Econometric Reviews, 29(5–6):622–641, 2010
Copyright © Taylor & Francis Group, LLC
ISSN: 0747-4938 print/1532-4168 online
DOI: 10.1080/07474938.2010.481972
ON SOME MODELS FOR VALUE-AT-RISK
Philip L. H. Yu1 , Wai Keung Li1 , and Shusong Jin2
1
Department of Statistics and Actuarial Science, The University of Hong Kong,
Hong Kong
2
School of Management, Fudan University, Shanghai, China
The idea of statistical learning can be applied in financial risk management. In recent
years, value-at-risk (VaR) has become the standard tool for market risk measurement and
management. For better VaR estimation, Engle and Manganelli (2004) introduced the
conditional autoregressive value-at-risk (CAViaR) model to estimate the VaR directly by quantile
regression. To entertain the nonlinearity and structural change in the VaR, we extend the
CAViaR idea using two approaches: the threshold GARCH (TGARCH) and the mixtureGARCH models. The estimation method of these models are proposed. Our models should possess
all the advantages of the CAViaR model and enhance the nonlinear structure. The methods are
applied to the S&P500, Hang Seng, Nikkei and Nasdaq indices to illustrate our models.
Keywords GARCH model; Mixtures; Threshold models; Value-at-risk.
JEL Classification
C22.
1. INTRODUCTION
Today, too many large data sets emerge and statistical learning theory is
an efficient and comprehensive way in digging out information from these
data set. Statistical learning has found many important applications in
economics. See, for example the review by Pau and Tan (1996). A popular
learning model is the finite mixture model (Bishop, 2006) where each
component can be regarded as an expert and training (learning) can
be regarded as a quasi-maximum likelihood problem. Estimation can
be done effectively using the Expectation Maximization (EM) algorithm.
The mixture approach can incorporate many useful learning models as
special cases including neural network models (Jordan and Jacobs, 1993).
Address correspondence to Wai Keung Li, Department of Statistics and Actuarial Science,
The University of Hong Kong, Pokfulam Road, Hong Kong; E-mail: [email protected]
On Some Models for Value-at-Risk
623
The threshold models of Tong and Lim (1980) can also be interpreted as
a kind of mixture model. A natural application of learning theory is to the
area of financial engineering.
In fact, most financial crises happened without warning. These extreme
price movements in the financial markets are rare, but they can bring
fatal results to some corporations and disasters to a country’s financial
market. Some examples are the crash at the New York stock market in
October 1987, and then, one decade later, the crash of the Asian stock
market. Recent scandals of Enron had also caused the Dow Jones Industrial
Average (DJIA) to drop sharply. These crises are the causes of ruin for
hundreds of companies with their stock value evaporated in a short time.
The recent U.S. subprime loan crisis also caused the DJIA to drop rapidly
and this crisis spreaded globally making the NIKKEI 225 index of Japan
to drop 5.4% on August 17, 2007. These market and credit risk issues had
caused panic among investors. Nowadays, in the trend of globalization,
market risk is of great importance and deserves more attention than
before. Therefore, to a risk manager, a good measure of market risk
is more than necessary. In particular, the Basel Committee on Banking
Supervision (1996) of the Bank for International Settlements imposes on
financial institutions such as banks and investment firms to meet capital
requirements based on value-at-risk (VaR) estimates. As a result, valueat-risk has come out as a standard measurement of an institution’s risk
exposure and has become a standard tool to evaluate and manage financial
risk.
Value-at-risk is mainly concerned with market risk. It can be defined
as the maximal loss of a position on a financial asset during a given time
period for a given probability. See Jorion (2001) for a comprehensive
review of VaR. Let xt +l = log Pt +l − log Pt be the logarithm return of
holding the asset from time t to t + l , where Pt is the asset price at time t .
Then the VaR over the time horizon with probability is given by
= P [xt +l ≤ VaR] ≡ Fl (VaR),
(1)
where Fl (x) is the CDF of xt +l , evaluated at xt +l = x. Although the
concept of VaR is not difficult to understand, the estimation of VaR is
a very challenging statistical problem. The RiskMetrics model, pioneered
by J. P. Morgan, assumes that returns of a financial asset follow a
conditional normal distribution with zero mean and variance being
expressed as an exponentially weighted moving average of historical
squared returns. However, as many articles have pointed out, the
RiskMetrics model fails to measure market risk accurately because this
model lacks any nonlinear property, which is a significant feature of
the financial market. In the finance literature, examples of nonlinearity
abound. For instance, DeBondt and Thaler (1985) pointed out that stock
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P. L. H. Yu et al.
returns always exhibit over-reaction and serial correlation over long time
horizons. French et al. (1987) found evidence that the expected market
risk premium is positively related to the predictable volatility of stock
returns, but the unexpected stock market returns are negatively related
to the unexpected change in the volatility of stock returns. Nam et al.
(2002) argued that the relative profitability of “loser” stocks is attributable
to an asymmetric reverting property of their return dynamics. Chan and
Maheu (2002) used conditional jump dynamics to explain the behavior
of the stock market returns. Furthermore, changing volatility (Schwert
and Stambaugh, 1987) is also an important stylized fact of stock returns.
