Uncertainty in Value-at-Risk Estimates under Parametric and ... - SSRN

Uncertainty in Value-at-Risk Estimates under Parametric and Non-parametric Modeling ∗ Wolfgang Aussenegg and Tatiana Miazhynskaia Vienna University of Technology Department of Finance and Corporate Control Favoritenstrasse 9-11, A-1040 Vienna, Austria Aussenegg: [email protected], phone:+43 1 58801-33082 Miazhynskaia: [email protected], phone:+43 1 58801-33087

April 2006 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 VaR distribution. We propose a new VaR approach based on Bayesian statistics in a GARCH volatility modeling environment. This Bayesian approach is compared with other parametric VaR methods (quasi-maximum likelihood and bootstrap resampling on the basis 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 and the uncertainty in VaR estimates. Within the parametric methods, the Bayesian approach is better able to produce adequate VaR estimates, and results mostly in a smaller VaR variability. The non-parametric methods imply more uncertain 99%-VaR estimates, but show good performance with respect to 95%-VaRs. K eywords: Value-at-Risk, Bayesian analysis, GARCH, Historical Simulation, Bootstrap resampling JEL classification code: C11, C50, G10 We thank an anonymous referee, Manuel Ammann, Kostas Giannopoulos, and Michael Halling, as well as participants of the 55th annual meeting of the Midwest Finance Association (Chicago, 2006), the 10th Symposium on Finance, Banking, and Insurance (Karlsruhe, 2005), the 12th annual meeting of the German Finance Association (Augsburg, 2005), and the 18th Workshop of the Austrian Working Group on Banking and Finance (Innsbruck, 2004) for their helpful comments and suggestions. We are also grateful to Reuters GesmbH, Vienna, for providing data. ∗

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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. However, a good risk management requires not only a point VaR estimate but it is also important to know how precise this VaR estimate is. 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 distributions. In this framework we consider a new VaR approach based on Bayesian statistics. 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 1

Dowd (2000b), e.g., shows in a theoretical example how VaR confidence bands increase with parameter uncertainty. 2 Haas and Kondratyev (2000) show how VaR confidence bands may be obtained in the case of a generalized Pareto distribution.

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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 several 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 normal and Student-t distributional specifications. Further we consider 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 approach, point estimates for parameters are replaced by distributions in the parameter space, which represent our knowledge about values of the parameters, and the complete posterior distribution of the parameters can be used for further analysis. We compare this Bayesian VaR approach with other parametric VaR methods, like quasimaximum 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 second, 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 evaluate the uncertainty in VaR estimates. To check how stable the relative behavior of the VaR models is we use in our empirical 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, our 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 on average mostly associated with lower uncertainty in VaR predictions. The proportion of failure test indicates that the Bayesian approach tends to be more often accurate than quasi-maximum likelihood and bootstrapping estimation methods. 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 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.

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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 method 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 allocation with consequences on the profitability or even financial stability of institutions (Manganelli and Engle, 2001). From a statistical point of view, the VaR computation requires the estimation of 4

a quantile of the return distribution. As soon as the probability distribution of the returns is specified, the VaR is calculated using the corresponding percentile of this distribution. The VaR corresponding to the p% percentile can be defined as the amount of capital to cover expected losses in (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 accurately 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 predictions 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 properly.

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Parametric VaR Models

VaR models that are based on standard statistical distributions include estimation of the standard deviation (or covariance matrix) of asset returns. 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 simplicity we omit the specification AR(1) further from our model designations): 5

GARCH-N :

            

rt = a0 + a1 rt−1 + et ,

t = 1, 2, . . . , N (1)

et | It−1 ∼ N(0, ht ), ht = α0 + α1 e2t−1 + β1 ht−1 ,

with the restrictions α0 , α1 , β1 ≥ 0 to ensure ht > 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 conditional 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 + a1 rt−1 + et ,

t = 1, 2, . . . , N (2)

et | It−1 ∼ Tν (0, ht ), ht = α0 + α1 e2t−1 + β1 ht−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 conditional 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 N Y

e2 q L(Y | θ1 ) = exp − t 2 . 2σt 2πσt2 t=1 1

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(

)

(3)

Under the assumption of a Student t-distribution, the likelihood for the sample Y is L(Y | θ2 ) =

N Y

t=1

) Γ( ν+1 2 q

Γ( ν2 ) π(ν − 2)σt2

e2t 1+ (ν − 2)σt2

!−(ν+1)/2

,

(4)

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 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 posterior distribution of the parameters can be used for further analysis. When models are estimated in the classical manner, the uncertainty in model parameters is estimated 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 observed 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 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).

