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Uncertainty in Value-at-Risk Estimatesunder Parametric and Non-parametric
Modeling
Tatiana Miazhynskaia and Wolfgang Aussenegg∗
Vienna University of Technology
Department of Finance and Corporate Control
Favoritenstrasse 9-11, A-1040 Vienna, Austria
fax: +43 1 58801-33098
Aussenegg: [email protected], phone:+43 1 58801-33082
Miazhynskaia: [email protected], phone:+43 1 58801-33087
March 2005
∗Corresponding author. In case of acceptance Wolfgang Aussenegg will present the paper andboth authors will attend the conference. We are grateful to participants of the 18th Workshop ofthe Austrian Working Group on Banking and Finance (Innsbruck, 2004) for their helpful commentsand thank Reuters GesmbH, Vienna, for providing data.
1
Uncertainty in Value-at-Risk Estimates under
Parametric and Non-parametric Modeling
March 2005
Abstract
This study evaluates a set of parametric and non-parametric Value-at-Risk(VaR) models that quantify the uncertainty in VaR estimates in form of a VaRdistribution. We propose a new VaR approach based on Bayesian statistics in aGARCH volatility modeling environment. This Bayesian approach is comparedwith other parametric VaR methods (quasi-maximum likelihood and bootstrapresampling on the basis of GARCH models) as well as with non-parametrichistorical simulation approaches (classical and volatility adjusted). All thesemethods are evaluated based on the frequency of failures and the uncertaintyin VaR estimates.
The parametric methods are found equal in their performance to produceadequate VaR estimates, while the Bayesian approach results mostly in a smallerVaR variability. The non-parametric methods imply more uncertain 99%-VaRestimates, but show good performance with respect to 95%-VaR estimates.
K eywords: Value-at-Risk, Historical Simulation, GARCH, Bayesian analysis, Bootstrap resampling
JEL classification code: C11, C50, G10
2
1 Introduction
In the last ten years the Value-at-Risk (VaR) concept has become world-wide the
major tool in market risk management. As proposed in 1995 by the Basle Committee
on Banking Supervision, banks are now (in most countries) allowed to calculate capital
requirements for their trading books based on a VaR concept. A large amount of
research effort has been and is devoted to produce better point VaR estimates. But a
good risk management requires not only a point VaR estimate but also some measure
of its accuracy. For risk managers it is therefore also important to know how precise
their VaR estimates are.
The variability in VaR estimates can have different sources. The first one is due to
data variability and structural changes in the data. Further, VaR model uncertainty
and uncertainty due to poorly characterized parameters in a specified mathematical
model are reflected in the VaR calculation.1 The aim of this paper is to compare
different methods to quantify the uncertainty in VaR estimates in form of VaR dis-
tributions.
The literature suggests to compute the uncertainty in VaR estimates in the form
of VaR confidence intervals, constructed mostly based on Monte Carlo simulations
and (or) some assumption about the profit and loss (P/L) distribution. Some authors
derive analytical formulas for VaR confidence bands (see e.g. Chappell and Dowd
(1999) for normal and Jorion (1996) for normal and Student-t distributed returns)
or VaR distributions under normality (Dowd, 2000a). Other authors show how to
estimate VaR confidence bands using the theory of order statistics (Dowd, 2001) or a
neural network framework (Prinzler, 1999).2
An important result documented by Bams, Lehnert and Wolff (2003) is that more
sophisticated tail-modeling approaches are associated with higher uncertainty in VaR
estimates. Jorion (1996) and Dowd (2001) report in this context that VaR confidence
bands of Student-t distributed returns are always larger than for normal distributed
1Dowd (2000b), e.g., shows in a theoretical example how VaR confidence bands increase withparameter uncertainty.
2Haas and Kondratyev (2000) show how VaR confidence bands may be obtained in the case of ageneralized Pareto distribution.
3
returns. In addition, the uncertainty in VaR estimates also depends on the sample size,
i.e. the number of observations used to calculate the VaR (see e.g. Dowd (2000a) and
Dowd (2001)). For normal distributed returns and a 95%-VaR point estimate Dowd
(2000a) reports a 95% confidence band of ± 20% for a sample size of 100 returns and
± 6% for a sample size of 1000 returns.
Our paper extends the existing literature on uncertainty in VaR estimates in sev-
eral ways: First, we compare parametric VaR models with non-parametric ones. As
a basis for our parametric VaR modeling we employ GARCH models for conditional
return distributions using a normal and Student-t distributional specifications.
Second, we propose a new approach based on Bayesian statistics to calculate VaR
distributions and the corresponding VaR point estimates, and exhibit how to calibrate
this VaR model in a GARCH environment to real financial data. In the Bayesian ap-
proach, point estimates for parameters are replaced by distributions in the parameter
space, which represent our knowledge about values of the parameters, and the com-
plete posterior distribution of the parameters can be used for further analysis. We
compare this Bayesian VaR approach with other parametric VaR methods, like quasi-
maximum likelihood and bootstrap resampling of GARCH models as well as with
non-parametric historical simulation approaches (classical and volatility adjusted).
All these methods are evaluated based on the frequency of failures (i.e. the frequency
of losses exceeding the VaR), and the uncertainty in VaR estimates.
And third, for every trading day we compute for all the methods mentioned above
not only a VaR point estimate but its whole distribution, which quantifies the one-day
VaR variability. This is important, as VaR distributions tend to differ significantly
from normality. Confidence bands are therefore often not sufficient to correctly eval-
uate the uncertainty in VaR estimates.
To check how stable the relative behavior of the VaR models is we use in our empir-
ical analysis financial data of different types, like foreign exchange rates, commodities,
stock indices, individual shares and interest rate sensitive instruments.
Our empirical results reveal that the uncertainty in VaR estimates highly depends
on the volatility level of the market. We can further document that this uncertainty
4
tends to increase the more we are going into the tails of return distributions. In
addition, non-parametric VaR models generate a much larger variability in 99%-VaR
estimates compared to parametric approaches. Between the parametric methods, the
Bayesian approach is associated with a lower uncertainty in VaR predictions. The
proportion of failure test finds no differences between the Bayesian, quasi-maximum
likelihood and bootstrapping estimation methods. The heavy-tailed GARCH-T model
provides in all considered cases an adequate fit, whereas the Gaussian GARCH-N
model tends to generate in some cases too low VaR estimates.
The paper is organized as follows. The following section briefly describes the
underlying basic VaR concept. Section 3 presents the Bayesian framework as well
as quasi-maximum likelihood and bootstrap GARCH frameworks. In the Section 4
we describe the non-parametric approaches and Section 5 describes the data used in
our empirical analysis. The empirical results are discussed in Section 6 and Section 7
concludes the paper.
2 Value at Risk
An important tool to quantify the market risk of a portfolio is the Value-at-Risk (VaR)
methodology. VaR is defined as the maximum potential loss in value of a portfolio of
financial instruments with a given probability over a certain horizon. In simpler words,
it is a number that indicates how much a financial institution can loose with some
probability over a given time horizon. The great popularity that this instrument has
achieved among financial practitioners is essentially due to its conceptual simplicity:
VaR reduces the (market) risk associated with any portfolio to just one number, that
is the loss associated with a given probability.
VaR measures can be used also to evaluate the performance of risk takers and for
regulatory requirements. Providing accurate estimates is of crucial importance. If the
underlying risk is not properly estimated, this may lead to a sub-optimal capital allo-
cation with consequences on the profitability or even financial stability of institutions
(Manganelli and Engle, 2001).
