Statistical Analysis of Financial Data

Transcription

1 ETH Zürich, WBL Angewandte Statistik 2017 Blockkurs Statistical Analysis of Financial Data Prof. Dr. Andreas Ruckstuhl Dozent für Statistische Datenanalyse Institut für Datenanalyse und Prozess Design IDP Zürcher Hochschule für Angewandte Wissenschaften ZHAW 30. Januar 2017 A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 1

2 Outline of Lecture 5 Lecture 1: Financial Data and Their Properties Lecture 2: Model for Conditional Heteroskedasticity and Risk Measures Lecture 3: Statistical Issues When Applying Portfolio Theory Lecture 4: (Financial) Factor Models Lecture 5: Copulas Basic Theory Gaussian and t-copulas Archimedean Copulas Rank Correlation Tail Dependence Calibrating Copulas (i.e., Estimating Copulas) *Vine Copula A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 2

3 8.1 Introduction The statistical analysis of multivariate Date is classically based on the multivariate normal distribution. As a consequence we need just to bother estimating the mean/expectation and the covariance matrix from the data. The stochastic dependence between two normally distributed variables is described by the covariance The covariance between two random variables depends on their variances as well as the strength of the linear relationship between them (see also slide 6, Part 3) To understand the relationship between variables, it is much better to examine their (sample) correlation matrix. A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 3

4 Example: Daily returns on two equities GE and IBM and on CRSP, a value-weighted index including dividends (Jan 3,1969 to Dec 31, 1998) > round(cor(stocks),3) > round(cov(stocks),6) ge ibm ge CRSP ibm CRSP ge e-05 ge ibm e-05 ibm CRSP e-05 CRSP Most people use the correlation matrix for examination: All sample correlations are positive The largest correlation is between CRSP and ge The correlation coefficients is only a summary of the linear relationship between variables. Interesting features, such as nonlinearity or the joint behaviour of extreme values, remain hidden A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 4

5 A way out of this shortcomings is the so-called scatterplot matrix: One sees little evidence of nonlinear relationships, but some structures caused by non-normal (i.e., non-gaussian) events. This lack of nonlinearities is typical of returns on equities, but it should not be taken for granted instead, one should always look at the scatterplot matrix. A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 5

6 8.2 Basic Theory Modelling multivariate distributions beyond normal distributions need an new approach to model dependencies (instead of correlations) A copula is a multivariate cumulative distribution function (CDF) whose univariate marginal distributions are all uniform [0,1] Remember: If Y has a continuous CDF F, then F(Y) has a Uniform(0,1) distribution T Suppose Y =(Y1,, Yd) has a multivariate CDF with continuous marginal univariate CDF Then the CDF of is a copula. This CDF is called the copula of Y and denoted by CY that is CY contains all information about dependencies among the components of Y but has no information about the marginal CDFs of Y. A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 6

7 Sklar s Theorem justifies the copula approach. It states that the joint CDF FY can be decomposed into the copula CY, which contains all information about the dependencies among (Y1,..., Yd), and the univariate marginal CDFs, which contain all information about the univariate marginal distributions. That is Motivation: For calibration we need an expression in terms of the joint density: The independence copula: A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 7

8 Example: Scatterplot of the three daily return series (GE, IBM and CRSP) (cf. slide 5) transformed to uniform marginals A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 8

9 8.3 Gaussian and t-copulas Using copulas to model multivariate dependencies, we need parametric families of copulas. The dependencies within Y is described by the correlation matrix Ω of Y since the copula does not depend on the univariate marginal distribution (including variance) Hence, the copula CY depends only on the correlation matrix Ω of Y The Gaussian copula with correlation matrix Ω will be denoted A distribution with a Gaussian copula is called a meta-gaussian distribution. This does not mean that Y has a multivariate Gaussian distribution, since the univariate distributions could be any distribution at all. Example: Meta-Gaussian distribution for 2 dimensions: Transform the marginal distribution to uniformity and apply a standard twodimensional Gaussian distribution with correlation ρ on it where Φ-1() is the inverse of the standard Gaussian distribution and Φ2(,, ρ) the bivariate distribution of two standard Gaussian distribution with correlation ρ A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 9

