Alexander J. McNeil

Risk Management: Correlation and Dependence in Risk Management: Properties and Pitfalls

Abstract Modern risk management calls for an understanding of stochastic dependence going beyond simple linear correlation. This article deals with the static (nontime- 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. Introduction Correlation in finance and insurance In financial theory the notion of correlation is central. The Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT) (Campbell, Lo & MacKinlay 1997) use correlation as a measure of dependence between different financial instruments and employ an elegant theory, which is essentially founded on an assumption of multivariate normally distributed returns, in order to arrive at an optimal portfolio selection. Although insurance has traditionally been built on the assumption of independence and the law of large numbers has governed the determination of premiums, the increasing complexity of insurance and reinsurance products has led recently to increased actuarial interest in the modelling of dependent risks (Wang 1997); an example is the emergence of more intricate multi-line products.

Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach

We propose a method for estimating Value at Risk VaR and related risk measures describing the tail of the conditional distribution of a heteroscedastic financial return series. Our approach combines pseudo-maximum-likelihood fitting of GARCH models to estimate . the current volatility and extreme value theory EVT for estimating the tail of the innovation distribution of the GARCH model. We use our method to estimate conditional . quantiles VaR and conditional expected shortfalls the expected size of a return exceeding . VaR , this being an alternative measure of tail risk with better theoretical properties than the quantile. Using backtesting of historical daily return series we show that our procedure gives better 1-day estimates than methods which ignore the heavy tails of the innovations or the stochastic nature of the volatility. With the help of our fitted models we adopt a Monte Carlo approach to estimating the conditional quantiles of returns over multiple-day horizons and find that this outperforms the simple square-root-of-time scaling method. q 2000 Elsevier Science B.V. All rights reserved.

Chapter 8 – Modelling Dependence with Copulas and Applications to Risk Management

The dependence between random variables is completely described by their joint distribution. However, dependence and marginal behavior can be separated. The copula of a multivariate distribution can be considered to be the part describing the dependence structure. Furthermore, strictly increasing transformations of the underlying random variables result in the transformed variables having the same copula. Hence copulas are invariant under strictly increasing transformations of the margins. This provides a way of studying scale-invariant measures of associations and also a starting point for construction of multivariate distributions. Scale-invariant measures of association such as Kendall’s tau and Spearman’s rho only depend on the copula and are thus invariant under strictly increasing transformations of the margins, which means that we can apply arbitrary continuous margins to our chosen copula leaving among other things the measures of association unchanged. Tail dependence and Kendall’s tau and Spearman’s rho are presented and evaluated for a large number of copula families. Among these copula families are families suitable for modelling extreme events, which are highly relevant as a basis for risk models in insurance and finance. The multivariate normal distribution and linear correlation are the basis of most models used to model dependence. Even though this distribution has a wide range of dependence it is quite seldom suitable for modelling real world situations in insurance and finance. We will show that using a model based on the multivariate normal distribution without knowledge of its limitations can prove very dangerous. Linear correlation is a natural measure of dependence in the context of the normal distribution. However, it should be noted that it is not invariant under strictly increasing transformations of the marginals and can be misleading as a measure of dependence. The problem of simulating dependent data arises naturally in Monte Carlo approaches to risk management. One main aim of this paper is to show that when addressing this problem knowledge of copulas and copula based dependence concepts is important, and also the usefulness of copula ideas in this approach to risk management. Another main aim of this paper is the construction of multivariate extensions of bivariate copula families. In particular we focus on multivariate extensions with a flexible and wide range of dependence for which efficient algorithms for random variate generation are presented. Acknowledgements This thesis was written during my stay at ETH in Zürich the fall 1999. I would like to express my gratitude and appreciation to Alexander McNeil for whom I had the pleasure of working with my thesis. I would also like to thank RiskLab for providing me with an office and Roger Kaufmann at RiskLab for valuable suggestions regarding the layout. Finally, I would like to thank Paul Embrechts at ETH and Jan Grandell, my supervisor at KTH in Stockholm, for giving me the opportunity to visit ETH.

Multivariate Archimedean copulas, d-monotone functions and ℓ1-norm symmetric distributions

It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a d-dimensional copula is that the generator is a d-monotone function. The class of d-dimensional Archimedean copulas is shown to coincide with the class of survival copulas of d-dimensional l 1 ; -norm symmetric distributions that place no point mass at the origin. The d-monotone Archimedean copula generators may be characterized using a little-known integral transform of Williamson [Duke Math. J. 23 (1956) 189-207] in an analogous manner to the well-known Bernstein-Widder characterization of completely monotone generators in terms of the Laplace transform. These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Archimedean copulas. They also yield useful expressions for the d-dimensional Kendall function and Kendalls rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for Archimedean copulas. The existence of a sharp lower bound for Archimedean copulas with respect to the positive lower orthant dependence ordering is shown.

Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory

Good estimates for the tails of loss severity distributions are essential for pricing or positioning high-excess loss layers in reinsurance. We describe parametric curve-fitting methods for modelling extreme historical losses. These methods revolve around the generalized Pareto distribution and are supported by extreme value theory. We summarize relevant theoretical results and provide an extensive example of their application to Danish data on large fire insurance losses.

Dependent defaults in models of portfolio credit risk

We analyse the mathematical structure of portfolio credit risk models with particular regard to the modelling of dependence between default events in these models. We explore the role of copulas in latent variable models (the approach that underlies KMV and CreditMetrics) and use non-Gaussian copulas to present extensions to standard industry models. We explore the role of the mixing distribution in Bernoulli mixture models (the approach underlying CreditRisk) and derive large portfolio approximations for the loss distribution. We show that all currently used latent variable models can be mapped into equivalent mixture models, which facilitates their simulation, statistical fitting and the study of their large portfolio properties. Finally we develop and test several approaches to model calibration based on the Bernoulli mixture representation; we find that maximum likelihood estimation of parametric mixture models generally outperforms simple moment estimation methods. J.E.L. Subject Classification: G31, G11, C15

Latent Curve Models: A Structural Equation Approach

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Sampling nested Archimedean copulas

We give algorithms for sampling from non-exchangeable Archimedean copulas created by the nesting of Archimedean copula generators, where in the most general algorithm the generators may be nested to an arbitrary depth. These algorithms are based on mixture representations of these copulas using Laplace transforms. While in principle the approach applies to all nested Archimedean copulas, in practice the approach is restricted to certain cases where we are able to sample distributions with given Laplace transforms. Precise instructions are given for the case when all generators are taken from the Gumbel parametric family or the Clayton family; the Gumbel case in particular proves very easy to simulate.

Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling

The idea of using common Poisson shock processes to model dependent event frequencies is well known in the reliability literature. In this paper we examine these models in the context of insurance loss modelling and credit risk modelling. To do this we set up a very general common shock framework for losses of a number of different types that allows for both dependence in loss frequencies across types and dependence in loss severities. Our aims are threefold: to demonstrate that the common shock model is a very natural way of approaching the modelling of dependent losses in an insurance or risk management context; to provide a summary of some analytical results concerning the nature of the dependence implied by the common shock specification; to examine the aggregate loss distribution that results from the model and its sensitivity to the specification of the model parameters.

VaR and expected shortfall in portfolios of dependent credit risks: Conceptual and practical insights

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 þ ) are presented as Bernoulli mixture models to facilitate their direct comparison. For homogeneous 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. � 2002 Elsevier Science B.V. All rights reserved.