Decay factor optimisation in time weighted simulation — Evaluating VaR performance

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International Journal of Forecasting 27 (2011) 1147–1159 www.elsevier.com/locate/ijforecast

Decay factor optimisation in time weighted simulation — Evaluating VaR performance ˇ Saˇsa Zikovi´ c a,1 , Bora Aktan b,∗ a University of Rijeka, Faculty of Economics, Rijeka, Croatia b Yasar University, Faculty of Economics and Administrative Sciences, Bornova, Izmir, Turkey

Abstract We propose an optimisation approach for determining the optimal decay factor in time weighted (BRW) simulation. The backtesting of the BRW simulation, which involves different decay factors, together with a broad range of competing VaR models, has been performed on a sample of seven stock indexes and two commodities: gold and WTI oil. The results obtained show that the BRW simulation with an optimised decay factor relative to the Lopez (1998) size-adjusted function is among the best performing VaR models, second only to the conditional extreme value approach (McNeil & Frey, 2000). The optimised decay factors are sufficiently stable over time, giving economic justification to the optimisation because they do not change over longer time periods. Unlike most of the VaR models tested, in the large majority of cases, the optimised BRW model passes the Basel II criteria but yields significantly lower VaR forecasts than the extreme value approaches, thus resulting in a lower idle capital, i.e. lower costs. c 2011 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. ⃝ Keywords: Value at Risk; Time weighted (BRW) simulation; Optimisation; Decay factor

1. Introduction In spite of the various advanced Value at Risk (VaR) estimation techniques which are at our disposal, there is a need for computationally faster, less demanding and less costly VaR models. These models are ∗ Corresponding author. Tel.: +90 232 411 5000; fax: +90 232 411 5020. ˇ E-mail addresses: [email protected] (S. Zikovi´ c), [email protected], [email protected] (B. Aktan). 1 Tel.: +385 51 355 179; fax: +385 51 212 268.

implemented by less conservative investors, and serve as a quick approximation of the true level of risk. One of the oldest, simplest and most widely used approaches to VaR calculation is historical simulation (HS). In historical simulation, potential changes in the risk factors are assumed to be identical to those that have occurred in the past. Modelling the risk factors that underlie the changes in the portfolio’s value shortens the computation time, because the number of relevant risk factors is smaller than the number of instruments in a portfolio. HS assumes that the historically observed factor changes used

c 2011 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. 0169-2070/$ - see front matter ⃝ doi:10.1016/j.ijforecast.2010.09.007

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in the simulation are independently and identically distributed (IID). The main strength of HS is that it is non-parametric, i.e., there are no ex ante explicit distributional assumptions about the data, except for the hidden assumption that the distribution of returns in the observation period will be identical to the distribution of future returns. In a study using simulated spot foreign exchange portfolios, Hendricks (1996) established that even with departures from normality, HS provides good estimates at the 99th percentile. In applying HS, there is a trade-off between long observation periods, which potentially violate the assumption of IID observations, and short observation periods, which reduce the forecasting precision. When relaxing the assumption that the returns are IID, it is reasonable to assume that simulated returns from the recent past represent the current portfolio risk better than those from the distant past. Boudoukh, Richardson, and Whitelaw (1998) (BRW hereafter) used this idea to generalize the HS by assigning a relatively higher probability to the returns from the more recent past. This small modification to the HS seems to make a significant difference to the model ˇ performance, see Pritsker (2001) and Zikovi´ c (2006). Boudoukh et al. (1998) tested the performance of their simulation on the USD/DEM exchange rate, the spot oil prices and the S&P500 stock index. They found that the BRW simulation performs better than both the parametric models and HS, and in doing so produces independent VaR exceptions. The BRW method appears to remedy the main problems of HS. Similarly, ˇ Zikovi´ c (2006) showed that the BRW simulation with a decay factor of 0.99 is superior to HS for a range of confidence levels in volatile, illiquid transitional markets. The most extensive study to date analysing the behaviour and characteristics of historical, BRW and filtered historical simulation (FHS) was provided by Pritsker (2001). Pritsker determined that the BRW and historical simulation adjust very slowly to the changes in risk at the 95% and 99% confidence levels. He found very little advantage from using the BRW simulation with standard decay factor values of 0.97 and 0.99 rather than HS. However, the correlation of the VaR estimates with the true VaR is fairly high for the BRW simulation relative to historical simulation. In the long run, the BRW forecasts change parallel to the true VaR, but are slow to respond to changes in risk. As a result, neither the VaR estimates based on

the historical simulation nor those based on the BRW model are very accurate. Because of the problematic features of both historical simulation and the BRW method, past research has focused on methodologies modelling the conditional heteroskedasticity and nonnormality in a theoretically more coherent fashion. One such semi-parametric approach which is widely used in practice involves weighting the historical data by the conditional volatility rather than time. The basic idea of this approach, which was originally suggested by Hull and White (1998), is to update the historical information set by scaling it by the current volatility, thus reflecting recent developments in the market. If we use bootstrapping on the historical residuals and a GARCH instead of EWMA volatility structure, we obtain the filtered historical simulation (FHS), a term coined by Barone-Adesi, Giannopoulos, and Vosper (1999), which is basically just a slightly improved version of the Hull and White (1998) idea. Apart from the studies mentioned above, the BRW simulation has not been analysed extensively in the mainstream VaR literature. In other words, we could not find any papers analysing or forecasting the risk using decay factor values other than those suggested originally by Boudoukh et al. (1998). We base our research on the premise that it is not correct to judge the performance of any model using ad hoc parameter values. The goal of this paper is to present an optimisation procedure for determining the decay factor in the time weighted BRW simulation, and explore the benefits of such an optimisation for VaR estimation. To the best of our knowledge, this is the first paper to apply an optimisation approach to the decay factor estimation within the BRW simulation. All other studies with which we are familiar take the values suggested by Boudoukh et al. (1998) as given. In that regard, we believe that this paper contributes to both science and practice. It presents an optimisation approach for estimating the decay factor in BRW simulation, identifies the value gained through the optimisation of the decay factor, analyses the stability of the optimal decay factor values, and compares the optimised BRW model with a wide range of VaR models. The VaR models that are analyzed in this paper include: (1) the normally distributed variancecovariance model (VCV), (2) the RiskMetrics system, (3) historical simulation (HS) with 100, 250 and 500 day rolling windows, (4) time weighted (BRW)

