K. C. Cheung and K. C. J. Sung are at the Department of Statistics, and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. S. C. P. Yam is with the Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. K. C. Cheung can be contacted via e-mail: kccg@hku.hk. The authors thank Jan Dhaene, Ludger RÜschendorf, and many seminar and conference participants, for pointing out more accurate references, and for their supportive comments and inspiring suggestions. K. C. Cheung acknowledges the financial support of the Research Grants Council of HKSAR (Project No. HKU701409P). S. C. P. Yam acknowledges financial support from The Hong Kong RGC GRF 404012 with the project title “Advanced Topics in Multivariate Risk Management in Finance and Insurance.” S. C. P. Yam also expresses his sincere gratitude to the hospitality of both Hausdorff Center for Mathematics of the University of Bonn and Mathematisches Forschungsinstitut Oberwolfach (MFO) in the German Black Forest during the preparation of the present work.

Feature Articles

# Risk-Minimizing Reinsurance Protection For Multivariate Risks

Article first published online: 14 FEB 2013

DOI: 10.1111/j.1539-6975.2012.01501.x

© The Journal of Risk and Insurance, 2014

Additional Information

#### How to Cite

Cheung, K. C., Sung, K. C. J. and Yam, S. C. P. (2014), Risk-Minimizing Reinsurance Protection For Multivariate Risks. Journal of Risk and Insurance, 81: 219–236. doi: 10.1111/j.1539-6975.2012.01501.x

#### Publication History

- Issue published online: 13 FEB 2014
- Article first published online: 14 FEB 2013

- Abstract
- Article
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- Cited By

### Abstract

- Top of page
- Abstract
- Introduction
- Risk Measures and Problem Formulation
- Relevance of Comonotonicity
- Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
- Further Discussion on the Optimal Deductibles
- Concluding Remarks
- Appendix
- References

In this article, we study the problem of optimal reinsurance policy for multivariate risks whose quantitative analysis in the realm of general law-invariant convex risk measures, to the best of our knowledge, is still absent in the literature. In reality, it is often difficult to determine the actual dependence structure of these risks. Instead of assuming any particular dependence structure, we propose the minimax optimal reinsurance decision formulation in which the worst case scenario is first identified, then we proceed to establish that the stop-loss reinsurances are optimal in the sense that they minimize a general law-invariant convex risk measure of the total retained risk. By using minimax theorem, explicit form of and sufficient condition for ordering the optimal deductibles are also obtained.

### Introduction

- Top of page
- Abstract
- Introduction
- Risk Measures and Problem Formulation
- Relevance of Comonotonicity
- Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
- Further Discussion on the Optimal Deductibles
- Concluding Remarks
- Appendix
- References

In spite of frequent substantial financial crises in the recent three decades, risk management has become the major focus in finance from both theoretical and practical perspectives. Artzner et al. (1999) proposed an axiomatic approach for coherent measures of risk that satisfies the properties of monotonicity, positive homogeneity, translation invariance, and subadditivity. Besides, they also provided a representation theorem for those risk measures; a similar result, in a different context, has been also obtained in Huber (1981). Since then, many scholars have made various important contributions along this direction. For example, as a generalization of coherent measures of risk, the notion of convex risk measures was introduced in the works by Frittelli and Rosazza Gianin (2002), Föllmer and Schied, 2002, 2005), and Heath and Ku (2004). As an important example, average value at risk (AV@R) is further explored in the works by Acerbi and Tasche (2002), Delbaen (2002), and Rockafellar and Uryasev (2001). If one imposes an additional axiom that the same value will be assigned to two risky positions with a common distribution, a property known as law-invariant, representation result was obtained by Kusuoka (2001) in the coherent case, and by Frittelli and Rosazza Gianin (2004) in the convex case. Further properties of law-invariant risk measures can also be found in Jouini et al. (2006) and Song and Yan (2009). Although the study of risk measures in the above works is essentially developed for bounded risks, generalizations to the unbounded risks have been investigated recently, such as in Filipovíc and Svindland (2012). In addition, as a special case of law-invariant convex risk measures, distortion risk measures enjoy a wide application in practice. For a survey of these results and their connections with comonotonicity, see, for example, Dhaene et al. (2006).