10 Example in R: Gaussian copula for 2 dim different marginals library(copula) nc1 <- normalcopula(0.8, dim=2) # define the copula model set.seed(5640) rnc1 <- rcopula(1000, nc1) # generate 1000 random numbers Margins: Gamma and beta distributed x <- qgamma(rnc1[,1],2) y <- qbeta(rnc1[,2],1,2) plot(x,y) # they correlate! A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 10

11 Meta-Gaussian distribution became enormously popular in the end causing problems because the market too strongly believed in it ( 2007 subprime crises) The Gauss-copula is the worst invention ever for credit risk management. (A risk Probabilist) see also t-copula Similarly, let be the copula of a multivariate t-distribution with correlation matrix Ω and degrees of freedom ν (The same for all dimensions). The shape parameter ν affects both the univariate marginal distribution and the copula, so ν is a parameter of the copula. A distribution with a t-copula is called a t-meta distribution. distribution A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 11

12 > library(copula) > tc1.3 <- tcopula(c(-.6,0.75,0), dim=3, dispstr="un", df=1) > set.seed(5640) > r.tc1.3 <- rcopula(500, tc1.3) > pairs(r.tc1.3) > cor(r.tc1.3) [,1] [,2] [,3] [1,] [2,] [3,] > tc25.3 <- tcopula(c(-0.6,0.75,0), dim=3, df=25) > set.seed(5640) > r.tc25.3 <- rcopula(500, tc25.3) > pairs(r.tc25.3) > cor(r.tc25.3) [,1] [,2] [,3] [1,] [2,] [3,] A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 12

13 8.4 Archimedean Copulas An Archimedean copula wit a strict generator has the form, ( ) where g() is the generator of the copula and satisfies g is continuous, strictly decreasing, convex mapping [0,1] onto [0, ) g(0)= and g(1)=0 Notice that in ( ), the value of C(u1,, ud) is unchanged if we permute u 1,, ud. One consequence of this property (called exchangeability) is: rank correlations (see later) are the same for all pairs of variables. Archimedian copulas are most useful in the bivariate case or in applications where we expect all pairs to have similar dependencies A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 13

14 Three generators for Archimedean copulas: Frank copula Generator: Inverse Generator: bivariate Frank copula: using ( ) θ=0 requires some care. Evaluate the limit of (bivariate) independence copula C0:, which yields the Clayton copula Generator Clayton copula The Clayton copula for θ=0 is defined as the independence copula C0. For -1 < θ < 0 the copula is not strict and there is no probability, e.g. for d=2: if u1 and u2 are both small then occurs. In theses cases, the Clayton copula is set equal to 0. A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 14

15 Gumbel copula Generator: Gumbel copula: The Gumbel copula is the independent copula when θ=1 but it cannot have negative dependence. Simulations On the next slides, simulation of bivariate samples are shown using scatterplots All simulations are with a sample size of 200 And with values of that give dependencies ranging from strongly negative (if possible) to strongly positive A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 15

16 Random samples from Frank copulas A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 16

17 Random samples from Clayton copulas A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 17

18 Random samples from Gumbel copulas A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 18

19 Some remarks to the simulation results: the Frank and Clayton copulas are rather different when the amount of dependence is somewhere between the extremes. In particular, the Clayton copula's exclusion of the region when -1 < θ < 0 is evident, especially in the example with θ=-07 The Frank copula is symmetric about the diagonal from (0, 1) to (1, 0), but the Clayton copula does not have this symmetry. It is clearly seen that the Gumbel copula cannot model negative dependencies. A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 19

20 8.5 Tail Dependence Tail dependence measures association between the extreme values of two random variables and depends only on their copula. Suppose that Y=(Y1, Y2)T is a bivariate random vector with copula CY. Lower Tail Dependence The coefficient of lower tail dependence is denoted by l and defined as ll := lim P Y2 FY-2 1 (q) Y1 FY-11 (q) = lim q 0 = lim q 0 P Y2 FY-2 1 (q ) and Y1 FY-11 (q ) P Y1 FY-11 (q ) q 0 P FY2 (Y2 ) q and FY1 (Y1 ) q P FY1 (Y1 ) q = lim q 0 CY (q, q ) q If Y1 and Y2 are independent, then Therefore, l = 0 implies that in the extreme left tail, Y1 and Y2 behave as they are independent. A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 20