5
From a statistical point of view, the VaR computation requires the estimation of
a quantile of the return distribution. As soon as the probability distribution of the
returns is specified, the VaR is calculated using the p% percentile of this distribution.
The VaR corresponding to the p% percentile can be defined as the amount of
capital to cover expected losses on (100-p)% of market scenarios. We therefore use
the notation (100-p)%-VaR. For more information on VaR and risk management issues
we refer to Duffie and Pan (1997), Dowd (1998), Wilson (1998), Brooks and Persand
(2000), McNeil and Frey (2000) and the book of Jorion (2000).
We discuss the 99%- and 95%-VaR levels. The first level has been selected by the
Basel Committee on Banking Supervision as the focus of attention, although the first
percentile of a distribution is more difficult to estimate than the fifth; and the second
level is employed by the popular RiskMetrics methodology of JP Morgan.
The quality of the VaR calculations can be controlled by backtesting: VaR predic-
tions are compared with the corresponding realized profit and losses. From the number
of cases where the losses exceed the VaR predictions one can evaluate, whether the
VaR estimates represent the chosen quantile.
3 Parametric VaR Models
VaR models that are based on standard statistical distributions determine the condi-
tional return distribution and estimate the standard deviation (or covariance matrix)
of the returns of a asset. For that reason good volatility forecasts are an integral part
of good VaR models. To find the VaR itself, one can take the corresponding percentile
of the predictive distribution of the returns.
3.1 Modeling the Volatility Process
One of the most widely used volatility models is the GARCH model (Bollerslev, 1986)
for which the conditional variance is governed by a linear autoregressive process of
past squared returns and variances. In our study we use the classical GARCH(1,1)
model with the conditional normal distribution and a AR(1) mean specification (for
6
short, we omit the specification AR(1) further from our model designations):
GARCH-N :
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
rt = a0 + a1rt−1 + et, t = 1, 2, . . . , N
et | It−1 ∼ N(0, ht),
ht = α0 + α1e2t−1 + β1ht−1,
with the restrictions α0, α1, β1 ≥ 0 to ensure σ2t > 0. N(0, ht) denotes the Gaussian
distribution with mean 0 and variance ht; It−1 denotes time series history up to time
t − 1. Stationarity in variance imposes that α1 + β1 < 1.
One well-known extension of the GARCH model above is to substitute the condi-
tional normal density by a Student-t density in order to allow for excess kurtosis in
the conditional distribution (see Bollerslev (1987) for details). The full specification
of our AR(1)-GARCH(1,1)-t model is
GARCH-T :
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
rt = a0 + a1rt−1 + et, t = 1, 2, . . . , N
et | It−1 ∼ Tν(0, ht),
ht = α0 + α1e2t−1 + β1ht−1,
where Tν(0, ht) denotes the Student t-distribution with mean 0, variance ht and ν
degrees of freedom. The new parameter - degrees of freedom ν - determines, among
other characteristics, the kurtosis of the conditional distribution.
The standard GARCH model based on a normal distribution captures several
”stylized facts” of asset return series, like heteroskedasticity (time-dependent con-
ditional variance), volatility clustering and excess kurtosis. The GARCH-T model
covers also fat tails in the conditional distribution of the returns.
The parameter vector to be estimated in the GARCH-N model is
θ1 = (a0, a1, α0, α1, β1) and the likelihood for a sample of N observations Y =
(r1, r2, . . . , rN) can be written as
L(Y | θ1) =N∏
t=1
1√2πσ2
t
exp
{− e2
t
2σ2t
}.
7
Under the assumption of a Student t-distribution, the likelihood for the sample Y is
L(Y | θ2) =N∏
t=1
Γ(ν+12
)
Γ(ν2)√
π(ν − 2)σ2t
(1 +
e2t
(ν − 2)σ2t
)−(ν+1)/2
,
where the parameter vector to be estimated is θ2 = (a0, a1, α0, α1, β1, ν).
Note that the standard formula for the t-density has been modified by the scale
factor ht(ν−2)ν
, where the degree-of-freedom adjustment is designed so that ht is exactly
equal to the conditional variance of the returns rt.
3.2 Estimation of the models
To estimate the models and to quantify the uncertainty in model parameters, we will
consider two fundamentally different frameworks: classical (maximum likelihood) and
Bayesian.
From a Bayesian viewpoint, there is no such thing as a true parameter value.
Point estimates for parameters are replaced by distributions in the parameter space,
which represent our knowledge about values of the parameters; and the complete pos-
terior distribution of the parameters can be used for further analysis. When models
are estimated in the classical manner, the uncertainty in model parameters is esti-
mated in two ways: within a quasi-maximum likelihood approach and by a bootstrap
resampling.
3.2.1 Bayesian approach
Basics of Bayesian inference. The distinctive feature of the Bayesian framework
(compared to the classical analysis) is its use of probability to express all forms of
uncertainty. In such a way, in addition to specifying a stochastic model for the ob-
served data Y given a vector of unknown parameters θ, we suppose that θ is a random
quantity as well. The dependency of Y on θ is defined in the form of the likelihood
8
L(Y |θ). Our subjective beliefs we may have about θ before having looked at the data
Y are expressed in a prior distribution π(θ).
At the center of the Bayesian inference is a simple and extremely important ex-
pression known as Bayes’ rule:
p(θ|Y ) =L(Y |θ)π(θ)∫L(Y |θ)π(θ)dθ
. (1)
Thus, having observed Y , our initial views about θ are updated by the data to get
the distribution of θ conditional on Y . It is called the posterior distribution of θ.
For many realistic problems, evaluation of p(θ | Y ) is analytically intractable, so
numerical or asymptotic methods are necessary. In this article we adopt the Markov
chain Monte Carlo (MCMC) sampling strategies as the tool to obtain posterior sum-
maries of interest. The idea is based on the construction of an irreducible and aperiodic
Markov chain with realizations θ(1), θ(2), . . . , θ(t), . . . in the parameter space, equilib-
rium distribution p(θ|Y ), and a transition probability K(θ′′, θ′) = π(θ(t+1) = θ′′ | θ(t) =
θ′), where θ′ and θ′′ are the realized states at time t and t + 1, respectively. Under
appropriate regularity conditions, asymptotic results guarantee that as t → ∞, θ(t)
tends in distribution to a random variable with density p(θ|Y ). For the underlying
statistical theory of MCMC see Tierney (1994).
The most known MCMC procedures are Gibbs sampling, when we have completely
specified full conditional distributions, and the Metropolis-Hastings (MH) algorithm
which provides a more general framework. For an introduction on MCMC simulation
methods we refer to Chib and Greenberg (1996) and Geweke (1999).
Bayesian estimation of the GARCH models. Due to the recurrent structure of
the variance equation in the GARCH model none of the full conditional distributions
(i.e., densities of each element or subvector of θ given all other elements) is of a known
form from which random numbers could easily be generated. There is no property of
conjugacy for GARCH model parameters. Therefore, we use the Metropolis-Hastings
algorithm which gives the easiest sampling strategy yielding the required realization
9
of p(θ|Y ) (see, e.g., Kim, Shephard and Chib (1998), Muller and Pole (1998) and
Nakatsuma (2000)).