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At the center of the Bayesian inference is a simple and extremely important expression known as Bayes’ rule:

p(θ|Y ) = R

L(Y |θ)π(θ) . L(Y |θ)π(θ)dθ

(5)

Thus, having observed Y , our initial views about θ are updated by the data to get the distribution of the parameter vector θ conditional on Y . This distribution 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 summaries 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, equilibrium distribution p(θ|Y ), and a transition probability

K(θ′′ , θ′ ) = π(θ(t+1) = θ′′ | θ(t) = θ′ ),

(6)

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, e.g., 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 8

conjugacy for GARCH model parameters. Therefore, we use the Metropolis-Hastings (MH) algorithm which gives the easiest sampling strategy yielding the required realization of p(θ|Y ) (see, e.g., Kim, Shephard and Chib (1998), M¨ uller and Pole (1998) and Nakatsuma (2000)). To sample the posterior, we adopt the random walk MH algorithm with the Gaussian proposal density: 1. Generate a candidate draw θ(new) ∼ N(θ(old) , c); 2. Accept θ(new) with probability

α(θ

(old)

,θ

(new)

L(Y |θ(new) )π(θ(new) ) ) = min ,1 ; L(Y |θ(old) )π(θ(old) ) (

)

(7)

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). At the beginning, 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 we refer to Miazhynskaia and Dorffner (2006). As priors we use Gaussian priors for mean parameters and lognormal priors for variance parameters. All priors are centered at the maximum likelihood estimate (MLE) of the corresponding parameter with a variance 10 times larger than the squared standard MLE parameter error after the maximum likelihood estimation:

L L L L ˆM ˆM aM a0 ∼ N(ˆ aM a1 ) a0 ), a1 ∼ N(ˆ 1 , 10 · ǫ 0 , 10 · ǫ

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L L α0 ∼ logN(log α ˆ 0M L , 10 · ǫˆM ˆ 1M L , 10 · ˆǫM α0 ), α1 ∼ logN(log α α1 ) L β1 ∼ logN(log βˆ1M L , 10 · ǫˆM β1 ).

These priors tend to be practically non-informative because their effective range is about 10 times larger than the effective range of the resulting posterior density. With respect to the Student-t degrees of freedom parameter ν we define an exponential prior Exp(λ). After tuning on synthetic data, we fix λ = 0.1. 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 GARCH 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 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 θˆM L and assume an asymptotic Gaussian distribution for the model parameters θ ∼ N(θˆM L , Θ).

(8)

The uncertainty about the parameters is quantified by the estimated covariance matrix (Davidson and MacKinnon, 1993), with ˆ −1(G ˆ T G) ˆ H ˆ −1 Θ=H

(9)

ˆ denotes the Hessian matrix evaluated at θˆM L , G ˆ is the score matrix where H ˆ = ( ∂ln(yt |θ) )t,i G ∂θi

(10)

evaluated at θˆM L and ln(Y |θ) denotes the logarithmic value of the likelihood function 10

L(Y | θ). In the next step we are using the parameter distribution in (8) to quantify the uncertainty in VaR estimates. 3.2.3

Bootstrap resampling

The third method to assess the uncertainty in the parameter estimation is the bootstrap resampling scheme proposed by Barone-Adesi, Bourgoin and Giannopoulos (1998).3 Once the maximum likelihood estimate of the model parameters is found, say θˆM L = (ˆ a0 , a ˆ1 , α ˆ0, α ˆ 1 , βˆ1 ), the conditional variances are estimated by the GARCH process

ˆ1 = with h

ˆt = α ˆ t−1 , t = 2, . . . , N, h ˆ0 + α ˆ 1 (rt−1 − µ ˆ t−1 )2 + βˆ1 h

(11)

µ ˆt = a ˆ0 + aˆ1 rt−1

(12)

α ˆ0 , 1−α ˆ 1 −βˆ1

the estimated unconditional variance. The standardized residuals

are then calculated as rt − µ ˆt ǫˆt = q , ˆt h

t = 1, . . . , N.

(13)

∗ To mimic the structure of the original series, bootstrap replicates {r1∗ , r2∗ , . . . , rN }

are obtained from the following recursion: ∗ h∗t = α ˆ0 + α ˆ 1 (rt−1 − µ∗t−1 )2 + βˆ1 h∗t−1 ,

(14)

∗ µ∗t = a ˆ0 + aˆ1 rt−1 ,

(15)

rt∗ = µ∗t +

q

h∗t · ǫ∗t , t = 1, . . . , N,

(16)

where ǫ∗t are random draws from the empirical distribution of the centered residuals ˆ 1 and µ∗ = ǫˆt − average(ˆ ǫt ) (see equation (13)) and the initial values are h∗1 = h 1 mean(rt ). ∗ Once the bootstrap pseudo series of returns {r1∗ , . . . , rN } are generated, one can ∗ compute the bootstrap MLE θˆBS on this data. This procedure that includes generating 3

See also Barone-Adesi, Giannopoulos and Vosper (1999).

