Risk Aggregation: Copula Approach Ken Seng Tan, Ph.D., ASA, CERA Canada Research Chair in Quantitative Risk Management April 18-19, 2009 kstan@uwaterloo.ca

kstan@uwaterloo.ca SOA CERA - EPP Introduction The goal of integrated risk management in a financial institution is to both measure and manage risk and capital across a diverse range of activities in the banking, securities, and insurance sectors This requires an approach for aggregating different risk types, and hence risk distributions a problem found in many applications in finance including risk management and portfolio choice. kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Risk Aggregation Risk Driver kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Topics Measures of Association Copulas Which Copula to Use? Applications Concluding Remarks kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Topic I Measures of Association Comovements (or dependence) between variables Pearson correlation Its potential pitfalls Comonotonic risks Rank correlations Tail dependence Copulas Which Copula to Use? Applications Concluding Remarks kstan@uwaterloo.ca SOA CERA - EPP

Pearson Correlation of Coefficient: ρ(X,Y) Most common measure of dependence Definition: Properties: -1 ≤ ρ(X,Y) ≤ 1 If X & Y are independent, then ρ(X,Y) = 0. If |ρ(X,Y)| = 1, then X and Y are said to be perfectly linearly dependent  X = aY + b, for nonzero a  linear correlation coefficient Invariant under strictly increasing linear transformations: kstan@uwaterloo.ca SOA CERA - EPP

Is Pearson Correlation a Good Measure of Dependence? Var(X) and Var(Y) must be finite Problems with heavy-tailed distributions Possible values of correlation depend on the marginal (and joint) distribution of X and Y All values between -1 and 1 are not necessary attainable Perfectly positively (negatively) dependent risks do not necessarily have a Pearson correlation of 1 (-1) Correlation is not invariant under non-linear transformations of risks kstan@uwaterloo.ca SOA CERA - EPP

Example: Attainable Correlations Suppose X ~ N(0,1), Y ~ N(0,σ2) For a given ρ(X ,Y), what can you say about ρ(eX , eY)? nonlinear transformation eX and eY are lognormally distributed Now assume σ = 4: If ρ(X ,Y) = 1 ρ(eX , eY) = 0.01372 = ρ(eZ , eσZ) for Z ~ N(0,1), If ρ(X ,Y) = -1 ρ(eX , eY) = ρ(eZ , e-σZ) = -0.00025 This implies for -1≤ ρ(X ,Y) ≤ 1  -0.00025 ≤ ρ (eX , eY) ≤ 0.01372 Implications? kstan@uwaterloo.ca SOA CERA - EPP

What can we conclude from the last example? Pearson correlation is an effective way to represent comovements between variables if they are linked by linear relationships, but it may be severely flawed in the presence of non-linear links Need better measures of dependence! kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Comonotonic risks Comonotonicity is an extension of the concept of perfect correlation to random variables with non-linear relations. Two risks X and Y are comonotonic if there exists a r.v. Z and increasing functions u and v such that X = u(Z) and Y = v(Z) X and Y are countermonotonic if u increasing and v decreasing, or vice versa. Example I: Last example with X ~ N(0,1) and Y ~ N(0,σ2) eX & eY are comonotoic when ρ(X,Y) = 1 eX & eY are countermonotoic when ρ(X,Y) = -1 Example II: ceding company’s risk and reinsurer’s risk Not perfectly (linearly) dependent but they are comonotonic kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Rank Correlations Non-parametric (or distribution-free) measures of association by looking at the ranks of the data Only need to know the ordering (or ranks) of the sample for each variable and not its actual numerical value Does not depend on marginal distributions Invariant under strictly monotone transforms Two variants of rank correlation: Kendall’s Tau (ρτ) Spearman’s Rho (ρS) kstan@uwaterloo.ca SOA CERA - EPP

Properties of Rank Correlations Rank correlation measures the degree of monotonic dependence between X and Y, whereas linear correlation measures the degree of linear dependence rank correlations are alternatives to the linear correlation coefficient as a measure of dependence for nonelliptical distributions kstan@uwaterloo.ca SOA CERA - EPP

Coefficients of Tail Dependence In risk management, we are often concerned with extreme values, particularly their dependence in the tails The concept of tail dependence relates to the amount of dependence in the upper-right-quadrant or lower-left-quadrant tail of a bivariate distribution Provide measures of extremal dependence A measure of joint downside risk or joint upside potential A bivariate distribution can have either upper tail dependence, or lower tail dependence, or both, or none (i.e. tail independence) kstan@uwaterloo.ca SOA CERA - EPP