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simulation with different decay factors, volatility weighted historical simulation (HW HS) (Hull & White, 1998) with EWMA conditional volatility, (5) the filtered historical simulation model (FHS) (Barone-Adesi et al., 1999), (6) a parametric GARCH model with the volatility specification which yields the highest value according to the Akaike Information Criterion (AIC), (7) an unconditional EVT approach using the Generalized Pareto distribution (GPD) (see Embrechts, Kluppelberg, & Mikosch, 1997), and (8) the conditional quantile EVT-GARCH approach developed by McNeil and Frey (2000). We tested the optimised decay factor BRW model and the other VaR models on seven stock market indexes (US — S&P500, DJIN, RTY; Japan — NIKKEI; Great Britain — FTSE; Germany — DAX; France — CAC), spot gold, and West Texas Intermediate (WTI) oil one-month futures, since it is a benchmark contract for oil. Gold is analysed because it is viewed as a safe harbour in times of crisis and is highly liquid: it can readily be bought or sold in large denominations and at narrow spreads, 24 h a day. Numerous research papers have also pointed out the benefits of including gold holdings, since they contribute to a more balanced portfolio (Aggarwal & Soenen, 1988; Draper, Faff, & Hillier, 2006; Johnson & Soenen, 1997). The rest of the paper is organised as follows: Section 2 discusses the characteristics of the time weighted (BRW) historical simulation approach to measuring VaR; Section 3 presents the characteristics and statistics of the analysed assets; Section 4 describes the optimisation procedure for obtaining decay factor values; and Section 5 discusses the optimisation results. In Section 6, we analyse and present the backtesting results of the optimal decay factor BRW model and other VaR models on the selected stock indexes and commodities. Our conclusions are summarized in Section 7. 2. Time weighted (BRW) historical simulation Institutional investors often rely on VaR figures calculated by historical simulation (HS VaR) because this drastically simplifies the procedure for computing the VaR, as it does not require any explicit distributional assumptions about portfolio returns. The value of the VaR is calculated as a percentile or order statistic of the set of portfolio returns. Generally

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speaking, it is easy to construct a time series of historical portfolio returns using current portfolio holdings and historical asset returns. A potential problem that is that historical asset prices for assets held at some point in time may not be available. In such cases, “pseudo” historical prices must be constructed using pricing models, factor models, or some ad hoc consideration. Assets without historical prices can also be matched to “similar” assets based on capitalization, industry, leverage and duration. Historical “pseudo” asset prices and returns can then be constructed using the historical prices of substitute assets: rw,t =

N −

wi,T ri,t ≡ WT′ Rt ,

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

i=1

The HS VaR can then be expressed as: HS − VaRαT +1|T ≡ rw ((T + 1)α),

(2)

where rw ((T + 1)α) is taken from the set of ordered “pseudo” returns {rw (1), rw (2), . . . , rw (T )}. If (T + 1)α is not an integer value, then the two adjacent observations can be interpolated to calculate the VaR. HS has several serious defects, which have been well documented, see Pritsker (2001). Most importantly, it does not incorporate conditionality into the VaR forecast properly. The only source of dynamics in the HS model lies in the observation window which is updated over time. In practice, this source of conditionality is minor. Another shortcoming of the HS is that it assigns an equal probability weight of 1/N to each observation, where N is the window length. This means that the HS estimate of VaR at the α confidence level corresponds to the N (1 − α) lowest return in the period N rolling sample. Since a crash is the lowest return in the N period sample, the N (1 − α) lowest return after the crash turns out to be the (N (1 − α) − 1) lowest return before the crash. If the N (1 − α) and (N (1 − α) − 1) lowest returns happen to be very close in magnitude, the crash actually has almost no impact on the HS estimate of VaR. The HS VaR only changes significantly if the observations around the order statistic rw ((T + 1)α) change significantly. Although HS makes no explicit assumptions about the distribution of the portfolio returns, an implicit assumption is hidden behind the procedure: the distribution of portfolio

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returns will not change within the window. Several problems may arise from this implicit assumption when using this method in practice. If we assume that all returns within the observation window used in HS have the same distribution, then all returns from the entire time series also have the same distribution: if yt−window , . . . , yt and yt+1−window , . . . , yt+1 are IID, then yt+1 and yt−window are also IID, by the transitive property. This means that the VaR forecasts under historical simulation are meaningful only if we assume that the entire historical data set used in the calculation has the same, constant distribution. Another problem connected with the HS lies in the fact that in order for the empirical quantile estimator to be consistent, the size of the observation window must be infinite. The length of the window must meet two contradictory requirements: it must be large enough to make statistical inference significant, but short enough to avoid the risk of taking observations outside the current volatility cluster. Clearly, there is no easy solution to this problem. If the market moves from a period of low volatility to a period of high volatility, VaR forecasts based on the HS will under-predict the true risk of a position, since it will be some time before the observations from the low volatility period drop out of the observation window. Finally, VaR forecasts based on HS may have predictable jumps due to the discreteness of extreme returns. If the VaR of a portfolio is computed using a rolling window of N days and today’s return is a large negative number, it is easy to predict that the VaR estimate will jump upward because of today’s observation. The reverse effect will reappear after exactly N days, when the large observation drops out of the observation window. When relaxing the assumption that the returns are IID, it is reasonable to assume that the simulated returns from the recent past will represent the current portfolio’s risk better than the returns from the distant past. Boudoukh et al. (1998) used this idea to introduce a generalization of HS, and assigned higher probabilities to the returns from the more recent past. The BRW approach combines exponential smoothing and HS, by applying exponentially declining weights to past returns of the portfolio. Each of the most recent N returns of the portfolio, yt ,yt−1 , . . . , yt−N +1 , is associated with 1−λ 1−λ 1−λ N −1 , provided a weight, 1−λ N , ( 1−λ N )λ, . . . , ( 1−λ N )λ that λ < 1. After the probability weights have

been assigned, the VaR is calculated based on the empirical cumulative distribution function of returns with modified probability weights. Under the BRW approach, the first probability weight asymptotically N →∞

