Multifractality and value-at-risk forecasting of exchange rates

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1 See discussions, stats, and author profiles for this publication at: Multifractality and value-at-risk forecasting of exchange rates Article in Physica A: Statistical Mechanics and its Applications May 2014 DOI: /j.physa CITATIONS 16 READS authors: Jonathan A. Batten Universiti Utara Malaysia 179 PUBLICATIONS 1,279 CITATIONS SEE PROFILE Harald Kinateder Universität Passau 21 PUBLICATIONS 69 CITATIONS SEE PROFILE Niklas F. Wagner Universität Passau 99 PUBLICATIONS 485 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Equity Premium Prediction View project Non Gaussian option pricing View project All content following this page was uploaded by Niklas F. Wagner on 17 February The user has requested enhancement of the downloaded file.

2 Multifractality and Value-at-Risk Forecasting of Exchange Rates Jonathan A. Batten Harald Kinateder Niklas Wagner Version: November 2013 The authors would like to thank Oliver Entrop, Michael King, Renatas Kizys, Thomas Wenger, an anonymous referee as well as participants at the 2012 conference of the Financial Engineering and Banking Society (FEBS) in London for helpful comments and suggestions. Chee-Jin Yap provided excellent assistance with the data set. All omissions and errors remain with the authors. Department of Finance, Monash University, Caulfield Campus PO Box 197, Caulfield East, Victoria 3145, Australia. Department of Business and Economics, Passau University, Passau, Germany. Corresponding author. Niklas Wagner, Department of Business and Economics, Passau University, Passau, Germany. Phone: , Fax: ,

3 Multifractality and Value-at-Risk Forecasting of Exchange Rates Abstract This paper addresses market risk prediction for high frequency foreign exchange rates under nonlinear risk scaling behaviour. We use a modified version of the multifractal model of asset returns (MMAR) where trading time is represented by the series of volume ticks. Our data set consists of 138,418 5-minute round-the-clock observations of EUR/USD spot quotes and trading ticks during the period January 5, 2006 to December 31, Considering fat-tails, long-range dependence as well as scale inconsistency with the MMAR, we derive out-of-sample value-at-risk (VaR) forecasts and compare our approach to 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 have notable risk management implications. The MMAR approach is a parsimonious model which produces admissible VaR forecasts at the 12-hour forecast horizon. For the daily horizon, the MMAR outperforms both alternatives based on conditional as well as unconditional coverage statistics. Keywords: High frequency exchange rates; Multifractality; MMAR; Valueat-risk; Foreign exchange risk forecasting; JEL Classification: C22

4 1 Introduction With daily estimated turnover in excess of US$1.3 trillion per day according to BIS [1], the EUR/USD spot foreign exchange (FX) rate is the most important currency pair traded in over-the-counter (OTC) spot markets. Given the volatility of these markets in recent years, the management of currency related asset positions is vital to financial intermediaries and international corporations alike. A widely used approach to financial risk measurement is Value-at-Risk (VaR). This approach enables regulators to determine the appropriate amount of risk capital necessary to ensure a financial intermediary is immune to the effects of adverse movements in asset prices, and also provides a yardstick for internal management decisions such as risk budgeting and performance evaluation. An important caveat to VaR estimation is the well-known fact that financial assets returns especially at higher frequencies do not display ideal statistical properties. Instead, multifractal or multiscaling return features that are characterized by a form of time-invariance may yield what is observed as fat-tailed returns with long-range dependence (or so-called long memory). Given these well-documented features, it is important to accurately forecast risk levels, which are consistent with the observed return properties. 1 Failure to correctly account 1 The empirical evidence is extensive. For example, Calvet and Fisher [5] find multifractality in a (Deutsche Mark) DMK/USD high frequency series and Xu and Gençay [32] also prove 5-minute USD/DMK returns are multifractal. Nekhili et al. [27] investigate scale properties of U.S. Dollar/Deutsche Mark FX returns and 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 [12] report multiscaling behaviour for a high frequency stock index series and high frequency 1

