Physica A 401 (2014) 71–81

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Physica A

journal homepage: www.elsevier.com/locate/physa

Multifractality and value-at-risk forecasting ofexchange rates✩

Jonathan A. Batten a, Harald Kinateder b, Niklas Wagner b,∗a Department of Finance, Monash University, Caulfield Campus PO Box 197, Caulfield East, Victoria 3145, Australiab Department of Business and Economics, Passau University, 94030 Passau, Germany

h i g h l i g h t s

• Wemodel the high frequency VaR for EUR/USD returns.• We use a modified version of the multifractal model of asset returns (MMAR).• We study the out-of-sample forecasting performance of the MMAR model and two alternative models.• Our dataset consists of 138,418 5-min round-the-clock observations of EUR/USD spot quotes and trading ticks.

a r t i c l e i n f o

Article history:Received 11 July 2013Received in revised form 20 November2013Available online 21 January 2014

Keywords:High frequency exchange ratesMultifractalityMMARValue-at-riskForeign exchange risk forecasting

a b s t r a c t

This paper addresses market risk prediction for high frequency foreign exchange ratesunder nonlinear risk scaling behaviour. We use a modified version of the multifractalmodel of asset returns (MMAR) where trading time is represented by the series of volumeticks. Our dataset consists of 138,418 5-min round-the-clock observations of EUR/USDspot quotes and trading ticks during the period January 5, 2006 to December 31, 2007.Considering fat-tails, long-range dependence as well as scale inconsistency with theMMAR, we derive out-of-sample value-at-risk (VaR) forecasts and compare our approachto historical simulation as well as a benchmark GARCH(1,1) location-scale VaR model.Our findings underline that the multifractal properties in EUR/USD returns in fact havenotable risk management implications. The MMAR approach is a parsimonious modelwhich produces admissible VaR forecasts at the 12-h forecast horizon. For the daily horizon,the MMAR outperforms both alternatives based on conditional as well as unconditionalcoverage statistics.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

With daily estimated turnover in excess of US$1.3 trillion per day according to BIS [1], the EUR/USD spot foreignexchange (FX) rate is the most important currency pair traded in over-the-counter (OTC) spot markets. Given the volatilityof these markets in recent years, the management of currency related asset positions is vital to financial intermediaries andinternational corporations alike. Awidely used approach to financial riskmeasurement is Value-at-Risk (VaR). This approachenables regulators to determine the appropriate amount of risk capital necessary to ensure a financial intermediary is

✩ The authors would like to thank Oliver Entrop, Michael King, Renatas Kizys, ThomasWenger, an anonymous referee as well as participants at the 2012conference of the Financial Engineering and Banking Society (FEBS) in London for helpful comments and suggestions. Chee-Jin Yap provided excellentassistance with the dataset. All omissions and errors remain with the authors.∗ Corresponding author. Tel.: +49 851 509 3240; fax: +49 851 509 3242.

E-mail addresses: niklas.wagner@uni-passau.de, nwagner@alum.calberkeley.org (N. Wagner).

0378-4371/$ – see front matter© 2014 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physa.2014.01.024

72 J.A. Batten et al. / Physica A 401 (2014) 71–81

immune to the effects of adversemovements in asset prices, and also provides a yardstick for internalmanagement decisionssuch as risk budgeting and performance evaluation.

An important caveat to VaR estimation is the well-known fact that financial assets returns – especially at higherfrequencies – do not display ideal statistical properties. Instead, multifractal or multiscaling return features that arecharacterized by a form of time-invariancemay yield what is observed as fat-tailed returns with long-range dependence (orso-called long memory). Given these well-documented features, it is important to accurately forecast risk levels, which areconsistent with the observed return properties.1 Failure to correctly account for these properties may result in insufficientcapital allocations. Systematic underestimation of appropriate levels of risk capital required may even lead to broadersystem-wide consequences.2