Because of the popularity in both interest in and actual use of various risk measures, many classical insurance-related problems have been retreated in the risk measure paradigm. For example, Dhaene et al. (2005) considered the optimal asset allocation problem, Schied (2007) dealt with portfolio selection problem, and Annaert et al. (2009) investigated the efficiency of the classical portfolio insurance problem under value at risk (V@R) or AV@R. Motivated by the seminal works of Arrow (1963) and Borch (1960), many researchers aim at seeking for optimal designs of (re)insurance contract so that the risk measure of the retained loss can be minimized. For example, Cai and Tan (2007), Cai et al. (2008), and Cheung (2010) considered the optimal increasing convex indemnity schedule which minimizes the V@R and the conditional tail expectation (CTE) of the retained cost. Another interesting study is given by Balbás et al. (2009) in which stop-loss contracts were shown to be optimal under certain scenarios; also see Balbás (2011) for the relevance of stop-loss contracts in optimal reinsurance problems involving coherent risk measures. Recently, Cheung et al. (Forthcoming) considered the optimal reinsurance decision problem under general law-invariant risk measures, including V@R, CTE, and general convex risk measures. Based on real-life experience and heuristics, truncated stop-loss reinsurance is obviously less welcome by both reinsurers and buyers because it may cause potential moral hazard or swindle. Unlike to some existing works that allow truncated stop-loss reinsurance to be optimal, similar to our previous work in Cheung et al. (Forthcoming), we now only consider a reinsurance contract to be feasible if it satisfies the two properties, namely: (1) an additional unit of loss cannot result in more than a unit increment of indemnity claim, and (2) for any additional loss claim, at least not lesser compensation could be requested.

The present research studies a single-period risk-measure-based optimal reinsurance decision problem for a basket of *n* insurable risks. Although there is some theoretical work on characterization of measures of multivariate risks, such as Ekeland et al. (2012) and Rüschendorf, 2006, 2012), apart from a few applications and implementations in the financial context (such as in Kiesel and Rüschendorf, 2008, 2001), similar consideration in the insurance literature is still rare. To the best of our knowledge, risk-minimizing reinsurance design in a multivariate setting is still absent in the literature. Whenever one studies a problem involving more than one risk, a knowledge in the joint distribution of the multivariate risks is essential. In reality, acquiring the dependence structure is usually difficult if not impossible; on the other hand, information of marginal distributions is relatively tractable. In this article, we assume that marginal distribution of each risk is known but not their dependence structure. Instead of assuming any particular dependence structure, we propose a minimax formulation in which the first step is to identify the least favorable dependence structure that would give rise to the maximum level of risk measure for any fixed reinsurance contracts. We then proceed to analyze the optimal reinsurance problem as if this were the actual dependency structure. This leads to a nonstandard (nonconvex) saddle-point problem.

By using a recently proposed geometric method (Cheung, 2010; Cheung et al., Forthcoming), we established the optimality of stop-loss reinsurance for each single risk; a similar result under utility framework for bivariate risks was obtained in Jouini et al. (2008). Under the constraint of fixed premium, by first identifying the special form of the optimal solution, a twist of the application of the classical minimax theorem is adopted to establish Equation (5) that characterizes the optimal values of deductibles by the marginal distribution functions of the risks. Our result is in the same spirit as the work by Bühlmann and Jewell (1979) in which Pareto-optimal allocation of deductibles was identified under the expected utility theory.