5 for these properties may result in insufficient capital allocations. Systematic underestimation of appropriate levels of risk capital required may even lead to broader system-wide consequences. 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 with the liquidation of positions due to adverse intraday price movements. We therefore use intraday data to support our VaR forecasts and employ a modified version of the multifractal model of asset returns (MMAR) proposed by Mandelbrot et al. [23]. The MMAR 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 accommodating the presence of fat-tails, which are both stylized facts of financial returns (see e.g. [9], [10], [20], [29] and [33]). Alternate approaches, such as fractionally integrated GARCH (generalized autoregressive conditional heteroskedasticity) models (or FIGARCH) have the same decay rate for all moments and are not scale-consistent. Moreover, the MMAR was previously found to be a suitable model for FX rate returns. 2 Following Clark [8], several studies have argued that trading volume could be observations of the 200 most liquid stocks at New York Stock Exchange (NYSE). Fillol [13] suggests a model for replicating the scaling properties observed in the French CAC-40 (Cotation Assistée en Continu) stock series. Mulligan and Koppl [26] find long-range dependence in U.S. macroeconomic data. 2 Calvet and Fisher [6] find that MMAR outperforms both GARCH and FIGARCH as models of foreign exchange rate series. 2

6 utilised to improve risk prediction. For example, King et al. [17] investigate the relationship between USD/CAD returns and order flow and argue that trading volume has strong out-of-sample predictive power for USD/CAD returns. Xue and Gençay [33] demonstrate that the existence of different market traders, with multiple trading frequencies, can increase volatility persistence. Intraday asset volatility also varies with the number of market traders. Given this evidence, we model MMAR trading time by the series of trading volume ticks and provide a modified MMAR approach for out-of-sample VaR forecasting. We thereby overcome limitations of previous multifractal model applications such as their combinatorial nature and their restriction to a bounded interval. 3 In order to test the forecasting ability of our novel VaR approach, we study the out-of-sample accuracy of VaR predictions for both 12-hour and daily (24-hour) forecast horizons. While these forecast periods are somewhat arbitrary they are consistent with the trading activities expected of global financial intermediaries with a subsidiary, or branch, that is always open during the 24-hour trading day. Our high frequency data set consists of round-the-clock EUR/USD spot exchange rate prices quoted by market participants on the Reuters trading platform during the period January 5, 2006 to December 31, These prices are bundled into 5-minute time stamped intervals with the spot price and the trading ticks recorded. We find that the EUR/USD returns are multifractal, with the moments 3 Calvet and Fisher [4] overcome these shortcomings by introducing a Markov-switching multifractal model, while Lux [21] provides a further model alternative. Note that McCulloch [25] models the intraday trading time using a unifractal rather than multifractal time. 3

7 showing different scaling exponents. Our MMAR approach is then compared with forecasts based on historical simulation and a benchmark location-scale VaR model based on GARCH(1,1). The results show that the MMAR approach produces admissible VaR forecasts for the 12-hour forecast horizon. For the daily horizon, we find 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-ofsample intraday VaR. Giot [16], for example, uses GARCH models with normal and Student-t innovations and RiskMetrics model for modeling intraday VaR of 15- and 30-minute returns of three stocks traded on the NYSE. The results show that a superior model is based on Student-t innovations. Sun et al. [30] try to take account of the stylised facts of 1-minute frequency DAX returns by using a GARCH model with Lévy stable and normal innovations. The authors find that the model with Lévy stable innovations outperforms the competing intraday VaR models. Dionne et al. [11] analyse a high frequency sample consisting of 63 trading days of three stocks traded on Toronto Stock Exchange. Their backtesting results imply that a logarithmic autoregressive, conditional duration, exponential GARCH model 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 stylised facts of intraday data, including leptokurtosis and long-range dependence. An alternative group of studies in the area deals with extreme value theory (EVT) 4

8 and VaR forecasting. Gençay and Selçuk [14], Gençay et al. [15] and Wagner [31], for example, investigate the performance of VaR models for daily stock returns based on EVT. The first two papers document that the EVT VaR model provides more accurate VaR estimates, as EVT VaR approaches are more robust while GARCH VaR models are more sensitive to excessive volatility fluctuations. The third paper further addresses the question of optimal VaR 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 literature and introduces the modified version of the multifractal model of asset returns. Section 3 presents the multiple-period VaR forecasting concept used in our MMAR model. Section 4 contains an empirical study of the various models forecasting ability using the EUR/USD foreign exchange rate series and finally, Section 5 concludes. 2 Multifractal Modeling of FX Returns 2.1 Multifractality Mandelbrot et al. [23] define a multifractal stochastic process as a process that possesses a nonlinear scaling function. By definition, a linear scaling function characterizes unifractal processes. The majority of financial time series exhibit nonlinear scaling functions, see [5], [9], and [32], for example. In the following, we briefly review multifractal processes and introduce the scaling function. 5