Hence, learning models that can reproduce nonlinearity and the changing
volatility structures are in great demand in risk management.
For a given value , 0 ≤ ≤ 1, the th quantile of the variable x is
defined as C (x) = infx | F (x) ≥ , where F is the CDF of x. Koenker
and Bassett (1978) suggested that based on a sample of iid realizations
xt of x the quantile b = C (x) can be estimated by solving the following
minimization problem
min
|xt − b| +
(1 − )|xt − b| (2)
b∈
t ∈t :xt ≥b
t ∈t :xt <b
With the quantile regression method, Wu and Xiao (2002) obtained a VaR
process for the AR model and suggested that this method can be applied
to the ARCH model.
Engle and Manganelli (2004) applied this method to the VaR problem
and proposed a conditional value-at-risk (CAViaR) model with a general
specification for VaR at time t ,
VaRt = f (xt , ) = 0 +
p
i VaRt −i + l (p+1 , , p+q ; t −1 ),
(3)
i=1
where l (·) is a pre-specified function. They suggested that the first order
CAViaR model: VaRt = 0 + 1 VaRt −1 + l (2 , xt −1 , VaRt −1 ) is sufficient for
practical use.
In this article, we will focus on conditional VaR estimation using the
threshold GARCH (TGARCH) and mixture-GARCH models as learning
models. We believe that these two models can better explain nonlinear
phenomena in the financial market. These two models can also be
regarded as extensions of the CAViaR model of Engle and Manganelli
(2004). Following Engle and Manganelli (2004), we call the new models
indirect-VaR models.
Only univariate VaR models are considered in this article. However, as
pointed out by Berkowitz and O’Brien (2002) large multivariate models
On Some Models for Value-at-Risk
625
need not outperform univariate time series models in their VaR forecast
performances. They also pointed out that univariate models based on the
profit and loss of a portfolio can be complementary to the large-scale
models in terms of advantages in forecasting and as a tool for identifying
the short comings of the (multivariate) structural model. In a more recent
study, Kuester et al. (2006) adopted a similar point of view. Bradley and
Taqqu (2004) also suggested that a univariate approach is preferred to a
bivariate approach for the asset allocation problem where investors seek
to minimize the probability of large losses. McAleer (2005) discussed
some issues in the modelling of univariate and multivariate volatility
models. More recently, McAleer and da Veiga (2008a) considered spillover
effects in VaR forecasting using multivariate GARCH models. They showed
that while spillover effects are statistically significant VaR forecasts are
in general insensitive to such effects. McAleer and da Veiga (2008b)
considered the pros and cons of single-index and portfolio models for
VaR forecasts. McAleer (2005) mentioned also a collection of multivariate
GARCH and stochastic volatility models. A thorough study of the relative
merits of these models in VaR forecasts is clearly of interest in the future.
The organization of this article is as follows. Section 2 reviews
TGARCH and mixture-GARCH models. Section 3 formulates two indirectVaR models based on either TGARCH or mixture-GARCH models. The
learning process for the proposed indirect-VaR models are introduced in
Section 4. Simulations are done in Section 5 to examine the effectiveness
of our proposed estimation methods. The proposed models are applied in
Section 6 to the S&P500 index to illustrate the capability of the models
in estimating VaR. For comparison, we also apply our models to Hang
Seng, Nikkei, and Nasdaq indices which can be regarded as indices for less
developed markets. Section 7 concludes by summarizing the advantages of
our models.
2. THE TGARCH AND MIXTURE-GARCH MODEL
To capture the asymmetric volatility characteristic of financial time
series, we extend the double threshold ARCH (DTARCH) of Li and Li
(1996) to a general k-regime threshold GARCH (TGARCH) model as
below:
xt = t ht ,
ht =
(j )
0
+
pj
r =1
r(j ) xt2−r
+
qj
(j )
i ht −i ,
j −1 ≤ xt −d < j ,
(4a)
j = 1, , k,
i=1
(4b)
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P. L. H. Yu et al.
where the regime number k and the delay parameter d are positive
integers, j = 1, , k. j ’s are real numbers such that −∞ = 0 < 1 < · · · <
k−1 < k = ∞, the superscript (j ) is used to signify the regime, t are iid
noise with mean 0 and unit variance. We denote model (4) as TGARCH(p1 ,
p2 , , pk ; q1 , q2 , , qk ). With some regularity conditions, Liu et al. (1997)
proved that model (4) is stationary and ergodic. One advantage of the
TGARCH model is that it can model asymmetric and limit cycle behavior
in a natural way and break the homoscedastic variance restriction as
ordinary SETAR-type models do.
