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.     

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.     

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.     

Upper comonotonicity

In this article, we study a new notion called upper comonotonicity, which is a generalization of the classical notion of comonotonicity. A random vector is upper-comonotonic if its components are moving in the same direction simultaneously when their values are greater than some thresholds. We provide a characterization of this new notion in terms of both the joint distribution function and the underlying copula. The copula characterization allows us to study the coefficient of upper tail dependence as well as the distributional representation of an upper-comonotonic random vector. As an application to financial economics, we show that the several commonly used risk measures, like the Value-at-Risk, the Tail Value-at-Risk, and the expected shortfall, are additive, not only for sum of comonotonic risks, but also for sum of upper-comonotonic risks, provided that the level of probability is greater than a certain threshold.     

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.     

On sums of two counter-monotonic risks

In risk management, capital requirements are most often based on risk measurements of the aggregation of individual risks treated as random variables. The dependence structure between such random variables has a strong impact on the behavior of the aggregate loss. One finds an extensive literature on the study of the sum of comonotonic risks but less, in comparison, has been done regarding the sum of counter-monotonic risks. A crucial result for comonotonic risks is that the Value-at-risk and the Tail Value-at-risk of their sum correspond respectively to the sum of the Value-at-risk and Tail Value-at-risk of the individual risks. In this paper, our main objective is to derive such simple results for the sum of counter-monotonic risks. To do so, we examine separately different contexts in the class of bivariate strictly continuous distributions for which we obtain closed-form expressions for the Value-at-risk and Tail Value-at-risk of the sum of two counter-monotonic risks. The expressions for the subadditive Tail Value-at risk allow us to quantify the maximal diversification benefit. Also, our findings allow us to analyze the tail of the distribution of the sum of two identically subexponentially distributed counter-monotonic random variables.     

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.     

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.     

Second order tail asymptotics for the sum of dependent, tail-independent regularly varying risks

In this paper we consider dependent random variables with common regularly varying marginal distribution. Under the assumption that these random variables are tail-independent, it is well known that the tail of the sum behaves like in the independence case. Under some conditions on the marginal distributions and the dependence structure (including Gaussian copula’s and certain Archimedean copulas) we provide the second-order asymptotic behavior of the tail of the sum.     

Improved convex upper bound via conditional comonotonicity

Comonotonicity provides a convenient convex upper bound for a sum of random variables with arbitrary dependence structure. Improved convex upper bound was introduced via conditioning by Kaas et al. [Kaas, R., Dhaene, J., Goovaerts, M., 2000. Upper and lower bounds for sums of random variables. Insurance: Math. Econ. 27, 151-168]. In this paper, we unify these results in a more general context using the concept of conditional comonotonicity. We also construct an approximating sequence of convex upper bounds with nice convergence properties.     

The weak tail dependence coefficient of the elliptical generalized hyperbolic distribution

The strong and the weak tail dependence coefficients are measures that quantify the probability of conjoint extreme events of two random variables. Whereas formulas for both tail dependence coefficients exist for the Gaussian and Student t distribution, only the strong tail dependence coefficient is known for their super-model, the elliptical generalized hyperbolic distribution, which is extremely popular in finance (see Schmidt 2003). In this work we derive a simple expression for the corresponding weak tail dependence coefficient using the mixture representation of the elliptical generalized hyperbolic distribution.     

Quantifying the error of convex order bounds for truncated first moments

The concepts of convex order and comonotonicity have become quite popular in risk theory, essentially since Kaas et al. [Kaas, R., Dhaene, J., Goovaerts, M.J., 2000. Upper and lower bounds for sums of random variables. Insurance: Math. Econ. 27, 151-168] constructed bounds in the convex order sense for a sum S of random variables without imposing any dependence structure upon it. Those bounds are especially helpful, if the distribution of S cannot be calculated explicitly or is too cumbersome to work with. This will be the case for sums of lognormally distributed random variables, which frequently appear in the context of insurance and finance.In this article we quantify the maximal error in terms of truncated first moments, when S is approximated by a lower or an upper convex order bound to it. We make use of geometrical arguments; from the unknown distribution of S only its variance is involved in the computation of the error bounds. The results are illustrated by pricing an Asian option. It is shown that under certain circumstances our error bounds outperform other known error bounds, e.g. the bound proposed by Nielsen and Sandmann [Nielsen, J.A., Sandmann, K., 2003. Pricing bounds on Asian options. J. Financ. Quant. Anal. 38, 449-473].     

An asymptotic expansion for the tail of compound sums of Burr distributed random variables

In this paper we show that it is possible to write the Laplace transform of the Burr distribution as the sum of four series. This representation is then used to provide a complete asymptotic expansion of the tail of the compound sum of Burr distributed random variables. Furthermore it is shown that if the number of summands is fixed, this asymptotic expansion is actually a series expansion if evaluated at sufficiently large arguments.     

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.     

Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables

In this paper we extend some results about the probability that the sum of n dependent subexponential random variables exceeds a given threshold u. In particular, the case of non-identically distributed and not necessarily positive random variables is investigated. Furthermore we establish criteria how far the tail of the marginal distribution of an individual summand may deviate from the others so that it still influences the asymptotic behavior of the sum. Finally we explicitly construct a dependence structure for which, even for regularly varying marginal distributions, no asymptotic limit of the tail of the sum exists. Some explicit calculations for diagonal copulas and t-copulas are given. Dominik Kortschak was supported by the Austrian Science Fund Project P18392.     

同单调相依结构下两重生命模型的概率分布

在寿险实务中,在处理涉及到多个生命的问题时往往假设各个生命之间是独立的,但事实上,因为受某些相同因素影响的生命之间总是存在一定的正相依性．本文证明了在给定边际分布的二维随机向量中,同单调相依结构是在相关序意义下最强的正相依结构,研究了在此相依结构下的两重生命模型的概率分布,并给出了随机序意义下两个状态消亡时间的随机上界和随机下界．     

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.     

Equality of Types for the Distribution of the Maximum for Two Values of n Implies Extreme Value Type

In this note, we prove a characterization of extreme value distributions. We show that, under some conditions, if the distribution of the maximum of n i.i.d. variables is of the same type for two distinct values of n then the distribution is one of the three extreme value types. This is an analogue of the well known result that if the sum of two i.i.d. random variables with finite second moment is of the same type as the original distribution then the distribution is Gaussian (Kagan et al., 1973). Our result was motivated by study of the m out of n bootstrap.     

ψ-混合随机变量序列的重对数律

本文讨论了同分布的 -混合序列其共同分布属于稳定分布(非高斯情形)吸引场部分和的Chover型重对数律.特别地当分布函数属于稳分布的正则吸引场时,得到了部分和及后置和更精细的结果,即积分检验的结果,由此立即可推出相应的Chover型重对数律.     

ψ-混合随机变量序列的重对数律

本文讨论了同分布的 -混合序列其共同分布属于稳定分布(非高斯情形)吸引场部分和的Chover型重对数律.特别地当分布函数属于稳分布的正则吸引场时,得到了部分和及后置和更精细的结果,即积分检验的结果,由此立即可推出相应的Chover型重对数律.