Paul Embrechts

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.

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.

Dependence structures for multivariate high-frequency data in finance

Stylized 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 deseasonalization of high-frequency data is addressed. In the following sections we analyse in detail the dependence structure as a function of the timescale. Particular emphasis is put on the tail behaviour, which is investigated by means of copulas.

Estimates for the probability of ruin with special emphasis on the possibility of large claims

Abstract The present paper investigates, for the general Andersen model, the asymptotic behaviour of the probability of ruin function when the initial risk reserve tends to infinity. Whereas the exponential (Cramer) case is well understood, in the past, less attention has been paid to a systematic study of a model taking big claim sizes into account. We give a thorough treatment of the latter and also review previously known but mostly scattered results to show how they all follow from essentially one mathematical model.

Extreme Value Theory as a Risk Management Tool

The financial industry, including banking and insurance, is undergoing major changes. The (re)insurance industry is increasingly exposed to catastrophic losses for which the requested cover is only just available. An increasing complexity of financial instruments calls for sophisticated risk management tools. The securitization of risk and alternative risk transfer highlight the convergence of finance and insurance at the product level. Extreme value theory plays an important methodological role within risk management for insurance, reinsurance, and finance.

Subexponentiality and infinite divisibility

SummaryLet ℒ denote the class of subexponential distribution functions. For F infinitely divisible on [0, ∞) with Lévy measure v, the following assertions are proved to be equivalent:(i)F∈ℒ,(ii)v(1,x]/v(1,∞)∈ℒ,(iii)1−F(x)∼v(x, ∞) as x→∞. In the proof of this theorem, some new results on ∞ are established.

An Academic Response to Basel 3.5

Recent crises in the financial industry have shown weaknesses in the modeling of Risk-Weighted Assets (RWAs). Relatively minor model changes may lead to substantial changes in the RWA numbers. Similar problems are encountered in the Value-at-Risk (VaR)-aggregation of risks. In this article, we highlight some of the underlying issues, both methodologically, as well as through examples. In particular, we frame this discussion in the context of two recent regulatory documents we refer to as Basel 3.5.

Using copulae to bound the Value-at-Risk for functions of dependent risks

Abstract. The theory of copulae is known to provide a useful tool for modelling dependence in integrated risk management. In the present paper we review and extend some of the more recent results for finding distributional bounds for functions of dependent risks. As an example, the main emphasis is put on Value-at-Risk as a risk measure.

On convolution tails

In proving limit theorems for some stochastic processes, the following classes of distribution functions were introduced by Chover--Ney--Wainger and Chistyakov F belongs to ([lambda]) if and only if: 1. (i) 2. (ii) for all yreal, 3. (iii)[integral operator][infinity]0 e[lambda][lambda]dF(y)

An academic response to Basel II

It is our view that the Basel Committee of Banking Supervision, in its Basel II proposals, has failed to address many of the key deficiencies of the global financial regulatory system and even created the potential for new sources of instability. This document highlights our concerns that the failure of the proposals to address key issues can have destabilising effects and thus harm the global financial system. In particular, there is considerable scope for under-estimation of financial risk, which may lead to complacency on the part of policy makers and insufficient understanding of the likelihood of a systematic crisis. Furthermore, it is unfortunate that the Basel Committee has not considered how financial institutions will react to the new regulations. Of special concern is how the proposed regulations would induce the harmonisation of investment decisions during the crises with the consequence of destabilising rather than stabilising the global financial system.