21 Upper Tail Dependence The coefficient of upper tail dependence is defined analogously For Gaussian and t-copula, Knowing whether or not there is tail dependence is important for risk management. If there are no tail dependencies among the returns on the assets in a portfolio, then there is little risk of clusters of very negative returns, and the risk of an extreme negative return on the portfolio is low. Conversely, if there are tail dependencies, then the likelihood of extreme negative returns occurring simultaneously on several assets in the portfolio can be high. A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 21

22 Tail Dependence for Gaussian and t-copulas Gaussian Copula: λl= 0, if correlation is ρ 1. That is, Gaussian copulas do not have tail dependence except in case of ρ=1 T-Copula with correlation ρ: where Ft,ν+1 is the CDF of the t-distribution with (υ+1) degree of freedom. Since Ft,ν+1(- ) = 0, we see that λl 0 if υ (i.e., consistent with Gaussian Copula) Left: Coefficients of tail-dependence for some t-copulas A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 22

23 Tail Dependance of Archimedean copulas In case of d=2, it can be shown that The Gumbel copula is able to model upper tail dependency, but no lower, whereas The Calyton copula can model lower tail dependence, but no upper: The Frank copula is not able to model either tail dependencies. Financial markets: losses occur more often jointly than gains do A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 23

24 8.6 Calibrating Copulas (i.e., Estimating Copulas) Assume that we have an i.i.d. sample and we wish to estimate the joint distribution using a copula approach. An important task is choosing a copula model. An appropriate copula model can be selected using graphical techniques as well as with AIC (see below) after fitting The fitting of the selected model is based on the log-likelihood of (cf. Slide 7) A direct maximum likelihood estimation may be too challenging because the large number of parameter cause numerical problems if any of the marginals are not well fitted by a convenient parametric family, this may cause bias in the estimated parameters of both the marginals and the copula A pseudo-maximum likelihood estimation may be more convenient. It is a two-step process, where each part (red, green) is estimated separately A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 24

25 In a first step, each of the d marginal distribution functions are estimated, one at a time (green part): There are two approaches, parametrically and nonparametrically. Parametrically: Fit convenient parametric distribution functions to each of the marginal distributions using maximum likelihood estimators Nonparametrically: The marginal distribution is estimated by the empirical CDF: [The divisor (n+1) ensures that the maximum value of is (n/(n+1) < 1] In a second step, is maximised over the parameter of the copula Also the maximisation of this step can be difficult when is highdimensional. E.g., if one uses a t-copula, then there are d(d-1) correlation parameters good starting values are needed (see next slide) A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 25

26 8.6.1 Estimation of in the Meta-t-Distribution The Pearson correlation coefficient ρ measures the linear dependency between two random variables. A such it is not convenient for the copulas approach because It is not invariant under nonlinear strictly increasing transformations Hence, it depends on the univariate marginal distributions as well as the copula. Marginal distribution and correlation do not determine the joint distribution except in case of Gaussian marginals Rank correlation coefficients remedy this problem, since they depend only on the copula Spearman s rank correlation depends only on the copula because it is invariant to any monotonically increasing transformation of the random variables; hence under CDF transformation as well. It is calculated as Pearson correlation of the ranks of the variables and estimates the Pearson correlation of the CDF transformed variables: A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 26

27 Another version which is based on ranks is Kendall s tau Let are two copies of independent bivariate of random variables. Then are called a concordant pair if either. In either case, Similarly, if are called a disconcordant pair Kendall's tau is the probability of a concordant pair minus the probability of a discordant pair. Therefore, Kendall s tau for is The sample Kendall s tau of a bivariate sample is A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 27

28 Kendall s tau has the following relation to Ω: Hence we can use as estimate (or starting value) Unfortunately, this matrix may not be positive (semi-) definite. must be adjusted Determine the eigenvalues "λ" and eigenvectors of Calculate where E is the matrix of the eigenvectors and ε a small number > 0, e.g., ε =0.001 But now the diagonal elements of may not be equal to 1 Hence, a easy fix yields the final solution: multiply the ith row and the ith column of The final result is called A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 28

