A Dynamic Grouped-t Copula Approach For High-dimensional Portfolios

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios Dean Fantazzini

April the 21th 2007, International Workshop on Computational and Financial Econometrics, Geneva (Switzerland)

Overview of the Presentation

1st Introduction

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

2

Overview of the Presentation

1st Introduction 2nd Dynamic Grouped-T Copula Modelling: Definition and Estimation

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

2-a

Overview of the Presentation

1st Introduction 2nd Dynamic Grouped-T Copula Modelling: Definition and Estimation 3rd Asymptotic Properties

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

2-b

Overview of the Presentation

1st Introduction 2nd Dynamic Grouped-T Copula Modelling: Definition and Estimation 3rd Asymptotic Properties 4th A Simulation Study

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

2-c

Overview of the Presentation

1st Introduction 2nd Dynamic Grouped-T Copula Modelling: Definition and Estimation 3rd Asymptotic Properties 4th A Simulation Study 5th Empirical Analysis

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

2-d

Overview of the Presentation

1st Introduction 2nd Dynamic Grouped-T Copula Modelling: Definition and Estimation 3rd Asymptotic Properties 4th A Simulation Study 5th Empirical Analysis 6th Conclusions

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

2-e

Introduction The increasing complexity of financial markets has pointed out the need for advanced dependence modelling in finance. Why? • Multivariate models with more flexibility than the multivariate normal distribution are needed; • When constructing a model for risk management, the study of both marginals and the dependence structure is crucial for the analysis. A wrong choice may lead to severe underestimation of financial risks. Recent developments in financial studies have tried to tackle these issues by using the theory of Copulas: see Cherubini et al. (2004) for a general review of copula methods in finance. However, mostly low-dimensional applications have been considered so far, while the rare high-dimensional cases present no dynamics at all.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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Introduction Daul, Giorgi, Lindskog, and McNeil (2003), Demarta and McNeil (2005) and Mc-Neil, Frey, and Embrechts (2005) underlined the ability of the grouped t-copula to model the dependence present in a large set of financial assets into account. We extend their methodology by allowing the copula dependence structure to be time-varying and we show how to estimate its parameters. Furthermore, we prove the consistency and asymptotic normality of this estimator under some special cases and we examine its finite samples properties via simulations. Finally, we apply this methodology for the estimation of the VaR of a portfolio composed of thirty assets.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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Dynamic Grouped-T Copula Modelling: Definition and Estimation Let Z|Ft−1 ∼ Nn (0, Rt ), t = 1, . . . T , given the conditioning set Ft−1 , where Rt is the n × n conditional linear correlation matrix which follows a ¯ is the unconditional correlation matrix. Furthermore DCC model, and R let U ∼ U nif orm(0, 1) be independent of Z . p

Let Gν denote the distribution function of ν/χν , where χν is a chi square distribution with ν degrees of freedom, and partition 1, . . . , n into m subsets of sizes s1 , . . . , sm . Set Wk = G−1 νk (U ) for k = 1, . . . , m and then Y|Ft−1 = (W1 Z1 , . . . , W1 Zs1 , W2 Zs1 +1 , . . . , W2 Zs1 +s2 , . . . , Wm Zn ), so that Y has a so-called grouped t distribution. Finally, define U|Ft−1 = (tν1 (Y1 ), . . . , tν1 (Ys1 ), tν2 (Ys1 +1 ), . . . , tν2 (Ys1 +s2 ), . . . , tνm (Yn )) (1) U has a distribution on [0, 1]n with components uniformly distributed on [0, 1]. We call its distribution function the dynamic grouped t-copula.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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Dynamic Grouped-T Copula Modelling: Definition and Estimation Note that (Y1 , . . . , Ys1 ) has a t distribution with ν1 degrees of freedom, and in general for k = 1, . . . , m − 1, (Ys1 +...+sk +1 , . . . , Ys1 +...+sk+1 ) has a t distribution with νk+1 degrees of freedom. Similarly, subvectors of U have a t-copula with νk+1 degrees of freedom, for k = 0, . . . , m − 1. In this case no elementary density has been given. However, there is a very useful correlation approximation, obtained by Daul et al. (2003) for the constant correlation case: ρi,j (zi , zj ) ≈ sin(πτij (ui , uj )/2)