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∗ pseudo returns and then estimating θˆBS , is repeated until a sufficiently large sample ∗ of parameter estimates θˆBS is collected.

3.3

Predictive VaR distribution

By 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) }M m=1 from the (empirical) distribution of the parameters. For every m, m = 1, . . . , M, we compute the predictive return distribution according to our GARCH specification (one step ahead). Then we calculate the corresponding 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 one point VaR estimate, we now have an entire distribution of VaR predictions for every day in the test set.

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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 non-parametric VaR modelling 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 (r1% ).

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 paramet12

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 historical 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 (historical) trading day th is therefore defined as

r(th )adj = r(th ) ·

σ(t0 ) , σ(th )

(17)

where σ(th ) and σ(t0 ) are EWMA (exponentially weighted moving average) forecasts of the return volatility for day th and t0 , respectively.4 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. For each trading day this approach involves random resampling, with replacement, from the return sample (past two years).5 This resampling is done 1000 times for every trading day yielding 1000 (artificial) return distributions and 1000 corresponding VaR 4

The EWMA volatilities are estimated with a decay factor 0.97. In the case of the volatility weighted historical simulation approach we resample from the volatility adjusted return sample generated according to equation (17). 5

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estimates for each trading day. The resulting VaR distribution enables us to analyze the uncertainty in our VaR estimates.6

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),7 (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.8 Daily closing prices are obtained from Reuters 3000 Xtra. In all cases we take the perspective of an investor whose home currency is the US-Dollar (USD). Our total database starts in January 1992 and ends in December 2003. This sample period of twelve years covers several volatility increasing events, like the Asian crisis (1997), the Russian crisis (1998), the burst of the interent bubble (2000-2002), 9/11 (2001), and the Irak War (2003). It also contains several subperiods of strongly increasing as well as decreasing prices with considerable price swings.9 The reason to use seven financial assets of four different asset classes (currencies, commodities, stocks, and fixed income instruments) is that we want to evaluate our VaR models in different time series environments. Overall, the choosen sample period (1992-2003) and the four different asset classes should contain most time series effects important to study the characteristics of our eight VaR models. To generate VaR estimates, a training period of two years of past returns is used. The years 1992 and 1993 are therefore applied only for training purposes. Our test 6

For more details on how to perform bootstrapping procedures see e.g. Dowd (2002). Reuters RIC: QBRT 8 One year US-Libor rates are used to calculate zero bond prices. 9 During the sample period of twelve years our financial assets experience the following minimum and maximum prices: (a) GBP: 0.50-0.73 GBP per USD, (b) JPY: 80-148 JPY per USD, (c) CHF: 1.12-1.82 CHF per USD, (d) Brent Crude Oil: 9-38 USD per barrel, (e) General Motors: 27-95 USD per share, (f) Standard and Poor’s 500 Stock Index: 394-1520 points, and (g) 1-year US Libor Rate: 1.0-7.8% p.a. 7

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period starts on January 3rd, 1994 (the first day for which VaRs are estimated) and ends on December 31st, 2003. In total we compute daily VaR estimates for ten years (1994 - 2003). The continuously compounded daily return ri,t for day t and asset i is defined as

ri,t = ln

Pi,t , Pi,t−1

(18)

where Pi,t denotes the closing price of asset 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 test period of ten years (see Figure 1). In this way, the parameters of the models are updated every quarter.10 The VaR calculations are performed for every day in the corresponding test period according to the estimated GARCH specification. *** Insert Figure 1 about here *** 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. Within the volatility adjusted historical simulation approach we estimate in a first step for every day in our test period (1994 - 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 1994) equation (17) is used to weight all past returns. 10

The reason for the quarterly updating is a compromise between computational time and the length of the test period (in our case ten years). To check whether this quarterly updating influences our results we estimate QML GARCH-N VaR distributions (our quickest approach) also with a monthly updating interval. Although the model parameters differ some what relative to the quarterly updating, our main results do not change. Because of this and the fact that all parametric approaches are treated in the same way, we do not think to discriminate some models in comparison to others by updating quarterly.

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6

Empirical Results

In this section we discuss our main empirical findings about VaR methods described above. The used VaR methods are summarized in Table 1.