In this paper we propose a parsimonious VaR prediction approach that yields improved FX risk forecasts. Typically,VaR is calculated based on daily return data, although doing so ignores the risk – and potential losses – associated withthe liquidation of positions due to adverse intraday price movements. We therefore use intraday data to support ourVaR forecasts and employ a modified version of the multifractal model of asset returns (MMAR) proposed by Mandelbrotet al. [14]. TheMMAR approach has the benefit that it parsimoniously addresses the complex properties of financial returns.It also allows the incorporation of various degrees of long memory at different powers of returns, while accommodatingthe presence of fat-tails, which are both stylized facts of financial returns (see e.g. [15–19]). Alternate approaches, such asfractionally integrated GARCH (generalized autoregressive conditional heteroskedasticity) models (or FIGARCH) have thesame decay rate for all moments and are not scale-consistent. Moreover, the MMAR was previously found to be a suitablemodel for FX rate returns.3

Following Clark [21], several studies have argued that trading volume could be utilized to improve risk prediction. Forexample, King et al. [22] investigate the relationship betweenUSD/CAD returns and order flowand argue that trading volumehas strong out-of-sample predictive power for USD/CAD returns. Xue and Gençay [19] demonstrate that the existence ofdifferent market traders, with multiple trading frequencies, can increase volatility persistence. Intraday asset volatilityalso varies with the number of market traders. Given this evidence, we model MMAR trading time by the series of tradingvolume ticks and provide a modified MMAR approach for out-of-sample VaR forecasting. We thereby overcome limitationsof previous multifractal model applications such as their combinatorial nature and their restriction to a bounded interval.4

In order to test the forecasting ability of our novel VaR approach, we study the out-of-sample accuracy of VaR predictionsfor both 12-h and daily (24-h) forecast horizons. While these forecast periods are somewhat arbitrary they are consistentwith the trading activities expected of global financial intermediarieswith a subsidiary, or branch, that is always open duringthe 24-h trading day. Our high frequency dataset consists of round-the-clock EUR/USD spot exchange rate prices quoted bymarket participants on the Reuters trading platform during the period January 5, 2006 to December 31, 2007. These pricesare bundled into 5-min time stamped intervals with the spot price and the trading ticks recorded.We find that the EUR/USDreturns are multifractal, with the moments showing different scaling exponents. Our MMAR approach is then comparedwith forecasts based on historical simulation and a benchmark location-scale VaR model based on GARCH(1,1). The resultsshow that the MMAR approach produces admissible VaR forecasts for the 12-h forecast horizon. For the daily horizon, wefind that the MMAR outperforms both alternatives based on conditional as well as unconditional coverage statistics.

Besides this investigation, there are several other studies that predict out-of-sample intraday VaR. Giot [26], for example,usesGARCHmodelswith normal and Student-t innovations andRiskMetricsmodel formodelling intradayVaRof 15- and30-min returns of three stocks traded on theNYSE. The results show that a superiormodel is based on Student-t innovations. Sunet al. [27] try to take account of the stylized facts of 1-min frequency DAX returns by using a GARCHmodel with Lévy stableand normal innovations. The authors find that the model with Lévy stable innovations outperforms the competing intradayVaRmodels. Dionne et al. [28] analyse a high frequency sample consisting of 63 tradingdays of three stocks tradedonTorontoStock Exchange. Their backtesting results imply that a logarithmic autoregressive, conditional duration, exponential GARCHmodel achieves better intraday VaR forecasts than ordinary GARCH and historical simulation. In contrast to these studies,our approach is not based on GARCH volatility and our MMAR VaR model is able to capture the stylized facts of intradaydata, including leptokurtosis and long-range dependence. An alternative group of studies in the area deals with extremevalue theory (EVT) and VaR forecasting. Gençay and Selçuk [29], Gençay et al. [30], Wagner [31] as well as Maghyereh andAl-Zoubi [32] for example, investigate the performance of VaR models for daily stock returns based on EVT. The papers