(2)

where i and j belong to different groups and τij is the pairwise Kendall’s tau. This approximation then allows for Maximum Likelihood estimation for each subgroup separately.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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Dynamic Grouped-T Copula Modelling: Definition and Estimation Definition 1 (Dynamic Grouped-T copula estimation). 1. Transform the standardized residuals (ˆ η1t , ηˆ2t , . . . , ηˆnt ) obtained from a univariate GARCH estimation, for example, into uniform variates (ˆ u1t , u ˆ2t , . . . u ˆnt ), using either a parametric cumulative distribution function (c.d.f.) or an empirical c.d.f.. 2. Collect all pairwise estimates of the unconditional sample Kendall’s tau given by  −1   X T ˜ ˜  τˆ ¯i,j (ˆ uj , u ˆk ) =  sign (ˆ ui,t − u ˆi,s )(ˆ uj,t − u ˆj,s ) (3) 2 1≤t<s
τ τ ˆ ˆ ¯ ¯ ¯(ˆ uj , u ˆk ), in an empirical Kendall’s tau matrix Σ defined by Σjk = τˆ and then construct the unconditional correlation matrix using this τ π ˆ ˆ ¯ ¯ relationship Rj,k = sin( 2 Σj,k ), where the estimated parameters are the q = n · (n − 1)/2 unconditional correlations [¯ ρ1 , . . . , ρ¯q ]′ .

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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Dynamic Grouped-T Copula Modelling: Definition and Estimation

3. Look for the ML estimator of the degrees of freedom νk+1 by maximizing the log-likelihood function of the T-copula density for each subvector of U, for k = 0, . . . , m − 1: νˆ1

=

arg max

T X

ˆ ¯ ν1 ), log ct−copula (ˆ u1,t , . . . , u ˆs1 ,t ; R,

(4)

t=1

νˆk+1

=

arg max

T X

ˆ ¯ νk+1 ), ˆs1 +...+sk+1 ,t ; R, log ct−copula (ˆ us1 +...sk +1,t , . . . , u

t=1

k = 1, . . . , m − 1

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

(5)

8

Dynamic Grouped-T Copula Modelling: Definition and Estimation ˆ t, 4. Estimate a DCC(1,1) model for the conditional correlation matrix R by using QML estimation with the normal copula density: α, β

=

arg max

T X

ˆ ¯ Rt ) = ˆn,t ; R, log cnormal (ˆ u1,t , . . . , u

(6)

t=1

=

arg max

T X t=1

1 |Rt |

1/2



1 − I)ζ exp − ζ ′ (R−1 t 2



(7)

where ζ = (Φ−1 (ˆ u1,t ), . . . , Φ−1 (ˆ un,t ))′ is the vector of univariate normal inverse distribution functions, and where we assume the ˆt following DCC(1,1) model for the correlation matrix R Rt

=

Qt

=

(diagQt )−1/2 Qt (diagQt )−1/2 (8) ! S L L S X X X X ¯+ βs Qt−s αl u ˆ t−l u ˆ ′t−l + 1− αl − βs Q l=1

s=1

l=1

s=1

¯ is the n × n unconditional correlation matrix of u where Q ˆ t , αl (≥ 0) PS PL and βs (≥ 0) are scalar parameters satisfying l=1 αl + s=1 βs < 1. A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

9

Asymptotic Properties Let us define a moment function of the type

E [ψ (Fi (ηi ), Fj (ηj ); ρ¯i,j )] = E [¯ ρ(zi , zj ) − sin(πτ (Fi (ηi ), Fj (ηj ))/2)] = 0 (9) where the marginal c.d.f.s Fi , i = 1, . . . , n can be estimated either parametrically or non-parametrically, we can easily define a q × 1 moments vector ψ for the parameter vector θ0 = [¯ ρ1 , . . . , ρ¯q ]′ as reported below:



  ψ (F1 (η1 ), . . . , Fn (ηn ); θ0 ) =  

E [ψ1 (F1 (η1 ), F2 (η2 ); ρ¯1 )] .. . E [ψq (Fn−1 (ηn−1 ), Fn (ηn ); ρ¯q )]

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios



  =0  (10)

