRα (X ) over VaRα (X ) is that CVaR is a coherent risk measure while VaR fails to satisfy the subadditivity property. CVaR is also known as the ‘‘average value at risk’’ or the ‘‘expected shortfall’’, but it is a little different from conditional tail expectation(CTE). Specifically, Föllmer and Schied (2004) show in Corollary 4.49 that CVaRα (X ) ≥ CTEα (X ) , E[X |X ≥ VaRα (X )], and demonstrate in Remark 4.50 that CTE is not a coherent risk measure. However, the above inequality is an identity when FX (x) is continuous and strictly increasing with a possible jump at 0 as assumed by Cai et al. (2008). See also Rockafellar and Uryasev (2002) for more details on the relationship between CVaR and CTE.





(3.6)

Proof. According to the definitions of {wi (d); i = 1, 2} in (3.1), we have



′ w2 (d) − w12 (d) = w2′ (d) − 2w1 (d)w1′ (d) = −2w1 (d)(1 − SX (d)),

a.s.

(3.7)

Thus, taking the derivatives of Q (d) in (3.4) with respect to (w.r.t.) d yields Q ′ (d) = 2(1 − SX (d))



1 2

   − w1 (d)g ′ w2 (d) − w12 (d) ,

a.s. (3.8)

If 0 ≤ d < ess inf X , then 1 − SX (d) = 0 such that Q (d) = 0. Thus, Q (d) = Q (ess inf X ). Consequently, we only need to find the ′

Y. Chi / Insurance: Mathematics and Economics 51 (2012) 310–321

infimum of Q (d) over (ess inf X , VaRα (X )). In this case, we have 1 − SX (d) > 0 and hence w2 (d)−w12 (d) is strictly decreasing according to (3.7). Further, recall that g (x) isincreasing withg ′′ (x) ≥ 0 and w1 (d) is decreasing, then w1 (d)g ′ w2 (d) − w12 (d) decreases and equals 0 when d = VaRα (X ). As a result, it follows from (3.8) that Q ′ (d) R 0

if

313

Proposition 4.1. Under the assumption of variance premium principle, the ceded loss function fv∗ that solves the optimal reinsurance model (1.1) with C (1.5) is given by fv∗ (x) = L(d∗v ,xα ] (x),

(4.2)

where

d R d˜ ∗ ,



where d˜ ∗ is given in (3.6), then the minimum value of Q (d) over [ess inf X , VaRα (X )] is attainable at d = d˜ ∗ . The final result follows from Theorem 3.1 and hence the proof is complete.  Below, we proceed to study the optimal reinsurance model (1.2) with variance related premium principles when the set C of admissible ceded loss functions is defined in (1.5). Theorem 3.3. Under the same assumptions as in Theorem 3.1, the layer reinsurance is optimal under the CVaR-based reinsurance model (1.2) in the sense that for any f ∈ C , there exists a layer reinsurance treaty fˆ (x) = L(d,c ] (x) for some 0 ≤ d ≤ VaRα (X ) ≤ c ≤ ∞ such that

d∗v , min x ≤ d ≤ xα : E[(X − d)+ ] ≤

(4.3)

The residual task of this subsection is to study the optimal reinsurance model (1.2) with variance principle for xα < x¯ . In order to simplify the analysis, we make the following assumption. Assumption 4.1. The survival distribution function SX (t ) is continuous on (0, ∞) and strictly decreases in a neighborhood of xα .2 First, we define G , [x, xα ] × [xα , x¯ ]

Thus, we have

  c 1 D , (d, c ) ∈ G o : SX (t )dt = , 2γ d

f ∈C

min

0≤d≤VaRα (X )≤c

W (d, c ),

(3.9)





1

α

 − 1 E[(X − c )+ ]

c

2(t − d)SX (t )dt −

+g d

c



and

SX (t )dt

2 

. (3.10)

2γ α

,

(4.4)

d

Proof. Under Assumption 4.1, the implicit function theorem in Rudin (1976) implies the equation c



Proof. The proof is lengthy and is also given in Appendix A.

a′ (c ) = SX (c )/SX (a(c )) < 1



4. Optimal reinsurance for variance and standard deviation premium principles By Theorems 3.1 and 3.3, we know that the layer reinsurance is always optimal over both the VaR and CVaR criteria and all the variance related premium principles. Hence, the study of optimal reinsurance models (1.1) and (1.2) is simplified to solving one-parameter and two-parameter minimization problems, respectively. Using variance and standard deviation premium principles as examples, the next two subsections are applied to illustrate how to solve these parameter-dependent optimization problems and derive the optimal layer reinsurance. In this section, for the sake of notational simplicity, we denote xα , VaRα (X ) and



1

Lemma 4.1. Under Assumption 4.1, we have |D | ≤ 1, where |D | is the cardinality of D .

Especially, if VaRα (X ) = ess sup X where ess sup X is defined in (2.5), then the layer reinsurance that solves the VaR-based reinsurance problem (1.1) in Theorem 3.1 is also a solution to the CVaR-based reinsurance model (1.2).

x , ess inf X ,

c−d=

where G is the Cartesian product of [x, xα ] and [xα , x¯ ], and G o represents the interior of G .

where W (d, c ) , E[max{X , d}] +

2γ

 + E[(X − xα )+ ] .

Moreover, fv∗ is also a solution to the optimal reinsurance model (1.2) if xα = x¯ .

CVaRα (Tfˆ (X )) ≤ CVaRα (Tf (X )). min CVaRα (Tf (X )) =

1

x¯ , ess sup X .

(4.1)

4.1. The variance principle

SX (t )dt = a

π(X ) = E[X ] + γ v ar (X ) for some γ > 0. Thus, Corollary 3.2 is applicable to the variance principle, and it together with Theorem 3.3 leads to the following proposition.