This article is organized as follows. In the “Risk Measures and Problem Formulation” section, we present some preliminary results in the theory of law-invariant convex risk measures and formulates the minimax optimal reinsurance decision problem. The “Relevance of Comonotonicity” section shows the comonotonicity of the optimal reinsurance contracts. The optimality of stop-loss contracts under law-invariant convex risk measure is shown in the “Optimal Reinsurance Under General Law-Invariant Convex Risk Measure” section. The “Further Discussion on the Optimal Deductibles” section investigates the magnitudes of the corresponding optimal deductibles that are characterized by Equation (5), and the “Concluding Remarks” section concludes the article.

### Risk Measures and Problem Formulation

- Top of page
- Abstract
- Introduction
- Risk Measures and Problem Formulation
- Relevance of Comonotonicity
- Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
- Further Discussion on the Optimal Deductibles
- Concluding Remarks
- Appendix
- References

A risk bearer faces a basket of *n* insurable risks that are modeled as nonnegative essentially bounded random variables defined on an atomless probability space . For each *k* = 1, 2, …, *n*, to avoid unnecessary technical details, we assume that the survival function *S _{k}* of

*X*is known, and is continuous and strictly decreasing on , where denotes the essential supremum of

_{k}*X*. The cumulative distribution function 1 −

_{k}*S*of

_{k}*X*is denoted by

_{k}*F*.

_{k}Fix any α ε (0, 1]. For any random variable *Y*, recall that the AV@R of *Y* at confidence level 1 − α is defined as:

where .

A functional ρ: is called a law-invariant convex risk measure if the following properties are satisfied for any *Y*_{1}, *Y*_{2} ε *L*^{∞}:

*Monotonicity:*If*Y*_{1}≤*Y*_{2}, then ρ (*Y*_{1}) ≤ ρ (*Y*_{2}).*Translation Invariance:*For any , ρ (*Y*_{1}+*m*) = ρ (*Y*_{1}) +*m*.*Convexity:*For any γ ε [0, 1], ρ (γ*Y*_{1}+ (1 − γ)*Y*_{2}) ≤ γ ρ (*Y*_{1}) + (1 − γ) ρ (*Y*_{2}).*Law Invariance:*If*Y*_{1}and*Y*_{2}have the same distribution under , then ρ (*Y*_{1}) = ρ (*Y*_{2}).

From (Föllmer and Schied, 2005; Jouini et al., 2006; Song and Yan, 2009), we have the following useful representation result for law-invariant convex risk measures:

###### Lemma 1

*If* *is atomless, then any law-invariant convex risk measure* ρ *can be expressed as*

- (1)

*where* *is the set of probability measures on* (0, 1], *and* *is a proper convex function*.1

###### Remark 1

For law-invariant convex risk measures on unbounded (i.e., ) but integrable risks, that is, in *L*^{1}, a recent work by Filipovíc and Svindland (2012) showed that under a mild condition of lower semicontinuity of a convex risk measure ρ, the same representation for ρ as stated in Lemma 1 still holds; hence, all our following results in the rest of this article would still be valid under this general setting.

Because β is proper and convex, the set is nonempty and convex. Hence, Equation (1) can be written as follows:

- (2)

To transfer and reduce the risk level, it is common for the risk bearer to seek reinsurance protection for each insurable risk. The objective of this article is to determine the optimal design of reinsurance contracts so that the risk exposure can be minimized. To this end, we let be a given reinsurance contract on *X _{k}*, where

*I*(

_{k}*x*) represents the payment received from the insurer if

_{k}*x*is the realized loss amount of

_{k}*X*. To avoid moral issues or insurance swindles, any feasible reinsurance contract should satisfy the following properties: (1) an additional unit of loss cannot result in more than a unit increment of indemnity claim and (2) as the buyer expected, for any additional loss claim, at least not lesser compensation that would be requested. Mathematically speaking, we impose the following assumptions on the feasible reinsurance contract

_{k}*I*:

_{k}*I*(_{k}*y*) −_{k}*I*(_{k}*x*) ≤_{k}*y*−_{k}*x*, for any 0 ≤_{k}*x*≤_{k}*y*,_{k}*I*(0) = 0 and_{k}*I*is increasing._{k}

