Market Risk Management For High-dimensional Portfolios: Evidence From Russian Stocks

Market Risk Management for High-Dimensional Portfolios: Evidence from Russian Stocks Dean Fantazzini

September 24th, 2008, Tsahkadzor

Overview of the Presentation

1st Introduction

2

Overview of the Presentation

1st Introduction 2nd The Benchmark Models so far: CCC and DCC models

2-a

Overview of the Presentation

1st Introduction 2nd The Benchmark Models so far: CCC and DCC models 3rd Advanced Multivariate Modelling: The Theory of Copulas

2-b

Overview of the Presentation

1st Introduction 2nd The Benchmark Models so far: CCC and DCC models 3rd Advanced Multivariate Modelling: The Theory of Copulas 4th Multivariate Modelling for High-Dimensional Portfolios: A Unified Approach with Copulas

2-c

Overview of the Presentation

1st Introduction 2nd The Benchmark Models so far: CCC and DCC models 3rd Advanced Multivariate Modelling: The Theory of Copulas 4th Multivariate Modelling for High-Dimensional Portfolios: A Unified Approach with Copulas 5th Empirical Application: Russian Stock Market

2-d

Overview of the Presentation

1st Introduction 2nd The Benchmark Models so far: CCC and DCC models 3rd Advanced Multivariate Modelling: The Theory of Copulas 4th Multivariate Modelling for High-Dimensional Portfolios: A Unified Approach with Copulas 5th Empirical Application: Russian Stock Market 6th Conclusions

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. ⇒ However, only low-dimensional applications have been considered so far, while high-dimensional studies have been quite rare in general, and there is no one dealing with Russian stocks.

3

Introduction

The most well known risk measure is the Value-at-Risk (VaR), which is defined as the maximum loss which can be incurred by a portfolio, at a given time horizon and at a given confidence level. ⇒ Our main purpose is to examine and compare different multivariate parametric models with the purpose of estimating the VaR for a high-dimensional portfolio composed of Russian financial assets. To achieve this aim, we unify past multivariate models by using a general copula framework and we propose many new extensions.

4

The Benchmark Models so far If we want to model portfolios with more than 2 assets, what can we do? • VaR estimation for a portfolio of assets can become very difficult due to the complexity of joint multivariate modelling. • Standard models for low-dimensional portfolios (2-5 assets) deal with the conditional variance-covariance matrix (BEKK, VEC models)... • ...unfortunately, positivity and stationarity constraints are difficult to impose. • Besides, they cannot be computed with high-dimensional portfolios ⇒ Recent proposal: models for the Conditional Correlation matrix...

5

The Benchmark Models so far • These models allow for some flexibility in the specifications of the variances: they need not be the same for each component. For example a GARCH(1,1) for one component, an EGARCH for another, ... • However, the specification of the correlations is less flexible... • But positivity conditions for Ht are easily imposed and estimation is facilitated (2 steps). As a consequence of this complexity, two models seem to have gained the greatest attention by practitioners and researchers so far: • The Constant Conditional Correlation (CCC) model by Bollerslev (1990); • The Dynamic Conditional Correlation (DCC) model by Engle (2002). 6

Models for the Conditional Correlation Matrix Let Yt be a vector stochastic process of dimension N × 1 and θ a finite vector of parameters. Yt = E [Yt |Ft−1 ] + εt

(1)

with 1/2

εt = Σt (θ)ηt

(2)

where •

Σt (θ) is a N × N positive definite matrix

•

Σt (θ) is the Cholesky Decomposition of Σt (θ)

•

ηt is a N × 1 random vector assumed to be i.i.d., with:

1/2

• E [ηt ] = 0 • V [ηt ] = In •

Ft is the information set available at time t. 7

Models for the Conditional Correlation Matrix The models for the conditional correlation matrix rely on the decomposition of the covariance matrix Σt as:

Σt = Dt Rt Dt

(3)

1/2

1/2

Dt = diag(σ11,t . . . σnn,t )

(4)

Rt = (ρij,t ),

(5)

with

ρii,t = 1

where Rt is the n × n matrix of conditional correlations, and σii,t is defined as a univariate GARCH model. Hence √ σij,t = ρij,t σii,t σjj,t

i 6= j

(6)

⇛ Positivity of Σt follows from positivity of Rt and of each σii,t for i = 1, . . . , n.