An alternative nonlinear time series model is the mixture AR-ARCH
model proposed by Wong and Li (2001). They assume that xt follows
a mixture of K -component models with each components being an ARARCH model. The general K -component model is given by
K
ek,t
(5a)
k F (xt | t −1 ) =
hk,t
k=1
(5b)
ek,t = xt − k0 − k1 xt −1 − · · · − kpk xt −pk ,
2
2
hk,t = k0 + k1 ek,t
−1 + · · · + kqk ek,t −qk ,
(5c)
where t −1 is the information set up to time t − 1; (·) is the (conditional)
cumulative distribution function of the standard Gaussian distribution; and
1 + · · · + K = 1, k > 0 (k = 1, , K ). Here F (xt | t −1 ) is the conditional
cumulative distribution function of xt given the past information. They
denote this model as the MAR-ARCH(K ; p1 , p2 , , pK ; q1 , q2 , , qK ). We
extend the ARCH formulation to GARCH and the mixture-GARCH model
with K components is defined as
qk
pk
K
2
(6)
ht =
Zkt k0 +
kik xt −ik +
kjk ht −jk ,
xt = t ht ,
k=1
ik =1
jk =1
where Zkt = 1 when the xt is drawn from the kth component (let k be the
probability that Zkt = 1), Zkt = 0 otherwise. Then 1 + 2 + · · · + K = 1,
k > 0, (k = 1, , K ); t is iid white noise following the standard normal
distribution. We call model (6) a mixture-GARCH(K ; p1 , p2 , , pK ; q1 ,
q2 , , qK ). As in the case for the GARCH(1,1) model, the most important
special case of the mixture-GARCH model is the (2;1,1;1,1) model,
√
xt = t ht ,
(7)
ht = Zt (10 + 11 xt2−1 + 11 ht −1 ) + (1 − Zt )(20 + 21 xt2−1 + 21 ht −1 ),
where Zt follows the Bernoulli distribution: Zt = 1 with probability ; and
Zt = 0 with probability (1 − ). We denote the model (7) as the mixtureGARCH(1,1) with two components.
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On Some Models for Value-at-Risk
During the preparation of this article many new mixture type models
have been proposed such as those in Mendes et al. (2006). It is clearly
of interest to compare the performance of these new models in VaR
forecasting with the proposed approach here. However, this will be beyond
the scope of the present article.
3. INDIRECT-VaR MODELS
By analogy with the popular GARCH(1,1) model, the GARCH(1,1)
indirect-VaR model (Engle and Manganelli, 2004, Section 3) is defined as
VaRt = (0 + 1 VaR2t −1 + 2 xt2−1 )1/2 (8)
Engle and Manganelli (2004) pointed out that if the underlying volatilities
were truly generated from a GARCH(1,1) process, the VaR follows the
GARCH(1,1) process (8). Note that (8) is not a GARCH(1,1) model in
the usual sense. Following Engle and Manganelli (2004) we will call (8)
a GARCH(1,1) indirect-VaR model. We can obtain a similar result for
indirect-VaR models when the underlying data are truly a TGARCH or
mixture-GARCH(1,1) model, respectively.
Specifically, if xt follows the TGARCH process (4a), then it can be
shown that the VaR2t process follows a TGARCH process (9). Similarly, if
xt follows the mixture-GARCH(1,1) process (7), then the VaR process also
follows an mixture-GARCH(1,1) process (10)
VaR2t
=
(j )
a0
+
pj
ar(j ) xt2−r
+
r =1
qj
(j )
bi VaR2t −i ,
j −1 ≤ xt −d < j ,
(9)
i=1
VaR2t = Zt a10 + a11 xt2−1 + b11 VaR2t −1
+ (1 − Zt ) a20 + a21 xt2−1 + b21 VaR2t −1 ,
(10)
where C is the lower (left-tail) th quantile of the standard normal
(j )
(j )
distribution, ar(j ) = C2 r(j ) , r = 0, , pj , bi = i , i = 1, , qj , j = 1, , k;
ak0 = C2 · k0 , ak1 = C2 · k1 and bk1 = k1 , k = 1, 2. The proof of (10) is
direct by noting that VaR2t = Zt C2 ht(1) + (1 − Zt )C2 ht(2) , where ht(1) = 10 +
11 xt2−1 + 11 ht −1 and ht(2) = 20 + 21 xt2−1 + 21 ht −1 .
We can also show that if the th VaR,t follows (9) or (10), then the
th VaR,t follows (11) or (12), respectively,
(j )
a0
=m ·
VaR2,t
= Zt (m · a10 + m ·
2
+
pj
VaR2,t
m ·
2
ar(j ) xt2−r
r =1
2
+
qj
(j )
bi VaR2,t −i ,
i=1
2
a11 xt2−1
+ b11 VaR2,t −1 )
j −1 ≤ xt −d < j ,
(11)
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P. L. H. Yu et al.
+ Ztc (m 2 · a20 + m 2 · a21 xt2−1 + b21 VaR2,t −1 ),
(12)
where m = C /C , Ztc = 1 − Zt and j = 1, , k.
In other words, if we have found the 100% VaR process and want to
know the 100% VaR process for = , we need not fit the model again.
4. LEARNING THRESHOLD AND MIXTURE
TYPE INDIRECT-VaR MODELS
4.1. The Indirect-VaR TGARCH Model
There are many ways to learn a threshold indirect-VaR model. One
of the most popular methods proposed by Tsay (1989) is to estimate the
parameters by the structural change method. However, in our case, it is
hard to apply. Note that Medeiros and Veiga (2009) proposed a sequential
testing approach for a threshold multiple regime GARCH type model.
