SEARCH

SEARCH BY CITATION

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Risk Measures and Problem Formulation
  5. Relevance of Comonotonicity
  6. Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
  7. Further Discussion on the Optimal Deductibles
  8. Concluding Remarks
  9. Appendix
  10. 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Risk Measures and Problem Formulation
  5. Relevance of Comonotonicity
  6. Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
  7. Further Discussion on the Optimal Deductibles
  8. Concluding Remarks
  9. Appendix
  10. 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Risk Measures and Problem Formulation
  5. Relevance of Comonotonicity
  6. Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
  7. Further Discussion on the Optimal Deductibles
  8. Concluding Remarks
  9. Appendix
  10. References

A risk bearer faces a basket of n insurable risks that are modeled as nonnegative essentially bounded random variables inline image defined on an atomless probability space inline image. For each k = 1, 2, …, n, to avoid unnecessary technical details, we assume that the survival function Sk of Xk is known, and is continuous and strictly decreasing on inline image, where inline image denotes the essential supremum of Xk. The cumulative distribution function 1 − Sk of Xk is denoted by Fk.

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:

  • display math

where inline image.

A functional ρ: inline image is called a law-invariant convex risk measure if the following properties are satisfied for any Y1, Y2 ε L:

  • Monotonicity:     If Y1Y2, then ρ (Y1) ≤ ρ (Y2).
  • Translation Invariance:     For any inline image, ρ (Y1 + m) = ρ (Y1) + m.
  • Convexity:     For any γ ε [0, 1], ρ (γ Y1 + (1 − γ) Y2) ≤ γ ρ (Y1) + (1 − γ) ρ (Y2).
  • Law Invariance:     If Y1 and Y2 have the same distribution under inline image, then ρ (Y1) = ρ (Y2).

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

  • display math(1)

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

Remark 1

For law-invariant convex risk measures on unbounded (i.e., inline image) but integrable risks, that is, in L1, 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 inline image is nonempty and convex. Hence, Equation (1) can be written as follows:

  • display math(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 inline image be a given reinsurance contract on Xk, where Ik(xk) represents the payment received from the insurer if xk is the realized loss amount of Xk. 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 Ik:

  1. Ik(yk) − Ik(xk) ≤ ykxk, for any 0 ≤ xkyk,
  2. Ik(0) = 0 and Ik is increasing.

Under these two conditions, it is seen that any feasible reinsurance contract Ik is continuous, and xkxkIk(xk) is increasing. The class of all feasible reinsurance contracts for the risk Xk is denoted by inline image. Define inline image. Suppose that the reinsurance premium is calculated under the expected value (actuarial pricing) principle with risk loading θ > 0. If inline image are bought for the n risks, the total reinsurance premium equals

  • display math(3)

and the total retained loss of the insured changes from X1 + ⋯ + Xn to

  • display math

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 PP0, where P0 is exogenously given and satisfies

  • display math

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 X1, …, Xn is that

  • display math

where inline image denotes the Fréchet space of all n-dimensional random vectors with marginal distributions F1, …, Fn, but the joint distribution is unknown. In view of this consideration, the risk bearer would naturally interest in the following optimization problem:

Problem 1
  • display math

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 Xk 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Risk Measures and Problem Formulation
  5. Relevance of Comonotonicity
  6. Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
  7. Further Discussion on the Optimal Deductibles
  8. Concluding Remarks
  9. Appendix
  10. 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 (Y1, …, Yn) is said to be comonotonic if there exist a random variable U and n increasing functions g1, …, gn such that

  • display math

In particular, we may choose U to be any uniform(0, 1) random variable, and each gi to be inline image, which is the left-continuous inverse distribution function of Yi. Given any distribution functions (F1, …, Fn), it follows from the definition that there always exists a comonotonic random vector inside inline image, which can be constructed explicitly as inline image, where U is a uniform (0, 1) random variable.

It is clear from the definition that comonotonicity is preserved upon increasing transformations: if (Y1, …, Yn) is comonotonic and if Γ1, …, Γn are increasing functions, then (Γ1(Y1), …, Γn(Yn)) 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 inline image is comonotonic additive:

  • display math

whenever (Y1, …, Yn) 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

  • display math

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

Proof

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

  • display math

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

  • display math

As each inline image has the same distribution as Xk and inline image is law-invariant, we can replace Zk by inline image in the previous expression. The right-hand side then equals

  • display math

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

Optimal Reinsurance Under General Law-Invariant Convex Risk Measure

  1. Top of page
  2. Abstract
  3. Introduction
  4. Risk Measures and Problem Formulation
  5. Relevance of Comonotonicity
  6. Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
  7. Further Discussion on the Optimal Deductibles
  8. Concluding Remarks
  9. Appendix
  10. References

From the proof of Proposition 1, we have

  • display math

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

  • display math

over inline image. Because the map xxIk(x) is increasing and continuous, we have

  • display math

where inline image.

