Volume 6, Issue 1

The family of Clayton copulas is one of the most widely used Archimedean copulas for dependency measurement. A major drawback of this copula is that when it accounts for negative dependence, the copula is nonstrict and its support is dependent on the parameter. The main motivation for this paper is to address this drawback by introducing a new two-parameter family of strict Archimedean copulas to measure exchangeable multivariate dependence. Closed-form formulas for both complete and d−monotonicity parameter regions of the generator and the copula distribution function are derived. In addition, recursive formulas for both copula and radial densities are obtained. Simulation studies are conducted to assess the performance of the maximum likelihood estimators of d−variate copula under known marginals. Furthermore, derivativefree closed-form formulas for Kendall’s distribution function are derived. A real multivariate data example is provided to illustrate the flexibility of the new copula for negative association.

Previous research has focused on the importance of modeling the multivariate distribution for optimal portfolio allocation and active risk management. However, existing dynamic models are not easily applied to high-dimensional problems due to the curse of dimensionality. In this paper, we extend the framework of the Dynamic Conditional Correlation/Equicorrelation and an extreme value approach into a series of Dynamic Conditional Elliptical Copulas. We investigate risk measures such as Value at Risk (VaR) and Expected Shortfall (ES) for passive portfolios and dynamic optimal portfolios using Mean-Variance and ES criteria for a sample of US stocks over a period of 10 years. Our results suggest that (1) Modeling the marginal distribution is important for dynamic high-dimensional multivariate models. (2) Neglecting the dynamic dependence in the copula causes over-aggressive risk management. (3) The DCC/DECO Gaussian copula and t-copula work very well for both VaR and ES. (4) Grouped t-copulas and t-copulas with dynamic degrees of freedom further match the fat tail. (5) Correctly modeling the dependence structure makes an improvement in portfolio optimization with respect to tail risk. (6) Models driven by multivariate t innovations with exogenously given degrees of freedom provide a flexible and applicable alternative for optimal portfolio risk management.

Motivated by the nice characterization of copulas A for which d ∞ (A, A t ) is maximal as established independently by Nelsen [11] and Klement & Mesiar [7], we study maximum asymmetry with respect to the conditioning-based metric D 1 going back to Trutschnig [12]. Despite the fact that D 1 (A, A t ) is generally not straightforward to calculate, it is possible to provide both, a characterization and a handy representation of all copulas A maximizing D 1 (A, A t ). This representation is then used to prove the existence of copulas with full support maximizing D 1 (A, A t ). A comparison of D 1 - and d ∞ -asymmetry including some surprising examples rounds off the paper.

Some empirical studies suggest that the computation of certain graph structures from a (large) historical correlation matrix can be helpful in portfolio selection. In particular, a repeated finding is that information about the portfolio weights in the minimum variance portfolio (MVP) from classical Markowitz theory can be inferred from measurements of centrality in such graph structures. The present article compares the two concepts from a purely algebraic perspective. It is demonstrated that this heuristic relationship between graph centrality and the MVP does not originate from a structural similarity between the two portfolio selection mechanisms, but instead is due to specific features of observed correlation matrices. This means that empirically found relations between both concepts depend critically on the underlying historical data. Repeated empirical evidence for a strong relationship is hence shown to constitute a stylized fact of financial return time series.

For a given d-dimensional distribution function (df) H we introduce the class of dependence measures μ(H, Q) = −E{n H(Z1, . . . , Zd)}, where the random vector (Z1, . . . , Zd) has df Q which has the same marginal dfs as H. If both H and Q are max-stable dfs, we show that for a df F in the max-domain of attraction of H, this dependence measure explains the extremal dependence exhibited by F. Moreover, we prove that μ(H, Q) is the limit of the probability that the maxima of a random sample from F is marginally dominated by some random vector with df in the max-domain of attraction of Q. We show a similar result for the complete domination of the sample maxima which leads to another measure of dependence denoted by λ(Q, H). In the literature λ(H, H), with H a max-stable df, has been studied in the context of records, multiple maxima, concomitants of order statistics and concurrence probabilities. It turns out that both μ(H, Q) and λ(Q, H) are closely related. If H is max-stable we derive useful representations for both μ(H, Q) and λ(Q, H). Our applications include equivalent conditions for H to be a product df and F to have asymptotically independent components.

In this paper, a generalized class of run shock models associated with a bivariate sequence {(X i , Y i )} i≥1 of correlated random variables is defined and studied. For a system that is subject to shocks of random magnitudes X 1 , X 2 , ... over time, let the random variables Y 1 , Y 2 , ... denote times between arrivals of successive shocks. The lifetime of the system under this class is defined through a compound random variable T = ∑ N t=1 Y t , where N is a stopping time for the sequence {X i } i≤1 and represents the number of shocks that causes failure of the system. Another random variable of interest is the maximum shock size up to N, i.e. M = max {X i , 1≤i≤ N}. Distributions of T and M are investigated when N has a phase-type distribution.

We review various methods for constructing bivariate copulas with given diagonal sections from seminal work to the most recent research on copulas with given diagonal and opposite diagonal sections. A survey on a generalization of copulas with given diagonal plane sections in higher dimensions and other sections that generalize the diagonal and opposite diagonal sections is of particular interest.

We prove that different conditional distributions can be represented as distorted distributions. These representations are used to obtain stochastic comparisons and bounds for them based on properties of the underlying copula. These properties can be used to explain the meaning of mathematical properties of copulas connecting them with dependence concepts. Some applications and illustrative examples are provided as well.

We present an analytical proof of the characterisation of bivariate Archimax copulas in terms of the properties of their generating functions.

