An Introduction to Copulas with Applications Svenska Aktuarieföreningen Stockholm 4-3- Boualem Djehiche, KTH & Skandia Liv Henrik Hult, University of Copenhagen I Introduction II Introduction to copulas III Measuring dependence IV Applications E-mail: boualem.djehiche@skandia.se, hult@math.ku.dk
INTRODUCTION Statistical models for portfolios in insurance and finance Understanding the uncertainty underlying variablitiy in claim sizes and price fluctuations of financial assets. Measuring the risk of insurance/financial portfolios held by departments within the company or the entire company. Take actions/decisions based on risk preferences. c 4 (B. Djehiche, H. Hult) 1
From observed data to measures of risk/pricing (1) Data collection. Observing insurance losses or financial losses. () Model building. Fit a statistical model to the observed data. (3) Compute risk. Evaluate a measure of the risk based on the fitted statistical model. (4) Pricing of insurance/financial contracts. c 4 (B. Djehiche, H. Hult)
Popular measures of risk Value-at-Risk (VaR) Value-at-Risk at the level p (p =.99) is the smallest number x such that the probability of losing more than x is less than 1 p. Expected Shortfall (ES) Expected Shortfall at the level p istheaveragesizeofthe losses that exceed VaR at level p. c 4 (B. Djehiche, H. Hult) 3
Value-at-Risk and Expected Shortfall.4.35.3.5..15.1.5 5 4 3 1 1 3 4 5 VaR ES c 4 (B. Djehiche, H. Hult) 4
Portfolio modelling 6 6 STORM FLOODING 5 5 4 4 3 3 (1) Data collection. 1 4 6 8 1 1 4 6 8 1 () Build portfolio model..4.35.3.5..15 (3) Compute the loss distribution for the portfolio. 1.1.5 8 6 4 4 6 8 1 (4) Compute measures of risk. Compute premiums of insurance contracts. c 4 (B. Djehiche, H. Hult) 5
Dependence modelling Consider the distribution of claims from two types of insurance, e.g. storm and flooding. 6 STORM 6 FLOODING 5 5 4 4 3 3 1 1 4 6 8 1 4 6 8 1 Is it enough to know the individual claim size distribution? What if the claims are somehow dependent? c 4 (B. Djehiche, H. Hult) 6
Dependence modelling Scatter plot of the two types of claims with different dependence structure but identical marginals. 9 9 8 8 7 7 6 6 5 5 4 4 3 3 1 1 4 6 8 1 4 6 8 1 Which effect does the dependence structure have on the distribution of large losses for the portfolio? c 4 (B. Djehiche, H. Hult) 7
Why advanced dependence modelling? New complex products in insurance and finance result in portfolios with complex dependence structures. Need for multivariate models with more flexibility than the multivariate normal distribution. Correlation is not a satisfactory dependence measure to capture the observed advanced dependence structures. Wrong dependence structure may lead to severe underestimation of the portfolio risk. Marginals + Dependence = Portfolio model c 4 (B. Djehiche, H. Hult) 8
INTRODUCTION TO COPULAS Brief historical background - Copulas The basic idea of a copula is to separate the dependence and the marginal distributions in a multivariate distribution (portfolio model). 194 s: Hoeffding studies properties of multivariate distributions 1959: The word copula appears for the first time (Sklar 1959). 1998: Academic literature on how to use copulas in risk management. 4: Some insurance companies and financial institutions have started to use copulas as a risk management tool. c 4 (B. Djehiche, H. Hult) 9
Copula - a definition Definition: A d-dimensional copula is a distribution function on [, 1] d with standard uniform marginal distributions. Example 1: C(u,v) = uv. If U U(, 1) and V U(, 1) are independent, then C(u, v) =uv = P(U u) P(V v) =P(U u, V v) =H(u, v), where H(u, v) is the distribution function of (U, V ). C is called the independence copula. c 4 (B. Djehiche, H. Hult) 1
Independence copula 1.9.8.7.6.5.4.3..1.1..3.4.5.6.7.8.9 1 c 4 (B. Djehiche, H. Hult) 11