8

The Constant Conditional Correlation (CCC) Model of Bollerslev (1990) The CCC model is defined as: √ Σt = Dt RDt = (ρij σiit σjjt )

(7)

where 1/2

1/2

Dt = diag (σ11t . . . σnnt )

(8)

σiit can be defined as any univariate GARCH model and Rt = R = (ρij )

(9)

is a symmetric positive definite matrix with ρii = 1, ∀i.. Therefore, the conditional correlations are constant (CCC). Hence, √ σij,t = ρij σii,t σjj,t

i 6= j

(10)

and thus the dynamics of the covariance is determined only by the dynamics of the two conditional variances. 9

Dynamic Conditional Correlation (DCC) of Engle (2002) Engle (2002) proposed a Dynamic Conditional Correlation (DCC) model defined as: Σt = Dt Rt Dt

(11)

where Dt is defined in (8), and Rt = (diagQt )−1/2 Qt (diagQt )−1/2

(12)

where the N × N symmetric positive definite matrix Qt is given by: ! L S L S X X X X ′ ¯+ Qt = 1 − αl − βs Q αl ηt−l ηt−l + βs Qt−s l=1

s=1

l=1

(13)

s=1

√ ¯ is the n × n unconditional variance matrix of ut , where ηit = εit / σii,t , Q αl (≥ 0) and βs (≥ 0) are scalar parameters satisfying PL PS α + l=1 l s=1 βs < 1, to have Qt > 0 and Rt > 0. Qt is the covariance matrix of ut , since qii,t is not equal to 1 by construction. Then, it is transformed into a correlation matrix by (12). If θ1 = θ2 = 0 and q¯ii = 1 the CCC model is obtained. 10

Dynamic Conditional Correlation (DCC) of Engle (2002) To show more how the DCC model worls, let us write the equation of the correlation coefficient in the bivariate case: ρ12,t = q

(1 − α − β)¯ q12 + αu1,t−1 u2,t−1 + βq12,t−1

2 2 q22 + αu2,t−1 + βq22,t−1 (1 − α − β)¯ q11 + αu1,t−1 + βq11,t−1 1 − α − β)¯

→ the conditional variance-covariance matrix Qt of the error terms is written like a GARCH equation, and then transformed to a correlation matrix.

⇒ So far, the CCC and DCC models seem to have gained the greatest attention by practitioners and researchers, given that they are the only models which can be estimated with high-dimensional portfolios. ⇒ However, they still assume that the error terms ηt follow a multivariate normal distribution. Can we do anything better? ... Well, maybe copulas can help! 11

Advanced Multivariate Modelling: The Theory of Copulas

→ A copula is a multivariate distribution function H of random variables X1 . . . Xn with standard uniform marginal distributions F1 , . . . , F n, defined on the unit n-cube [0,1]n with the following properties: 1. The range of C (u1 , u2 , ..., un ) is the unit interval [0,1]; 2. C (u1 , u2 , ..., un ) = 0 if any ui = 0, for i = 1, 2, ..., n. 3. C (1, ..., 1, ui , 1, ..., 1) = ui , for all ui ∈ [0, 1] The previous three conditions provides the lower bound on the distribution function and ensures that the marginal distributions are uniform. The Sklar’s theorem justifies the role of copulas as dependence functions...

12

Advanced Multivariate Modelling: The Theory of Copulas

(Sklar’s theorem): Let H denote a n-dimensional distribution function with margins F1 . . . Fn . Then there exists a n-copula C such that for all real (x1 ,. . . , xn ) H(x1 , . . . , xn ) = C(F1 (x1 ), . . . , Fn (xn ))

(14)

If all the margins are continuous, then the copula is unique. Conversely, if C is a copula and F1 , . . . Fn are distribution functions, then the function H defined in (14) is a joint distribution function with margins F1 , . . . Fn . → A copula is a function that links univariate marginal distributions of two or more variables to their multivariate distribution. → F1 and Fn need not to be identical or even to belong to the same distribution family.

