Journal of Banking & Finance 35 (2011) 1158–1169

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Journal of Banking & Finance

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

How accurate is the square-root-of-time rule in scaling tail risk: A global study

Jying-Nan Wang a,⇑, Jin-Huei Yeh b, Nick Ying-Pin Cheng c,d

a Department of Finance, Minghsin University of Science and Technology, Taiwanb Department of Finance, National Central University, Taiwanc Department of Finance, Vanung University, Taiwand Department of Finance, Yuan Ze University, Taiwan

a r t i c l e i n f o

Article history:Received 9 March 2010Accepted 18 September 2010Available online 25 September 2010

JEL classification:G18G20C20

Keywords:Value at riskSquare-root-of-time ruleJump diffusionSerial dependenceHeavy-tailVolatility clusteringSubsampling-based test

0378-4266/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.jbankfin.2010.09.028

⇑ Corresponding author. Tel.: +886 3 559 3143142x3412.

E-mail addresses: jnwang@must.edu.tw (J.-N. WaYeh), nickcheng@mail.vnu.edu.tw (N.Y.-P. Cheng).

a b s t r a c t

The square-root-of-time rule (SRTR) is popular in assessing multi-period VaR; however, it makes severalunrealistic assumptions. We examine and reconcile different stylized factors in returns that contribute tothe SRTR scaling distortions. In complementing the use of the variance ratio test, we propose a new intu-itive subsampling-based test for the overall validity of the SRTR. The results indicate that serial depen-dence and heavy-tailedness may severely bias the applicability of SRTR, while jumps or volatilityclustering may be less relevant. To mitigate the first-order effect from time dependence, we suggest asimple modified-SRTR for scaling tail risks. By examining 47 markets globally, we find the SRTR to be leni-ent, in that it generally yields downward-biased 10-day and 30-day VaRs, particularly in Eastern Europe,Central-South America, and the Asia Pacific. Nevertheless, accommodating the dependence correction is anotable improvement over the traditional SRTR.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Following several serious financial crises in little more than adecade, including the Asian Financial Crisis of 1997, the Dot-ComBubble of 2000, and the Global Financial Tsunami of 2008, riskmanagement, particularly in relation to tail risks, has recently in-creased considerably in importance in numerous subfields of fi-nance. Value at Risk (VaR), defined as a worst case scenario interms of losses on a typical day, is a popular measure of tail riskmanagement that is not only recommended by banking supervi-sors (BCBS, 1996a), but is also widely used throughout the financialindustry, including by banks and investment funds, see Pérignonand Smith (2010a,b). It is even used by nonfinancial corporationsin supervising in-house financial risks following the success ofthe J.P. Morgan RiskMetrics system.

Operationally, tail risk such as VaR is generally assessed using a1-day horizon, and short-horizon risk measures are converted to

ll rights reserved.

2x1890; fax: +886 3 559

ng), jhyeh@ncu.edu.tw (J.-H.

longer horizons. A common rule of thumb, borrowed from the timescaling of volatility, is the square-root-of-time rule (hereafter theSRTR), according to which the time-aggregated financial risk isscaled by the square root of the length of the time interval, justas in the Black–Scholes formula where the T-period volatility is gi-ven by r

ffiffiffiTp

. Regulators also advocate the routine use of the SRTR.For example, to avoid duplication of risk measurement systems,financial institutions are allowed to derive their two-week VaRmeasure by scaling up the daily VaR by SRTR; see, for example,BCBS (1996b). In fact, horizons of up to a year are not uncommon;many banks link trading volatility measurement to internal capitalallocation and risk-adjusted performance measurement schemes,which rely on annual volatility estimates by scaling 1-day volatilityby

ffiffiffiffiffiffiffiffiffi252p

.If the SRTR is to serve as a good approximation of all quantiles

and horizons, it not only requires the iid property of zero-mean re-turns, but also that of the Normality of the returns. These pre-assumptions are far from being realized in real world financial as-set returns, provided the numerous documented stylized facts thatare conflict with these properties. Accordingly, numerous studieshave attempted to identify how these different effects give riseto bias in SRTR approximation. The first attempt is based on the

1 Finding that using SRTR to estimate Sharpe Ratios causes bias when returnsexhibit serial dependence, Lo (2002) suggests using the variance ratio test as a pretest.Other related works include Huang (1985) and Ayadi and Pyun (1994), among manyothers.

J.-N. Wang et al. / Journal of Banking & Finance 35 (2011) 1158–1169 1159

fact that asset returns may be weakly dependent, both in levels andhigher moments. As illustrated in Jorion (2001), the SRTR tends tounderstate long-term tail risk when the return follows a persistentpattern, but tends to overstate the tail risk of temporally-aggre-gated returns if it displays mean-reverting behavior. Similarly,the presence of volatility clustering, as well-documented in the caseof most financial assets since Engle (1982), Bollerslev et al. (1992),Bollerslev et al. (1994), under the dynamic setup, has been demon-strated using detailed examples of how the common practice ofconverting 1-day volatility estimates to h-day estimates by SRTRscaling is inappropriate and yields overestimates of the variabilityof long-horizon volatility. On this, see Diebold et al. (1997) andMüller et al. (1990).

Numerous extant studies have demonstrated that asset returnsexhibit heavy-tails (Fama, 1965; Jansen and de Vries, 1991; Pagan,1996). Although allowing for dynamic dependence in the condi-tional variance partially contributes to the leptokurtic nature, theGARCH effect alone does not explain the excess kurtosis in finan-cial asset returns. On the one hand, this motivates studies to em-ploy their empirical GARCH modeling with student-t orgeneralized error distributions to account for heavier tails. Onthe other hand, researchers have turned to models that generateprice discontinuities to resolve the empirical regularity. Research-ers have long realized that financial time series exhibit certainunusual and extreme violent movements, known as jumps andmodeled using jump diffusions developed by Merton (1976) thatcreate discontinuous sample paths. See Andersen et al. (2002),Pan (2002), Eraker et al. (2003), Becker et al. (2009), Câmara(2009) for recent evidence on the prevailing phenomena of jumpsin price processes. Nonetheless, how the underlying jumps influ-ence the SRTR approximation of longer-term tail risks remainedunclear until the work of Danielsson and Zigrand (2006). Theyintuitively and clearly show that SRTR tends to underestimatethe time-aggregated VaR and the downward bias deteriorates withthe time horizon owing to the existence of negative jumps. How-ever, it remains unseen if in general price jumps are not confinedto downside extreme losses only, would the SRTR-induced down-ward-bias move in the other direction instead or becomenegligible?