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

Proposition 2

For any inline image, there is a scalar  inline image such that

  • display math

Furthermore, the inequality is strict if akdk and inline image.

Proof

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

  • display math

which is the point where the curve x − (xdk)+ upcrosses xIk(x). Obviously, inline image. If inline image, then we must have inline image and there is nothing to prove. Suppose that inline image (and hence dk > 0), we have to consider the following cases:

Case 1: If inline image, then

  • display math

Case 2: If inline image, then

  • display math

Moreover, the last inequality is strict if akdk.        Q.E.D.

Using Proposition 1 and Proposition 2, we can now solve Problem 1.

Proposition 3

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

  • display math
Proof

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

  • display math

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

  • display math

which implies that

  • display math

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 inline image if inline image.

From Proposition 3, we see that for any given reinsurance contracts (I1, …, In), we can always achieve a lower risk by replacing each Ik by the corresponding stop-loss contract inline image 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.

Theorem 1

There is an optimal solution inline image of the following minimization problem:

  • display math

Moreover, inline image is an optimal solution to Problem 1.

Proof

By Proposition 1,

  • display math

which is lower semicontinuous in (d1, …, dn). As the feasible set is compact, the result follows.        Q.E.D.

Further Discussion on the Optimal Deductibles

  1. Top of page
  2. Abstract
  3. Introduction
  4. Risk Measures and Problem Formulation
  5. Relevance of Comonotonicity
  6. Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
  7. Further Discussion on the Optimal Deductibles
  8. Concluding Remarks
  9. Appendix
  10. 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 inline image 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 ξ1, ξ2 ε Ξ and any two nonnegative numbers γ1, γ2 with γ1 + γ2 = 1, there exists an element ξ0 ε Ξ such that for every θ ε Θ, f0, θ) ≤ γ1f1, θ) + γ2f2, θ). The concavity of f on Θ can be defined in a similar way.

Theorem 2

Given a compact Hausdorff space Ξ1 and an arbitrary set Ξ2, let inline image be a real-valued function such that for every ξ2 ε Ξ2, f(*, ξ2) is lower semi-continuous on Ξ1. If f is convex on Ξ1 and concave on Ξ2, then we have

  • display math

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
  • display math

subject to inline image for some fixed P ε [0, P0].

The original minimization Problem 1 is equivalent to a two-step optimization: (1) solve Problem 2 for each fixed P ε [0, P0]; (2) then optimize the value function with respect to P over [0, P0].

Define inline image for any fixed P ε [0, P0]. To solve Problem 2 by applying Theorem 2, we define Ξ1 ≜ Λ and inline image and identify inline image 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 Ξ1 and Ξ2 are convex. Second, inline image is compact because each inline image is compact by Tychonoff Theorem; indeed, the family of all 1-Lipschitz functions on each compact interval inline image is a compact set under the usual supremum norm. Therefore, as a closed subset of inline image, Ξ1 is also compact Hausdorff. Finally, the functional

  • display math

is continuous and convex in (I1, …, In) ε Ξ1 for each μ ε Ξ2, and concave in μ ε Ξ2 for each (I1, …, In) ε Ξ1 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
  • display math

subject to inline image, where inline image is a given probability measure.

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

  • display math

and define the scalars inline image by

  • display math(4)

Clearly, we have the relation:

  • display math(5)
Theorem 3

inline image 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 inline image as in Equation (4), then inline image and

  • display math

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

  • display math

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 inline image to denote the family of all deductibles satisfying:

  1. S1(d1) = S2(d2) = … = Sn(dn);
  2. inline image;
  3. inline image for each k.

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

  • display math

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

Theorem 4

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

  • display math

Moreover, inline image is an optimal solution to Problem 1.

Proof

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

The following corollary says that if any two of these n insurable risks X1, …, Xn 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 Y1 is said to be smaller than Y2 in stochastic dominance, denoted as Y1ST Y2, if for any 0 < α < 1, we have inline image.

Corollary 1

Given that any two risks Xi and Xj satisfy

  • display math

their corresponding optimal deductibles inline image and inline image as obtained in Theorem 4 satisfy

  • display math
Proof

The result follows from the following inequalities:

  • display math

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

Remark 4

For a fixed premium P, we have

  • display math

where the minimum is taken over all feasible deductibles dk. Let inline image 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 inline image satisfies Equation (5).

