Testing the hypothesis that international equity market correlation increases in volatile times is a difficult exercise and misleading results have often been reported in the past because of a spurious relationship between correlation and volatility. This paper focuses on extreme correlation, that is to say the correlation between returns in either the negative or positive tail of the multivariate distribution. Using “extreme value theory ” to model the multivariate distribution tails, we derive the distribution of extreme correlation for a wide class of return distributions. Using monthly data on the five largest stock markets from 1958 to 1996, we reject the null hypothesis of multivariate normality for the negative tail, but not for the positive tail. We also find that correlation is not related to market volatility per se but to the market trend. Correlation increases in bear markets, but not in bull markets.

Abstract. Fundamental properties of conditional value-at-risk, as a measure of risk with significant advantages over value-at-risk, are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence of models based on scenarios and finite sampling. Conditional value-at-risk is able to quantify dangers beyond value-at-risk, and moreover it is coherent. It provides optimization shortcuts which, through linear programming techniques, make practical many large-scale calculations that could otherwise be out of reach. The numerical efficiency and stability of such calculations, shown in several case studies, are illustrated further with an example of index tracking. Key Words: Value-at-risk, conditional value-at-risk, mean shortfall, coherent risk measures, risk sampling, scenarios, hedging, index tracking, portfolio optimization, risk management

We examine "realized" daily equity return volatilities and correlations obtained from high-frequency intraday transaction prices on individual stocks in the Dow Jones

In an insurance c o text,o ne isoB interested in thedistributio functio o a sum o rando variables. Such a sum appears when coBBthe aggregate claimso f an insurance po rtfo o ver a certain reference perio d. Italso appears when coBdisco ted payments related to a single po licyo a poBat di#erent future po ints in time. Theassumptio o mutual independence between the coB oB ts o the sum is very co venient fro a coB- po int o view, butsoB no a realistico ne. In The Concept of Comonotonicity in Actuarial Science and Finance: Theory, we determined appro ximatio fo sumso f rando variables, when thedistributio o f the coB oB ts are kno wn, but thesto chastic dependence structure between them isunkno wn oto o cumberso to wo rk with. Practical applicatio o f thistheo will becoBin this paper. o. papers are to a large extent ano verviewo recent research resultsos tained by theautho butalso newtheoBand practical results are presented. 1

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.

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.

In this paper we present a new approach to incorporate dynamic default dependency in intensity-based default risk models. The model uses an arbitrary default dependency structure which is specified by the Copula of the times of default, this is combined with individual intensity-based models for the defaults of the obligors without loss of the calibration of the individual default-intensity models. The dynamics of the survival probabilities and credit spreads of individual obligors are derived and it is shown that in situations with positive dependence, the default of one obligor causes the credit spreads of the other obligors to jump upwards, as it is experienced empirically in situations with credit contagion. For the

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www.math.ethz.ch/finance 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 bulk of the paper 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 and spectral measures. 1

This paper inv estigates the potential for extreme co-mov ements between financial assets by directly testing the underlying dependence structure. In particular, a t-dependence structure, deriv ed from the Student t distribution, is used as a proxy to test for this extremal behav#a(0 Tests in three di#erent markets (equities, currencies, and commodities) indicate that extreme co-mov ements are statistically significant. Moreov er, the "correlation-based" Gaussian dependence structure, underlying the multiv ariate Normal distribution, is rejected with negligible error probability when tested against the t-dependencealternativ e. The economic significance of these results is illustratedv ia three examples: co-mov ements across the G5 equity markets; portfoliov alue-at-risk calculations; and, pricing creditderiv ativ es. JEL Classification: C12, C15, C52, G11. Keywords: asset returns, extreme co-mov ements, copulas, dependence modeling, hypothesis testing, pseudo-likelihood, portfolio models, risk management. # The authorsw ould like to thankAndrew Ang, Mark Broadie, Loran Chollete, and Paul Glasserman for their helpful comments on an earlier version of this manuscript. Both authors arewS; the Columbia Graduate School of Business, e-mail: {rm586,assaf.zeevi}@columbia.edu, current version available at www.columbia.edu\# rm586 1 Introducti7 Specification and identification of dependencies between financial assets is a key ingredient in almost all financial applications: portfolio management, risk assessment, pricing, and hedging, to name but a few. The seminal work of Markowitz (1959) and the early introduction of the Gaussian modeling paradigm, in particular dynamic Brownian-based models, hav e both contributed greatly to making the concept of co rrelatio almost synony...