The Asymmetric t-copula with Individual Degrees of Freedom

Transcription

1 The Asymmetric t-copula with Individual Degrees of Freedom d-fine GmbH Christ Church University of Oxford A thesis submitted for the degree of MSc in Mathematical Finance Michaelmas 2012

2 Abstract This thesis investigates asymmetric dependence structures of multivariate asset returns. Evidence of such asymmetry for equity returns has been reported in the literature. In order to model the dependence structure, a new t-copula approach is proposed called the skewed t-copula with individual degrees of freedom (SID t-copula). This copula provides the flexibility to assign an individual degree-of-freedom parameter and an individual skewness parameter to each asset in a multivariate setting. Applying this approach to GARCH residuals of bivariate equity index return data and using maximum likelihood estimation, we find significant asymmetry. By means of the Akaike information criterion, it is demonstrated that the SID t-copula provides the best model for the market data compared to other copula approaches without explicit asymmetry parameters. In addition, it yields a better fit than the conventional skewed t-copula with a single degree-of-freedom parameter. In a model impact study, we analyse the errors which can occur when modelling asymmetric multivariate SID-t returns with the symmetric multivariate Gauss or standard t-distribution. The comparison is done in terms of the risk measures value-at-risk and expected shortfall. We find large deviations between the modelled and the true VaR/ES of a spread position composed of asymmetrically distributed risk factors. Going from the bivariate case to a larger number of risk factors, the model errors increase.

3 Contents 1 Introduction 1 2 Asymmetric t-copula approach Copulas The Gauss copula The standard t-copula The skewed t-copula with individual degrees of freedom (SID t-copula) Simulation of the SID t-copula Calibration of the SID t-copula Application of the SID t-copula to bivariate equity returns Univariate equity index returns Tail dependence Calibration to market data Modelling the risk of asymmetric multivariate returns Simulation of SID t-distributed returns Results for d = Results for d > Summary and Outlook 34 References 37 A Generalized hyperbolic skew Student-t distribution 39 B SID t-copula density 41 C ARMA(r,m)-GARCH(p,q) model 43 D Binomial statistical error 44 E Notes on numerics 45 i

4 Chapter 1 Introduction Copulas have become the subject of intense research in the field of statistics over the last decades. A key benefit of copulas lies in the separation of the dependence structure between stochastic variables and their marginal distributions, thus being a more general concept than multivariate distributions where dependence and the univariate distribution are inextracably linked. Besides allowing for more flexible modelling approaches, copulas overcome well-known limitations and pitfalls of linear correlation approaches by providing more general measures of dependence like rank correlations [1]. An important field of application of the copula concept is financial risk management. However, for reasons of convenience, mainly the Gauss copula and the standard Student-t copula with a single degree-of-freedom parameter are used in practice. These two copulas belong to the class of elliptical copulas and are the unique copulas of the related multivariate distributions, namely the multivariate Gauss distribution and the multivariate Student-t distribution [2]. They are symmetric in the sense that they do not have a parameter which captures asymmetric features with respect to the upper and lower tail of the joint probability distribution. For example, the asymptotic tail dependence is in both cases symmetric, and goes to the same finite value for the standard Student-t copula and to zero for the Gauss copula [2]. Among the variety of copulas and multivariate distributions which have been studied in connection with risk management, the t-copula and the multivariate t- distribution belong to the most intensively studied besides the Gaussian counterparts. There is strong empirical evidence showing that the Gaussian assumption for modelling multivariate financial return data is questionable (see [3] for an overview). The probably most important advantage of the t-distribution over the Gaussian distribution is that it embodies important stylized features like heavy tails while being analytically and numerically relatively tractable. In fact, the t-copula and distribution comprises the Gauss copula and distribution since the latter are recovered from the former by letting the degree-of-freedom parameter go to infinity. Various generalizations of the standard t-copula have been proposed. The skewed t-copula approach in Ref. [4] generalizes the standard t-copula in order to model asymmetric correlation and tail dependence in multidimensional return data. This was motivated by the empirical evidence of asymmetric multivariate equity or eq- 1