Concluding Remarks

  1. Top of page
  2. Abstract
  3. Introduction
  4. Risk Measures and Problem Formulation
  5. Relevance of Comonotonicity
  6. Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
  7. Further Discussion on the Optimal Deductibles
  8. Concluding Remarks
  9. Appendix
  10. 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Risk Measures and Problem Formulation
  5. Relevance of Comonotonicity
  6. Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
  7. Further Discussion on the Optimal Deductibles
  8. Concluding Remarks
  9. Appendix
  10. References

Proof of Theorem 3

We divide the proof into two parts. We shall first show that there are scalars (d1, …, dn) satisfying Equation (5) and inline image is an optimal solution of Problem 3. Second, we shall establish the explicit form of inline image as shown in (4). (1) For P = 0, the only admissible Ik are zero, and so (5) holds with every inline image. (2) For 0 < PP0, we first note that the function:

  • display math

is a continuous function, inline image and inline image. As an immediate consequence of intermediate-value theorem, one can find an α0 ε (0, 1) such that inline image. And hence, there are scalars inline image with inline image and inline image for each k such that inline image is a feasible solution of Problem 3 that satisfies the premium constraint inline image.

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:

  • display math(A1)

subject to the premium constraint

  • display math

and the constraint inline image for each k.

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

  • display math

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

  • display math

Therefore,

  • display math

and it now suffices to consider

  • display math

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

  • display math

In other words, minimizing inline image is then equivalent to maximizing

  • display math

As ψ is continuous, increasing and concave, there is a scalar λ ≥ 1 such that ψ (α) ≥ λ α on [0, α0] and ψ (α) ≤ λ α on [α0, 1]; see Rockafellar (1970) for details. Now, in accordance with the definitions of α0 and λ, we deduce that for any feasible (d1, …, dn),