9 Definition 2.1 A time series process {Y t } 1 t T is called multifractal, if it has stationary increments and satisfies E ( Y t,h Y t q) = c(q)h τ(q)+1, (1) for all 1 h T and 0 q Q. The constant Q denotes the highest finite moment, the h-period ahead realisation, Y t,h, is given by Y t,h = h τ=1 Y t+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 τ BM (q) = q 1, (2) 2 and the one for FBM is τ F BM (q) = qh 1, (3) for 0 < H < 1. In order to infer τ(q) from observed data, Calvet and Fisher [5] propose to estimate the parameters in a regression model which follows from Definition 2.1: log S(q, h) = τ(q) log h. (4) 6

10 The partition function S(q, h) of the time series process {Y t } 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 1 i=0 Y i h+h Y i h q. (5) 2.2 Multifractal Model of Asset Returns Based on the MMAR, we model FX returns, {R t } 1 t T, by compounding a FBM with a multifractal stochastic trading time. The FX return is defined as R t log X t log X t 1, where X t denotes a spot FX rate. Assumption 2.1 The MMAR assumes that the FX returns {R t } 1 t T follow a compound process of the form R t = B (H) [θ t ], (6) where B (H) [ ] is a FBM with self-affinity index 0 < H < 1, which operates on a multifractal stochastic trading time {θ t } 1 t T. As shown by Clark [8], the stochastic trading time relates to the tails of the process. The most important feature of FBM, as discussed in Mandelbrot and van Ness [24], is that FBM is able to model different degrees of persistence. For H = 0.5, FBM is ordinary BM with independent and identically distributed (i.i.d.) increments. Yet, B (H) [ ] is antipersistent when 0 < H < 0.5 and displays 7

11 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 θ t operate independently of each other. Assumption 2.3 The trading time θ t is given by the normalized volume ticks V t, V t = V t T t=1 (V t), (7) where V t 0 denotes the number of volume ticks at time t. It is assumed that V t is multifractal as given by the scaling law in equation (1). In our setting θ t is not a trading time based on a model assumption, but reflects the normalized number of occurring trades. The motivation for this is twofold. First, empirical work e.g. by Ma [22] supports the hypothesis that observable trading time confirms the multifractal trading time hypothesis suggested by Mandelbrot et al. [23]. Second, the actual volume ticks reflect the behaviour of the market participants more accurately than an unobservable trading time. The use of volume ticks distinguishes our work from that of previous MMAR approaches. 8

12 3 MMAR Market Risk Prediction Our MMAR market 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, we simulate MMAR sample paths under the assumptions of Section 2.2. The simulated sample paths are used for out-of-sample VaR prediction. 3.1 Value-at-Risk Given some probability 0 < α < 1, the F t 1 -measurable VaR for the h-period ahead time interval (t 1, t + h 1], V art,h α, is defined as V ar α t,h = inf {r : F t,h (r) α}, (8) which yields P(R t,h V ar α t,h F t 1) = α. The h-period ahead FX return, R t,h = log X t+h 1 log X t 1, equals the sum of one-period returns R t,h = h τ=1 R t+τ 1 and F t,h (r) = P(R t,h r F t 1 ) is its conditional distribution. 3.2 Simulation The first step of our MMAR VaR simulation approach is to simulate D MMAR sample paths R (d) R 1 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 9

13 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] = ( g T A ) σ (9) with g N (0, 1) (10) and K = AA T, (11) where K R T T is a positive-definite matrix and g R T 1 is a vector of standard Gaussian numbers, while σ is the average of the unconditional standard deviations of R t and V t. The symmetric matrix A R T T is the square root of the matrix K. The elements of the matrix K are k s,t = 1 2 ( Θ 2H t + Θ 2H s Θ t Θ s 2H), (12) for each given time 1 s t T, where {Θ t } 1 t T is the cumulative trading time that increases with time t Θ t = t θ i. i=1 Since the MMAR creates a scale-consistent sample path, which relates sim- 10