Under these two conditions, it is seen that any feasible reinsurance contract *I _{k}* is continuous, and

*x*↦

_{k}*x*−

_{k}*I*(

_{k}*x*) is increasing. The class of all feasible reinsurance contracts for the risk

_{k}*X*is denoted by . Define . Suppose that the reinsurance premium is calculated under the expected value (actuarial pricing) principle with risk loading θ > 0. If are bought for the

_{k}*n*risks, the total reinsurance premium equals

- (3)

and the total retained loss of the insured changes from *X*_{1} + ⋯ + *X _{n}* to

It is natural and rational for the risk bearer to choose reinsurance contracts in an optimal way, according to a certain optimality criterion. In this article, we assume that reinsurance contracts are chosen so as to minimize a general law-invariant convex risk measure, subject to the premium constraint *P* ≤ *P*_{0}, where *P*_{0} is exogenously given and satisfies

However, this problem is not well posed without an explicit identification of the dependence structure among the *n* risks. Unfortunately, the exact dependence structure is often unknown, or too difficult to work with even if it is known. So we assume that the only information available to the risk bearer about the risks *X*_{1}, …, *X _{n}* is that

where denotes the Fréchet space of all *n*-dimensional random vectors with marginal distributions *F*_{1}, …, *F _{n}*, but the joint distribution is unknown. In view of this consideration, the risk bearer would naturally interest in the following optimization problem:

###### Problem 1

where ρ is a given general law-invariant convex risk measure, and *P* satisfies (3).

There is a natural and economic justification behind this minimax formulation. Because the marginal distribution of each *X _{k}* is known but not the dependence structure (joint distribution), the insured prudently assumes the worst scenario, that is, the dependence structure that gives rise to the largest level of risk for any reinsurance contracts. Such a minimax formulation was also adopted in other actuarial and financial contexts, such as in Cheung, 2006, 2007).

### Relevance of Comonotonicity

- Top of page
- Abstract
- Introduction
- Risk Measures and Problem Formulation
- Relevance of Comonotonicity
- Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
- Further Discussion on the Optimal Deductibles
- Concluding Remarks
- Appendix
- References

Before solving Problem 1, and in particular its inner maximization part, it is necessary to recall briefly the notion of comonotonicity. A random vector (*Y*_{1}, …, *Y _{n}*) is said to be

*comonotonic*if there exist a random variable

*U*and

*n*increasing functions

*g*

_{1}, …,

*g*such that

_{n}In particular, we may choose *U* to be any uniform(0, 1) random variable, and each *g _{i}* to be , which is the left-continuous inverse distribution function of

*Y*. Given any distribution functions (

_{i}*F*

_{1}, …,

*F*), it follows from the definition that there always exists a comonotonic random vector inside , which can be constructed explicitly as , where

_{n}*U*is a uniform (0, 1) random variable.

It is clear from the definition that comonotonicity is preserved upon increasing transformations: if (*Y*_{1}, …, *Y _{n}*) is comonotonic and if Γ

_{1}, …, Γ

_{n}are increasing functions, then (Γ

_{1}(

*Y*

_{1}), …, Γ

_{n}(

*Y*)) is comonotonic. Comonotonicity represents the strongest possible positive dependence structure in that components of a comonotonic random vector are always moving in the same direction. Finally, we remark that is comonotonic additive:

_{n}whenever (*Y*_{1}, …, *Y _{n}*) is comonotonic. More details about the theory of comonotonicity can be found in Denuit et al. (2005) and Dhaene et al. (2002).

Now we are ready to state our next result, which solves the maximization part of Problem 1.

###### Proposition 1

*Problem 1 is equivalent to*

*where* *is comonotonic. In other words, comonotonicity is the worst dependence structure that solves the “max” part of Problem 1*.