5 uity/fx portfolio returns which were shown to be more strongly correlated in bearish than in bullish markets [5, 6, 7]. However, only a single degree-of-freedom parameter can be calibrated in the approach proposed in [4]. The limitation of the standard t- copula to a single degree-of-freedom parameter for all risk factors, on the other hand, was relieved by the grouped t-copula with a common degree-of-freedom parameter for pre-specified groups of risk factors [8] or even individual degree-of-freedom parameters for every risk factor [9]. However, in these two approaches the t-copula is symmetric. This thesis focuses on the investigation how asymmetric correlations can be modelled in terms of a suitably chosen and calibrated t-copula. To this end, we propose a new approach called the skewed t-copula with individual degrees of freedom (SID-t copula) which essentially is a combination of the t-copulas used in [4] and [9]. With the SID t-copula, we will not only be able to describe asymmetric dependence in multivariate asset returns explicitly, but can also investigate other versions of the t-copula, symmetric or asymmetric, which are contained as sub-models in the SID t-copula, i.e. which are recovered by putting restrictions on the parameters of the SID t-copula. Value-at-risk (VaR) and expected shortfall (ES) are two standard measures to quantify risk [2, 10]. Both refer to quantiles of the profit and loss distribution of a risky asset or portfolio. It is not only of theoretical but also practical interest to know the error one makes when using symmetric multivariate distributions to model asymmetric returns. We address the issue in this thesis in a Monte Carlo model study which not only treats the bivariate case but also comprises calculations for more than two risk factors. We conclude the introduction by giving an outline of the chapters of this thesis. In Chapter 2, a brief introduction to the concept of copulas is given with the main definitions and theorems. Using the stochastic representation, the SID t-copula is constructed by extending the representation of the standard t-copula. Some examples of different copulas are simulated and discussed in terms of the bivariate dependence structure before concluding the chapter with a section on the calibration of the SID t-copula. In Chapter 3 the SID t-copula is applied to bivariate stock index return data. Rather than taking the distributions of log-returns directly as marginal input for the copula, we will follow standard practice and apply our copula model to ARMA- GARCH-filtered residual returns. The residuals are obtained from a fit of the empirical data under some distributional assumption, which we choose in our case to be the same marginal distribution as the one implied by the multivariate SID t-distribution. After inspecting the univariate distribution of the residual returns, the empirical tail dependence is analysed for signs of asymmetry. Eventually the SID t-copula is calibrated to the joint returns of the two German stock indexes DAX and TECDAX. Chapter 4 contains a Monte Carlo study of the model errors in terms of value-atrisk and expected shortfall which occur when SID-t multivariate distributed returns are modelled using symmetric multivariate distributions, namely the Gaussian or the symmetric t-distribution. Rather than fitting the copulas, this analysis is entirely in the context of multivariate distributions. VaR and ES are calculated for various 2

6 spread positions and the deviations between the true SID-t VaR/ES and the approximative Gaussian and symmetric t-vars/es s are discussed. While most of the study deals with bivariate spread positions, we also consider spread positions composed of up to eight risk factors. The last chapter summarizes the main results and conclusions of this thesis and gives an outlook on possible further research on the topic. 3

7 Chapter 2 Asymmetric t-copula approach 2.1 Copulas For the definition of a copula, we restrict ourselves to the bivariate case and summarize the main properties of d-dimensional copulas which are important for the present work. For the generalization to more than two dimensions, proofs, and detailed discussions of a number of further properties, we refer to standard textbooks on the topic by Nelsen [11] and McNeil et al. [2]. Following Nelsen [11], a function with the properties C : [0, 1] 2 [0, 1], (u, v) C(u, v) (2.1) 1. For all u, v [0, 1] it holds: C(u, 0) = C(0, v) = 0 and C(u, 1) = u and C(1, v) = v 2. For all u 1, u 2, v 1, v 2 [0, 1] with u 1 u 2 and v 1 v 2 it holds: C(u 2, v 2 ) C(u 2, v 1 ) C(u 1, v 2 ) + C(u 1, v 1 ) 0 is called a (bivariate) copula (function). More useful than the formal definition is the link that can be established between copula functions and multivariate distribution functions. Consider a random variate X = (X 1,..., X d ). It was shown by Sklar that any joint probability distribution function F with marginal distributions F i (i = 1,..., d) can be written as F (x 1,..., x d ) = C(F 1 (x 1 ),..., F d (x d )) (2.2) for all x i IR. C is called the copula of F (Sklar s Theorem). Thus, a copula in d dimension is a d-dimensional probability distribution function on the hypercube [0, 1] d with uniform marginal distributions. Two important properties are: 1. If the marginal distributions of F are continuous then C is uniquely determined on [0, 1] d. 4

8 2. The copula remains invariant under strictly increasing transformations of the components x i. Thus, for strictly increasing marginal distribution functions for X as used throughout this work, the copula of the joint distribution function can be expressed as C(u) := C(u 1,..., u d ) = F (F 1 1 (u 1 ),..., F 1 d (u d)) (2.3) It can be directly seen in 2.2 that the copula concept allows to extricate dependence and marginal behavior. From the point of view of modelling this is very appealing: due to the generality of Sklar s Theorem, one can apply any copula on given marginal distribution functions to obtain a multivariate distribution function which may be appropriate for the problem at hand. In particular, one can take the empirical marginal distribution functions as input for the modelling of the dependence structure. It may be necessary to work with the copula density rather than with the copula itself, e.g. if one wishes to fit a copula to empirical data using the maximum likelihood method. The copula probability density is generally given by If 2.3 holds, the density can be written according to c(u) := c(u) := d C(u 1,..., u d ) u 1... u d (2.4) f ( F 1 1 (u 1 ),..., F 1 d (u d) ) f 1 ( F 1 1 (u 1 ) )...f d ( F 1 d (u d) ) (2.5) with the density f of the joint distribution and the densities f i of the marginal distributions. In the following, some examples of copulas are discussed. The aim is to gradually develop the skewed t-copula with individual degree-of-freedom parameters starting with the Gauss and standard Student-t copulas. These two copulas belong to the elliptical class. It should be mentioned that there is also the important class of Archimedean copulas, which is however not considered in this work. 2.2 The Gauss copula The most common example of a copula is the Gauss copula defined as C G (u; R) = Φ R ( Φ 1 (u 1 ),..., Φ 1 (u d ) ) (2.6) with u (0, 1) d and where Φ 1 is the inverse of the univariate standard normal distribution function and Φ R is the multivariate standard normal distribution function parametrized by the linear correlation matrix R. In general the multivariate normal distribution reads 5