Although we sound different alarms from distinct perspectivesby disclosing SRTR scaling as being inappropriate and misleading,with documented upward biases for some effects and downwardbiases for others, it is unclear after all whether the overall validityof the SRTR is appropriate or not for practical risk implementationgiven that all these effects coexist in a given asset. However, thispaper is not merely concerned with individual effects, such as aweak dependence of returns, volatility scaling, price discontinu-ities or leptokurticity, as is the case for the literature on the timescaling performance of the SRTR. Instead, we are interested inthe interactions among these stylized facts on the scaling of tailrisks via the application of the SRTR. To our knowledge, no previ-ous investigation has reconciled the quality of approximation intime-aggregated tail risks using the SRTR under various confound-ing factors.

This study fills this void by first devising a general frameworkfor disentangling and separately estimating the sensitivity towardeach systematic risk factor. To examine the overall performance ofthe SRTR approximation and characterize the potential bias, we de-fine a bias function using a benchmark VaR based on averaging aset of subsampled non-overlapping temporal aggregated VaRs.Based on Monte Carlo experiments, this investigation demon-strates that dependence at the return level is the dominant biasfactor. The SRTR leads to a systematic underestimation (overesti-mation) of risk when the return follows a persistent (mean-revert-ing) process, and can do so by a substantial margin. Moreover, themagnitude of downward (upward) bias increases with the time

horizon. However, volatility clustering tends to drive the time-aggregated VaR to slightly underestimate its true value. Alterna-tively, the heavy-tailed nature of the underlying return overstatesthe time-aggregated VaR via the SRTR. Perhaps surprisingly, unlikethe solely unilateral downside jumps specified by Danielsson andZigrand (2006) that indicate a severe underestimation bias, theMonte Carlo allowing for both sided jumps with Poisson arrivalperformed in this study suggests that there is a slight overestima-tion when scaling with the SRTR.

In view of these results, proper tests for a preliminary verifica-tion of the applicability of the SRTR in practice are required. Thisstudy first recommends a new informal but informative subsam-pling-based test, complementing the variance ratio test developedby Lo and MacKinlay (1988),1 for empirical studies. Moreover, it alsocontributes to the literature by suggesting a simple modified-SRTRthat is robust to the time dependence-induced biases. By utilizing47 markets included in the MSCI index, including both developedand emerging markets, this study demonstrates that the SRTRunderestimates 10-day and 30-day VaRs by an average of approxi-mately 5.7% and 13%, respectively. We also observe that the severityof downward bias is greater for emerging markets in Eastern Europe,Central and South America, and the Asia Pacific. For some developedmarkets, even when the model assumptions are violated, the SRTRscaling yields results that are correct on average, as shown in theglobal investigation. This occurs because the underestimation result-ing from the dynamic dependence structure is counterbalanced bythe overestimation resulting from the excess kurtosis and jumps.Hence SRTR scaling can be appropriate in some cases. Although itswidespread use as a tool for approximate horizon conversion isunderstandable, caution is, however, necessary. We believe thatthe use of certain pretests as we proposed beforehand is importantand may illuminate the applicability of SRTR in the practical approx-imation of tail risks. Our newly-proposed modified-SRTR approach isshown to be effective in alleviating the bias attributable to the first-order effect from time dependence and the dependence correction isa notable improvement over the traditional unadjusted raw SRTR.

The remainder of this paper is organized as follows. Section 2formally defines the time-aggregated VaR and SRTR scaling. Sec-tion 3 then performs algebraic analysis, in conjunction with MonteCarlo simulations, to disentangle each isolated different stylized ef-fect on the SRTR. This section also briefly reviews the variance ratiotest devised by Lo and MacKinlay (1988). Section 4 introduces thesuggested variance ratio test and a newly-developed subsample-based test for pretesting the applicability of the SRTR. More impor-tantly, we introduce a new tail risk scaling rule–the Modified-SRTR. Section 5 subsequently summarizes the global empiricalstudy based on data from 47 developed and emerging markets in-cluded in the MSCI index. Finally, Section 6 presents theconclusions.

2. Time-aggregated value at risk

The 1-day VaR, defined as VaR (1), measures the maximum pos-sible loss over one trading day under a given confidence level100 � (1 � c). Supposing that the initial investment of the assetis$1 and R is the random rate of return, then, the asset value atthe end of this trading day is v = 1 + R. Then, the one-day VaR,VaR (1), under 100 � (1 � c) confidence level is defined as

VaRð1Þ ¼ � inffrjP½R 6 r� > cg: ð1Þ

1160 J.-N. Wang et al. / Journal of Banking & Finance 35 (2011) 1158–1169

Following the concern of 718 (Lxxvi) in the Basel II Accord, we de-note the confidence level as 99% in this paper.

The VaR (1) of an asset can simply be estimated through thequantile function of the historical returns. Supposing that a se-quence of T daily log prices of an asset fptg

Tt¼0 is available, then

its daily returns are frtgTt¼1, where rt = pt � pt�1. By letting q(�) de-

note the quantile function, given P(rt 6 q(0.01)) = 0.01, the valueof VaR (1) is defined as q(0.01). However, in practice, it is usuallyhard to estimate the regulatory h-day VaR, VaR (h), since the timehorizon needed for the VaR (h) is quite long, especially when h islarge. For example, if we want to obtain the VaR (10) of an asset,10 years of stock data may generate only 250 observations of 10-day returns (250 trading days per year). Therefore, the Basel Com-mittee on Banking Supervision suggests that banks scale VaR (1) upto 10 days using SRTR. More generally, it says that VaR (1) could beconverted to VaR (h) by multiplying by

ffiffiffihp

. Under the I.I.D. andzero mean assumptions, without doubt,

ffiffiffihp

VaR (1) is equal toVaR (h).

The purpose of this paper is to investigate the validity of theSRTR in time-aggregated VaR. Before proceeding with our algebraicanalysis, we need a true VaR (h) as a benchmark for the compari-son. In practice it is usually difficult if not possible to estimatethe regulatory benchmark VaR. In this study, we recommend find-ing the benchmark h-period VaR, VaR (h), through a subsamplingscheme on the return series, and then use it for further character-ization of the biases in our Monte Carlo experiments as well as inempirical studies. This quantity is also employed to develop aninformative pretest for examining the overall applicability of theSRTR in reality in a later section.

