Vrije Universiteit Amsterdam Faculty of Sciences MASTER THESIS. Michal Rychnovský Portfolio Credit Risk Models. Department of Mathematics

Transcription

1 Vrije Universiteit Amsterdam Faculty of Sciences MASTER THESIS Michal Rychnovský Portfolio Credit Risk Models Department of Mathematics Supervisor: Dr. P.J.C. Spreij Program of Study: Stochastics and Financial Mathematics 2010

2 Acknowledgments I would like to thank Dr. Peter Spreij for all his help, discussions and comments in writing this thesis. Michal Rychnovský ii

3 Abstract Title: Portfolio Credit Risk Models Author: Michal Rychnovský Author s address: michal@rychnovsky.cz Supervisor: Dr. P.J.C. Spreij Supervisor s university: Universiteit van Amsterdam Supervisor s department: Korteweg-de Vries Institute for Mathematics Supervisor s address: p.j.c.spreij@uva.nl Abstract: The aim of the present work is to describe possible models for portfolio credit risk. Our main interest is put on such models where the correlation structure is described by a copula. We start by introducing the terms of probability of default and expected loss, together with some single obligor models. Then we give an example of a duo basket model, followed by a discussion about copulas and dependence measures. Finally, we focus on threshold models and their limit behavior for number of loans going to infinity; here we derive the tail expression of the limit distribution of average losses. Keywords: Portfolio credit risk, copula, dependence measure, threshold model, limit behavior. iii

4 Contents Introduction 1 1 Single Obligor Credit Risk Models Expected Loss Scoring Models Dynamic Models Structural Models Duo Basket Example Threshold Model Default Correlation First-to-Default and Second-to-Default Probability Copula Functions Definition of Copula Sklar s Theorem Frechét-Hoeffding s bounds Comonotonicity and Countermonotonicity Some Examples of Copulae Dependence Measures Linear Correlation Rank Correlation Tail Dependence Multivariate Tail Dependence Portfolio Credit Risk Models Threshold Model Industry Models iv

5 5.3 Properties of Models Limit Behavior of Large Portfolios Expected Average Loss Polynomially Declining Tails Exponentially Declining Tails Conclusions 56 Bibliography 57 A Probability 60 A.1 Quantile Function A.2 Gamma and Inverse Gamma Distribution A.3 Multivariate t-distribution B Extreme Value Theory 63 B.1 Regularly Varying Functions B.2 Class M a θ v

6 Introduction The banking sector is one of the most profitable but also one of the riskiest industries, where most risk is connected to the event called default. Default is usually defined as a violation of debt contract conditions, such as a lack of will or a disability to pay a loan back. In the case of default, the creditor e.g. a bank or other financial institution suffers a loss. The risk is usually called credit risk. In order to quantify credit risk, banks try to develop analytical models to estimate default probabilities and expected losses, and calculate an adequate capital reserve to cover all unexpected losses. This is in practice done under supervision of regulatory institutions such as the Basel Committee on Banking Supervision. In this work we describe the principles of several credit risk models. In the first chapter we summarize the most common approaches to single obligor credit risk modeling. Then we start with an introductory example of a simple duo basket model. Our main interest is put on credit risk of a portfolio of loans, where its correlation structure is described by a copula. Therefore in Chapters 3 and 4 we describe the mathematical foundations of copula functions and dependence measures. In Chapter 5 we define two of the most common industry models based on copula functions. Finally, in the last chapter we focus on the average loss, when the number of losses goes to infinity, and describe the tail expression for the limit distribution. 1

7 Chapter 1 Single Obligor Credit Risk Models In the first chapter we introduce several common approaches used for credit risk modeling. Throughout this chapter we assume that we have one client with certain characteristics asking for a loan. The first section is mainly based on Rychnovský 2008 and the second section on Pazdera et al In the following chapters we then focus on the multiple loans portfolios. 1.1 Expected Loss Consider a probability space Ω, F, P with an F-measurable random variable L = 1 D, where D is the event that default occurs in a one-year horizon. Then we can define the one-year Probability of Default P D as P D = PL = What is important in the field of credit risk, is the distribution of potential losses, and in particular, the expected loss. In order to quantify the expected loss caused by the client s default, Basel II 2001 defines the following characteristics: Exposure at Default EAD is the amount of money owed at the moment of default, and Loss Given Default LGD is the percentage of EAD which has not been recovered after default. 1 LGD is usually assumed to be discounted and corrected for recovery costs. 1 At the moment of default the bank begins to recover its money back. This process is often partially successful; sometimes even the whole sum is recovered. 2

8 Then the expected loss can be calculated as EL = P D LGD EAD. 1.2 The unexpected loss is according to Basel II 2001 defined as the Value at Risk, i.e. the 1 α quantile of the loss distribution see Figure 1.1, where X is a potential loss and α is a predefined level. PX > VaR = α, 1.3 Figure 1.1: Value at Risk according to Basel For specification of Value at Risk or other quantiles Basel II 2001 uses Vašíček formula as the mapping function. For more details see the original articles Vašíček 1987, The three characteristics in 1.2 are usually modeled separately. In further sections we focus on different approaches for Probability of Default modeling. For more information about Loss Given Default and Exposure at Default modeling we refer to Engelmann and Rauhmeier 2006, Kim and Kim 2006 or Witzany et al Scoring Models There are several approaches for Probability of Default modeling, based on different information about the obligor. Usually we suppose we have a database of previous clients and their defaults to calibrate the model. For the scoring model we want to evaluate score the obligor according to his characteristics in order to quantify his ability to pay the money back and in consequence his P D. 3

9 Before presenting some of these scoring functions used in practice, we introduce some useful notation. Suppose that our database contains s explanatory categorical variables regressors for each client, where the i-th variable consists of s i categories. Then put Z = { i, j : i {1,..., s}, j {1,... s i, } } 1.4 the set of all ordered pairs i, j of variables i and their categories j. Then for each client k we have the vector x k = x i j k : i, j Z 1.5 of dummy variables i.e. x i j k = 1 if the client k lies in the category j of the variable i, and x i j k = 0 otherwise. Then we denote by G the index set of all non-defaulted clients and B the index set of all defaulted clients, and in the same spirit we define G i j = { k : k {1,..., n}, k G, x i j k = 1 } as the index set of all non-defaulted clients k lying in the category j of the variable i; and B i j = { k : k {1,..., n}, k B, x i j k = 1 } as the index set of all defaulted clients k lying in the category j of the variable i. Now, based on the database of the observed clients, we define the total odds as the ratio of the number of non-defaulted vs. defaulted clients in the sample, odds = G B, 1.6 and also for individual categories j of variable i the odds i j for certain category, odds i j = Gi j 1.7 Bj i. Finally we define the odds ratio OR i j as the ratio of categorical odds i j and total odds, OR i j = oddsi j odds. 1.8 Now, we are ready to introduce three scoring functions used in the financial practice. In all cases the scoring function represents an estimate of the client s 1 P Dx oddsx =, conditional to his regressors. The derivation and practical implementation of these models can be found in Rychnovský P Dx