Table 1: VaR methods

non-parametric models

historical simulation (HS)

Method 1

adjusted historical simulation (HSA)

Method 2

GARCH-N parametric models

model

GARCH-T model

Bayesian approach (BA)

Method 3

quasi-maximum likelihood (QML)

Method 4

bootstrap (BS)

Method 5

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 impression 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).

*** Insert Figure 2 about here ***

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

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estimates are associated with higher uncertainty (see Figure 3). This effect is most pronounced in (more complex) models that react faster to volatility changes 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.

*** Insert Figure 3 about here ***

Another important finding of the analysis in Figures 2 and 3 is that VaR distributions typically deviate from normality. To test whether the deviation from normality is significant a Jarque-Bera test is performed. Table 2 presents the percentage 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 40% 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 2 also shows that VaR distributions for the 95% quantile tend to depart less from normality than for the 99% quantile.

*** Insert Table 2 about here ***

Table 3 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 deviation (variation coefficient) is defined as the absolute standard deviation normalized by the mean of the corresponding VaR distribution (in percent).


17

First, Table 3 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.11 This discrepancy can not be observed for 95%-VaRs, where the uncertainty generated by 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.12 Fourth, within the class of parametric models, the Bayesian framework generates on average in three out of four cases a smaller VaR variability, followed by the quasimaximum likelihood approach (see Table 3). 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 that 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 11

Apart from GARCH-T Bootstrap resampling approach. This observation is in line with evidence provided in the literature (see, e.g., Bams et al. (2003), Jorion (1996) or Dowd (2001)). 12

18

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. The results for the proportion of failure test are given in Tables 4 and 5. Median VaRs as well as 30% and 40% percentiles (PT) of the corresponding VaR distributions are taken to represent point VaR estimates.


With respect to 99% median VaRs, the parametric GARCH-N modelling is rejected for six out of seven positions (see Table 4). Even the 30% percentile of the corresponding VaR distributions yields significantly too low point VaR estimates. This supports the common opinion in literature 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 when the Bayesian approach is used, and in six out of seven cases for the quasi-maximum likelihood and the bootstrap resampling approach. Furthermore, also the classical historical simulation approach (HS) generates acceptable proportion of failures for 99% median VaR estimates, whereas the volatility adjusted historical simulation approach (HSA) has to be rejected for the JPY, the CHF and the SP500 positions. Opposite to the evidence for 99%-VaRs, fat tails in return distributions are no longer relevant for 95%-VaR estimates. As Table 5 reveals, GARCH-N models never underestimate 95%-VaRs. But in some cases, as for the General Motors (the position with the highest return volatility) GARCH-N models are too conservative (too high VaR). In contrast, GARCH-T models are not always accurate when we take median 95%-VaRs as point estimate. Median VaR estimates are too low for the CHF position (all three GARCH-T models) as well as for Brent Cruide Oil and the one year zero bond position (quasi-maximum likelihood and bootstrap resampling). Only 19

percentiles below the median deliver adequate 95%-VaR point estimates for these positions.


Overall, we can summarize that, first, for 99%-VaR estimates the Bayesian approach within the GARCH-T framework tends to be a better method taking into account accuracy of VaR estimates and their uncertainty. The GARCH-T quasimaximum likelihood approach also results in lower uncertainty of VaR estimates but suffers with respect to VaR accuracy. The classical historical simulation method proved to be accurate enough but its VaR point estimates are associated with much larger uncertainty. Second, for 95%-VaR estimates only the volatility adjusted historical simulation approach is always accurate for our positions. Within parametric approaches, the Bayesian approach tends to be preferable considering accuracy and uncertainty of VaR estimates.

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 modeling we discuss 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 total VaR distribution for a particular trading day (instead of a VaR point estimate) and provides 20

therefore a natural way to quantify the uncertainty in VaR estimates. We compare this Bayesian 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 financial assets with a ten year test period (1994-2003). 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 pronounced 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 normality. Within our ten year test period this is nearly always the case for our nonparametric methods and can on average be observed in 50 to 90% of all trading days for our parametric approaches. The heavy-tailed GARCH-T models generate more often non-Gaussian VaR distributions than GARCH-N models. 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. Furthermore, the uncertainty generated by the non-parametric models is comparable with those of the parametric models for 95%VaRs. But, with respect to 99%-VaR the historical simulation delivers on average much more uncertainty in VaR estimates. Compared to the GARCH-T model, the normal GARCH model shows lower uncertainty 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, although the distinction to the quasimaximum likelihood approach is small.