,

x < a < xα

for any c ∈ (xα , x¯ ) has at most one solution. Furthermore, if (a(c ), c ) ∈ D , then a(c ) is differentiable with the derivative such that c − a(c ) is strictly increasing. Consequently, if |D | ̸= 0, D must contain only one element. The proof is therefore complete.  Further, we introduce some additional notations:



cv , sup c ∈ [xα , x¯ ] :

c



FX (t )dt ≤ x



 (1/α − 1) or c = xα ; 2γ

dv , sup d ∈ (x, xα ) : E[(X − d)+ ] = x¯ − d ≤

1 2γ α



1 2γ

(4.5)

and

,

where sup ∅ = −∞. It is necessary to point out that dv = −∞ if x¯ = ∞. Proposition 4.2. Under Assumption 4.1, the ceded loss function that solves the optimal reinsurance problem (1.2) with variance premium principle is given by

In the Introduction, the variance premium principle is defined by

1 2γ

L(d,c ] (x), (d, c ) ∈ D = (x − dv )+ min{x, cv }



fv∗C (x)

if |D | > 0, if |D | = 0 and otherwise.

dv ̸= −∞,

2 Under the assumption, it is trivial that x < x¯ for 0 < α < S (0). α X

314

Y. Chi / Insurance: Mathematics and Economics 51 (2012) 310–321

Proof. By Theorem 3.3, the study of optimal reinsurance model (1.2) with variance premium principle is simplified to solving the two-parameter minimization problem: min

0≤d≤xα ≤c

W (d, c ),

where W (d, c ) is now given by W (d, c ) = d + E[(X − d)+ ] +

+γ





1

α

α

 − 1 E[(X − c )+ ]

c

2(t − d)SX (t )dt −



c

SX (t )dt

2 

With the help of the former equation, the latter inequality can be simplified as x¯ − d ≤ 2γ1α . Thus, we have d = dv according to (4.5). • Finally, we demonstrate that W (d, c ) has no minimum points ∂ W (xα ,xα ) on G3 . Specifically, as > 0, W (d, c ) is not minimal ∂d at the point (xα , xα ). Moreover, if (xα , c ) ∈ G3 \ {(xα , xα )} is c a minimum point, then (4.6) implies x SX (t )dt ≥ 21γ such that

(c − xα ) > .

Consequently, we have

   c  ∂ W (d, c ) 1   SX (t )dt , = 2γ (1 − SX (d)) −  ∂d 2γ d  c   ∂ W (d, c ) 1/α − 1   FX (t )dt − = 2 γ SX ( c ) , ∂c 2γ d

Thus, all the stationary points of W (d, c ) over G o constitute the set D in (4.4). Next, we proceed to identify the minimum points of W (d, c ) located on the boundary of G . We split the boundary into three disjoint subsets, i.e. ∂ G = ∪3i=1 Gi , where G2 = (x, xα ) × {¯x},

G3 = (x, xα ) × {xα } ∪ {xα } × [xα , x¯ ],

(4.7)

and divide the following analysis into three cases.

• If (x, c ) ∈ G1 is a minimum point of W (d, c ), then it follows c ∂ W (d,c ) from the expression of ∂ d in (4.6) that 21γ − x SX (t )dt ≥ 0. Thus, c ∈ A where    c 1 A , c ∈ [xα , x¯ ] : SX (t )dt ≤ . (4.8) 2γ x Furthermore, c must be a solution to the minimization problem: miny∈[xα ,¯x] W (x, y). As a result, it follows from (4.6) that c = cv , where cv is given in (4.5). • If (d, x¯ ) ∈ G2 is a minimum point, then Fermat’s theorem, together with (4.6), implies x¯



1 2γ

SX (t )dt = E[(X − d)+ ]

=

FX (t )dt ≤ d

1/α − 1 2γ

,

x < d < xα .

xα

2γ α

.

FX (t )dt > FX (xα )(c − xα ) > ∂ W (x ,c )

1/α−1 , 2γ

and

2γ

1/α − 1 2γ

,

x < d < xα .

Straightforward algebra leads to xα − d contradicted to the following result



1

xα

=

2γ

1 , 2γ α

≥

that is

SX (t )dt > SX (xα )(xα − d) = α(xα − d).

d

Collecting all the above arguments, we get that the minimum value of W (d, c ) can appear in the set Kv ,



z ∈ R2+ : I(z = (x, cv ), cv ∈ A )

 + I(z ∈ D ) + I(z = (dv , x¯ )) > 0 , where I(.) is the indicator function. Further, if D ̸= ∅, then Lemma 4.1 shows |D | = 1. Without loss of generality, we let D = {(dD , cD )}. In this case, we demonstrate dv = −∞ and cv ̸∈ A such that the minimum value of W (d, c ) is attainable at (dD , cD ). Specifically, if dv ̸= −∞, then it follows from the definition of dv in (4.5) that x¯



SX (t )dt = dv



1

cD

=

2γ

SX (t )dt

and x¯ − dv ≤

dD

1 2γ α

.

However, as stated in the proof of Lemma 4.1, c − a(c ) is strictly increasing, then we have x¯ − dv > cD − dD = 2γ1α , which is contradicted to the above equation. On the other hand, if A ̸= ∅, then there exists a cˆ ∈ [xα , cD ) such that

 cˆ x

SX (t )dt =

1 2γ

and

A = [xα , cˆ ]. Due to the strictly increasing property of c − a(c ), we have c



FX (t )dt ≤

cˆ



x

FX (t )dt x

= cˆ − x −

1

< cD − dD −

2γ

1 2γ

=

1/α − 1 2γ

,

∀c ∈ A .

Thus, the definition of cv in (4.5) leads to cv > cˆ such that cv ̸∈ A . Else if |D | = 0 and dv ̸= −∞, then using the similar arguments, we can get cv ̸∈ A . Specifically, if A ̸= ∅, recall that 21γ =

 x¯

E[(X − dv )+ ] = that

 c˜ x

SX (t )dt =

d

x¯



and

FX (t )dt ≥

1

d

for

∂ W (d, c ) ∂ W (d, c ) = = 0. ∂d ∂c As 1 − SX (d) > 0 and SX (c ) > 0, straightforward algebra yields  c 1 1 SX (t )dt = and c − d = . 2 γ 2 γ α d

xα



for

for any d < x. Consequently, the minimum of W (d, c ) over 0 ≤ d ≤ xα ≤ c is attainable in a compact subset of G , where G is given in (4.4). Furthermore, we know from Fermat’s theorem that the minimum of W (d, c ) over G must be attainable at some stationary point or must lie on the boundary. If (d, c ) ∈ G o is a stationary point of W (d, c ), then we have

SX (t )dt =

d

(4.6)

G1 = {x} × [xα , x¯ ],

xα



∂ W (d,c ) If x¯ = ∞, then the above equation implies > 0 ∂c ∂ W (d,c ) =0 sufficiently large c; otherwise, if x¯ < ∞, we have ∂ c ∂ W (d,c ) any c ≥ x¯ . Moreover, the above equation leads to ∂ d =0

c

1

α then using the expression of in (4.6), we get W (xα , c ) > ∂c miny∈(xα ,¯x] W (xα , y), that is contradicted to the assumption that (xα , c ) is a minimal point. Similarly, if (d, xα ) ∈ G3 \ {(xα , xα )} is a minimum point of W (d, c ), then Fermat’s theorem, together with the partial derivatives of W (d, c ) in (4.6), implies

Under Assumption 4.1, W (d, c ) is differentiable with partial derivatives

0 ≤ d ≤ xα ≤ c .