14 ulated returns over different sampling frequencies and exhibits the simplicity of self-affine processes (see e.g. [5]), we derive the corresponding h-period MMAR return of the dth simulation run, R (d) t,h, as R (d) t,h = hh R (d) t, (13) 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 estimates F (d) t,h (α). For large numbers of simulation runs D, the Law of Large Numbers assures convergence and the average of D quantile estimates converges to a stable quantile, which is the multiple-period predicted VaR: ( 1 V art,h α = D D d=1 ) F (d) t,h (α). (14) 4 Empirical Analysis 4.1 Dataset and Descriptive Statistics The 5-minute round-the-clock EUR/USD series is obtained from the Reuters trading platform, and covers the period January 5, 2006, through to December 31, 2007, for a total of 138,418 high frequency FX observations. The data is bundled into 5-minute sequential time stamped intervals with the spot price and the ticks recorded. A tick is a trade through the platform for the standard 11

15 minimum amount of about 3-5 million euro. Since the highest frequency is 5- minutes (h = 1, R t,1 = R t ), one hour corresponds to h = 12 and one day is equal to h = 288. Table 1 shows that as the sampling frequency increases, the unconditional volatility rises. Additionally, one can clearly observe the declining kurtosis leading to approximately Gaussian data. For example, 5-minute EUR/USD returns are fat-tailed (compare with Figure 2) and are highly shortterm dependent since one can observe the presence of significant autocorrelation in the absolute EUR/USD returns. Yet, this effect declines, when the sampling frequency increases. [Insert Table 1 about here] The EUR/USD returns at 5-minute frequency are illustrated in Figure 1. In Figure 2 Quantile-Quantile (QQ) plots of the sample EUR/USD returns for 5- minute and daily sampling are provided. 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 points of the empirical distribution should lie on the line. The Q-Q plot shows that 5-minute EUR/USD returns are fat-tailed. With an increasing sampling frequency (daily) the EUR/USD returns are less tailed, but still have fatter tails than a normal distribution. This finding is not uncommon for FX returns. For example, Lipton- Lifschitz [18] find a kurtosis of five in daily USD/DEM returns from 1986 to [Insert Figure 1 about here] 12

16 [Insert Figure 2 about here] Figure 3 illustrates 5-minute EUR/USD trading ticks V t and the cumulative trading time Θ t. [Insert Figure 3 about here] 4.2 Evidence of Multifractality Figure 4 presents EUR/USD return partition functions for moments 1 q 5 in the scaling region up to one week. The partition functions are plotted against h using logarithmic scales. For q [1, 3] the lines are approximately linear. Generally higher 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 time horizon of about 100 minutes. [Insert Figure 4 about here] Figure 5 illustrates estimated scaling functions of EUR/USD returns and Brownian motion. The scaling function of ordinary Brownian motion (dashed line) as defined in equation (2) is linear, which is typical of unifractal processes. The scaling function of EUR/USD however, is completely different and obviously nonlinear, which is a characteristic of multifractal processes (see Definition 2.1). [Insert Figure 5 about here] 13

17 4.3 Out-of-Sample VaR Forecasts Model Estimation In order to examine out-of-sample performance, we calculate the 1% VaR from January 6, 2006 to December 31, 2007 for 12-hour (h = 144) and daily (h = 288) forecast horizons. The matrix A which is the square root of the matrix K is computed by the Cholesky decomposition. The self-affinity index H of R t is estimated using the variance of residuals approach of Peng et al. [28]. We simulate D = 5, 000 MMAR sample paths to predict the multiple-period VaR. The VaR results of our MMAR approach are compared to two alternatives. All parameters of our VaR models (MMAR and the two benchmark models) are calculated using a moving window, which consists of minute high frequency EUR/USD returns and volume ticks. The time-t VaR forecasts are updated every time h, which results in 138, successive VaR predictions. h Benchmark Models First, instead of simulated multifractal return series, we use historical simulation of EUR/USD returns. The multiple-period VaR based on this historical simulation is: where F t (α) denotes the empirical α-quantile of R t. V ar α t,h = ( hf t (α) ), (15) 14