9 Φ Σ (x) = x 1... x d f N;Σ (x) = f N;Σ(x)dx ( 1 (2π) d detσ exp 1 ) 2 (x µ) Σ 1 (x µ) (2.7) Here µ is a location parameter vector and Σ is the covariance matrix with elements σ i,j = E [(X i µ i ) (X j µ j )]. The elements of Σ are connected with the components of the corresponding correlation matrix R via ρ i,j σ i,j σ i σ j (2.8) where σi 2 is the variance of X i. It is due to the invariance property with respect to the marginal distributions mentioned in the last section that one can work with a standardized copula parametrized by the correlation matrix rather than by the covariance matrix [12]. 2.3 The standard t-copula The t-copula and likewise the multivariate t-distribution may be viewed as generalizations of the Gauss copula and distribution since the latter represents a limiting case of the former. Ref. [12] gives an overview of the t-copula and the extentions to multiple degree-of-freedom parameters and to the standard skewed t-copula. The standard t-copula in d dimensions reads ( C t (u; R, ν) = t R,ν t 1 ν (u 1 ),..., t 1 ν (u d ) ) (2.9) where ν is the degree-of-freedom parameter, t 1 ν is the inverse of the univariate standard Student-t distribution function, and t R,ν is the multivariate standard Student-t distribution parametrized by the correlation matrix R and ν. The density of the multivariate Student t distribution is given by [13] f t;d,ν (x) = Γ( ν+d) ( 2 Γ( (πν) ν ) 1 + (x µ) D 1 ) ν+d 2 (x µ) 2 d D ν (2.10) where Γ is the Gamma function. D = ν 2 cov(x) is the dispersion matrix which ν has to be positive-definite and is only defined for ν > 2. It can be shown that the multivariate t-distribution approaches the Gaussian distribution for ν. Using 2.10, the unique t-copula 2.9 explicitly reads C t (u; R, ν) = Γ( ν+d 2 ) Γ( ν 2 ) (πν) d R t 1 ν (u 1 )... t 1 ν (u d ) ( 1 + x R 1 x ν ) ν+d 2 dx (2.11) 6

10 From 2.5 on easily gets the density of the standard t-copula: c t (u; R, ν) = f R,ν(t 1 ν (u 1 ),..., t 1 ν (u d )) di=1 (2.12) f ν (t 1 ν (u i )) with the joint density f R,ν of the multivariate t-distribution with ν degrees of freedom, zero mean and correlation matrix R, and the density f ν of the univariate standard t-distribution. In order to generalize the standard t-copula, it is useful at this point to introduce the stochastic representations of the distributions discussed so far. The multivariate t-distribution belongs to the class of multivariate normal variance mixtures representing elliptically symmetric distributions. 1 The stochastic representation of a random variable X which is multivariate t-distributed is X = µ + W Z (2.13) where the random variables Z are normally distributed with covariance matrix Σ, Z N(0, Σ), and W is a random variable independent of Z which is distributed according to the inverse gamma distribution, W Ig( ν, ν 2 2 ).2 Obviously, one recovers the Gaussian multivariate stochastic representation by setting W 1. Next the representation 2.13 will be extended to the so-called normal mean-variance mixture which adds asymmetry to the so-far symmetric distributions. 2.4 The skewed t-copula with individual degrees of freedom (SID t-copula) The two main shortcomings of the standard t-copula when considering real market returns are the following: It is symmetric with respect to extreme joints events, i.e. inherently assigns the same dependency structure at low and high quantiles of the joint distribution function. This is reflected by an identical upper and lower tail dependent coefficient. 3 The Gauss copula exhibits zero asymptotic tail dependence. There is only one parameter ν for all risk factors. This must be viewed critically when considering multivariate settings with marginals possessing rather different degrees of freedom. In such a case it would be desirable to assign degree-of-freedom parameters ν i to every risk factor X i. To overcome these deficiencies, we propose a new approach called the skewed t-copula with individual degrees of freedom (SID t-copula). Having the stochastic representation 2.13 in mind, we 1 For a detailed discussion of variance mixtures, see [2]. 2 This is equivalent to ν W χ2 ν. 3 The upper and lower tail dependent coefficient is 2t ν+1 ( (ν+1)(1 ρ) 1+ρ ). 7

11 introduce a skewness parameter for every risk factor X i and attribute an own parameter for the degrees of freedom ν i to each risk factor. The stochastic representation for X that fulfills these two requirements reads X = µ + ( W 1 Z 1 + γ 1 W 1,..., W d Z d + γ d W d ) (2.14) where as before Z N(0, Σ) and W j = G 1 ν j (S) is the inverse of the distribution function G νj of the univariate Ig( ν j, ν j ) distribution and S U(0, 1) is a uniform variate independent of Z. 4 The d-dimensional parameter vector γ governs the skewness 2 2 of the marginal and joint distributions. 5 From 2.14 one easily recovers the limiting cases: For γ= 0 the t-copula with individual degrees of freedom ν i for the random variable X = µ + ( W 1 Z 1,..., W d Z d ) For ν 1 = ν 2 =... = ν d the skewed t-copula for the random variable X = µ + W Z + γw For γ= 0 and ν 1 = ν 2 =... = ν d the standard t-copula for the random variable X = µ + W Z. Note that no a-priori grouping of risk factors with identical degree-of-freedom parameters within each group is necessary as it was proposed in [8]. The above approach is essentially a combination of the approach using a t-copula with multiple degrees of freedom proposed in [9] and of the skewed t-copula approach proposed in [4]. The random vector 2.14 is said to be distributed according to the multivariate skewed t-distribution with individual degrees of freedom, and the random vector u = (t 1 (x 1 ),..., t d (x d )) (2.15) is said to be distributed according to the skewed t-copula with individual degrees of freedom. t i t νi,γ i is the univariate distribution function for the density of the generalized hyperbolic skew Student-t distribution which is given in Appendix A. One can write down an expression for the SID t-copula and calculate the associated probability density the according to 2.4. The details are given in Appendix B. Here we give the result for the SID t-copula density which represents the probability density of the uniformly transformed risk factors X i : 1 ( d d 1 c R ν,γ(u) = f N,R (z 1 (u 1, s),..., z d (u d, s)) (w k (s)) 1/2 ds f tk (x k )) (2.16) 0 k=1 k=1 where 4 Note that the W j are perfectly dependent through their common variable S. 5 The upper tail dependent coefficient of a bivariate version of 2.14 is studied analytically in [14]. However, we are not aware of any published work in which a copula density based on this stochastic representation is derived and calibrated to market data. 8