SX (t )dt /SX (xα ) ≥ xα

d

d

c



c



FX (t )dt ≤ x

dv 1 2γ

SX (t )dt, then there exists a c˜ ∈ [xα , x¯ ) such and A = [xα , c˜ ]. As a result,

c˜



FX (t )dt = c˜ − x − x

1 2γ

< x¯ − dv −

1 2γ

Y. Chi / Insurance: Mathematics and Economics 51 (2012) 310–321

≤

1/α − 1 2γ

,

Proof. According to the definitions of {wj (d); j = 1, 2} in (3.1), we have

∀c ∈ A

such that cv ̸∈ A . Therefore, W (d, c ) can attain the minimum at (dv , x¯ ) in this case. Moreover, as L(dv ,¯x] (X ) = (X − dv )+ , a.s., the stop-loss reinsurance with the deductible dv is optimal. Otherwise, if D = ∅ and dv = −∞, then we have Kv = {(x, cv )} such that the minimum of W (d, c ) is attainable at (x, cv ). Furthermore, as W (x, cv ) = W (0, cv ), we can choose the optimal reinsurance strategy fv∗C (x) = L(0,cv ] (x) = min{x, cv }. The proof is thus complete.  Example 4.1. The loss X is assumed to follow the Pareto distribution with probability density function p(x) =

2

(x + 1)3

, x > 0.

x=0

w12 (d) =

and

xα



xα



d

SX (x)SX (y)dxdy

d xα





xα

d



d  xα

y d

d

≤ 2SX (d)

xα



2γ

FX (t )dt =

= 0

c2 c+1

(4.10)

As a consequence, taking the derivatives of ψ(d) in (4.9) w.r.t. d yields 2

w13 (d)

  w2 (d)SX (d) − w12 (d) ≥ 0,

w12 (d) ≥ 2SX (xα −)



xα

a.s.

we have cv = 38.975 such that



1dx SX (y)dy = SX (xα −)w2 (d).

d

The above equation, together with (4.10), leads to 1 SX (d)

≤

1 w2 (d) ≤ 2 SX (xα −) w1 (d)

such that limd↑xα ψ(d) complete. 

,

y



d

c

c 1. If γ = 0.25, since 0 SX (t )dt = 1+ < 21γ for any c ≥ xα , then c we have D = ∅. By solving c

(y − d)SX (y)dy = SX (d)w2 (d).

d

√





SX (x)dx SX (y)dy

=2

Thus, ψ(d) is increasing over [0, xα ). Moreover, taking the similar arguments as that for (4.10) yields

x¯ = ∞

such that xα = 1/ α − 1 and dv = −∞. We set α = 5%, then xα = 3.472. The following analysis is divided into two cases: γ = 0.25 and γ = 0.75.

1/α − 1

I(x ≤ y)SX (x)SX (y)dxdy

=2

ψ ′ (d) =

Then we have SX (t ) = 1/(t + 1)2 ,

315

=

1 . SX (xα −)

The proof is therefore

By virtue of the above lemma, we provide a solution to the VaR-based reinsurance model with standard deviation principle in the following proposition.

fv∗C (x) = min{x, 38.975}. 2. If γ = 0.75, by solving the equations

Proposition 4.3. When the reinsurance premium is calculated by the standard deviation principle, the ceded loss function fs∗ that solves the optimal reinsurance model (1.1) is given by

 c  c−d 1   = SX (t )dt = ,  2γ ( d + 1)(c + 1) d 1    = c − d, 2γ α 0 < d < xα < c < ∞,

fs∗ (x) = L(d∗s , xα ] (x),

(4.11)

where d∗s is given by

we have d = 0.361 and c = 13.694 such that D = {(0.361, 13.694)}. In this case, the optimal reinsurance is given by

d∗s , min x ≤ d ≤ xα : ψ(d) ≥ θ 2 + 1 or d = xα .

fv∗C (x) = (x − 0.361)+ − (x − 13.694)+ . 4.2. The standard deviation principle

Proof. If x = xα , we have L(d,xα ] (X ) = xα − d, a.s. and hence Q (d) = xα for any 0 ≤ d ≤ xα according to (3.4); otherwise, if then (4.10) implies ψ(d) > 1. As the loading function x < xα , √ g (x) = θ x for the standard deviation principle, we have

In this subsection, we study the VaR and CVaR based optimal reinsurance models (1.1) and (1.2) with standard deviation premium principle, which is defined in the Introduction by

w12 (d) Q (d) = θ (1 − SX (d)) − θ w2 (d) − w12 (d)   





Moreover, fs (x) is also a solution to the optimal reinsurance model (1.2) if xα = x¯ .



for some θ > 0. It is trivial that Corollary 3.2 is not applicable to the standard deviation principle. To solve the optimal reinsurance model (1.1), we need to introduce a useful lemma. Lemma 4.2. Denote

ψ(d) ,

w2 (d) , w12 (d)

′

= θ (1 − SX (d))

 π(X ) = E[X ] + θ v ar (X )

0 ≤ d < xα ,

(4.9)

where {wj (d); j = 1, 2} are given in (3.1) and xα is defined in (4.1), then ψ(d) is an increasing function with the supremum S (x1 −) . X

α

(4.12)

∗



1

1

θ

−

1

ψ(d) − 1

,



a.s.

Since Q ′ (d) = 0 for any 0 ≤ d < x, then we have Q (d) = Q (x). On the other hand, the above equation implies Q ′ (d) ≥ 0

if and only if

ψ(d) ≥ θ 2 + 1,

x < d < xα .

As shown in Lemma 4.2, ψ(d) is an increasing function, then the minimum value of Q (d) is attainable at d = d∗s . Consequently, it follows from Theorem 3.1 that fs∗ (x) is a solution to the optimal reinsurance model (1.1) with standard deviation principle. Moreover, Theorem 3.3 implies that fs∗ (x) is also a solution to the CVaR-based reinsurance model (1.2) when xα = x¯ . The proof is finally complete. 