13

Advanced Multivariate Modelling: The Theory of Copulas

Main consequences: • For continuous multivariate distributions, the univariate margins and the multivariate dependence can be separated; • Copula is invariant under strictly increasing and continuous transformations: no matter whether we work with price series or with log-prices. Example. Independent copula: C(u, v) = u · v What is the probability that both returns in market A and B are in their lowest 10th percentiles? C(0.1; 0.1) = 0.1 · 0.1 = 0.01

14

Advanced Multivariate Modelling: The Theory of Copulas

By applying Sklar’s theorem and using the relation between the distribution and the density function, we can derive the multivariate copula density c(F1 (x1 ),, . . . , F n (xn )), associated to a copula function C(F1 (x1 ),, . . . , F n (xn )): n n Y ∂ n [C(F1 (x1 ), . . . , Fn (xn ))] Y · fi (xi ) = c(F1 (x1 ), . . . , Fn (xn ))· fi (xi ) f (x1 , ..., xn ) = ∂F1 (x1 ), . . . , ∂Fn (xn ) i=1 i=1

Therefore, we get c(F1 (x1 ), ..., Fn (xn )) =

f (x1 , ..., xn ) · , n Q fi (xi )

(15)

i=1

15

Advanced Multivariate Modelling: The Theory of Copulas By using this procedure, we can derive the Normal copula density:

c(u1 , . . . , un )

=

=

f

N ormal

(x1 , ..., xn )

n Q

1

=

(2π)n/2 |Σ|1/2 n Q √1 2π i=1

fiN ormal (xi ) i=1   1 1 ′ −1 ζ (Σ − I)ζ exp − 1/2 2 |Σ|

exp




− 12 x′ Σ−1 x


1 2 exp − 2 xi

=

(16)

where ζ = (Φ−1 (u1 ), ..., Φ−1 (un ))′ is the vector of univariate Gaussian inverse distribution functions, ui = Φ (xi ), while Σ is the correlation matrix. The log-likelihood is then given by l

gaussian

(θ) =

− T2

ln |Σ| −

1 2

T P

t=1

′

ςt (Σ−1 − I)ςt

16

Advanced Multivariate Modelling: The Theory of Copulas

If the log-likelihood function is differentiable in θ and the solution of the equation ∂θ l(θ) = 0 defines a global maximum, we can recover the ˆ for the Gaussian copula: θˆM L = Σ ∂ ∂Σ−1

l

gaussian

(θ )

= T2

Σ−

1 2

T P

′

ςt ςt = 0

t=1

and therefore T X ′ 1 ˆ Σ= ςt ςt T t=1

(17)

17

Advanced Multivariate Modelling: The Theory of Copulas We can derive the Student’s T-copula in a similar way: c(u1 , u2 , . . . , un ; Σ) = f

student N Q

i=1

(x1 ,...,xN )

=

fistudent (xi )

ν+N 1 Γ 2 1 ν Γ 2 |Σ| 2

(

)

( )



Γ ν 2 ν+1 Γ 2

( )

(

)

lStudent (θ ) = Γ −T ln



ν+N 2   ν Γ 2



Γ −N T ln



ν+1 2   ν Γ 2



−

T 2

ln |Σ|−

N Q

i=1



ς2 1+ νt

 − ν+1 2

   ′ −1 2 T X N X Σ ς ς ν + 1 ς t  t ln  1 + ln  1 + it  + ν 2 ν t=1 t=1 i=1

T ν+N X 2

N

! − ν+N ′ 2 ςt Σ−1 ςt 1+ ν



In this case, we don’t have an analytical formula for the ML estimator and a numerical maximization of the likelihood is required. However, this can become computationally cumbersome, if not impossible, when the number of assets is very large. This is why multi-step parametric or semi-parametric approaches have been proposed. 18

Multivariate Modelling for High-Dimensional Portfolios: A Unified Approach Given the previous background, the CCC and DCC models can be easily represented as special cases within a more general copula framework! Particularly, the joint normal density function is simply the by-product of a normal copula with correlation matrix Σ = Rt together with normal marginals: Yt

=

E [Yt |Ft−1 ] + Dt ηt

ηt

∼

H(η1 , . . . , ηn ) ≡ C N ormal (F1N ormal (η1 ), . . . , FnN ormal (ηn ); Rt ) 1/2

(18)

1/2

where Dt = diag(σ11,t . . . σnn,t ) and the Sklar’s Theorem was used. → Rt = R for the CCC model. → As for the DCC model, Rt has a dynamic structure of this type: Rt Qt