Although the data may actually have a structural change, an accurate
numerical procedure is difficult to implement. We resort to use a more
direct but slightly more time consuming method for the TGARCH model
estimation. Firstly, we sort the data points into an ascending order. Then
we remove the smallest a and the largest b data points and consider each
data point in between as a possible candidate value for the threshold
parameter. Note that this is merely a practical device to ensure that
there are enough observations for each regime. With a sample size of n
√
observations, n → ∞, a and b can be allowed to go 0 at rate O(1/ n), say.
We use a = b = 25% for the numerical works in this article. As an example,
we illustrate the method to fit an indirect-VaR TGARCH(1,1;1,1) model for
the underlying series xt with two regimes. Following (9), let
(1)
xt −d < ,
a0 + a1(1) xt2−1 + b1(1) VaR2t −1 ,
VaR2t =
(13)
(2)
(2) 2
(2)
2
a0 + a1 xt −1 + b1 VaRt −1 ,
xt −d ≥ The initial value VaR1 is determined by C , the th quantile of the standard
normal distribution, times the sample standard derivation. Other VaRt can
be determined iteratively by (13). If C < 0, we take the negative square
root. If C > 0, we choose the positive square root. For a given threshold
candidate , we obtain the conditional th quantile by minimizing
|xt − VaRt | + xt <VaRt (1 − )|xt − VaRt | · I (xt −d < )
n
L=
t =d+1 I (xt −d < )
xt ≥VaRt |xt − VaRt | +
xt <VaRt (1 − )|xt − VaRt | · I (xt −d ≥ )
n
,
+
t =d+1 I (xt −d ≥ )
xt ≥VaRt
629
On Some Models for Value-at-Risk
where I (·) is the indicator function. The first term can be interpreted as
the objective function for the conditional quantile function corresponding
to the regime xt −d < normalized by the number of observations in
that regime. A similar interpretation holds for the second term which
corresponds to the regime xt −d ≥ . The parameter d is chosen from
natural numbers. Since in practice d seldom exceeds 2, so we choose 1, 2,
and 3 as its candidates. We repeat the method above for all the candidates
and choose the threshold and parameter values that minimize L globally.
4.2. The Mixture-GARCH(1,1) Indirect-VaR Model
Motivated by the Expectation–Maximization (EM) algorithm, we
suggest a two-stage estimation to find the th quantile for VaR estimation.
Define the objective function L by
L=
xt ≥VaRt
|xt − VaRt | +
(1 − )|xt − VaRt | (14)
xt <VaRt
As in White (1994), we minimize L in a way similar to an EM style
algorithm so that it can be applied to obtain the quasi maximum likelihood
estimates (QMLE). We assume that xt also follows a mixture-GARCH(1,1)
model stated in (7) and the corresponding VaR process follows (10). It
is clear that the Zt s in (10) are unobserved and hence can be regarded
as missing information. Therefore, the two-stage algorithm can be used
to handle this type of parameter estimation. We divide all the parameters
into two groups, the parameters containing missing information and the
other parameters not containing it. We call them group 1 and group
2, respectively. After that, our algorithm contains two stages. In stage
1, we suppose that all the parameters in group 2 are given by the
previous step, and use them as initial values to estimate the missing
information parameters in group 1 by taking conditional expectations of
these parameters in group 1 with respect to the values of parameters in
group 2. In stage 2, we use the conditional expectation of parameters
under the mixture-GARCH model for xt in group 1 as the real value
of the missing information and estimate the parameters in group 2 by
maximizing the likelihood function. These two steps are repeated until all
the parameters have converged. Here, the parameter corresponds to the
missing information, so it is in group 1. The other parameters ak0 , ak1 and
bk1 , k = 1, 2 are in group 2.
Stage 1. Given parameter estimates âk0 , âk1 , b̂k1 , k = 1, 2, from the
previous step, we estimate ˆ k as follows. We denote Tkt as the conditional
630
P. L. H. Yu et al.
probability of xt in the kth component, k = 1, 2. Since
following Wong and Li (2001, Section 3), we obtain
Tkt =
k
VaR(k)
t
C · xt
2
VaR(k)
t
l =1
l
VaR(lt )
C · xt
√
ht = VaRt /C ,
−1
VaR(lt )
,
where (·) is the density of the standard normal distribution,
VaR(k)
= Sgn(C )(âk0 + âk1 xt2−1 + b̂k1 VaR2t −1 )1/2 ,
t
and Sgn(x) is 1, 0, −1 if x > 0, x = 0 and x < 0 respectively. Tkt can be
computed by using the estimates of 1 and 2 obtained from the previous
iteration. Therefore, we can estimate k , k = 1, 2 by
n
Tkt
ˆ k = t =3
,
k = 1, 2
n−2
Stage 2. Now, using ˆ 1 and ˆ 2 obtained in Stage 1, ak0 , ak1 , bk1 , k = 1, 2
are updated by minimizing L where
t| +
t| ,
|xt − VaR
(1 − )|xt − VaR
L=
t
xt ≥VaR
t
xt <VaR
and
t = Sgn(C ) ˆ 1 (a10 + a11 xt2−1 + b11 VaR
2t −1 )1/2
VaR
2t −1 )1/2 + ˆ 2 (a20 + a21 xt2−1 + b21 VaR
(15)
Since we cannot determine the source component of xt , t = 1, , n, we
have to use the expectation of Zt and 1 − Zt , ˆ 1 and ˆ 2 , to estimate VaRt .
t is in fact the conditional expectation of the true VaRt . We
Therefore, VaR
repeat these two steps until the parameter estimates converge.