29 8.6.2 Example: Bivariate data set of returns on IBM and the CRSP index library(ecdat) data(crspday, package="ecdat") IBM <- as.numeric(crspday[,5]) CRSP <- as.numeric(crspday[,7]) ## Fitting marginal t-distributions using package QRM > library(qrm) > IBM.tfit <- fit.st(ibm) ## fit a "scale-location" t-distribution > (IBM.est <- IBM.tfit$par.ests) nu mu sigma > CRSP.tfit <- fit.st(crsp) >(CRSP.est <- CRSP.tfit$par.ests) nu mu sigma ## Transformation of the data to uniform margins pstd <- function (q, mean=0, sigma=1, nu = 5) { # X U ## sd=sigma*s with s=sqrt(nu/(nu - 2)) z <- (q mean)/sigma ## i.e., sd(z) = sqrt(nu/(nu - 2)) pt(q = z, df = nu) } UD <- cbind(pstd(ibm, IBM.est["mu"], IBM.est["sigma"], IBM.est["nu"]), pstd(crsp, CRSP.est["mu"], CRSP.est["sigma"], CRSP.est["nu"])) A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 29

30 ## Are the marginal distribution uniformly distributed? > h.n1 <- length(ibm)/10 > h.t <- h.n1 + 2*c(1,-1)*sqrt(h.n1) ## "confidence band" for height ## of bars in histogram > hist(ibm, col=gray(0.9)) > hist(crsp, col=gray(0.9)) > hist(ud[,1], col=gray(0.9)) > abline(h=h.t, col="gray") > hist(ud[,2], col=gray(0.9)) > abline(h=h.t, col="gray") A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 30

31 ## Fitting t copula in several steps ## from Kendall s tau to an estimation of Omega > (rhotau <- cor(ibm, CRSP, method="kendall")) [1] > (omega <- sin(pi/2*rhotau)) [1] ## Define the t-copula using Omega as correlation parameter > require(copula) > tc2 <-tcopula(omega, dim=2) ## fit by "maximum pseudo-likelihood" > (UD.tc2 <- fitcopula(tc2, data=ud, method="mpl",start=c(omega,8), + optim.method="l-bfgs-b", lower=c(0.1, 5.5), upper=c(0.7, 15))) fitcopula() estimation based on 'maximum pseudo-likelihood' and a sample of size Estimate Std. Error z value Pr(> z ) rho <2e-16 *** df NA NA NA --The maximized loglikelihood is Optimization converged A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 31

32 ## Fitting Gumbel, Frank and Clayton copulas > (UD.gc2 <- fitcopula(copula=gumbelcopula(3,dim=2), data=ud, + method="mpl", optim.method="bfgs")) fitcopula() estimation based on 'maximum pseudo-likelihood' and a sample of size Estimate Std. Error z value Pr(> z ) param <2e-16 *** --The maximized loglikelihood is Optimization converged ## Fitting Frank and Clayton copulas without R output (UD.fc2 <- fitcopula(copula=frankcopula(3,dim=2), data=ud, + method="mpl", optim.method="bfgs")) (UD.cc2 <- fitcopula(copula=claytoncopula(3,dim=2), data=ud, + method="mpl", optim.method="bfgs")) A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 32

33 ## Compare the empirical copula with the estimated ones ## using simulations > n <- nrow(ud) ## define the 4 copulas > ht <- tcopula(param=ud.tc2@estimate[1], df=round(ud.tc2@estimate[2])) > hg <- gumbelcopula(ud.gc2@estimate, dim=2) > hf <- frankcopula(ud.fc2@estimate, dim=2) > hc <- claytoncopula(ud.cc2@estimate, dim=2) > par(mfrow=c(2,3)) ## display data > plot(ud, main="data") ## simulate from the fitted copulas > set.seed(4711) > htr <- rcopula(n, ht); > hfr <- rcopula(n, hf); hgr <- rcopula(n, hg) hcr <- rcopula(n, hc) ## display simulated data > h.pplot <- function(x, main=""){ plot(x, main=main, xlab=expression(u[1]),ylab=expression(u[2]), col=rgb(0,0,1, alpha=0.2), pch=16) } > > > > h.pplot(htr, h.pplot(hgr, h.pplot(hfr, h.pplot(hcr, main="from main="from main="from main="from t copula") Gumbel copula") Frank copula ) Clayton copula") A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 33