10

Asymptotic Properties

ˆ Let assume that the standardized Theorem 1.1.1 (Consistency of θ). errors (η1t , . . . , ηnt ) are i.i.d random variables with dependence structure given by (1). Suppose that (i) the parameter space Θ is a compact subset of Rq , (ii) the q-variate moment vector ψ (F1 (η1 ), . . . , Fn (ηn ); θ0 ) defined in (10) is continuous in θ0 for all ηi , (iii) ψ (F1 (η1 ), . . . , Fn (ηn ); θ) is measurable in ηi for all θ in Θ, (iv) E [ψ (F1 (η1 ), . . . , Fn (ηn ); θ)] 6= 0 for all θ 6= θ0 in Θ, (v) E [supθ∈Θ kψ (F1 (η1 ), . . . , Fn (ηn ); θ) k] < ∞, (vi) ρ¯i,j = 0 or ρ¯ij = o(1), p Then θˆ → θ0 = [0]q×1 as n → ∞.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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Asymptotic Properties

Theorem 1.1.2 (Consistency of νˆk+1 , k = 0, . . . , m − 1). Let the assumptions of the previous theorem hold, as well as the regularity conditions reported in Proposition A.1 in Genest et al.(1995) with respect to all the m t − copulas included in the grouped-t copula defined in (1). p Then νˆk+1 → νk+1 as n → ∞.

Theorem 1.1.3 (Consistency of the DCC(1,1) parameters α ˆ and ˆ Let the assumptions of the previous theorem hold, as well as the β). assumptions A1 - A5 in Engle and Sheppard (2001) with respect to the p p normal copula density (6). Then α ˆ → 0 and βˆ → 0, as n → ∞.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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Asymptotic Properties

The asymptotic normality is not straightforward, since we use a multi-stage procedure where we perform a different kind of estimation at every stage. A possible solution is to consider the ML used in the 3rd and 4th stages in Definition 1 as special method-of-moment estimators. Let define the sample moments Ψ for the parameter vector ˆ ′ as reported below: ˆ = [ρˆ Ξ ¯1 , . . . ρˆ ¯q , νˆ1 , . . . , νˆm , α, ˆ β]

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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Asymptotic Properties



 ˆ = Ψ F1 (η1,t ), . . . , Fn (ηn,t ); Ξ 

              =             

1 T

1 T 1 T

1 T 1 T 1 T

T P

t=1

ψ1 F1 (η1,t ), F2 (η2,t ); ρˆ ¯1






   ..  .   T 
P  ˆ ψq Fn−1 (ηn−1,t ), Fn (ηn,t ); ρ¯q  t=1     T P  ˆ ¯ νˆ1 ψν1 F1 (η1,t ), . . . , Fs1 (ηs1,t ); R,   t=1 =0 ..   .     T P ˆ ¯ νˆm   ψνm Fs1 +...+sm−1 +1 (ηs1 +...+sm−1 +1,t ), . . . , Fn (ηn,t ); R,  t=1  T  P ˆ  ˆ ¯ ˆ β) ψα (F1 (η1,t ), . . . , Fn (ηn,t ); R, α,  t=1   T P ˆ ˆ ¯ α, ψβ (F1 (η1,t ), . . . , Fn (ηn,t ); R, ˆ β) t=1

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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Asymptotic Properties Let also define the population moments vector with a correction to take the non-parametric estimation of the marginals into account, together with its variance (see Genest et al. (1995), § 4): 

            ∆0 =             



ψ1 (F1 (η1 ), F2 (η2 ); ρ¯1 ) . ..

      ψq (Fn−1 (ηn−1 ), Fn (ηn ); ρ¯q )   s 1
P  ¯ ν1 + Wi,ν1 (ηi ) ψν1 F1 (η1 ), . . . , Fs1 (ηs1 ); R,   i=1   ..  .   n
P  ¯ νm + ψνm Fs1 +...+sm−1 +1 (η1 ), . . . , Fn (ηn ); R, Wi,νm (ηi )   i=s1 +...+sm−1 +1  n  P ¯  ψα (F1 (η1 ), . . . , Fn (ηn ); R, α, β) + Wi,α (ηi )  i=1  n  P ¯ α, β) + ψβ (F1 (η1 ), . . . , Fn (ηn ); R, Wi,β (ηi ) i=1

(11)



Υ0 ≡ var [∆0 ] = E ∆0 ∆0

′



A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

(12)

15

Asymptotic Properties

where Wi,ν1 (ηi )

Wi,νm (ηi )

=

=

Z

Z

1l Fi (ηi )≤ui

∂2 ¯ ν1 )dC(u1 , . . . , us ) log c(u1 , . . . us1 ; R, 1 ∂ν1 ∂ui .. .