1 The empirical evidence is extensive. For example, Calvet and Fisher [2] find multifractality in a (Deutsche Mark) DMK/USD high frequency series andXu and Gençay [3] also prove 5-min USD/DMK returns are multifractal. Nekhili et al. [4] investigate scale properties of US Dollar/Deutsche Mark FX returnsand state that the co-existence of short-term as well as long-term traders indicates different time scales for different market traders. Eisler and Kertész [5]report multiscaling behaviour for a high frequency stock index series and high frequency observations of the 200 most liquid stocks at New York StockExchange (NYSE). Fillol [6] suggests amodel for replicating the scaling properties observed in the French CAC-40 (Cotation Assistée en Continu) stock series.Mulligan and Koppl [7] find long-range dependence in US macroeconomic data.2 This is especially important for emerging markets, where domestic political instability (e.g. [8]) may lead to region-wide contagion effects, and

eventually broader-world wide problems. To some extent these risks can be reduced through diversification [9,10]. Note that recent evidence using fractalbased measures links the efficiency of financial markets to their level of institutional development [11–13].3 Calvet and Fisher [20] find that MMAR outperforms both GARCH and FIGARCH as models of foreign exchange rate series.4 Calvet and Fisher [23] overcome these shortcomings by introducing a Markov-switching multifractal model, while Lux [24] provides a further model

alternative. Note that McCulloch [25] models the intraday trading time using a unifractal rather than multifractal time.

J.A. Batten et al. / Physica A 401 (2014) 71–81 73

document that the EVT VaR model provides more accurate VaR estimates, as EVT VaR approaches are more robust whileGARCH VaRmodels aremore sensitive to excessive volatility fluctuations. The third paper addresses the question of optimalVaR capital allocation when changes in the time-varying behaviour of extreme returns are considered.

The remainder of this paper is organized as follows. Section 2 contains a brief review of the multifractality literatureand introduces the modified version of the multifractal model of asset returns. Section 3 presents the multiple-period VaRforecasting concept used in ourMMARmodel. Section 4 contains an empirical study of the variousmodels’ forecasting abilityusing the EUR/USD foreign exchange rate series and finally, Section 5 concludes.

2. Multifractal modelling of FX returns

2.1. Multifractality

Mandelbrot et al. [14] define a multifractal stochastic process as a process that possesses a nonlinear scaling function. Bydefinition, a linear scaling function characterizes unifractal processes. Themajority of financial time series exhibit nonlinearscaling functions, see Refs. [2,15,3], for example. In the following, we briefly reviewmultifractal processes and introduce thescaling function.

Definition 2.1. A time series process {Yt}1≤t≤T is called multifractal, if it has stationary increments and satisfies

E|Yt,h − Yt |

q= c(q)hτ(q)+1, (1)

for all 1 ≤ h ≤ T and 0 ≤ q ≤ Q . The constant Q denotes the highest finite moment, the h-period ahead realization, Yt,h,is given by Yt,h =

hτ=1 Yt+τ−1, and the scaling function τ(q) as well as the prefactor c(q) are both deterministic functions

of q.

According to Definition 2.1, unifractal processes such as Brownian motion (BM) and fractional Brownian motion (FBM),for example, have linear scaling functions. The scaling law of BM is

τBM(q) =q2− 1, (2)

and the one for FBM is

τFBM(q) = qH − 1, (3)

for 0 < H < 1. In order to infer τ(q) from observed data, Calvet and Fisher [2] propose to estimate the parameters in aregression model which follows from Definition 2.1:

log S(q, h) = τ(q) log h. (4)

The partition function S(q, h) of the time series process {Yt}1≤t≤T is obtained by partitioning the series into N = ⌊T/h⌋non-overlapping subintervals of length h ∈ [1, T ]:

S(q, h) =N−1i=0

|Yi·h+h − Yi·h|q. (5)

2.2. Multifractal model of asset returns

Based on theMMAR ofMandelbrot et al. [14], wemodel FX returns, {Rt}1≤t≤T , by compounding a FBMwith amultifractalstochastic trading time. The FX return is defined as Rt ≡ log Xt − log Xt−1, where Xt denotes a spot FX rate.

Assumption 2.1. The MMAR assumes that the FX returns {Rt}1≤t≤T follow a compound process of the form

Rt = B(H)[θt ], (6)

where B(H)[·] is a FBMwith self-affinity index 0 < H < 1, which operates on amultifractal stochastic trading time {θt}1≤t≤T .