Simulated Samples of Some Bivariate Distributions kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Topic II Measures of Association Copulas What is a copula? Key results Some examples of copulas Other properties of copulas Which Copula to Use? Applications Concluding Remarks kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Basic Copula Primer Copulas provide important theoretical insights and practical applications in multivariate modeling The key idea of the copula approach is that a joint distribution can be factored into the marginals and a dependence function called a copula. The dependence structure is entirely determined by the copula Using a copula, marginal risks that are initially estimated separately can then be combined in a joint risk (or aggregate) distribution that preserves the original characteristics of the marginals. facilitate a bottom-up approach to multivariate model building; Given marginal distributions, the joint distribution is completely determined by its copula. kstan@uwaterloo.ca SOA CERA - EPP

Basic Copula Primer (cont’d) This implies that the multivariate modeling can be decomposed into two steps: Define the appropriate marginals and Choose the appropriate copula The separation of marginal and dependence is also useful from a practical (or calibration) point of view; Copulas express dependence on a quantile scale allow us to define a number of useful alternative dependence measures useful for describing the dependence of extreme outcomes the concept of quantile is also natural in risk management (e.g. VaR) kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP What is a Copula? A copula C is a multivariate uniform distribution function (d.f.) with standard uniform marginals We focus on bivariate case Bivariate copula: C(u,v) = Pr( U ≤ u, V ≤ v ) where U, V ~ Uniform(0,1) Connection between (bivariate) d.f., marginals and copula function: kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Does such a copula always exist? kstan@uwaterloo.ca SOA CERA - EPP

Mathematical Foundation: Sklar’s Theorem (1959) Suppose X and Y are r.v. with continuous d.f. FX & FY. If C is any copula, then is a joint d.f. with marginals FX & FY. Conversely, if FX,Y(x,y) is a joint d.f. with marginals FX & FY , then there exists a unique copula C such that Key result: Decomposition of multivariate d.f. Marginal information is embedded in FX & FY and the dependence structure is captured by the copula C(·,·) kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Examples of Copula: kstan@uwaterloo.ca SOA CERA - EPP

Copulas: Gaussian vs Student t kstan@uwaterloo.ca SOA CERA - EPP

Other Properties of Copulas Flexibility Useful when “off-the-shelf” multivariate distributions inadequately characterize the joint risk distribution Easy to simulate Invariant property Dependence measures Offers important insights to modeling dependence via rank correlations tail dependence kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP (1) Flexibility: The power of copula lies in its flexibility in creating multivariate d.f. via arbitrary marginals Useful when “off-the-shelf” multivariate distributions inadequately characterize the joint risk distribution Example: In credit risk modeling, the default time may be modeled as X1 ~ Inverse Gaussian The recovery rate may be modeled as X2 ~ Beta Interested in the joint distribution of X1 and X2 see Jouanin, Riboulet and Roncalli (2004) kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP 2) Easy to simulate Offers Monte Carlo risk studies risk measures economic capital stress testing … Simulated samples: Gaussian copula ρ = 0.7 Gumbel copula: θ = 2.0 Clayton copula: θ = 2.2 t-copula: v = 4 ρ =0.71 kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP 3) Invariant Property: Invariant under strictly increasing transformations of the marginals Let C be a copula for X & Y, If g(.) and h(.) are strictly increasing functions Then C is also the copula for g(X) and h(Y) This is due to the fact that copula relates the quantiles of the two distributions rather than the original variables Example: Consider two standard normals X & Y and let their dependence be represented by the Gaussian copula. Under increasing transforms, eX & eY still have the Gaussian copula Useful with confidentiality of banks’ or insurers’ data. Copulas can be estimated even if data is transformed appropriately. kstan@uwaterloo.ca SOA CERA - EPP

4a) Kendall’s Tau and Spearman’s Rho via Copula kstan@uwaterloo.ca SOA CERA - EPP

4b) Tail Dependence via Copula Recall that tail dependence relates to the magnitude of dependence in the upper-right-quadrant or lower-left-quadrant tail of a bivariate distribution the joint exceedance (tail) probabilities at high (and low) quantiles examine tail dependence either for a fixed quantile or asymptotically. kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Comparison of Tail dependence: Gaussian vs t copulas (std normal marginals) kstan@uwaterloo.ca SOA CERA - EPP