Before considering VaR (h), we need to generate the daily priceswhere the sample size may not be too short and construct an h-dayreturn series from the original data. By leaving the first h prices asseeds, one may begin by subsampling the price series with a fixedlength of h days with one of the seeds as the starting points. In thisregard, we confront a total of h � 1 different subsamples of h-hori-zon return series and the kth subsample time-aggregated returnfrom a non-overlapping interval, denoted by fRk

s ðhÞgbðT=hÞ�1cs¼1 , is de-

fined as

phþk � pk�1|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}Rk

1ðhÞ

;p2hþk � phþk�1|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}Rk

2ðhÞ

; . . . ; pshþk � pðs�1Þhþk|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}Rk

s ðhÞ

; . . . ;

where k = 1,2, . . . ,h � 1. Given PðRks ðhÞ 6 Qkð0:01ÞÞ ¼ 0:01, a specific

h-day VaR denoted by VaRk(h) can be computed intuitively byQk(0.01) for the kth series. For each k, there will be a correspondingVaRk(h). As these fVaRkðhÞgh�1

k¼1 are obtained from different priceswithout overlapping return periods, a benchmark naturally arisesto be defined as

VaRðhÞ ¼ 1h� 1

Xh�1

k¼1

VaRkðhÞ: ð2Þ

Supposing the SRTR is correct, the scaledffiffiffihp

VaR (1) shall be equalto VaR (h). To examine whether the SRTR is tenable to serve as agood approximation for the multi-horizon VaR, given our subsam-pled and averaged benchmark VaR (h), we define a bias functionf(h) to measure the approximation error of the SRTR in scaling tailrisks by,2

fðhÞ ¼ffiffiffihp

VaRð1ÞVaRðhÞ � 1: ð3Þ

If f(h) is positive (negative), using SRTR produces an overestimated(underestimated) time-aggregated VaR. In the next section, we con-

2 Danielsson and Zigrand (2006) calculate VaR (h)/(ffiffiffihp

VaR (1)) as an indicator ofbias.

struct the Monte Carlo explorations for different non-I.I.D. returnsfeatures to investigate the influences of serial dependence, heavy-tailed distributions, jumps, and volatility clustering on the time-aggregated VaR, respectively.

3. Characterizing biases: algebraic analysis with Monte Carlos

Different biases arise from different data generating processes,except for the pre-assumed I.I.D. Gaussian case with zero mean.To accommodate a wide spectrum of stylized facts documentedin the literature, we consider a fairly general data generation pro-cess (DGP) for daily return rt that follows a non-zero mean ARMA(1,1)-GARCH (1,1) model with Poisson jumps as

rt ¼ lþ /1rt�1 þ at þ h1at�1 þ Jt;

at ¼ rt�t;

r2t ¼ a0 þ b1r2

t�1 þ a1a2t�1;

ð4Þ

where t = 1,2, . . . ,T and Jt is a compound Poisson process with jumpsize distributed as Nð0;r2

j Þ and constant jump intensity k. We allowGARCH (1,1) to govern the evolution of the conditional variance ofat over time. {�t} is a sequence of I.I.D. N (0,1), a0 > 0, a1 P 0, b1 P 0,and a1 + b1 < 1. Assuming there are 250 trading days per year, we let

a0 ¼0:15� ð1� a1 � b1Þ

250; ð5Þ

simply to control the annualized volatility to be roughly about 15%.Through the Monte Carlo simulation, a sequence of 5000 daily

returns, r1,r2, . . . ,r5000, is constructed, which amounts to a samplingperiod of about 20 years. Then we subsample (h � 1) sequences ofh-day temporal aggregated returns from the above daily returns.The VaR (1) of the daily returns is defined as the 1%-quantile forthis simulation. The time-aggregated VaR (h), h = 10 or 30, is com-puted through the average of the subsampled quantities. With2000 replications, we denote the true VaR (1) and VaR (h) as themeans of the 1-day and h-day VaRs, respectively. We examinethe following specific DGPs by restricting certain parameters toisolate the different effects that might have an impact on thetime-aggregated VaR. We will come back and reconcile all theseimpacts on the SRTR scaling of a tail risk later.

3.1. Non-zero mean

As the validity of SRTR scaling for quantiles hinges on a series ofassumptions, this subsection presents the bias arising from the pri-mary factor of a non-zero mean for the underlying return. By let-ting /1 = h1 = a1 = b1 = 0, rt = r and assume a zero jump (rj = 0), anon-zero mean model is

rt ¼ lþ at ; ð6Þ

where at = r�t. Assuming Normality for simplicity, the daily VaR (1)is

VaRð1Þ ¼ �l� rU�1ð0:01Þ; ð7Þ

where U�1(�) is the inverse normal cumulative distribution func-tion. Straightforwardly, we can find the VaR under a longer holdingperiod h to be

VaRðhÞ ¼ �h� l�ffiffiffihp� rU�1ð0:01Þ: ð8Þ

Therefore, the bias indicator f is

fðhÞ ¼ lþ rU�1ð0:01Þffiffiffihp� lþ rU�1ð0:01Þ

� 1: ð9Þ

We let l = 0.08%, 0.04%, 0.02%, �0.02%, �0.04%, and �0.08%, whichimply that the means of their annualized returns are 20%, 10%, 5%,

Table 1Temporally-Aggregated VaR under Different Scenarios. We consider different DGPs of returns as shown in Eq. (4) where t = 1,2, . . . ,5000. For each simulation, we generate the 10-day and 30-day VaRs through the SRTR and the subsampling approach. We repeat 2000 times and report the means of

ffiffiffiffiffiffi10p

VaR (1), VaR (10), f(10),ffiffiffiffiffiffi30p

VaR (1), VaR (30), andf(30). ffiffiffiffiffiffi

10p

VaR (1) VaR (10) f(10)%ffiffiffiffiffiffi30p

VaR (1) VaR (30) f(30)%

Panel A: Non-zero mean models with different l%0.080 0.178 0.172 3.340 0.308 0.287 7.3030.040 0.179 0.176 1.586 0.310 0.299 3.6830.020 0.179 0.178 0.923 0.311 0.304 2.181�0.020 0.181 0.182 �0.671 0.313 0.318 �1.385�0.040 0.181 0.184 �1.433 0.314 0.322 �2.429�0.080 0.183 0.188 �2.836 0.317 0.335 �5.614

Panel B: AR (1) models with different /1

0.700 0.252 0.513 �50.876 0.437 0.986 �55.7140.500 0.208 0.335 �37.969 0.360 0.606 �40.6100.200 0.184 0.220 �16.534 0.318 0.385 �17.394�0.200 0.184 0.153 20.114 0.319 0.261 21.917�0.500 0.208 0.128 62.641 0.360 0.212 70.004�0.700 0.252 0.119 111.642 0.437 0.190 129.847

Panel C: MA (1) models with different h1

0.700 0.220 0.298 �26.229 0.380 0.521 �27.0610.500 0.202 0.264 �23.587 0.349 0.462 �24.4370.200 0.184 0.213 �13.789 0.318 0.372 �14.376�0.200 0.184 0.149 23.601 0.318 0.252 26.486�0.500 0.201 0.107 88.856 0.349 0.166 110.253�0.700 0.220 0.086 155.105 0.381 0.115 232.305

Panel D: GARCH (1,1) models with different (a1,b1)(0.130,0.820) 0.193 0.198 �2.746 0.333 0.338 �1.431(0.150,0.800) 0.195 0.201 �2.931 0.338 0.343 �1.549(0.130,0.840) 0.198 0.203 �2.526 0.342 0.349 �1.987(0.150,0.820) 0.201 0.207 �2.842 0.348 0.358 �2.794

