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


1st Introduction

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

2-a


1st Introduction


3rd Advanced Multivariate Modelling: The Theory of Copulas

2-b


1st Introduction



4th Multivariate Modelling for High-Dimensional Portfolios: A

Unified Approach with Copulas

2-c


1st Introduction





5th Empirical Application: Russian Stock Market

2-d


1st Introduction





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

εt = Σ1/2t (θ)ηt (2)

where

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

• Σ1/2t (θ) is the Cholesky Decomposition of Σt(θ)

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

• 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 = DtRtDt (3)

Dt = diag(σ1/211,t . . . σ

1/2nn,t) (4)

Rt = (ρij,t), with ρii,t = 1 (5)

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,tσjj,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 ofBollerslev (1990)

The CCC model is defined as:

Σt = DtRDt = (ρij√

σiitσjjt) (7)

where

Dt = diag (σ1/211t . . . σ

1/2nnt) (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,tσjj,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 = DtRtDt (11)

where Dt is defined in (8), and

Rt = (diagQt)−1/2Qt(diagQt)

−1/2 (12)

where the N × N symmetric positive definite matrix Qt is given by:

Qt =

(1 −

L∑

l=1

αl −S∑

s=1

βs

)Q +

L∑

l=1

αlηt−lη′t−l +

S∑

s=1

βsQt−s (13)

where ηit = εit/√

σii,t, Q is the n × n unconditional variance matrix of ut,

αl (≥ 0) and βs (≥ 0) are scalar parameters satisfying∑Ll=1 αl +

∑Ss=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 qii = 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 =(1 − α − β)q12 + αu1,t−1u2,t−1 + βq12,t−1√(

(1 − α − β)q11 + αu21,t−1 + βq11,t−1

) (1 − α − β)q22 + αu2

2,t−1 + βq22,t−1

)

→ 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, . . . , Fn,

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


(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


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


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),, . . . , Fn(xn)), associated to a copula function

C(F1(x1),, . . . , Fn(xn)):

f(x1, ..., xn) =∂n [C(F1(x1), . . . , Fn(xn))]

∂F1(x1), . . . , ∂Fn(xn)·

n∏

i=1

fi(xi) = c(F1(x1), . . . , Fn(xn))·n∏

i=1

fi(xi)

Therefore, we get

c(F1(x1), ..., Fn(xn)) =f(x1, ..., xn)

n∏i=1

fi(xi)· , (15)

15


By using this procedure, we can derive the Normal copula density:

c(u1, . . . , un) =fNormal(x1, ..., xn)

n∏i=1

fNormali (xi)

=

1

(2π)n/2|Σ|1/2 exp(− 1

2x′Σ−1x

)

n∏i=1

1√2π

exp(− 1

2x2

i

) =

=1

|Σ|1/2exp

(−1

2ζ′(Σ−1 − I)ζ

)(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

lgaussian(θ) = −T2

ln |Σ| − 12

T∑t=1

ς′

t(Σ−1 − I)ςt

16


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

θML = Σ for the Gaussian copula:

∂∂Σ−1 lgaussian (θ ) =T

2Σ − 1

2

T∑t=1

ς′

t ςt = 0

and therefore

Σ =1

T

T∑

t=1

ς′

t ςt (17)

17


We can derive the Student’s T-copula in a similar way:

c(u1, u2, . . . , un; Σ) =

fstudent(x1,...,xN )N∏

i=1fstudent

i (xi)

= 1

|Σ|12

Γ( ν+N2 )

Γ( ν2 )

[Γ( ν

2 )Γ( ν+1

2 )

]N

(1+

ς′tΣ−1ςt

ν

)− ν+N2

N∏i=1

(1+

ς2tν

)− ν+12

lStudent (θ ) =

−T lnΓ(

ν+N2

)

Γ(

ν2

) −NT lnΓ(

ν+12

)