To sample the posterior, we adopt the random walk MH algorithm with the
Gaussian candidate density:
1. Generate a candidate draw θ(new) ∼ N(θ(old), c);
2. Accept θ(new) with probability
α(θ(old), θ(new)) = min
{L(Y |θ(new))π(θ(new))
L(Y |θ(old))π(θ(old)), 1
};
3. Repeat until a sufficiently large sample is collected.
The variance c of the proposal distribution was tuned such as to be near the optimal
acceptance rate in the range of 25-40% (Carlin and Louis, 1996).
Simulations are performed for a single-parameter block. After initial exploratory
runs, correlations between the parameters are calculated and the blocked update of
highly correlated parameters is implemented in order to increase the efficiency and to
improve the convergence of the Markov chain. Moreover, it appears that it is more
computationally convenient to work with a logarithmic transformation of the variance
parameters (α0, α1, β1) onto a subvector taking values in (−∞, +∞). For more details
on the simulation scheme see Miazhynskaia and Dorffner (2005).
As the priors, we use the Gaussian priors for the mean parameters and the log-
normal priors for the variance parameters. All priors are centered at the MLE of
the corresponding parameter and with a variance 10 times larger than the squared
standard MLE parameter error after the maximum likelihood estimation:
a0 ∼ N(aML0 , 10 · εML
a0), a1 ∼ N(aML
1 , 10 · εMLa1
)
α0 ∼ logN(log ˆαML0 , 10 · εML
α0), α1 ∼ logN(log ˆαML
1 , 10 · εMLα1
)
β1 ∼ logN(log ˆβML1 , 10 · εML
β1)
ν ∼ Exp(0.1).
In this way, such priors turned out to be practically non-informative because their ef-
10
fective range is about 10 times larger than the effective range of the resulting posterior
density.
The Bayesian approach is often subject to criticism because of the ’subjective’
choice of the parameter priors. We repeated the Bayesian procedure, varying the
prior informativity, and found no significant influence on the results.
Note that the need to impose stationarity conditions in a Bayesian context is not
well understood and not broadly accepted (see Vrontos, Dellaportas and Politis (2000)
for further comments). In our analysis, we relaxed these conditions and just checked
the stationarity of the GARCH models posteriori.
3.2.2 Quasi-Maximum Likelihood approach
In this approach we follow Bams et al. (2003) to reflect parameter uncertainty in
VaR calculations. We begin with the maximum likelihood estimate (MLE) of the
model parameters θML and assume an asymptotic Gaussian distribution for the model
parameters
θ ∼ N(θML, Θ). (2)
The uncertainty about the parameters is quantified by the estimated covariance matrix
(Davidson and MacKinnon, 1993)
Θ = H−1(GT G)H−1
where H denotes the Hessian matrix evaluated at θML. G is the score matrix (∂l(yt|θ)∂θi
)t,i
evaluated at θML and l(Y |θ) is the logarithmic value of the likelihood function L(Y | θ).We are using the parameter distribution in (2) to quantify the uncertainty in the VaR.
3.2.3 Bootstrap resampling
The third method to assess the uncertainty in the parameter estimation is the boot-
strap resampling scheme by Pascual, Romo and Ruiz (2000). Once the maximum
likelihood estimate of the model parameters is found, say θML = (a0, a1, α0, α1, β1),
the conditional variances are estimated by the GARCH process
ht = α0 + α1(rt−1 − µt−1)2 + β1ht−1, t = 2, . . . , N,
µt = a0 + a1rt−1
11
with h1 = α0
1−α1−β1, the estimated unconditional variance.
The standardized residuals are then calculated as
εt =rt − µt√
ht
, t = 1, . . . , N. (3)
To mimic the structure of the original series, bootstrap replicates {r∗1, r∗2, . . . , r∗N} are
obtained from the following recursion:
h∗t = α0 + α1(r
∗t−1 − µ∗
t−1)2 + β1h
∗t−1,
µ∗t = a0 + a1r
∗t−1,
r∗t = µ∗t +
√h∗
t · ε∗t , t = 1, . . . , N,
where ε∗t are random draws from the empirical distribution of the centered residuals
εt − ¯εt (see equation (3)) and the initial values are h∗1 = h1 and µ∗
1 = mean(rt).
Once the bootstrap pseudo series of returns {r∗1, . . . , r∗N} are generated, one can
compute the bootstrap MLE θ∗BS on this data. This procedure, which generates
pseudo returns and then estimates θ∗BS), is repeated until a sufficiently large sample
of parameter estimates θ∗BS is collected.
3.3 Predictive VaR distribution
In estimating the parametric VaR models using the methods described in Section
3.2, we get not a point parameters estimate, but the whole (empirical) parameter
distribution, incorporating the model (parameter) uncertainty. This distribution is
used to quantify the uncertainty in the VaR estimates.
Consider M samples of the parameters {θ(m)}Mm=1 from the distribution of the pa-
rameters. For every m, m = 1, . . . , M , we compute the predictive return distribution
according to our GARCH specification (one step ahead). Then we calculate the corre-
sponding percentile of this predictive distribution which we take as a measure of VaR.
Altogether we get a sample of M values for the VaR estimate for every day. This
procedure is repeated for all days in the test set. In this way, instead of arriving at
12
one point VaR estimate, we now have an entire sample of VaR predictions for every
day in the test set.
4 Non-parametric VaR Models
In addition to the parametric VaR models discussed in section 3 we also apply - mainly
for comparison purposes - the historical simulation approach. It is a non-parametric
VaR model that is widely used by financial institutions to compute VaR estimates.
As non-parametric methodology the historical simulation approach does not require
any assumptions about the return distribution of risk factors or P/Ls. It is solely
based on the historical return distribution of the corresponding risk factors. This
implies, e.g., that fat tails are automatically included in VaR estimates. The VaR
at the 99% confidence level (99%-VaR) can be defined as the 1% percentile of the
portfolio’s empirical return distribution (r∗1%).
The advantages of the classical historical simulation approach are especially that
it is conceptually simple, easy to implement and that it does not depend on paramet-
ric assumptions about return distributions. One of it’s main disadvantages is that
volatility clustering effects are not captured. This means that if the current return
volatility is above (below) the average return volatility in the sample period, the his-
torical simulation approach will produce a VaR estimate that is too low (too high)
for the actual risk.
We therefore use in addition to the classical approach a volatility adjusted version
proposed by Hull and White (1998). In this second approach all daily returns in the
sample period (in our case about two years) are adjusted by comparing the return
volatility of each trading day in the sample period with the current volatility (i.e.
the volatility at the end of the sample period). The return r(th) for a particular
(historical) trading day th in the sample period is therefore weighted by the ratio of
the volatility forecast σ(t0) for the current trading day t0 and the volatility forecast
σ(th) for the historical trading day th. The current trading day t0 is the trading day
for which we want to estimate the VaR. The volatility adjusted return r(th)adj for the
13
(historical) trading day th is therefore defined as
r(th)adj = r(th) · σ(t0)
σ(th), (4)
where σ(th) and σ(t0) are EWMA (exponentially weighted moving average) forecasts
of the return volatility for day th and t0, respectively.3 After adjusting all returns in
the sample period, the (100-p∗)%-VaR is defined as the p∗ percentile of the distribution
of adjusted returns.
To measure the uncertainty in VaR estimates generated by our two historical
simulation methods, VaR distributions are estimated using a bootstrapping approach.