Panel E: Student-t models with different l3.000 0.352 0.331 6.328 0.610 0.563 8.3275.000 0.260 0.240 8.707 0.451 0.407 10.8597.000 0.233 0.217 7.392 0.403 0.370 8.7949.000 0.219 0.206 5.987 0.379 0.354 7.049

Panel F: Jump models with different (k,rj)(0.058,0.020) 0.185 0.184 0.511 0.320 0.317 1.006(0.058,0.030) 0.191 0.189 1.098 0.331 0.326 1.557(0.082,0.020) 0.187 0.185 0.992 0.324 0.319 1.499(0.082,0.030) 0.196 0.193 1.586 0.339 0.331 2.257

J.-N. Wang et al. / Journal of Banking & Finance 35 (2011) 1158–1169 1161

�5%, �10%, and �20%, respectively.3 The simulation results are re-ported in Panel A of Table 1. Interestingly, SRTR tends to underesti-mate (overestimate) VaR (10) or VaR (30) due to the effect of anegative (positive) l. However, as the readers will see in the follow-ing subsection, we show that this non-zero-mean-induced bias isslight and only of second-order importance in the presence of serialdependence.

3.2. Series dependence

By fixing l = a1 = b1 = 0, rt = r, and assuming a zero jump(rj = 0), a weakly stationary ARMA (1,1) model for rt is

rt ¼ /1rt�1 þ at þ h1at�1; ð10Þ

where at = r�t.The k-order autocorrelation of rt is

q1 ¼ /1 þh1r2

var½rt�; qk ¼ /1qk�1; for k > 1; ð11Þ

where the variance of rt is

var½rt � ¼ð1þ 2/1h1 þ h2

1Þr2

1� /21

: ð12Þ

3 The average annual return of the examed 47 markets is 6.61%. Only two amongthem, namely, Portugal and Turkey, have higher average annual returns that exceed20%.

Then the variance of Rks ðhÞ is written as

var½Rkt ðhÞ� ¼

Xh�1

i¼0

Xh�1

j¼0

Covðrt�i; rt�jÞ ¼ var½rt � hþ 2Xh�1

k¼1

ðh� kÞqk

( )

¼ var½rt � hþ 2� h1�/1

� 1�/h1

ð1�/1Þ2

!/1 þ

h1ð1�/21Þ

1þ 2/1h1 þ h21

!( ):

ð13Þ

Therefore, based on the above result it is straightforward to show

VaRðhÞ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivar½Rk

t ðhÞ�q

U�1ð0:01Þ; ð14Þ

where U�1(�) is the quantile function from the standard Normal dis-tribution. Then we can find the bias f through (3), (12)–(14), i.e.,

fðhÞ ¼ffiffiffihp

hþ2� h1�/1

� 1�/h1

ð1�/1Þ2

!/1þ

h1ð1�/21Þ

1þ2/1h1þ h21

!( )�1=2

�1:

ð15Þ

We let h1 = 0 for the AR (1) models and consider /1 = 0.7, 0.5, 0.2,�0.2, �0.5, �0.7. The MA (1) models, /1 = 0 and h1 = 0.7, 0.5, 0.2,�0.2, �0.5, �0.7 are also examined. The results in panel B and panelC of Table 1 show that the SRTR yields severe overestimation (in thecase of negative serial correlation) or severe underestimation re-sults (in the case of positive serial correlation) for VaR (10) andVaR (30). According to (15), the patterns of bias function acrosshorizons, f(10), f(30), and f(90), are plotted in Figs. 1 and 2.

1162 J.-N. Wang et al. / Journal of Banking & Finance 35 (2011) 1158–1169

3.3. Tail behavior

If rt is normally distributed, Rks ðhÞ is still normally distributed

due to the additive of the normal distribution. To obtain the 1%-quantile of the normal distribution for the VaR calculation is nothard. We can then calculate VaR (1) or VaR (h) using (14). Underthe Normal case, it is easy to find that the bias function is only re-lated to variances, i.e.,

fðhÞ ¼ h� var½rt �var½RtðhÞ�

� �1=2

: ð16Þ

Nevertheless, without the Normal assumption, not only the vari-ance, but also the tail behavior before/after temporal aggregation,may affect f(h).

While the work of Dacorogna et al. (2001) has demonstratedthat, except for the boundary case of Normality, any other heavy-tailed distribution under a stable law leads the SRTR to underesti-mate the VaR, to our knowledge, there is no general theoretical

−0.7 −0.5 −0.3 −0.1 0.1 0.3 0.5 0.7−0.6

−0.2

0.2

0.6

1

1.4

φ1

ζ

h=10h=30h=90

Fig. 1. Bias functions of aggregated VaR with AR (1) models. Consider the modelfrom (10) given /1 = 0. We plot different time horizons of bias functions, f(10),f(30), and f(90), which are determined by (15).

−0.7 −0.5 −0.3 −0.1 0.1 0.3 0.5 0.7−0.5

0

0.5

1

1.5

2

2.5

3

θ1

ζ

h=10h=30h=90

Fig. 2. Bias functions of aggregated VaR with MA (1) models. Consider the modelfrom (10) given h1 = 0. We plot the different time horizons of bias functions, f(10),f(30), and f(90), which are determined by (15).

model that can provide a good interpretation of the relationshipbetween the time-aggregated VaR and tail behavior. In particular,the question is a little involved from the distributional perspectivesince there are some heavy-tailed distributions that may not beclosed under temporal aggregation, even if they are iid generated.For instance, a student-t with 2 degrees of freedom would scalelike the SRTR in the tails; however, the fat-tail may no longer existafter aggregation and thus this case gives rise to another source ofSRTR approximation error beyond the discussions in the literature.In addition, while the leptokurticity of the observed returns may beattributable to its underlying distributional property, it can also bethe consequence of a higher moment dependence such as volatilityclustering, as well as some occasional price discontinuities orjumps. We will therefore investigate these issues through thenumerical analysis of three popular economic phenomena thathave contributed to the excess kurtosis in the stylized facts: vola-tility clustering, heavy-tailed distributions and price jumps.

3.3.1. Volatility clusteringBy letting l = /1 = h1 = 0 and assuming no jump, we consider

the following GARCH (1,1) model

rt ¼ rt�t ;

r2t ¼ a0 þ b1r2

t�1 þ a1r2t�1:

ð17Þ

Drost and Nijman (1993) derive the temporal aggregation of theGARCH processes and show, under regularity conditions, that thecorresponding sample path of Rt(h) follows a similar GARCH (1,1)process with different parameters. The results have been suggestedto convert short-run volatility into long-run volatility in Christoffer-sen and Diebold (1997) and Diebold et al. (1998). They do, however,point out that using SRTR is inappropriate and produces overesti-mates of the variability of long-horizon volatility. While theseworks highlight the dangers of SRTR in the scaling of time-varyingvolatilities into longer horizons, we take a different route. We con-duct a series of Monte Carlo experiments to explore the robustnessof the SRTR in scaling VaR in the presence of GARCH effects in theunderlying return series.