10 1.2.1 Independence Model The Independence model is based on the assumption that all the values of the regressors are independent. Due to its simplicity, this model is very often used, although this assumption is in practice seldom fulfilled. In the independence model there are no parameters to be estimated and the conditional odds can be represented in the terms of a scoring function in the following way, S IM x = odds ORj i xi j, 1.9 i,j Z where x = x i j : i, j Z is the set of dummy variables representing the client. Sometimes also a logarithm of this function is used as a scoring function, 2 ln S IM x = lnodds + i,j Z x i j lnor i j The main disadvantage of this model is the assumption of independence and the fact that all categories have the same weights. Therefore, there are two more models generalizing this approach by adding a nonnegative weight to each variable or even to each category WOE Model The first generalization of the function 1.9 is the WOE model for Weight of Evidence, where to all variables a weight is assigned according to their statistical importance. 3 This way we get a scoring function in the form S W OE x, w = odds ORj i wi x i j, 1.11 i,j Z or a logarithm of the scoring function in the form ln S W OE x, w = lnodds + i,j Z w i x i j lnor i j According to Aspey et al. 2003, the WOE model is suitable for databases with at least 150 defaults. 2 In this case the score value corresponds to logit of the function 1 P D. 3 The weights are then estimated using logistic regression. For more information about logistic regression we refer to Agresti 1990 or Hosmer and Lemeshow

11 1.2.3 Full Logistic Model Finally, in the full logistic model we put a certain weight to each category of the categorical variables i.e. to each dummy variable. 4 The scoring function is then in the form S F LM x, w = odds ORj i wi j xi j, 1.13 or in its logarithmic form, i,j Z ln S F LM x, w = lnodds + i,j Z w i jx i j lnor i j This model is the most flexible but also the most complicated from the three introduced models. From the form 1.14 we can see that due to the used logit function this approach is equivalent to the common logistic regression used for PD estimation. According to Aspey et al. 2003, the full logistic model is suitable for databases with at least 1200 defaults. 1.3 Dynamic Models Whereas the scoring models estimate only the one-year Probability of Default based on defaults occurring in a one year horizon, dynamic models understand the time until default as a random variable. One of the most common dynamic approaches is based on survival analysis. This section is based mainly on Kalbfleisch et al and Collett Survival Analysis Survival analysis deals with modeling of the time elapsed until some particular event occurs it is called exit or end-point, conditional on the specific characteristics of the subject. In the case of P D modeling we model the time until default of the client with given characteristics. First we introduce several terms of survival analysis. Assume that X is an absolutely continuous nonnegative random variable representing the time to default of a client. Denote F the distribution function and f the density of X. Then we define a hazard function or intensity of the client as 1 λt = lim Pt X < t + h X t h 0+ h 4 The weights are then again estimated using logistic regression. 6

12 By a survivor function St we denote the probability that the client will not default until time t will survive, i.e. St = 1 F t. Using this relation we can rewrite the hazard function 1.15 into the form λt = lim h 0+ F t + h F t h From 1.16 we get also a converse relation [ St = exp 1 St = ft St = d log St dt t Finally, we define a cumulative hazard function as Λt = t 0 0 ] λudu λudu = log St Cox Model D. R. Cox in Cox 1972 assumed the hazard function of subject k at time t in the form λt; x k = λ 0 t expx kβ, 1.19 where x k is the vector of characteristics of subject k and β is a vector of parameters. The function λ 0 t is then called a baseline hazard function, independent of the client s characteristics. Due to the fact that the relation λt; x k λt; x l = expx k β expx l β depends only on clients characteristics, the Cox model is often called the proportional hazards model. In Cox 1975 Cox introduced a generalization of 1.19 by implementing time dependent explanatory characteristics x k t. This model then assumes the hazard function in the form λt, x k t = λ 0 t expx k t β The corresponding survivor function is then [ St, x k t = P T > t x k t = exp t 0 ] λ 0 u expx k u βdu. Then for the case of discrete time and no multiple defaults at any time, we can derive the Breslow-Crowley maximum likelihood estimator of the baseline hazard function. If n is the number of clients in our database, Y k t is an indicator that 7

13 client k has not defaulted until time t, and dn k t is an indicator that client k defaulted in the time interval t 1, t], we can estimate the baseline hazard function as n k=1 λ 0 t = dn kt n k=1 exp x k tβ Y k t For β we then substitute an estimate β. If m is the number of defaulted clients and t 1 < < t m are the observed default times, we can define auxiliary functions M and N as expx k t k β Mβ, t k = n i=1 Y kt k expx k t k β and N β, t k = m Y k t k x k t k Mβ, t k. i=1 Using this notation, the estimate of β can be computed from the expression for the partial likelihood function by solving the equation m [x k t k N β, t k ] = k=1 The t-year Probability of Default can be in terms of 1.17 and 1.20 then estimated as [ t ] P Dx, t = 1 exp λ 0 t expx βdu, where x is the vector of clients characteristics. For further details as well as formulas for multiple defaults we refer to Collett As a parametric alternative to the Cox model also the Accelerated Failure Time AFT model can be used see e.g. Kalbfleisch et al Structural Models The models presented in the previous sections all belong to the class of so called reduced form models. These models are easily calibrated to estimate the Probability of Default but give no information about the circumstances of the default. This is on the other hand the main aim of so called structural or firm-value models. These approaches are designed to model the underlying structure of the firm s value in time. In the following text we describe the Merton model as the original concept for many of the present structural models. 8

