基于情景分析的多维拟凸风险统计量

本文定义了三类特殊的多维风险统计量,分别是多维共单调拟凸风险统计量、多维拟凸风险统计量和多维经验分布不变拟凸风险统计量,并采用对偶方法给出了它们的表示定理.本文的结果既是一维拟凸风险统计量的推广,也是多维凸风险统计量的拓展.     

Characterizations of counter-monotonicity and upper comonotonicity by (tail) convex order

In this paper, we characterize counter-monotonic and upper comonotonic random vectors by the optimality of the sum of their components in the senses of the convex order and tail convex order respectively. In the first part, we extend the characterization of comonotonicity by  Cheung (2010) and show that the sum of two random variables is minimal with respect to the convex order if and only if they are counter-monotonic. Three simple and illuminating proofs are provided. In the second part, we investigate upper comonotonicity by means of the tail convex order. By establishing some useful properties of this relatively new stochastic order, we prove that an upper comonotonic random vector must give rise to the maximal tail convex sum, thereby completing the gap in  Nam et al. (2011)’s characterization. The relationship between the tail convex order and risk measures along with conditions under which the additivity of risk measures is sufficient for upper comonotonicity is also explored.     

Risk measures with comonotonic subadditivity or convexity on product spaces

In this paper, by an axiomatic approach, we propose the concepts of comonotonic subadditivity and comonotonic convex risk measures for portfolios, which are extensions of the ones introduced by Song and Yan(2006)Representation results for these new introduced risk measures for portfolios are given in terms of Choquet integralsLinks of these newly introduced risk measures to multi-period comonotonic risk measures are representedFinally, applications of the newly introduced comonotonic coherent risk measures to capital allocations are provided.     

An overview of representation theorems for static risk measures

In this paper, we give an overview of representation theorems for various static risk measures: coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity, law-invariant coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity and respecting stochastic orders. This work was supported by National Natural Science Foundation of China (Grant No. 10571167), National Basic Research Program of China (973 Program) (Grant No. 2007CB814902), and Science Fund for Creative Research Groups (Grant No. 10721101)     

Characterization of upper comonotonicity via tail convex order

In this paper, we show a characterization of upper comonotonicity via tail convex order. For any given marginal distributions, a maximal random vector with respect to tail convex order is proved to be upper comonotonic under suitable conditions. As an application, we consider the computation of the Haezendonck risk measure of the sum of upper comonotonic random variables with exponential marginal distributions.     

Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders

This paper proposes some new classes of risk measures, which are not only comonotonic subadditive or convex, but also respect the (first) stochastic dominance or stop-loss order. We give their representations in terms of Choquet integrals w.r.t. distorted probabilities, and show that if the physical probability is atomless then a comonotonic subadditive (resp. convex) risk measure respecting stop-loss order is in fact a law-invariant coherent (resp. convex) risk measure.     

Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders

This paper proposes some new classes of risk measures, which are not only comonotonic subadditive or convex, but also respect the (first) stochastic dominance or stop-loss order. We give their representations in terms of Choquet integrals w.r.t. distorted probabilities, and show that if the physical probability is atomless then a comonotonic subadditive (resp. convex) risk measure respecting stop-loss order is in fact a law-invariant coherent (resp. convex) risk measure.     

Characterizing a comonotonic random vector by the distribution of the sum of its components

In this article, we characterize comonotonicity and related dependence structures among several random variables by the distribution of their sum. First we prove that if the sum has the same distribution as the corresponding comonotonic sum, then the underlying random variables must be comonotonic as long as each of them is integrable. In the literature, this result is only known to be true if either each random variable is square integrable or possesses a continuous distribution function. We then study the situation when the distribution of the sum only coincides with the corresponding comonotonic sum in the tail. This leads to the dependence structure known as tail comonotonicity. Finally, by establishing some new results concerning convex order, we show that comonotonicity can also be characterized by expected utility and distortion risk measures.     

Upper comonotonicity and convex upper bounds for sums of random variables

It is well-known that if a random vector with given marginal distributions is comonotonic, it has the largest sum with respect to convex order. However, replacing the (unknown) copula by the comonotonic copula will in most cases not reflect reality well. For instance, in an insurance context we may have partial information about the dependence structure of different risks in the lower tail. In this paper, we extend the aforementioned result, using the concept of upper comonotonicity, to the case where the dependence structure of a random vector in the lower tail is already known. Since upper comonotonic random vectors have comonotonic behavior in the upper tail, we are able to extend several well-known results of comonotonicity to upper comonotonicity. As an application, we construct different increasing convex upper bounds for sums of random variables and compare these bounds in terms of increasing convex order.     

Reducing risk by merging counter-monotonic risks

In this article, we show that some important implications concerning comonotonic couples and corresponding convex order relations for their sums cannot be translated to counter-monotonicity in general. In a financial context, it amounts to saying that merging counter-monotonic positions does not necessarily reduce the overall level of risk. We propose a simple necessary and sufficient condition for such a merge to be effective. Natural interpretations and various characterizations of this condition are given. As applications, we develop cancelation laws for convex order and identify desirable structural properties of insurance indemnities that make an insurance contract universally marketable, in the sense that it is appealing to both the policyholder and the insurer.     