1−λ approaches the value of 1−λ, 1−λ N −→ 1−λ. In real life situations, with limited data sets, this means that the most recent return receives a probability weight of somewhat over 1% for λ = 0.99 and over 3% for λ = 0.97, i.e., the shorter the observation period, the higher the probability weight. In both cases, if the most recent observation is the highest loss in N days, it will automatically become the VaR estimate at the 99% confidence level, which is not the case for either HS or FHS. The BRW method appears to remedy one of the main problems of HS and FHS, since large losses are immediately reflected in the VaR forecasts. The simplest way to implement the BRW approach is to construct a history of N hypothetical returns that a portfolio would have earned if it had been held for each of the previous N days, rt−1 , . . . , rt−N , and then assign exponentially declining probability weights wt−1 , . . . , wt−N to the return series. Given the probability weights, the α% confidence level VaR can be approximated from G(.; t; N ), the empirical cumulative distribution function of r based on the return observations:

G(r ; t, N ) =

N −

1{rt−i ≤cl} wt−i ,

i=1 N −

wt−i = 1,

(3) wt−i−1 = λwt−i .

i=1

The empirical cumulative distribution function, unless smoothed, is discrete; therefore, in most cases, the solution for the VaR at the α confidence level does not correspond to a particular return from the return history. Instead, the BRW solution for VaR at the α percent confidence level is between the return with a cumulative distribution lower than α and the return with a cumulative distribution higher than α. These returns can be used as estimates of the BRW VaR model at the confidence level α. The estimate that understates VaR at the α% confidence level (upper limit) is given by Pritsker (2001): BRW u (t|λ, N , α) = inf(r ∈ {rt−1 , . . . , rt−1−N }|G(r ; t, N ) ≥ α), (4)

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whereas the estimator of the lower limit is given by: BRW o (t|λ, N , α) = sup(r ∈ {rt−1 , . . . , rt−1−N }|G(r ; t, N ) ≤ α), (5) where λ is the exponential weight factor, N is the length of the history of returns used to compute VaR, and α is the VaR confidence level. An alternative approach to solving this problem is the kernel smoothing technique suggested by Butler and Schachter (1998). Although the BRW approach also suffers from logical inconsistencies, it still represents a significant improvement over HS, see Pritsker (2001). It drastically simplifies the assumptions needed in parametric models and incorporates a more flexible specification than HS. In order to fully understand the link between HS and the assumptions behind the BRW approach, the BRW quantile estimator can be expressed as: V a Rt+1,α =

t −

rj I

j=t−N +1

×

 N −

 f i (λ; N )I (rt+1−i ≤ r j ) = α ,

(6)

i=1

where f i (λ; N ) are the weights associated with return ri and I (·) is the indicator function. If f i (λ; N ) = 1/N , the BRW quantile estimator equals the HS estimator. The main difference between the BRW approach and HS is the specification of the quantile process. With HS, each return is given the same weight, while in the BRW approach, the returns are assigned different weights, depending on the date of their occurrence. Strictly speaking, none of these models is completely nonparametric, since a parametric specification is proposed for the quantile. Boudoukh et al. (1998) set λ equal to 0.97 and 0.99, but did not propose any method for estimating this key unknown parameter. The lack of an appropriate method for estimating the optimal decay factor is our main issue. 3. Assets analysed The data used in the analyses of the VaR models include daily log return series from seven stock indexes: US — S&P500, DJIN, RTY; Japan — NIKKEI; Great Britain — FTSE; Germany — DAX;

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and France — CAC; and two commodities: spot gold and WTI oil one-month futures. The returns are collected from the Bloomberg web site for the period 04/01/2000–02/01/2009. The VaR figures calculated are for a one-day horizon at the 99% confidence level. To secure the same out-of-sample VaR backtesting period for all of the assets tested, the out-of-sample data sets were formed by taking out 1500 of the latest observations for each asset (approximately 6 years of daily data). The rest of the observations were used as presample observations for the VaR starting values and the volatility model calibration. Table 1 provides a summary of the descriptive statistics and normality tests for the daily log returns of the assets analysed. All of the assets analysed show a significant degree of leptokurtosis, i.e. fatter tails than would be assumed under normality, ranging from 7 in the case of WTI oil futures to almost 12 in the case of the S&P500 index. This characteristic, together with asymmetry, which is very pronounced in some cases, such as that of the NIKKEI index, shows that investors can expect negative extremes of a higher frequency and magnitude than would be assumed under a normal distribution. In line with the basic statistics, both normality tests show that the data generating processes behind the assets analysed are not normally distributed. All of the assets analysed exhibit heteroskedasticity and autoregression in their returns, with the exception of the NIKKEI index. These characteristics represent a problem for parametric VaR models, as well as for nonparametric and semi-parametric approaches that are based on the IID assumption, such as HS and the BRW approach. Since the elementary assumptions of the majority of VaR models are not met, the VaR figures obtained from such models should be used with caution. To estimate VaR models that use the GARCH volatility, specifically the FHS, GARCH and EVT GARCH models, we first fit the ARMA( p, q) model to daily returns. We find that, in all cases, the ARMA( p, q) process provides a satisfactory representation of the analysed asset returns’ dynamic behaviour. In the second step, we consider several specifications for the conditional variance of the residual terms, namely GARCH, EGARCH and GJR-GARCH, with normal, Student-t and GED innovations. The ARMAGARCH parameters were calculated using daily rolling windows of 1.000 trading days. In order to

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Table 1 Summary statistics for log returns for the period 04/01/2000–02/01/2009. SP500