Γ(

ν2

) −T

2ln |Σ|−

ν + N

2

T∑

t=1

ln

1 +

ς′tΣ−1ςt

ν

+

ν + 1

2

T∑

t=1

N∑

i=1

ln

1 +ς2it

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 ofa normal copula with correlation matrix Σ = Rt together with normalmarginals:

Yt = E [Yt|Ft−1] + Dtηt (18)

ηt ∼ H(η1, . . . , ηn) ≡ CNormal(F Normal1 (η1), . . . , F Normal

n (ηn); Rt)

where Dt = diag(σ1/211,t . . . σ

1/2nn,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 = (diagQt)−1/2Qt(diagQt)

−1/2

Qt =

(1 −

L∑

l=1

αl −S∑

s=1

βs

)Q +

L∑

l=1

αlηt−lη′t−l +

S∑

s=1

βsQt−s

19


• A multivariate model that allows for marginal kurtosis and normaldependence can be expressed as follows:

Yt = E [Yt|Ft−1] + Dtηt (19)

ηt ∼ H(η1, . . . , ηn) ≡ CNormal(F Student′s−t1 (η1), . . . , F Student′s−t

n (ηn); Rt)

(20)

where F Student′s−ti 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’sT copula, instead,

Yt = E [Yt|Ft−1] + Dtηt (21)

ηt ∼ H(η1, . . . , ηn) ≡ CStudent′s t(F Student′s−t1 (η1), . . . , F Student′s−t

n (ηn); Rt, ν)

(22)

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

20


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 beseparated in m distinct groups:

Xt = E [Xt|Ft−1] + Dtηt

ηt ∼ H(η1, . . . , ηn) ≡ CGrouped t(F Student′s−t1 (η1), . . . , F Student′s−t

n (ηn); Rt, ν1, . . . , νm)

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

latter case).

21


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 = ηtσt, ηti.i.d.∼ f(0, 1)

σ2t = ω + αε2

t−1 + βσ2t−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


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


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


0200

400

600

800

1000

L2 d

issi

mila

rity

measu

re

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

Figure 5: Dendogram for the 30-asset portfolio

25


Marginal

Distribution

Moment

specification

Copula Copula Parameters

Specification

Model 1)NORMAL

AR(1) GARCH(1,1)NORMAL

Constant Correlation

Model 2)NORMAL

AR(1) GARCH(1,1)NORMAL

DCC(1,1)

Model 3)NORMAL

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


Const. D.o.F.s

Model 4)NORMAL

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

DCC(1,1)

Const. D.o.F.

Model 5)NORMAL

AR(1) GARCH(1,1)GROUPED T


Const. D.o.F.s

Model 6)NORMAL

AR(1) GARCH(1,1)GROUPED T

DCC(1,1)

Const. D.o.F.

Model 7)Student’s t

AR(1) GARCH(1,1)

Constant D.o.F.NORMAL



AR(1) GARCH(1,1)

Constant D.o.F.NORMAL

DCC(1,1)


AR(1) GARCH(1,1)

Constant D.o.F.T-COPULA


Const. D.o.F.s


AR(1) GARCH(1,1)

Constant D.o.F.T-COPULA

DCC(1,1)

Const. D.o.F.


AR(1) GARCH(1,1)

Constant D.o.F.GROUPED T


Const. D.o.F.s


AR(1) GARCH(1,1)

Constant D.o.F.GROUPED T

DCC(1,1)

Const. D.o.F.

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


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


1. Kupiec’s test: Following binomial theory, the probability of

observing N failures out of T observations is (1-p)T−NpN , so that the

test of the null hypothesis H0: p = p∗ is given by a LR test statistic:

LR = 2 · ln[(1 − p∗)T−Np∗N] + 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.