This approach involves for each trading day random resampling, with replacement,
from the return sample (past two years).4 This resampling is done 1000 times for every
trading day yielding 1000 (artificial) return distributions and 1000 corresponding VaR
estimates for each trading day. The resulting VaR distribution function enables us to
analyze the uncertainty in our VaR estimates.5
5 Data and Empirical Research Design
In our empirical study daily returns of seven different financial assets are used. These
assets are: (i) a cash position in British Pound (GBP), (ii) a cash position in Japanese
Yen (JPY), (iii) a cash position in Swiss France (CHF), (iv) a position in Brent Crude
Oil delivery today (Brent),6 (v) a position in General Motors shares (GM), (vi) a
position in a portfolio exposed to the Standard and Poor’s 500 Stock Index (SP500),
and (vii) a position in a zero bond (ZB) with a (constant) maturity of one year.7 In
all cases we take the perspective of an investor whose home currency is the US-Dollar
(USD). Our total database starts in January 1997 and ends in December 2003. To
3The EWMA volatilities are estimated with a decay factor 0.97.4In the case of the volatility weighted historical simulation approach we resample from the volatil-
ity adjusted return sample generated according to equation (4).5For more details on how to best perform bootstrapping procedures see e.g. Dowd (2002).6Reuters RIC: QBRT7One year US-Libor rates are used to calculate zero bond prices.
14
generate VaR estimates, a training period of two years of past returns is used. The
years 1997 and 1998 are therefore only applied for training purposes. Our test period
starts on January 4th, 1999 (the first day for which VaRs are estimated) and ends on
December 31st, 2003. In total we compute daily VaR estimates for five years (1999 -
2003) or 1301 trading days. Daily closing prices are obtained from Reuters 3000 Xtra.
The daily return ri,t for day t and asset i is defined as
ri,t = lnPi,t
Pi,t−1, (5)
where Pi,t denotes the closing price of currency i on trading day t in USD.
For our parametric modeling the data is structured in the following way: Two
years are used as training set to estimate the models. Then the following quarter
is used as test period in which for every trading day VaR predictions are generated.
In the next step this segment is moved by one quarter, so that the test data are not
overlapping and we get continuous VaR estimates for our period of 5 years (see Figure
1).8 In this way, the parameters of the models are updated every quarter. The VaR
calculations are performed for every day in the corresponding test period according
to the estimated GARCH specification.
To generate VaR estimates based on our two historical simulation approaches a
rolling sample of two years is used. This sample is updated every trading day by one
observation. For VaR estimates based on the volatility adjusted historical simulation
approach we estimate in a first step for every day in our test period (1999 - 2003) an
EWMA volatility forecast based on daily returns from the last two years and a decay
factor of 0.97. To generate the VaR estimates for a particular day (starting with the
first trading day in 1999) equation (4) is used to weight all past returns.
8This procedure is motivated by significant computational time for model estimation.
15
6 Empirical Results
In this section we discuss our main empirical findings about VaR methods discussed
above. In short, these methods are:
non-parametric
models
historical simulation (HS) Method 1
adjusted historical simulation (HSA) Method 2
parametric
models
GARCH-N
model
Bayesian approach (BA) Method 3
Quasi-maximum likelihood (QML) Method 4
bootstrap (BS) Method 5
GARCH-T
model
Bayesian approach (BA) Method 6
Quasi-maximum likelihood (QML) Method 7
bootstrap (BS) Method 8
In a first step we want to demonstrate how the variability in returns influences the
predictive VaR distribution. In this respect Figures 2 and 3 exhibit the distributions
of the 99%- and 95%-VaRs for the JPY/USD position, predicted by our eight VaR
methods, for two trading days, January 8th, 2002, and March 8th, 2002, respectively.
VaRs are plotted in return scale. We use these two trading days to provide an im-
pression how our eight VaR methods react to (i) a small last return of around zero
(see Figure 2) and (ii) a large negative last return (see Figure 3).
First of all, Figures 2 and 3 document that the uncertainty in VaR estimates
strongly depends on the volatility in the market. A more volatile return environment
(as on March 8th, 2002) leads to significantly wider VaR distributions, i.e. VaR
point estimates are associated with a higher uncertainty (see Figure 3). This effect is
most pronounced in (more complex) models that react faster to volatility changes in
their VaR estimates, as it is the case for all our parametric models and the volatility
adjusted historical simulation approach. On the other hand, the classical historical
simulation approach does not react significantly on the one-day increase in the market
variability.
16
Another important finding of the analysis in Figures 2 and 3 is that VaR distribu-
tions typically deviate from normality. To test whether the deviation from normality
is significant a Jarque-Bera test of normality is performed. Table 1 presents the per-
centage of trading days for which we reject the null hypothesis of normality for the
predicted VaR distributions at the 5% significance level. This happens in nearly 100%
of all cases when non-parametric models are used and varies between 20% and 97%
for the parametric approaches. The heavy-tailed GARCH-T models tend to generate
more often non-Gaussian VaR distributions than GARCH-N models. Table 1 also
shows that VaR distributions for the 95% quantile tend to depart less from normality
than for the 99% quantile.
Figure 4 presents for the JPY/USD position actual daily losses (in percent) and
the corresponding 99%-VaR distributions (upper panel) and the 95%-VaR distribu-
tions (lower panel) predicted by the parametric Bayesian approach. The two other
parametric approaches (QML and Bootstrap resampling) deliver similar results and
are therefore not presented.
The pictures show how sensitive the VaR reacts to movements in the return series.
While there is no significant difference between the GARCH-N and the GARCH-
T models concerning the 95%-VaR, GARCH-T provides larger 99%-VaR estimates
than GARCH-N. This evidence is consistent with the fat tailed nature of GARCH-T
models.
The 95% confidence interval of the predicted VaR distributions can be taken as a
measure of uncertainty in VaR estimation. One can see that the uncertainty increases
the further we are going into the tails of a return distribution (see Figure 4). In
addition, the dispersion of the 99%-VaR estimates is much larger than for the 95%-
VaR estimates, and the GARCH-T model generates wider VaR confidence intervals
than the Gaussian GARCH model.
Figure 5 provides the evidence for the two historical simulation approaches. The
volatility adjusted historical simulation approach (HSA) tends to generate smaller
VaR distributions for 99%-VaR estimates than the classical historical simulation ap-
proach (HS), i.e. it implies lower uncertainty in VaR estimates. So if there is volatility
17
clustering, the volatility adjusted historical simulation approach generates less VaR
uncertainty and seems in this sence to outperform the classical historical simulation
approach. In this context, Figure 5 also reveals that the volatility adjusted historical
simulation approach reacts much stronger to volatility changes in returns than the
classical historical simulation approach (HS) does.
Figures 4 and 5 indicate further that the parametric models produce less uncer-
tainty in VaR estimates compared to the historical simulation models. This conclu-
sion is also supported by the results in Table 2. This table presents VaR uncertainty
characteristics in form of the relative standard deviation of the 99%- and 95%-VaR
predictive distributions averaged over all test points. The relative standard devia-
tion is defined as the absolute standard deviation normalized by the mean of the
corresponding VaR distribution (in percent).
First, in line with the evidence presented above, Table 2 shows that the variability
generated by non-parametric models is significantly larger for 99%-VaRs than for 95%-
VaRs. For VaR estimates generated by parametric models the difference between 99%
and 95%-VaR estimates is less pronounced. While the GARCH-T model exhibits a
slightly lower uncertainty in 95%-VaR estimates, the opposite is true for the GARCH-
N model.
Second, the non-parametric historical simulation models deliver on average, over
all data sets, more uncertainty in 99%-VaR estimates than the parametric models.
This discrepancy can not be observed for 95%-VaRs. The uncertainty in 95%-VaR
estimates generated by the non-parametric models is much lower and comparable with
the (best) parametric models.