34 Which simulated copula is most similar to the data? A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 34

35 u1 <- UD[,1]; u2 <- UD[,2] Udex <- (1:n)/(n+1) ## h.l <- c(0.02, 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 0.9) ## contour levels ## The function 'C.n()' from package copula computes the empirical copula Cn <- C.n(u=cbind(rep(Udex,n), rep(udex,each=n)), U=UD, offset=0,method="c") ## prepare the contours of the empirical copula: EmpCop <- expression(contour(udex, Udex, matrix(cn,n,n), col=2, add=t, levels=h.l)) par(mfrow=c(2,3), las=1, mar=c(3.5,4,2,1), cex.lab=1.2, mgp=c(2.2,0.8,0)) ## "raw" data plot(u1, u2, main="uniform...", xlab=expression(hat(u)[1]), ylab = expression(hat(u)[2])) ## t contour(tcopula(param=ud.tc2@estimate[1], dim=2, df=round(ud.tc2@estimate[2])), pcopula, main=expression(hat(c)[t]), levels=h.l, xlab=...) eval(empcop) ## Frank contour(frankcopula(param= UD.fc2@estimate[1], dim=2), pcopula, main=expression(hat(c)[fr]), levels=h.l, xlab=...) eval(empcop) ## Clayton contour(claytoncopula(param=ud.cc2@estimate[1], dim = 2), pcopula, main=expression(hat(c)[cl]), levels=h.l, xlab=...) eval(empcop) ## Gumbel contour(claytoncopula(param=ud.gc2@estimate[1], dim = 2), pcopula, main = expression(hat(c)[gu]), levels=h.l, xlab=...) eval(empcop) A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 35

36 A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 36

37 Assess the fit by AIC Akaike s information criterion AIC = -2log-likelihood + 2p ## with R > h.aic <- function(obj){ + np <- length(obj@estimate) + h <- obj@loglik + c(loglik=h, AIC=-2*h+2*np) } > rbind(h.aic(ud.tc2), h.aic(ud.fc2), h.aic(ud.gc2), h.aic(ud.cc2)) loglik AIC [1,] [2,] [3,] [4,] ## According to the AIC the t copula fits the data best A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 37

38 8.7* Vine Copula There is a wide range of parametric copula families (Gauss, t, Frank, Clayton,...) But: Standard multivariate copulas can become inflexible in high dimensions. do not allow for different dependency structures between pairs of variables. Way out: E.g., Vine copulas for higher-dimensional data Vine Copulas: Copulas The dependency structure is determined by bivariate copulas and a nested set of trees. Example in d=3 dimensions: One possible decomposition of f(x1,x2,x3) is: Since We obtain A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 38

39 The decomposition is not unique A graphical strucure called regular vine structure has been introduced to help organize them: See also R Package VineCopula more research is needed, particularly towards application A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 39

40 Take Home Message Lecture 5 Copulas are a general concept to model dependencies between random variables A copula is a multivariate cumulative distribution function (CDF) whose univariate marginal distributions are all uniform (0,1) To use copulas to model multivariate dependencies, we need parametric families of copulas Gaussian copula, t-copula, Archimedean copulas (e.g., Frank, Gumbel and Clayton copulas) In finance, the feature of modelling tail dependence is an advantageous characteristic of copulas t-copula, Clayton copula, Gumbel copula Practically, use the introduced copulas mainly to model 2 dimensional r.v. vine copulas enables a flexible modelling of the dependence structure for portfolios of large dimensions A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 40

41 Lecture 5 in the books: Chapter 8 Copulas Sections marked by * are not examined A. Ruckstuhl -- WBL 2017, Lecture 5 of SAoFD -- Page 41