Joint Exceedance Probabilities at High Quantitles Joint exceedance probabilities are given for Normal copula For t-copula, we report the ratio of the joint exceedance probabilities of t-copula to normal-copula From Table 5.2 of McNeil, Frey and Embrechts (2005) kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Joint 99% (or equivalently 1%) Exceedance Probabilities in High Dimensions Consider daily returns on five stocks with constant ρ = 0.5. Impact on the choice of copula? Prob. on any day all returns are below 1% quantile How often does such an event happen on average? Gaussian 7.48 x 10-5 once every 53.1 years t (4 d.f.) (7.48 x 10-5) x 7.68 once every 6.9 years kstan@uwaterloo.ca SOA CERA - EPP

Asymptotic Tail Dependence Limiting probability Asymptotic upper tail dependence is obtained by taking the limit α-quantile  1 Asymptotic lower tail dependence is obtained by taking the limit α-quantile  0 limiting probability > 0 implies tail dependence Gaussian Asymptotic tail independence (ρ < 1) t Asymptotic tail dependence (ρ > -1) Gumbel Asymptotic upper tail dependence (θ > 1) Clayton Asymptotic lower tail dependence (θ > 0) kstan@uwaterloo.ca SOA CERA - EPP

Simulated Copulas with Standard Normal Marginals In all cases, linear correlation is around 0.7 Gumbel copula: θ = 2.0 Clayton copula: θ = 2.2 t copula: v = 4 ρ =0.71 kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Topic III Measures of Association Copulas Which Copula to Use? Applications Concluding Remarks kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Which Copula to Use? Given observed data set: { (x1,y1), …, (xT,yT) } how do we select a copula that reflects the underlying characteristics of the data? Model selection Principle of parsimony Akaike’s Information Criterion (AIC) Schwartz Bayesian Criterion (SBC) Klugman, Panjer and Willmot (2008) Loss Models: From Data to Decisions. Venter (2002) “Tails of Copulas” Genest, Remillard and Beaudoinc (in press): “Goodness-of-fit tests for copulas: A review and a power study” Parameter estimation One-step approach Two-step approach Model validation Goodness-of-fit test Kolmogorov-Smirnov test Anderson-Darling test … Examine tail dependence kstan@uwaterloo.ca SOA CERA - EPP

Parameter Estimation: One-Step Approach Direct Maximum Likelihood (ML) method Estimate jointly the marginals and the copula function using the method of ML nC + nX + nY dimensions optimization problem kstan@uwaterloo.ca SOA CERA - EPP

Parameter Estimation: Two-Step Approach Inference-functions for Margins (IFM) method Step 1: for each risk factor, independently determine parametric form of marginal, say, using method of ML nX parameters for 1st factor and nY parameters for 2nd factor Step 2: given marginals, determine copula using method of ML nC dimensions optimization problem Pseudo-likelihood method/Semi-parametric Approach Similar to IFM except that the marginals are the empirical cdf Rank-correlation-based Method of Moments Calibrating copula by matching to the empirical rank correlations, independent of marginals kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Topic IV Measures of Association Copulas Choosing the Right Copula Applications Concluding Remarks kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Frees, Carriere, and Valdez (1996): “Annuity Valuation with Dependent Mortality” Gompertz marginals (for both males and females) and Frank's copula are calibrated to the joint lives data from a large Canadian insurer. The estimation results show strong positive dependence between joint lives with real economic significance. The study shows a reduction of approximately 5% in annuity values when dependent mortality models are used, compared to the standard models that assume independence. kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Klugman and Parsa (1999): “Fitting Bivariate Loss Distributions with Copulas” Calibrate Frank’s copula to the joint distribution of loss and allocated loss adjustment expense (ALAE) for a liability line using 1,500 claims supplied by Insurance Services Office. Marginals: Examine a number of severity distributions Loss data: 2-parameter inverse paralogistic distribution ALAE: 3-parameter inverse Burr distribution Discuss ML inference for copulas and bivariate goodness-of-fit tests Frees and Valdez (1997) “Understanding relationships using copulas” Using similar data, they adopt Pareto marginals for both distributions and consider Frank’s copula and Gumbel copula kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Kole, Koedijk and Verbeek (2007): “Selecting Copulas for Risk Management” They show the importance of selecting an accurate copula for risk management. They extend standard goodness-of-fit tests to copulas. Using a portfolio consisting of stocks, bonds and real estate, these tests provide clear evidence in favor of the Student's t copula, and reject both Gaussian copula and Gumbel copula. Gaussian copula underestimates the probability of joint extreme downward movements, while the Gumbel copula overestimates this risk. Gaussian copula is too optimistic on diversification benefits, while the Gumbel copula is too pessimistic. These differences are significant. They also conclude that both dependence in the center and dependence in the tails are important kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Rosenberg and Schuermann (2006): “A general approach to integrated risk management with skewed, fat-tailed risks” A comprehensive study of banks’ returns driven by credit , market, and operational risks They propose a copula-based methodology to integrate a bank’s distributions of credit, market, and operational risk-driven returns. Their empirical analysis uses information from regulatory reports, market data, and vendor data most of them are publicly available, industry-wide data They examine the sensitivity of risk estimates to business mix, dependence structure, risk measure, and estimation method kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Fig 2 of Rosenberg and Schuermann (2006) kstan@uwaterloo.ca SOA CERA - EPP