DJIN

RTY

NIKKEI

FTSE

DAX

CAC

GOLD

OIL

Mean Median Min Max Standard dev. Skewness Kurtosis

−0.0002 0.0004 −0.0947 0.1096 0.0136 −0.1222 11.8657

−0.0001 0.0004 −0.0820 0.1051 0.0129 0.0024 11.5056

0.0003 0.0005 −0.1998 0.1777 0.0265 0.0905 8.8155

−0.0003 −0.0001 −0.1211 0.1323 0.0162 −0.3374 10.0597

−0.0002 0.0002 −0.0927 0.0938 0.0133 −0.1223 9.8494

−0.0001 0.0006 −0.0887 0.1080 0.0167 0.0580 7.6937

−0.0002 0.0002 −0.0947 0.1059 0.0157 0.0312 8.4946

0.0005 0.0006 −0.0676 0.0739 0.0115 −0.1205 7.5100

0.0003 0.0013 −0.1654 0.1641 0.0254 −0.2719 7.0072

Normality tests* Shapiro/Francia (%) Lilliefors test (%) Autoregresion Heteroskedasticity

0 0 AR(1) EGARCH-t

0 0 AR(1) EGARCH-t

0 0 0 0 AR(1) GARCH-t EGARCH-t

0 0 AR(1) EGARCH-t

0 0 ARMA(1, 1) EGARCH-t

0 0 ARMA(1, 1) EGARCH-t

0 0 AR(1) EGARCH-t

0 0 AR(1) GARCH-t

For all the time series, the descriptive statistics for daily returns are expressed in percentages. ∗ The values are p-values.

capture the dynamics of the data generating process and the presence of the “leverage effect” in almost all of the assets analysed, with the exception of the RTY stock index and oil futures, we had to employ the EGARCH model with T distributed innovations. As expected, the asymmetry parameter, which controls the asymmetric impact of positive and negative shocks on the conditional variance, indicates the presence of a higher conditional volatility after negative shocks. The diagnostic tests of the standardized and squared residuals indicate that the GARCH(1, 1) processes employed are able to take into account the heteroskedasticity present in the data set analysed, since the Ljung-Box test statistic for squared standardized residuals shows no serial correlation. The ARCH-LM test also finds no ARCH effects in the residuals. 4. The BRW decay factor optimization There are many metrics in the VaR backtesting framework relative to which the BRW decay factor parameter can be optimised. The backtesting literature offers a wide spectrum of backtesting procedures to choose from: see Berkowitz (2001), Blanco and Ihle (1998), Christoffersen (1998), Crnkovic and Drachman (1996), Kupiec (1995) and Lopez (1998), among others. For the purpose of optimising the decay factor in the BRW simulation, we opted for the Lopez (1998) size-adjusted loss function. The Lopez (1998) test

is based on the forecast evaluation approach to backtesting and is derived from the evaluation methods which are often used in ranking macroeconomic models. The flexible nature of the test allows the analyst to emphasize any particular concern one might have. For example, extreme losses can be given a greater weight if they are of greater concern. Besides taking into account the frequency of VaR exceedances, it also accounts for the size of these exceedances. Thus, it penalizes the VaR models that allow higher excesses, i.e. larger losses. For investors and company stakeholders, the size of the VaR exceedances is just as important as their frequency, or maybe even more important. In practice, if exceedances are more frequent than desired but are only marginal in size, they represent far less of a problem than a lower frequency of losses with large magnitudes. The first input in a forecast evaluation model is a set of paired observations of returns and their associated VaR forecasts. The second input is a loss function that assigns a score to each observation, depending on how the observed return compares to the VaR forecast for a given period. To evaluate the forecasts, it is necessary to specify the loss function. Lopez (1998) suggested a size-adjusted loss function:  1 + (L t − V a Rt )2 if L t > V a Rt Ct = (7) 0 if L t ≤ V a Rt , where L t represents the loss and V a Rt the calculated VaR values at time t. This loss function allows the sizes of the tail losses to influence the final rating of the

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VaR model in a simple and straightforward manner. The VaR model generating higher tail losses generates higher values under this loss function than the VaR model generating lower tail losses, ceteris paribus. We performed the optimisation of the decay factor for BRW simulation with regard to minimising the Lopez (1998) size-adjusted function. The decay factor that minimizes the Lopez size adjusted function for a given time series is chosen as optimal, since it minimizes the deviation (positive or negative) between the observed and expected VaR exceedances, while taking their size into account. Thus, we treat the overconservative and inadequate VaR forecasts equally. The optimisation procedure can be written as follows: λopt is the decay factor for which |Ct (λopt )| = minλ |Ct (λ)|, where  1 + (L t − V a Rt,α )2 if L t > V a Rt,α Ct (λ) = (8) 0 if L t ≤ V a Rt,α , V a Rt,α =

t −

rj I

j=t−N

×

 N −

 f i (λ; N )I (rt−i ≤ r j ) = α ,

i=1

L t represents the loss and V a Rt the forecasted VaR values at time t. The optimisation procedure is straightforward. The proposed algorithm runs through the decay factor values, calculates the VaR at each step and records the VaR performance in the backtesting period at the selected confidence level. After finishing its runs through the decay factor values and recording the VaR performance for each decay factor, it searches for the BRW VaR model with the lowest Lopez size-adjusted value in absolute terms. The decay factor used in the BRW VaR model with the lowest Lopez sizeadjusted value is chosen as optimal, since it produces the lowest possible deviation from the realised level of risk, i.e. the frequency of exceedances and their magnitude. 5. The BRW decay factor optimisation results Based on the presented optimisation procedure, we calculated the optimal values of the decay factors in non-overlapping periods of 250, 500 and 1500