We entertain pairs of (a1,b1) with a1 + b1 � 95 or 97 for theGARCH (1,1) models. Panel D of Table 1 shows that the SRTR tendsto yield only a slightly underestimated VaR (10) or VaR (30) in thepresence of volatility clustering in terms of the negative fs rangingfrom �1.43% to �2.93%. These downward biases are intuitivelyreasonable for overlooking the time-varying risks.

3.3.2. Heavy-tailed distributionTo demonstrate the effect of different distributional consider-

ations from the literature, we let the return process be

rt ¼ rxt ; ð18Þ

where {xt} is a sequence of independent and identical student-tdistributions with m degrees of freedom. On the variance of theaggregated entity, since {xt} is independent, var [Rt(h)] is equal tovar [rt] multiplied by h. Nevertheless, it is intuitive that if we addh daily returns to a h-day return, the long-tailedness appearing inthe daily return shortens as h increases. To see this, Fig. 3 showsthe probability density functions of daily and 10-day returns withm = 5 and it is rather obvious that the tail part has been diluted afteraggregation.

By setting m = 3, 5, 7, and 9 to obtain different heavy-tailed re-turn distributions from (18), we find that an overestimated aggre-gated VaR based on the SRTR is due to the heavier tail in thedaily return distribution. To probe further into what may havegone wrong with the SRTR scaling approximation, we report boththe variance and kurtosis of the daily, 10-day and 30-day returnsunder different DGPs in Table 2. By cross inspecting across the

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5Standardized 10−day return densityStandardized 1−day return density

Fig. 3. Probability densities of 1-day and 10-day returns. We generate the dailyreturns which follow model (18) with m = 5. Then, we depict the probabilitydensities of standardized 1-day and 10-day returns.

J.-N. Wang et al. / Journal of Banking & Finance 35 (2011) 1158–1169 1163

Panel E in Table 1 and panel D in Table 2, it is readily seen that theupward bias comes solely from the decrease in kurtosis, and notfrom the scaling of volatility, since var[Rk

t ð10Þ] ’ 10 � var[rt]whereas kurt[Rk

t ð10Þ] < kurt[rt]. Similar patterns can be found inRk

s ð30Þ. Therefore, the SRTR overestimates VaR (10) and VaR (30)by about 5–11% with different m for failing to take into accountthe change in tail behavior among the original high frequency re-turns and the temporally-aggregated ones in panel E of Table 1.

3.3.3. JumpsWhile fat-tailed distributions may be more suited for daily inter-

nal risk management, they may not be suited for the modeling ofuncommon and unexpected systemic events. We consider a general

Table 2Variance and kurtosis of returns. We generate rt from the DGPs as (18) and (19). ThusRt(10) can be calculated by rt. To characterize the potential biases attributable to thescaling of variance or aggregation bias due to changing tail behavior, for each DGP wereport the corresponding variance and kurtosis of rt, Rt(10), and Rt(30).

Variance (%) Kurtosis

1-day 10-day 30-day 1-day 10-day 30-day

Panel A: AR (1) models with different /1

0.700 0.117 4.881 18.148 2.998 2.991 2.9700.500 0.080 2.080 6.876 2.998 2.979 2.9520.200 0.063 0.899 2.776 3.000 2.987 2.958-0.200 0.063 0.434 1.270 2.997 2.986 2.956-0.500 0.080 0.303 0.838 3.003 2.990 2.964-0.700 0.117 0.262 0.680 2.992 2.979 2.957

Panel B: MA (1) models with different h1

0.700 0.089 1.648 5.090 3.002 2.990 2.9680.500 0.075 1.288 3.985 2.998 2.996 2.9750.200 0.062 0.841 2.576 2.999 2.980 2.966-0.200 0.062 0.408 1.175 2.998 2.987 2.929-0.500 0.075 0.209 0.508 2.998 2.975 2.945-0.700 0.090 0.139 0.246 3.001 2.979 2.943

Panel C: GARCH (1, 1) models with different (a1,b1)(0.130,0.820) 0.060 0.596 1.788 4.381 4.844 4.247(0.150,0.800) 0.060 0.599 1.800 5.037 5.603 4.748(0.130,0.840) 0.060 0.599 1.790 5.703 6.111 5.280(0.150,0.820) 0.060 0.601 1.792 6.871 7.305 5.895

Panel D: Student-t models with different l3.000 0.180 1.793 5.393 78.455 10.396 5.2935.000 0.100 1.002 3.016 9.011 3.584 3.1347.000 0.084 0.842 2.525 5.065 3.190 3.0359.000 0.077 0.770 2.324 4.183 3.093 3.005

Panel E: Jump models with different (k,rj)(0.058,0.020) 0.062 0.624 1.867 3.097 2.991 2.961(0.058,0.030) 0.066 0.655 1.970 3.445 3.026 2.979(0.082,0.020) 0.064 0.634 1.904 3.143 3.004 2.969(0.082,0.030) 0.068 0.681 2.044 3.657 3.054 2.987

jump diffusion process4 by letting l = /0 = /1 = h1 = a1 = b1 = 0 andrt = r to isolate the effect from jumps

rt ¼ r�t þ Jt ; ð19Þ

where Jt is a compound Poisson process with constant jump inten-sity k and random jump size distributed as Nð0;r2

j Þ. The aggregatedvariance of rt can be written as

var½RtðhÞ� ¼ varXh�1

i¼0

r�t�i þXh�1

i¼0

Jt�i

" #¼ h� var½rt�: ð20Þ

We let k = 0.058, 0.082 and rj = 2%, 3% to allow for a variety of com-binations of jump intensities and sizes.5 by cross inspecting acrossthe Panel F in Table 1 and panel E in Table 2, readers may find theresults are similar to previous heavy-tailed cases: the problem withthe SRTR in jump models is not due to their variance scaling but totheir changing tail behavior. It is intuitive that a return distributionwith jumps has heavier tails. Furthermore, summing over h daily re-turns smoothes out the infrequent jump effects. Thus, the SRTR pro-vides an overestimated aggregated VaR.

We also find var[Rkt ð10Þ] ’ 10 � var[rt] while kurt[Rk

t ð10Þ] <kurt[rt]. These simulation results shown in panel F of Table 1 indi-cate that jumps indeed let the SRTR produce an overestimatedtime-aggregated VaR. Nevertheless, under reasonable k and rj set-tings, we also find that the size of the systematic overestimationbias from the SRTR only has a slight impact, as they are less than1.6% in approximating VaR (10) and less than 2.3% in approximat-ing VaR (30). Our result is largely different from that of Danielssonand Zigrand (2006) who allow for downside jumps only in theirsetup and it is reasonable to document downward bias in theircase. Instead, we not only find upward bias via the SRTR scalingin the presence of Poisson jumps, but, we document that the biasesmay not be that much even after we consider some sizable jumpintensities and jump sizes.