Remark. In probability theory, the parameters of model (10) should
be estimated based on the information t −1 , which is the -field
generated by observations (x1 , , xt −1 ). In our estimation method, at first
glance, the parameters seem to be estimated based on the information
(x1 , , xt −1 , z1 , , zt −1 ). However, the conditional expectation of the
missing information z1 , , zt −1 is replaced by the conditional expectation
and can be treated as a function of x1 , , xt −1 from the method of
our estimation. Therefore, our method can be regarded as a generalized
estimation method based on t −1 .
On Some Models for Value-at-Risk
631
For the threshold indirect VaR model, given known d and , estimators
for the other model parameters are consistent and asymptotically normally
distributed under the conditions C0–C7 and AN1–AN4 of Engle and
Manganelli (2004). This is because given known d and the indirect-VaR
TGARCH model reduces to the indirect-VaR GARCH model in Engle and
Manganelli (2004) and the large sample results there carry over to the
present situation directly. When d and are unknown the problem is far
more complicated as has been demonstrated by Chan (1993) in the case
of least squares estimation of the threshold autoregressive models. The
mathematical challenge to obtain a similar result as in Chan (1993) even
for quantile threshold autoregression is clearly worthwhile but beyond the
scope of the present work. It is, however, only natural to conjecture that
the estimators of and d are actually super-consistent as in the classical
threshold autoregressive case (Chan, 1993).
Note that the results in Engle and Manganelli (2004) are extensions
of earlier works of Weiss (1991) and Powell (1984, 1986, 1991). In
particular, it is required that f (xt , ) in (3) is continuous given the past
information set t −1 , assumption C1 in Engle and Manganelli (2004),
which is clearly met in the threshold case but not in the mixture case.
Nevertheless, it is conjectured that estimators for the mixture indirect VaR
model is consistent and asymptotically normally distributed under similar
regularity conditions including the identifiability condition that 1 ≤ 2 .
A proof is clearly beyond the scope of this article as the condition of
Engle and Manganelli (2004) needs to be suitably modified. Note that
very little on the distribution of the error terms have been assumed,
hence reducing the risk of misspecification. Moreover, as point out by
Manganelli and Engle (2001), even if the quantile process is misspecified,
the minimization of the regression quantile objective function can still
be interpreted as the minimization of the Kullback–Leibler information
criterion, which measures the discrepancy between the true model and
the one under study. Our models inherit all these advantages and it
is expected that after incorporating a flexible nonlinear structure on
VaR, the VaR can be estimated more accurately. In a recent article,
Komunjer (2005) considered quasi-maximum likelihood estimation for
the conditional quantiles based on the so called “tick-exponential” family
of densities. The large sample results of his article cover the CAViaR
models of Engle and Manganelli (2004). It may also be possible to extend
Komunjer’s result to the present situation.
5. SIMULATIONS
In this section we simulate two time series data sets from the following
two models and fit the indirect-VaRt process with TGARCH and mixtureGARCH model by quantile regression. The models are the following ones.
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P. L. H. Yu et al.
Model 1.
02 + 025xt2−1 + 03ht −1 ,
ht =
01 + 015xt2−1 + 04ht −1 ,
xt = t ht ,
xt −1 < 0,
xt −1 ≥ 0
(16)
Model 2.
√
xt = t ht ,
ht = Zt (01 + 01xt2−1 + 04ht −1 )1/2 + (1 − Zt )(05 + 02xt2−1 + 03ht −1 )1/2 ,
(17)
where Zt = 1 with probability 0.3 and Zt = 0 with probability 0.7.
Model 1 is a TGARCH(1, 1; 1, 1) model with two regimes and Model
2 is a two-component mixture-GARCH(1, 1) model. For each model, we
generated white noise t , t = 1, , 1001, and simulated a time series
realization of length 1001 using (16) or (17). Let the series be xt ,
t = 1, , 1001. The xt , t = 1, , 1000 were used for model fitting. The
last one, x1001 , was used to test the accuracy of the model. Initial values
for estimation were drawn uniformly from a small interval containing the
true parameters in the center. By repeating the above procedure 1500
times, we obtained 1500 one-step VaR’s for testing the accuracy of the VaR
prediction. The results are listed in Table 1. The upper section contains
the result for TGARCH(1,1;1,1) model (16). We found that the means
of the estimated parameters were very close to the true value. The number
of times x1001 smaller than the estimated 5% VaR is 72. That is about 4.8%
TABLE 1 Five percent quantile regression results of two models
Model
TGARCH
(1,1;1,1)
Indirect-VaR
Mixture
-GARCH
Indirect
-VaR
Mean of estimation
(std. dev.)
