Modern risk management calls for an understanding of stochastic dependence going beyond simple linear correlation. This paper deals with the static (non-time-dependent) case and emphasizes the copula representation of dependence for a random vector. Linear correlation is a natural dependence measure for multivariate normally and, more generally, elliptically distributed risks but other dependence concepts like comonotonicity and rank correlation should also be understood by the risk management practitioner. Using counterexamples the falsity of some commonly held views on correlation is demonstrated; in general, these fallacies arise from the naive assumption that dependence properties of the elliptical world also hold in the non-elliptical world. In particular, the problem of finding multivariate models which are consistent with prespecified marginal distributions and correlations is addressed. Pitfalls are highlighted and simulation algorithms avoiding these problems are constructed. ...

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The recent accounting scandals at Enron, WorldCom, and Tyco were related to the misrepresentation of liabilities. We provide a structural model of correlated multi-firm default, in which public bond investors are uncertain about the liability-dependent barrier at which individual firms default. In lack of complete information, investors form prior beliefs on the barriers, which they update with the default status information of firms arriving over time. Whenever a firm suddenly defaults, investors learn about the default threshold of closely associated business partner firms. Due to the unpredictable nature of defaults in our model, this updating leads to "contagious" jumps in credit spreads of business partner firms, which correspond to re-assessments of these firms' health by investors. We characterize joint default probabilities and the default dependence structure as assessed by imperfectly informed investors, where we emphasize the the modeling of dependence with copulas.

The t copula and its properties are described with a focus on issues related to the dependence of extreme values. The Gaussian mixture representation of a multivariate t distribution is used as a starting point to construct two new copulas, the skewed t copula and the grouped t copula, which allow more heterogeneity in the modelling of dependent observations. Extreme value considerations are used to derive two further new copulas: the t extreme value copula is the limiting copula of componentwise maxima of t distributed random vectors; the t lower tail copula is the limiting copula of bivariate observations from a t distribution that are conditioned to lie below some joint threshold that is progressively lowered. Both these copulas may be approximated for practical purposes by simpler, better-known copulas, these being the Gumbel and Clayton copulas respectively.

We consider the modelling of dependent defaults using latent variable models (the approach that underlies KMV and CreditMetrics) and mixture models (the approach underlying CreditRisk+). We explore the role of copulas in the latent variable framework and present results from a simulation study showing that even for fixed asset correlation assumptions concerning the dependence of the latent variables can have a large effect on the distribution of credit losses. We explore the effect of the tail of the mixing-distribution for the tail of the credit-loss distributions. Finally, we discuss the relation between latent variable models and mixture models and provide general conditions under which these models can be mapped into each other. Our contribution can be viewed as an analysis of the model risk associated with the modelling of dependence between credit losses.

Stylised facts for univariate high–frequency data in finance are well–known. They include scaling behaviour, volatility clustering, heavy tails, and seasonalities. The multivariate problem, however, has scarcely been addressed up to now. In this paper, bivariate series of high–frequency FX spot data for major FX markets are investigated. First, as an indispensable prerequisite for further analysis, the problem of simultaneous deseasonalisation of high–frequency data is addressed. In the following sections we analyse in detail the dependence structure as a function of the time scale. Particular emphasis is put on the tail behaviour, which is investigated by means of copulas.

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Building on the work of Bedford, Cooke and Joe, we show how multivariate data, which exhibit complex patterns of dependence in the tails, can be modelled using a cascade of pair-copulae, acting on two variables at a time. We use the pair-copula decomposition of a general multivariate distribution and propose a method to perform inference. The model construction is hierarchical in nature, the various levels corresponding to the incorporation of more variables in the conditioning sets, using pair-copulae as simple building blocs. Paircopula decomposed models also represent a very exible way to construct higher-dimensional coplulae. We apply the methodology to a nancial data set. Our approach represents the rst step towards developing of an unsupervised algorithm that explores the space of possible pair-copula models, that also can be applied to huge data sets automatically.

In the first part of this paper we address the non-coherence of value-at-risk (VaR) as a risk measure in the context of portfolio credit risk, and highlight some problems which follow from this theoretical deficiency. In particular, a realistic demonstration of the non-subadditivity of VaR is given and the possibly nonsensical consequences of VaR-based portfolio optimisation are shown. The second part of the paper discusses VaR and expected shortfall estimation for large balanced credit portfolios. All standard industry models (Creditmetrics, KMV, CreditRisk+) are presented as Bernoulli mixture models to facilitate their direct comparison. For homogeneuous groups it is shown that measures of tail risk for the loss distribution may be approximated in large portfolios by analysing the tail of the mixture distribution in the Bernoulli representation. An example is given showing that, for portfolios of lower quality, choice of model has some impact on measures of extreme risk.