12 z k (u k, s) = t 1 k (u k) γ k w k (s) w k (s) (k = 1, 2,..., d) w k (s) = G 1 ν k (s) x k = t 1 k (u k) (k = 1, 2,..., d) and f N,R (z) is the multivariate standard normal probability density with correlation matrix R. Note that an integration has to be done d times when calculating the density due to the increased flexibility regarding the individual degrees of freedom. This makes the simulation and calibration of the SID t-copula computationally more demanding than the standard t-copula, since for a data sample of size N the integration has to be done N d times. Formula 2.16 is the key theoretical result in this work. In the next two sections, we discuss the simulation and calibration of the SID t-copula. 2.5 Simulation of the SID t-copula The simulation of the SID t-copula is straightforward given the stochastic representation It consists of the following steps: 1. Draw a random vector Z from N(0, R) for a given correlation matrix R. 2. Simulate a random number S from the standard uniform distribution U(0, 1) and calculate W k (S) = G 1 ν k (S) for k = 1,..., d. 3. With these random numbers and for a given skewness vector γ, calculate X = ( W 1 Z 1 + γ 1 W 1,..., W d Z d + γ d W d ). 4. The vector u = (t 1 (x 1 ),..., t d (x d )) is a sample from the SID t-copula. Fig. 2.1 shows the dependence structure u = (u 1, u 2 ) generated by steps 1 to 4 for different t-copulas including the limiting case of the Gauss copula. Starting with the Gauss copula, one observes a symmetric shift of weight to the lower tail (u (0, 0)) and the upper tail (u (1, 1)) with decreasing number of degrees of freedom of the t-copula. At the same time, the density of the distribution is tightened around the diagonal and more weight is also found in the upper left and lower right corner. The situation is quite different in the asymmetric case. Here one notices stronger correlations in the lower tail as can be seen in the smaller variance of the points with respect to their distance from the diagonal for u (0, 0). Note that taking different values for the skewness (γ 1 γ 2 ) introduces additional asymmetry with respect to the diagonal, i.e. the density of points with u 1 near 1 and u 2 near 0 is different from the density of points with u 1 near 0 and u 2 near 1. 9

13 (a) Symmetric copulas (γ = 0): Gaussian (left) and t-copula (right). (b) Symmetric t-copulas (γ = 0). (c) Asymmetric t-copulas (γ 0). Figure 2.1: Dependence structure of different bivariate copulas (ρ = 0.8). 10

14 For further discussion of the copulas, we show the tail dependence coefficients in Fig. 2.2 for some of the copulas shown in Fig The lower (upper) tail dependence coefficient is defined as the probability of the conditional probability that X 1 lies below (above) the quantile q given that X 2 lies below (above) it, divided by the unconditional probability of X 2 lying below (above) it and in the limit q 0 (q 1). The tail dependence coefficients thus represent measures for the coincidence of extremely unlikely joint events. Formally they read: Ptail lower P (F 1 (X 1 ) < q F 2 (X 2 ) < q) = lim q 0 P (F 2 (X 2 ) < q) P upper P (F 1 (X 1 ) > q F 2 (X 2 ) > q) tail = lim q 1 P (F 2 (X 2 ) > q) (2.17) (2.18) where F 1, F 2 denote the marginal distribution functions. A symmetric dependence structure would mean Ptail lower = P upper tail. The quantiles of the upper tail dependence coefficients have been mapped onto the lower quantiles (q 1 q) for better comparison. As a consequence of the Monte Carlo runs for each copula which were performed to obtain the results, the statistical errors are rather small (see Appendix D). The coefficients from the Gauss and the standard t-copula are both symmetric, with the t-copula producing higher tail dependence towards lower quantiles just as expected. The SID t-copula is the only copula which generates asymmetry of the tail dependence coefficients. 2.6 Calibration of the SID t-copula A standard problem in financial risk management is the calibration of a multivariate model to real return time series, say of length N, where the individual time series of the d risk factors x i,j (i = 1,..., d, j = 1,..., N) are assumed to be independently and identically distributed. The calibration is usually done in two steps, first by transforming the sample onto the [0, 1] d domain and then by estimating the copula parameters. The transformation to the unit hypercube is commonly achieved by one of three methods, either a parametric fit of the marginal distributions (also known as inference function for margins method ) or using the empirical distribution method ( pseudo-likelihood method ) or a combination of the two. For the fit to real market data, we employ the pseudo-likelihood method in this work and make use of the empirical distribution function. The ith marginal distribution function is estimated via F i (x) = 1 N + 1 N 1 {xi,j x} (2.19) j=1 where the pre-factor ensures that transformed points do not hit the boundary of 11