316

Y. Chi / Insurance: Mathematics and Economics 51 (2012) 310–321

1 ∗ Remark 4.1. If SX (xα −) > 1+θ 2 , we have ds = xα according to (4.12). In this case, the optimal reinsurance fs∗ (x) = 0. In other words, under the standard deviation premium principle and the VaR criterion, if the loading coefficient θ is too large, which is equivalent to saying that the reinsurance is too costly, then the insurer would not cede any risk.

By the above two equations, taking the derivatives of ϕ 2 (d, c ) in (4.13) w.r.t. c yields

c

2(t − d)SX (t )dt

d

 c d

SX (t )dt

2

d



d

=   t c



(4.13)

for x ≤ d ≤ xα ≤ c ≤ x¯ , then we have φ(d, xα ) = ψ(d) where ψ(d) is given in (4.9). Lemma 4.3. Under Assumption 4.1, both φ(d, c ) and ϕ(d, c ) are continuous and strictly increasing in each argument. Proof. By straightforward algebra, the first-order partial derivatives of φ(d, c ) are given by 2(t − d)SX (t )dt d

∂ϕ 2 (d, c ) =   ∂d c t d c



d

FX (d)



FX (t ) (c − t )

×

d

(4.15)

c

SX (t )dt

2

(SX (d) − SX (t ))dtSX (y)dy > 0

such that

(4.16)

d

∂φ(d,c ) ∂d

> 0. On the other hand, we have

∂φ(d,c ) ∂c

> 0 as

c

 (d, c ) ∈ G :  c o

c

(c + d − 2t ) (SX (t ) − SX ((c + d)/2)) dt > 0,

φ(d, c ) = θ 2 + 1,

FX (t )dt

 c d

 c 

t

FX (t )FX (s)dtds

d



FX (s)ds FX (t )dt

=2 d

c

=

d

d

c

2(t − d)SX (t )dt −

c



d

SX (t )dt

t

2



d

d

SX (t )dt

= 1/α, φ(d, c ) = θ + 1 , 2

SX (t )dt

x < d < xα



d(c )

=  c d

2

c

×

(SX (t ) − SX (c ))dt + d′ (c )



FX (s)ds SX (t )dt .

=2



c−d

1

c d(c )

d

 c 

>

where the second equality is implied by (4.14) and (4.15). As a consequence, we have  ′ c − d(c ) c S (t )dt d(c ) X

= 

and



∂ϕ(d,c ) ∂d

∂φ(d(c ), c ) ∂d c SX (c ) d(c ) (c + d(c ) − 2t ) SX (t )dt =−  2 , c c SX (d(c )) d(c ) 2(t − d(c ))SX (t )dt − d(c ) SX (t )dt

c

2

SX (s)ds/(c − t )

has at most one solution. Furthermore, if (d(c ), c ) ∈ H , then d(c ) is differentiable with the derivative

(4.17)

where the equality is implied by d (c + d − 2t )dt = 0. Furthermore, the similar argument as that of (4.10) would imply c

t

Proof. By the proof of Lemma 4.3, we know that φ(d, c ) is ∂φ(d,c ) differentiable over G o with ∂ d > 0, then the implicit function theorem in Rudin (1976) shows that for any c ∈ (xα , x¯ ), the equation

d



c

Lemma 4.4. Under Assumption 4.1, we have |H | ≤ 1.

′

d

=

SX (s)ds dt > 0, t

where G is defined in (4.4).

∂φ(d(c ), c ) d (c ) = − ∂c

(c + d − 2t )SX (t )dt 



(4.18)

d

d

SX (s)ds − (c − d) d

c



where the inequality is derived by the fact that

H ,

y

=2

2

c



d

(4.14)

Further, taking the similar proof as that of (4.10), we have





FX (s)ds SX (t )dt

Building upon φ(d, c ), we define

c 2SX (c ) d (c + d − 2t )SX (t )dt ∂φ(d, c ) c = . ∂c ( d SX (t )dt )3

 c

FX (s)ds (FX (c ) − FX (t ))dt > 0.

The above equation, together with the fact ϕ(d, c ) > 0, implies ∂ϕ(d,c ) > 0. On the other hand, taking the derivatives of ϕ 2 (d, c ) in ∂c (4.13) w.r.t. d, straightforward algebra leads to

d





d

and

2(t − d)SX (t )dt −

2

decreases in t. Consequently, due to ϕ(d, c ) > 0, we have 0. The proof is therefore complete. 

c



d

c

t

×

F (t )dt d X c ϕ(d, c ) , √ φ(d, c ) − 1 d SX (t )dt



FX (s)ds

FX (s)ds SX (t )dt

d

 c 

c

SX (d)



FX (s)ds (FX (c )SX (t ) − FX (t )SX (c ))dt

c d

2

d

d

d

2 ∂φ(d, c ) = c SX (d) ∂d ( d SX (t )dt )3  c 2  − SX (t )dt

t

 c  ×

and



FX (s)ds

FX (s)ds SX (t )dt

d

d

Now, the residual task of this subsection is to study the optimal reinsurance model (1.2) with standard deviation principle for xα < x¯ . Like Section 4.1, the optimization problem will be solved under Assumption 4.1 for simplicity. First, we introduce two auxiliary functions:

φ(d, c ) ,

c

∂ϕ 2 (d, c ) =   ∂c t c

1

SX (t )dt

2



c

d(c )

(SX (d(c )) − SX (t ))dt



c c   V (d, c ) × SX (d) d 2(t − d)SX (t )dt − ( d SX (t )dt )2

Y. Chi / Insurance: Mathematics and Economics 51 (2012) 310–321

 c  c d

= c d



(SX (t ) − SX (c ))dt (SX (d) − SX (y))dy c y > 0, SX (t )dt × d d (SX (d) − SX (t ))dtSX (y)dy y

(4.19)

where d(c ) is written by d for notational simplicity and the last equality is derived by (4.16) and the following simplification V (d, c ) ,

c



(SX (t ) − SX (c ))dt

SX (d)

×

c



2(t − d)SX (t )dt −

d

− SX (c )

(SX (d) − SX (t ))dt ×  

d c



SX (d)