Convex order and comonotonic conditional mean risk sharing

Using a standard reduction argument based on conditional expectations, this paper argues that risk sharing is always beneficial (with respect to convex order or second degree stochastic dominance) provided the risk-averse agents share the total losses appropriately (whatever the distribution of the losses, their correlation structure and individual degrees of risk aversion). Specifically, all agents hand their individual losses over to a pool and each of them is liable for the conditional expectation of his own loss given the total loss of the pool. We call this risk sharing mechanism the conditional mean risk sharing. If all the conditional expectations involved are non-decreasing functions of the total loss then the conditional mean risk sharing is shown to be Pareto-optimal. Explicit expressions for the individual contributions to the pool are derived in some special cases of interest: independent and identically distributed losses, comonotonic losses, and mutually exclusive losses. In particular, conditions under which this payment rule leads to a comonotonic risk sharing are examined.     

Characterization of comonotonicity using convex order

It is well known that if a random vector with given marginal distributions is comonotonic, it has the largest sum with respect to the convex order. In this paper, we prove that the converse is also true, provided that each marginal distribution is continuous.     

Characterization of comonotonicity using convex order

It is well known that if a random vector with given marginal distributions is comonotonic, it has the largest sum with respect to the convex order. In this paper, we prove that the converse is also true, provided that each marginal distribution is continuous.     

A new proof of Cheung’s characterization of comonotonicity

It is well known that if a random vector with given marginal distributions is comonotonic, it has the largest sum in the sense of the convex order. Cheung (2008) proved that the converse of this assertion is also true, provided that all marginal distribution functions are continuous and that the underlying probability space is atomless. This continuity assumption on the marginals was removed by Cheung (2010). In this short note, we give a new and simple proof of Cheung’s result without the assumption that the underlying probability space is atomless.     

Optimal reinsurance with positively dependent risks

In the individual risk model, one is often concerned about positively dependent risks. Several notions of positive dependence have been proposed to describe such dependent risks. In this paper, we assume that the risks in the individual risk model are positively dependent through the stochastic ordering (PDS). The PDS risks include independent, comonotonic, conditionally stochastically increasing (CI) risks, and other interesting dependent risks. By proving the convolution preservation of the convex order for PDS random vectors, we show that in individualized reinsurance treaties, to minimize certain risk measures of the retained loss of an insurer, the excess-of-loss treaty is the optimal reinsurance form for an insurer with PDS dependent risks among a general class of individualized reinsurance contracts. This extends the study in Denuit and Vermandele (1998) on individualized reinsurance treaties to dependent risks. We also derive the explicit expressions for the retentions in the optimal excess-of-loss treaty in a two-line insurance business model.     

Comonotonic convex upper bound and majorization

When the dependence structure among several risks is unknown, it is common in the actuarial literature to study the worst dependence structure that gives rise to the riskiest aggregate loss. A central result is that the aggregate loss is the riskiest with respect to convex order when the underlying risks are comonotonic. Many proofs were given before. The objective of this article is to present a new proof using the notions of decreasing rearrangement and the majorization theorem, and give clear explanation of the relation between convex order, the theory of majorization and comonotonicity.     

Unifying practical uncertainty representations. II: Clouds

There exist many simple tools for jointly capturing variability and incomplete information by means of uncertainty representations. Among them are random sets, possibility distributions, probability intervals, and the more recent Ferson’s p-boxes and Neumaier’s clouds, both defined by pairs of possibility distributions. In the companion paper, we have extensively studied a generalized form of p-box and situated it with respect to other models. This paper focuses on the links between clouds and other representations. Generalized p-boxes are shown to be clouds with comonotonic distributions. In general, clouds cannot always be represented by random sets, in fact not even by two-monotone (convex) capacities.     

A note on natural risk statistics

Recently Heyde, Kou and Peng [C.C. Heyde, S.G. Kou, X.H. Peng, What is a good external risk measure: Bridging the gaps between robustness, subadditivity, and insurance risk measures, 2007, preprint.] proposed the notion of a natural risk statistic associated with a finite sample that relaxes the subadditivity assumption in the classical coherent risk statistics. In this note we use convex analysis to provide alternate proofs of the representation results regarding natural risk statistics.     

Comonotonic bounds on the survival probabilities in the Lee–Carter model for mortality projection

In the Lee–Carter framework, future survival probabilities are random variables with an intricate distribution function. In large homogeneous portfolios of life annuities, value-at-risk or conditional tail expectation of the total yearly payout of the company are approximately equal to the corresponding quantities involving random survival probabilities. This paper aims to derive some bounds in the increasing convex (or stop-loss) sense on these random survival probabilities. These bounds are obtained with the help of comonotonic upper and lower bounds on sums of correlated random variables.     

Ordering Results for Risk Bounds and Cost-efficient Payoffs in Partially Specified Risk Factor Models

Motivated by the problem of sharp risk bounds in partially specified risk factor models and by the method of cost-efficient payoffs with given payoff structure we introduce and describe some stochastic odering problems for conditionally comonotonic resp. antimonotonic random variables. The aim is to describe the influence of the specified dependence of the components of the random vector X with a benchmark Z on the risk bounds in a risk portfolio resp. on the gain of cost efficiency of the optimal payoffs. We obtain in particular explicit results in dependence on distributional parameters for elliptical models in the case of risk bounds and for the multivariate Samuelson model in the case of cost efficient payoffs.