21

Within the class of non-parametric models the volatility adjusted historical simulation approach generates a 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 for 99%-VaR estimates our GARCH-T model nearly always provides an adequate fit, whereas the GARCH-N model tends to generate too low 99%-VaR estimates. In contrast, with respect to 95%-VaRs, the GARCH-T model is less accurate compared to the GARCH-N model. Non-parametric methods differ in their accuracy level. While the classical historical simulation approach provides accurate 99% but not 95%-VaR estimates, the opposite is true for the volatility adjusted historical simulation. Overall, the Bayesian VaR approach provides in the GARCH-T modelling environment an adequate VaR framework with less uncertainty in VaR estimates. This approach can therefore be considered as an interesting alternative to existing VaR methods. Open questions for future research are how the total VaR distribution 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. It might also be interesting to combine VaR uncertainty and VaR accuracy measures.

22

References Bams, D., Lehnert, T. and Wolff, C. (2003). An evaluation framework for alternative VaR models. CEPR Working Paper, University of Maastricht. Barone-Adesi, G., Bourgoin, F. and Giannopoulos, K. (1998). Don’t look back, Risk 11: 100– 103. Barone-Adesi, G., Giannopoulos, K. and Vosper, L. (1999). VaR without correlations for portfolios of derivative securities, Journal of Futures Markets 19(5): 583–602. Bollerslev, T. (1986). A generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31: 307–327. Bollerslev, T. (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return, Review of Economics and Statistics 69: 542–547. Brooks, C. and Persand, G. (2000). Value at risk and market crashes, Journal of Risk 2(4): 5–26. Carlin, B. and Louis, T. (1996). Bayes and Empirical Bayes Methods for Data Analysis, Chapman & Hall, London. Chappell, D. and Dowd, K. (1999). Confidence intervals for VaR, Financial Engineering News 9: 1–2. Chib, S. and Greenberg, E. (1996). Markov chain Monte Carlo simulation methods in econometrics, Econometric Theory 12: 409–431. Davidson, R. and MacKinnon, J. G. (1993). Estimation and Inference in Econometrics, Oxford University Press, Oxford. Dowd, K. (1998). Beyond Value at Risk: the New Science of Risk Management, John Willey & Sons, Chichester. Dowd, K. (2000a). Assessing VaR accuracy, Derivatives Quarterly 6(3): 61–63. Dowd, K. (2000b). Estimating value at risk: A subjective approach, Journal of Risk Finance 1(4): 43–46. Dowd, K. (2001). Estimating VaR with order statistics, Journal of Derivatives 8(3): 23–30. Dowd, K. (2002). Measuring market risk, John Willey & Sons, Chichester. Duffie, D. and Pan, J. (1997). An overview of value at risk, Journal of Derivatives 4: 7–49. Geweke, J. (1999). Using simulation methods for Bayesian econometric models: Inference, development and communication, Econometric Reviews 18: 1–126. Haas, M. and Kondratyev, A. (2000). Value-at-Risk and expected shortfall with confidence bands: An extreme value theory approach. Working Paper, Group of Financial Engineering Research Center CAESAR. Hull, J. and White, A. (1998). Incorporating volatility updating into the historical simulation method for value-at-risk, Journal of Risk 1(1): 5–19.

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Jorion, P. (1996). Risk2: Measuring the risk in value at risk, Financial Analysts Journal 52: 47–56. Jorion, P. (2000). Value at Risk, McGraw Hill, New York. Kim, S., Shephard, N. and Chib, S. (1998). Stochastic volatility: Likelihood inference and comparison with ARCH models, Review of Economic Studies 65: 361–393. Kupiec, H. (1995). Techniques for verifying the accuracy of risk management models, Journal of Derivatives 3: 73–84. Manganelli, S. and Engle, R. (2001). Value at risk models in finance. Working paper NN 75, European Central Bank. McNeil, A. and Frey, R. (2000). Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach, Journal of Empirical Finance 7: 271– 300. Miazhynskaia, T. and Dorffner, G. (2006). A comparison of Bayesian model selection based on MCMC with application to GARCH-type models, forthcoming: Statistical Papers . M¨ uller, P. and Pole, A. (1998). Monte Carlo posterior integration in GARCH models, Sankhya - The Indian Journal of Statistics 60: 127–144. Nakatsuma, T. (2000). Bayesian analysis of ARMA-GARCH models: a Markov chain sampling approach, Journal of Econometrics 95: 57–69. Prinzler, R. (1999). Reliability of neural network based value-at-risk estimates. Working Paper, TU Dresden. Tierney, L. (1994). Markov chains for exploring posterior distributions, Annals of Statistics 21: 1701–1762. Vrontos, I., Dellaportas, P. and Politis, D. (2000). Full Bayesian inference for GARCH and EGARCH models, Journal of Business & Economic Statistics 18(2): 187–198. Wilson, T. (1998). Calculating risk capital, in C. Alexander (ed.), The Handbook of Risk Management and Analysis, John Willey & Sons, New York, pp. 193–232.