As shownbyClark [21], the stochastic trading time relates to the tails of the process. Themost important feature of FBM, asdiscussed inMandelbrot and van Ness [33], is that FBM is able tomodel different degrees of persistence. ForH = 0.5, FBM isordinary BMwith independent and identically distributed (i.i.d.) increments. Yet, B(H)

[·] is antipersistentwhen 0 < H < 0.5and displays long memory for 0.5 < H < 1.

Additionally, we impose the assumption of independence and that of an observable trading time as follows.

Assumption 2.2. The FBM B(H)[·] and the trading time θt operate independently of each other.

74 J.A. Batten et al. / Physica A 401 (2014) 71–81

Assumption 2.3. The trading time θt is given by the normalized volume ticks V t ,

V t =Vt

Tt=1

(Vt)

, (7)

where Vt ≥ 0 denotes the number of volume ticks at time t . It is assumed that V t is multifractal as given by the scaling lawin Eq. (1).

In our setting θt is not a trading time based on a model assumption, but reflects the normalized number of occurringtrades. The motivation for this is twofold. First, empirical work e.g. by Ma [34] supports the hypothesis that observabletrading time confirms the multifractal trading time hypothesis suggested by Mandelbrot et al. [14]. Second, the actualvolume ticks reflect the behaviour of the market participants more accurately than an unobservable trading time. The useof volume ticks distinguishes our work from that of previous MMAR approaches.

3. MMARmarket risk prediction

Our MMARmarket risk prediction approach is based on Monte Carlo simulation. The MMAR allows us to simulate scale-consistent returns that parsimoniously address fat-tails and long memory in returns. Applying a factorization approach, wesimulate MMAR sample paths under the assumptions of Section 2.2. The simulated sample paths are used for out-of-sampleVaR prediction.

3.1. Value-at-risk

Given some probability 0 < α < 1, the Ft−1-measurable VaR for the h-period ahead time interval (t − 1, t + h− 1],VaRα

t,h, is defined as

VaRαt,h = − inf

r : Ft,h(r) ≥ α

, (8)

which yields P(Rt,h ≤ −VaRαt,h|Ft−1) = α. The h-period ahead FX return, Rt,h = log Xt+h−1 − log Xt−1, equals the sum of

one-period returns Rt,h =h

τ=1 Rt+τ−1 and Ft,h(r) = P(Rt,h ≤ r|Ft−1) is its conditional distribution.

3.2. Simulation

The first step of ourMMAR VaR simulation approach is to simulateDMMAR sample paths R(d)∈ R1×T for d ∈ {1, . . . ,D},

where {R(d)t }1≤t≤T is the simulated time-t one-period MMAR return of the dth simulation run. Given Assumptions 2.1, 2.2

and 2.3, we use a method based on the factorization of the autocovariance matrix, K , to simulate MMAR sample paths R(d):

R(d)= B(H)

[θ ] =gTA

σ (9)

with

g ∼ N (0, 1) (10)

and

K = AAT, (11)

where K ∈ RT×T is a positive-definite matrix and g ∈ RT×1 is a vector of standard Gaussian numbers, while σ is the averageof the unconditional standard deviations of Rt and V t . The symmetric matrix A ∈ RT×T is the square root of the matrix K .The elements of the matrix K are

ks,t =12

Θ2H

t +Θ2Hs − |Θt −Θs|

2H, (12)

for each given time 1 ≤ s ≤ t ≤ T , where {Θt}1≤t≤T is the cumulative trading time that increases with time t

Θt =

ti=1

θi.

Since the MMAR creates a scale-consistent sample path, which relates simulated returns over different samplingfrequencies and exhibits the simplicity of self-affine processes (see e.g. [2]), we derive the corresponding h-period MMARreturn of the dth simulation run, R(d)

t,h , as

R(d)t,h = hHR(d)

t , (13)

J.A. Batten et al. / Physica A 401 (2014) 71–81 75

Table 1EUR/USD return statistics: Summary statistics of EUR/USD returns for various levels of aggregation h. The Phillips–Perron unit root test statistics indicatethat the null has to be rejected in favour of the stationarity alternative. Sample period is from January 5, 2006 to December 31, 2007.