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trading days for each of the analysed stock indexes and commodities for the period 2003–2009 (Table 2). The results obtained for the 250- and 500-day intervals are also presented in Fig. 1. The values of the optimal decay factor from the analysed stock markets show a lot of common characteristics. In the first 500 trading days (between 2003 and 2005), the optimal decay values were significantly below 0.99 (mostly between 0.97 and 0.98), and they gradually increased in the second and third 500-day periods (from 2005 to 2009). In this four-year period, the values of the optimal decay factors were uniformly above 0.99. Within the 500day time frame, the co-movement in the values of the optimal decay factors is clearly visible. At a higher frequency (250 days), the behaviour of the optimal decay factor values is more erratic, but shows the same trend. For all of the stock markets analysed, the optimal decay factor values grew gradually from the beginning of 2003, reaching a peak in 2007 and then falling during 2008. This behaviour of the values of the optimal decay factor can be connected to the global financial crisis. The interruption in the growth of the optimal decay factor values coincides with the beginning of the major shocks in the markets around the world. The BRW model shows its responsiveness to the ongoing crisis by lowering the values of the decay factors (putting more weight on the newest shocks), thus increasing the forecasted VaR figures sharply. With the lowering of the optimal decay factors, the model tried to update its risk forecasts of the stock indexes more quickly, whereas for gold and WTI oil futures, the optimal decay factors continued to increase, perceiving a lower downside risk. The difference between the behaviours of the optimal decay factors of the two commodities and the stock market indexes, which was especially pronounced during 2008, is in line with the empirical finding that commodities exhibit different price patterns to stock indexes, especially in times of crisis (see Draper et al., 2006). There is economic justification for optimising the decay factors for each time series, since, once calculated, these values rarely change. The testing showed that the optimal decay factors remain stable over annual time frames, and often even over twoyear periods. This is a very useful characteristic, allowing the optimisation procedure to be performed

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Table 2 Optimal decay factor values for assets tested at the 99% confidence level and for different time intervals over the period 04/01/2000–02/01/2009. Time period*

SP500

DJIN

RTY

NIKKEI

FTSE

DAX

CAC

GOLD

OIL

1-250 days

Optimal λ Lopez score Average VaR (%)

0.979 0.007 −2.65

0.973 0.015 −2.23

0.974 0.009 −2.61

0.999 0.021 −3.85

0.977 0.034 −5.56

0.973 0.012 −4.66

0.971 0.042 −3.68

0.993 0.033 −2.81

0.998 0.084 −7.39

2-250 days

Optimal λ Lopez score Average VaR (%)

0.984 0.002 −1.57

0.981 0.006 −1.53

0.992 0.003 −2.68

0.973 0.036 −3.42

0.976 0.014 −1.77

0.974 0.017 −2.66

0.981 0.010 −2.57

0.999 0.008 −3.17

0.994 0.036 −5.85

1-500 days

Optimal λ Lopez score Average VaR (%)

0.984 0.009 −2.20

0.974 0.015 −1.89

0.981 0.011 −2.67

0.987 0.042 −3.80

0.976 0.032 −2.31

0.973 0.034 −3.60

0.981 0.042 −3.32

0.994 0.036 −3.13

0.998 0.110 −6.72

3-250 days

Optimal λ Lopez score Average VaR (%)

0.994 0.002 −1.58

0.987 0.006 −1.52

0.985 0.013 −2.14

0.986 0.026 −2.39

0.993 0.005 −1.57

0.989 0.007 −2.17

0.989 0.005 −1.97

0.994 0.017 −2.30

0.991 0.034 −5.56

4-250 days

Optimal λ Lopez score Average VaR (%)

0.993 0.006 −1.63

0.988 0.014 −1.65

0.988 0.014 −2.48

0.997 0.016 −2.99

0.995 2.026 −2.08

0.988 0.015 −2.50

0.986 0.017 −2.40

0.991 1.052 −3.73

0.998 0.007 −4.67

2-500 days

Optimal λ Lopez score Average VaR (%)

0.995 0.027 −1.61

0.995 0.013 −1.68

0.996 0.021 −2.46

0.994 0.030 −2.94

0.998 0.032 −2.23

0.992 0.023 −2.40

0.993 0.022 −2.30

0.995 2.074 −3.08

0.995 0.014 −4.84

5-250 days

Optimal λ Lopez score Average VaR (%)

0.994 4.040 −2.26

0.994 2.034 −2.20

0.987 2.032 −3.02

0.993 0.032 −3.01

0.995 2.036 −2.80

0.997 0.011 −2.49

0.994 1.015 −2.84

0.992 0.010 −3.70

0.991 0.012 −4.48

6-250 days

Optimal λ Lopez score Average VaR (%)

0.993 9.186 −4.33

0.991 6.149 −3.98

0.991 0.004 −2.67

0.993 4.198 −5.88

0.991 3.144 −4.60

0.992 2.121 −4.70

0.992 4.131 −4.71

0.993 2.053 −4.06

0.996 6.172 −5.97

3-500 days

Optimal λ Lopez score Average VaR (%)

0.993 14.231 −3.29

0.994 10.181 −3.08

0.996 10.203 −3.83

0.993 5.230 −4.45

0.991 6.183 −3.67

0.992 1.130 −3.65

0.992 6.147 −3.79

0.993 2.065 −3.87

0.996 6.175 −5.22

1500 days

Optimal λ Lopez score Average VaR (%)

0.995 10.201 −2.64

0.994 4.196 −2.50

0.995 5.245 −3.08

0.993 5.304 −3.71

0.995 4.214 −3.14

0.991 0.183 −3.48

0.995 1.183 −3.55

0.995 5.175 −3.36

0.996 4.324 −5.58

∗ Non-overlapping time periods represent the time intervals used to calculate the optimal decay factors, covering the period from January 2003 to January 2009. For example, 1-250 indicates the first 250 trading days, i.e. the period between January 2003 and January 2004. The average VaR values and Lopez (1998) scores are calculated over the time periods indicated, using BRW simulation with the obtained optimal decay factors at the 99% confidence level.