1l Fi (ηi )≤ui

(13)

∂2 ¯ νm ) log c(ui=s1 +...+sm−1 +1 , . . . un ; R, ∂νm ∂ui dC(ui=s1 +...+sm−1 +1 , . . . , un ) (14)

Wi,α (ηi ) Wi,β (ηi )

= =

Z

Z

1l Fi (ηi )≤ui 1l Fi (ηi )≤ui

∂2 ¯ α, β)dC(u1 , . . . , un ) log c(u1 , . . . un ; R, ∂α∂ui ∂2 ¯ α, β)dC(u1 , . . . , un ) log c(u1 , . . . un ; R, ∂β∂ui (15)

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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Asymptotic Properties

Theorem 1.1.4 (Asymptotic Distribution). Consider the general case where the marginals are estimated non-parametrically by using the empirical distributions functions. Let the assumptions of the previous theorems hold. Assume further that ∂Ψ(·;Ξ) is O(1) and uniformly negative ∂Ξ′ definite, while Υ0 is O(1) and uniformly positive definite. Then, the multi-stages estimator of the dynamic grouped-t copula verifies the properties of asymptotic normality:  √  ∂Ψ −1′   ∂Ψ −1 d ˆ − Ξ0 ) −→ N 0, E (16) Υ0 E ∂Ξ′ T (Ξ ∂Ξ′

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

17

Simulation Study

The previous asymptotic properties hold only under the very special case when zi , zj are uncorrelated and R is the identity matrix. When this restriction does not hold, the estimation procedure previously described may not deliver consistent estimates. Daul et al. (2003) performed a Monte-Carlo study with a grouped-t copula with constant R, employing an estimation procedure equal to the first three steps of definition 1. They showed that the correlations parameters present a bias that increases nonlinearly in Rj,k , but the magnitude of the error is rather low. Instead, no evidence is reported for the degrees of freedom.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

18

Simulation Study

We consider the following possible DGPs: 1. We examine the case that four variables have a Grouped-T copula ¯ of the with m = 2 groups, with unconditional correlation matrix R underlying multivariate normal random vector Z equal to 1

0.30

-0.20

0.50

0.30

1

-0.25

0.40

-0.20

-0.25

1

0.10

0.50

0.40

0.10

1

2. We examine different values for the DCC(1,1) model parameters, equal to [α = 0.10, β = 0.60] and [α = 0.01, β = 0.95]. The former corresponds to a case of low persistence in the correlations, while the latter implies strong persistence in the correlation structure, instead.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

19

Simulation Study

3. We examine two cases for the degrees of freedom νk for the m = 2 groups: • ν1 = 3 and ν2 = 4; • ν1 = 6 and ν2 = 15; The first case corresponds to a situation of strong tail dependence, that is there is a high probability to observe an extremely large observation on one variable, given that the other variable has yielded an extremely large observation. The last exhibit low tail dependence, instead. 4. We consider three possible data situations: n = 500, n = 1000 and n = 10000.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

20

Simulation Study ¯ j,k : there is a general negative • Unconditional correlation parameters R bias that stabilize after n = 1000. However, this bias is quite high when there is strong tail dependence among variables (νk are low), while it is much lower when the tail dependence is rather weak (νk are high). Besides, it almost disappears when correlations are lower than 0.10, thus confirming the previous asymptotics results. The effects of different dynamic structure in the correlations are negligible, instead. • DCC(1,1) parameters (α, β): the higher is the persistence in the correlations structure (high β), the quicker βˆ converges to the true values. In general, the effects of different DGPs on β are almost negligible. The parameter α describing the effect of past shocks shows positive biases, instead, that are higher in magnitude when high tail dependence and high persistence in the correlations are considered.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

21

Simulation Study • Degrees of freedom νk : the speed of convergence towards the true values is, in general, very low and changes substantially according to the magnitude of νk and the dynamic structure in the correlations. Particularly, when there is high tail dependence (νk are low) the convergence is much quicker than when there is low tail dependence (νk are high). Furthermore, the convergence is quicker when there is strong persistence in the correlations structure (β is high), rather than the persistence is weak (β is low). This is good news since financial assets usually show high tail dependence and high persistence in the correlations (see Mcneil et al. (2005) and references therein). Besides, it is interesting to note that the biases are negative for all the considered DGPs, i.e. the estimated νˆk are lower than the true values νk .