Frequency 5 min 1 h 1 day 1 week

Size 138,417 11,534 480 96Mean 1.4× 10−6 0.00002 0.00040 0.00205Std. Dev. 0.0003 0.0009 0.0045 0.0109Skewness 0.85 0.42 0.05 −0.48Kurtosis 124.73 16.78 4.08 3.23AC of Rt (1) 0.068 −0.001 −0.045 −0.179AC of Rt (2) −0.023 0.005 −0.001 0.083AC of Rt (3) −0.015 0.016 0.023 −0.118AC of |Rt |(1) 0.176 0.133 −0.017 0.047AC of |Rt |(2) 0.160 0.075 0.024 0.136AC of |Rt |(3) 0.135 0.062 0.031 0.049

Fig. 1. Continuously compounded 5-min EUR/USD returns. Sample period is from January 5, 2006 to December 31, 2007.

for h > 0 and 0 < H < 1. The conditional distribution of R(d)t,h is given by F (d)

t,h (r (d)) and we obtain D quantile estimatesF (d)←t,h (α). For large numbers of simulation runs D, the Law of Large Numbers assures convergence and the average of Dquantile estimates converges to a stable quantile, which is the multiple-period predicted VaR:

VaRαt,h = −

1D

Dd=1

F (d)←t,h (α)

. (14)

4. Empirical analysis

4.1. Dataset and descriptive statistics

The 5-min round-the-clock EUR/USD series is obtained from the Reuters’ trading platform, and covers the period January5, 2006, to December 31, 2007, for a total of 138,418 high frequency FX observations. The data is bundled into 5-minsequential time stamped intervals with the spot price and the ticks recorded. A tick is a trade through the platform forthe standard minimum amount of about 3–5 million euro. Since the highest frequency is 5-min (h = 1, Rt,1 = Rt ), onehour corresponds to h = 12 and one day is equal to h = 288. Table 1 shows that as the sampling frequency increases, theunconditional volatility rises. Additionally, one can clearly observe the declining kurtosis leading to approximately Gaussiandata. For example, 5-min EUR/USD returns are fat-tailed (compare with Fig. 2) and are highly short-term dependent sinceone can observe the presence of significant autocorrelation in the absolute EUR/USD returns. Yet, this effect declines, whenthe sampling frequency increases.

The EUR/USD returns at 5-min frequency are illustrated in Fig. 1. In Fig. 2 Quantile–Quantile (Q–Q) plots of the sampleEUR/USD returns for 5-min and daily sampling are provided. In both panels the empirical quantiles (points) are plottedagainst the theoretical quantiles of a normal distribution (straight line). If the two distributions are similar, the points ofthe empirical distribution should lie on the line. The Q–Q plot shows that 5-min EUR/USD returns are fat-tailed. With anincreasing sampling frequency (daily) the EUR/USD returns are less tailed, but still have fatter tails than anormal distribution.

76 J.A. Batten et al. / Physica A 401 (2014) 71–81

(a) 5-min EUR/USD returns. (b) Daily EUR/USD returns.

Fig. 2. Quantile–Quantile (Q–Q) plots of EUR/USD returns are provided in (a) for 5-min and (b) daily sampling. In both panels the empirical quantiles(points) are plotted against the theoretical quantiles of a normal distribution (straight line). If the two distributions are similar, the point of the empiricaldistribution should lie on the line. With an increasing sampling frequency the returns are less tailed, but have still fatter tails than a normal distribution.Sample period is from January 5, 2006 to December 31, 2007.

(a) 5-min volume ticks. (b) cumulative trading time.

Fig. 3. Panel (a) illustrates 5-min EUR/USD trading ticks Vt . A tick is a trade through the platform for the standard minimum amount of between 3–5million euro. Panel (b) contains the cumulative trading time Θt . Sample period is from January 5, 2006 to December 31, 2007.

This finding is not uncommon for FX returns. For example, Lipton–Lifschitz [35] find a kurtosis of five in daily USD/DEMreturns from 1986 to 1996.