24

Table 2: Percentage of trading days with non-Gaussian VaR distributions using the Jarque-Bera test at the 5% significance level. We use two non-parametric models (HS = classical historical simulation, 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 3rd, 1994 and ends on December 31st, 2003. The seven positions analyzed are: a cash position in Japanese Yen (JPY), a cash position in Swiss France (CHF), a cash position in British Pound (GBP), a position in Brent Crude Oil delivery today (Brent), a position in General Motors shares (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. All results are based on prices in USD and daily log. returns (see equation (18)). Panel A: Percentage of trading days with non-Gaussian VaR distributions: 99%-VaR JPY

CHF

GBP

Brent

GM

SP500

ZB

Average

HS HSA

99.2 97.7

94.3 97.4

96.4 95.8

99.0 98.9

99.6 98.0

99.0 98.2

97.9 98.8

97.9 97.8

BA QMLE BS

51.7 46.7 87.3

59.1 48.8 80.2

60.0 50.6 85.8

61.8 60.5 92.4

56.0 65.9 81.4

52.4 70.5 88.4

63.2 60.4 97.7

57.7 57.6 87.6

BA QMLE BS

94.3 51.3 89.0

96.1 58.1 82.8

97.1 61.1 88.3

90.5 55.5 91.7

93.8 62.7 81.6

86.1 57.7 89.6

74.1 81.3 98.2

90.3 61.1 88.7

GARCH-T

GARCH-N

Method

Panel B: Percentage of trading days with non-Gaussian VaR distributions: 95%-VaR JPY

CHF

GBP

Brent

GM

SP500

ZB

Average

HS HSA

93.5 92.2

95.3 80.8

95.8 93.1

85.3 95.1

98.5 94.5

97.8 92.6

99.2 97.3

95.1 92.2

BA QMLE BS

43.3 41.5 84.9

51.5 44.1 75.3

52.2 38.4 82.9

51.7 54.7 90.1

49.2 61.1 77.2

45.5 63.7 85.7

53.7 57.0 96.9

49.6 51.5 84.7

BA QMLE BS

61.6 58.8 92.6

61.8 57.4 76.5

64.0 52.6 81.0

64.3 55.4 90.3

59.7 62.8 72.7

57.4 57.8 90.2

70.3 78.8 96.7

62.7 60.5 85.7

GARCH-T

GARCH-N

Method

25

Table 3: Relative standard deviation of the 99%- and 95%-VaR predictive distributions averaged over all test points. The relative standard deviation is defined as the absolute standard deviation divided by the mean of the corresponding VaR distribution (in percent). We use two non-parametric models (HS = classical historical simulation, 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 3rd, 1994 and ends on December 31st, 2003. The seven positions analyzed are: a cash position in Japanese Yen (JPY), a cash position in Swiss France (CHF), a cash position in British Pound (GBP), a position in Brent Crude Oil delivery today (Brent), a position in General Motors shares (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. All results are based on prices in USD and daily log. returns (see equation (18)). Panel A: Relative standard deviation (in %) of the 99%-VaR predictive distributions. JPY

CHF

GBP

Brent

GM

SP500

ZB

Average

HS HSA

16.07 10.98

11.02 7.16

10.40 8.53

14.73 10.58

13.30 11.09

13.80 11.10

13.12 12.43

13.20 10.27

BA QMLE BS

5.32 6.23 10.26

5.46 6.43 8.09

5.73 4.33 9.91

5.90 4.89 9.48

5.69 10.90 8.56

6.22 5.10 8.75

6.01 7.08 12.57

5.76 6.42 9.66

BA QMLE BS

8.78 9.40 10.33

7.97 9.97 8.97

8.54 4.95 10.75

8.30 5.16 10.56

7.79 10.71 8.55

8.38 6.60 9.76

10.12 8.13 16.48

8.55 7.85 10.77

GARCH-T

GARCH-N

Method

Panel B: Relative standard deviation (in %) of the 95%-VaR predictive distributions. JPY

