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This is the presentation of the paper "Market Risk Management for Emerging Markets: Evidence from Russian Stock Market" held at the VII-th International School Seminar "Multivariate Statistical Analysis and Econometrics", Tsahkadzor, Armenia, September 24th 2008, which is now forthcoming in the book "Emerging Markets: Performance, Analysis and Innovation", published by Chapman-Hall/CRC Finance
Citation preview
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
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
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),, . . . , 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
Advanced Multivariate Modelling: The Theory of Copulas
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
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
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
Advanced Multivariate Modelling: The Theory of Copulas
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
Multivariate Modelling for High-Dimensional Portfolios:A Unified Approach
• 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
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 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
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 = 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
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
Multivariate Modelling for High-Dimensional Portfolios:A Unified Approach
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
Multivariate Modelling for High-Dimensional Portfolios:A Unified Approach
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
Constant Correlation
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
Constant Correlation
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
Constant Correlation
Model 8)Student’s t
AR(1) GARCH(1,1)
Constant D.o.F.NORMAL
DCC(1,1)
Model 9)Student’s t
AR(1) GARCH(1,1)
Constant D.o.F.T-COPULA
Constant Correlation
Const. D.o.F.s
Model 10)Student’s t
AR(1) GARCH(1,1)
Constant D.o.F.T-COPULA
DCC(1,1)
Const. D.o.F.
Model 11)Student’s t
AR(1) GARCH(1,1)
Constant D.o.F.GROUPED T
Constant Correlation
Const. D.o.F.s
Model 12)Student’s t
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
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−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
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 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
Empirical Application: Russian Stock Market
→ 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
Empirical Application: Russian Stock Market
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
Empirical Application: Russian Stock Market
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
Empirical Application: Russian Stock Market
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
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 re-
ported in bold font.
35
Empirical Application: Russian Stock Market
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).
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