###### Proof

Fix any . From Lemma 1 and the translation invariance property of ρ,

According to the subadditivity of , the right-hand side is less than or equal to

As each has the same distribution as *X _{k}* and is law-invariant, we can replace

*Z*by in the previous expression. The right-hand side then equals

_{k}where the first equality follows from the comonotonicity of and the comonotonic additivity of . A similar result can be found in Rüschendorf (2012). Q.E.D.

### Optimal Reinsurance Under General Law-Invariant Convex Risk Measure

- Top of page
- Abstract
- Introduction
- Risk Measures and Problem Formulation
- Relevance of Comonotonicity
- Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
- Further Discussion on the Optimal Deductibles
- Concluding Remarks
- Appendix
- References

From the proof of Proposition 1, we have

To find optimal reinsurance contracts that minimize the left-hand side, we first consider the problem of minimizing

over . Because the map *x* ↦ *x* − *I _{k}*(

*x*) is increasing and continuous, we have

where .

To simplify our notation, we denote the stop-loss reinsurance contract with a deductible of *d* by , so that for *x* ≥ 0.

###### Proposition 2

*For any* , *there is a scalar* *such that*

*Furthermore, the inequality is strict if* *a _{k}* ≥

*d*

_{k}*and*.

###### Proof

For any , there is a unique such that by intermediate value theorem. We define

which is the point where the curve *x* − (*x* − *d _{k}*)

_{+}upcrosses

*x*−

*I*(

_{k}*x*). Obviously, . If , then we must have and there is nothing to prove. Suppose that (and hence

*d*> 0), we have to consider the following cases:

_{k}**Case 1:** If , then

**Case 2:** If , then

Moreover, the last inequality is strict if *a _{k}* ≥

*d*. Q.E.D.

_{k}Using Proposition 1 and Proposition 2, we can now solve Problem 1.

###### Proposition 3

*For any* , *there is a scalar* *for each* *k* = 1, 2, …, *n* *such that* *and* *for all* *k*, *where*

###### Proof

It follows from Proposition 2 that for each *k* = 1, 2, …, *n*, there is a scalar such that and

for any α ε (0, 1]. Hence, for any probability measure μ *on* (0, 1], we have

which implies that

as required. Q.E.D.

###### Remark 2

From Propositions 2 and 3, if β (μ) = ∞ whenever μ is not equivalent to the Lebesgue measure on (0, 1], then if .

From Proposition 3, we see that for any given reinsurance contracts (*I*_{1}, …, *I _{n}*), we can always achieve a lower risk by replacing each

*I*by the corresponding stop-loss contract without increasing the premium. Thus, it is sufficient to restrict our attention to stop-loss reinsurance contracts, and hence our original infinite dimensional optimization problem (1) is now reduced to a finite dimensional problem.

_{k}###### Theorem 1

*There is an optimal solution* *of the following minimization problem*:

*Moreover*, *is an optimal solution to Problem 1*.

###### Proof

By Proposition 1,

which is lower semicontinuous in (*d*_{1}, …, *d _{n}*). As the feasible set is compact, the result follows. Q.E.D.

### Further Discussion on the Optimal Deductibles

- Top of page
- Abstract
- Introduction
- Risk Measures and Problem Formulation
- Relevance of Comonotonicity
- Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
- Further Discussion on the Optimal Deductibles
- Concluding Remarks
- Appendix
- References

Although the results in the previous sections conclude that stop-loss reinsurance contracts are optimal for the multivariate Problem 1, further properties of the level of the optimal deductibles can be obtained by applying the minimax theorem in view of Proposition 1 and the representation theorem in Lemma 1. We first recall the following classical minimax theorem by Fan (1953), which does not require any canonical convexity structure on the underlying set but on the function only.

Given two arbitrary sets Ξ and Θ (which are not necessarily topologized), let be a real-valued function defined on the product set Ξ × Θ. We say that *f* is convex on Ξ if for any two elements ξ_{1}, ξ_{2} ε Ξ and any two nonnegative numbers γ_{1}, γ_{2} with γ_{1} + γ_{2} = 1, there exists an element ξ_{0} ε Ξ such that for every θ ε Θ, *f*(ξ_{0}, θ) ≤ γ_{1}*f*(ξ_{1}, θ) + γ_{2}*f*(ξ_{2}, θ). The concavity of *f* on Θ can be defined in a similar way.