(c + d − 2t )SX (t )dt d

2(t − d)SX (t )dt −

d



d

+ SX (c )

c



SX (t )dt ×

SX (t )dt

2 

c

2

d

(c − t )SX (t )dt − SX (d)(c − d)2



d

c



c

d

 

 c  d

d

d

d

d

y

∂ W (d,c ) ∂c ∂ W (d,c ) ∂d

in (4.4). Furthermore, Fermat’s theorem shows that the minimum of W (d, c ) over G must be attainable at some stationary point or must lie on the boundary. If (d, c ) ∈ G o is a stationary point of W (d, c ), then we have ∂ W (d,c ) = ∂ W∂(cd,c ) = 0. By (4.21), they are equivalent to ∂d

φ(d, c ) = 1 + θ 2 and  c d

c−d

φ(x,c1 )−1

Moreover, c1 should satisfy



cs , min xα ≤ c ≤ x¯ : ϕ(x, c ) ≥ 1

α √  v ar ((X − d)+ ) =θ . E[(X − d)+ ]

E[(X − d)+ ] and

(4.20)

if if if

    

1 + θ2 >

|H | > 0, |H | = 0 and ds ̸= −∞, |H | = 0, ds = −∞ and

otherwise.

1

α

,

Proof. The proof is a slight modification to that of Proposition 4.2. By Theorem 3.3, the study of optimal reinsurance model (1.2) with standard deviation principle is simplified to deriving the minimum of W (d, c ) over 0 ≤ d ≤ xα ≤ c, where

+θ

 

1

α

− 1)E[(X − c )+ ]

c

2(t − d)SX (t )dt − d

Consequently, the (4.21), together with the increasing properties of ϕ(d, c ) in Lemma 4.3, implies c1 = cs , where cs is given by (4.20). • If (d, x¯ ) ∈ G2 is a minimum point of W (d, c ), then Fermat’s ∂ W (d,¯x) theorem, together with (4.21), implies ∂ d = 0 and ϕ(d, x¯ )− 1/α−1

 L(d,c ] (x), (d, c ) ∈ H    (x − ds )+  min{x, cs }





c ∈[xα ,¯x]

Proposition 4.4. Under Assumption 4.1, the ceded loss function that solves the optimal reinsurance problem (1.2) with standard deviation premium principle is given by

0

B , c ∈ [xα , x¯ ] : φ(x, c ) ≥ θ 2 + 1 .

W (x, c1 ) = min W (x, c ).

(1/α − 1) or c = x¯ ; θ 

It should be pointed out that ds = −∞ if x¯ = ∞.

W (d, c ) , d + E[(X − d)+ ] + (

.

• If W (d, c ) is minimal at (x, c1 ) ∈ G1 , then it follows from the ∂ W (x,c1 ) expression of in (4.21) that θ1 − √ 1 ≥ 0 such ∂d

We define

fs∗C (x) =

α

Therefore, all the stationary points of W (d, c ) over G o constitute the set H in (4.18). Next, we proceed to identify the minimum points of W (d, c ) located on the boundary of G . Recall that ∂ G = ∪3i=1 Gi where {Gi ; 1 ≤ i ≤ 3} are defined in (4.7), then we investigate the properties of the minimum points of W (d, c ) located in these three sets, separately.

that c1 ∈ B where



1

=

SX (t )dt

Thus, H must contain only one element if it is not empty. The proof is finally complete. 

ds , sup x < d < xα : x¯ − d ≤

(4.21)

= 0 for any c ≥ x¯ . Moreover, = 0 for any 0 ≤ d < x. the above equation implies Consequently, the minimum value of W (d, c ) over 0 ≤ d ≤ xα ≤ c is attainable in a compact subset of G , where G is given

t

(SX (d) − SX (y))dySX (t )dt   c − SX (c ) (c − y)(SX (d) − SX (y))dy d  c  c   c =2 SX (t )dt × SX (t )dt (SX (d) − SX (y))dy d d y   c − SX ( c ) (c − y)(SX (d) − SX (y))dy d   c  c  c =2 SX (t )dt × (SX (t ) − SX (c ))dt (SX (d) − SX (y))dy. SX (t )dt ×

=2

   1 1 ∂ W (d, c )   = θ (1 − SX (d)) −√ ;  ∂d θ φ(d, c) − 1  1/α − 1 ∂ W (d, c )   = θ SX (c ) ϕ(d, c ) − ,  ∂c θ

otherwise, if x¯ < ∞, we have

c



c

SX (t )dt ×

=

SX (t )dt

2 

d

c



c



Under Assumption 4.1, W (d, c ) is differentiable with partial derivatives

where φ(d, c ) and ϕ(d, c ) are defined in (4.13). If x¯ = ∞, we have 1 ∂ W (d,c ) ϕ(d, c ) > 1/α− for sufficiently large c such that > 0; θ ∂c

d



317

c



SX (t )dt d

≤ 0. By simplifying, we have d = ds where ds is defined θ in (4.20). • We demonstrate that no minimum points of W (d, c ) are located in G3 \ {(xα , xα )}. Specifically, if W (d, c ) is minimal at (d, xα ) ∈ G3 \ {(xα , xα )}, then Fermat’s theorem, together with (4.21), ∂ W (d,xα ) implies = 0 and ∂ W (∂dc,xα ) ≥ 0, which are equivalent ∂d to 1 xα − d φ(d, xα ) = 1 + θ 2 and  xα ≥ . α S ( t ) dt X d This leads to a contradiction: 1 xα − d 1

1

α

α

≤  xα d

SX (t )dt

.

SX (xα )

=

.

On the other hand, if (xα , c ) ∈ G3 \{(xα , xα )} is a minimum point ∂ W (xα ,c ) of W (d, c ), then from the first argument, we have ≤0 ∂d such that √φ(x 1,c )−1 ≥ θ1 . Consequently, we have α

ϕ(xα , c ) ≥ 2

<

=

1

θ



c − xα

c xα

SX (t )dt

1/α − 1

θ

,

 −1 >

1

θ



c − xα −1 (c − xα )SX (xα )



318

Y. Chi / Insurance: Mathematics and Economics 51 (2012) 310–321

that leads to W (xα , c ) > miny∈[xα ,¯x] W (xα , y). It is contradicted to the assumption that (xα , c ) is a minimum point. Finally, if 1 + θ 2 < α1 , then Lemma 4.2 implies

φ(xα , xα ) = ψ(xα ) =

1

> 1 + θ2

α

∂ W (xα ,xα ) ∂d

> 0. Thus, (xα , xα ) is not a minimum point of W (d, c ); otherwise, if 1 +θ 2 ≥ α1 , the above arguments show ∂ W (xα ,xα ) ≤ 0 and ∂ W (∂xαc ,xα ) ≥ 0. Consequently, the increasing ∂d properties of φ(d, c ) and ϕ(d, c ) in Lemma 4.3, together with the partial derivatives of W (d, c ) in (4.21), imply that W (d, c ) may attain the minimum at (xα , xα ). such that

Collecting all the above arguments, we get min CVaRα (Tf (X )) = f ∈C

min W (d, c ),

(d,c )∈Ks

where Ks , z ∈ R2+ : Ks (z) > 0 and





Ks (z) = I z = (x, cs ), cs ∈ B + I (z = (ds , x¯ )) + I (z ∈ H )







+ I z = (xα , xα ), 1 + θ ≥ 2

1

α



.