Third, compared to the GARCH-T model, the normal GARCH model provides
lower uncertainty in the 99%-VaR estimates over all estimation methods. Thus,
the more complex heavy-tailed GARCH-T model results in wider VaR distributions.
These differences in the VaR predictive variability are not so pronounced for 95%-
VaRs. But, still, a simpler model (GARCH-N) tends to show lower VaR standard
deviations.9
9This observation is in line with evidence provided in the literature (see, e.g., Bams et al. (2003),Jorion (1996) or Dowd (2001)).
18
Fourth, within the class of parametric models, the Bayesian framework results in
a smaller VaR variability, followed by the Quasi-maximum likelihood approach and
the Bootstrap resampling.
Besides a low variability in VaR estimates, a good VaR approach should also
generate proportion of failures comparable with the chosen quantile. In a second step
we therefore compare the estimated VaRs with actual losses to determine whether
the VaR estimates represent the chosen quantile properly. A well-known evaluation
method is the proportion of failures test, discussed by Kupiec (1995). This test
examines the frequency with which losses greater than VaR estimates are observed.
The outcome of the binomial event ”success-failure” is distributed as a series of draws
from an independent Bernoulli distribution and the verification test is based on the
proportion of failures (PF) in the sample. Ideally, the frequency of failures, i.e. the
number of trading days where the actual loss exceeds the predicted (100− p∗)%-VaR
level, should be close to p∗%. Following Kupiec (1995), we apply a likelihood ratio test
to examine whether the observed frequency deviates significantly from the predicted
level.
Since we have not a point VaR estimate but its whole probability distribution, a
question appears which statistic to take in order to make inference about adequacy of
the VaR models. We calculate the number of violations for the mean and the median
as well as the 5%, 20%, 40%, 60%, and 80% percentiles of the VaR predictive distri-
butions. The proportion of failures and the corresponding p-values of the likelihood
ratio test for the mean and median statistics are given in Tables 3 and 4, and p-values
of additional quantiles are plotted in Figures 6 to 8. Proportion of failures outside
the region where a VaR model is considered adequate are marked bold (i.e. p-values
of 0.05 and below).
With respect to the 99%-VaR (see Table 3), the parametric GARCH-N modelling
is rejected for the GBP/USD, the Brent Crude Oil and the Zero Bond position. The
generated VaR estimates are significantly too low for these positions. For the Brent
Crude Oil position only percentiles of the corresponding VaR distributions below the
median deliver adequate VaR point estimates (see Figure 6). For the GBP/USD
19
position and the Zero Bond position we have to go even further into the left part
of the VaR distributions to find adequate VaR point estimates. Percentiles below
20% for the GBP/USD and below 5% for the Zero Bond position produce adequate
VaRs. This indicates that the assumption of conditional normality is in many cases not
adequate when modelling returns of financial assets. On the other hand, the GARCH-
T model passes the testing successfully for all seven positions. Note that there are no
significant differences between the three parametric VaR models. Furthermore, the
non-parametric historical simulation models mostly generate acceptable proportion of
failures for the 99%-VaR estimates if we take mean or median of VaR distributions
as point estimate.
In contrast to the evidence for the 99%-VaRs, fat tails in return distributions are
no longer relevant for 95%-VaR estimates. As Table 4 and the right panels of Figures
6-8 reveal, nearly no model generates too low VaRs.10 But in some cases, as for
General Motors position (the position with the highest return volatility) GARCH-N
models are too conservative (too high VaR).
Note that for the parametric methods the median and the mean statistics are in
all cases very similar, but this is not the case for the historical simulation models.
The VaR distributions in the latter case are mostly unsymmetric.
Overall we can summarize that, first, for 99%-VaR estimates the GARCH-N model
is often not adequat and the parametric GARCH-T model is better than the non-
parametric models. Both pass the proportion of failure test but the uncertainty in
VaR estimates is significantly higher for non-parametric models. Within the class of
parametric models the Bayesian estimation approach is preferable, as the variability
in VaR estimates tends to be lower.
Second, for the 95%-VaR the quality difference between the eight approaches is less
pronounced. As overall best models (adequate portion of failures and low variability
in VaR estimates) we can state the GARCH-T model (under Bayesian approach) and
the volatility adjusted historical simulation method.
10An exception is the GARCH-T Quasi-maximum likelihood approach for the Zero Bond position.
20
7 Conclusion
The Value-at-Risk (VaR) of a portfolio is (only) an estimate and is thus associated
with errors. The resulting uncertainty in VaR estimates can have different sources,
like the volatility level in the data or specific characteristics of particular VaR models.
For risk managers it is therefore important to know how large this uncertainty is and
which factors determine it.
The aim of this study is to analyze the magnitude of this uncertainty for a set
of parametric and non-parametric VaR models. Our parametric VaR modeling is
based on a GARCH framework for modeling volatility. Within the parametric mod-
eling we propose a new approach based on Bayesian statistics to calculate the VaR.
This approach generates - in contrast to other VaR models - in the first place a to-
tal VaR distribution for a particular trading day (instead of a VaR point estimate)
and is therefore a natural way to quantify the uncertainty in VaR estimates. We
compare this Bayesian VaR estimation framework with two other parametric VaR
estimation approaches, like quasi-maximum likelihood and bootstrap resampling of
GARCH models, as well as with non-parametric historical simulations (classical and
volatility adjusted).
The empirical part of this study is based on seven different financial assets with
a five year test period (1999-2003). In a first step we analyze the effect of the return
volatility on the uncertainty of VaR estimates. Our empirical results reveal that the
uncertainty in VaR estimates highly depends on the volatility level in the market. A
more volatile return environment leads to significantly wider VaR distributions, i.e.
VaR point estimates are associated with higher uncertainty. This effect is most pro-
nounced in (more complex) models that react faster to volatility changes, as it is the
case for our GARCH models and the volatility adjusted historical simulation method.
Another important finding is that VaR distributions typically deviate from normal-
ity. This is nearly always the case in our non-parametric models and varies between
20% and 97% for the parametric approaches. The heavy-tailed GARCH-T models
generate more often non-Gaussian VaR distributions than GARCH-N models.
21
We can further document that the uncertainty in VaR estimates tends to increase
the more we are going into the tails of our return distributions. This is especially the
case for the non-parametric models, where the dispersion of the 99%-VaR estimates is
much larger than for the 95%-VaR estimates. Furthermore, the uncertainty generated
by the non-parametric models is comparable with those of parametric models for 95%-
VaRs. But, with respect to 99%-VaR the non-parametric historical simulation models
deliver on average much more uncertainty in VaR estimates.
Compared to the GARCH-T model, the normal GARCH model shows lower uncer-
tainty in VaR estimates. This conclusion is stable over the two VaR percentiles (95%
and 99%). Thus, the more complex heavy-tailed GARCH-T model results in wider
VaR distributions. Within the three estimation frameworks, the Bayesian method
generates on average a smaller VaR variability and seems therefore to be more suit-
able than the other parametric models.
Within the class of non-parametric models the volatility adjusted historical simula-
tion approach generates a somewhat lower uncertainty in VaR estimates and therefore
tends to outperform the classical historical simulation approach.
The uncertainty in VaR estimates is of course not the only quality criteria. A
good VaR model should also represent the chosen quantile properly. A proportion
of failure test reveals that our GARCH-T model nearly always provides an adequate
fit, whereas the GARCH-N model tends to generate too low 99%-VaR estimates. We
found no significant differences between the three parametric estimation frameworks
with respect to the quality of VaR methods. The non-parametric models pass the
proportion of failure test in most cases and never generate too low VaRs.