###### Theorem 2

*Given a compact Hausdorff space* Ξ_{1} *and an arbitrary set* Ξ_{2}, *let* *be a real-valued function such that for every* ξ_{2} ε Ξ_{2}, *f*(*, ξ_{2}) *is lower semi-continuous on* Ξ_{1}. *If* *f* *is convex on* Ξ_{1} *and concave on* Ξ_{2}, *then we have*

To solve Problem 1, we first fix the premium *P* and consider the following optimization problem in accordance with Lemma 1 and Proposition 1:

###### Problem 2

subject to for some fixed *P* ε [0, *P*_{0}].

The original minimization Problem 1 is equivalent to a two-step optimization: (1) solve Problem 2 for each fixed *P* ε [0, *P*_{0}]; (2) then optimize the value function with respect to *P* over [0, *P*_{0}].

Define for any fixed *P* ε [0, *P*_{0}]. To solve Problem 2 by applying Theorem 2, we define Ξ_{1} ≜ Λ and and identify with the objective functional in Problem 2. We now check that all the conditions in the statement of Theorem 2 are satisfied. First, both feasible sets Ξ_{1} and Ξ_{2} are convex. Second, is compact because each is compact by Tychonoff Theorem; indeed, the family of all 1-Lipschitz functions on each compact interval is a compact set under the usual supremum norm. Therefore, as a closed subset of , Ξ_{1} is also compact Hausdorff. Finally, the functional

is continuous and convex in (*I*_{1}, …, *I _{n}*) ε Ξ

_{1}for each μ ε Ξ

_{2}, and concave in μ ε Ξ

_{2}for each (

*I*

_{1}, …,

*I*) ε Ξ

_{n}_{1}by the convexity of the penalty function β.

By applying Theorem 2, the minimum sign and the supremum sign in Problem 2 can be interchanged and it suffices to first consider the following minimization problem:

###### Problem 3

subject to , where is a given probability measure.

The next theorem suggests an optimal solution of Problem 3. Let *d** satisfy the equation

and define the scalars by

- (4)

Clearly, we have the relation:

- (5)

###### Theorem 3

*is an optimal solution of Problem 3*.

For the proof of this theorem, see the Appendix.

###### Remark 3

Recall that for any given *d**, if we define as in Equation (4), then and

It follows that is the optimal solution to the following optimal capital allocation problem:

For a proof of and further discussion on this result, see Dhaene et al. (2002), Meilijson and Nadas (1979), and Rüschendorf (2005). In a broad sense, Theorem 3 is a nonlinear generalization of this classical optimal capital allocation problem.

We use the notation to denote the family of all deductibles satisfying:

*S*_{1}(*d*_{1}) =*S*_{2}(*d*_{2}) = … =*S*(_{n}*d*);_{n}- ;
- for each
*k*.

Note that, is a singleton for a fixed premium *P*. By taking , , and to be the same objective functional as that in Problem 2, we can interchange back the minimum sign and the supremum sign, namely:

Based on this observation, we now obtain the next key result of this section.

###### Theorem 4

*There is an optimal solution* , *which also satisfies Equation* (5), *of the following minimization problem*:

*Moreover*, *is an optimal solution to Problem 1*.

###### Proof

The proof is similar to that of Theorem 1 and uses the fact that is compact. Q.E.D.

The following corollary says that if any two of these *n* insurable risks *X*_{1}, …, *X _{n}* can be ordered stochastically, the corresponding optimal deductibles preserve the same order. The result complements one of the key results (Proposition 3, which is in the framework of expected utility theory) in Cheung (2007). Recall that

*Y*

_{1}is said to be smaller than

*Y*

_{2}in stochastic dominance, denoted as

*Y*

_{1}≼

_{ST}

*Y*

_{2}, if for any 0 < α < 1, we have .