15 (a) Symmetric copulas: Gauss copula (left) and t-copula (right). (b) Asymmetric t-copulas. Figure 2.2: Tail dependence coefficient for different t-copulas (ρ = 0.8). The quantiles of the upper tail dependence coefficients have been mapped onto the lower quantiles. The error bars represent the binomial statistical error. 12

16 the unit cube. Then one builds a pseudo-sample from the copula by constructing vectors û j = (u 1,j,..., u d,j ) = ( F 1 (x 1,j ),..., F d (x 1,d ) ) (2.20) Once the pseudo-copula data has been generated, the second step consists of estimating the copula parameters. This is usually done by the method of maximum likelihood estimation (see e.g. Refs. [15, 16] or standard textbooks on statistics for introductions to the maximum likelihood method). In our case the parameters R, ν, γ must be estimated. To this end, one minimizes the negative log-likelihood function N logl(r, ν, γ; û 1,..., û N ) = logc R ν,γ(û j ) (2.21) j=1 with respect to R, ν, γ. The optimization problem posed by may be quite demanding in higher dimensions due to the additional boundary condition that the correlation has to be positive-definite. In practice, alternative methods are therefore often used for estimating the correlation matrix. Among those we mention the moment method using Kendall s tau for the estimation of pairwise correlations (see Ref. [12] and references therein). This method, however, cannot be employed in the present case since it is only applicable when dealing with elliptical distributions. A third method for the estimation of R makes use of the stochastic representation For one common degree-of-freedom parameter ν, one obtains the following formula for the covariance matrix: Σ = ν 2 ν ( cov(x) 2ν 2 ) (ν 2) 2 (ν 4) γγ (2.22) This relation can be generalized to the case of individual degree-of-freedom parameters for the risk factors, where additional calculation of the mean and covariance of the random variables W i have to be performed. However, we found this approach for the SID t-copula to be statistically not stable enough to apply it to real return time series. We conclude this chapter by adding two remarks. The first is on the estimation of the model parameters. There are different ways of estimating the parameters. It is important to be aware to the implicit model assumption connected with the different estimation methods. On the one hand, fitting all parameters using the copula focuses on the dependence structure only and one does not have to make assumptions about the underlying stochastic process. On the other hand, using a relation like 2.22 which is derived from the corresponding stochastic representation (for 2.22, from the skewed-t stochastic process with one degree-of-freedom parameter), one implicitly assumes the associated multivariate distribution function (e.g. the multivariate skewed t-distribution for 2.22). The marginal parameters ν i and γ i can be estimated by fitting the marginal distributions or, for 13

17 better inference, by fitting both the marginal distributions and the copula (with fixed correlation matrix). Since the copula fits are numerically expensive, we will use the first method, i.e. fitting the marginal distributions only, in the model study in chapter 4. The second remark is on the possibilities to model asymmetry in the dependence structure. In principle, for modelling an asymmetric dependence structure one does not necessarily have to use a copula which includes an asymmetry parameter (vector). The simulation results for the dependence structure and the tail dependence shown in Section 2.5 are surely representatives of the indicated copulas, but particular ones since they were generated from the respective multivariate distributions. However, it may be sufficient to combine empirical asymmetric marginal distributions with a symmetric copula like the Gauss copula using It should be remembered that this separation of marginal behavior and dependence is one the key benefits of the copula approach in general. It is one of the main issues adressed in the next chapter if the asymmetric parametrization of the SID t -copula is actually required to model the dependence structure of real multivariate asset returns or if it may be sufficient to use symmetric t-copulas or even the Gauss copula. 14

18 Chapter 3 Application of the SID t-copula to bivariate equity returns 3.1 Univariate equity index returns Before discussing the bivariate case, we briefly discuss the univariate daily returns of the DAX which is one of the two equity risk factors whose dependence structure is studied later. The DAX equity index comprises the performance of the 30 largest German companies. The time series with daily returns covers the period from 2 April 2003 to 15 October 2012 (2315 data points). 1 As is commonly done for equity returns, the returns r t on day t were modelled in a first step as log-normally distributed, r t = log ( St S t 1 ) (3.1) where S t denotes the index value. Rather than considering the distribution of log-returns directly, however, we produce in a second step the residuals (also called innovations) of an ARMA(1,1)-GARCH(1,1) fit of the empirical returns. The residuals are obtained from the empirical returns by jointly fitting a mean filtration process and a conditional variance equation for the residuals (see Appendix C) [2, 4, 10]. For the fit one has to specify a distribution for the residuals, which we choose in our case to be the GHST distribution with the density A.1. The ARMA-GARCH fit was carried out using the R package rugarch [17]. Fig. 3.1a shows the histogram of the ARMA-GARCH residuals calibrated to the DAX log-normal returns (A.1. The GHST probability density has been added using the best fit parameters. For comparison, the symmetric Student-t and Gaussian probability densities are also shown. 2 The ARMA-GARCH fit yields γ = 0.5 ± 0.3 for the skewness and ν = 11 ± 3 for the degree-of-freedom parameter, thus supporting the negative skew one recognizes 1 All stock index data are adjusted close quotations (i.e. without the effects of corporate action) and were downloaded from the Yahoo Finance webpage. 2 Note that the fits under these two distributional assumptions produces different residuals. 15