= =

(diagQt )−1/2 Qt (diagQt )−1/2 ! L S L X X X ¯+ 1− αl − βs Q αl ηt−l η ′

t−l

l=1

s=1

l=1

+

S X

βs Qt−s

s=1

19

Multivariate Modelling for High-Dimensional Portfolios: A Unified Approach • A multivariate model that allows for marginal kurtosis and normal dependence can be expressed as follows: Yt

=

E [Yt |Ft−1 ] + Dt ηt

(19)

ηt

∼

H(η1 , . . . , ηn ) ≡ C N ormal (F1Student

′

s−t

′

(η1 ), . . . , FnStudent

s−t

(ηn ); Rt ) (20)

Student′ s−t Fi

where is the cumulative distribution function of the marginal Student’s-t, and Rt can be made constant or time-varying, as in the standard CCC and DCC models, respectively. • If the financial assets present tail dependence, we can use a Student’s T copula, instead, Yt

=

E [Yt |Ft−1 ] + Dt ηt

(21)

ηt

∼

H(η1 , . . . , ηn ) ≡ C Student

′

st

′

(F1Student

s−t

′

(η1 ), . . . , FnStudent

s−t

(ηn ); Rt , ν) (22)

where ν are the Student’s t copula degrees of freedom. 20

Multivariate Modelling for High-Dimensional Portfolios: A Unified Approach Daul, Giorgi, Lindskog, and McNeil (2003), Demarta and McNeil (2005) and Mc-Neil, Frey, and Embrechts (2005), Fantazzini (2009a) underlined the ability of the grouped t-copula to model the dependence present in a large set of financial assets into account. ⇒ The grouped-t copula can be considered as a copula imposed by a kind of multivariate-t distribution where m distinct groups of assets have m different degrees of freedom. ⇒ Therefore, we can use a Grouped t copula if the financial assets may be separated in m distinct groups: Xt

=

E [Xt |Ft−1 ] + Dt ηt

ηt

∼

H(η1 , . . . , ηn ) ≡ C Grouped t (F1Student

′

s−t

′

(η1 ), . . . , FnStudent

s−t

(ηn ); Rt , ν1 , . . . , νm )

where Rt can be constant or time-varying (see Fantazzini (2009a) for the latter case). 21

Multivariate Modelling for High-Dimensional Portfolios: A Unified Approach Particularly, we considered different parameterizations by changing the following four elements: 1. Marginals distribution: Normal, Student’s T; 2. Conditional Moments of the Marginals: • AR(1)-GARCH(1,1) model for the continuously compounded returns yt = 100 × [log(Pt ) − log(Pt−1 )]: yt

=

µ + φ1 yt−1 + εt

εt

=

σt2

ηt σt , ηt ∼ f (0, 1)

=

i.i.d.

2 ω + αε2t−1 + βσt−1

→ Other GARCH models (like FIGARCH, FIEGARCH, APARCH, etc.) as well as other marginal distributions (Skewed t, Laplace, etc.) were not considered due to poor numerical convergence properties (Russian stocks are more noisy and less liquid than European or American stocks). 22

Multivariate Modelling for High-Dimensional Portfolios: A Unified Approach

3. Type of Copulas: • Normal copula • T - copula • Grouped - T ; 4. Constant / Dynamic copula parameters: • Constant Correlation Matrix R (for Normal, T-copulas, or Grouped-T copulas) • Dynamic Correlation Matrix Rt : DCC(1,1) model (for Normal, T-copulas, or Grouped-T copulas).

23

Multivariate Modelling for High-Dimensional Portfolios: A Unified Approach REMARK: Grouped t Copula The variables at hand can be classified in different groups , according to different criteria: • Geographical location, like in Daul et al. (2003); • Credit Rating, like in Fantazzini (2009a); • If none of the previous criteria is available (or there is only partial information), one may resort to hierarchical cluster analysis based on L2 dissimilarity measure and “dendrograms”: → Dendrograms graphically present the information concerning which observations are grouped together at various levels of (dis)similarity. → The height of the vertical lines and the range of the (dis)similarity axis give visual clues about the strength of the clustering. 24

0

200

L2 dissimilarity measure 400 600 800

1000

Multivariate Modelling for High-Dimensional Portfolios: A Unified Approach

1 9 4 261718 3 131425 2 29302411 5 6 12151610212219 7 8 20232827

Figure 5: Dendogram for the 30-asset portfolio

25

Multivariate Modelling for High-Dimensional Portfolios: A Unified Approach Marginal Distribution Model 1) Model 2) Model 3) Model 4) Model 5) Model 6)

