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: waussen@pop.tuwien.ac.at, phone:+43 1 58801-33082

Miazhynskaia: tmiazhyn@pop.tuwien.ac.at, 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

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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

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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).

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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

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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

}.

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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

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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

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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

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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

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(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

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GB

P

mean median 5% 20% 40% 60% 80% 0

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Bre

nt

mean median 5% 20% 40% 60% 80% 0

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Bre

nt

mean median 5% 20% 40% 60% 80% 0

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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