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The Use of Copulas in Risk Management

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The Use of Copulas in Risk Management THE USE OF COPULAS IN RISK MANAGEMENT by YOLANDA SOPHIA STANDER SHORT DISSERTATION submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE in STATISTICS in the FACULTY OF SCIENCE at the UNIVERSITY OF JOHANNESBURG SUPERVISOR: PROF. F LOMBARD JANUARY 2006 .5TA THE USE OF COPULAS IN RISK MANAGEMENT by YOLANDA SOPHIA STANDER SHORT DISSERTATION submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE in STATISTICS in the FACULTY OF SCIENCE at the UNIVERSITY OF JOHANNESBURG SUPERVISOR: PROF. F LOMBARD JANUARY 2006 TABLE OF CONTENTS INTRODUCTION 3 CHAPTER 1 : BASIC COPULA THEORY 5 1.1. Overview 5 1.1.1. Definition 5 1.2. Elliptical Copulas 6 1.2.1. Definition 6 1.22 Multivariate Gaussian copula 6 1.2.3. Multivariate Student's copula 7 1.2.4. Drawbacks to using elliptical copulas 7 1.3. Archimedean Copulas 8 1.3.1. Definition 8 1.3.2. Properties 8 1.3.3. Copula Distribution Function 9 1.4. Estimating Copula Functions 9 1.4.1. Nonparametric Estimation 9 1.4.2. Parametric Estimation Methods 10 1.4.3. Comparison of Parametric Estimation Methods 11 1.5. Goodness-of-Fit Testing 11 1.5.1. Conditional Distribution Function 11 1.5.2 Parametric Distribution Function of an Archimedean Copula 12 1.5.3. Nonparametric Distribution Function of an Archimedean Copula 12 1.5.4. Kolmogorov-Smimov Test 13 1.5.5. Akaike's Information Criteria (AIC) 14 1.6. Practical Example 15 1.7. Concluding Remarks 20 CHAPTER 2: DEPENDENCE MEASURES 21 2.1. Background 21 2.1.1. Overview 21 2.1.2. Definition of Dependence Measure 21 2.2. Linear Correlation 22 2.2.1. Definition 22 222. Shortcomings 22 2.3. Rank Correlation 23 1 23.1. Kendall's tau 23 2.3.2. Spearman 's rho 24 2.3.3. Advantages and shortcomings 24 2.4. The Copula as a Dependence Measure 25 2.4.1. Background 25 2.4.2. Example: Comparison of dependence measures 25 2.5. Concluding Remarks 30 CHAPTER 3: RISK MANAGEMENT AND EXTREMES 32 3.1. Overview 32 3.2. Extreme Value Theory 33 3.2.1. Block-Maxima Approach 33 3.22 Peaks-over-Threshold Approach 35 3.2.3. Extremal Index 36 3.24. Practical application 37 3.3. Multivariate Extreme Value Distribution 37 3.4. Extreme Copulas - The Bivariate Case 38 3.4.1. Background 38 3.4.2 Estimating the parameters 40 3.5. Application in Risk Management 40 3.6. Concluding Remarks 44 CONCLUDING REMARKS 45 REFERENCES 47 APPENDIX A: LOGLIKELIHOOD FUNCTIONS FOR VARIOUS COPULA 50 FAMILIES Overview 50 Gumbel Copula 50 Summary 51 2 INTRODUCTION In this dissertation we take a closer look at how copulas can be used to improve the risk measurement at a financial institution. The focus is on market risk in a trading environment. In practice risk numbers are calculated with very basic measures that are easy to explain to senior management and to traders. It is important that traders understand the risk measure as that helps them to understand the risk inherent in any deal and may assist them in deciding on the optimal hedge. The purpose of a hedge is to reduce the risk in a portfolio. As senior management is responsible for deciding on the optimal risk limits and risk appetite of the financial institution, it is important for them to understand what the risks are and how to measure these. The simplicity of the risk measures leads to certain inadequacies that can have very negative consequences for a financial institution. If the risk measure does not adequately capture the risk of a deal, the financial institution may suffer big losses when there are stress events in the market. Alternatively, when the risk measure overestimates the risk of a deal, too much economic capital is tied up in the deal. This inhibits the trader from adding more deals to a portfolio that may potentially lead to big profits. Economic capital is the capital that has to be held against positions to protect the financial institution if and when extreme market moves occur. In this dissertation the focus is on how copulas can be used to improve current risk measures. We focus on bivariate copulas. Bivariate copulas are easier to depict graphically than multivariate copulas with more than two dimensions. It is also easier to prove that the fitted bivariate copulae do adequately describe the underlying dependence structure between risk factors. Even though the focus is on the bivariate case, all methodologies can easily be extended to higher dimensions. In Chapter 1 copulas are defined and some basic copula properties are shown. We consider the definition of elliptical copulas and discuss some