Model 7) Model 8) Model 9) Model 10) Model 11) Model 12)

NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL

Moment specification AR(1) GARCH(1,1) AR(1) GARCH(1,1) AR(1) GARCH(1,1) AR(1) GARCH(1,1) AR(1) GARCH(1,1) AR(1) GARCH(1,1)

Copula

NORMAL NORMAL

Copula Parameters Specification Constant Correlation DCC(1,1)

T-COPULA

Constant Correlation Const. D.o.F.s

T-COPULA

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

GROUPED T

Constant Correlation Const. D.o.F.s

GROUPED T

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

Student’s t

AR(1) GARCH(1,1) Constant D.o.F.

NORMAL

Constant Correlation

Student’s t

AR(1) GARCH(1,1) Constant D.o.F.

NORMAL

Student’s t

AR(1) GARCH(1,1) Constant D.o.F.

T-COPULA

Constant Correlation Const. D.o.F.s

Student’s t

AR(1) GARCH(1,1) Constant D.o.F.

T-COPULA

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

Student’s t

AR(1) GARCH(1,1) Constant D.o.F.

GROUPED T

Constant Correlation Const. D.o.F.s

Student’s t

AR(1) GARCH(1,1) Constant D.o.F.

GROUPED T

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

DCC(1,1)

26

Empirical Application: Russian Stock Market In order to compare the different multivariate models, we measured the Value at Risk of a high-dimensional portfolio composed of 30 Russian assets. → We chose the 30 most liquid assets with at least 2000 historical daily data quoted at the RTS and MICEX Russian markets. → Time period: 5/01/2000 - 23/05/2008 → We use a rolling forecasting scheme of 1000 observations, because it may be more robust to a possible parameter variation. → In our case we have 2000 observations, so we split the sample in this way: 1000 observations for the estimation window and 1000 for the out-of-sample evaluation. 27

Empirical Application: Russian Stock Market

We assessed 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.

28

Empirical Application: Russian Stock Market 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 29

Empirical Application: Russian Stock Market 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

(23)

\ 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 pre-specified loss function and the “best” forecast model is the model that produces the smallest loss... 30

Empirical Application: Russian Stock Market → Let L(Yt ; Yˆt ) denote the loss if one had made the prediction, Yˆt , when the realized value turned out to be Yt . → The performance of model k relative to the benchmark model (at time t), can be defined as: Xk (t) = L(Yt , Yˆ0t ) − L(Yt , Yˆkt ),

k = 1, . . . , l;

t = 1, . . . , n.

→ The question of interest is whether any of the models k = 1, . . . , l is better than the benchmark model: µk = E [Xk (t)] ≤ 0,    µ1     ..   µ= .  =E    µl H0 : µ ≤ 0

k = 1, . . . , l.  X1 (t)  ..   .  Xl (t)

or in compact notation:

31

Empirical Application: Russian Stock Market One way to test this hypothesis is to consider the test statistic Tnsm

¯k n1/2 X = max k σ ˆk

where n X 1 ¯k = Xk (t), X n t=1

¯ k ). σ ˆk2 = var(n c 1/2 X

The superscript “sm“ refers to standardized maximum. Under some regularity condition, Hansen (2005) shows that Tnsm

¯k p µ X → max k = max k k σ ˆk σk

which is greater than zero if and only if µk > 0 for some k. So one can test H0 using the test statistic Tnsm . → Hansen gets a consistent estimate of the p-using a bootstrap procedure 32

Empirical Application: Russian Stock Market

Long positions 0.25% M.