Loss greater
than VaR
Parameter
True value
a0(1)
a1(1)
b1(1)
a0(2)
a1(2)
b1(2)
d
02 × 1642 = 05279
025 × 1642 = 06724
0.3
01 × 1642 = 02689
015 × 1642 = 04034
0.4
1
0.0
0.5028
0.6802
0.2820
0.2438
0.4112
0.3916
1.6946
0.2217
(0.0456)
(0.0426)
(0.0624)
(0.0550)
(0.0457)
(0.0678)
(0.8118)
(0.1714)
72 (4.8%)
a10
a11
b11
a20
a21
b21
01 × 1642 = 02689
01 × 1642 = 02689
0.4
05 × 1642 = 13448
02 × 1642 = 05379
0.3
0.3
0.3729
0.3122
0.3678
1.5723
0.6012
0.2785
0.2603
(0.0514)
(0.0579)
(0.0714)
(0.0417)
(0.0479)
(0.0372)
(0.0819)
76 (5.1%)
On Some Models for Value-at-Risk
633
of the times that the 5% quantile value was exceeded. This is very close to
the theoretical value of 5%.
The lower part of Table 1 contains the results for mixture-GARCH
model. Although the estimated parameters for the VaR process were
slightly different from the true value, only 76 x1001 are outside of the
estimated VaR. That is about 51% of the 5% quantile value was exceeded.
From Table 1, we can find that the quantile regression method for the
indirect-VaR estimation in TGARCH and mixture-GARCH(1,1) cases can
provide accurate VaR estimates.
6. APPLICATIONS
In this section, we applied our model to four real data sets: S&P500
daily closing indices from Jan. 2, 1998 to Aug. 31, 2004, Hang Seng daily
closing indices of Hong Kong from Jan. 2, 1997 to Dec. 31, 2003, Nasdaq
daily closing indices from Jan. 2, 1997 to Dec. 31, 2003, and Nikkei daily
closing indices from Jan. 6, 1998 to Dec. 30, 2003. We will discuss the study
of S&P500 in details and other three examples will be explained briefly.
All these data sets are obtained from Yahoo.
The first example is the daily S&P500 closing indices from Jan. 2, 1998
to Aug. 31, 2004. Days that the market was closed were removed, and there
are totally 1674 observations in the series. We denoted them as yt and
used these observations for our empirical research.
Let the daily log-returns of the S&P500 rt be defined by rt =
log yt − log yt −1 . With the log-transformation, all rt are around zero.
Figure 1 shows the S&P500 series and its return series respectively. The
autocorrelation (ACF) and partial autocorrelation (PACF)1 functions of
the return series rt do not show any apparent autoregressive structure
and therefore, we will use the return series rt to fit the model directly
without any linear filter. To illustrate the capability of our method,
we choose the first 1173 log-returns, r1 , , r1173 , for model fitting. The
remaining 500 data points, r1174 , , r1673 are used to check the number of
data points outside the predicted VaRt .
The first indirect-VaR model we fit to the r1 , , r1173 was the
TGARCH(1,1;1,1) indirect-VaR. The method was the same as what had
been described in Subsection 4.1. As mentioned before, we sorted the logreturn in ascending order, discard the smallest and largest 25% and tried
every rt in the middle 50% as candidates for the threshold and obtained
a TGARCH(1,1;1,1) model (18). For the sake of checking accuracy of
long position VaR estimation, we did not update the model as new
data points arrive during the checking stage. The second indirect-VaR
model considered was the mixture-GARCH(1,1) indirect-VaR model (19).
1
The plots of the ACF and PACF can be found in Jin (2005).
634
P. L. H. Yu et al.
FIGURE 1 Plot of S&P500 (Jan. 2, 1998–Aug. 31, 2004). The upper panel is the plot of the real
series and the lower panel is the plot of the log-return series.
We applied the two-stage quantile regression method which had been
described in Subsection 4.2 to fit the mixture-GARCH(1,1) indirect-VaR
model. We summarized the result as below.
TGARCH(1,1;1,1) Indirect-VaR:

2413 × 10−6 + 7938 × 10−2 rt2−1 + 9072 × 10−1 VaR2t −1 ,




rt −1 ≤ −1343 × 10−2 ,
VaR2t =

4176 × 10−4 + 2135 × 10−2 rt2−1 + 1472 × 10−1 VaR2t −1 ,



rt −1 > −1343 × 10−2 (18)
On Some Models for Value-at-Risk
635
Since this is the left tail VaR, we choose the negative square roots. There
are 31 return values lying outside of the 5% VaR. This amounts to an
empirical coverage of 62%.
Mixture-GARCH(1,1) Indirect-VaR:
VaR2t = zt 62713 × 10−4 + 01896rt2−1 + 005369VaR2t −1 )
+ (1 − zt )(66137 × 10−4 + 025093rt2−1 + 06733VaR2t −1 ), (19)
FIGURE 2 Plot of Hang Seng Index (Jan. 2, 1997–Dec. 31, 2003). The upper panel is the plot
of the real series and the lower panel is the plot of the log-return series.
636
P. L. H. Yu et al.
where zt = 1 with probability 0.1435 and zt = 0 with probability 0.8565.
We also choose the negative square root for VaRt . There are 28 return
values lying outside of the 5% VaR. This amounts to an empirical coverage
of 56%.