Fig. 3 illustrates 5-min EUR/USD trading ticks Vt and the cumulative trading time Θt .

4.2. Evidence of multifractality

Fig. 4 presents EUR/USD return partition functions for moments 1 ≤ q ≤ 5 in the scaling region up to one week. Thepartition functions are plotted against h using logarithmic scales. For q ∈ [1, 3] the lines are approximately linear. Generallyhigher moments are more sensitive to deviations from scaling as they capture information in the tails of EUR/USD returns.The partition function for moments q = 4 and q = 5 shows linear behaviour up to log h = 3, which corresponds to a timehorizon of about 100 min.

Fig. 5 illustrates estimated scaling functions of EUR/USD returns and Brownian motion. The scaling function of ordinaryBrownian motion (dashed line) as defined in Eq. (2) is linear, which is typical of unifractal processes. The scaling function of

J.A. Batten et al. / Physica A 401 (2014) 71–81 77

Fig. 4. EUR/USD return partition functions for moments 1 ≤ q ≤ 5. For each moment q the curves represent the ranges log h from 5-min up to one week.5-min correspond to log 1 = 0 and one week is log 1440 = 7.27. The renormalization for all partition functions is log S(q, 5-min) = 0. This procedureallows us to plot all curves on the same graph. Sample period is from January 5, 2006 to December 31, 2007.

Fig. 5. Estimated scaling functions of EUR/USD returns and Brownian motion. For each partition function S(q, h), we estimate the slope by OLS regression(4) to obtain τ(q). The scaling function of ordinary Brownian motion (dashed line) is linear which is typical for unifractal processes. Multifractal processeshave nonlinear scaling functions (see Definition 2.1). The estimated τ(q) (solid line) of EUR/USD returns is nonlinear and concave. Sample period is fromJanuary 5, 2006 to December 31, 2007.

EUR/USD however, is completely different and obviously nonlinear, which is a characteristic of multifractal processes (seeDefinition 2.1).

4.3. Out-of-sample VaR forecasts

Model estimationIn order to examine out-of-sample performance, we calculate the 1% VaR from January 6, 2006 to December 31, 2007 for

12-h (h = 144) and daily (h = 288) forecast horizons. The matrix A which is the square root of the matrix K is computedby the Cholesky decomposition. The self-affinity index H of Rt is estimated using the variance of residuals approach of Penget al. [36]. We simulate D = 5000 MMAR sample paths to predict the multiple-period VaR. The VaR results of our MMARapproach are compared to two alternatives.

All parameters of our VaRmodels (MMAR and the two benchmarkmodels) are calculated using amovingwindow, whichconsists of 288 5-min high frequency EUR/USD returns and volume ticks. The time-t VaR forecasts are updated at every timeh, which results in ⌊ 138,417−288h ⌋ successive VaR predictions.

78 J.A. Batten et al. / Physica A 401 (2014) 71–81

Table 2VaR Forecasts: Results of 1%-VaR predictions for 12-h (h = 144) and daily (h = 288) forecast horizons. VaR denotes average sample VaR. All VaR forecastsare based on a moving window which consists of 288 5-min high frequency EUR/USD returns and volume ticks. MMAR denotes the multifractal modelof asset returns, HS is the historical simulation and GARCH refers to the generalized autoregressive conditional heteroskedasticity model. The forecastingperiod is from January 6, 2006 to December 31, 2007.

Horizon Size MMAR HS GARCH

12-h 959 VaR 0.0098 0.0070 0.0080α 1.04 2.60 2.19LRuc 0.02 17.34** 10.24**

[0.895] [0.000] [0.001]LRind 0.21 1.34 0.94

[0.646] [0.247] [0.332]LRcc 0.23 18.68** 11.18**

[0.892] [0.000] [0.004]

1 day 479 VaR 0.0137 0.0098 0.0109α 0.84 1.46 1.25LRuc 0.14 10.24** 0.29

[0.709] [0.001] [0.593]LRind 0.07 0.07 0.15

[0.795] [0.795] [0.696]LRcc 0.21 10.31** 0.44

[0.902] [0.006] [0.803]

LR-statistics are as defined in the Appendix.** denote rejection of the null hypothesis at the 1% significance level (corresponding p-values in parenthesis).

