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.     

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.     

Distorted Mix Method for constructing copulas with tail dependence

This paper introduces a method for constructing copula functions by combining the ideas of distortion and convex sum, named Distorted Mix Method. The method mixes different copulas with distorted margins to construct new copula functions, and it enables us to model the dependence structure of risks by handling the central and tail parts separately. By applying the method we can modify the tail dependence of a given copula to any desired level measured by tail dependence function and tail dependence coefficients of marginal distributions. As an application, a tight bound for asymptotic Value-at-Risk of order statistics is obtained by using the method. An empirical study shows that copulas constructed by this method fit the empirical data of SPX 500 Index and FTSE 100 Index very well in both central and tail parts.     

Applications of conditional comonotonicity to some optimization problems

In this article, we study two optimization problems. The first is finding the best L₁-approximant of a given random vector on some affine subspaces subject to a measurability condition. The second is finding the optimal allocation of policy limits such that the expected retained loss is minimized. Explicit solutions of both problems are constructed by utilizing the notion of conditional comonotonicity.     

Vector-Valued Tail Value-at-Risk and Capital Allocation

Enterprise risk management, actuarial science or finance are practice areas in which risk measures are important to evaluate for heterogeneous classes of homogeneous risks. We present new measures: bivariate lower and upper orthant Tail Value-at-Risk. They are based on bivariate lower and upper orthant Value-at-Risk, introduced in Cossette et al. (Insurance: Math Econ 50(2):247–256, 2012). Many properties and applications are derived. Notably, they are shown to be positive homogeneous, invariant under translation and subadditive in distribution. Capital allocation criteria are suggested. Moreover, results on the sum of random pairs are presented, allowing to use a more accurate model for dependent classes of homogeneous risks.     

Worst VaR scenarios: A remark

Theorem 15 of Embrechts et al. [Embrechts, Paul, Höing, Andrea, Puccetti, Giovanni, 2005. Worst VaR scenarios. Insurance: Math. Econom. 37, 115-134] proves that comonotonicity gives rise to the on-average-most-adverse Value-at-Risk scenario for a function of dependent risks, when the marginal distributions are known but the dependence structure between the risks is unknown. This note extends this result to the case where, rather than no information, partial information is available on the dependence structure between the risks. A result of Kaas et al. [Kaas, Rob, Dhaene, Jan, Goovaerts, Marc J., 2000. Upper and lower bounds for sums of random variables. Insurance: Math. Econom. 23, 151-168] is also generalized for this purpose.     

随机变量的和、积、商分布的一种新表示

应用Feller提出的点-集函数并结合二元copula,对二元连续型正值随机变量的和、积、商的分布进行了研究,得到了和、积、商分布的一种新的计算方法.最后给出一个应用实例.     

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.     

On the distribution of the (un)bounded sum of random variables

We propose a general treatment of random variables aggregation accounting for the dependence among variables and bounded or unbounded support of their sum. The approach is based on the extension to the concept of convolution to dependent variables, involving copula functions. We show that some classes of copula functions (such as Marshall-Olkin and elliptical) cannot be used to represent the dependence structure of two variables whose sum is bounded, while Archimedean copulas can be applied only if the generator becomes linear beyond some point. As for the application, we study the problem of capital allocation between risks when the sum of losses is bounded.     

Approximation of bivariate copulas by patched bivariate Fréchet copulas

Bivariate Fréchet (BF) copulas characterize dependence as a mixture of three simple structures: comonotonicity, independence and countermonotonicity. They are easily interpretable but have limitations when used as approximations to general dependence structures. To improve the approximation property of the BF copulas and keep the advantage of easy interpretation, we develop a new copula approximation scheme by using BF copulas locally and patching the local pieces together. Error bounds and a probabilistic interpretation of this approximation scheme are developed. The new approximation scheme is compared with several existing copula approximations, including shuffle of min, checkmin, checkerboard and Bernstein approximations and exhibits better performance, especially in characterizing the local dependence. The utility of the new approximation scheme in insurance and finance is illustrated in the computation of the rainbow option prices and stop-loss premiums.     

Tail order and intermediate tail dependence of multivariate copulas

In order to study copula families that have tail patterns and tail asymmetry different from multivariate Gaussian and t copulas, we introduce the concepts of tail order and tail order functions. These provide an integrated way to study both tail dependence and intermediate tail dependence. Some fundamental properties of tail order and tail order functions are obtained. For the multivariate Archimedean copula, we relate the tail heaviness of a positive random variable to the tail behavior of the Archimedean copula constructed from the Laplace transform of the random variable, and extend the results of Charpentier and Segers [7] [A. Charpentier, J. Segers, Tails of multivariate Archimedean copulas, Journal of Multivariate Analysis 100 (7) (2009) 1521–1537] for upper tails of Archimedean copulas. In addition, a new one-parameter Archimedean copula family based on the Laplace transform of the inverse Gamma distribution is proposed; it possesses patterns of upper and lower tails not seen in commonly used copula families. Finally, tail orders are studied for copulas constructed from mixtures of max-infinitely divisible copulas.     

