An analysis of the literature on systemic financial risk: A survey

Selection of Value at Risk Models for Energy Commodities

One of the most useful tools for handling multivariate distributions with givenunivariatemarginals is the copula function. Using it, any multivariate distribution function can be represented in a way that emphasizes the separate roles of the marginals and of the dependence structure. The goal of the present paper is to introduce an analogous tool, called the linkage function, that can be used for the study of multivariate distributions with givenmultivariatemarginals by emphasizing the separate roles of the dependence structurebetweenthe given multivariate marginals, and the dependence structurewithineach of the nonoverlapping marginals. Preservation of some setwise positive dependence properties, from the linkage functionLto the joint distributionFand vice versa, are studied. When two different distribution functions are associated with the same linkage function (that is, have the same setwise dependence structure) we show that strong stochastic dominance order among the corresponding multivariate marginal distributions implies an overall stochastic dominance between the two underlying distribution functions.

Linkages: A tool for the construction of multivariate distributions with given nonoverlapping multivariate marginals

CoVaR of families of copulas

This paper deals with the problem of estimating the level sets of an unknown distribution function $F$. A plug-in approach is followed. That is, given a consistent estimator $F_n$ of $F$, we estimate the level sets of $F$ by the level sets of $F_n$. In our setting no compactness property is a priori required for the level sets to estimate. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric difference. Our results are motivated by applications in multivariate risk theory. In this sense we also present simulated and real examples which illustrate our theoretical results.

Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory

CoVaR is a systemic risk measure proposed by Adrian and Brunnermeier (2011) able to measure a financial institution’s contribution to systemic risk and its contribution to the risk of other financial institutions. CoVaR stands for conditional Value-at-Risk, i.e. it indicates the Value at Risk for a financial institution that is conditional on a certain scenario. In this paper, two alternative extensions of the classic univariate Conditional Value-at-Risk are introduced in a multivariate setting. The two proposed multivariate CoVaRs are constructed from level sets of multivariate distribution functions ( resp. of multivariate survival distribution functions). These vector-valued measures have the same dimension as the underlying risk portfolio. Several characterizations of these new risk measures are provided in terms of the copula structure and stochastic orderings of the marginal distributions. Interestingly, these results are consistent with existing properties on univariate risk measures. Furthermore, comparisons between existent risk measures and the proposed multivariate CoVaR are developed. Illustrations are given in the class of Archimedean copulas. Estimation procedure for the multivariate proposed CoVaRs is illustrated in simulated studies and insurance real data.

On multivariate extensions of the conditional Value-at-Risk measure

In this paper, some new properties of the upper-corrected orthant of a random vector are proved. The univariate right-spread or excess wealth function, introduced by Fernandez-Ponce et al. (1996) , is extended to multivariate random vectors, and some properties of this multivariate function are studied. Later, this function was used to define the excess wealth ordering by Shaked and Shanthikumar (1998) and Fernandez-Ponce et al. (1998) . The multivariate excess wealth function enable us to define a new stochastic comparison which is weaker than the multivariate dispersion orderings. Also, some properties relating the multivariate excess wealth order with stochastic dependence are described.

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A characterization of the multivariate excess wealth ordering

On a new NBUE property in multivariate sense: an application

The test for exponentiality of a dataset in terms of a specific aging property constitutes an interesting problem in reliability analysis. To this end, a wide variety of tests are proposed in the literature. In this paper, the excess-wealth function is recalled and new asymptotic properties are studied. By using the characterization of the exponential distribution based on the excess-wealth function, a new exponentiality test is proposed. Through simulation techniques, it is shown that this new test works well on small sample sizes. The exact null distribution and normality asymptotic is also obtained for the statistic proposed. This test and a new empirical graph based on the excess-wealth function are applied to extreme-value examples.

Testing exponentiality against NBUE distributions with an application in environmental extremes

Recently, some well-known univariate aging classes of lifetime distributions have been characterized by means of properties of their quantile functions and excess-wealth functions. The generalization of the univariate aging notions to the multivariate case involve, among other factors, appropriate definitions of multivariate quantiles or regression representation and related notions, which are able to correctly describe the intrinsic characteristic of the concepts of aging that should be generalized. The multivariate versions of these notions, which are characterized by using the multivariate u-quantiles and the multivariate excess-wealth function, are considered in this paper. Relationships between such multivariate aging classes are studied, and examples are provided.

M. R. Rodríguez-Griñolo

Papers

On multivariate extensions of the conditional Value-at-Risk measure

A characterization of the multivariate excess wealth ordering

On a new NBUE property in multivariate sense: an application

Testing exponentiality against NBUE distributions with an application in environmental extremes

New multivariate aging notions based on the corrected orthant and the standard construction