Elena Di Bernardino

On Multivariate Extensions of Value-at-Risk

In this paper, we introduce two alternative extensions of the classical univariate Value-at-Risk (VaR) in a multivariate setting. The two proposed multivariate VaR are vector-valued measures with the same dimension as the underlying risk portfolio. The lower-orthant VaR is constructed from level sets of multivariate distribution functions whereas the upper-orthant VaR is constructed from level sets of multivariate survival functions. Several properties have been derived. In particular, we show that these risk measures both satisfy the positive homogeneity and the translation invariance property. Comparison between univariate risk measures and components of multivariate VaR are provided. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Illustrations are given in the class of Archimedean copulas.

Risk Management with Expectiles

Expectiles (EVaR) are a one-parameter family of coherent risk measures that have been recently suggested as an alternative to quantiles (VaR) and to expected shortfall (ES). In this work we review their known properties, we discuss their financial meaning, we compare them with VaR and ES and we study their asymptotic behaviour, refining some of the results in Bellini et al. [(2014). “Generalized Quantiles as Risk Measures.” Insurance: Mathematics and Economics, 54:41–48]. Moreover, we present a real-data example for the computation of expectiles by means of simple Garch(1,1) models and we assess the accuracy of the forecasts by means of a consistent loss function as suggested by Gneiting [(2011). “Making and Evaluating Point Forecast.” Journal of the American Statistical Association, 106 (494): 746–762]. Theoretical and numerical results indicate that expectiles are perfectly reasonable alternatives to VaR and ES risk measures.

On certain transformations of Archimedean copulas: Application to the non-parametric estimation of their generators

Abstract We study the impact of certain transformations within the class of Archimedean copulas. We give some admissibility conditions for these transformations, and define some equivalence classes for both transformations and generators of Archimedean copulas. We extend the r-fold composition of the diagonal section of a copula, from r ∈ N to r ∈ R. This extension, coupled with results on equivalence classes, gives us new expressions of transformations and generators. Estimators deriving directly from these expressions are proposed and their convergence is investigated. We provide confidence bands for the estimated generators. Numerical illustrations show the empirical performance of these estimators.

On tail dependence coefficients of transformed multivariate Archimedean copulas

This paper presents the impact of a class of transformations of copulas in their upper and lower multivariate tail dependence coefficients. In particular we focus on multivariate Archimedean copulas. In the first part of this paper, we calculate multivariate transformed tail dependence coefficients when the generator of the considered transformed copula exhibits some regular variation properties, and we investigate the behaviour of these coefficients in cases that are close to tail independence. We obtain new results under specific conditions involving regularly varying hazard rates of components of the transformation. These results are also valid for non-transformed Archimedean copulas. In the second part we deal with transformations presented by Di Bernardino and Rulliere 20. We show the utility of using transformed Archimedean copulas, as they permit to build Archimedean generators exhibiting any chosen couple of lower and upper tail dependence coefficients. Finally, we detail the extreme behaviour of the transformed radial part of Archimedean copulas (using results in Larsson and Neslehova 52) and we explain possible applications with transformed Markov chains.

Estimation of multivariate conditional-tail-expectation using Kendall's process

This paper deals with the problem of estimating the multivariate version of the Conditional-Tail-Expectation, proposed by Di Bernardino et al. [(2013), ‘Plug-in Estimation of Level Sets in a Non-Compact Setting with Applications in Multivariable Risk Theory’, ESAIM: Probability and Statistics, (17), 236–256]. We propose a new nonparametric estimator for this multivariate risk-measure, which is essentially based on Kendalls process [Genest and Rivest, (1993), ‘Statistical Inference Procedures for Bivariate Archimedean Copulas’, Journal of American Statistical Association, 88(423), 1034–1043]. Using the central limit theorem for Kendalls process, proved by Barbe et al. [(1996), ‘On Kendalls Process’, Journal of Multivariate Analysis, 58(2), 197–229], we provide a functional central limit theorem for our estimator. We illustrate the practical properties of our nonparametric estimator on simulations and on two real test cases. We also propose a comparison study with the level sets-based estimator introduced in Di Bernardino et al. [(2013), ‘Plug-In Estimation of Level Sets in A Non-Compact Setting with Applications in Multivariable Risk Theory’, ESAIM: Probability and Statistics, (17), 236–256] and with (semi-)parametric approaches.

Estimating a bivariate tail: A copula based approach

This paper deals with the problem of estimating the tail of a bivariate distribution function. To this end we develop a general extension of the POT (peaks-over-threshold) method, mainly based on a two-dimensional version of the Pickands–Balkema–de Haan Theorem. We introduce a new parameter that describes the nature of the tail dependence, and we provide a way to estimate it. We construct a two-dimensional tail estimator and study its asymptotic properties. We also present real data examples which illustrate our theoretical results.

Cross-Correlations and Joint Gaussianity in Multivariate Level Crossing Models

A variety of phenomena in physical and biological sciences can be mathematically understood by considering the statistical properties of level crossings of random Gaussian processes. Notably, a growing number of these phenomena demand a consideration of correlated level crossings emerging from multiple correlated processes. While many theoretical results have been obtained in the last decades for individual Gaussian level-crossing processes, few results are available for multivariate, jointly correlated threshold crossings. Here, we address bivariate upward crossing processes and derive the corresponding bivariate Central Limit Theorem as well as provide closed-form expressions for their joint level-crossing correlations.

A test of Gaussianity based on the Euler characteristic of excursion sets

In the present paper, we deal with a stationary isotropic random field X : R d → R and we assume it is partially observed through some level functionals. We aim at providing a methodology for a test of Gaussianity based on this information. More precisely, the level func-tionals are given by the Euler characteristic of the excursion sets above a finite number of levels. On the one hand, we study the properties of these level functionals under the hypothesis that the random field X is Gaussian. In particular, we focus on the mapping that associates to any level u the expected Euler characteristic of the excursion set above level u. On the other hand, we study the same level functionals under alternative distributions of X, such as chi-square, harmonic oscillator and shot noise. In order to validate our methodology, a part of the work consists in numerical experimentations. We generate Monte-Carlo samples of Gaussian and non-Gaussian random fields and compare, from a statistical point of view, their level functionals. Goodness-of-fit p−values are displayed for both cases. Simulations are performed in one dimensional case (d = 1) and in two dimensional case (d = 2), using R.

On an asymmetric extension of multivariate Archimedean copulas based on quadratic form

Abstract An important topic in Quantitative Risk Management concerns the modeling of dependence among risk sources and in this regard Archimedean copulas appear to be very useful. However, they exhibit symmetry, which is not always consistent with patterns observed in real world data. We investigate extensions of the Archimedean copula family that make it possible to deal with asymmetry. Our extension is based on the observation that when applied to the copula the inverse function of the generator of an Archimedean copula can be expressed as a linear form of generator inverses. We propose to add a distortion term to this linear part, which leads to asymmetric copulas. Parameters of this new class of copulas are grouped within a matrix, thus facilitating some usual applications as level curve determination or estimation. Some choices such as sub-model stability help associating each parameter to one bivariate projection of the copula. We also give some admissibility conditions for the considered copulas. We propose different examples as some natural multivariate extensions of Farlie-Gumbel-Morgenstern or Gumbel-Barnett.