far less frequently, resulting in a reduction in computation time and costs. The decay factor values have been rounded to three decimal places, since further refinements of the decay factors did not yield any significant improvements. 6. Comparison of VaR backtesting results In order to see whether there is any practical advantage of optimising the decay factor, we tested the performance of the optimised BRW simulation against the commonly assumed decay factors, as well

as against an array of popular VaR models, including Hull and White’s (1998) EWMA volatility weighted historical simulation, filtered historical simulation (Barone-Adesi et al., 1999), and the conditional (McNeil & Frey, 2000) and unconditional (GPD) extreme value approaches. All of the VaR models analysed were tested in several ways in order to determine their statistical characteristics and their ability to measure the market risk in the analysed markets adequately. The first test used was the Kupiec test, a simple expansion of the failure rate, which is

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Fig. 1. Optimal decay factor values for the tested assets at the 250- and 500-day time intervals at the 99% confidence level over the period 04/01/2000–02/01/2009. The x-axis represents non-overlapping time periods from January 2000 to January 2009; the y-axis represents the optimal decay factor values at the 99% confidence level, as used in the BRW VaR model. Source: Table 2.

a requirement of the Basel Committee on Banking Supervision. The second test was the Christoffersen (IND) independence test, which tests whether the VaR exceedances are IID. The Christoffersen unconditional (UC) and conditional (CC) tests were also calculated, but in our opinion they provide a somewhat distorted image of the relative performances of VaR models. Since the Christoffersen UC test is distributed as chi-squared with one degree of freedom, deviations from the expected value of the test that occur on the conservative side (i.e. the number of exceedances is lower than the excepted value) are treated more severely, a characteristic which is not compatible with regulators’ desires to increase the safety of the financial system. Kupiec and Christoffersen independence (IND) backtesting results, at the 5% significance level, for tested VaR models at the 99% confidence level, are presented in Tables 3 and 4, respectively. The backtesting results show that the popular and widely used VaR models, such as historical simulation, VCV and RiskMetrics, perform unsatisfactorily across all of the tested stock indexes and commodities. The FHS model shows a significant improvement over the basic historical simulation approach, the BRW simulation with the usual decay factors and

Hull and White’s (1998) volatility weighted historical simulation (HW HS) model. The GARCH, EGARCH and GJR-GARCH models with Gaussian, T , skewed T and GED distributions did not yield any significant improvements over the more simplistic models, and only provided satisfactory results for DAX, CAC and WTI oil futures. Such weak performances of the widely used VaR models could be attributed to the fact that the time period considered included the ongoing global financial crisis. However, since we used a sufficiently long backtesting period of 1500 days (almost six years of daily data), the global financial crisis should not be used as an excuse. Therefore, we can safely state that these VaR models should not be used in the tested markets for the purpose of risk measurement. The supreme performance was recorded for conditional and unconditional EVT models that satisfied both tests for all of the assets tested, with the exception of the unconditional GPD model, which failed the Christoffersen independence test for FTSE and CAC. The BRW simulation with the optimised decay factor failed only in the case of the S&P500 index, and thus surpassed even the unconditional EVT approach, which satisfied the Kupiec test for all of the assets but produced dependent VaR exceedances. Overall, the results are consistent, in that they all lead to the

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Table 3 Kupiec backtesting results at the 99% confidence level, 5% significance level; period: 1500 days up to January 2, 2009.

HS 100 HS 250 HS 500 BRW λ = 0.97 BRW λ = 0.99 BRW λ = opt HW EWMA FHS VCV RiskMetrics GARCH EVT GARCH GPD

SP500

DJIN

RTY

NIKKEI

FTSE

DAX

CAC

GOLD

OIL

0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.08* 1.00**

0.00 0.00 0.00 0.00 0.05* 0.12** 0.00 0.18** 0.00 0.00 0.00 0.34** 1.00**

0.00 0.00 0.00 0.00 0.00 0.08* 0.08* 0.01 0.00 0.00 0.18** 0.43** 1.00**

0.00 0.00 0.00 0.00 0.05* 0.08* 0.00 0.08* 0.00 0.00 0.00 0.96** 0.93**

0.00 0.00 0.00 0.00 0.02 0.12** 0.18** 0.08* 0.00 0.00 0.01 0.53** 0.98**

0.00 0.01 0.01 0.00 0.34** 0.43** 0.01 0.01 0.00 0.00 0.05* 0.93** 0.34**

0.00 0.00 0.00 0.00 0.12** 0.34** 0.05* 0.34** 0.00 0.00 0.12** 0.18** 0.96**

0.00 0.00 0.01 0.01 0.03 0.08* 0.53** 0.18** 0.00 0.00 0.00 0.82** 0.98**

0.00 0.00 0.00 0.00 0.01 0.12** 0.43** 0.98** 0.00 0.00 0.12** 1.00** 0.96**

The reported values represent Kupiec (1995) test p-values. ∗ Indicates VaR models which satisfy the Kupiec backtesting criterion at the 5% significance level. ∗∗ Indicates VaR models which satisfy the Kupiec backtesting criterion at the 10% significance level. Table 4 Christoffersen independence (IND) backtesting results at the 99% confidence level, 5% significance level; period: 1500 days up to January 2, 2009. HS 100 HS 250 HS 500 BRW λ = 0.97 BRW λ = 0.99 BRW λ = opt HW EWMA FHS VCV RiskMetrics GARCH EVT GARCH GPD

SP500

DJIN

RTY

NIKKEI

FTSE

DAX

CAC

GOLD

OIL

0.93** 0.76** 0.37** 0.99** 0.34** 0.36** 0.01 0.32** 0.20** 0.18** 0.30** 0.46** 0.91**

0.04 0.25** 0.25** 0.55** 0.46** 0.48** 0.08* 0.42** 0.18** 0.01 0.34** 0.56** 0.88**

0.84** 0.27** 0.63** 0.22** 0.32** 0.27** 0.27** 0.40** 0.88** 0.47** 0.51** 0.58** 0.85**