Benchmark modelsFirst, instead of simulatedmultifractal return series,weusehistorical simulation of EUR/USD returns. Themultiple-period

VaR based on this historical simulation is:

VaRαt,h = −

√hF←t (α)

, (15)

where F←t (α) denotes the empirical α-quantile of Rt .Second, we compare the VaR results of our MMAR approach to a VaR model in the location-scale class. The multiple-

period VaR based on GARCH is:

VaRαt,h = −

µt,h + σt,h F←(α)

, (16)

where F← is the inverse of the innovations distribution. The multiple-period mean, µt,h, as well as the multiple-periodvolatility, σt,h, are determined by the GARCH(1,1)model of Bollerslev [37]. Estimation of the GARCH(1,1)modelwith skewedStudent-t innovations is carried out by the maximum likelihood method.

Summary resultsThe out-of-sample VaR forecast results are summarized in Table 2. We provide the average value of the VaR estimates,

VaR, and the empirical coverage rate, α. We use the Christoffersen [38] LR statistics for VaR performance evaluation.E.g. Berkowitz and O’Brien [39] also use the Christoffersen [38] LR statistics to study the VaR performance of six US banks.Sun et al. [27] evaluate high-frequency VaR predictions with the Christoffersen [38] approach. Appendix contains a detailedexplanation.5

We consider a violation of a backtesting statistic to have occurred when a corresponding p-value of the respective teststatistic is below 5%. The LRuc statistic tests for the correct violation level, which is α = 1% in our setting. VaR forecasts maybe correct on average, but produce violation clusters, a phenomenon ignored by unconditional coverage as it assumes thatviolations are independent (see e.g. [38,27]). In case VaR violations occur in clusters then cumulative losses may turn out tobe much higher than under the case of independence. These violation clusters often occur when a VaR model does not or isrelatively slow in adopting to changing market conditions.

The LRind statistic tests for independence of the VaR violations, while the conditional coverage statistic, LRcc , combinesboth concepts. As a result, the LRcc tests not only if violations occur in α percent of time, but also tests if they areindependent over time (see e.g. [39,38,27]). Consequently, we consider LRcc as the most important backtesting statistic.The VaR predictions of the three models for the 12-h horizon from January 6, 2006 to December 31, 2007 are plotted inFig. 6. The dash dotted line refers to the MMAR approach, the dashed line is historical simulation and the solid line is theGARCH based VaR model. In the following, we analyze the models’ performance in detail.

5 All models are validated based on non-overlapping h-period EUR/USD returns in order to provide independent out-of-sample tests.

J.A. Batten et al. / Physica A 401 (2014) 71–81 79

Fig. 6. 12-h ahead VaR forecasts for our MMAR approach (dash dotted line), historical simulation (dashed line) and GARCH (solid line). The forecastingperiod is from January 6, 2006 to December 31, 2007.

Unconditional coverage versus conditional coverageThe unconditional coverage test indicates that for the 12-h horizon of GARCH and for all horizons of historical simulation,

the predicted VaR exceeds the actual loss levels. This leads to insufficient coverage of potential losses since the VaR level istoo low. Our MMAR approach passes all backtesting statistics for p-values above 0.05. For both horizons, we find that ourMMARmodel achieves better conditional coverage than historical simulation and the GARCH based VaR. All models pass theindependence test indicating that there is no severe clustering of VaR violations over time. As a result, backtesting violationsresult due to VaR predictions exceeding actual levels.

12-h versus daily forecast horizonComparing the two forecast horizons, we find that historical simulation and GARCH produce better VaR results (and

therefore higher p-values of all backtesting statistics) for the daily horizon. This is because returns sampled at a higherfrequency often obtain more extreme returns than lower frequency returns. Our MMARmodel has less problems in such anenvironment. First, the MMAR is characterized by a special form of time-invariance, which combines extreme returns withlong memory [2]. Second, our MMAR approach uses trading volume. This is important, as increased trading activities canproduce extreme returns.