19 (a) Histogram of the GHST residuals and residual probability densities for different distributional assumptions for the residuals : skewed Student-t (straight line), symmetric Student-t- (dashed line), Gauss (dotted line). (b) Magnified lower tail of (a). The arrows indicate the 1%- quantile of the respective residual distribution model. Figure 3.1: Residual distributions from an ARMA(1,1)-GARCH(1,1) fit of DAX logreturns. 16

20 by visual inspection of the distribution. However, the numerical evidence is somewhat spoiled by the relatively large statistical standard error for γ. Comparing the GHST density with the symmetric Student-t and the Gaussian density, both the Student-t and the Gaussian density are shifted to the left compared to the GHST density, thereby accounting for the increased empirical probability density in the lower part of the histogram. Naturally, a region of particular interest in risk management is the extreme lower part of the distribution where one finds the probability of large losses. Fig. 3.1b shows a magnified part of 3.1a for low quantiles. The arrows indicate the 1%-quantile of the different residual distributions, i.e. the (negative) VaR for a confidence level of 99%. Due to the skewness of the DAX residuals, the VaR computed with a GHST fit is almost 10% larger than the VaR computed using a symmetric Student-t assumption for the residuals. The difference between both, skewed and conventional, Student-t VaR and the Gauss VaR can be attributed to the fat tails of the empirical distribution which are captured better by the Student-t densities than by the Gaussian density. 3.2 Tail dependence We now turn to the bivariate case and analyse the tail dependence of the joint residual distribution the equity index pair DAX/TECDAX. The TECDAX contains the 30 largest German companies in the technology sector. The time period of the joint calibration data of DAX and TECDAX to obtain the ARMA-GARCH residuals is the same as used in the previous section (April October 2012). In Fig. 3.2, the empirical upper and lower tail dependence ocefficients using finite q s are shown, covering the lower and upper 10% of the joint empirical probability distribution function. The quantiles of the upper tail dependence coefficients have been mapped onto the lower quantiles (q 1 q) for better comparison. The error bars indicate the binomial statistical error (see Appendix D). Rather than being interested in the finiteness of the asymptotic tail dependence, which cannot be proven or disproven upon the presented data, we look at differences between the empirical upper and lower tail dependences for finite q. One observes that the empirical lower tail dependence is larger than the upper one. The finding is statistically significant for most of the quantiles of the DAX/TECDAX asset pair and thus represents empirical evidence for an asymmetric dependence structure. More empirical evidence on this issue has been reported by Sun et al. [4]). We proceed next with the calibration of the SID t-copula for a model description of the empirically found asymmetry in the DAX/TECDAX asset pair. 3.3 Calibration to market data We investigate next what type of model from the t-copula family is required to describe asymmetry in the dependence structure. To this end, we will calibrate the SID t-copula proposed in Section 3 (the full model) to market data by means of a 17

21 Figure 3.2: Upper and lower tail dependence for DAX/TECDAX. The quantiles of the upper tail dependence coefficients have been mapped onto the lower quantiles. maximum likelihood fit and compare the relative quality of the reduced models by inspecting the likelihood function in the (γ 1, γ 2 )- and (ν 1, ν 2 )- parameter sub-spaces and by means of an information criterion. We take again the German equity index pair DAX/TECDAX as an example for the application of the SID t-copula method proposed in Section 2.4. The data foundation is the same as reported in the previous section. Fig. 3.3 shows the dependence structure of the asset pair. Before proceding to the presentation of the results, we summarize the main questions that we want to address in the remaining part of this chapter and give a summary of the main computation steps. The analysis of asymmetric dependence structure can be further subdivided into the following questions: 1. Is there statistically significant asymmetry in the dependence structure of the bivariate asset return data? 2. If asymmetry can be found, is a skewed Gauss copula sufficient to capture the asymmetry? 3. If the skewed Gauss copula cannot describe the dependence structure adequately, do we need to include individual degree-of-freedom parameters in our model or is it sufficient to assign one common parameter ν to both risk factors? 18

22 Figure 3.3: Dependence structure of the equity index pair DAX/TECDAX. The last point addresses the new aspect treated in the study of asymmetric dependence structures using skewed Student-t copula versions. The main computation steps of the analysis are as follows: 1. Perform an ARMA(1,1)-GARCH(1,1) fit of the daily log-returns of DAX and TECDAX using the generalized hyperbolic skewed t-distribution (see Appendix A). 2. Transform the residuals obtained by step 1 to the unit plane (0, 1) (0, 1) by means of Minimize the negative log-likelihood function 2.21 of the density of the SID-t copula. 4. Determine the statistical errors of the estimated parameters. 5. Calculate information criteria of the SID t-copula fit and the fits using symmetric copulas (symmetric t-copula with individual degrees of freedom, standard-t copula with one common degree-of-freedom parameter, and Gauss copula). The maximum likelihood fit of the biariate dependence structures was performed by scanning the parameter grid (ρ, ν 1, ν 2, γ 1, γ 2 ) and locating the minimum of the log-likelihood function. In order to assess the statistical errors of the parameter estimations, additional scans were carried out for some of the parameters while holding the others fixed, namely of ρ with (ν 1, ν 2, γ 1, γ 2 ) fixed, (ν 1, ν 2 ) with (ρ, γ 1, γ 2 ) fixed 19