Rosenberg and Schuermann (2006) (cont’d) Their findings: Given a risk type, total risk is more sensitive to differences in business mix or risk weights than to differences in inter-risk correlations The choice of copula (normal versus t ) has a modest effect on total risk Assuming perfect correlation overestimates risk by more than 40%. Assuming joint normality of the risks, underestimates risk by a similar amount kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP Concluding Remarks In this presentation, we discussed various dependence measures, highlighted pitfalls with the commonly used linear correlation; we introduced copula, particularly its role in modeling dependence and joint risk distributions; we reviewed various ways of calibrating copula to empirical data; we also examined some of its applications in insurance, finance, and risk management, kstan@uwaterloo.ca SOA CERA - EPP

Concluding Remarks (cont’d) A quote from Embrechts (2008) “Copulas: A personal view” : “… the question “which copula to use?” has no obvious answer. There definitely are many problems out there for which copula modeling is very useful. … Copula theory does not yield a magic trick to pull the model out of a hat.” Nevertheless copula has some obvious advantages: the separation of marginals and dependence modeling is appealing, particularly for problems with a large number of risk drivers it can still be a powerful tool, providing a simple way of coupling marginal d.f. while inducing dependence Tail dependence is important, especially for risk management “One of my probability friends, at the height of the copula craze to credit risk pricing, told me that “The Gauss–copula is the worst invention ever for credit risk management.” ” Embrechts (2008) Numerous studies have supported the use of the t-copula, as opposed to the Gaussian copula “All models are wrong but some are useful” George E.P. Box kstan@uwaterloo.ca SOA CERA - EPP

kstan@uwaterloo.ca SOA CERA - EPP References P. Embrechts (2008) “Copulas: A personal view” www.math.ethz.ch/~embrechts/ A. McNeil, R. Frey, P. Embrechts (2005) “Quantitative Risk Management” Princeton University Press. J. Yan (2007) “Enjoy the Joy of Copulas: With a Package copula”. Journal of Statistical Software vol. 21 issue #4. Copula R package (freeware) cran.r-project.org C. Genest, B. Remillard, and D. Beaudoin “Goodness-of-fit tests for copulas: A review and a power study” forthcoming in Insurance, Mathematics and Economics. E.W. Frees and E.A. Valdez (1997) “Understanding relationships using copulas” North American Actuarial Journal 2(1):1-25 E.W. Frees, J. Carriere, and E.A. Valdez (1996) “Annuity Valuation with Dependent Mortality.” Journal of Risk and Insurance, 63(2):229-261. J-F Jouanin, G. Riboulet and T. Roncalli (2004) “Financial Applications of Copula Functions” in Risk Measures for the 21st Century editor G. Szego. E. Kole, K. Koedijk, and M. Verbeek (2007) “Selecting copulas for risk management”, J of Banking & Finance 31:2405-2423. S.A. Klugman, H.H. Panjer, and G.E. Willmot (2008) Loss Models: From Data to Decisions. 3rd edition. Wiley. S.A. Klugman and R. Parsa (1999) “Fitting bivariate loss distributions with copulas” Insurance: Mathematics and Economics 24:139-148. J.V. Rosenberg and T. Schuermann (2006) “A general approach to integrated risk management with skewed, fat-tailed risks”, J of Financial Economics 79:569-614 G. Venter (2002) “Tails of Copulas”. Proceedings of the Casualty Actuarial Society, LXXXIX 2:68–113. kstan@uwaterloo.ca SOA CERA - EPP