14 1.4.1 Merton Model The original model was introduced in Merton Consider a firm with a stochastic value process V t. Assume that the value V t of the firm s assets at time t consists of its equity value S t and its debt value B t the value in time t of a single debt obligation with maturity T and face value B. Thus, V t = S t + B t for t [0, T ] At time T two situations may occur. 1. V T > B. In this case the value of the firm s assets exceeds the value of the debt. Here the debt is fully recovered and the shareholders get the residual value. Then B T = B and S T = V T B. 2. V T B. In this case the value of the firm s assets is less than its liabilities, and the firm falls into default. Here all the value of the firm s assets is paid to the bondholders. Then B T = V T and S T = 0. Summarizing these two situations, we get similar expressions as we know from derivatives pricing models, S T = V T B B T = B B V T Therefore, to develop the Black-Scholes-type pricing model, Merton 1974 makes the following assumptions: 1. There are no transactions costs, taxes, or problems with indivisibilities of assets. 2. There is a sufficient number of investors with comparable wealth levels so that each investor believes that he can buy and sell as much of an asset as he wants at the market price. 3. There exists an exchange market for borrowing and lending at the same rate of interest. 4. Short-sales of all assets, with full use of the proceeds, is allowed. 5. Trading in assets takes place continuously in time. 6. The Modigliani-Miller theorem that the value of the firm is invariant to its capital structure obtains. 9

15 7. The Term-Structure is flat and known with certainty. I.e., the price of a riskless discount bond which promises a payment of one dollar at time T in the future is P T = exp[ rt ] where r is the instantaneous riskless rate of interest, the same for all time. 8. The dynamics for the value of the firm, V t, through time can be described by a diffusion-type stochastic process with stochastic differential equation dv t = αv t Cdt + σv t dw t, 1.27 where α is the instantaneous expected rate of return on the firm per unit time, C is the total dollar payouts by the firm per unit time to either its shareholders or liabilities-holders e.g., dividends or interest payments if positive, and it is the net dollars received by the firm from new financing if negative; σ 2 is the instantaneous variance of the return on the firm per unit time; dw t is a standard Wiener process. According to 1.25, the equity value at the terminal time T corresponds to a European call option on V t with strike price B and maturity T. Then the value of the equity today can be expressed in the Black-Scholes-type formula, S 0 = V 0 Φd 1 Be rt Φd 2, 1.28 where Φ is the cumulative distribution function of the standard normal distribution, and d 1 = log V 0 + r + 1 B 2 σ2 T σ, 1.29 T d 2 = d 1 σ T Moreover, under the risk neutral measure Q we have log V T N log V 0 + r 1 2 σ2 T, σ 2 T. And thus at time t = 0 we get the Probability of Default as log V 0 B P D = QV T B = 1 Φ + r σ2 T σ σ T, 1.31 T i.e. in the form P D = 1 Φd 2. For more information about structural models together with some applications we refer to McNeil et al or Schönbucher

16 Chapter 2 Duo Basket Example In this chapter we start with credit risk valuation for a portfolio consisting of more loans than one. For better understanding the basics of portfolio dependence structure, we start with an introductory example of a particular duo basket, i.e. a portfolio of two loans. This example is taken from Bluhm and Overbeck Threshold Model Suppose we have two loans in our basket. Loan A has its probability of default p A and loan B has its probability of default p B. For both loans we expect EAD being 100% of the face value and LGD = 1, i.e. in the case of default all the investment will be lost. In the case of threshold models we suppose that for each obligor there exists a random variable X representing a creditworthiness index, and that default occurs if and only if this index falls bellow a certain bound. In our example we follow an approach of CreditMetrics, see Gupton et al For a one-year horizon we introduce Bernoulli variables L A and L B, L A = 1 {XA <d A }, 2.1 L B = 1 {XB <d B }, 2.2 where X A and X B are standard normal random variables with correlation coefficient ρ, and d A and d B are real bounds. Then we get for the probabilities of default the relations p A = PL A = 1 = PX A < d A = Φd A, p B = PL B = 1 = PX B < d B = Φd B, 11

17 where Φ is the cumulative distribution function of the standard normal distribution. We also get the converse relations d A = Φ 1 p A, d B = Φ 1 p B. According to CreditMetrics, a specific structure is assumed. Suppose that X A and X B are defined as follows: X A = ρ Y + 1 ρ ε A, 2.3 X B = ρ Y + 1 ρ ε B, 2.4 where Y, ε A and ε B are independent, standard normal random variables. In this structure, Y is called a common factor representing the state of the economy, and ε A and ε B are specific factors for individual loans. It is not difficult to check that according to this definition, X A and X B are again standard normally distributed with correlation coefficient ρ. Moreover, we can compute the joint distribution of X A and X B. Proposition 2.1. Let Y, ε A and ε B be independent, standard normal random variables. Then X A and X B defined in 2.3 and 2.4 have a bivariate normal joint distribution, XA 0 1 ρ N 2, ρ 1 X B Proof. If we rewrite equations 2.3 and 2.4 into a matrix form, we get Y XA ρ 1 ρ 0 =, 2.6 ρ 0 1 ρ X B where due to the independence of its components, the random vector Y, ε A, ε B has a joint normal distribution, Y Z = N 3 0, We denote B = ε A ε B ρ 1 ρ 0 ρ 0 1 ρ normal distribution with the expectation XA E = B E Z = X B and get that 0 0 ε A ε B XA and the covariance matrix XA Var = VarBZ = BVar ZB = BB = X B X B = BZ has also a 1 ρ. ρ 1 12

18 2.2 Default Correlation In the previous section we defined ρ as the correlation coefficient of the random variables X A and X B. However, the real subject of our concern is the default correlation, i.e. the correlation between the default indicators L A and L B. Proposition 2.2. Let A and B be two loans with default probabilities p A and p B and correlation ρ = CorrX A, X B. Then under the CreditMetrics model the default correlation of the loans A and B is of the form CorrL A, L B = Φ 2,ρ Φ 1 p A, Φ 1 p B p A p B, 2.7 pa 1 p A p B 1 p B where Φ 2,ρ denotes the standard bivariate normal distribution function with correlation ρ, i.e. a b 1 Φ 2,ρ a, b = 2π 1 ρ exp x2 + y 2 ρxy dxdy ρ 2 Proof. First recall that L A and L B are Bernoulli variables with parameters p A and p B. Thus, we have E L A = p A, E L B = p B and Var L A = p A 1 p A, Var L B = p B 1 p B. Furthermore, we can compute E L A L B = PL A = 1, L B = 1 = PX A < d A, X B < d B = P X A < Φ 1 p A, X B < Φ 1 p B = Φ 2,ρ Φ 1 p A, Φ 1 p B, where the last step is a result of Proposition 2.1. Then we can compute the default correlation as CorrL A, L B = CovL A, L B Var LA Var LB = E L AL B E L A E L B Var LA Var LB = Φ 2,ρ Φ 1 p A, Φ 1 p B p A p B. pa 1 p A p B 1 p B 2.3 First-to-Default and Second-to-Default Probability In our explanatory example we try to simulate the concept of Collateralized Debt Obligations CDOs. In this concept, a portfolio of loans is sold to a special 13