18 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: V ar α t,h = ( µ t,h + σ t,h F (α) ), (16) where F is the inverse of the innovations distribution. The multiple-period mean, µ t,h, as well as the multiple-period volatility, σ t,h, are determined by the GARCH(1,1) model of Bollerslev [3]. Estimation of the GARCH(1,1) model with skewed Student-t innovations is carried out by the maximum likelihood method. Summary Results The out-of-sample VaR forecast results are summarized in Table 2. We provide the average value of the VaR estimates, V ar, and the empirical coverage rate, α. We use the Christoffersen [7] LR statistics for VaR performance evaluation. E.g. Berkowitz and O Brien [2] also use the Christoffersen [7] LR statistics to study the VaR performance of six U.S. banks. Sun et al. [30] evaluate highfrequency VaR predictions with the Christoffersen [7] approach. Appendix A contains a detailed explanation. 4 [Insert Table 2 about here] We consider a violation of a backtesting statistic to have occurred when a corresponding p-value of the respective test statistic is below 5 percent. 4 All models are validated based on non-overlapping h-period EUR/USD returns in order to provide independent out-of-sample tests. 15 The

19 LR uc statistic tests for the correct violation level, which is α = 1% in our setting. VaR forecasts may be correct on average, but produce violation clusters, a phenomenon ignored by unconditional coverage as it assumes that violations are independent (see e.g. [7], [30]). In case VaR violations occur in clusters then cumulative losses may turn out to be much higher than under the case of independence. These violation clusters often occur when a VaR model does not or is relatively slow in adopting to changing market conditions. The LR ind statistic tests for independence of the VaR violations, while the conditional coverage statistic, LR cc, combines both concepts. As a result, the LR cc tests not only if violations occur in α percent of time, but also tests if they are independent over time (see e.g. [2], [7], [30]). Consequently, we consider LR cc as the most important backtesting statistic. The VaR predictions of the three models for the 12-hour horizon from January 6, 2007 to December 31, 2008 are plotted in Figure 6. The dash dotted line refers to the MMAR approach, the dashed line is historical simulation and the solid line is the GARCH based VaR model. In the following, we analyze the models performance in detail. Unconditional Coverage versus Conditional Coverage The unconditional coverage test indicates that for the 12-hour 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 is too low. Our MMAR approach passes all backtesting statistics for p- 16

20 values above For both horizons, we find that our MMAR model achieves better conditional coverage than historical simulation and the GARCH based VaR. All models pass the independence test indicating that there is no severe clustering of VaR violations over time. As a result, backtesting violations result due to VaR predictions exceeding actual levels. 12-hour versus Daily Forecast Horizon Comparing 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 higher frequency often obtain more extreme returns than lower frequency returns. Our MMAR model has less problems in such an environment. First, the MMAR is characterized by a special form of time-invariance, which combines extreme returns with long memory [5]. Second, our MMAR approach uses trading volume. This is important, as increased trading activities can produce extreme returns. The predicted VaR using a GARCH estimation approach does not violate any backtesting statistic for the daily horizon. The MMAR performs best for the 12-hour horizon: We find that the p-value of LR cc (which is 0.892) and the empirical violation level α = 1.04% both report excellent VaR forecasting. Concerning the daily horizon, we report good backtesting results for the MMAR, although this is inferior to the 12-hour horizon. To conclude, for both horizons, our MMAR model dominates the VaR results of both the historical simulation 17

21 and the GARCH approach and never violates a backtesting statistic. [Insert Figure 6 about here] Implications Regulatory 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 lack sufficient risk capital. In our setting for example, consider the case of a bank that takes a one million euro long position against the US dollar and intends to close it after 12 hours. MMAR would estimate 9,800 euro as an average risk capital level, historical simulation and GARCH based VaR would predict average levels of about 7,000 and 8,000 euro, respectively. Given this and the empirical violation rates reported in Table 2, the bank with an oversimplified risk model could end up considerably undercapitalised. While the empirical violation rate of MMAR is roughly 1 percent and hence close to the target rate, the predicted capital 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 requirements but also for internal risk management tasks. 18

22 We propose a novel MMAR approach to intraday VaR forecasts for EUR/USD FX returns. The approach provides superior forecast results when compared with historical simulation and a standard GARCH model. The MMAR model not only considers multifractality in FX returns, it is also a convenient model, which is parsimonious in its specification. 19

23 References [1] Bank for International Settlements, Triennial Central Bank Survey: Foreign exchange turnover in April 2013: preliminary global results, Monetary and Economic Department, September 2013: [2] J. Berkowitz, J. O Brien, How accurate are value-at-risk models at commercial banks?, Journal of Finance 57 (2002) [3] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31 (1986) [4] L. Calvet, A. Fisher, Forecasting multifractal volatility, Journal of Econometrics 105 (2001) [5] L. Calvet, A. Fisher, Multifractality in asset returns: Theory and evidence, Review of Economics and Statistics 84 (2002) [6] L. Calvet, A. Fisher, How to forecast long-run volatility: Regimeswitching and the estimation of multifractal processes, Journal of Financial Econometrics 2 (2004) [7] P. Christoffersen, Evaluating interval forecasts, International Economic Review 39 (1998) [8] P.K. Clark, A subordinated stochastic process model with finite variance for speculative prices, Econometrica 41 (1973)