N/T

1)

0.50%

0.40%

pU C 0.38

pCC 0.67

2)

0.40%

0.38

3)

0.40%

4)

N/T

1%

0.70%

pU C 0.40

pCC 0.67

0.67

0.70%

0.40

0.38

0.67

0.80%

0.40%

0.38

0.67

5)

0.50%

0.16

6)

0.50%

7)

N/T

5%

1.40%

pU C 0.23

pCC 0.20

0.67

1.50%

0.14

0.22

0.44

1.80%

0.70%

0.40

0.67

0.37

1.00%

0.05

0.16

0.37

0.60%

0.10%

0.28

0.56

8)

0.10%

0.28

9)

0.10%

10)

N/T 6.50%

pU C 0.04

pCC 0.02

0.16

6.50%

0.04

0.02

0.02

0.05

7.70%

0.00

0.00

1.60%

0.08

0.11

6.70%

0.02

0.02

0.13

1.90%

0.01

0.03

7.70%

0.00

0.00

0.66

0.88

1.70%

0.04

0.07

6.70%

0.02

0.00

0.30%

0.33

0.62

1.20%

0.54

0.71

7.90%

0.00

0.00

0.56

0.30%

0.33

0.62

1.10%

0.75

0.84

7.80%

0.00

0.00

0.28

0.56

0.30%

0.33

0.62

1.10%

0.75

0.84

8.00%

0.00

0.00

0.20%

0.74

0.94

0.30%

0.33

0.62

1.00%

1.00

0.90

7.80%

0.00

0.00

11)

0.10%

0.28

0.56

0.30%

0.33

0.62

1.20%

0.54

0.71

7.90%

0.00

0.00

12)

0.20%

0.74

0.94

0.30%

0.33

0.62

1.10%

0.75

0.84

7.90%

0.00

0.00

Table 1: Actual VaR exceedances N/T , Kupiec’s and Christoffersen’s tests p-values: Long positions.

33

Empirical Application: Russian Stock Market

Short positions 0.25% M.

N/T

1)

0.50%

0.70%

pU C 0.02

pCC 0.06

2)

0.80%

0.01

3)

0.80%

4)

N/T

1%

1.00%

pU C 0.05

pCC 0.07

0.02

1.00%

0.05

0.01

0.02

1.00%

0.70%

0.02

0.06

5)

0.80%

0.01

6)

0.70%

7)

N/T

5%

1.30%

pU C 0.36

pCC 0.24

0.07

1.30%

0.36

0.05

0.08

1.40%

1.00%

0.05

0.07

0.02

1.10%

0.02

0.02

0.06

1.00%

0.30%

0.76

0.95

8)

0.30%

0.76

9)

0.20%

10)

N/T 3.40%

pU C 0.01

pCC 0.00

0.56

3.50%

0.02

0.01

0.23

0.20

4.00%

0.13

0.02

1.30%

0.36

0.56

3.50%

0.02

0.01

0.08

1.50%

0.14

0.16

4.10%

0.18

0.01

0.05

0.07

1.30%

0.36

0.56

3.50%

0.02

0.01

0.50%

1.00

0.98

1.00%

1.00

0.90

4.70%

0.66

0.02

0.95

0.50%

1.00

0.98

0.90%

0.75

0.87

4.70%

0.66

0.02

0.74

0.94

0.50%

1.00

0.98

0.90%

0.75

0.87

4.80%

0.77

0.03

0.30%

0.76

0.95

0.50%

1.00

0.98

1.00%

1.00

0.90

4.80%

0.77

0.03

11)

0.30%

0.76

0.95

0.50%

1.00

0.98

1.10%

0.75

0.84

4.80%

0.77

0.03

12)

0.30%

0.76

0.95

0.50%

1.00

0.98

1.00%

1.00

0.90

4.90%

0.88

0.03

Table 2: Actual VaR exceedances N/T , Kupiec’s and Christoffersen’s tests p-values: Short positions.

34

Empirical Application: Russian Stock Market Long position

Short position

0.25%

0.50%

1%

5%

0.25%

0.50%

1%

5%

Model 1)

2.360

4.275

7.830

29.852

10.408

13.276

18.107

45.527

Model 2)

2.332

4.239

7.811

29.918

10.430

13.269

18.047

45.451

Model 3)

2.329

4.283

7.956

30.412

10.500

13.373

18.170

45.343

Model 4)

2.334

4.257

7.807

29.957

10.421

13.288

18.046

45.453

Model 5)

2.376

4.347

8.089

30.399

10.702

13.492

18.245

45.349

Model 6)

2.346

4.267

7.844

29.939

10.384

13.260

18.008

45.405

Model 7)

2.480

4.428

7.858

30.135

9.648

12.471

17.178

44.059

Model 8)

2.480

4.414

7.837

30.142

9.681

12.531

17.273

44.001

Model 9)