19 purpose vehicle SPV and the cash flows from the loans are allocated to tranches with different risks and adequate returns. In our case we consider a first-to-default tranche and a second-to-default tranche, and compute their default probabilities. Definition 2.3. We define a first-to-default probability as the probability that at least one obligor defaults, i.e. p 1st = P {X A < d A } {X B < d B }, 2.9 and a second-to-default probability as the probability that both obligors default, i.e. p 2nd = P {X A < d A } {X B < d B } Lemma 2.4. Suppose the CreditMetrics model given by 2.3 and 2.4. Then the random variables L A and L B defined in 2.1 and 2.2 are conditionally independent for a given state of economy Y = y, and the conditional probabilities of default g pa,ρy = PL A = 1 Y = y and g pb,ρy = PL B = 1 Y = y are given by Φ 1 p A ρy g pa,ρy = Φ, ρ Φ 1 p B ρy g pb,ρy = Φ ρ Proof. The conditional independence of L A and L B follows from the conditional independence of X A and X B defined in 2.3 and 2.4, where ε A and ε B are independent. The conditional probability of default for loan A can be computed as g pa,ρy = P L A = 1 Y = y = P X A < d A Y = y ρ = P Y + 1 ρ εa < d A Y = y = P ε A < d A ρ Y Y = y 1 ρ da ρy = Φ 1 ρ Φ 1 p A ρy = Φ. 1 ρ Then g pb,ρ is computed analogously. Proposition 2.5. The one-year first-to-default probability equals p 1st = g pa,ρy + g pb,ρy 1 g pa,ρy dφy where g pa,ρ and g pb,ρ are the one-year conditional probabilities of default, defined in 2.11 and

20 Proof. Denote A = {X A < d A } and B = {X B < d B }. Due to the conditional independence from Lemma 2.4 we can write the one-year first-to-default probability from Definition 2.3 as p 1st = P A B A B \ A = P = = = P A B \ A Y = y dφy B P A Y = y + P A C Y = y dφy g pa,ρy + g pb,ρy 1 g pa,ρy dφy. Proposition 2.6. The one-year second-to-default probability equals p 2nd = g pa,ρyg pb,ρy dφy where g pa,ρ and g pb,ρ are the one-year conditional probabilities of default, defined in 2.11 and Proof. Again denote A = {X A < d A } and B = {X B < d B }. Then due to the conditional independence from Lemma 2.4 we can write the one-year second-todefault probability from Definition 2.3 as p 2nd = P A B = = = P A B Y = y dφy P A Y = y P B Y = y dφy g pa,ρyg pb,ρy dφy. Proposition 2.7. From Definition 2.3 we can write the one-year second-to-default probability in the equivalent form p 2nd = Φ 2,ρ Φ 1 p A, Φ 1 p B, 2.15 where Φ 2,ρ again denotes the standard bivariate normal distribution function with correlation ρ, as defined in

21 Proof. Using the definition of the second-to-default probability and the joint distribution of X A, X B from Proposition 2.1, we can write p 2nd = P {X A < d A } {X B < d B } = P X A < Φ 1 p A, X B < Φ 1 p B = Φ 2,ρ Φ 1 p A, Φ 1 p B. The probability of the form 2.15 is often referred as the joint default probability. As we will see in the next chapter, this form can be understood as a Gaussian copula. 16

22 Chapter 3 Copula Functions As it is seen in the previous example, the concept of copula functions arises naturally when we speak about joint default probabilities. Although there are several sophisticated methods to specify the distribution of potential losses of a single obligor see Chapter 1, this is usually not sufficient to determine the joint distribution. Therefore, a copula approach brings a way to combine given marginal distributions to get possible joint distributions. In this chapter, based mainly on McNeil et al. 2005, Embrechts et al. 2002, Embrechts 2009 and Bluhm and Overbeck 2006, we present a mathematical background. 3.1 Definition of Copula In this section we present two equivalent definitions of copula together with some basic properties. Definition 3.1. A copula is the distribution function of a random vector in [0, 1] d with standard uniform marginal distributions. Remark 3.2. Alternatively, a copula is any function C : [0, 1] d [0, 1] which has the following three properties. 1. Cx 1,..., x d is increasing in each component x i. 2. C1,..., 1, x i, 1,..., 1 = x i for all i {1,..., d}, x i [0, 1]. 3. For all a 1,..., a d, b 1,..., b d [0, 1] d with a i b i we have: 2 i 1 =1 2 1 i 1+ +i d Cx 1i1,..., x did 0, 3.1 i d =1 where x j1 = a j and x j2 = b j for all j {1,..., d}. 17

23 Definition 3.3. A copula C is called exchangeable if it is the distribution function of an exchangeable random vector of standard uniform variates, i.e. Cu 1,..., u d = Cu Π1,..., u Πd for any permutation Π1,..., Πd of 1,..., d. Definition 3.4. A copula C is called radially symmetric if it is the distribution function of a radially symmetric random vector U, i.e. if U d = 1 U for 1 = 1,..., 1. Definition 3.5. Suppose that a copula C is the distribution function of a random vector U. Then the distribution function of 1 U is called survival copula of U, and is denoted by Ĉ. Proposition 3.6. If a copula C is radially symmetric, then for the corresponding survival copula Ĉ we have C = Ĉ. Proof. The radial symmetry of C gives the relation U d = 1 U. It follows that the distribution functions of U and 1 U equal. Proposition 3.7. Suppose that a bivariate copula C is the distribution function of a random vector U 1, U 2. Then for the corresponding survival copula Ĉ we have the relation Ĉu 1, u 2 = C1 u 1, 1 u 2 + u 1 + u 2 1. Proof. We start from the definition of survival copula and compute, Ĉu 1, u 2 = P1 U 1 u 1, 1 U 2 u 2 = PU 1 1 u 1, U 2 1 u 2 = 1 PU 1 < 1 u 1 PU 2 < 1 u 2 + PU 1 < 1 u 1, U 2 < 1 u 2 = 1 1 u 1 1 u 2 + C1 u 1, 1 u 2 = C1 u 1, 1 u 2 + u 1 + u Sklar s Theorem Whereas the definition of copula is rather formal, Sklar s theorem offers the key properties used in practice. First we present a proposition and then the theorem. Proposition 3.8. Let F be a distribution function and F 1 denote the left-continuous quantile function, i.e. F 1 u = inf{x : F x u} from Definition A.3. Then the following two properties hold. 1. Quantile transformation If U has the standard uniform distribution, then F 1 U has distribution function F. 18