24 [9] R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues, Quantitative Finance 1 (2001) [10] Z. Ding, C.W.J. Granger, R.F. Engle, A long memory property of stock market returns and a new model, Journal of Empirical Finance 1 (1993) [11] G. Dionne, P. Duchesne, M. Pacurar, Intraday Value at Risk (IVaR) using tick-by-tick data with application to the Toronto Stock Exchange, Journal of Empirical Finance 16 (2009) [12] Z. Eisler, J. Kertész, Multifractal model of asset returns with leverage effects, Physica A 343 (2004) [13] J. Fillol, Multifractality: Theory and evidence an application of the French stock market, Economics Bulletin 3 (2003) [14] R. Gençay, F. Selçuk, Extreme value theory and Value-at-Risk: Relative performance in emerging markets, International Journal of Forecasting 20 (2004) [15] R. Gençay, F. Selçuk, A. Ulugülyaǧci, High volatility, thick tails and extreme value theory in value-at-risk estimation, Insurance: Mathematics and Economics 33 (2003) [16] P. Giot, Market risk models for intraday data, European Journal of Finance 11 (2005)

25 [17] M. King, L. Sarno, E. Sojli, Timing exchange rates using order flow: The case of the Loonie, Journal of Banking and Finance 34 (2010) [18] A. Lipton-Lifschitz, Predictability and unpredictability in financial markets, Physica D 133 (1999) [19] R. Liu, T. Lux, Long memory in financial time series: Estimation of the bivariate multi-fractal model and its application for value-at-risk, Working Paper, University of Kiel, [20] I.N. Lobato, N.E. Savin, Real and spurious long-memory properties of stock market data, Journal of Business and Economic Statistics 16 (1998) [21] T. Lux, The Markov-Switching multi-fractal model of asset returns: GMM estimation and linear forecasting of volatility, Journal of Business and Economic Statistics 26 (2008) [22] G. Ma, Multiscaling trading time, Working Paper, Brandeis University, [23] B.B. Mandelbrot, A. Fisher, L. Calvet, A multifractal model of asset returns, Working Paper, Cowles Foundation, [24] B.B. Mandelbrot, J.W. Van Ness, Fractional Brownian motion, fractional noises and applications, SIAM Review 10 (1968)

26 [25] J. McCulloch, Fractal market time, Journal of Empirical Finance 19 (2012) [26] R.F. Mulligan, R. Koppl, Monetary policy regimes in macroeconomic data: An application of fractal analysis, Quarterly Review of Economics and Finance 51 (2011) [27] R. Nekhili, A. Altay-Salih, R. Gençay, Exploring exchange rate returns at different time horizons, Physica A 313 (2002) [28] C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Mosaic organization of DNA nucleotides, Physical Review E 49 (1994) [29] D. Sornette, V.F. Pisarenko, Properties of a simple bilinear stochastic model: Estimation and predictability, Physica D 237 (2008) [30] W. Sun, S. Rachev, F.J. Fabozzi, A new approach for using Lévy processes for determining high frequency value-at-risk predictions, European Financial Management 15 (2009) [31] N. Wagner, Estimating financial risk under time-varying extremal return behavior, OR Spectrum 25 (2003) [32] Z. Xu, R. Gençay, Scaling, self-similarity and multifractality in FX markets, Physica A 323 (2003)

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28 A Backtesting VaR Market risk models predict VaR with random error. The validity of a VaR prediction model is measured based on predicted versus actual loss levels. To evaluate the out-of-sample performance of the proposed models we follow the concept of Christoffersen [7]. The indicator (or hit) function I t = 1 {Rt,h < V art,h α } represents the history of observations, t = 1,..., T, for which losses in excess of the predicted VaR occur. A.1 Unconditional Coverage When a VaR model is designed perfectly, the number of observations that fall outside the predicted VaR should be exactly in line with the given VaR level, such that E(I t F t ) = α holds. Hence, the test of unconditional coverage is H 0 : E(I t F t ) = α vs. H 1 : E(I t F t ) α. Under the null hypothesis, the likelihood-ratio (LR) test statistic follows as LR uc = 2 ln[l(α)/l( α)] χ 2 (1), (A.1) where L(α) is the binomial likelihood with parameter α and α = 1 T T t=1 I t is the maximum likelihood estimator of α.

