Dependence structures and limiting results with applications in finance and insurance

Transcription

1 Dependence structures and limiting results with applications in finance and insurance Arthur Charpentier, prix scor 2006 Institut des Actuaires, Juin

2 Présentation Petite introduction générale Contenu de la thèse: problèmes de dépendance en assurance Motivation Les copules et copules conditionnelles Application en risque de crédit Application en réassurance Estimation d une densité de copule Dépendance (temporelle) pour les risques environmentaux 2

4 Prix SCOR-Tillinghast, jeune docteur, décerné en Décembre 2006 par Denis Kessler. Obtenue en 2006 par Stéphane Loisel (ISFA) ou en 2004 par Pauline Barrieu (LSE). 4

5 Thèse intitulée Dependence structures and limiting results, with applications in finance and insurance. Thèse soutenue en juin 2006, sous la direction de Jan Beirlant et Michel Denuit Jan Beirlant Michel Denuit 5

6 Leuven: Jan Dhaene, Marc Goovaerts & Jef Teugels 6

7 Michel Denuit, Jan Dhaene, Marc Goovaerts & Rob Kaas (2005). Actuarial Theory for Dependent Risks: Measures, Orders and Models. Wiley. Etienne De Vylder Marc Goovaerts & Jean Haezendonck. (1984) Premium Calculation in Insurance. NATO Science Series C. Nick Bingham, Charles Goldie & Jozef Teugels (1987). Regular Variation. Encyclopedia of Mathematics and its Applications. Tomasz Rolski, Hanspeter Schmidli, Volker Schmidt & Jozef Teugels (2001). Stochastic Processes for Insurance and Finance. Wiley. Wim Schoutens (2003). Lévy Processes in Finance: Pricing Financial Derivatives. Wiley. Revue Insurance: Mathematics and Economics 7

8 8

10 Some motivations Everybody who opens any journal on stochastic processes, probability theory, statistics, econometrics, risk management, finance, insurance, etc., observes that there is a fast growing industry on copulas [...] The International Actuarial Association in its hefty paper on Solvency II recommends using copulas for modeling dependence in insurance portfolios [...] Since Basle II copulas are now standard tools in credit risk management. Are copulas suitable for modeling multivariate extremes? Copulas generate any multivariate distribution. If one wants to make an honest analysis of multivariate extremes the distributions used should be related to extreme value theory in some way. Mikosch (2005). 10

11 Motivations in non-life insurance In reinsurance, it could be interesting to split a claim cost into several components, e.g. the loss amount (paid to the insured) and the allocated expenses (lawyers, expertise...), see e.g. Frees & Valdez (1998) or Klugman & Parsa (1999). In that case, X i = (X 1,i, X 2,i ) s are the amounts associated with i-th claim. In the context of reinsurance pricing, assuming that allocated expenses are prorata capita, the reimbursement of the reinsurer, with an excess-of-loss treaty (with infinite limit, and deductible d), when a claim expressed as X = (X 1, X 2 ) occurred, is 0, if X 1 d, g(x 1, X 2 ) = X 1 d + X 1 d X 2, if X 1 > d. X 1 The pure premium per claim is then E(g(X)) which is based on the joint distribution of X since g is nonlinear. 11

12 Motivations in life insurance In life insurance, analogous of those financial derivatives can be considered. Consider a husband and his wife, and denote by T x and T y the survival life lengths, assuming that the man has age x and his wife y when they buy a life-insurance contract. Several contracts can be considered, where capital C k is due each year k, as long as the spouses are both still alive, g(t x, T y ) = v k C k 1(T x > k and T y > k), k=1 as long as there is a survivor, g(t x, T y ) = v k C k 1(T x > k or T y > k). Note that C k can be stochastic if the capital is indexed on a financial asset, or if the income is indexed by some stochastic interest rate. k=1 12

13 Motivations in life insurance The associated pure premium, called annuities when C k = 1, can be written respectively (with standard actuarial notations) a xy = v k P(T x > k, T y > k) and a xy = k=1 v k P(T x > k or T y > k). Those contracts are usually built for an husband and his wife, i.e. contracts with more risks can be considered if children are involved, or even higher when dealing with collective insurance contracts. k=1 13

14 Motivations in credit risk Applications with a high number of risks can also be considered, in credit risk for instance. Let X = (X 1,..., X d ) denote the vector of indicator variables, indicating if the i-th contract defaulted during a given period of time. If a credit derivative is based on the occurrence of k defaults among d companies, and thus, the pricing is related to the distribution of the number of defaults, N, defined as N = X X d. Under the assumption of possible contagious risks, the distribution of N should integrate dependencies. 14

15 Motivations in risk management Consider some risk measure R, i.e. a mapping from the set of random risks to R (so that R(X) is a capital amount). As a capital allocation criterion in finance, R( ) is computed for the different business units of the firm, and for the different desks within each business unit R(X i ), in order to set the appropriate trading limits. The overall capital needed is then R(X X d ). Similarly, in non-life insurance, claims reserves are estimated independently on each line of business. But the overall reserves needed might exceed the sum, since R(X X d ) can be higher than R(X 1 ) R(X d ). 15