24 2. Probability transformation If Y has a continuous distribution function F, then F Y has the standard uniform distribution. Proof. For all y R and u [0, 1], we have 1. As F is increasing, it follows from Proposition A.2 iv that P F 1 U y = P U F y = F y. 2. Since F is continuous, F 1 is strictly increasing from Proposition A.2 ii. Now using Proposition A.4 and Proposition A.2 viii, we get P F Y u = P Y F 1 u = F F 1 u = u. Theorem 3.9. Sklar Let F be a joint distribution function with univariate marginal distribution functions F 1,..., F d. Then there exists a copula C : [0, 1] d [0, 1] such that, for all x 1,..., x d in R = [, ], F x 1,..., x d = C F 1 x 1,..., F d x d. 3.2 If the margins are continuous, then C is unique. Conversely, if C is a copula and F 1,..., F d are univariate distribution functions, then the function F defined in 3.2 is a joint distribution function with margins F 1,..., F d. Proof. We show the existence and uniqueness of the copula for the case of continuous margins F 1,..., F d. The complete proof can be found for instance in Nelsen For all x 1,..., x d in R = [, ] we have F x 1,..., x d = PX 1 x 1,..., X d x d = P F 1 X 1 F 1 x 1,..., F d X d F d x d = C F 1 x 1,..., F d x d, where C is a copula because F 1 X 1,..., F d X d U0, 1 according to Proposition 3.8 for continuous margins. If we now substitute x i = F 1 i u i for u i [0, 1], i {1,..., d} to 3.2 and use Proposition A.2 viii, we get a unique expression of the copula, Cu 1,..., u d = F F1 1 u 1,..., F 1 d u d. 3.3 Conversely, for a copula C and univariate distribution functions F 1,..., F d, we construct a random vector X 1,..., X d = F1 1 U 1,..., F 1 U d, where U 1,..., U d d 19

25 are standard uniform random variables with the joint distribution function C. Then for the distribution function F of the random vector X 1,..., X d we get F x 1,..., x d = PX 1 x 1,..., X d x d = P F1 1 U 1 x 1,..., F 1 d U d x d = P U 1 F 1 x 1,..., U d F d x d = C F 1 x 1,..., F d x d, where we again used the equivalence of Proposition A.2 iv. As we see, Sklar s theorem gives us a way to construct a joint distribution with given margins using a chosen copula. In the next sections we present some more properties of copula functions. 3.3 Frechét-Hoeffding s bounds Theorem Frechét-Hoeffding s bounds For any copula Cu 1,..., u d we have the bounds { d } max u i + 1 d, 0 Cu 1,..., u d min{u 1,..., u d }. 3.4 i=1 Proof. Let U 1,..., U d U0, 1. The first inequality is given by d d Cu 1,..., u d = P {U i u i } = 1 P {U i > u i } 1 i=1 d P U i > u i = i=1 i=1 i=1 d u i + 1 d. And the second inequality follows from the expression d Cu 1,..., u d = P {U i u i } P U k u k i=1 for all k {1,..., d}. Proposition Let U be a standard uniformly distributed random variable. Then a the bivariate Frechét-Hoeffding lower bound is the copula of random vector U, 1 U, and 20

26 b the bivariate Frechét-Hoeffding upper bound is the copula of U, U. Proof. As both U and 1 U are standard uniform margins, the distribution function of U, 1 U resp. U, U is a copula according to Definition 3.1. a The distribution function of U, 1 U is F x 1, x 2 = PU x 1, 1 U x 2 = P1 x 2 U x 1 { x 1 + x 2 1 for 1 x 2 < x 2, = 0 for 1 x 2 x 2 = max {x 1 + x 2 1, 0}, which is the Frechét-Hoeffding lower bound. b The distribution function of U, U is F x 1, x 2 = PU x 1, U x 2 = PU min{x 1, x 2 } = min{x 1, x 2 }, which is the Frechét-Hoeffding upper bound. 3.4 Comonotonicity and Countermonotonicity In this section we introduce the definitions of comonotonicity and countermonotonicity and their most important properties Comonotonicity Definition Random variables X 1,..., X d are called comonotonic if they admit as copula the Frechét-Hoeffding upper bound comonotonic copula C u u 1,..., u d = min{u 1,..., u d }. 3.5 Proposition Random variables X 1,..., X d are comonotonic if and only if X 1,..., X d d = v 1 Z,..., v d Z 3.6 for some random variable Z and increasing functions v 1,..., v d. 21

27 Proof. Suppose that X 1,..., X d are comonotonic random variables with a joint distribution function F and marginal distribution functions F 1,..., F d. Let U be a standard uniform random variable. From comonotonicity we get F x 1,..., x d = min { F 1 x 1,..., F d x d } = P U min { F 1 x 1,..., F d x d } = P U F 1 x 1,..., U F d x d where F 1 i = P F 1 1 U x 1,..., F 1 d U x d, denote the quantile function, F 1 i u = inf{x : F i x u}, and the last equality follows from the equivalence in Proposition A.2 iv. Then the last expression gives X 1,..., X d = d F1 1 U,..., F 1 d U, where F1 1,..., F 1 d are increasing functions and U is a random variable, i.e. an expression of the form 3.6. Conversely, assume that X 1,..., X d d = v 1 Z,..., v d Z for some random variable Z and increasing functions v 1,..., v d. Then F x 1,..., x d = PX 1 x 1,..., X d x d = P v 1 Z x 1,..., v d Z x d = PZ A 1,..., Z A d, with A i = v 1 i, x i ] for all i {1,..., d}. From monotonicity of all v i we get that A i are intervals of the form, k i or, k i ]. Then there exists such k that d i=1 A i = A k ; and thus, d F x 1,..., x d = PZ A 1,..., Z A d = PZ A i i=1 = min { PZ A 1,..., PZ A d } { = min P } v 1 Z x 1,..., P vd Z x 1 = min { PX 1 x 1,..., PX d x d } = min { F 1 x 1,..., F d x d }, which is the Frechét-Hoeffding upper bound copula Countermonotonicity Definition Random variables X 1 and X 2 are called countermonotonic if they admit as copula the Frechét-Hoeffding lower bound countermonotonic copula C l u 1, u 2 = max {u 1 + u 2 1, 0}