LRCC = −2 ln[(1−p)T−NpN ]+2 ln[(1−π01)n00πn01

01 (1−π11)n10πn11

11 ]

where nij is the number of observations with value i followed by j for

i, j = 0, 1 and

πij =nij∑j nij

29


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 aRt+1|t, Lt+1 is the realized loss, while

V aRt+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


→ Let L(Yt; Yt) denote the loss if one had made the prediction, Yt,

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, Y0t) − L(Yt, Ykt), 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, k = 1, . . . , l. or in compact notation:

µ =

µ1

...

µl

= E

X1(t)

...

Xl(t)

H0 : µ ≤ 0

31


One way to test this hypothesis is to consider the test statistic

T smn = max

k

n1/2Xk

σk

where

Xk =1

n

n∑

t=1

Xk(t), σ2k = var(n1/2Xk).

The superscript “sm“ refers to standardized maximum. Under some

regularity condition, Hansen (2005) shows that

T smn = max

k

Xk

σk

p→maxk

µk

σk

which is greater than zero if and only if µk > 0 for some k. So one can

test H0 using the test statistic T smn .

→ Hansen gets a consistent estimate of the p-using a bootstrap

procedure

32


Long positions

0.25% 0.50% 1% 5%

M. N/T pUC pCC N/T pUC pCC N/T pUC pCC N/T pUC pCC

1) 0.40% 0.38 0.67 0.70% 0.40 0.67 1.40% 0.23 0.20 6.50% 0.04 0.02

2) 0.40% 0.38 0.67 0.70% 0.40 0.67 1.50% 0.14 0.16 6.50% 0.04 0.02

3) 0.40% 0.38 0.67 0.80% 0.22 0.44 1.80% 0.02 0.05 7.70% 0.00 0.00

4) 0.40% 0.38 0.67 0.70% 0.40 0.67 1.60% 0.08 0.11 6.70% 0.02 0.02

5) 0.50% 0.16 0.37 1.00% 0.05 0.13 1.90% 0.01 0.03 7.70% 0.00 0.00

6) 0.50% 0.16 0.37 0.60% 0.66 0.88 1.70% 0.04 0.07 6.70% 0.02 0.00

7) 0.10% 0.28 0.56 0.30% 0.33 0.62 1.20% 0.54 0.71 7.90% 0.00 0.00

8) 0.10% 0.28 0.56 0.30% 0.33 0.62 1.10% 0.75 0.84 7.80% 0.00 0.00

9) 0.10% 0.28 0.56 0.30% 0.33 0.62 1.10% 0.75 0.84 8.00% 0.00 0.00

10) 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


Short positions

0.25% 0.50% 1% 5%

M. N/T pUC pCC N/T pUC pCC N/T pUC pCC N/T pUC pCC

1) 0.70% 0.02 0.06 1.00% 0.05 0.07 1.30% 0.36 0.24 3.40% 0.01 0.00

2) 0.80% 0.01 0.02 1.00% 0.05 0.07 1.30% 0.36 0.56 3.50% 0.02 0.01

3) 0.80% 0.01 0.02 1.00% 0.05 0.08 1.40% 0.23 0.20 4.00% 0.13 0.02

4) 0.70% 0.02 0.06 1.00% 0.05 0.07 1.30% 0.36 0.56 3.50% 0.02 0.01

5) 0.80% 0.01 0.02 1.10% 0.02 0.08 1.50% 0.14 0.16 4.10% 0.18 0.01

6) 0.70% 0.02 0.06 1.00% 0.05 0.07 1.30% 0.36 0.56 3.50% 0.02 0.01

7) 0.30% 0.76 0.95 0.50% 1.00 0.98 1.00% 1.00 0.90 4.70% 0.66 0.02

8) 0.30% 0.76 0.95 0.50% 1.00 0.98 0.90% 0.75 0.87 4.70% 0.66 0.02

9) 0.20% 0.74 0.94 0.50% 1.00 0.98 0.90% 0.75 0.87 4.80% 0.77 0.03

10) 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


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 re-

ported in bold font.

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Long position Short Position

Benchmark 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 Rus-

sian 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. Ane 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).

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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.

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

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