Since S&P500 is an index of a mature stock market, some performance
checking of our model in less developed markets and their indices are
also needed. We choose three of these indices, Hang Seng index (Fig. 2)
of Hong Kong, Nikkei index of Japan and the Nasdaq index for stocks
of high technology industries. The periods for these three indices are
seven years, from the beginning of 1997 to the end of 2003. Because of
the difference between the holidays of these markets, the length of the
three series are slightly different. Note that the period we chose covered
the Asian financial crisis, this posed a challenge to our model. As usual,
we first check the ACF and PACF for these three series and find that
the log-return series of Hang Seng and Nikkei indices contain apparent
autoregressive components. From the plot of ACF’s and PACFs of the three
indices, we found that Hang Seng and Nikkei indices should have their
linear components removed before model fitting since our model only
focuses on the residuals. After removing the autoregressive components,
we fit the residuals to our indirect-VaR models. The linear part for Hang
Seng index is estimated to be rt − 00381rt −1 + 00665rt −2 − 01124rt −3 +
0061rt −4 = xt . Then we fit to xt a TGARCH indirect-VaR model:
VaR2t =
135 × 10−5 + 0127xt2−1 + 0712 VaR2t −1 ,
142 × 10−5 + 0229xt2−1 + 0771 VaR2t −1 ,
xt −1 ≤ 1126 × 10−2 ,
xt −1 > 1126 × 10−2 We use the last 500 residuals to check the performance of our model. We
do not update our model when new data points feed in and find that there
are 21 points outside the 5% VaR. This amounts to an empirical coverage
of 4.2%. We also fit to the xt a mixture GARCH indirect-VaR model:
VaR2t = zt 203 × 10−4 + 0141xt2−1 + 0671 VaR2t −1
+ (1 − zt ) 133 × 10−5 + 0213xt2−1 + 0692 VaR2t −1 (20)
The Bernoulli variable zt = 1 with probability 0.2895. Similar to TGARCH,
the out-sample checking shows that there are 29 points outside the 5%
VaR. This amounts to an empirical coverage of 5.8%.
The Nikkei index also shows a significant autoregressive component.
We skip the time series and ACF plots to save space. The linear component
On Some Models for Value-at-Risk
637
is estimated to be rt + 00462rt −1 + 0687rt −2 = xt . Fitting to the residuals xt
a TGARCH indirect-VaR model gives:

302 × 10−5 + 190 × 10−1 xt2−1 + 808 × 10−1 VaR2t −1 ,




xt −1 ≤ −543 × 10−2 ,
VaR2t =

602 × 10−5 + 213 × 10−1 xt2−1 + 643 × 10−1 VaR2t −1 ,



xt −1 > −543 × 10−2 ,
and we find that there are 22 points lying outside the 5% VaR, about 4.4%
coverage. If we fit to the residuals xt a mixture-GARCH indirect-VaR model,
we got,
VaR2t = zt (4127 × 10−5 + 0168xt2−1 + 0687 VaR2t −1 )
+ (1 − zt )(4067 × 10−5 + 0315xt2−1 + 0415 VaR2t −1 ),
where zt = 1 with probability 0.4129 and find that 23 observations lie
outside the 5% VaR, about 4.6% empirical coverage.
The ACF of the Nasdaq does not show any apparent pattern and again
we skip the time series and ACF plots to save space. Therefore, we fit to the
log-return data our TGARCH and mixture GARCH indirect-VaR models
directly. The TGARCH indirect-VaR model is

136 × 10−5 + 213 × 10−1 rt2−1 + 876 × 10−1 VaR2t −1 ,




rt −1 ≤ −1925 × 10−3 ,
VaR2t =

124 × 10−5 + 1791 × 10−1 rt2−1 + 732 × 10−1 VaR2t −1 ,



rt −1 > −1925 × 10−3 There are 24 points outside the 5% VaR which amount to a 4.8% coverage.
The mixture GARCH model for Nasdaq is
VaR2t = zt 1337 × 10−5 + 02018rt2−1 + 05279 VaR2t −1
+ (1 − zt ) 1309 × 10−5 + 01364rt2−1 + 06183 VaR2t −1 ,
where zt = 1 with probability 0.4220. There are 27 points of the last 500
out-sample lying outside the 5% VaR, about 5.4% empirical coverage.
For comparison purpose, we also fitted some traditional VaR models
for all these series. These models are the GARCH model with normal
innovation, GARCH model with student-t innovation, integrated GARCH
(IGARCH) model with normal innovation, IGARCH model with studentt innovation and the popular RiskMetrics method. We summarize the
results in Table 2. From the table, in terms of coverage accuracy, the
mixture indirect-VaR models is among the best for two indices (S&P500
638
P. L. H. Yu et al.
TABLE 2 Percentage of coverage of some commonly used models for the nominal 5% VaR
S&P500
HSI
Nikkei
Nasdaq
GARCH
3.2%
3.2%
5.8%
5.4%
S&P500
HSI
Nikkei
Nasdaq
RiskMetrics
4.2%
5.6%
5.8%
4.0%
GARCH(t )
2.8% (11.696)
3.4% (6.882)
6.0% (11.038)
5.6% (12.853)
IGARCH
4.0%
3.4%
4.6%
5.4%
IGARCH(t )
3.6% (10.684)
4.0% (6.127)
4.6% (8.977)
5.6% (12.500)
Mixture indirectVaR model
5.6%
5.8%
4.6%
5.4%
Threshold indirect-VaR
6.2%
4.2%
4.4%
4.8%
Inside the parentheses are the degrees of freedom of the student-t distributions.
and Nikkei) and second best in the other two indices (HSI and Nasdaq).