The main objective of this paper is to non-parametrically estimate the quantiles of a conditional distribution in the censorship model when the sample is considered as an -mixing sequence. First of all, a kernel type estimator for the conditional cumulative distribution function (cond-cdf) is introduced. Afterwards, we estimate the quantiles by inverting this estimated cond-cdf and state the asymptotic properties when the observations are linked with a single-index structure. The pointwise almost complete convergence and the uniform almost complete convergence (with rate) of the kernel estimate of this model are established. This approach can be applied in time series analysis.

We investigate the properties of a new transformation of copulas based on the co-copula and an univariate function. It is shown that several families in the copula literature can be interpreted as particular outputs of this transformation. Symmetry, association, ordering and dependence properties of the resulting copula are established.

This paper proposes competing procedures to the tests of symmetry for bivariate copulas of Genest, Nešlehová and Quessy (2012). To this end, the null hypothesis of symmetry is expressed in terms of the copula characteristic function that uniquely determines the copula of a given bivariate population with continuous marginal distributions. Then, test statistics based on L2 weighted distances computed from an empirical version of the copula characteristic function are proposed. Their asymptotic behavior is derived under the null hypothesis as well as under general alternatives. In particular, it is established that these rank statistics behave asymptotically as first-order degenerate V-statistics under the null hypothesis and this large-sample representation is exploited in order to provide suitably adapted multiplier bootstrap versions for the computation of p-values. The simulations that are reported show that the new tests are more powerful than the competing methods based on the empirical copula introduced by Genest, Nešlehová and Quessy (2012).

Williamson’s integral representation of n-monotone functions on the half-line is generalized to several dimensions. This leads to a characterization of multivariate survival functions with multiply ℓ 1 - symmetry. We then introduce a new class of generalized Archimedean copulas, where in contrast to nested Archimedean copulas no extra compatibility conditions for their generators are required.

We derive a new (lower) inequality between Kendall’s τ and Spearman’s ρ for two-dimensional Extreme-Value Copulas, show that this inequality is sharp in each point and conclude that the comonotonic and the product copula are the only Extreme-Value Copulas for which the well-known lower Hutchinson-Lai inequality is sharp.

Because of its many advantages, the use of decision trees has become an increasingly popular alternative predictive tool for building classification and regression models. Its origins date back for about five decades where the algorithm can be broadly described by repeatedly partitioning the regions of the explanatory variables and thereby creating a tree-based model for predicting the response. Innovations to the original methods, such as random forests and gradient boosting, have further improved the capabilities of using decision trees as a predictive model. In addition, the extension of using decision trees with multivariate response variables started to develop and it is the purpose of this paper to apply multivariate tree models to insurance claims data with correlated responses. This extension to multivariate response variables inherits several advantages of the univariate decision tree models such as distribution-free feature, ability to rank essential explanatory variables, and high predictive accuracy, to name a few. To illustrate the approach, we analyze a dataset drawn from the Wisconsin Local Government Property Insurance Fund (LGPIF)which offers multi-line insurance coverage of property, motor vehicle, and contractors’ equipments.With multivariate tree models, we are able to capture the inherent relationship among the response variables and we find that the marginal predictive model based on multivariate trees is an improvement in prediction accuracy from that based on simply the univariate trees.

We consider the problem of determining risk bounds for the Value at Risk for risk vectors X where besides the marginal distributions also information on the distribution or on the expectation of some functionals T j (X), 1 ≤ j ≤ m, is available. In particular this formulation includes the case where information on subgroup sums or maxima or on the correlations or covariances is available. Based on the method of dual bounds we obtain improved risk bounds compared to the marginal case. In general the explicit calculation of the dual bounds poses a challenge. We discuss various forms of relaxation of these bounds which are accessible and in some cases even lead to sharp bounds.

In this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property,whichwas introduced in [19], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on a rearrangement invariant space X with the strong Fatou property is (X, L1) lower semicontinuous and that the converse is true on a wide range of rearrangement invariant spaces. We also study inf-convolutions of law-invariant or surplus-invariant risk measures that preserve the (strong) Fatou property.

Aone-to-one correspondence is drawnbetween lawinvariant risk measures and divergences,which we define as functionals of pairs of probability measures on arbitrary standard Borel spaces satisfying a few natural properties. Divergences include many classical information divergence measures, such as relative entropy and convex f -divergences. Several properties of divergence and their duality with law invariant risk measures are characterized, such as joint semicontinuity and convexity, and we notably relate their chain rules or additivity properties with certain notions of time consistency for dynamic law risk measures known as acceptance and rejection consistency. The examples of shortfall risk measures and optimized certainty equivalents are discussed in detail.

Conditionally comonotonic risk vectors have been proved in [4] to yield worst case dependence structures maximizing the risk of the portfolio sum in partially specified risk factor models. In this paper we investigate the question how risk bounds depend on the specification of the pairwise copulas of the risk components X i with the systemic risk factor. As basic toolwe introduce a new ordering based on sign changes of the derivatives of copulas. This together with discretization by n-grids and the theory of supermodular transfers allows us to derive concrete ordering criteria for the maximal risks.

In the present paper we consider the Default Risk Charge (DRC) measure as an effective alternative to the Incremental Risk Charge (IRC) one, proposing its implementation by a quasi exhaustive-heuristic algorithm to determine the minimum capital requested to a bank facing the market risk associated to portfolios based on assets issued by several financial agents. While most of the banks use the Monte Carlo simulation approach and the empirical quantile to estimate this risk measure, we provide new computational approaches, exhaustive or heuristic, currently becoming feasible because of both the new regulation and to the high speed - low cost technology available nowadays. Concrete algorithms and numerical examples are provided to illustrate the effectiveness of the proposed techniques.