Tail negative dependence and its applications for aggregate loss modeling

Tail order of copulas can be used to describe the strength of dependence in the tails of a joint distribution. When the value of tail order is larger than the dimension, it may lead to tail negative dependence. First, we prove results on conditions that lead to tail negative dependence for Archimedean copulas. Using the conditions, we construct new parametric copula families that possess upper tail negative dependence. Among them, a copula based on a scale mixture with a generalized gamma random variable (GGS copula) is useful for modeling asymmetric tail negative dependence. We propose mixed copula regression based on the GGS copula for aggregate loss modeling of a medical expenditure panel survey dataset. For this dataset, we find that there exists upper tail negative dependence between loss frequency and loss severity, and the introduction of tail negative dependence structures significantly improves the aggregate loss modeling.     

Multivariate dependence concepts through copulas

In this paper, multivariate dependence concepts such as affiliation, association and positive lower orthant dependent are studied in terms of copulas. Relationships among these dependent concepts are obtained. An affiliation is a notion of dependence among the elements of a random vector. It has been shown that the affiliation property is preserved using linear interpolation of subcopula. Also our results are applied to the multivariate skew-normal copula. As an application, the dependence concepts used in auction with affiliated signals are discussed. Several examples are given for illustration of the main results.     

Ordering optimal deductible allocations for stochastic arrangement increasing risks

The insurer usually solicits the insured through granting a certain amount of deductible to multiple risks according to his/her own will. Due to the nonlinear nature of the concerned optimization problem, in the literature on the optimal allocations of deductibles researchers usually assume independence or comonotonicity among concerned risks and ignore the impact due to frequency. In this study we build two sufficient conditions for the decreasing optimal allocation of deductibles, relaxing the stochastic arrangement increasing or right tail weakly stochastic arrangement increasing discount factors in Cai and Wei (2014, Theorems 6.3 and 6.6) to the conditionally upper orthant arrangement increasing or weak conditionally upper orthant arrangement increasing frequencies.     

Characterizing mutual exclusivity as the strongest negative multivariate dependence structure

Mutual exclusivity is an extreme negative dependence structure that was first proposed and studied in Dhaene and Denuit (1999) in the context of insurance risks. In this article, we revisit this notion and present versatile characterizations of mutually exclusive random vectors via their pairwise counter-monotonic behaviour, minimal convex sum property, distributional representation and the characteristic function of the sum of their components. These characterizations highlight the role of mutual exclusivity in generalizing counter-monotonicity as the strongest negative dependence structure in a multi-dimensional setting.     

依随机序正相依风险下的最优再保险简

In the classic bivariate compound Poisson models, the numbers of claims are assumed to be correlated through a common Poisson distribution, while the claim sizes are independent. In this paper, we assume that both the numbers of claims and claim sizes are positively dependent through the stochastic ordering. Through comparing, we find that the condition of positive dependence through the stochastic ordering is weaker than correlating through a common Poisson distribution. In fact, the assumption of positive dependence through the stochastic ordering is weaker than independence, comonotonicity, conditionally stochastically increasing et al.. With the positively dependent risks through the stochastic ordering, we get the optimal reinsurance strategy. In addition, with the mixed two-dimensional and stochastic-dimensional dependent risks, we give the explicit expressions of retention vector under the criterion of minimizing the variance of the total retained loss and maximizing the quadratic utility, which partially solves the problem, proposed by Cai and Wei (2012a), of getting such expressions with multi-dimensional dependent risks.     

Risk aggregation with dependence uncertainty

Risk aggregation with dependence uncertainty refers to the sum of individual risks with known marginal distributions and unspecified dependence structure. We introduce the admissible risk class to study risk aggregation with dependence uncertainty. The admissible risk class has some nice properties such as robustness, convexity, permutation invariance and affine invariance. We then derive a new convex ordering lower bound over this class and give a sufficient condition for this lower bound to be sharp in the case of identical marginal distributions. The results are used to identify extreme scenarios and calculate bounds on Value-at-Risk as well as on convex and coherent risk measures and other quantities of interest in finance and insurance. Numerical illustrations are provided for different settings and commonly-used distributions of risks.