The predicted VaR using a GARCH estimation approach does not violate any backtesting statistic for the daily horizon. TheMMAR performs best for the 12-h horizon: we find that the p-value of LRcc (which is 0.892) and the empirical violation levelα = 1.04% both report excellent VaR forecasting. Concerning the daily horizon, we report good backtesting results for theMMAR, although this is a little bit inferior to the 12-h horizon. To conclude, for both horizons, our MMAR model dominatesthe VaR results of both the historical simulation and the GARCH approach and never violates a backtesting statistic.

ImplicationsRegulatory authorities require that financial intermediaries perform stress tests to ensure they possess sufficient capital

to remain solvent in the event of adverse price movements. Financial intermediaries which use poor VaR models may lacksufficient risk capital. In our setting for example, consider the case of a bank that takes a onemillion euro longposition againstthe US dollar and intends to close it after 12 h. MMAR would estimate 9800 euro as an average risk capital level, historicalsimulation and GARCH based VaR would predict average levels of about 7000 and 8000 euro, respectively. Given this andthe empirical violation rates reported in Table 2, the bank with an oversimplified risk model could end up considerablyundercapitalized. While the empirical violation rate of MMAR is roughly 1% and hence close to the target rate, the predictedcapital levels of the other two parsimonious models imply higher empirical violation rates as well as a lack of capital.

5. Conclusion

Accurate forecasts of intraday VaR are vital to many financial intermediaries, not only due to regulatory requirementsbut also for internal risk management tasks. We propose a novel MMAR approach to intraday VaR forecasts for EUR/USDFX returns. The approach provides superior forecast results when compared with historical simulation and a standardGARCH model. The MMAR model not only considers multifractality in FX returns, it is also a convenient model, which isparsimonious in its specification.

Appendix. Backtesting VaR

Market riskmodels predict VaRwith randomerror. The validity of a VaR predictionmodel ismeasured based on predictedversus actual loss levels. To evaluate the out-of-sample performance of the proposed models we follow the concept of

80 J.A. Batten et al. / Physica A 401 (2014) 71–81

Christoffersen [38]. The indicator (or hit) function It = 1{Rt,h<−VaRαt,h}

represents the history of observations, t = 1, . . . , T ,for which losses in excess of the predicted VaR occur.

A.1. Unconditional coverage

When a VaRmodel is designed perfectly, the number of observations that fall outside the predicted VaR should be exactlyin line with the given VaR level, such that E(It |Ft) = α holds. Hence, the test of unconditional coverage is

H0 : E(It |Ft) = α vs. H1 : E(It |Ft) = α.

Under the null hypothesis, the likelihood-ratio (LR) test statistic follows as

LRuc = −2 ln[L(α)/L(α)] ∼ χ2(1), (A.1)

where L(α) is the binomial likelihood with parameter α andα = 1T

Tt=1 It is the maximum likelihood estimator of α.

A.2. Independence

Besides the above requirement, VaR violations should be independent, which requires an additional test. Let nij denotethe number of observations for which It = j occurred following It−1 = i and assume that {It} is a first-order Markov chainwith transition probabilities πij = P(It = j|It−1 = i). This yields the likelihood

L(Π) = (1− π01)n00π

n0101 (1− π11)

n10πn1111 .

Maximum likelihood estimators for the transition probabilities are:

π01 =n01

n00 + n01, and π11 =

n11

n10 + n11.

Under the null hypothesis of independence, P(It = 0) = π0 = π01 = π11, which implies

L(π0) = (1− π0)n00+n10π

n01+n110 .

The maximum likelihood estimate for π0 is

π0 =n01 + n11

n00 + n10 + n01 + n11.

Based on π0 and Π , the independence LR test statistic is

LRind = −2 ln[L(π0)/L(Π)] ∼ χ2(1). (A.2)

A.3. Conditional coverage

The LRind statistic (A.2) tests for independence, but it does not take coverage into account. Christoffersen [38] thereforeproposes a combined test statistic:

LRcc = LRuc + LRind

= −2 ln[L(α)/L(Π)] ∼ χ2(2). (A.3)

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