Further, if |H | > 0, then Lemma 4.4 implies H = {(dH , cH )} for some (dH , cH ) ∈ G o . In this case, we demonstrate α1 > 1 + θ 2 , ds = −∞ and cs ̸∈ B such that the minimum of W (d, c ) is attainable at (dH , cH ). Specifically, it follows from the definition of H in (4.18) that 1

α

cH − dH

− ( 1 + θ 2 ) =  cH dH

 cH =

dH

SX (t )dt

− φ(dH , cH )

(cH + dH − 2t )SX (t )dt c > 0, ( dHH SX (t )dt )2

(4.22)

where the inequality is derived by (4.17). Moreover, if ds ̸= −∞, then φ(ds , x¯ ) = 1 + θ 2 . As stated in the proof of Lemma 4.4,  c c −d(c ) is strictly increasing, then we have d(c ) SX (t )dt

cH − dH x¯ − ds >  cH = 1/α,  x¯ S (t )dt S (t )dt dH X ds X

ϕ(x, c ) ≥ ϕ(x, c0 ) = >



θ 1

θ

c0 − x

 c0 x



SX (t )dt

cH − dH

 cH dH

SX (t )dt

 −1  −1 =

1/α − 1

θ

, ∀c ∈ B ,

then it follows from the definition of cs in (4.20) that cs < c0 such that cs ̸∈ B . Else if H = ∅ and ds ̸= −∞, we have 1

α



1

ϕ(x, c ) =  φ(x, c ) − 1 ≥θ≥

1/α − 1

θ

c−x

c x

SX (t )dt

 −1 >

 φ(x, c ) − 1

, ∀c ∈ B ,

where the first inequality can be derived by taking the similar proof as that of (4.22). Moreover, we have xα ̸∈ B as φ(x, xα ) < φ(xα , xα ) = ψ(xα ) = 1/α ≤ 1 + θ 2 . Thus, we get cs ̸∈ B . Consequently, Ks = {(xα , xα )} and hence the optimal reinsurance is fs∗C (x) = 0. Otherwise, we have Ks = {(x, cs )} such that the minimum of W (d, c ) is attainable at (x, cs ). Furthermore, as W (x, cs ) = W (0, cs ), we can choose the optimal reinsurance policy fs∗C (x) = L(0,cs ] (x) = min{x, cs }. The proof is thus complete.  Example 4.2. Under the same assumptions as in Example 4.1, we have

α = 5%, SX (t ) = 1/(t + 1)2 , xα = 3.472 < x¯ = ∞.

ds = −∞,

We further set the loading coefficient θ equations

x=0

and

= 3. By solving the

     c+1 c−d 2(d + 1)2 (c + 1)2   ln − , 1 + θ 2 = φ(d, c ) = (c − d)2 d+1 c+1 1 c−d   = (c + 1)(d + 1),  = c α S (t )dt d X 0 < d < xα < c < ∞, we have d = 0.977 and c = 9.115 such that H = {(0.977, 9.115)}. Consequently, the ceded loss function that solves the optimal reinsurance model (1.2) with standard deviation premium principle is given by fs∗C (x) = min {(x − 0.977)+ , 8.138} . 5. Concluding remarks

that is contradicted to the definition of ds in (4.20). Furthermore, if B ̸= ∅, then there exists a c0 ∈ (cH , x¯ ] such that φ(x, c0 ) = 1 + θ 2 and B = [c0 , x¯ ]. Consequently, we have 1

have Ks = {(ds , x¯ )} and W (d, c ) attains the minimum at (ds , x¯ ). As L(ds ,¯x] (X ) = (X − ds )+ , a.s., the stop-loss reinsurance with the deductible ds is optimal. Else, if H = ∅, ds = −∞ and 1 + θ 2 ≥ α1 , we can show cs ̸∈ B . Specifically, if B ̸= ∅, we have

x¯ − ds − (1 + θ 2 ) ≥  x¯ − φ(ds , x¯ ) S (t )dt ds X  x¯ (¯x + ds − 2t )SX (t )dt d = s  > 0. 2 x¯ S ( t ) dt ds X

Moreover, the fact φ(ds , x¯ ) = 1 + θ 2 , together with the strictly increasing properties of φ(d, c ) in Lemma 4.3, leads to φ(x, c ) < 1 + θ 2 for any c ∈ [xα , x¯ ] such that B = ∅. Consequently, we

In this paper, we have investigated the VaR and CVaR based optimal reinsurance models with variance related premium principles, where both the ceded and retained loss functions are constrained to be increasing for the purpose of eliminating the moral hazard. We have found that the layer reinsurance is very robust in the sense that it is always optimal over both the VaR and CVaR criteria and all the variance related premium principles. To illustrate the applicability of our results, we have further shown how to derive the optimal deductible and the upper limit of the layer reinsurance by using the variance and standard deviation premium principles as examples. Due to the popularity of layer reinsurance in reinsurance market, we reaffirm that VaR and CVaR are suitable risk measures for an insurer’s risk-transfer decision if moral hazard is excluded from the design of the reinsurance contracts. Acknowledgments The author thanks a reviewer for his/her constructive suggestions that have significantly improved the paper. The author is also grateful to Sheldon X. Lin and Ming Zhou for valuable comments. This research was supported by National Natural Science Foundation of China under grant 11001283, the MOE (China) Project of Key Research Institute of Humanities and Social Sciences at Universities (No. 11JJD790004), and the 2011 research grant from China Institute for Actuarial Science at Central University of Finance and Economics.