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

22

Simulation Study

We explore the consequences of our multi-step estimation procedure of the dynamic grouped-t copula on Value at Risk (VaR) estimation, by using the same DGPs previously discussed. As we want to study only the effects of the estimated dependence structure, we consider the same marginals for all DGPs, as well as the same past shocks u ˆ t−1 . For sake of simplicity, we suppose to invest an amount Mi = 1$, i = 1, . . . , n = 4 in every asset. We consider eight different quantiles to better highlight the overall effects of the estimated copula parameters on the joint distribution of the losses: 0.25%, 0.50%, 1.00%, 5.00%, 95.00%, 99.00%, 99.50%, 99.75%.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

23

Simulation Study

In general, the estimated quantiles show a very small underestimation, which can range between 0 and 3%. Particularly, we can observe that • the error in the approximation of the quantiles is lower the lower the tail dependence between assets is, i.e. when νk are high, ceteris paribus. As a consequence, when estimating the quantile they tend to offset the effect of lower correlations, which would decrease the computed quantile, instead. • the error in the approximation of the quantiles is lower the higher is the persistence in the correlations, ceteris paribus. This result is due to the much smaller biases of the parameters α, β when the true DGPs are characterized by high persistence in the correlations.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

24

Simulation Study • the error in the approximation of the quantiles tend to slightly increase as long as the sample dimension increases. Such a result can be explained considering that the computed degrees of freedom νˆk slowly converge to the true values when the dimension of the dataset increases, while the negative biases in the correlations tend to stabilize. As a consequence, the computed νˆk do not offset any more the effect of lower correlations, and therefore the underestimation in the VaR increases. • the approximations of the extreme quantiles are much better than those of the central quantiles, while the analysis reveals no major difference between left tail and right tail. • It is interesting to note that up to medium-sized datasets consisting of n = 1000 observations, the effects of the biases in the degrees of freedom and the biases in the correlations tend to offset each other and the error in approximating the quantile is close to zero.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

25

Empirical Analysis In order to compare our approach with previous multivariate models proposed in the literature, we measured the Value at Risk of a high-dimensional portfolio composed of 30 assets . Marginal Distribution

Moment specification

Copula

Copula Parameters Specification

Model 1)

NORMAL

AR(1) T-GARCH(1,1)

NORMAL

Constant Correlation

Model 2)

NORMAL

AR(1) T-GARCH(1,1)

NORMAL

DCC(1,1)

Model 3)

SKEW-T

AR(1) T-GARCH(1,1)

T-COPULA

Constant Correlation Const. D.o.F.

Model 4)

SKEW-T

AR(1) T-GARCH(1,1)

T-COPULA

DCC(1,1) Constant D.o.F.

Model 5)

SKEW-T

AR(1) T-GARCH(1,1)

GROUPED T

Constant Correlation Constant D.o.F.s

Model 6)

SKEW-T

AR(1) T-GARCH(1,1)

GROUPED T

DCC(1,1) Constant D.o.F.s

As for the grouped-t copula, we classify the assets in 5 groups according to their credit rating: 1) AAA; 2) AA (AA+,AA,AA−); 3) A (A+,A,A−); 4) BBB (BBB+,BBB,BBB−); 5) BB (BB+,BB,BB−).

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

26

Empirical Analysis

We will assess the performance of the competing multivariate models using the following back-testing techniques • Kupiec (1995) unconditional coverage test; • Christoffersen (1998) conditional coverage test; • Loss functions to evaluate VaR forecast accuracy; • Hansen and Lunde (2005) and Hansen’s (2005) Superior Predictive Ability (SPA) test.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

27

Empirical Analysis 1. Kupiec’s test: Following binomial theory, the probability of observing N failures out of T observations is (1-p)T −N pN , so that the test of the null hypothesis H0 : p = p∗ is given by a LR test statistic: N