0.08* 0.14** 0.03 0.25** 0.03 0.27** 0.08* 0.46** 0.25** 0.02 0.34** 0.77** 0.04

0.01 0.30** 0.08* 0.30** 0.38** 0.44** 0.21** 0.38** 0.04 0.23** 0.36** 0.61** 0.02

0.76** 0.38** 0.43** 0.40** 0.58** 0.58** 0.43** 0.43** 0.25** 0.14** 0.30** 0.74** 0.16**

0.02 0.08* 0.47** 0.25** 0.48** 0.56** 0.30** 0.56** 0.03 0.14** 0.48** 0.21** 0.03

0.22** 0.34** 0.38** 0.36** 0.40** 0.44** 0.61** 0.51** 0.99** 0.28** 0.32** 0.69** 0.80**

0.09* 0.02 0.00 0.67** 0.36** 0.24** 0.14** 0.80** 0.01 0.01 0.24** 0.85** 0.77**

The reported values represent Christoffersen (1998) independence test p-values. ∗ Indicates VaR models which satisfy the Christoffersen independence backtesting criterion at the 5% significance level. ∗∗ Indicates VaR models which satisfy the Christoffersen independence backtesting criterion at the 10% significance level.

conclusion that only the conditional EVT GARCH model performed satisfactorily for all of the assets tested, while the other VaR models showed a tendency to seriously underpredict the true level of risk. The backtesting shows that the performance of the BRW simulation depends, to a very large extent, on the choice of the decay factor, and that even small adjustments to the decay factor, for example 0.003 in the case of NIKKEI, can make the difference between an acceptable and unacceptable VaR model.

Our results are similar to those obtained by Pritsker (2001), in that the BRW simulation with the commonly employed decay factors is a deficient VaR model under Basel II criteria. As a result, regulators will require the usage of a higher scaling factor if it is used to calculate capital requirements. On the other hand, the optimised BRW simulation is an acceptable VaR model, and, if employed by a financial institution, the minimal scaling factor can be used, leading to lower capital requirements. Based on our

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findings, we can state that studies evaluating the performance of the BRW model are flawed if they do not optimise the decay factor in some way. Taking ad hoc values is certainly not a reliable way of testing the performance of any VaR model. In this paper, we suggest optimising the decay factor with regard to the Lopez function, although there is no reason to think that optimising the decay factor using some other target function could not yield even better results. With regard to the independence of VaR exceedances, the results of the Christoffersen (1998) independence test are slightly better, but are still unsatisfactory for widespread VaR models. This means that their VaR exceptions are not IID, i.e. they tend to cluster together, making them unusable in real life situations. In addition to knowing whether a given VaR model passes or fails certain criteria, it is also useful to know which model gives the closest fit to the true level of risk, and which is the most acceptable for financial institutions, based on the average VaR values they forecast. By producing a lower but reliable VaR value, a financial institution can gain a competitive advantage through holding lower capital reserves, i.e. reduced idle capital. Our findings on this issue are presented in Tables 5 and 6. With regard to the Lopez test, the BRW model with optimised decay factors, together with the Hull and White (1998) model, is second only to the EVT GARCH model, which shows its superiority to the classical VaR models in this respect as well. For assets where the optimal decay factor BRW simulation was not ranked first, it was ranked very high — second or third. When looking at the Kupiec, independence and Lopez tests, the performances of the non-EVT models are far worse than those reported by other similar studies. This can be attributed to the increased market stress, and the occurrence of extreme loses that cannot be accounted for by classical models. The optimised BRW model and the FHS model showed similar backtesting performances. However, the former performed better with respect to the Kupiec test and required far fewer parameters, thus minimizing the model risk and speeding up the calculation. Our findings show that the optimal decay factor BRW simulation could be used as a viable alternative when the EVT VaR models are not being

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used, since, as with the conditional EVT approach, it satisfies the backtesting criteria in many cases, but at a significantly lower cost in capital reserves. 7. Conclusion We have presented an optimisation approach to determining the optimal decay factor in BRW VaR simulation based on minimising the Lopez (1998) size adjusted function. The optimal decay factors obtained in this manner show consistency over different time frames, providing a economic justification for their optimisation for each asset, since, once calculated, they remain unchanged for longer periods of time. The values of the optimal decay factor for the stock market indexes analysed show a lot of common characteristics. Within the 500-day time frame, the co-movement in the values of the optimal decay factors is clearly visible. At the higher frequency (250 days), the behaviour of the optimal decay factor values is more erratic, but demonstrates the same trend. In the case of the stock markets analysed, the optimal decay factor values grew gradually from 2003, reaching a peak in 2007 and then falling slightly during 2008, coinciding with the beginning of the major shocks in the markets around the world. The backtesting shows that the popular and widely used VaR models performed poorly across all of the stock indexes and commodities tested. Such weak performances of the widely used VaR models could be attributed to the fact that the time period considered included the ongoing global financial crisis. Since we used six years of daily data, the global financial crisis should not be used as an excuse. We conclude that, for the time period under consideration, only the conditional EVT GARCH model performed satisfactorily for all of the assets tested. The performance of the BRW simulation depends, to a very large extent, on the choice of the decay factor values. The BRW simulation with optimised decay factors is second only to the EVT GARCH approach, and fails in only one of the nine assets tested. These results show that the optimal decay factor BRW simulation could be used as a viable alternative when the EVT VaR models are not being used, since, as with the conditional EVT approach,

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Table 5 Ranking of competing VaR models at the 99% confidence level, for the period of 1500 days up to January 2, 2009, according to the Lopez test. SP500 HS 100 HS 250 HS 500 BRW λ = 0.97 BRW λ = 0.99 BRW λ = opt HW EWMA FHS VCV RiskMetrics GARCH EVT GARCH GPD