Y. Chi / Insurance: Mathematics and Economics 51 (2012) 310–321

319

Appendix A. The proofs of Theorems 3.1 and 3.3

by Chi and Tan (2011a), the admissible ceded loss function defined in (1.5) is increasing and Lipschitz continuous, i.e.

In this section, we provide the proofs of Theorems 3.1 and 3.3 using the following approach: for any given ceded loss function f ∈ C , we construct a layer reinsurance treaty such that it is better than f in the sense to minimize the total risk exposure of an insurer under VaR or CVaR risk measure. Before giving the details of the proofs, we introduce several useful lemmas. First, we show that both the mean and variance of a random variable can be reduced by imposing an upper limit.

0 ≤ f (x2 ) − f (x1 ) ≤ x2 − x1 ,

Lemma A.1. For any random variable ξ with finite expectation, we have

E[min{ξ , m}] ≤ E[ξ ],

v ar (min{ξ , m}) ≤ v ar (ξ ),

∀m ∈ R.

(A.1)

Proof. It is trivial that the former equation in (A.1) holds. Thus, we only need to prove the latter one. The result is trivial for the case v ar (ξ ) = ∞. On the other hand, if v ar (ξ ) < ∞, note that x = min{x, m} + (x − m)+ for any m ∈ R, then we have

 k˜ 0 (x) = (x − (VaRα (X ) − f (VaRα (X ))))+ ≤ f (x); k˜ (x) = x − (x − f (VaRα (X )))+  f (VaRα (X )) = min{x, f (VaRα (X ))} ≥ f (x).

P(kf (X ) ≤ y) = P(X ≤ a∗ , kf (X ) ≤ y)

= P(X ≤ y) ≤ P(f (X ) ≤ y). On the other hand, for any y ≥ a∗ , we have

P(kf (X ) > y) = P(X ≥ a0 , kf (X ) > y)

= P(X > VaRα (X ), f (X ) > y) + P(a0 ≤ X ≤ VaRα (X ), X + f (VaRα (X )) − VaRα (X ) > y) ≤ P(X > VaRα (X ), f (X ) > y) + P(a0 ≤ X ≤ VaRα (X ), f (X ) > y) ≤ P(f (X ) > y),

cov(min{ξ , m}, (ξ − m)+ ) , E[min{ξ , m}(ξ − m)+ ] − E[min{ξ , m}]E[(ξ − m)+ ]

= mE[(ξ − m)+ ] − E[min{ξ , m}]E[(ξ − m)+ ] = E[(m − ξ )+ ]E[(ξ − m)+ ] ≥ 0.

where a0 , a∗ + VaRα (X ) − f (VaRα (X )) and the first inequality is derived by (A.5). The final result follows from Lemma A.2. The proof is therefore complete. 



To proceed, we rewrite Theorem 3.2.4 in Rolski et al. (1999) as the following lemma. Lemma A.2. Provided that the random variables Y1 and Y2 have finite expectations, if they satisfy

SY1 (t ) ≤ SY2 (t ),

FY1 (t ) ≤ FY2 (t ),

t < t0

and

t ≥ t0

(A.2)

for some t0 ∈ R, then Y1 ≤cx Y2 , i.e.

E[G(Y1 )] ≤ E[G(Y2 )] for any convex function G(x) provided the expectations exist. Using the above lemma, we obtain the following result. Lemma A.3. For any ceded loss function f ∈ C , denote x − (x − a)+ + (x − (a + VaRα (X ) − f (VaRα (X ))))+ , 0 ≤ x ≤ VaRα (X ), (A.3) f (x), x > VaRα (X ),

 kf (x) ,

(A.5)

Furthermore, since k˜ a (x) is increasing and continuous in a, then there exists an a∗ satisfying 0 ≤ a∗ ≤ f (VaRα (X )) such that E[k˜ a∗ (X )] = E[f (X )]. We set kf (x) = k˜ a∗ (x) and hence kf ∈ C . Next, we show kf (X ) ≤cx f (X ) by verifying (A.2). Specifically, for any y < a∗ , we have

where the last inequality is derived by

E[Y1 ] = E[Y2 ],

(A.4)

then for any 0 ≤ x ≤ VaRα (X ), we have

v ar (ξ ) = v ar (min{ξ , m}) + v ar ((ξ − m)+ ) + 2cov(min{ξ , m}, (ξ − m)+ ) ≥ v ar (min{ξ , m}),

The proof is therefore complete.

∀ 0 ≤ x1 ≤ x2 ,

where 0 ≤ a ≤ f (VaRα (X )) is determined by E[kf (X )] = E[f (X )], then kf ∈ C and kf (X ) ≤cx f (X ).

Now, we are ready to carry out the proof of Theorem 3.1. Proof of Theorem 3.1. For any f ∈ C , let f1 (x) , min{f (x), f (VaRα

(X ))}, then we have f1 ∈ C

and f1 (VaRα (X )) = f (VaRα (X )).

Building upon the ceded loss function f1 , we follow the way of (A.3) to construct kf1 (x), then Lemma A.3 implies kf1 (VaRα (X )) = f (VaRα (X )) and kf1 (X ) ≤cx f1 (X ). Thus, we have v ar (kf1 (X )) ≤ v ar (f1 (X )) < ∞ because f1 (X ) ≤ f (VaRα (X )) ≤ VaRα (X ) < ∞. According to the definition of kf (x) in (A.3), we have kf1 (VaRα (X )) − kf1 (x)

 =

b − min{x, a}, (VaRα (X ) − x)+ ,

0 ≤ x ≤ VaRα (X ) − b + a, x > VaRα (X ) − b + a

for some a satisfying 0 ≤ a ≤ f (VaRα (X )) and b = f (VaRα (X )), then we can find a b∗ satisfying b − a ≤ b∗ ≤ b such that the layer reinsurance treaty f˜ (x) = L(VaRα (X )−b∗ ,VaRα (X )] (x) has the following property





E f˜ (VaRα (X )) − f˜ (X ) = E kf1 (VaRα (X )) − kf1 (X ) .





Furthermore, by verifying (A.2) with t0 = b∗ , Lemma A.2 implies f˜ (VaRα (X )) − f˜ (X ) ≤cx kf1 (VaRα (X )) − kf1 (X ).

