Volume 4, Issue 1

Integral representations of the exact distributions of order statistics are derived in a geometric way when three or four random variables depend on each other as the components of continuous ln,psymmetrically distributed random vectors do, n ∈ {3,4}, p > 0. Once the representations are implemented in a computer program, it is easy to change the density generator of the ln,p-symmetric distribution with another one for newly evaluating the distribution of interest. For two groups of stock exchange index residuals, maximum distributions are compared under dependence and independence modeling.

A measure-of-cone representation of skewed continuous ln,p-symmetric distributions, n ∈ N, p > 0, is proved following the geometric approach known for elliptically contoured distributions. On this basis, distributions of extreme values of n dependent random variables are derived if the latter follow a joint continuous ln,p-symmetric distribution. Light, heavy, and extremely far tails as well as tail indices are discussed, and new parameters of multivariate tail behavior are introduced.

We study the integration of a copula with respect to the probability measure generated by another copula. To this end, we consider the map [. , .] : C × C → R given by where C denotes the collection of all d–dimensional copulas and Q D denotes the probability measures associated with the copula D. Specifically, this is of interest since several measures of concordance such as Kendall’s tau, Spearman’s rho and Gini’s gamma can be expressed in terms of the map [. , .]. Quite generally, the map [. , .] can be applied to construct and investigate measures of concordance.

In thiswork,we extend some parameters built on a probability distribution introduced before to the casewhere the proximity between real numbers is measured by using a Bregman divergence. This leads to the definition of the Bregman superquantile (thatwe can connect with severalworks in economy, see for example [18] or [9]). Axioms of a coherent measure of risk discussed previously (see [31] or [3]) are studied in the case of Bregman superquantile. Furthermore,we deal with asymptotic properties of aMonte Carlo estimator of the Bregman superquantile. Several numerical tests confirm the theoretical results and an application illustrates the potential interests of the Bregman superquantile.

We construct new multivariate copulas on the basis of a generalized infinite partition-of-unity approach. This approach allows, in contrast to finite partition-of-unity copulas, for tail-dependence as well as for asymmetry. A possibility of fitting such copulas to real data from quantitative risk management is also pointed out.

We propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. Considering two Brownian motions B1t and B2t, the main result is that the range of possible values for is the same for Markovian pairs and all pairs of Brownian motions, that is with φ being the cumulative distribution function of a standard Gaussian random variable.

We derive a model based on the structure of dependence between a Brownian motion and its reflection according to a barrier. The structure of dependence presents two states of correlation: one of comonotonicity with a positive correlation and one of countermonotonicity with a negative correlation. This model of dependence between two Brownian motions B1 and B2 allows for the value of to be higher than 1/2 when x is close to 0, which is not the case when the dependence is modeled by a constant correlation. It can be used for risk management and option pricing in commodity energy markets. In particular, it allows to capture the asymmetry in the distribution of the difference between electricity prices and its combustible prices.

For a high dimensional field of random variables, global correlation is defined as the ratio of average covariance and average variance, and its elementary properties are studied. Global correlation is used to harmonize uncertainty assessments at global and local scales. It can be estimated by the correlation of random aggregations of fixed size of disjoint sets of random variables. Illustrative applications are given using crop loss per county per year and forest carbon.

A weak version of the joint hazard rate order, useful to stochastically compare not independent random variables, has been recently defined and studied in [4]. In the present paper, further results on this order are proved and discussed. In particular, some statements dealing with the relationships between the jointweak hazard rate order and other stochastic orders are generalized to the case of non symmetric copulas, and its relations with some multivariate aging notions (studied in [2]) are presented. For this purpose, the new notions of Generalized Supermigrative and Generalized Submigrative copulas are defined. Other new results, examples and discussions are provided as well.

We study measures of concordance for multivariate copulas and copulas that induce measures of concordance. To this end, for a copula A, we consider the maps C → R given by where C denotes the collection of all d–dimensional copulas, M is the Fréchet–Hoeffding upper bound, Π is the product copula, [. , .] : C × C → R is the biconvex form given by [C, D] := ∫ [0,1]d C(u) dQD(u) with the probability measure QD associated with the copula D, and ψΛ C → C is a transformation of copulas. We present conditions on ψΛ and on A under which these maps are measures of concordance. The resulting class of measures of concordance is rich and includes the well–known examples Spearman’s rho and Gini’s gamma.

The aim of this manuscript is to determine the relative size of several functions (copulas, quasi– copulas) that are commonly used in stochastic modeling. It is shown that the class of all quasi–copulas that are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi– copulas. Moreover, it is proved that copulas are nowhere dense in the class of quasi-copulas. The results are obtained via a checkerboard approximation of quasi–copulas.

Building upon earlier work in which axioms were formulated for multivariate measures of concordance, we examine properties of such measures. In particular,we examine the relations between the measure of concordance of an n-copula and the measures of concordance of the copula’s marginals.