29 A.2 Independence Besides the above requirement, VaR violations should be independent, which requires an additional test. Let n ij denote the number of observations for which I t = j occurred following I t 1 = i and assume that {I t } is a first-order Markov chain with transition probabilities π ij = P(I t = j I t 1 = i). This yields the likelihood L(Π) = (1 π 01 ) n 00 π n (1 π 11 ) n 10 π n Maximum likelihood estimators for the transition probabilities are: π 01 = n 01 n 00 + n 01, and π 11 = n 11 n 10 + n 11. Under the null hypothesis of independence, P(I t = 0) = π 0 = π 01 = π 11, which implies L(π 0 ) = (1 π 0 ) n 00+n 10 π n 01+n The maximum likelihood estimate for π 0 is π 0 = n 01 + n 11 n 00 + n 10 + n 01 + n 11. Based on π 0 and Π, the independence LR test statistic is LR ind = 2 ln[l( π 0 )/L( Π)] χ 2 (1). (A.2)

30 A.3 Conditional Coverage The LR ind statistic (A.2) tests for independence, but it does not take coverage into account. Christoffersen [7] therefore proposes a combined test statistic: LR cc = LR uc + LR ind = 2 ln[l(α)/l( Π)] χ 2 (2). (A.3)

31 B Tables Table 1: EUR/USD Return Statistics Summary statistics of EUR/USD returns for various levels of aggregation h. The Phillips- Perron unit root test statistics indicate that the null has to be rejected in favour of the stationarity alternative. Sample period is from January 5, 2006 to December 31, Frequency 5 minutes 1 hour 1 day 1 week Size 138,417 11, Mean Std. Dev Skewness Kurtosis AC of R t (1) AC of R t (2) AC of R t (3) AC of R t (1) AC of R t (2) AC of R t (3)

32 Table 2: VaR Forecasts Results of 1%-VaR predictions for 12-hour (h = 144) and daily (h = 288) forecast horizons. V ar denotes average sample VaR. LR-statistics are as defined in Appendix A, where * and ** denote rejection of the null hypothesis at the 5 and 1 percent significance levels, respectively (corresponding p-values in parenthesis). All VaR forecasts are based on a moving window which consists of minute high frequency EUR/USD returns and volume ticks. MMAR denotes the multifractal model of asset returns, HS is the historical simulation and GARCH refers to the generalized autoregressive conditional heteroskedasticity model. Horizon Size MMAR HS GARCH 12-hour 959 V ar α LR uc ** 10.24** [0.895] [0.000] [0.001] LR ind [0.646] [0.247] [0.332] LR cc ** 11.18** [0.892] [0.000] [0.004] 1 day 479 V ar α LR uc ** 0.29 [0.709] [0.001] [0.593] LR ind [0.795] [0.795] [0.696] LR cc ** 0.44 [0.902] [0.006] [0.803]

33 C Figures Figure 1: Continuously compounded 5-minute EUR/USD returns. January 5, 2006 to December 31, Sample period is from

34 (a) 5-minute EUR/USD Returns (b) Daily EUR/USD Returns Figure 2: Quantile-Quantile (Q-Q) plots of EUR/USD returns are provided in (a) for 5 minutes 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 empirical distribution 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, (a) 5-minute Volume Ticks (b) Cumulative Trading Time Figure 3: Panel (a) illustrates 5-minute EUR/USD trading ticks V t. A tick is a trade through the platform for the standard minimum amount of between 3-5 million euro. Panel (b) contains the cumulative trading time Θ t. Sample period is from January 5, 2006 to December 31, 2007.

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

36 Figure 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 processes have nonlinear scaling functions (see Definition 2.1). The estimated τ(q) (solid line) of EUR/USD returns is nonlinear and concave. Sample period is from January 5, 2006 to December 31, 2007.

37 View publication stats Figure 6: 12-hour ahead VaR forecasts for our MMAR approach (dash dotted line), historical simulation (dashed line) and GARCH (solid line). The forecasting period is from January 6, 2006 to December 31, 2007.