3.4. A summary

The preceding analysis yields the following findings. First, theweak dependence in returns dominates among all the confoundingfactors considered in this study when the SRTR is used to estimateVaR (h). Positive serial dependence leads to a severe underestima-tion in the SRTR’s approximation of VaR (h), while severe overesti-mation occurs in the case of negative serial dependence. Giventhese results, this study proposes using the variance ratio test, typ-ically employed to test for market efficiency, to examine the syn-thetic underlying serial dependence in empirical studies as apretest of the applicability of VaR scaling using the SRTR.

Second, using the SRTR may produce an overestimate or under-estimate of VaR (h) because of the changes in tails. In cases of over-estimates of the student-t distribution and jumps, the heavy-tailednature is smoothed out by aggregating daily returns. However, vol-atility clustering may lead to the SRTR resulting in a slight down-ward bias owing to neglecting the time-varying nature. Tosummarize, this study carefully uses the SRTR to estimate thetime-aggregated VaR, when the real data exhibits serial depen-dence, volatility clustering, a heavy-tailed distribution, or jumps.

4 Danielsson and Zigrand (2006) demonstrate that the square-root-of-time ruleleads to systematic underestimation of risks, and their setup allows for downsidejumps that represent losses only. However, there is no a priori theoretical reason torestrict, let alone expect, the prices to jump down only and therefore we entertain ourjump component to jump symmetrically and our parameters to be in line with thejump diffusion literature.

5 Andersen et al. (2002) show that the jump intensity is about 0.014, that is, 14times every thousand trading days on average, for the daily S&P 500 cash index. Theyalso estimate the jump size parameter rj to be at 1.5%. In the simulation conducted byHuang and Tauchen (2005), their rj varies from 0 to 2.5%.

6 The Central Limit Theorem holds for weakly dependent data as long it isappropriately standardized by a standard error that accounts for the underlyingdependence.

7 We thank the anonymous referee for pointing this suggestion out to us.

1164 J.-N. Wang et al. / Journal of Banking & Finance 35 (2011) 1158–1169

4. Pretesting SRTR applicability and a modified-SRTR

4.1. Variance ratio test

From the viewpoint of empirical exploration, Lo and MacKinlay(1988) propose a statistic to test the hypothesis of a random walk.The general h-period variance ratio statistic VR (h) is denoted as

VRðhÞ � var½RtðhÞ�h � var½rt �

¼ 1þ 2Xh�1

k¼1

1� kh

� �qk; ð21Þ

where qk is the kth order autocorrelation coefficient of {rt}. WhenVR (h) = 1, this means that {rt} follows the random walk hypothesis;when VR (R) – 1, {rt} exhibits serial dependence. Moreover, we re-gard VR (h) as an indicator which measures the synthetical effectson different degrees of serial dependence. If VR (h) is significantlylarger (smaller) than one, we say this series is characterized by asynthetically positive (negative) serial dependence of {rt}. It is intu-itive that the positive (negative) serial dependence causes the SRTRto be underestimated (overestimated). Lo and MacKinlay (1988) de-fine the following statistic to estimate the VR (h) of Eq. (21),

VRðhÞ :¼ 1þ 2Xh�1

k¼1

1� kq

� �q̂k; ð22Þ

where q̂k denotes the autocorrelation coefficient estimators. Underthe random walk hypothesis, VRðhÞ still approaches one. For thestandard inferences, it is necessary to compute its asymptotic vari-ance. First denote a heteroskedasticity-consistent estimator of theasymptotic variance of qk,

d̂k ¼ T �XT

j¼kþ1

ðrj � l̂Þ2ðrj�k � l̂Þ2" # XT

j¼1

ðrj � l̂Þ2" #�2

ð23Þ

where l̂ � 1T

PTk¼1rk. Then, the following is a heteroskedasticity-con-

sistent estimator of the asymptotic variance of VRðhÞ,

#̂h � 4Xh�1

k¼1

1� kh

� �2

d̂k: ð24Þ

Regardless of the presence of general heteroskedasticity, the stan-dardized statistic w*(h) can be used to test the hypothesis of a ran-dom walk, i.e.,

w�ðhÞ ¼ffiffiffiTpðVRðhÞ � 1Þffiffiffiffiffi

#̂h

q a Nð0;1Þ; ð25Þ

where a denotes for ‘‘asymptotically distributed as”.

4.2. A new subsample-based test for overall SRTR applicability

While the variance ratio test is in spirit in line with the SRTR, itoffers only a partial picture since it is informative in detecting onlyspecifically the dependence structure of the return series. As thedependence structure is of first-order importance, this paper offersa new and simply complementary approach to test the overallvalidity of applying the SRTR to scale VaR to a specific asset byutilizing the subsamples we used to construct our benchmarkVaR (h). Since the way fVaRkðhÞgh�1

k¼1 is constructed is based onfRk

s ðhÞgbðT=hÞ�1cs¼1 , the h-period return from the non-overlapping sub-

sampling of the original prices, the subsampled fVaRkðhÞgh�1k¼1 is em-

ployed to compute the benchmark VaR (h) by taking the subsampleaverage.

Before we arrive at a formal test for the SRTR’s validity, a prop-erly computed standard error of the bias term,

ffiffiffihp

VaRð1Þ � VaRðhÞ,is needed. Nonetheless, the well-documented time dependence

may carry over to the time-aggregated returns and thus the simplevariance estimator as defined by

R0ðhÞ ¼1

h� 2

Xh�1

k¼1

VaRkðhÞ � VaRðhÞð Þ2" #1=2

;

may not be sufficient to accommodate the generality of the returnprocess. Moreover, the R0(h) defined above is also subject to thesmall sample bias since h is commonly limited to 10 or 30.

To accommodate both of these concerns, we choose to rely onthe block bootstrapped samples to produce a reliable standard er-ror for a formal test. This is done by taking into account the poten-tial time dependence by retaining the dynamic structure of theunderlying returns by randomly drawing subsamples using blocksof consecutive returns, thereby alleviating the small sample biasproblem and improving the testing performance. The implementa-tion procedure is as follows.

We choose a block length of 10 and these blocks could be over-lapping. Specifically, the data are divided into T � 9 blocks, withthe first block being {r1,r2, . . . ,r10}, the second block being{r2,r3, . . . ,r11}, . . . , . . . , and the last block being {rT�9,rT�8, . . . ,rT}.We then randomly resample T/10 blocks to construct a new boot-strapped sample of T days of returns. For each bootstrapped resam-ple, we calculate and save the values of VaR (1), the subsampledand averaged VaR (h), and

ffiffiffihp

VaRð1Þ � VaRðhÞ. By repeating theabove procedure for 5000 replications, we obtain a bootstrappedsampling distribution of the bias,

ffiffiffihp

VaRð1Þ � VaRðhÞ, based onthe 5000 bootstrapped resamples. Therefore, the bootstrappedstandard deviation of

ffiffiffihp

VaRð1Þ � VaRðhÞ, defined as R(h), cannow be calculated.