23 Figure 3.4: Log-likelihood as a function of the correlation parameter ρ in units of standard deviations from the minimum log-likelihood which has zero value. and (γ 1, γ 2 ) with (ρ, ν 1, ν 2 ) fixed. The numerical calculations were done using a selfwritten R program. The ARMA-GARCH fit was performed using the R package rugarch (see also Appendix E). Fig. 3.4 shows the variation of the log-likelihood with the correlation parameter where the other parameters are fixed. The estimation error of ρ is about 0.02 within one standard deviation, where the s-standard-deviation error is approximately obtained from the relation logl(θ ) = logl max 1 2 s2 (3.2) which holds in the large sample limit [16]. Using 3.2 one can calculate the values of the parameter vector θ for given s which yield a contour in the parameter space indicating the s-standard-deviation error. Figs. 3.5 and 3.6 show contour plots of the log-likelihood function for varying skewness and varying degree-of-freedom parameters fot the stock index pair DAX/TECDAX. In order get a visualization of the estimation error of the parameters, differently coloured regions are shown where each colour corresponds to one standard deviation of the log-likelihood. The darkest region represents the 1σ-zone. Regarding the log-likelihood differences with respect to the maximum log-likelihood in units of the standard deviation of the estimation in the plane (γ DAX, γ T ECDAX ) (Fig. 3.5), the parameter γ DAX is statistically significantly different from zero given a distance of about 4σ from the zero axis. Comparing this to the univariate fit of the 20

24 Figure 3.5: Contour plots of the log-likelihood function 2.21 in the γ DAX, γ TECDAX - plane for fixed degree-of-freedom parameters and fixed correlation. The negative loglikelihood function was shifted such that the minimum has zero value. Each colour corresponds to one standard deviation of the log-likelihood. residual DAX log-return with a rather large estimation error discussed in Section 3.1, we obtain more conclusive evidence of asymmetry in the bivariate case than in the univariate case. Although within a smaller significance level than the DAX parameter, γ T ECDAX is also negative (within 3σ). A point of particular interest is if the modelling flexibility of the SID t-approach with individual degree-of-freedom parameters is required. Remember that this is the new aspect in the description of dependence structures compared with the t-copula approaches that have been reported so far (cf. Section 2.16). In Fig. 3.6 inference on this can be gained by inspecting the log-likelihood values along the diagonal representing a maximum likelihood fit of a standard skewed t-copula (ν DAX = ν TECDAX ). The difference between the minimum on this line and the log-likelihood minimum obtained with the SID t-copula falls into the 2σ-region, thus moderately indicating the need for modelling flexibility as regards the degree-of-freedom parameters. The use of a Gauss copula, which corresponds formally to the end of the diagonal 21

25 Figure 3.6: Contour plots of the log-likelihood function 2.21 in the ν DAX, ν TECDAX - plane for fixed skewness parameters and fixed correlation. The negative log-likelihood function was shifted such that the minimum has zero value. Each colour corresponds to one standard deviation of the log-likelihood. (ν DAX = ν TECDAX ) must be strongly rejected in view of the results since the log-likelihood lies far outside the 4σ region. The parameter ν TECDAX is smaller than 10 with a significance of 2σ. For a better assessment of the model goodness we use the Akaike information criterion (AIC) (see e.g. [18]). This quantity combines the log-likelihood function with the number of parameters of a model and is defined as follows: AIC = 2k 2logL (3.3) with the number of parameters k. Obviously, the use of more parameters is punished via the first term. Note that the AIC provides a relative measure of model goodness but does not represent a test of the absolute model goodness in the sense of testing a null hypothesis. Considering a candidate set of n models, the quantity P AIC i ( ) AICmin AIC i = exp 2 22 (i = 1,..., n) (3.4)

26 Copula model # of parameters AIC Pi AIC SID 5 (ρ, ν 1, ν 2, γ 1, γ 2 ) Skewed t-copula (1 dof) 4 (ρ, ν, γ 1, γ 2 ) Skewed Gauss 3 (ρ, γ 1, γ 2 ) Unskewed ID t-copula 3 (ρ, ν 1, ν 2 ) Table 3.1: Akaike information criterion and relative probability of the models to be the actual model of smallest information loss with respect to the best model for some copula models. is the relative probability that the ith model minimizes the estimated information loss due to the modelling with respect to model with AIC min [18]. Table 3.1 summarizes AIC and Pi AIC values for some models for the dependence structure with respect to the SID t-copula which has the minimum AIC. The relative probability that the skewed t-copula with a common degree-of-freedom parameter minimizes the information loss due to modelling is 10% with respect to the best model, the SID t-copula. The other two models, namely the skewed Gauss copula and the unskewed t-copula with individual degree-of-freedom parameters, have a very small relative probability compared to the SID t-copula. We finish this section with a note on the impact of the observed asymmetry on the risk measures value-at-risk and expected shortfall. While the calibration of the copula yield significant evidence for the need to include asymmetry parameters in the description of the dependence structure of the asset pair DAX/TECDAX, it is a priori unclear how this asymmetry affects VaR and expected shortfall compared to the other t-copula approaches. To analyse this, one would have to generate pseudo-random variates from the copula and transform them back using the inverse marginal distribution functions. Thus, asymmetry enters the calculation via the empirical inverse marginal distributions, possibly making up for the lack of an explicit parametric modelling of asymmetry in the copula. This analysis would be rewarding, however we considered it to be out of scope for the present work. Instead, we chose a numerically more feasible analysis to gain insight into the VaR and ES impact. In the next chapter we use multivariate t-distributions to quantify the model error when modelling skewed multivariate returns using symmetric multivariate distributions. The results of this study may give some qualitative understanding of the modelling errors when using symmetric t-copulas for the description of asymmetric multiariate returns. 23