LR = 2 · ln[(1 − p∗ )T −N p∗ ] + 2 · ln[(1 − N/T )T −N (N/T )N ] 2. Christoffersen’s test: . Its main advantage over the previous statistic is that it takes account of any conditionality in our forecast: for example, if volatilities are low in some period and high in others, the VaR forecast should respond to this clustering event. n11 n01 ] (1−π11 )n10 π11 LRCC = −2 ln[(1−p)T −N pN ]+2 ln[(1 −π01 )n00 π01

where nij is the number of observations with value i followed by j for i, j = 0, 1 and nij πij = P j nij A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

28

Empirical Analysis 3. Loss functions: As noted by the Basle Committee on Banking Supervision (1996), the magnitude as well as the number of exceptions are a matter of regulatory concern. Since the object of interest is the conditional α-quantile of the portfolio loss distribution, we use the asymmetric linear loss function proposed in Gonzalez and Rivera (2006) and Giacomini and and Komunjer (2005), and defined as Tα (et+1 ) ≡ (α − 1l (et+1 < 0))et+1

(17)

\ where et+1 = Lt+1 − V aR t+1|t , Lt+1 is the realized loss, while \ V aR t+1|t is the VaR forecast at time t + 1 on information available at time t. 4. Hansen’s (2005) Superior Predictive Ability (SPA) test: The SPA test is a test that can be used for comparing the performances of two or more forecasting models. The forecasts are evaluated using a prespecified loss function and the “best” forecast model is the model that produces the smallest loss. A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

29

Empirical Analysis

Long position

Short position

0.25%

0.50%

1%

5%

0.25%

0.50%

1%

5%

Model 1)

5.275

8.963

14.512

47.131

5.722

8.855

13.987

44.517

Model 2)

4.843

8.281

13.967

45.584

7.598

11.006

16.521

47.243

Model 3)

3.610

7.603

13.929

46.574

4.777

8.118

Model 4)

4.462

8.354

14.381

46.386

4.974

8.265

13.644

44.138

Model 5)

3.880

7.942

14.304

47.101

4.870

8.082

13.797

44.432

Model 6)

3.374

7.143

13.448

45.553

4.901

8.447

13.661

44.329

13.508

44.314

Table 1: Asymmetric loss functions (17). The smallest value is reported in bold font.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

30

Empirical Analysis

Long position Benchmark

Short Position

0.25%

0.50%

1%

5%

0.25%

0.50%

1%

5%

Model 1)

0.012

0.003

0.013

0.115

0.113

0.133

0.113

0.113

Model 2)

0.009

0.015

0.132

0.780

0.299

0.300

0.279

0.248

Model 3)

0.380

0.165

0.093

0.005

0.999

0.951

0.999

0.994

Model 4)

0.239

0.221

0.239

0.171

0.276

0.300

0.297

0.591

Model 5)

0.096

0.091

0.093

0.016

0.875

0.990

0.735

0.866

Model 6)

0.979

0.970

0.967

0.917

0.832

0.155

0.800

0.959

Table 2: Hansen’s SPA test for the portfolio consisting of thirty stocks. P-values smaller than 0.10 are reported in bold font.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

31

Empirical Analysis

Long positions 0.25% M.

N/T

1)

0.50%

1.40%

pU C 0.00

pCC 0.00

2)

1.30%

0.00

3)

0.90%

4)

N/T

1%

1.90%

pU C 0.00

pCC 0.00

0.00

1.60%

0.00

0.00

0.01

1.40%

0.60%

0.06

0.17

5)

0.80%

0.01

6)

0.50%

0.16

N/T

5%

2.30%

pU C 0.00

pCC 0.00

0.00

1.90%

0.01

0.00

0.00

2.00%

1.40%

0.00

0.00

0.02

1.30%

0.00

0.37

1.10%

0.02

N/T 6.30%

pU C 0.07

pCC 0.19

0.03

5.80%

0.26

0.49

0.01

0.01

6.60%

0.03

0.08

1.90%

0.01

0.03

6.20%

0.09

0.24

0.00

1.90%

0.01

0.03

6.10%

0.12

0.18

0.02

1.80%

0.02

0.05

6.00%

0.16

0.35

Table 3: Actual VaR exceedances N/T , Kupiec’s and Christoffersen’s tests for the portfolio consisting of thirty stocks (Long positions).

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

32

Empirical Analysis

Short positions 0.25% M.