Intuitively, under certain regularity conditions, it is easy to ar-gue from the Central Limit Theorem6 thatffiffiffi

hp

VaRð1Þ � VaRðhÞRðhÞ Nð0;1Þ: ð26Þ

This statistic serves as our benchmark pretest for the overall valid-ity of the SRTR in our subsequent analysis, after considering differ-ent confounding dynamic and distributional properties that prevailin real asset returns.

4.3. Scaling tail risk with a new modified-SRTR

As we have shown, the dynamic serial dependence in the returnprocess, among the other stylized features, serves as the first-ordereffect that biases the validity of the SRTR in scaling quantiles. Inview of this, the subsection moves one step further to propose asimple and robust correction to the existing SRTR. It is a well-ac-cepted fact that a variance ratio greater than 1 suggests the exis-tence of a positive dependence in the underlying return series,and the opposite situation holds true for the case of a VR (h) of lessthan 1. This simple correction thus mainly makes use of the esti-mated variance ratio as indicated in Eq. (22) to adjust the raw SRTRby taking the time dependence structure into consideration.7

Accordingly, we formally define an estimator which estimates VaR(h) through this robust rule as

MVaRðhÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih� VRðhÞ

q� VaRð1Þ: ð27Þ

We thus refer to it as the modified-SRTR (MVaR). Note that if a timeseries is serially uncorrelated, the variance ratio is 1 and thereforeMVaR (h) will simply reduce to

ffiffiffihp

VaR (1), which is essentially

Table 3Global Evidence on SRTR Scaling of Tail Risk. We consider 47 markets globally with both developed markets and emerging markets listed in the MSCI. The sample period extendsfrom January 2, 1997 to December 31, 2009 (3391 trading days). For each market, the time-aggregated VaR (10 days and 30 days) computed through both the simple SRTR andsubsampling approach are reported. We also report the overall bias, f, along with its significance based on the block-bootstrapped standard error, as well as the statistic of thevariance ratio test, w*. ffiffiffiffiffiffi

10p

VaR (1) VaR (10) f(10)% w*(10)ffiffiffiffiffiffi30p

VaR (1) VaR (30) f(30)% w*(30)

Panel A: AfricaMorocco 7.72 9.27 �16.76 4.60a 13.37 14.26 �6.27 4.80a

South Africa 12.16 13.27 �8.32 1.50 21.07 28.25 �25.43a 1.90c

Turkey 24.97 23.02 8.46 0.68 43.25 42.91 0.79 1.58

Average 14.95 15.19 �5.54 2.26 25.89 28.47 �10.30 2.76

Panel B: EuropeAustria 14.60 15.63 �6.60 0.45 25.28 34.04 �25.73a 1.37Belgium 10.80 12.55 �13.93c 1.26 18.71 23.31 �19.73b 1.21Denmark 11.82 12.05 �1.96 0.11 20.47 23.44 �12.68 0.59Finland 17.78 18.40 �3.40 �0.80 30.79 31.04 �0.81 0.29France 13.76 13.85 �0.59 �2.20b 23.84 22.62 5.38 �1.26Germany 15.71 15.46 1.63 �1.16 27.21 26.84 1.39 �0.57Greece 16.46 17.02 �3.34 2.99a 28.50 28.78 �0.97 3.05a

Ireland 13.61 16.53 �17.69a 0.93 23.57 30.27 �22.13a 1.26Italy 12.58 14.14 �11.05 0.81 21.79 24.83 �12.27 1.15Luxembourg 14.75 16.16 �8.70 2.40b 25.55 36.37 �29.75a 3.29a

Netherlands 13.38 14.66 �8.73 �0.14 23.17 29.19 �20.62a 0.84Norway 15.90 15.45 2.93 �0.31 27.54 31.56 �12.74 0.55Portugal 10.40 11.34 �8.22 2.38b 18.02 22.00 �18.09c 3.04a

Spain 12.91 13.29 �2.86 �0.99 22.37 22.74 �1.64 0.14Sweden 14.02 13.27 5.63 �2.17b 24.29 24.02 1.09 �0.87Switzerland 11.55 12.20 �5.37 �0.39 20.00 21.16 �5.51 0.25UK 11.07 10.99 0.78 �1.64 19.18 19.56 �1.95 �1.00

Average 13.59 14.29 �4.79 0.09 23.55 26.58 �10.40 0.79

Panel C: Eastern EuropeCzech R. 13.23 15.70 �15.77b 1.06 22.91 31.81 �28.00a 1.63Hungary 17.81 18.59 �4.18 0.24 30.85 44.36 �30.46a 1.16Poland 13.61 14.59 �6.75 1.80 23.57 29.14 �19.09b 2.23b

Russia 28.21 31.16 �9.47 0.80 48.86 55.44 �11.88 0.47

Average 18.21 20.01 �9.04 0.97 31.55 40.19 �22.36 1.38

Panel D: Central and South AmericaArgentina 21.55 21.72 �0.77 1.50 37.33 43.24 �13.67b 1.82c

Brazil 20.03 22.18 �9.66 �1.28 34.70 38.19 �9.14 �0.16Chile 9.57 11.70 �18.20b 4.28a 16.58 23.49 �29.41a 2.89a

Mexico 14.18 14.29 �0.80 0.60 24.56 27.01 �9.07 0.69Peru 15.63 18.27 �14.44c 2.86a 27.07 38.48 �29.65a 3.15a

Venezuela 18.08 22.65 �20.15a 0.63 31.32 47.28 �33.75a 1.31

Average 16.51 18.47 �10.67 1.43 28.59 36.28 �20.78 1.62

Panel E: North AmericaCanada 11.08 11.10 �0.14 �0.87 19.20 22.26 �13.75 �0.06US 11.29 10.98 2.82 �2.40a 19.55 21.38 �8.55 �1.32

Average 11.19 11.04 1.34 �1.64 19.37 21.82 �11.15 �0.69

Panel F: Asia PacificAustralia 9.31 9.22 1.02 �0.72 16.12 17.75 �9.17 �0.22Hong Kong 16.02 16.38 �2.23 �0.53 27.74 31.27 �11.27 0.13India 15.49 15.77 �1.78 0.42 26.82 28.16 �4.73 0.95Israel 10.71 10.86 �1.35 0.81 18.55 18.90 �1.85 1.88c