CHF

GBP

Brent

GM

SP500

ZB

Average

HS HSA

7.78 8.57

8.09 7.81

8.05 6.97

6.94 6.51

7.60 6.53

7.84 7.56

11.56 9.95

8.26 7.70

BA QMLE BS

5.97 6.66 10.75

6.13 6.94 8.60

6.37 4.77 10.35

6.51 5.23 9.95

6.32 11.18 9.07

6.86 5.48 9.37

6.63 7.47 12.98

6.40 6.82 10.15

BA QMLE BS

7.73 8.73 9.45

6.87 8.96 8.50

7.34 5.26 10.03

7.54 5.38 9.80

7.20 9.90 7.52

7.71 7.49 9.24

9.72 8.50 16.09

7.73 7.74 10.09

GARCH-T

GARCH-N

Method

26

Table 4: Results of the proportion of failure test for adequacy of the 99%-VaR using the likelihood ratio test proposed by Kupiec (1995). The median as well as 30% and 40% percentiles (PT) of the corresponding VaR distribution are taken as point estimates. Proportion of failures significantly different from 1% at the 5% and the 1% significance level are marked with * and **, respectively. We use two non-parametric models (HS = classical historical simulation, HSA = volatility adjusted historical simulation) and three parametric models (BA = Bayesian approach, QMLE = Quasimaximum likelihood approach, BS = Bootstrap resampling). The test period starts on January 3rd, 1994 and ends on December 31st, 2003. The seven positions analyzed are: a cash position in Japanese Yen (JPY), a cash position in Swiss France (CHF), a cash position in British Pound (GBP), a position in Brent Crude Oil delivery today (Brent), a position in General Motors shares (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. All results are based on prices in USD and daily log. returns (see equation (18)).

JPY

CHF

GBP

Brent

GM

SP500

ZB

median 40% PT 30% PT median 40% PT 30% PT median 40% PT 30% PT median 40% PT 30% PT median 40% PT 30% PT median 40% PT 30% PT median 40% PT 30% PT

HS 0.85 0.81 0.62 1.08 1.08 0.92 0.92 0.88 0.88 1.21 1.17 1.02 0.99 0.99 0.87 1.35 1.27 1.07 1.11 1.07 0.91

HSA 1.42* 1.38 1.23 1.61** 1.58** 1.35 1.31 1.31 1.19 1.37 1.33 1.05 1.15 1.15 0.95 1.43* 1.39 0.99 1.39 1.31 1.23

BA 1.83** 1.72** 1.64** 1.87** 1.79** 1.64** 1.60** 1.48* 1.44* 2.02** 1.86** 1.70** 1.17 1.13 1.09 1.65** 1.65** 1.53* 2.26** 2.18** 2.01**

GARCH-N QMLE 1.95** 1.87** 1.83** 1.72** 1.60** 1.52* 1.72** 1.64** 1.52* 1.98** 1.82** 1.70** 1.13 1.05 1.05 1.69** 1.57** 1.53* 2.18** 2.05** 1.97**

27

BS 1.87** 1.79** 1.60** 1.87** 1.68** 1.52* 1.68** 1.56** 1.40 2.02** 1.94** 1.82** 1.29 1.09 1.01 1.86** 1.65** 1.53* 2.22** 2.10** 1.89**

BA 1.38 1.29 1.21 1.31 1.17 1.09 1.17 1.01 0.98 1.41 1.27 1.15 0.97 0.93 0.89 1.39 1.29 1.29 0.93 0.89 0.77

GARCH-T QMLE BS 1.42* 1.42* 1.37 1.40 1.29 1.21 1.21 1.17 1.09 1.13 0.98 1.01 1.17 1.09 1.17 1.05 1.01 0.94 1.21 1.29 1.19 1.23 1.15 1.11 0.97 0.93 0.93 0.93 0.89 0.89 1.31 1.31 1.33 1.25 1.21 1.21 1.05 1.01 1.01 0.89 1.01 0.85

Table 5: Results of the proportion of failure test for adequacy of the 95%-VaR using the likelihood ratio test proposed by Kupiec (1995). The median as well as 30% and 40% percentiles (PT) of the corresponding VaR distribution are taken as point estimates. Proportion of failures significantly different from 5% at the 5% and the 1% significance level are marked with * and **, respectively. We use two non-parametric models (HS = classical historical simulation, HSA = volatility adjusted historical simulation) and three parametric models (BA = Bayesian approach, QMLE = Quasimaximum likelihood approach, BS = Bootstrap resampling). The test period starts on January 3rd, 1994 and ends on December 31st, 2003. The seven positions analyzed are: a cash position in Japanese Yen (JPY), a cash position in Swiss France (CHF), a cash position in British Pound (GBP), a position in Brent Crude Oil delivery today (Brent), a position in General Motors shares (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. All results are based on prices in USD and daily log. returns (see equation (18)).