16 Visualization of dependence and marginal problem Looking at the joint distribution of a random pair (X, Y ) (e.g density or scatterplot) is not enough to understand dependence. Random vector with exponential margins Random vector with exponential margins Figure 1: Visualization of dependence looking at (X, Y )? 16

17 Correlation versus independence? Consider the following case (credit risk): X 1,..., X 1000 are default indicators in a portfolio, X i B(p), where p is the default probability = 10%. In the case where the X i s are independent, N = X X 1000 B(n, p), a binomial distribution. The line in red on Figure 2 is obtained with a small correlation, i.e. corr(x i, X j ) =

18 Number of defaults, n=1000, p=10% Number of defaults, n=1000, p=10% Figure 2: Impact of correlation, r = 0 versus r = 0.005, in credit risk. 18

20 Z Y Arthur CHARPENTIER - Prix scor-tillinghast 2006 Copulas, an introduction (in dimension 2) Definition 1. A copula C is a joint distribution function on [0, 1] 2, with uniform margins on [0, 1]. Fonction de répartition à marges uniformes X Figure 3: Graphical representation of a copula. 20

21 z x Arthur CHARPENTIER - Prix scor-tillinghast 2006 If C is twice differentiable, define its density as c(u, v) = 2 C(u, v). u v Densité d une loi à marges uniformes x Figure 4: Density of a copula. 21

22 Fonction de répartition à marges uniformes Densité d une loi à marges uniformes Fonction de répartition à marges uniformes Densité d une loi à marges uniformes Figure 5: Distribution functions and densities. 22

23 Fonction de répartition à marges uniformes Densité d une loi à marges uniformes Fonction de répartition à marges uniformes Densité d une loi à marges uniformes Figure 6: Distribution functions and densities. 23

24 Sklar s theorem Theorem 2. (Sklar) Let C be a copula, and F X and F Y two marginal distributions, then F (x, y) = C(F X (x), F Y (y)) is a bivariate distribution function, with F F(F X, F Y ). Conversely, if F F(F X, F Y ), there exists C such that F (x, y) = C(F X (x), F Y (y)). Further, if F X and F Y are continuous, then C is unique, and given by C(u, v) = F (F 1 X 1 (u), F (v)) for all (u, v) [0, 1] [0, 1] We will then define the copula of F, or the copula of (X, Y ). Y In that case, the copula of (X, Y ) is the distribution function of (F X (X), F Y (Y )). Proposition 3. If (X, Y ) has copula C, the copula of (g(x), h(y )) is also C for any increasing functions g and h. 24

25 Copulas and ranks The copula of X = (X 1,..., X d ) is the distribution function of U = (F 1 (X 1 ),..., F d (Y d )). In practice, since marginal distributions are unknown, the idea is to substitute empirical distribution function, F i (x i ) = #{observations X i,j s lower than x i } #{observations } = 1 n n 1(X i,j x i ). j=1 Note that F i (X i,j0 ) = #{observations X i,j s lower than X i,j0 } #{observations } = 1 n n j=1 1(X i,j X i,j0 ) = R i,j 0 n, where R i,j0 denotes the rank of X i,j0 within {X i,1,..., X i,n }. On a statistical point of view, studying the copula means studying ranks. 25

26 Scatterplot of (X,Y) Scatterplot of the ranks of (X,Y) Y (raw data) Ranks of the Yi s X (raw data) Ranks of the Xi s Scatterplot of the ranks of (X,Y), divided by n Scatterplot of (U,V), the copula type tranform of (X,Y) Ranks of the Yi s/n Vi=Ranks of the Yi s/n Ranks of the Xi s/n Ui=Ranks of the Xi s/n+1 Figure 7: Copulas, ranks and parametric inference, from (X i, Y i ) to (U i, V i ). 26

27 Limiting dependence structures for tail events, with applications to credit derivatives, (2006), Journal of Applied Probability, 43, , with A. Juri, Compare (X, Y ) and (X, Y ) given X > t and Y > t. Lower tail dependence for Archimedean copulas: characterizations and pitfalls, (2007), Insurance Mathematics and Economics, 40, , with J. Segers, Convergence of Archimedean Copulas, (2006), to appear, Probability and Statistical Letters, with J. Segers, Tails of Archimedean Copulas, (2006), submitted, with J. Segers, Compare (X, Y ) and (X, Y ) given X > F 1 X 1 (p) and Y > F (p). Y 27