2.546

4.448

7.870

30.212

9.695

12.512

17.238

44.075

Model 10)

2.551

4.491

7.853

30.134

9.614

12.448

17.198

44.012

Model 11)

2.498

4.415

7.841

30.268

9.611

12.473

17.241

44.111

Model 12)

2.505

4.432

7.852

30.226

9.686

12.530

17.230

44.067

Table 3: Asymmetric loss functions (23). The smallest value is reported in bold font. 35

Empirical Application: Russian Stock Market Long position Benchmark

Short Position

0.25%

0.50%

1%

5%

0.25%

0.50%

1%

5%

Model 1)

0.138

0.172

0.730

0.981

0.461

0.318

0.061

0.011

Model 2)

0.864

1.000

0.902

0.186

0.364

0.337

0.064

0.003

Model 3)

0.990

0.537

0.120

0.025

0.076

0.146

0.060

0.002

Model 4)

0.898

0.429

0.957

0.065

0.400

0.288

0.048

0.005

Model 5)

0.238

0.213

0.060

0.023

0.065

0.093

0.067

0.006

Model 6)

0.196

0.259

0.188

0.167

0.341

0.325

0.040

0.005

Model 7)

0.268

0.274

0.680

0.580

0.911

0.808

0.892

0.427

Model 8)

0.304

0.298

0.937

0.628

0.506

0.234

0.192

0.909

Model 9)

0.000

0.155

0.589

0.233

0.390

0.600

0.477

0.327

Model 10)

0.000

0.000

0.723

0.594

0.797

0.858

0.709

0.725

Model 11)

0.000

0.296

0.867

0.057

0.906

0.945

0.446

0.227

Model 12)

0.077

0.235

0.816

0.180

0.180

0.390

0.560

0.373

Table 4: Hansen’s SPA test for the portfolio consisting of thirty Russian stocks. P-values smaller than 0.05 are reported in bold font. 36

Conclusions • Empirical analysis 1: If one is interested in forecasting the extreme quantiles, particularly at the 1% and 99% levels, (which is the usual case for regulatory purposes), then using a Student’s t GARCH model with any copula does a good job. • Empirical analysis 2: The fact that the type of copula plays a minor role is not a surprise, given previous empirical evidence with American and European stocks, see e.g. An´ e and Kharoubi (2003), Junker and May (2005) and Fantazzini (2008). → Simulation evidence in Fantazzini (2008b) highlights that copula misspecification is overcome by marginal misspecification when dealing with small-to-medium sized samples. → Besides, copula misspecification is large only in case of negative dependence, while much smaller with positive dependence. In the latter case, different models may deliver quite close VaR estimates (given the same marginals are used). 37

Conclusions

• Empirical analysis 3: It is interesting to note that if normal marginals are used, then models with dynamic dependence deliver statistically significant (and more precise) VaR estimates than models with constant dependence. → If Student’s t marginals are used, the differences are much smaller and not significant! → This confirms that marginal misspecification may result in significant misspecified dependence structure. • 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.

38

References [1] Cherubini, U., Luciano, E., Vecchiato, W. (2004). Copula Methods in Finance. Wiley. [2] Christoffersen, P. (1998). Evaluating Interval Forecats. International Economic Review, 39, 841-862. [3] Fantazzini, D. (2008). Dynamic copula Modelling for Value at Risk. Frontiers in Finance and Economics, 5(2),1-36. [4] Fantazzini, D. (2008b). The Effects of Misspecified Marginals and Copulas on Computing the Value at Risk: A Monte Carlo Study, Computational Statistics and Data Analysis, forthcoming. [5] Fantazzini, D. (2009a). A Dynamic Grouped-T Copula Approach for Market Risk Management, (in) A VaR Implementation Handbook, McGraw-Hill, New York [6] Fantazzini, D. (2009b). Market Risk Management for Emerging Markets: Evidence from Russian Stock Market, (in) Financial Innovations in Emerging Markets, Chapman Hall-CRC/Taylor and Francis, London [7] Giacomini, R., Komunjer, I. (2005). Evaluation and Combination of Conditional Quantile Forecasts. Journal of Business and Economic Statistics, 23, 416-431. [8] Hansen, P. (2005). A Test for Superior Predictive Ability. Journal of Business and Economic Statistics, 23(4), 365-380.

39