Overall, our Bayesian VaR approach provides, compared to non-parametric and
other parametric VaR models, an adequate VaR framework with less uncertainty in
VaR estimates. This new approach can therefore be considered as an interesting
alternative to existing VaR methods.
Open questions for future research are how VaR distributions can be used in market
risk management, and how to account for VaR uncertainty in choosing traditional VaR
point estimates used to calculate capital requirements for financial institutions.
22
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24
Table 1:Percentage of trading days with non-Gaussian VaR distributions using a Jarque-Bera testat the 5% significance level. We use two non-parametric models (HS = classical historicalsimulation, HSA = volatility adjusted historical simulation) and three parametric models(BA = Bayesian approach, QMLE = Quasi-maximum likelihood approach, BS = Bootstrapresampling). The test period starts on January 4th, 1999 and ends on December 31st, 2003.The seven positions analyzed are: a cash position in British Pound (GBP), a cash position inJapanese Yen (JPY), a cash position in Swiss France (CHF), a position in Brent Crude Oildelivery today (Brent), a position in General Motors shares (GM), a position in a portfolioexposed to the Standard and Poor’s 500 Stock Index (SP500), and a position in a zero bond(ZB) with a (constant) maturity of one year. All results are based on prices in USD anddaily log. returns (see equation (5)).
Panel A: Percentage of trading days with non-Gaussian VaR distributions: 99%-VaR
Method JPY CHF GBP Brent GM SP500 ZB
HS 99.5 97.2 97.5 99.0 99.7 100.0 100.0HSA 96.5 99.9 97.6 98.3 98.1 100.0 97.6
GA
RC
H-N
BA 49.6 54.3 61.9 54.8 54.4 50.3 62.0QMLE 26.6 49.3 48.4 89.6 86.4 95.2 85.4BS 77.5 74.3 78.5 88.5 94.7 90.4 96.8
GA
RC
H-T
BA 94.9 96.9 97.6 89.6 91.1 83.7 78.7QMLE 57.0 70.0 63.0 86.9 81.6 87.4 87.4BS 83.1 76.6 85.3 90.8 88.1 89.4 98.6
Panel B: Percentage of trading days with non-Gaussian VaR distributions: 95%-VaR
Method JPY CHF GBP Brent GM SP500 ZB
HS 99.3 92.9 94.9 76.7 97.1 99.2 98.5HSA 91.4 84.1 96.0 95.2 91.3 91.2 96.3
GA
RC
H-N
BA 42.5 48.1 54.8 47.0 48.1 42.4 56.0QMLE 20.2 38.5 37.7 81.1 78.7 88.0 80.4BS 73.4 67.5 73.8 84.6 93.7 88.1 95.3
GA
RC
H-T
BA 52.1 53.7 53.0 65.4 67.1 52.3 74.6QMLE 48.5 72.5 35.5 95.8 85.0 93.4 91.6BS 89.8 65.0 75.3 87.9 83.5 90.5 97.3
25
Table 2:Standard deviation of the 99%- and 95%-VaR predictive distributions averaged over all testpoints. The relative standard deviation is defined as the absolute standard deviation dividedby the mean of the corresponding VaR distribution (in percent). We use two non-parametricmodels (HS = classical historical simulation, HSA = volatility adjusted historical simula-tion) and three parametric models (BA = Bayesian approach, QMLE = Quasi-maximumlikelihood approach, BS = Bootstrap resampling). The test period starts on January 4th,1999 and ends on December 31st, 2003. The seven positions analyzed are: a cash positionin British Pound (GBP), a cash position in Japanese Yen (JPY), a cash position in SwissFrance (CHF), a position in Brent Crude Oil delivery today (Brent), a position in GeneralMotors shares (GM), a position in a portfolio exposed to the Standard and Poor’s 500 StockIndex (SP500), and a position in a zero bond (ZB) with a (constant) maturity of one year.All results are based on prices in USD and daily log. returns (see equation (5)).
Panel A: Relative standard deviation (in %) of the 99%-VaR predictive distributions.
Method JPY CHF GBP Brent GM SP500 ZB Average
HS 14.66 7.35 7.71 17.34 15.48 14.83 10.94 12.62HSA 12.92 6.06 6.43 12.05 12.15 12.66 12.37 10.66
GA
RC
H-N
BA 5.00 5.07 5.58 5.84 6.22 6.07 6.17 5.72QMLE 7.32 6.84 4.56 6.37 13.67 5.89 8.51 7.59BS 8.50 6.51 7.60 7.99 10.48 7.85 11.32 8.61
GA
RC
H-T
BA 7.48 7.12 7.42 8.25 8.78 7.52 10.05 8.09QMLE 10.04 11.59 5.68 7.18 7.49 8.08 8.67 8.39BS 9.44 7.52 8.26 9.67 9.65 8.70 16.99 10.03
Panel B: Relative standard deviation (in %) of the 95%-VaR predictive distributions.
Method JPY CHF GBP Brent GM SP500 ZB Average
HS 7.26 7.41 6.73 6.42 8.10 5.71 8.36 7.14HSA 8.23 6.72 5.75 6.38 6.70 6.15 9.11 7.01
GA
RC
H-N
BA 5.67 5.76 6.20 6.49 6.81 6.67 6.86 6.35QMLE 7.98 7.50 5.25 6.96 14.36 6.45 9.15 8.24BS 9.03 7.07 8.08 8.53 10.96 8.42 11.79 9.13
GA
RC
H-T
BA 6.55 6.14 6.56 7.57 8.09 7.18 9.39 7.35QMLE 9.27 8.97 6.02 7.20 6.97 9.18 8.90 8.07BS 8.64 7.31 8.29 8.73 8.22 8.30 15.96 9.49
26
Table 3:Results of the portion of failure test for adequacy of the 99%-VaR. The table presents portion offailures with p-values in parenthesis. Proportion of failures significantly different from 1% (at the5% significance level) are marked bold. We use two non-parametric models (HS = classical historicalsimulation, HSA = volatility adjusted historical simulation) and three parametric models (BA =Bayesian approach, QMLE = Quasi-maximum likelihood approach, BS = Bootstrap resampling).The test period starts on January 4th, 1999 and ends on December 31st, 2003. The seven positionsanalyzed are: a cash position in British Pound (GBP), a cash position in Japanese Yen (JPY), a cashposition in Swiss France (CHF), a position in Brent Crude Oil delivery today (Brent), a positionin General Motors shares (GM), a position in a portfolio exposed to the Standard and Poor’s 500Stock Index (SP500), and a position in a zero bond (ZB) with a (constant) maturity of one year. Allresults are based on prices in USD and daily log. returns (see equation (5)).