28 Proposition Random variables X 1 and X 2 are countermonotonic if and only if X 1, X 2 = d v 1 Z, v 2 Z 3.8 for some random variable Z and functions v 1 and v 2, where v 1 is increasing and v 2 decreasing, or vice versa. Proof. Suppose that X 1 and X 2 are countermonotonic. According to Proposition 3.11 a, the random vector X 1, X 2 admits as copula the Frechét-Hoeffding lower bound the distribution function of U, 1 U, where U has the standard uniform distribution. Then for a corresponding joint distribution function F and marginal distribution functions F 1 and F 2 we get F x 1, x 2 = C l F 1 x 1, F 2 x 2 = P U F 1 x 1, 1 U F 2 x 2 From the last expression we can conclude that = P F 1 1 U x 1, F U x 2. X 1, X 2 d = F 1 1 U, F U, where the function F 1 1 x is increasing and F x is decreasing. Conversely, assume that X 1, X 2 d = v 1 Z, v 2 Z for v 1 increasing and v 2 decreasing. If we now again define the events A 1 = { v 1 Z x 1 } = { Z v 1 1, x 1 ] }, A 2 = { v 2 Z x 2 } = { Z v 1 2, x 2 ] }, we get for the distribution function of v 1 Z, v 2 Z the expression P v 1 Z x 1, v 2 Z x 2 = PA1 A 2. Then there are two cases. a A 1 A 2 =. In this case PA 1 A 2 = 0 and F 1 x 1 + F 2 x 2 = PA 1 + PA 2 < 1. b A 1 A 2. In this case we get PA 1 A 2 = 1 due to the monotonicity of v 1 and v 2. From the formula PA 1 A 2 = PA 1 + PA 2 PA 1 A 2 we get the expression PA 1 A 2 = PA 1 + PA 2 1 = F 1 x 1 + F 2 x 2 1. Therefore, combining both options, we conclude that P v 1 Z x 1, v 2 Z x 2 = PA1 A 2 = max { F 1 x 1 + F 2 x 2 1, 0 }. In the next section we present several copula functions used in credit risk management. 23

29 3.5 Some Examples of Copulae Independence Copula Let U 1,..., U d be independent, standard uniform random variables. Then their joint distribution function is the independence copula of the form C u 1,..., u d = d u i for all u 1,..., u d [0, 1]. 3.9 i= Comonotonic Copula As an opposite to the independence copula we present the comonotonic copula, C u u 1,..., u d = min{u 1,..., u d } for all u 1,..., u d [0, 1] According to Theorem 3.10, this copula represents the upper Frechét-Hoeffding bound, and for any copula C we get Cu 1,..., u d C u u 1,..., u d for all u 1,..., u d [0, 1] Gaussian Copula Probably the most used copula in risk management is the Gaussian copula or normal copula, given by C d,γ u 1,..., u d = Φ d,γ Φ 1 u 1,..., Φ 1 u d 3.11 for all u 1,..., u d [0, 1], where Φ d,γ denotes the cumulative distribution function of the multivariate standard normal distribution with correlation 1 matrix Γ. As we have seen in Proposition 2.7, this copula arises naturally from the CreditMetrics approach Student-t Copula If we use the student-t distribution with m degrees od freedom instead of the normal distribution, we get the definition of the student-t copula, C d,γ,m u 1,..., u d = Θ d,γ,m Θ 1 m u 1,..., Θ 1 m u d Due to the assumed unit variances, Γ is also a covariance matrix. 24

30 for all u 1,..., u d [0, 1], where Θ d,γ,m denotes the cumulative distribution function of the multivariate standard t-distribution with m degrees of freedom and correlation 2 matrix Γ see Definition A.8 for µ = 0 and Γ = Σ, and Θ 1 m denotes the quantile function of the univariate t-distribution with m degrees of freedom. The student-t copula carries a similar relation to the Gaussian copula as the t-distribution to the normal distribution we can observe fatter tails as compared to the Gaussian copula and convergence to the Gaussian copula for m Archimedean Copulae Let ϕ : [0, 1] [0, ] be a continuous, strictly decreasing, convex function satisfying ϕ1 = 0 and ϕ0 =. Then we say that C ϕ is an Archimedean copula, generated by ϕ, if C ϕ u 1,..., u d = ϕ 1 ϕu ϕu d 3.13 for all u 1,..., u d [0, 1]. Such a ϕ is then called a strict generator of the Archimedean copula. Using different ϕ we can generate the whole class of Archimedean copulae. In the following paragraphs we introduce two examples Gumbel Copula If we take the Archimedean generator ϕ Guθ x = log x θ for any θ [1,, we get the Gumbel copula of the form C ϕguθ u 1,..., u d = exp log u 1 θ + + log u d θ 1 θ 3.14 for all u 1,..., u d [0, 1]. For θ = 1 we obtain the independence copula as a special case, and for the limit θ we obtain the comonotonic copula Clayton Copula Finally, for the Archimedean generator ϕ Clη x = x η 1 for any η 0,, we get the Clayton copula, C ϕclη u 1,..., u d = u η u η d + 1 d 1 η 3.15 for all u 1,..., u d [0, 1]. For the limit θ 0 we obtain the independence copula and for θ we obtain the comonotonic copula. 2 Due to the assumed unit variances, Γ is also a covariance matrix. 25

31 Chapter 4 Dependence Measures In this chapter, based mainly on Embrechts et al and McNeil et al. 2005, we introduce several approaches to dependence measures. We start with the usual Pearson linear correlation, followed by rank correlation and coefficient of tail dependence. In the last section we propose several approaches for multivariate tail dependence coefficients. 4.1 Linear Correlation Definition 4.1. The Pearson correlation coefficient ρ linear correlation between random variables X 1 and X 2 with 0 < VarX 1 < and 0 < VarX 2 < is defined as ρx 1, X 2 = CorrX 1, X 2 = CovX 1, X 2 Var X2 Var X Lemma 4.2. Let X 1, X 2 be a random vector with a joint distribution function F and marginal distribution functions F 1 and F 2. Then, if the covariance of X 1 and X 2 is finite, it is given by CovX 1, X 2 = F x 1, x 2 F 1 x 1 F 2 x 2 dx 1 dx Proof. Introduce a random vector X 1, X 2 with a joint distribution function F and marginal distribution functions F 1 and F 2, independent of X 1, X 2. Then we can compute E X 1 X 1X 2 X 2 = EX 1 X 2 EX 1 X 2 EX 1X 2 + EX 1X 2 = 2 EX 1 X 2 2E X 1 E X 2 = 2 CovX 1, X 2. 26