In this a paper a non-linear macro stress testing methodology with focus on early warning is developed. The methodology builds on a variant of Random Forests and its proximity measures. It is embedded in a framework, in which naturally defined contagion and feedback effects transfer the impact of stressing a relatively small part of the observations on the whole dataset, allowing to estimate a stressed future state. It will be shown that contagion can be directly derived from the proximities while iterating the proximity based contagion leads to naturally defined feedback effects. Since the methodology is Random Forests based the framework can be estimated on large numbers of risk indicators up to big data dimensions, fostering the stability of the results while reducing inaccuracies in estimated stress scenarios by only stressing a small part of the observations. This procedure allows accurate forecasting of events under stress and the emergence of a potential macro crisis. The framework also estimates a set of the most influential economic indicators leading to the potential crisis, which can then be used as indications of remediation or prevention.

In this paper, we present a method to obtain upper and lower bounds on integrals with respect to copulas by solving the corresponding assignment problems (AP’s). In their 2014 paper, Hofer and Iacó proposed this approach for two dimensions and stated the generalization to arbitrary dimensons as an open problem. We will clarify the connection between copulas and AP’s and thus find an extension to the multidimensional case. Furthermore, we provide convergence statements and, as applications, we consider three dimensional dependence measures as well as an example from finance.

The paper uses Lévy processes and bivariate Lévy copulae in order to model the behavior of intraday log-returns. Based on assumptions about the form of marginal tail integrals and a Clayton Lévy copula, the model allows for capturing intraday cross-dependency. The model is applied to VaR of the portfolios constructed on stock returns as well as on cryptocurrencies. The proposed method shows fair performance compared to classical time series models.

The purpose of this paper is twofold. First we consider a ruin theory approach along with risk measures in order to determine the solvency capital of long-term guarantees such as life insurances or pension products. Secondly, for such products,we challenge the definition of the Solvency Capital Requirement (SCR) under the Solvency II (SII) regulatory framework based on a yearly viewpoint. Several methods for the calculation of the solvency capital are presented. We start our study with risk measures as considered in the SII framework and then proceed with the ruin theory approach. Instead of considering the continuous time setting of the ruin theory,we consider the discrete time—the yearly basis—of the accounting viewpoint.We finally give an illustration with a fixed guaranteed rate product along with the equity, interest rate and longevity risks. The latter risk brings us to consider zero-coupon longevity bonds in which we invest the capital. We show that long-term guarantees might be overloaded under the SII regulation.

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.

One of risk measures’ key purposes is to consistently rank and distinguish between different risk profiles. From a practical perspective, a risk measure should also be robust, that is, insensitive to small perturbations in input assumptions. It is known in the literature [14, 39], that strong assumptions on the risk measure’s ability to distinguish between risks may lead to a lack of robustness. We address the trade-off between robustness and consistent risk ranking by specifying the regions in the space of distribution functions, where law-invariant convex risk measures are indeed robust. Examples include the set of random variables with bounded second moment and those that are less volatile (in convex order) than random variables in a given uniformly integrable set. Typically, a risk measure is evaluated on the output of an aggregation function defined on a set of random input vectors. Extending the definition of robustness to this setting, we find that law-invariant convex risk measures are robust for any aggregation function that satisfies a linear growth condition in the tail, provided that the set of possible marginals is uniformly integrable. Thus, we obtain that all law-invariant convex risk measures possess the aggregation-robustness property introduced by [26] and further studied by [40]. This is in contrast to the widely-used, non-convex, risk measure Value-at-Risk, whose robustness in a risk aggregation context requires restricting the possible dependence structures of the input vectors.

Based on a novel extension of classical Hoeffding-Fréchet bounds, we provide an upper VaR bound for joint risk portfolios with fixed marginal distributions and positive dependence information. The positive dependence information can be assumed to hold in the tails, in some central part, or on a general subset of the domain of the distribution function of a risk portfolio. The newly provided VaR bound can be interpreted as a comonotonic VaR computed at a distorted confidence level and its quality is illustrated in a series of examples of practical interest.

Variable annuities contain complex guarantees, whose fair market value cannot be calculated in closed form. To value the guarantees, insurance companies rely heavily on Monte Carlo simulation, which is extremely computationally demanding for large portfolios of variable annuity policies. Metamodeling approaches have been proposed to address these computational issues. An important step of metamodeling approaches is the experimental design that selects a small number of representative variable annuity policies for building metamodels. In this paper, we compare empirically several multivariate experimental design methods for the GB2 regression model, which has been recently discovered to be an attractive model to estimate the fair market value of variable annuity guarantees. Among the experimental design methods examined, we found that the data clustering method and the conditional Latin hypercube sampling method produce the most accurate results.