JPY

CHF

GBP

Brent

GM

SP500

ZB

median 40% PT 30% PT median 40% PT 30% PT median 40% PT 30% PT median 40% PT 30% PT median 40% PT 30% PT median 40% PT 30% PT median 40% PT 30% PT

HS 4.61 4.58 4.34 5.11 5.04 4.73 4.84 4.65 4.23 5.74 5.51 5.23 5.05 5.01 4.89 6.03* 5.92* 5.56 4.92 4.64 4.09

HSA 5.27 5.15 4.96 5.34 5.11 4.92 5.38 5.19 5.07 5.20 4.96 4.65 5.20 5.13 4.89 5.16 5.01 4.89 4.88 4.64 4.44

BA 4.99 4.68 4.53 5.58 5.35 5.15 4.56 4.56 4.37 5.19 5.07 4.83 3.73** 3.55** 3.39** 5.12 4.80 4.56 4.27 4.23 4.07

GARCH-N QMLE 4.96 4.68 4.56 5.58 5.42 5.23 4.56 4.53 4.45 5.15 5.03 4.91 3.81** 3.47** 3.23** 4.96 4.88 4.56 4.23 4.19 4.11

28

BS 4.99 4.76 4.29 5.42 5.31 5.03 4.72 4.49 4.25 5.47 5.27 4.79 3.93* 3.51** 3.27** 5.41 4.88 4.64 4.35 4.23 4.15

BA 5.54 5.42 5.11 6.22** 5.81 5.46 5.27 5.15 4.96 5.67 5.55 5.07 4.21 3.83** 3.59** 5.69 5.32 4.76 5.56 5.44 5.16

GARCH-T QMLE 5.58 5.38 5.23 6.22** 6.05* 5.85 5.46 5.23 5.11 6.01* 5.71 5.51 3.97* 3.83** 3.71** 5.57 5.32 4.96 6.07* 5.76 5.48

BS 5.74 5.54 5.19 6.11** 5.93* 5.58 5.35 5.11 4.88 6.25* 5.86 5.71 3.85** 3.67** 3.47** 5.88* 5.69 5.08 6.11* 5.80 5.44

Figure 1: Training data and test periods used in the parametric VaR calculations.

29

Figure 2:

99% VaR predictive distribution for 08.01.2002 on JPY/USD 20 BA − VaR QML − VaR 15 BS − VaR

GARCH−N model

GARCH−N model

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 distributions generated by the GARCH-N model are plotted in the upper two figures; the results of the GARCH-T are plotted in the middle figures; the distributions generated by the historical simulation approaches are plotted in the lower figures. The return of the position in JPY/USD on January 7th, 2002, was 0.05%. VaRs are plotted in return scale. We use two non-parametric models (HS = classical historical simulation, HSA = volatility adjusted historical simulation) and three parametric models (BA = Bayesian approach, QMLE = Quasi-maximum likelihood approach, BS = Bootstrap resampling).

10 5 0 −3

−2.5

−2

−1.5 99%VaR

−1

5

−1.3 −1.2 −1.1

−1 −0.9 −0.8 −0.7 −0.6 −0.5 95%VaR

20

BA − VaR QML − VaR BS − VaR

GARCH−T model

GARCH−T model

10

0

−0.5

10

5

0 −3


−2.5

−2

−1.5 99%VaR

−1

15 10 5 0

−0.5

3


−1.3 −1.2 −1.1

−1 −0.9 −0.8 −0.7 −0.6 −0.5 95%VaR

10

HS − VaR HSA − VaR

8

HS

HS

2


6 4

1

2 0 −3

−2.5

−2

−1.5 99%VaR

−1

0

−0.5

30

−1.3 −1.2 −1.1

−1 −0.9 −0.8 −0.7 −0.6 −0.5 95%VaR

Figure 3:


GARCH−N model

GARCH−N model

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 generated by the GARCH-N model are plotted in the upper two figures; the results of the GARCH-T are plotted in the middle figures; the distributions generated by the historical simulation approaches are plotted in the lower figures. The return of the position in JPY/USD on March 7th, 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) and three parametric models (BA = Bayesian approach, QMLE = Quasi-maximum likelihood approach, BS = Bootstrap resampling).

2 1 0 −4

−3.5

−3

−2.5 −2 99%VaR

−1.5

−1

3

GARCH−T model

GARCH−T model

1

−1.5

−1 95%VaR

−0.5

0

−1 95%VaR

−0.5

0

−1 95%VaR

−0.5

0

4


2 1

−3.5

−3

−2.5 −2 99%VaR

−1.5

−1

3


2 1 0 −2

−0.5

2

−1.5

10


8

HS

1.5

HS

2

0 −2

−0.5

4

0 −4


1


6 4

0.5 0 −4

2 −3.5

−3

−2.5 −2 99%VaR

−1.5

−1

0 −2

−0.5

31

−1.5