32 Now, if we apply the identity a b = 1{b x} 1 {a x} dx for a = X 1 X1 and b = X 2 X2 and use the notation A 1 = {X 1 x 1 }, A 1 = {X1 x 1 }, A 2 = {X 2 x 2 } and A 2 = {X2 x 2 }, we get 2 CovX 1, X 2 = E X 1 X1X 2 X2 = E 1 A 1 1 A1 1 A 2 1 A2 dx 1 dx 2 = E 1 A 1 A 1 2 A 1 A 2 1 A1 A A 1 A 2 dx 1 dx 2 = = 2 2 PA1 A 2 2 PA 1 PA 2 dx 1 dx 2 F x 1, x 2 F 1 x 1 F 2 x 2 dx 1 dx 2. Theorem 4.3. Let X 1, X 2 be a random vector with marginal distribution functions F 1 and F 2 and an unspecified joint distribution function. If 0 < VarX 1 < and 0 < VarX 2 <, then the following statements hold. 1 The attainable correlations of X 1 and X 2 form a closed interval [ρ min, ρ max ] with ρ min < 0 < ρ max. 2 The correlation is maximal, ρ = ρ max, if and only if X 1 and X 2 are comonotonic, and minimal, ρ = ρ min, if and only if X 1 and X 2 are countermonotonic. 3 The maximal attainable correlation is ρ max = 1 if and only if X 1 and X 2 are of the same type, i.e. if there exist constants α > 0 and β R such that X d 2 = αx 1 + β. The minimal attainable correlation is ρ min = 1 if and only if X 1 and X 2 are of the same type, i.e. if there exist constants α > 0 and β R such that X d 2 = αx 1 + β. Proof. By combining 4.1 and 4.2, we see that linear correlation is dependent only on the unspecified joint distribution function F considering F 1, F 2, VarX 1 and VarX 2 fixed. Now it follows from Theorem 3.10 that correlation is maximal for the Frechét-Hoeffding upper bound C u i.e. X 1 and X 2 comonotonic and minimal for the lower bound C l i.e. X 1 and X 2 countermonotonic. Moreover, if we take as copula the function λc l + 1 λc u, for any λ [0, 1], we get the correlation of λρ min + 1 λρ max. Thus, any ρ [ρ min, ρ max ] is attainable by setting λ = ρmax ρ ρ max ρ min. As the variances are assumed to be positive, the variables X 1 and X 2 are not degenerate, and for its distribution functions we get max { F 1 x 1 + F 2 x 2 1, 0 } < F 1 x 1 F 2 x 2 < min { F 1 x 1, F 2 x 2 }. 27

33 Then it follows from 4.2 that ρ min < 0 < ρ max. The last statement is then a standard result, as ρ = ±1 if and only if there is a linear relationship between X 1 and X Rank Correlation Rank correlations are based on orderings of the sample rather than on the values themselves. Therefore, it does not depend on the marginal distributions but only on the mixing copula see Theorem 4.5 and Theorem Spearman s Rho One of the commonly used rank correlations is Spearman s rho, which is the correlation coefficient of the values of the distribution functions of the data. 1 We introduce its definition and some basic properties. Definition 4.4. Let X 1 and X 2 be random variables with distribution functions F 1 and F 2. Then Spearman s rank correlation coefficient ρ S Spearman s rho for random variables X 1 and X 2 is defined as ρ S X 1, X 2 = Corr F 1 X 1, F 2 X Theorem 4.5. Let X 1 and X 2 be random variables with continuous distribution functions F 1 and F 2 and a unique copula C. Then Spearman s correlation coefficient is given by 1 1 ρ S X 1, X 2 = 12 Cu1, u 2 u 1 u 2 du1 du Proof. By Definition 4.4 we can write 0 0 ρ S X 1, X 2 = Corr F 1 X 1, F 2 X 2 = 12 Cov F 1 X 1, F 2 X 2, as F 1 X 1, F 2 X 2 U0, 1 according to Proposition 3.8, and their variances are 1 thus equal. Then we apply Lemma 4.2 for random variables U 12 1 = F 1 X 1 and U 2 = F 2 X 2 to get ρ S X 1, X 2 = Cu1, u 2 u 1 u 2 du1 du 2. 1 In practice, this is represented by ranks of the sample data. 28

34 Proposition 4.6. Spearman s rho is a symmetric measure taking values in [ 1, 1]. Furthermore, if X 1 and X 2 are random variables with continuous distribution functions F 1 and F 2 and a unique copula C, then the following statements hold. 1 If X 1 and X 2 are independent, then ρ S X 1, X 2 = 0. 2 X 1 and X 2 are comonotonic if and only if ρ S X 1, X 2 = 1. 3 X 1 and X 2 are countermonotonic if and only if ρ S X 1, X 2 = 1. Proof. Symmetry and range follow straight from the definition. If X 1 and X 2 are independent, then their copula is the independence copula C u 1, u 2 = u 1 u 2 and the statement follows from Theorem 4.5. Similarly, the value of formula 4.4 is maximal for the Frechét-Hoeffding upper bound C u i.e. X 1 and X 2 comonotonic and minimal for the lower bound C l i.e. X 1 and X 2 countermonotonic Kendall s Tau Kendall s tau is a measure of concordance. 2 It is defined as the probability that two independent random vectors from the same distribution are concordant minus the probability that they are discordant. In a short form, this can be defined as follows. Definition 4.7. Let X 1, X 2 and X1, X2 be two independent, identically distributed random vectors. Then Kendall s correlation coefficient ρ τ Kendall s tau for random variables X 1 and X 2 is defined as ρ τ X 1, X 2 = E sign X 1 X1X 2 X Theorem 4.8. Let X 1 and X 2 be random variables with continuous distribution functions F 1 and F 2 and a unique copula C. Then Kendall s correlation coefficient is given by ρ τ X 1, X 2 = Cu 1, u 2 dcu 1, u Two pairs x 1, x 2, x 1, x 2 R 2 are said to be concordant if x 1 x 1x 2 x 2 > 0, and discordant if x 1 x 1x 2 x 2 < 0. 29