DJIN

RTY

NIKKEI

FTSE

DAX

CAC

GOLD

OIL

22.24 18.28 24.40 24.25 12.24 10.20 12.09

18.22 16.25 20.34 17.25 6.21 4.20 11.09

20.29 15.32 15.41 18.31 12.26 5.10 5.10

23.44 15.39 17.45 20.40 6.33 5.30 11.14

20.31 13.26 11.33 13.24 8.21 4.21 3.05

18.29 9.22 10.31 12.25 1.19 0.18 10.16

24.35 11.25 11.33 20.26 4.20 1.12 6.12

18.29 11.23 9.25 9.25 7.19 5.18 −0.87

24.46 15.47 18.53 21.42 10.38 4.32 0.10

12.16 30.42 17.16 13.17 5.13 −11.97

7.12 29.35 13.15 11.14 1.07 −10.94

9.14 21.39 11.13 3.09 0.07 −9.88

5.16 32.56 14.30 11.20 −6.95 −5.82

9.13 30.39 19.20 10.14 −0.93 −7.90

10.20 20.36 15.21 6.17 −5.94 1.24

1.18 26.39 15.19 4.16 3.13 −6.89

3.18 24.39 21.31 12.24 −3.90 −7.93

−7.91 19.48 11.31 4.21 −9.92 −6.87

The reported values represent Lopez (1998) test scores. The grey areas indicate the VaR models with the lowest Lopez scores, i.e. the smallest deviation from the expected value. Table 6 Average VaR values (in %) at the 99% confidence level, for the period of 1500 days up to January 2, 2009. SP500

DJIN

RTY

NIKKEI

FTSE

DAX

CAC

GOLD

OIL

HS 100 HS 250 HS 500 BRW λ = 0.97 BRW λ = 0.99 BRW λ = opt

2.24* 2.35 2.47 2.29 2.44 2.64

2.09* 2.27 2.42 2.12 2.36 2.50

2.79* 2.89 2.96 2.84 2.98 5.82

3.15 3.48 3.47 3.41 3.68 3.71

2.44 2.71 3.00 2.52 2.84 3.14

2.98 3.29 3.60 3.00 3.43 3.48

2.83 3.16 3.45 2.86 3.26 3.55

2.88 3.14 3.15 3.04 3.30 3.36

4.98 5.02 5.18 5.24 5.50 5.58

HW EWMA FHS VCV RiskMetrics GARCH EVT GARCH GPD

6.18 2.31 2.24 2.35 2.37 2.52** 8.18

5.76 2.32** 2.16 2.23 2.26 2.67 6.28

6.28 3.01 2.93 3.10 3.23** 3.32 7.13

7.11 3.33** 3.06* 3.19 3.22 4.22 6.41

5.89 2.37** 2.37 2.39 2.36* 2.70 5.34

5.81 2.72* 3.03 2.87 2.88** 3.67 4.34

5.75 2.93 2.79 2.75 2.72* 2.80 5.76

5.19 3.08** 2.51* 2.65 2.83 3.72 4.28

7.96 6.24 4.91* 5.17 5.36** 6.79 8.99

The grey areas indicate VaR models which satisfy the Kupiec (1995) and Christoffersen (1998) independence tests at the 5% significance level. ∗ Indicates the lowest average VaR value. ∗∗ The lowest average VaR value for a model which satisfies the Kupiec and Christoffersen independence tests.

they satisfy the backtesting criteria in many cases, but at a significantly lower cost in capital reserves. Studies evaluating the performance of the BRW model are flawed if they do not optimise the decay factor in some way. Taking ad hoc parameter values is certainly not a reliable way of testing the performance of any VaR model. In this paper, we have suggested optimising the decay factor with regard to the Lopez (1998) size adjusted function, but there is no reason to think that the optimisation of the decay factor using

some other target function could not yield even better results. Acknowledgements We are grateful to the Editor, Michael P. Clements, for his immense support and encouragement, and to the anonymous referees for their helpful suggestions, which significantly improved the paper. We are especially grateful to our inspirer, Kevin Dowd, for his continuous generosity with his knowledge and amity.

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Pritsker, M. (2001). The hidden dangers of historical simulation. Board of Governors of the Federal Reserve System, Economics discussion series. Working paper no. 27. ˇ Zikovi´ c, S. (2006). Applying hybrid approach to calculating VaR in Croatia. In Proceedings of the international conference of the faculty of economics in Sarajevo — from transition to sustainable development: the path to European integration, Sarajevo, Bosnia and Herzegovina. ˇ Saˇsa Zikovi´ c has studied economics at universities in Rijeka (Croatia) and Ljubljana (Slovenia). He holds a Ph.D. in Economics from the University of Ljubljana, Slovenia. He is currently an Assistant Professor of Finance at the Faculty of Economics, Rijeka. He has published a book and 49 scientific papers in both international and domestic academic journals. His research interests are in volatility forecasting, risk management and energy. He teaches Banking, Bank Management, Foreign Exchange Management and Risk Management. He also works as a consultant for financial institutions in the field of risk management and holds training courses in risk management for private bankers and banking regulators. He is member of the Scientific Society of Economists and the American Finance Association. Bora Aktan Bora Aktan is an Assistant Professor of Finance and Deputy Chair of International Trade and Finance Department at Yasar University, Faculty of Economics and Administrative Sciences; and Visiting Fellow at Middle East Technical University, Department of Business Administration in NC. Prior to joining the University, Dr. Aktan has worked as both Corporate Strategy and Business Development Directors in international firms largely active in Mexico, England and Turkey. He serves as a board member on the Economics and Financial Affairs Committee for the Aegean Region Chamber of Industry. Dr. Aktan’s current research activities focus on emerging capital markets, banking and global investing. He has published in different refereed journals such as International Journal of Business, Journal of Business Economics and Management, and Journal of Property Investment and Finance. His teaching specialties include corporate finance, investment analysis and portfolio management, financial markets and institutions. He is a member of various professional bodies such as the International Institute of Forecasters (IIF), the American Finance Association (AFA), the Financial Management Association (FMA), and the Society for Financial Econometrics (SOFIE). Dr. Aktan is also on the editorial boards of several international scholarly journals, such as Qualitative Research in Financial Markets, Actual Problems of Economics, Investment Management and Financial Innovation, and the Asian Journal of Mathematics and Statistics.