GARCH-N GARCH-THS HSA BA QMLE BS BA QMLE BS
JPY
mean 0.69 1.15 1.40 1.40 1.40 1.17 1.17 1.17(0.236) (0.590) (0.171) (0.171) (0.171) (0.551) (0.551) (0.551)
median 0.46 1.15 1.40 1.40 1.40 1.17 1.17 1.17(0.029) (0.590) (0.171) (0.171) (0.171) (0.551) (0.551) (0.551)
CHF
mean 0.84 1.46 1.56 1.40 1.56 1.01 1.01 1.01(0.563) (0.119) (0.063) (0.171) (0.063) (0.960) (0.960) (0.960)
median 0.84 1.54 1.56 1.40 1.56 1.01 1.01 1.01(0.563) (0.072) (0.063) (0.171) (0.063) (0.960) (0.960) (0.960)
GBP
mean 1.23 1.54 1.79 1.87 1.72 1.40 1.33 1.33(0.423) (0.072) (0.010) (0.005) (0.019) (0.171) (0.264) (0.264)
median 1.23 1.54 1.72 1.87 1.79 1.48 1.33 1.33(0.423) (0.072) (0.019) (0.005) (0.010) (0.106) (0.264) (0.264)
Brent
mean 0.86 1.25 1.67 1.51 1.59 1.11 0.96 0.96(0.612) (0.381) (0.029) (0.090) (0.052) (0.688) (0.873) (0.873)
median 0.86 1.18 1.67 1.59 1.67 1.11 0.96 0.96(0.612) (0.540) (0.029) (0.052) (0.029) (0.688) (0.873) (0.873)
GM
mean 1.11 1.04 1.29 1.21 1.38 1.13 1.05 1.05(0.688) (0.901) (0.320) (0.465) (0.209) (0.646) (0.856) (0.856)
median 1.11 1.04 1.29 1.21 1.38 1.13 1.13 1.05(0.688) (0.901) (0.320) (0.465) (0.209) (0.646) (0.646) (0.856)
SP500
mean 0.80 0.72 0.97 0.81 1.05 0.73 0.89 0.73(0.452) (0.288) (0.918) (0.485) (0.856) (0.313) (0.692) (0.313)
median 0.96 1.11 0.97 0.97 1.05 0.73 0.73 0.73(0.873) (0.688) (0.918) (0.918) (0.856) (0.313) (0.313) (0.313)
ZB
mean 1.43 1.58 1.93 1.77 1.77 0.89 1.05 0.89(0.153) (0.054) (0.003) (0.014) (0.014) (0.680) (0.870) (0.680)
median 1.51 1.51 2.01 1.77 1.93 0.89 1.05 0.89(0.093) (0.093) (0.002) (0.014) (0.003) (0.680) (0.870) (0.680)
27
Table 4:Results of portion of failure test for adequacy of 95%-VaR. The table presents proportion of fail-ures with p-values in parenthesis. Proportion of failures significantly different from 5% (at the 5%significance level) are marked bold. We use two non-parametric models (HS = classical historicalsimulation, HSA = volatility adjusted historical simulation) and three parametric models (BA =Bayesian approach, QMLE = Quasi-maximum likelihood approach, BS = Bootstrap resampling).The test period starts on January 4th, 1999 and ends on December 31st, 2003. The seven positionsanalyzed are: a cash position in British Pound (GBP), a cash position in Japanese Yen (JPY), a cashposition in Swiss France (CHF), a position in Brent Crude Oil delivery today (Brent), a positionin General Motors shares (GM), a position in a portfolio exposed to the Standard and Poor’s 500Stock Index (SP500), and a position in a zero bond (ZB) with a (constant) maturity of one year. Allresults are based on prices in USD and daily log. returns (see equation (5)).
GARCH-N GARCH-THS HSA BA QMLE BS BA QMLE BS
JPY
mean 3.46 5.07 4.06 3.98 4.06 4.21 4.21 4.29(0.007) (0.909) (0.109) (0.082) (0.109) (0.184) (0.184) (0.233)
median 3.53 5.22 4.06 4.06 3.98 4.21 4.21 4.29(0.011) (0.714) (0.109) (0.109) (0.082) (0.184) (0.184) (0.233)
CHF
mean 4.92 5.30 5.38 5.23 5.62 5.69 5.85 5.77(0.888) (0.623) (0.535) (0.712) (0.320) (0.264) (0.173) (0.215)
median 4.99 5.22 5.46 5.38 5.54 5.69 5.85 5.69(0.990) (0.714) (0.456) (0.535) (0.384) (0.264) (0.173) (0.264)
GBP
mean 5.38 5.68 5.38 5.38 5.38 5.38 5.62 5.15(0.538) (0.268) (0.535) (0.535) (0.535) (0.535) (0.320) (0.809)
median 5.45 5.68 5.38 5.38 5.46 5.38 5.69 5.15(0.459) (0.268) (0.535) (0.535) (0.456) (0.535) (0.264) (0.809)
Brent
mean 4.70 4.86 4.46 4.22 4.62 4.78 5.49 5.41(0.622) (0.816) (0.370) (0.193) (0.529) (0.715) (0.429) (0.506)
median 4.78 4.94 4.38 4.22 4.62 4.78 5.41 5.41(0.717) (0.918) (0.303) (0.193) (0.529) (0.715) (0.506) (0.506)
GM
mean 5.25 4.78 3.56 3.16 3.40 3.88 3.88 3.80(0.681) (0.715) (0.015) (0.001) (0.006) (0.061) (0.061) (0.044)
median 5.18 4.86 3.48 3.48 3.64 3.96 3.96 3.80(0.777) (0.815) (0.010) (0.010) (0.021) (0.084) (0.084) (0.044)
SP500
mean 5.10 5.02 4.94 4.53 5.10 5.26 5.26 5.74(0.877) (0.979) (0.917) (0.442) (0.876) (0.679) (0.679) (0.240)
median 5.02 5.02 4.85 4.53 5.10 5.34 5.26 5.58(0.979) (0.979) (0.813) (0.442) (0.876) (0.588) (0.679) (0.356)
ZB
mean 5.47 4.75 4.11 3.95 3.95 5.31 6.20 5.56(0.453) (0.687) (0.136) (0.077) (0.077) (0.615) (0.061) (0.377)
median 5.39 4.99 4.19 3.95 4.11 5.31 6.28 5.80(0.532) (0.990) (0.177) (0.077) (0.136) (0.615) (0.046) (0.208)
28
Figure 1:Training data and test periods used in the parametric VaR calculations.
Return series
29
Figure 2:99%- and 95%-VaR distributions predicted for the position in JPY/USD for January 8th,2002, are depicted on the left- and on the right-hand side, respectively. The distributionsgenerated by the GARCH-N model are plotted in the upper two figures; the results of theGARCH-T are plotted in the middle figures; the distributions generated by the histori-cal simulation approaches are plotted in the lower figures. The return of the position inJPY/USD on January 7th, 2002, was 0.05%. VaRs are plotted in return scale. We usetwo non-parametric models (HS = classical historical simulation, HSA = volatility adjustedhistorical simulation) and three parametric models (BA = Bayesian approach, QMLE =Quasi-maximum likelihood approach, BS = Bootstrap resampling).
−3 −2.5 −2 −1.5 −1 −0.50
5
10
15
20
GA
RC
H−
N m
odel
99%VaR
99% VaR predictive distribution for 08.01.2002 on JPY/USD
−3 −2.5 −2 −1.5 −1 −0.50
5
10
GA
RC
H−
T m
odel
99%VaR
−3 −2.5 −2 −1.5 −1 −0.50
1
2
3
HS
99%VaR
−1.3 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.50
5
10
15
20
GA
RC
H−
N m
odel
95%VaR
95% VaR predictive distribution for 08.01.2002 on JPY/USD
−1.3 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.50
5
10
15
20
GA
RC
H−
T m
odel
95%VaR
−1.3 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.50
2
4
6
8
10
HS
95%VaR
BA − VaRQML − VaRBS − VaR
BA − VaRQML − VaRBS − VaR
HS − VaRHSA − VaR
BA − VaRQML − VaRBS − VaR
BA − VaRQML − VaRBS − VaR
HS − VaRHSA − VaR
30
Figure 3:99%- and 95%-VaR distributions predicted for the position in JPY/USD for March 8th, 2002,are depicted on the left- and on the right-hand side, respectively. The distributions generatedby the GARCH-N model are plotted in the upper two figures; the results of the GARCH-Tare plotted in the middle figures; the distributions generated by the historical simulationapproaches are plotted in the lower figures. The return of the position in JPY/USD on March7th, 2002, was -2.7%. VaRs are plotted in return scale. We use two non-parametric models(HS = classical historical simulation, HSA = volatility adjusted historical simulation) andthree parametric models (BA = Bayesian approach, QMLE = Quasi-maximum likelihoodapproach, BS = Bootstrap resampling).