###### Corollary 1

*Given that any two risks* *X _{i}*

*and*

*X*

_{j}*satisfy*

*their corresponding optimal deductibles* *and* *as obtained in Theorem 4 satisfy*

###### Proof

The result follows from the following inequalities:

the second last equality follows from the fact that *in* (5). Q.E.D.

###### Remark 4

For a fixed premium *P*, we have

where the minimum is taken over all feasible deductibles *d _{k}*. Let and μ* be the respective minimizer and maximizer on the left- and right-hand sides of the last equation. If the function ψ associated to μ*, as defined in the proof of Theorem 3, is strictly concave, then it is necessary that satisfies Equation (5).

### Concluding Remarks

- Top of page
- Abstract
- Introduction
- Risk Measures and Problem Formulation
- Relevance of Comonotonicity
- Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
- Further Discussion on the Optimal Deductibles
- Concluding Remarks
- Appendix
- References

In this work, we formulated a minimax optimal reinsurance decision problem with an objective functional being an arbitrary law-invariant risk measure. We showed that under the minimax criterion, comonotonicity is the least favorable dependence structure among the insurable risks. By using geometric arguments, we then established the optimality of stop-loss contracts. As an application of the minimax theory (Theorem 2), we also deduced that all the optimal deductibles satisfy Equation (5), from which an interesting ordering property of these optimal deductibles was obtained.

### Appendix

- Top of page
- Abstract
- Introduction
- Risk Measures and Problem Formulation
- Relevance of Comonotonicity
- Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
- Further Discussion on the Optimal Deductibles
- Concluding Remarks
- Appendix
- References

#### Proof of Theorem 3

We divide the proof into two parts. We shall first show that there are scalars (*d*_{1}, …, *d _{n}*) satisfying Equation (5) and is an optimal solution of Problem 3. Second, we shall establish the explicit form of as shown in (4). (1) For

*P*= 0, the only admissible

*I*are zero, and so (5) holds with every . (2) For 0 <

_{k}*P*≤

*P*

_{0}, we first note that the function:

is a continuous function, and . As an immediate consequence of intermediate-value theorem, one can find an α_{0} ε (0, 1) such that . And hence, there are scalars with and for each *k* such that is a feasible solution of Problem 3 that satisfies the premium constraint .

From the proof of Proposition 3, we know that it suffices to restrict to the class of stop-loss contracts in the solution of Problem 3. In other words, we aim to minimize the functional:

- (A1)

subject to the premium constraint

and the constraint for each *k*.

By applying Fubini's Theorem to each summand in (A1),

Using Lemma 4.63 in Föllmer and Schied (2005), there is a continuous increasing concave function such that ψ (0) = 0, ψ (1) = 1, and

Therefore,

and it now suffices to consider

By applying the integration by parts to the last integral, it becomes

In other words, minimizing is then equivalent to maximizing

As ψ is continuous, increasing and concave, there is a scalar λ ≥ 1 such that ψ (α) ≥ λ α on [0, α_{0}] and ψ (α) ≤ λ α on [α_{0}, 1]; see Rockafellar (1970) for details. Now, in accordance with the definitions of α_{0} and λ, we deduce that for any feasible (*d*_{1}, …, *d _{n}*),

the first claim then follows. Note that

where the last equality follows from Dhaene et al. (2002). Therefore, our second claim follows from the definition of *d** and Equation (5). Q.E.D.

- 1
For a given locally convex space

*E*,*a convex function**is called a proper convex function if**f*(*x*) < ∞*for some**x*ε*E*.

### References

- Top of page
- Abstract
- Introduction
- Risk Measures and Problem Formulation
- Relevance of Comonotonicity
- Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
- Further Discussion on the Optimal Deductibles
- Concluding Remarks
- Appendix
- References

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