Proof. First, we show that kf (x) is well-defined. Denote the func-

(A.6)

tion on the right-hand side of (A.3) by k˜ a (x), as it is parameterized by a. Then we demonstrate

Finally, recall that the reinsurance premium is calculated by a variance related principle (1.4), then the translation invariance property of VaR risk measure implies

k˜ 0 (x) ≤ f (x) ≤ k˜ f (VaRα (X )) (x),

VaRα (Tf (X )) = VaRα (Rf (X )) + E[f (X )] + g (v ar (f (X )))

∀x ≥ 0.

Specifically, the result is trivial for x > VaRα (X ), then we only need to prove the above inequality for 0 ≤ x ≤ VaRα (X ). As pointed out

= VaRα (X ) − f (VaRα (X )) + E[f (X )] + g (v ar (f (X ))) ≥ VaRα (X ) − f1 (VaRα (X )) + E[f1 (X )] + g (v ar (f1 (X )))

320

Y. Chi / Insurance: Mathematics and Economics 51 (2012) 310–321

≥ VaRα (X ) − kf1 (VaRα (X )) + E[kf1 (X )] + g (v ar (kf1 (X ))) = VaRα (X ) − E[kf1 (VaRα (X )) − kf1 (X )]    + g v ar kf1 (VaRα (X )) − kf1 (X ) ≥ VaRα (X ) − E[f˜ (VaRα (X )) − f˜ (X )]    + g v ar f˜ (VaRα (X )) − f˜ (X )

Then taking the similar arguments as that for (A.6), we can construct a layer reinsurance treaty fˆ (x) , L(bˆ ∗ ,c ] (x) for some bˆ ∗ satisfying f2 (VaRα (X )) − a ≤ VaRα (X ) − bˆ ∗ ≤ f2 (VaRα (X )) such that fˆ (VaRα (X )) − fˆ (X ) ≤cx f2 (VaRα (X )) − f2 (X ).

= VaRα (Tf˜ (X )) = Q (VaRα (X ) − b∗ ), where Q (d) is defined in (3.4), the second equality is derived by (2.4) and (A.4), and the three inequalities are implied by Lemmas A.1 and A.3 and (A.6), respectively. The proof is therefore complete. 

As a result, recall that the reinsurance premium is calculated by a variance related principle (1.4), then the translation invariance property of CVaR risk measure implies CVaRα (Tf (X )) = CVaRα (X ) − CVaRα (f (X ))

+ E[f (X )] + g (v ar (f (X ))) ≥ CVaRα (X ) − CVaRα (f2 (X )) + E[f2 (X )] + g (v ar (f2 (X ))) = CVaRα (X ) + VaRα (X )  α 1 − min{VaRs (X ), c }ds α 0 − E[f2 (VaRα (X )) − f2 (X )] + g (v ar (f2 (VaRα (X )) − f2 (X ))) ≥ CVaRα (X ) + VaRα (X )  α 1 − min{VaRs (X ), c }ds α 0 − E[fˆ (VaRα (X )) − fˆ (X )]

Before giving the proof of Theorem 3.3, we introduce another useful lemma. Lemma A.4. For any f ∈ C , let f (x), 0 ≤ x ≤ VaRα (X ), min {x + f (VaRα (X )) − VaRα (X ), M } , x > VaRα (X ),

 hf (x) ,

(A.7)

where M ≥ f (VaRα (X )) is determined by CVaRα (f (X )) = CVaRα (hf (X )), then we have hf (X ) ≤cx f (X ). Proof. First, the existence of hf (x) can be verified by taking the same proof as that of Theorem 3.2 in Chi and Tan (2011b). Next, we demonstrate E[hf (X )] = E[f (X )]. Specifically,

E[f (X )] = E[VaRU (f (X ))] = E[f (VaRU (X ))] α

 =

f (VaRs (X ))ds +

α

0

= α × CVaRα (f (X )) +

= α × CVaRα (hf (X )) +

f (VaRs (X ))ds hf (VaRs (X ))ds

CVaRα (fˆ (X )) =

1

 α

α

hf (VaRs (X ))ds = E[hf (X )],

Now, we close this section by providing the proof of Theorem 3.3. Proof of Theorem 3.3. For any f ∈ C , it is trivial that hf (x) defined in (A.7) is an admissible ceded loss function, i.e. hf ∈ C . Further, building upon hf (x), we define f2 (x) , khf (x) by following the way of (A.3), then we have CVaRα (f2 (X )) = CVaRα (hf (X )) = CVaRα (f (X )) f2 (x) = L(0,a] (x) + L(b,c ] (x), (A.8)

Moreover, using Lemmas A.3 and A.4, we have f2 (X ) ≤cx hf (X ) ≤cx f (X ).

(A.9)

Further, by virtue of the representation of f2 (x) in (A.8), we have f2 (VaRα (X )) − f2 (x)

=

f2 (VaRα (X )) − min{x, a}, VaRα (X ) − min{x, c },

0 ≤ x ≤ b, x > b.



α

α

0

= CVaRα (X ) − bˆ ∗ −

1

α

E[(X − c )+ ], d

where the last equality is derived by X ∼ VaRU (X ). Thus, by straightforward algebra, we have CVaRα (Tfˆ (X )) = W (bˆ ∗ , c ), where W (d, c ) is defined in (3.10). Finally, if VaRα (X ) = ess sup X , we have W (d, c ) = W (d, VaRα (X )) = d + w1 (d) + g (w2 (d) − w12 (d))

= Q (d) for any 0 ≤ d ≤ VaRα (X ) ≤ c, where Q (d) is given by (3.4). As a consequence, the final result follows from Theorem 3.1 and (3.9). The proof is therefore complete. 

and f2 ∈ C is of the form for some 0 ≤ a ≤ b ≤ VaRα (X ) ≤ c .

1

(VaRs (X ) − bˆ ∗ )+ ds  α 1 − (VaRs (X ) − c )+ ds α 0  1 1 ∗ ˆ = CVaRα (X ) − b − (VaRs (X ) − c )+ ds α 0

1



where U is uniformly distributed on [0, 1], the first equality is derived by the fact that f (X ) and VaRU (f (X )) are equal in distribution, and the second equality is implied by (2.4). Finally, by verifying (A.2) with t0 = M , we get the result from Lemma A.2. The proof is therefore complete. 



+ g (v ar (fˆ (VaRα (X )) − fˆ (X ))) = CVaRα (Tfˆ (X )), where the second equality is implied by (A.10), and the two inequalities are derived by (A.9) and (A.11), respectively. Furthermore, since bˆ ∗ ≤ VaRα (X ) ≤ c, then we have

1



(A.11)

(A.10)

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