N/T

1)

0.50%

0.80%

pU C 0.01

pCC 0.02

2)

0.70%

0.02

3)

0.20%

4)

N/T

1%

1.00%

pU C 0.05

pCC 0.06

0.06

0.90%

0.11

0.74

0.94

0.70%

0.30%

0.76

0.95

5)

0.30%

0.76

6)

0.30%

0.76

N/T

5%

1.50%

pU C 0.14

pCC 0.16

0.06

1.30%

0.36

0.40

0.07

0.90%

0.70%

0.40

0.06

0.95

0.80%

0.22

0.95

0.70%

0.40

N/T 5.30%

pU C 0.67

pCC 0.71

0.56

5.00%

1.00

0.95

0.75

0.87

5.90%

0.20

0.43

0.90%

0.75

0.87

5.50%

0.47

0.77

0.06

0.90%

0.75

0.87

4.80%

0.77

0.54

0.35

0.90%

0.75

0.87

5.20%

0.77

0.94

Table 4: Actual VaR exceedances N/T , Kupiec’s and Christoffersen’s tests for the portfolio consisting of thirty stocks (Short positions).

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

33

Conclusions • Introduction of the dynamic grouped-t copula for the joint modelling of high-dimensional portfolios, where we use the DCC model to specify the time evolution of the correlation matrix of the grouped-t copula. • Consistency and asymptotic normality of the estimator under the special case of a correlation matrix equal to the identity matrix. • Monte Carlo simulations to study the properties of this estimator under different data generating processes where such a strong restriction on the correlation matrix does not hold. • We investigated the effects of such biases and finite sample properties on conditional quantile estimation, given the increasing importance of the Value-at-Risk as risk measure. We found that the error in the approximation of the quantile can range between 0 and 3%.

A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

34

Conclusions • Empirical analysis 1: When long positions were of concern, we found that the dynamic grouped-T copula (together with skewed-t marginals) outperformed both the constant grouped-t copula and the dynamic student’s T copula as well as the dynamic multivariate normal model proposed in Engle (2002). • Empirical analysis 2: As for short positions, we found out that a multivariate normal model with dynamic normal marginals and constant normal copula was already a proper choice. This last result confirms previous evidence in Junker and May (2005) and Fantazzini (2008) for bivariate portfolios. • Avenue for future research 1: more sophisticated methods to separate the assets into homogenous groups when using the grouped-t copula. • Avenue for future research 2: look for alternatives to DCC modelling A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

35

References

References [1] Basle Committee on Banking Supervision (1996). Supervisory Framework for the Use of Backtesting in Conjunction with the Internal Models Approach to Market Risk Capital Requirements, Basel, January. [2] Basle Committee on Banking Supervision (2005). Amendment to the Capital Accord to Incorporate Market Risks, BIS. [3] Cherubini, U. and Luciano, E. and W. Vecchiato W. (2004). Copula Methods in Finance, Wiley. [4] Christoffersen, P. (1998). Evaluating Interval Forecats, International Economic Review, 39, 841-862. [5] Daul, S. and De Giorgi, E. and Lindskog, F. and McNeil, A. (2003). The grouped t-copula with an application to credit risk, Risk,1, 73–76. [6] Demarta, S. and McNeil, A. J. (2005). The t Copula and Related Copulas, International Statistical Review, 73, 111 –129. [7] Fantazzini, D. (2008). Dynamic Copula Modelling for Value at Risk, Frontiers in Finance and Economics, 5(2), 72-108. [8] Fantazzini, D. (2009a). A Dynamic Grouped-T Copula Approach for A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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References Market Risk Management, (in) The Var Implementation Handbook, p. 253-282, McGraw-Hill, New York [9] Genest, C. and Ghoudi, K. and Rivest, L. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions, Biometrika, 82, 543 - 552. [10] Giacomini, R. and Komunjer, I. (2005). Evaluation and Combination of Conditional Quantile Forecasts, Journal of Business and Economic Statistics, 23, 416-431. ´lez-Rivera, G. and Lee, T. and Santosh, M. (2006). Forecasting [11] Gonza volatility: A reality check based on option pricing, utility function, value-at-risk, and predictive likelihood, I.J.F., 20, 629-645. [12] Hansen, P. (2005). A Test for Superior Predictive Ability, Journal of Business and Economic Statistics, 23(4), 365-380. [13] Kupiec, P. (1995). Techniques for verifying the accuracy of risk measurement models, Journal of Derivatives, 2, 173–184. [14] McNeil, A. and Frey, R. and Embrechts P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press. A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios

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