35 Proof. We can rewrite the definition as ρ τ X 1, X 2 = P X 1 X 1X 2 X 2 > 0 P X 1 X 1X 2 X 2 < 0 = 2 P X 1 X 1X 2 X 2 > 0 1 = 4 PX 1 < X 1, X 2 < X 2 1 = 4 E PX 1 < X 1, X 2 < X 2 X 1, X 2 1 = 4 = 4 = 4 = PX 1 < x 1, X 2 < x 2 df x 1, x 2 1 F x 1, x 2 df x 1, x 2 1 C F 1 x 1, F 2 x 2 dc F 1 x 1, F 2 x 2 1 Cu 1, u 2 dcu 1, u 2 1, where in the last step we used the substitution u 1 = F 1 x 1 and u 2 = F 2 x 2. Proposition 4.9. Kendall s tau is a symmetric measure taking values in [ 1, 1]. Furthermore, if X 1 and X 2 are random variables with continuous distribution functions F 1 and F 2 and a unique copula C, then the following statements hold. 1 If X 1 and X 2 are independent, then ρ τ X 1, X 2 = 0. 2 X 1 and X 2 are comonotonic if and only if ρ τ X 1, X 2 = 1. 3 X 1 and X 2 are countermonotonic if and only if ρ τ X 1, X 2 = 1. Proof. Symmetry and range follow straight from the definition. If X 1 and X 2 are independent, then the probabilities of concordance and discordance equal and Kendall s tau is zero. Now suppose that X 1 and X 2 are comonotonic. According to Proposition 3.6, we can write X 1 = v 1 Z and X 2 = v 2 Z, where v 1 and v 2 are increasing functions. Then for some random variables Z and Z we get [ v1 ρ τ X 1, X 2 = E sign Z v 1 Z v 2 Z v 2 Z ] = 1, since v 1 and v 2 are deterministic increasing functions and F 1 and F 2 are assumed to be continuous thus PX 1 = X 1 = PX 2 = X 2 = 0. Conversely, suppose that ρ τ X 1, X 2 = 1, i.e. P X 1 X 1X 2 X 2 > 0 = 1. For individual random events this means that P P { ω, ω Ω Ω : X 1 ω X 1 ω X 2 ω X 2 ω > 0 } = 1. If we now denote A 1 = {X 1 x 1 } = { ω Ω : X 1 ω x 1 }, A 2 = {X 2 x 2 } = { ω Ω : X 2 ω x 2 }, 30

36 we need to show that X 1 and X 2 admit the comonotonic copula, i.e. PA 1 A 2 = min { PA 1, PA 2 } for all x 1 and x 2. Without loss of generality, assume that PA 1 < PA 2. Now, if PA 1 \A 2 > 0 then also PA 2 \A 1 > 0 and P X 1 X 1X 2 X 2 < 0 > 0, which is a contradiction. Therefore, PA 1 A 2 = PA 1 = min { PA 1, PA 2 }. Similarly, we prove the last statement. First suppose that X 1 and X 2 are countermonotonic. According to Proposition 3.8, we can write X 1 = v 1 Z and X 2 = v 2 Z, where v 1 is increasing and v 2 is decreasing. Then for some random variables Z and Z we get [ v1 ρ τ X 1, X 2 = E sign Z v 1 Z v 2 Z v 2 Z ] = 1, due to monotonicity of v 1 and v 2, and continuity of F 1 and F 2. Conversely, suppose that ρ τ X 1, X 2 = 1, i.e. P X 1 X 1X 2 X 2 < 0 = 1. In this case, we need to show that PA 1 A 2 = max { PA 1 + PA 2 1, 0 } for all x 1 and x 2. Now we distinguish the following two cases. Either PA 1 A 2 = 0 and PA 1 + PA 2 < 1, or PA 1 A 2 > 0. In the latter case we get PX 1 x 1, X 2 x 2 > 0. However, this is only possible when PX 1 > x 1, X 2 > x 2 = 0, which is equivalent to PA 1 A 2 = 1, since 0 = PX 1 > x 1, X 2 > x 2 = PA C 1 A C 2 = 1 PA 1 A 2. Using the formula PA 1 A 2 = PA 1 +PA 2 PA 1 A 2, we then get PA 1 A 2 = PA 1 + PA 2 1. Therefore, combining both options, we conclude that PA 1 A 2 = max { PA 1 + PA 2 1, 0 }. 4.3 Tail Dependence The coefficient of tail dependence is a measure providing information about dependence in a tail of a bivariate distribution. As well as rank correlations, in the case of continuous marginal distribution functions, tail dependence only depends on the mixing copula see Theorem The coefficient of tail dependence is defined as the conditional probability that one variable has an extremal value provided that the other has an extremal value. We present the following formal definition. Definition Let X 1 and X 2 be random variables with distribution functions F 1 and F 2. The coefficient of lower tail dependence of X 1 and X 2 is given by λ l X 1, X 2 = lim P X 2 F2 1 q X 1 F1 1 q,

37 provided the limit exists. If λ l 0, 1], we say that X 1 and X 2 show lower tail dependence; if λ l = 0, we say that X 1 and X 2 are asymptotically independent in the lower tail. Analogously, the coefficient of upper tail dependence of X 1 and X 2 is given by λ u X 1, X 2 = lim P X 2 > F2 1 q X 1 > F1 1 q, 4.8 q 1 provided the limit exists. If λ u 0, 1], we say that X 1 and X 2 show upper tail dependence; if λ u = 0, we say that X 1 and X 2 are asymptotically independent in the upper tail. Theorem Let X 1 and X 2 be random variables with continuous distribution functions F 1 and F 2. Then there exists a unique copula C such that both λ l X 1, X 2 = lim Cq, q q 4.9 and Ĉq, q λ u X 1, X 2 = lim, 4.10 q where Ĉ is the survival copula of C from Definition 3.5. If C is radially symmetric, then λ l = λ u. Proof. We start from the definition of the coefficient of tail dependence and apply Proposition 3.8 and Sklar s theorem. λ l X 1, X 2 = lim P X 2 F2 1 q X 1 F1 1 q P X 1 F1 1 q, X 2 F2 1 q = lim P X 1 F1 1 q P F 1 X 1 q, F 2 X 2 q = lim P F 1 X 1 q Cq, q = lim, q 32