29 Let U = (U 1,..., U n ) be a random vector with uniform margins, and distribution function C. Let C r denote the copula of random vector where r 1,..., r d (0, 1]. (U 1,..., U n ) U 1 r 1,..., U d r d, (1) If F i r ( ) denotes the (marginal) distribution function of U i given {U 1 r 1,..., U i r i,..., U d r d } = {U r}, and therefore, the conditional copula is F i r (x i ) = C(r 1,..., r i 1, x i, r i+1,..., r d ) C(r 1,..., r i 1, r i, r i+1,..., r d ), C r (u) = C(F 1 r(u 1 ),..., F d r (u d )). (2) C(r 1,..., r d ) Frank copula has independence in tails (C = C ) and the copula has dependence in tails (C C ). The associated limited copula is Clayton. 29

31 univariate case bivariate case limiting distribution dependence structure of of X n:n (G.E.V.) componentwise maximum when n (X n:n, Y n:n ) (Fisher-Tippet) limiting distribution of X X > x (G.P.D.) when x dependence structure of (X, Y ) X > x, Y > y when x, y (Balkema-de Haan-Pickands) 31

33 Example Loss-ALAE: consider the following dataset, were the X i s are loss amount (paid to the insured) and the Y i s are allocated expenses. Denote by R i and S i the respective ranks of X i and Y i. Set U i = R i /n = ˆF X (X i ) and V i = S i /n = ˆF Y (Y i ). Figure 8 shows the log-log scatterplot (log X i, log Y i ), and the associate copula based scatterplot (U i, V i ). Figure 9 is simply an histogram of the (U i, V i ), which is a nonparametric estimation of the copula density. Note that the histogram suggests strong dependence in upper tails (the interesting part in an insurance/reinsurance context). 33

34 Log log scatterplot, Loss ALAE Copula type scatterplot, Loss ALAE log(alae) Probability level ALAE log(loss) Probability level LOSS Figure 8: Loss-ALAE, scatterplots (log-log and copula type). 34

35 Figure 9: Loss-ALAE, histogram of copula type transformation. 35

36 The basic idea to get an estimator of the density at some point x is to count how many observation are in the neighborhood of x (e.g. in [x h, x + h) for some h > 0). Therefore, consider the moving histogram or naive estimator as suggested by Rosenblatt (1956), f(x) = 1 2nh n I(X i [x h, x + h)). i=1 Note that this can be easily extended using other definitions of the neighborhood of x, f(x) = 1 n ( ) x Xi K, nh h i=1 where K is a kernel function (e.g. K(ω) = I( ω 1)/2). 36

37 Estimation of Frank copula Figure 10: Theoretical density of Frank copula. 37

38 Estimation of Frank copula Figure 11: Estimated density of Frank copula, using standard Gaussian (independent) kernels, h = h. 38

39 Estimation of the copula density (Beta kernel, b=0.1) Estimation of the copula density (Beta kernel, b=0.1) Figure 12: Estimated density of Frank copula, Beta kernels, b =

40 Estimation of the copula density (Beta kernel, b=0.05) Estimation of the copula density (Beta kernel, b=0.05) Figure 13: Estimated density of Frank copula, Beta kernels, b =

42 Dependence in environmental risks dependence for windspeed: how to model seasons with many storms? dependence for temperature: what is the return period of the heat wave of 2003? dependence for flood events: Hurst versus Gumbel? 42

43 Modeling temperature Minimal daily temperature in Paris ( ) Maximal daily temperature in Paris ( ) Temperature in 0.1 C Temperature in 0.1 C July and August July and August Figure 14: Daily temperature in Paris, years 1997 to

44 Arthur CHARPENTIER - Prix scor-tillinghast 2006 Modeling temperature Daily minima in Paris detrended (in C) x 0 10 Daily Minimum Temperatures in Paris Time ACF 5 10 Autocorrelation of daily time series Lag Series: x Smoothed Periodogram spectrum 10 Temperature ( C) date frequency bandwidth = 7.63e 05 Figure 15: Trend of the series, and analysis of the series of residuals. 44

45 heat wave, type (A): 11 consecutive days with temperature exceeding 19 C, short memory short memory long memory short tail noise heavy tail noise short tail noise optimistic 88 years 69 years 53 years pessimistic 79 years 54 years 37 years Table 1: Periods of return (expected value, in years) before the next heat wave similar with August 2003 (type (A)). 45

46 heat wave, type (B): 3 consecutive days with temperature exceeding 24 C, short memory short memory long memory short tail noise heavy tail noise short tail noise optimistic 115 years 59 years 76 years pessimistic 102 years 51 years 64 years Table 2: Periods of return (expected value, in years) before the next heat wave similar with August 2003 (type (B)). 46

47 Modeling flood events Two classical approaches, based on annualized maxima, Hurst (1951), long memory Gaussian process Gumbel (1958), i.i.d. observations Gumbel distributed Based on 100 years of observations, not enough to assess if observations are, or not, independent: impact on return periods. Another idea, Todorovic & Zelenhasic (1970) and Todorovic & Rousselle: consider flood event as a marked Poisson process. Financial ACD (Autoregressive Conditional Duration) approach of Engle & Russell (1998). 47

48 ... to go further For additional information, the thesis can be download from internet or 48