drawbacks to using them in a financial application. Some useful Archimedean copula properties are discussed and it is shown how to generate the copula function for n 2 dimensions. The various ways in which to estimate the parameters of a copula are also discussed as well as goodness-of-fit tests that are used to test whether the copula fits the underlying data adequately. Finally the chapter ends with an example that illustrates the theory. A back-test is done to establish whether the copula adequately describes the dependence structure over time. It is also shown how the fitted 3 copula can be used to generate stress scenarios that are used as an alternative to historical scenarios when calculating a value-at-risk (VaR) number. In chapter 2 the properties of a dependence measure are discussed and it is argued that linear correlation does not conform to these desired properties. Rank correlation measures have some additional properties that make them more efficient than linear correlation measures in certain instances. We also consider their relationship to copulas. Finally it is shown how copulas can be used in practice to get another view on the dependence structure between risk factors. In risk measurement we are mainly concerned with extreme moves that market variables may show. In chapter 3 some of the techniques used in risk management are discussed as well as some of their shortcomings. The shortcomings are addressed by applying extreme value theory to calculate stress factors and using copulas to model the dependence structure between risk factors. The theory underlying bivariate extreme copulas is discussed and illustrated with a practical example. 4 CHAPTER 1 : BASIC COPULA THEORY 1.1. OVERVIEW 1.1.1. Definition A copula can be defined as a multivariate distribution function on [0,1] N with uniformly distributed marginals. Sklar's Theorem states that if F is a *dimensional distribution function with marginals F1,...,FN, then there exists an *copula C such that for all x in 9l" F(X1,...,XN)=C(Fi(X1),...,FN(XN)). (1.1) The converse is also true, in other words if C is a N-copula and are univariate distribution functions, then the function F is an N-dimensional distribution function with margins Equation (1.1) can be restated as follows: C(ui , uN ) = F(F1-1 (ui , FN 1 (UN )) (1.2) for any (ui,...,uN) in [0,1]N. Cis unique when are continuous, otherwise Cis uniquely determined on Ran F1 x x Ran F, where Ran F denotes the range of the function F (Cherubini et al, 2004 pp.135-136; Embrechts et al., 2001 p.4). Similarly the survival copula C is defined by (1.3) where =1- F(.). The copula C has the property that C(1,...,1,u,1,...,1)= u for all u in [0,1] (Embrechts et al, 2001 p.3). The density c of a copula is given by: C(1/1,...,UN)= au,...auN (1.4) The relationship between the copula density and the density f of the /dimensional distribution F is: N f(xi ,...,xN )=c(Fi (xi ),...,FN (XN))nfn (xn ) n =1 (1.5) 5 where f,, is the density of the distribution function F,,. 1.2. ELLIPTICAL COPULAS 1.2.1. Definition Suppose we have a N-dimensional random vector X a vector p E 9I N with the locality parameters, and a Nx N positive definite symmetric matrix E If the characteristic function (t) of X-p is a function of the quadratic form trit then we say that Xhas an elliptical distribution. An alternative definition of an elliptical copula is to note that X has an elliptical distribution with the rank(2)=k if and only if there exists a positive random variable R, independent of a k-dimensional uniformly distributed random vector U, and a n x k matrix A where AAT = I such that X =d ,u+RAU Elliptical copulas are the copulas of elliptical distributions like the multivariate normal and Student t (Embrechts et al., 2001 pp.22-30). 1.2.2. Multivariate Gaussian copula A multivariate Gaussian copula is defined by: C(ui ,...uN )= 10171 0/1 , ON-1 (UN )] where alp denote the standard multivariate normal distribution with correlation matrix p. The multivariate normal density function is given by Op (x)= exp(- x p x NI 1 2 (2r01 where x= while the univariate standard normal density function is given by fn (x n ) = exp(-- x 2 L n . Thus, from (1.5) 1 r -1. exp(- x p x) N 1 2 1 (27r)T Ipli c(01. (xi ), . - - tOn (X n), --- 1 0 N(X N).1= N 1 1 2 1-1—,_ exp(- - xn ) n=-142ir 2 and by setting u „ = Pn( ) n ) it follows that the copula density (1.4) is 6 N 1(--exp 2 [0-1(4 P-ik-1 (u)l) (2707 I pl c(u)= (2707 expi— 2 (or01)[0-1( 1 exp( — 2 (ro [p -i 1191 2 (1.6) where u = (4,...,u,v ) and IN denotes the N x N identity matrix (Bouye et al., 2001b pp.14-17).
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