The threshold indirect-VaR is the second best in two indices (HSI and
Nikkei) but is the best in the case of Nasdaq. The RiskMetrics is better than
the mixture indirect-VaR model in one index but is inferior in three other
cases. The IGARCH model ties with the mixture indirect-VaR in one index
(Nikkei) but performs not as well in the other three. On the other hand,
the GARCH-type models perform the worst in all cases except that the
GARCH with normal innovation performs the second best in the Nasdaq
case. Table 3 shows the ranks of the results in Table 2. In the case of ties,
the average of the respective ranks is used. The smaller the sum of ranks
the better is the performance. According to the sum of ranks, it seems
that the mixture indirect-VaR model (sum of ranks is 8.5) gives the most
stable and best performance in terms of coverage accuracy and it could be
a valuable toolkit for financial risk managers. The threshold indirect-VaR
TABLE 3 Ranks for some commonly used models for the nominal 5% VaR
S&P500
HSI
Nikkei
Nasdaq
Sum
GARCH
6
7
55
3
215
S&P500
HSI
Nikkei
Nasdaq
Sum
RiskMetrics
2
1
55
7
155
GARCH(t )
7
55
7
55
25
IGARCH
3
55
2
3
135
Threshold indirect-VaR
4
2.5
4
1
11.5
IGARCH(t )
5
4
2
55
165
Mixture indirectVaR model
1
2.5
2
3
8.5
639
On Some Models for Value-at-Risk
TABLE 4 Percentage of coverage for some commonly used models for the nominal
1% VaR
S&P500
HSI
Nikkei
Nasdaq
GARCH
0.60%
0.60%
1.60%
1.60%
S&P500
HSI
Nikkei
Nasdaq
RiskMetrics
0.80%
1.20%
1.20%
0.80%
GARCH(t )
0.40% (11.696)
0.60% (6.882)
1.20% (11.038)
1.00% (12.853)
IGARCH
0.80%
0.60%
0.80%
1.00%
Threshold indirect-VaR
1.20%
0.80%
0.80%
0.80%
IGARCH(t )
0.80% (10.684)
0.80% (6.127)
1.20% (8.977)
1.00% (12.500)
Mixture indirectVaR model
1.00%
1.00%
0.80%
0.80%
Inside the parentheses are the degrees of freedom of the student-t distributions.
model performs the second best (sum of ranks is 11.5) and can be another
good choice for risk management.
Risk managers also like to calculate the VaR at the 1% level. Table 4
shows the 1% results obtained by reestimating all the models at the 1%
level. Table 5 shows the ranks of the results at the 1% level. From the table,
in terms of coverage accuracy, the mixture indirect-VaR models are still
among the best for two indices (S&P and HSI). The average performance
of the mixture indirect-VaR models according to the sum of the ranks is
still superior to the other models. Since the data points lying outside the
1% VaR are very few, missing one datum point could change the accuracy
dramatically. Therefore, in the 1% VaR case, that the performance of the
threshold indirect-VaR model is not outstanding does not mean this model
is not good. Note that ranks of the threshold indirect-VaR model are stable,
this model could still be a good choice for risk management.
TABLE 5 Ranks for some commonly used models for the nominal 1% VaR
S&P500
HSI
Nikkei
Nasdaq
Sum
GARCH
6
6
7
7
26
S&P500
HSI
Nikkei
Nasdaq
Sum
RiskMetrics
35
3
35
5
15
GARCH(t )
7
6
35
2
185
IGARCH
35
6
35
2
15
Threshold indirect-VaR
3.5
3
3.5
5
15
IGARCH(t )
35
3
35
2
12
Mixture indirectVaR model
1
1
3.5
5
10.5
640
P. L. H. Yu et al.
7. CONCLUSION
In this article, we extended the CAViaR model of Engle and Manganelli
(2004) using a threshold or a mixture approach, respectively. The quantile
regression method is used in the estimation for TGARCH and mixtureGARCH indirect-VaR models. A full modelling methodology is suggested
in both the threshold and mixture cases. From the results of the
simulations and the post-sample prediction results in the real-data studies,
we find that these new extensions are potentially useful. One advantage is
that we can use the approaches to study the nonlinear phenomenon due to
positive and negative shocks in the financial market and the performance
of our model appears to be more stable than other methods.
In real life, financial portfolio often contain many financial assets. If we
regard the portfolio as a single asset, then our models can be applied to it.
As pointed out by Berkowitz and O’Brien (2002) and Kuester et al. (2006),
this approach may actually provide better VaR forecasts than a multivariate
approach and can serve as a benchmark for evaluating and identifying an
appropriate multivariate model.
ACKNOWLEDGMENTS
Philip L. H. Yu would like to thank a small project fund from
the University of Hong Kong for partial support. W. K. Li would like
to thank the Croucher Foundation for awarding a Senior Research
Fellowship (2003–2004) and the Hong Kong Research Grant Council grant
HKU7036/06P for partial support. The authors would also like to thank
two referees for their valuable advices.
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