  • display math

the first claim then follows. Note that

  • display math

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. 1

    For a given locally convex space E, a convex function inline image is called a proper convex function if f(x) < ∞ for some x ε E.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Risk Measures and Problem Formulation
  5. Relevance of Comonotonicity
  6. Optimal Reinsurance Under General Law-Invariant Convex Risk Measure
  7. Further Discussion on the Optimal Deductibles
  8. Concluding Remarks
  9. Appendix
  10. References
  • Acerbi, C., and D.Tasche, 2002, On the Coherence of Expected Shortfall, Journal of Banking and Finance, 26(7): 1487-1503.
  • Annaert, J., S. vanOsselaer, and B.Verstraete, 2009, Performance Evaluation of Portfolio Insurance Strategies Using Stochastic Dominance Criteria, Journal of Banking and Finance, 33(2): 272-280.
  • Artzner, P., F.Delbaen, J. M.Eber, and D.Heath, 1999, Coherent Measures of Risk, Mathematical Finance, 9(3): 203-228.
  • Arrow, K. J., 1963, Uncertainty and the Welfare Economics of Medical Care, American Economic Review, 53: 941-973.
  • BalbÁs, A., B.BalbÁs, and A.Heras, 2009, Optimal Reinsurance with General Risk Measures, Insurance: Mathematics and Economics, 44(3): 374-384.
  • BalbÁs, A., B.BalbÁs, and A.Heras, 2011, Stability of the Optimal Reinsurance with Respect to the Risk Measure, European Journal of Operational Research, 214: 796-804.
  • Borch, K., 1960, An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance, Transactions of the 16th International Congress of Actuaries, 597-610.
  • BÜhlmann, H., and W. S.Jewell, 1979, Optimal Risk Exchanges, ASTIN Bulletin, 10: 243-262.
  • Cai, J., and K. S.Tan, 2007, Optimal Retention for a Stop-Loss Reinsurance Under the VaR and CTE Risk Measures, ASTIN Bulletin, 37(1): 93-112.
  • Cai, J., K. S.Tan, C.Weng, and Y.Zhang, 2008, Optimal Reinsurance Under VaR and CTE Risk Measures, Insurance: Mathematics and Economics, 43(1): 185-196.
  • Cheung, K. C., 2006, Optimal Portfolio Problem with Unknown Dependency Structure, Insurance: Mathematics and Economics, 38: 167-175.
  • Cheung, K. C., 2007, Optimal Allocation of Policy Limits and Deductibles, Insurance: Mathematics and Economics, 41: 382-391.
  • Cheung, K. C., 2010, Optimal Reinsurance Revisited—Geometric Approach, ASTIN Bulletin, 40: 221-239.
  • Cheung, K. C., K. C. J.Sung, S. C. P.Yam, and S. P.Yung, forthcoming, Optimal Reinsurance Under General Law Invariant Risk Measures, Scandinavian Actuarial Journal. DOI:10.1080/03461238.2011.636880
  • Delbaen, F., 2002, Coherent Risk Measures on General Probability Spaces, in: Advances in Finance and Stochastics (Berlin: Springer-Verlag), pp. 1-37.
  • Denuit, M., J.Dhaene, M. J.Goovaerts, and R.Kaas, 2005, Actuarial Theory for Dependent Risks: Measures, Orders and Models(Chichester, UK: John Wiley and Sons).
  • Dhaene, J., M.Denuit, M. J.Goovaerts, R.Kaas, and D.Vyncke, 2002, The Concept of Comonotonicity in Actuarial Science and Finance: Theory, Insurance: Mathematics and Economics, 31: 3-33.
  • Dhaene, J., S.Vanduffel, M. J.Goovaerts, R.Kaas, and D.Vyncke, 2005, Comonotonic Approximations for Optimal Portfolio Selection Problems, Journal of Risk and Insurance, 72(2): 253-301.
  • Dhaene, J., S.Vanduffel, Q.Tang, M.Goovaerts, R.Kaas, and D.Vyncke, 2006, Risk Measures and Comonotonicity: a Review, Stochastic Models, 22: 573-606.
  • Ekeland, I., A.Galichon, and M.Henry, 2012, Comonotonic Measures of Multivariate Risks, Mathematical Finance, 22(1): 109-132.
  • Fan, K., 1953, Minimax Theorems, Proceedings of the National Academy of Sciences of the United States of America, 39: 42-47.
  • FilipoviĆ, D., and G.Svindland, 2012, The Canonical Model Space for Law-Invariant Convex Risk measures in L', Mathematical Finance, 22(3): 585-589.
  • FÖllmer, H., and A.Schied, 2002, Convex Measures of Risk and Trading Constraints, Finance and Stochastics, 6(4): 429-447.
  • FÖllmer, H., and A.Schied, 2004, Stochastic Finance: An Introduction in Discrete Time (Berlin: Walter de Gruyter).
  • Frittelli, M., and E. RosazzaGianin, 2002, Putting Order in Risk Measures, Journal of Banking and Finance, 26(7): 1473-1486.
  • Frittelli, M., and E. RossazaGianin, 2005, Law Invariant Convex Risk Measures, Advances in Mathematical Economics, 7: 33-46.
  • Heath, D., and H.Ku, 2004, Pareto Equilibria with Coherent Measures of Risk, Mathematical Finance, 14(2): 163-172.
  • Huber, P. J., 1981, Robust Statistics (New York: Wiley).
  • Jouini, E., W.Schachermayer, and N.Touzi, 2006, Law Invariant Risk Measures have the Fatou Property, Advances in Mathematical Economics, 9: 49-71.
  • Jouini, E., W.Schachermayer, and N.Touzi, 2008, Optimal Risk Sharing for Law Invariant Monetary Utility Functions, Mathematical Finance, 18: 269-292.
  • Kiesel, S., and L.RÜschendorf, 2008, Characterization of Optimal Risk Allocations for Convex Risk Functionals, Statistics and Decisions, 26: 303-319.
  • Kiesel, S., and L.RÜschendorf, 2010, On Optimal Allocation of Risk Vectors, Insurance: Mathematics and Economics, 47: 167-175.
  • Kusuoka, S., 2001, On Law Invariant Coherent Risk Measures, Advances in Mathematical Economics, 3: 83-95.
  • Meilijson, I., and A.Nadas, 1979, Convex Majorization with an Application to the Length of Critical Paths, Journal of Applied Probability, 16: 671-677.
  • Rockafellar, R. T., 1970, Convex analysis (Princeton, NJ: Princeton University Press).
  • Rockafellar, R. T., and S.Uryasev, 2001, Conditional Value-At-Risk for General Loss Distributions, Research Report 2001-5, ISE Department, University of Florida.
  • RÜschendorf, L., 2005, Stochastic Ordering of Risks, Influence of Dependence and a.s. Constructions, in: Advances on Models, Characterizations and Applications, N.Balakrishnan, I. G.Bairamov, and O. L.Gebizlioglu, eds. (Boca Raton, FL: Chapman and Hall/CRC Press), pp. 19-56.
  • RÜschendorf, L. 2006, Law Invariant Convex Risk Measures for Portfolio Vectors, Statistics and Decisions, 24: 97-108.
  • RÜschendorf, L. 2012, Worst Case Portfolio Vectors and Diversification Effects, Finance and Stochastics, 16(1): 155-175.
  • Schied, A. 2007, Optimal Investments for Risk- and Ambiguity-Averse Preferences: A Duality Approach, Finance and Stochastics, 11(1): 107-129.
  • Song, Y., and J. A.Yan, 2009, An Overview of Representation Theorems for Static Risk Measures, Science in China Series A: Mathematics, 52(7): 1412-1422.