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The t copula and related copulas
 INTERNATIONAL STATISTICAL REVIEW
, 2005
"... 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 m ..."
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Cited by 52 (0 self)
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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, betterknown copulas, these being the Gumbel and Clayton copulas respectively.
On the OutofSample Importance of Skewness and Asymmetric Dependence for Asset Allocation
 Journal of Financial Econometrics
, 2004
"... Recent studies in the empirical finance literature have reported evidence of two types of asymmetries in the joint distribution of stock returns. The first is skewness in the distribution of individual stock returns. The second is an asymmetry in the dependence between stocks: stock returns appear t ..."
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Cited by 47 (3 self)
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Recent studies in the empirical finance literature have reported evidence of two types of asymmetries in the joint distribution of stock returns. The first is skewness in the distribution of individual stock returns. The second is an asymmetry in the dependence between stocks: stock returns appear to be more highly correlated during market downturns than during market upturns. In this article we examine the economic and statistical significance of these asymmetries for asset allocation decisions in an outofsample setting. We consider the problem of a constant relative risk aversion (CRRA) investor allocating wealth between the riskfree asset, a smallcap portfolio, and a largecap portfolio. We use models that can capture timevarying moments up to the fourth order, and we use copula theory to construct models of the timevarying dependence structure that allow for different dependence during bear markets than bull markets. The importance of these two asymmetries for asset allocation is assessed by comparing the performance of a portfolio based on a normal distribution model with a portfolio based on a more flexible distribution model. For investors with no shortsales constraints, we find that knowledge of higher moments and asymmetric dependence leads to gains that are economically significant and statistically significant in some cases. For short salesconstrained investors the gains are limited.
PairCopula Constructions of Multiple Dependence
"... 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 paircopulae, acting on two variables at a time. We use the paircopula decomposition of a general multivariate distribution an ..."
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Cited by 42 (12 self)
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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 paircopulae, acting on two variables at a time. We use the paircopula 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 paircopulae as simple building blocs. Paircopula decomposed models also represent a very exible way to construct higherdimensional 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 paircopula models, that also can be applied to huge data sets automatically.
Copula goodnessoffit testing: an overview and power comparison
, 2007
"... Abstract. Several copula goodnessoffit approaches are examined, three of which are proposed in this paper. Results are presented from an extensive Monte Carlo study, where we examine the effect of dimension, sample size and strength of dependence on the nominal level and power of the different app ..."
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Cited by 18 (1 self)
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Abstract. Several copula goodnessoffit approaches are examined, three of which are proposed in this paper. Results are presented from an extensive Monte Carlo study, where we examine the effect of dimension, sample size and strength of dependence on the nominal level and power of the different approaches. While no approach is always the best, some stand out and conclusions and recommendations are made. A novel study of pvalue variation due to permuation order, for approaches based on Rosenblatt’s transformation is also carried out. Results show significant variation due to permutation order for some of the approaches based on this transform. However, when approaching rejection regions, the additional variation is negligible.
Portfolio credit risk with extremal dependence: Asymptotic Analysis and Efficient Simulation
, 2006
"... We consider the risk of a portfolio comprised of loans, bonds, and financial instruments that are subject to possible default. In particular, we are interested in performance measures such as the probability that the portfolio incurs large losses over a fixed time horizon and the expected excess los ..."
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Cited by 11 (1 self)
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We consider the risk of a portfolio comprised of loans, bonds, and financial instruments that are subject to possible default. In particular, we are interested in performance measures such as the probability that the portfolio incurs large losses over a fixed time horizon and the expected excess loss given that large losses are incurred during this horizon. Contrary to the normal copula that is commonly used in practice (e.g., in the CreditMetrics system), we assume a portfolio dependence structure that is semiparametric, does not hinge solely on correlation, and supports extremal dependence among obligors. A particular instance within the proposed class of models is the socalled tcopula model that is derived from the multivariate Student t distribution and hence generalizes the normal copula model. The size of the portfolio, the heterogenous mix of obligors, and the fact that default events are rare and mutually dependent makes it quite complicated to calculate portfolio credit risk either by means of exact analysis or naïve Monte Carlo simulation. The main contributions of this paper are twofold. We first derive sharp asymptotics for portfolio credit risk that illustrate the implications of extremal dependence among obligors. Using this as a stepping stone, we develop importance sampling algorithms that are shown to be asymptotically optimal and can be used to efficiently compute portfolio credit risk via Monte Carlo simulation.
Extreme Events and MultiName Credit Derivatives
, 2003
"... The dependence structure of asset returns lies at the heart of a class of models that is widely employed for the valuation of multiname credit derivatives. In this work, we study the dependence structure of asset returns using copula functions. In particular, using a statistical methodology that ..."
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Cited by 5 (0 self)
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The dependence structure of asset returns lies at the heart of a class of models that is widely employed for the valuation of multiname credit derivatives. In this work, we study the dependence structure of asset returns using copula functions. In particular, using a statistical methodology that relies on a minimal amount of distributional assumptions, we compare the dependence structures of asset and equity returns to provide some insight into the common practice of estimating the former using equity data. Our results show that the presence of joint extreme events in the data is not compatible with the assumption of Normal dependence, and support the use of equity returns as proxies for asset returns. Furthermore, we present evidence that the likelihood of joint extreme events does not diminish as we decrease the sampling frequency of our observations. Building on our empirical findings, we then describe how to capture the e#ects of joint extreme events by means of a simple and computationally e#cient timetodefault simulation. Using a tcopula model, we analyze the impact of extreme events on the fair values and risk measures of popular multiname credit derivatives such as n baskets and synthetic loss tranches.
Estimation and Forecasting of Dynamic Conditional Covariance: A Semiparametric Multivariate Model
, 2007
"... The existing parametric multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) model could hardly capture the nonlinearity and the nonnormality, which are widely observed in financial data. We propose semiparametric conditional covariance (SCC) model to capture the informat ..."
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Cited by 5 (0 self)
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The existing parametric multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) model could hardly capture the nonlinearity and the nonnormality, which are widely observed in financial data. We propose semiparametric conditional covariance (SCC) model to capture the information hidden in the standardized residuals and missed by the parametric MGARCH models. Our twostage SCC estimator incorporates the parametric and nonparametric estimators of the conditional covariance in a multiplicative way. We prove the consistency and asymptotic normality of our semiparametric estimator. We conduct a small set of Monte Carlo experiments to demonstrate the advantage of our SCC estimators over their parametric counterparts in terms of mean squared error. For both insample fitting and outofsample forecasting conditional covariance matrix, our SCC models also outperform the parametric ones in empirical applications on bivariate stock indices and two stock portfolios with thirty underlying stocks.
PairCopula Constructions of Multiple Dependence
"... 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 paircopulae, acting on two variables at a time. We use the paircopula decomposition of a general multivariate distribution an ..."
Abstract

Cited by 4 (3 self)
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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 paircopulae, acting on two variables at a time. We use the paircopula 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 paircopulae as simple building blocs. Paircopula decomposed models also represent a very exible way to construct higherdimensional 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 paircopula models, that also can be applied to huge data sets automatically.
Conditional Monte Carlo estimation of quantile sensitivities
, 2009
"... doi 10.1287/mnsc.1090.1090 ..."