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.50
1
2
3
4
GA
RC
H−
N m
odel
99%VaR
99% VaR predictive distribution for 08.03.2002 on JPY/USD
BA − VaRQML − VaRBS − VaR
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.50
1
2
3
4
GA
RC
H−
T m
odel
99%VaR
BA − VaRQML − VaRBS − VaR
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.50
0.5
1
1.5
2
HS
99%VaR
HS − VaRHSA − VaR
−2 −1.5 −1 −0.5 00
1
2
3
4
GA
RC
H−
N m
odel
95%VaR
95% VaR predictive distribution for 08.03.2002 on JPY/USD
BA − VaRQML − VaRBS − VaR
−2 −1.5 −1 −0.5 00
1
2
3
4
GA
RC
H−
T m
odel
95%VaR
BA − VaRQML − VaRBS − VaR
−2 −1.5 −1 −0.5 00
2
4
6
8
10
HS
95%VaR
HS − VaRHSA − VaR
31
Figure 4:VaRs estimated for a cash position in JPY/USD in 2002 using the Bayesian approach. Inthe upper panel we depicted the 99%-VaR predictions and in the lower panel the 95%-VaRsare plotted. The VaRs estimated by the normal GARCH model are exhibited by the thicklines. The results of the GARCH-T model are plotted by the thin lines. The solid linedenotes for every day the mean of the predicted VaR distribution and the dotted line isused to plot the 95% confidence interval (CI) of the VaR distribution. VaRs are plotted inreturn scale.
2002 2003−3
−2.5
−2
−1.5
−1
−0.5
0Bayesian VaRs
2002 2003−3
−2.5
−2
−1.5
−1
−0.5
0
years
real loss95%VaR(mean) − GARCH−N95%VaR(CI) − GARCH−N95%VaR(mean) − GARCH−T95%VaR(CI) − GARCH−T
real loss99%VaR(mean) − GARCH−N99%VaR(CI) − GARCH−N99%VaR(mean) − GARCH−T99%VaR(CI) − GARCH−T
32
Figure 5:VaR estimates for a cash position in JPY/USD in 2002 using the historical simulationapproach. In the upper panel we depicted the 99%-VaR predictions and in the lower panelthe 95%-VaRs are plotted. The VaRs estimated by the the volatility adjusted historicalsimulation approach (HSA) are exhibited by the thick lines. The results of the classicalhistorical simulation approach (HS) are plotted by the thin lines. The solid line denotes forevery day the mean of the predicted VaR distribution and the dotted line is used to plotthe 95% confidence interval (CI) of the VaR distribution. VaRs are plotted in return scale.
2002 2003−3
−2.5
−2
−1.5
−1
−0.5
0HS VaRs
real loss99%VaR(mean) − HS99%VaR(CI) − HS99%VaR(mean) − HSA99%VaR(CI) − HSA
2002 2003−3
−2.5
−2
−1.5
−1
−0.5
0
years
real loss95%VaR(mean) − HS95%VaR(CI) − HS95%VaR(mean) − HSA99%VaR(CI) − HSA
33
Figure 6:p-values of the proportion of failure test for GARCH-N models. Mean, median and 5%,20%, 40%, 60%, 80% percentiles are taken to represent a point VaR estimates. We usethree parametric models: BA = Bayesian approach, QML = Quasi-maximum likelihoodapproach, BS = Bootstrap resampling. The test period starts on January 4th, 1999 andends on December 31st, 2003. The seven positions analyzed are: a cash position in BritishPound (GBP), a cash position in Japanese Yen (JPY), a cash position in Swiss France(CHF), a position in Brent Crude Oil delivery today (Brent), a position in General Motorsshares (GM), a position in a portfolio exposed to the Standard and Poor’s 500 Stock Index(SP500), and a position in a zero bond (ZB) with a (constant) maturity of one year. Allresults are based on prices in USD and daily log. returns (see equation (5)).
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
JP
Y
99% VaR, GARCH−N
VaR − Bayesian
VaR − QMLE
VaR − bootstrap
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
JP
Y
95% VaR, GARCH−N
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
CH
F
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
CH
F
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GB
P
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GB
P
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
Bre
nt
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
Bre
nt
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GM
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GM
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
SP
500
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
SP
500
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
ZB
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
ZB
34
Figure 7:p-values of the proportion of failure test for GARCH-T models. Mean, median and 5%,20%, 40%, 60%, 80% percentiles are taken to represent a point VaR estimates. We usethree parametric models: BA = Bayesian approach, QML = Quasi-maximum likelihoodapproach, BS = Bootstrap resampling. The test period starts on January 4th, 1999 andends on December 31st, 2003. The seven positions analyzed are: a cash position in BritishPound (GBP), a cash position in Japanese Yen (JPY), a cash position in Swiss France(CHF), a position in Brent Crude Oil delivery today (Brent), a position in General Motorsshares (GM), a position in a portfolio exposed to the Standard and Poor’s 500 Stock Index(SP500), and a position in a zero bond (ZB) with a (constant) maturity of one year. Allresults are based on prices in USD and daily log. returns (see equation (5)).
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
JP
Y
99% VaR, GARCH−T
VaR − Bayesian
VaR − QMLE
VaR − bootstrap
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
JP
Y
95% VaR, GARCH−T
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
CH
F
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
CH
F
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GB
P
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GB
P
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
Bre
nt
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
Bre
nt
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GM
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GM
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
SP
500
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
SP
500
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
ZB
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
ZB
35
Figure 8:p-values of the proportion of failure test for for two historical simulations methods. Mean,median and 5%, 20%, 40%, 60%, 80% percentiles are taken to represent a point VaR esti-mates. HS = classical historical simulation, HSA = volatility adjusted historical simulation.The test period starts on January 4th, 1999 and ends on December 31st, 2003. The sevenpositions analyzed are: a cash position in British Pound (GBP), a cash position in JapaneseYen (JPY), a cash position in Swiss France (CHF), a position in Brent Crude Oil deliverytoday (Brent), a position in General Motors shares (GM), a position in a portfolio exposedto the Standard and Poor’s 500 Stock Index (SP500), and a position in a zero bond (ZB)with a (constant) maturity of one year. All results are based on prices in USD and dailylog. returns (see equation (5)).
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
JP
Y
99% VaR
VaR − HS
VaR − HSA
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
JP
Y
95% VaR
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
CH
F
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
CH
F
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GB
P
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GB
P
mean median 5% 20% 40% 60% 80% 0
0.05
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Bre
nt
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
Bre
nt
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GM
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
GM
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
SP
500
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
SP
500
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
ZB
mean median 5% 20% 40% 60% 80% 0
0.05
0.2
ZB
36