27 Chapter 4 Modelling the risk of asymmetric multivariate returns 4.1 Simulation of SID t-distributed returns In the previous two chapters the skewed t-copula with individual degrees of freedom (SID t-copula) was proposed by appropriately modifying the stochastic multivariate-t representation and was calibrated to market data. We now want to turn our attention to the multivariate SID-t distribution. In this chapter, the effects of multivariate skewness on the risk measures value-at-risk (VaR) and expected shortfall (ES) are analysed by means of a Monte Carlo model study. It is a question not only of theoretical, but also of practical interest given the fact that the risk measurement based on symmetric multivariate distributions is standard practice in market risk controlling. The value-at-risk of an asset or a portfolio is the estimated loss over some time horizon at a given confidence level α. The VaR of a unit of investment is defined as the smallest number l such that the probability that the loss L exceeds l is no larger than (1 α) ([2]): VaR α := inf {l IR : P (L > l) 1 α}. (4.1) The VaR is thus simply a quantile of the loss distribution of a portfolio. The loss distribution L is also called the return distribution. Typical values for α are α = 0.95 or α = 0.99 and 1 or 10 days for the time horizon in market risk. For the calculations in this chapter we chose a confidence level of The expected shortfall 1 is closely related to the VaR ([2]): ES α := 1 1 VaR u (L)du. (4.2) 1 α α Thus expected shortfall is the average over the VaRs for all levels u α, meaning that it is more sensitive to the shape of the tail of the loss distribution. Moreover 1 Expected shortfall as defined here is also known under the names conditional value-at-risk (CVaR), average value-at-risk (AVaR), and expected tail loss (ETL). 24

28 ES is, in contrast to VaR, a coherent risk measure. 2 An obvious relation between the two risk measures is ES α VaR α. While the VaR has been used extensively in risk management for some time, the expected shortfall has recently attracted increasing attention as a possible alternative measure for market risk in the framework of the fundamental review of the trading book [19]. The basic idea of the model study presented next is to assume that the risk factor returns in the real world are distributed according to the multivariate SID t-distribution, i.e. they are generated using the recipe given by the four steps at the beginning of Section 2.5 In particular this means that they have a dependence structure according to the SID t-copula (cf. remarks at the end of chapter 2). These synthetic returns are then fitted using models without skewness, namely the symmetric t- and the Gaussian multivariate distributions. The parameters were estimated using 2.22 for the covariance matrix and using the marginal GHST distribution with the density A.1 for the maximum likelihood estimation of the degree-of-freedom parameters. The different approaches are compared in terms of VaR and ES for a synthetic portfolio consisting of one long and one short position: V = X L X S. (4.3) In terms of the formulas 4.1 and 4.2, we have L V for this simple linear portfolio. On an elementary level, this spread position represents the typical case of a hedged portfolio. To minimize statistical uncertainty, we performed multiple MC evaluations of sample sizes of The statistical simulation errors are smaller than the symbol size in the graphical representation of the results presented further below. A point of particular interest is the impact of skewness in the presence of a larger number of risk factors. Therefore, in addition to the 2d-simulations, calculations for d > 2 were carried out. The portfolios in this case were constructed with pairwise spread positions. 4.2 Results for d = 2 The results of the bivariate simulations presented in this section have all been obtained by the same procedure. We summarize the main steps before beginning with the discussion : 1. Simulate the SID t-bivariate stochastic process according to 2.14 in order to produce asymmetric synthetic returns. 2. Calculate the ( true ) VaR and ES according to 4.1 and 4.2, respectively, for the spread position V = X L X S built by the two simulated risk factors. 2 VaR lacks the property of sub-additivity and thus fails to fulfill all requirements of a coherent risk measure, the others being positive homogeneity, monotony, and translation invariance [2]. 25

29 Figure 4.1: Absolute VaR (left) and VaR relative to the true VaR (right) of a spread position with asymmetrically distributed risk factors for two values of ρ. Both risk factor distributions have the same negative skew γ L = γ S = γ. The degree-of-freedom parameters are ν L = ν S = 6, the correlation parameter is ρ = For each sample simulated in step 1, estimate the parameters of the two symmetric copulas using relation 2.22 and maximum likelihood fits of the marginal distributions: (a) Gaussian fit: only the covariance matrix Σ is estimated. (b) Student-t fit: the covariance matrix Σ and the degree-of-freedom parameters ν L and ν S are estimated. Fig. 4.1 shows the results for two negatively skewed risk factors with equal degree-offreedom parameters (ν L = ν S ). One observes that the true VaR and the Gauss VaR are practically insensitive to the variation of γ, with the Gauss VaR being about 10% smaller. To discuss this, we recall the dependence structures shown in Fig. 2.1 and interprete the behavior of the VaR in terms of the copula density which is the density of points in the scatter plots. An increasing negative skew leads to a shift of weight from the bottom left to the top right corner of the diagram. For γ L = γ S, this shift occurs along the diagonal due to symmetry reasons. In addition, the distribution of points becomes tighter around the diagonal. For low (and high) quantiles, the uniform probability transformation is approximately linear and thus one can study the qualitative behavior of the VaR in the (u L, u S )-plane. For the VaR of a spread position, one has to consider lines in the direction of the diagonal, u L u S = const, indicating lines of (approximately) constant spread VaR where the location of the line depends on the chosen quantile. The quantile can be visualized approximately as the straight line parallel to the diagonal 26