A generic example If (X, Y ) is a pair of random variables with distribution function H(x, y) and marginal distributions F X (x) andf Y (y) respectively, then U = F X (X) U(, 1) and V = F Y (Y ) U(, 1) and the distribution function of (U, V ) is a copula. P(U u) =P(F X (X) u) =P(X FX 1 (u)) = F X(FX 1 (u)) = u. and C(u, v) =P(U u, V v) =P(X F 1 X = H(F 1 X (u),f 1 Y (v)). (u),y F 1 (v)) Y c 4 (B. Djehiche, H. Hult) 1
Illustration of Sklar s theorem 5 4 1.8 3 1 1 1.6 1.4 1. 1.8.6.4 3. 4 4 4.5 1 Left: Simulation of bivariate normal distribution. Right: The associated copula. c 4 (B. Djehiche, H. Hult) 13
Sklar s Theorem Theorem Let H be a joint df with marginal dfs F 1,...,F d Then there exists a copula C such that H(x 1,...,x d )=C(F 1 (x 1 ),...,F d (x d )). (1) Conversely, if C is a copula and F 1,...,F d are distribution functions, then H defined by (1) is a joint distribution function with marginal dfs F 1,...,F d. c 4 (B. Djehiche, H. Hult) 14
Important consequences H(x 1,...,x d )=C(F 1 (x 1 ),...,F d (x d )) A copula describes how the marginals are tied together in the joint distribution. In this way the joint df is decomposed into the marginal dfs and a copula. The marginal dfs and the copula can be estimated separately. Given a copula we can obtain many multivariate distributions by selecting different marginal dfs, H(x 1,...,x d )=C(G 1 (x 1 ),...,G d (x d )) c 4 (B. Djehiche, H. Hult) 15
Illustration 8 8 6 6 4 4 4 4 6 6 8 5 8 5 4 4 15 1 5 5 5 5 1 15 5 1 15 14 1 1 8 6 4 4 4 4 4 4 4 Left: Gaussian copula with different marginals. Middle: t copula with different marginals. Right: Gumbel copula with different marginals. c 4 (B. Djehiche, H. Hult) 16
Example of copulas 6 6 Gumbel 4 4 4 4 6 6 4 4 6 6 6 6 4 4 6 4 6 6 t(5) 4 4 4 4 6 6 Normal 4 4 6 6 6 Clayton 4 All simulations with standard normal marginal distributions. Top left: Gumbel copula; Top right: Normal copula; Down left: t5 copula; Down right: Clayton copula. c 4 (B. Djehiche, H. Hult) 17
MEASURING DEPENDENCE Correlation Linear correlation: ϱ(x, Y )=E((X E(X))(Y E(Y ))), ϱ(x, Y )= 1 n 1 n i=1 (x i x)(y i y). Sensitive to outliers. Measures the average dependence between X and Y. Invariant under strictly increasing linear transformations. c 4 (B. Djehiche, H. Hult) 18
Examples Gaussian X1 Y1-4 - 4-4 - 4 t X Y -4-4 -4-4 Gumbel X1 Y1 4 6 8 1 1 14 4 6 8 1 1 14 Clayton X Y 4 6 8 1 1 14 4 6 8 1 1 14 Left: Samples from two distributions with standard normal margins, ϱ =.8 Right: Samples from two distributions with Gamma(3, 1) marginal dfs, linear correlation ϱ =.5. c 4 (B. Djehiche, H. Hult) 19
Rank correlation Two pairs (x, y) and( x, ỹ) are said to be concordant if (x x)(y ỹ) > and discordant if (x x)(y ỹ) <. Kendall s tau: τ(x, Y )=P((X X)(Y Ỹ ) > ) P((X X)(Y Ỹ ) < ) where ( X,Ỹ ) is an independent copy of (X, Y ). It is estimated by #concordant pairs #discordant pairs τ(x, Y )=. #pairs c 4 (B. Djehiche, H. Hult)
Properties of Kendall s tau Insensitive to outliers. Measures the average dependence between X and Y. Invariant under strictly increasing transformations. Kendall s tau depends only on the copula of (X, Y ). c 4 (B. Djehiche, H. Hult) 1
Emphasizing the tails Gaussian X1 Y1-4 - 4-4 - 4 t X Y -4-4 -4-4 Left: Samples from two distributions with standard normal margins, and τ =.6. c 4 (B. Djehiche, H. Hult)
Tail dependence Definition: Let (X, Y ) be a random vector with marginal distribution functions F X and F Y. The coefficient of upper tail dependence of (X, Y )is defined as λ U (X, Y ) = lim P(Y > F 1 u 1 Y (u) X>F 1 X (u)), provided that the limit λ U [, 1] exists. The coefficient of lower tail dependence is defined as λ L (X, Y ) = lim P(Y F 1 1 u Y (u) X FX (u)), provided that the limit λ L [, 1] exists. If λ U > (λ L > ), then we say that (X, Y ) has upper (lower) tail dependence. c 4 (B. Djehiche, H. Hult) 3
Illustration of upper tail dependence -1-5 5 1-1 -5 5 1 c 4 (B. Djehiche, H. Hult) 4
Example Gaussian X1 Y1-4 - 4-4 - 4 t X Y -4-4 -4-4 Left: Samples from two distributions with standard normal margins, τ =.6. The left distribution (normal copula) has λ U (X, Y )= and the right distribution (t copula) has λ U (X, Y )=.3. c 4 (B. Djehiche, H. Hult) 5
Example 3 3 1 1 1 1 3 3 1 1 3 3 3 1 1 3 Left: Samples from two distributions with t 3 margins, τ =.6. The left distribution (normal copula) has λ U (X, Y )=andthe right distribution (t copula) has λ U (X, Y )=.3. c 4 (B. Djehiche, H. Hult) 6
APPLICATIONS Applications of copulas in risk management Model fitting. Finding a reasonable model for computing risk measures. Stress testing. Under given conditions on marginal dfs and measures of dependence we can change copula to understand the sensitivity of the portfolio risk with respect to the dependence structrure. Dynamic financial analysis. Copulas are useful when building large simulation models for long time horizons. c 4 (B. Djehiche, H. Hult) 7
An example from insurance Portfolio with n risks (X 1,...,X n ) representing potential losses in different lines of business. Company seeks protection against simultaneous big losses in different lines of business. Reinsurance contract: payout f((x i,k i ); i =1,...,l)= ( l )( l 1 {Xi >k i } i=1 i=1 ) (X i k i ). c 4 (B. Djehiche, H. Hult) 8
Example cont. The reinsurer will price the contract by computing E(f((X i,k i ); i = 1,...,l)). For this the reinsurer needs the joint distribution H(x 1,...,x l ) which is difficult to obtain. In particular, data for big losses are rare. He may be able to estimate marginal dfs and pairwise rank correlations. Typically, Kendall s tau estimates can be transformed into a estimates of the copula parameters. c 4 (B. Djehiche, H. Hult) 9
Example Deciding upon a class of copulas (e.g. Normal or Gumbel) a calculation of the price of the contract can be performed. Possibly by simulation. The choice of copula family is crucial for pricing the contract. Uncertainty in choosing the right copula family. Choice of copula family depend on the partcular application and the underlying loss causing mechanism. c 4 (B. Djehiche, H. Hult) 3
Example Consider payout probabilities of f((x i,k i ); i =1,...,l)= ( l )( l 1 {Xi >k i } i=1 i=1 for l =5,X i LN(, 1), τ(x i,x j )=.5 ) (X i k i ). Mirrored Clayton copula and Gaussian copula. c 4 (B. Djehiche, H. Hult) 31
Simulations.35 16.3 14.5 1. 1 8.15 6.1 4.5 5 1 15 5 1 15 Left: Payout probabilities for the reinsurance contract with normal copula (solid) and mirrored Clayton copula (dashed). Right: Qoutient between payout probabilities (Clayton/Normal). c 4 (B. Djehiche, H. Hult) 3
Price of the reinsurance contract 4 11 3.5 1 9 3 8.5 7 6 1.5 5 4 1 3.5 5 1 15 1 5 1 15 Left: Price for the reinsurance contract with normal copula (solid) and mirrored Clayton copula (dashed). Right: Qoutient between prices (Clayton/Normal). c 4 (B. Djehiche, H. Hult) 33
References Embrechts, P., Lindskog, F., McNeil A. (3): Modelling Dependence with Copulas and Applications to Risk Management. In: Handbook of Heavy Tailed Distributions in Finance, ed. S. Rachev, Elsevier, Chapter 8, pp. 39-384. Also available at: www.math.ethz.ch/ baltes/ftp/papers.html Embrechts, P., McNeil, A., Straumann, D. () Correlation and dependence in risk management: properties and pitfalls. In: Risk Management: Value at Risk and Beyond, ed. M.A.H. Demptser, Cambridge University Press, Cambridge, pp. 176-3. Also available at: www.math.ethz.ch/ baltes/ftp/papers.html Joe, H. (1997): Multivariate Models and Dependence Cocnepts. Chapman & Hall, London. Nelsen, R. (1999): An introduction to copulas. Springer, New York. c 4 (B. Djehiche, H. Hult) 34