Japan 13.75 11.30 21.71b �1.71c 23.82 24.88 �4.25 �1.03Jordan 11.55 11.11 3.92 0.44 20.00 20.81 �3.86 1.24Korea 18.84 16.65 13.13c �0.21 32.63 31.31 4.23 0.71Malaysia 12.42 15.90 �21.92a 0.44 21.51 28.58 �24.75a 0.97New Zealand 7.15 8.49 �15.73 0.45 12.39 15.49 �19.98 1.01Pakistan 16.41 22.56 �27.26a 3.82a 28.42 44.73 �36.46a 3.82a

Philippines 14.56 14.70 �0.96 2.44b 25.21 28.84 �12.59 3.29a

Shanghai 16.51 15.45 6.81 0.52 28.59 24.88 14.90 1.53Singapore 12.27 16.94 �27.56a 2.10b 21.25 29.44 �27.82a 2.33b

Thailand 16.01 19.36 �17.30a 2.04b 27.73 35.59 �22.08a 2.93a

Taiwan 13.74 14.09 �2.54 1.12 23.79 25.67 �7.32 1.93c

Average 13.65 14.59 �4.80 0.76 23.64 27.09 �11.13 1.43

Total average 14.36 15.32 �5.70 0.62 24.87 29.05 �13.00 1.21

a Those test statistics that are statistically significant at the confidence levels of 99%.b Those test statistics that are statistically significant at the confidence levels of 95%.c Those test statistics that are statistically significant at the confidence levels of 90%.

J.-N. Wang et al. / Journal of Banking & Finance 35 (2011) 1158–1169 1165

1166 J.-N. Wang et al. / Journal of Banking & Finance 35 (2011) 1158–1169

the case for the typical raw SRTR. However, it should be noted thatthe new scaling rule corrects for serial dependence only and thusthere remain other potential bias factors that might distort the scal-ing of multi-horizon tail risks.

5. Empirical evidence

5.1. Data

To examine whether the SRTR is an appropriate method forobtaining the time-aggregated VaR, this study employs daily stockmarket index returns from 47 markets listed in the Morgan StanleyCapital International (MSCI) index from Datastream, which in-cludes most developed and emerging markets. The sample periodranges from January 2, 1997 to December 31, 2009 (3391 tradingdays). The markets are divided into six regions, Africa, Europe,Eastern Europe, Central and South America, North America, andthe Asia Pacific. For each market, VaR (1) is obtained from the1%-quantile of daily returns and the corresponding VaR (10) andVaR (30) are calculated by (2). This study also reports the com-puted biases from (3). Because the degree of series dependencedominates in the SRTR approximation, we also examine its exis-tence via the variance ratio tests for each market. Furthermore, thisstudy applies the newly-proposed test statistic in (26) to testwhether the overall SRTR induced biases are significantly overesti-

0 10 20 30 40 50 60−40

−30

−20

−10

0

10

20

30

40

VaR (h) %

ζ (h)

%

h=10h=30

Fig. 4. Scatter Plot of VaR (h) with the corresponding f(h). Based on 47 markets, wesketch two scatter plots of VaR (10) and VaR (30) with their corresponding f(10)and f(30), respectively, from January 2, 1997 to December 31, 2009.

0 20 40 60 80 1000

20

40

60

80

100

kurt[

Rk t(1

0)]

kurt [rt]

Fig. 5. Scatter Plot of kurt[rt] with its corresponding scaled kurt[Rkt ðhÞ]. Using 47 market

kurt[Rkt ð30Þ] from January 2, 1997 to December 31, 2009.

mated or underestimated. The test statistics that are statisticallysignificant at confidence levels of 99%, 95% and 90% are markedwith ***,** and *, respectively in Table 3.

5.2. Main findings

Table 3 lists the empirical results for each market. The means ofVaR (10) and VaR (30) are 15.32% and 29.05%, respectively. In pre-senting this table in visual form, f(10) denotes the approximationbias obtained using the SRTR to estimate VaR (10), which generallyyields downward bias of about 5.7%. Usually, severe downwardbiases are associated with positive serial dependence, as indicatedby a positive and significant w* statistic. Meanwhile, the bias growsrapidly with an increasing horizon. For instance, when the timehorizon is increased to 30 days, the averaged understated biasgrows to 13%. Fig. 4 illustrates the results of Table 3 graphically,and sketches the scatter plot of the benchmarks VaR (10) andVaR (30) with their corresponding biases, f(10) and f(30), in per-centage terms for each market. Except for Japan and Korea forthe 10-day horizon, the upward biases f are generally below 10%,and are insignificant among the other nine markets experiencingSRTR overestimation. However, the markets experiencing SRTRunderestimation of over 10% include 13 markets for VaR (10) and26 markets for VaR (30).

To be specific, according to the subsample-based test statisticpresented in (26), while the SRTR-scaled 10-day VaR significantlyunderestimates the benchmark VaR (10) in 10 markets, usuallydue to persistent returns, two markets (namely, Japan and Korea)experience significant SRTR overestimation that may be attribut-able to their different mean-reverting behavior in terms of returndependence. When considering a 30-day horizon, 18 markets aresignificantly underestimated, and none are significantly overesti-mated. As a whole, the above preliminary results suggest that SRTRis a lenient rule for scaling longer-term tail risks, corresponding to asituation of insufficient prudence and financial institutions facing acombination of extreme risk and inadequate capital requirements.

Most notably, the results differ among the six geographicalareas surveyed. Interestingly, North America displays the lowestaverage tail risks. Meanwhile, while Eastern Europe and Centraland South America have larger VaRs than the other areas, theiraverage bias is roughly double that of the other areas. Althoughthe average SRTR bias, f(10), in the Asia Pacific is just �4.8%,removing the overestimated outliers of Japan and Korea turns theaverage f(10) in the Asia Pacific into �8.22%, equaling Eastern Eur-ope and Central and South America. Within the Asia Pacific, thereare five markets, namely, Malaysia, New Zealand, Pakistan, Singa-pore, and Thailand, with the VaRs being underestimated by over15%.

0 20 40 60 80 1000

20

40

60

80

100

kurt [rt]

kurt[

Rk t(3

0)]

s, we sketch two scatter plots of kurt[rt] with their corresponding kurt[Rkt ð10Þ] and

Table 4Global Evidence of Modified-SRTR Scaling. We recompute the time-aggregated VaR (10 days and 30 days) using the newly-proposed modified-SRTR scaling rule, MVaR (h), for the47 countries regions and the overall bias f, along with its significance based on the block-bootstrapped standard error, and the statistic of variance ratio test, w*.

MVaR (h) VaR (10) f(10)% w*(10) MVaR (h) VaR (30) f(30)% w*(30)

Panel A: AfricaMorocco 9.53 9.27 2.76 4.60a 18.01 14.26 26.26 4.80a

South Africa 13.04 13.27 �1.69 1.50 24.06 28.25 �14.83c 1.90c

Turkey 25.72 23.02 11.71 0.68 47.92 42.91 11.69 1.58

Average 16.10 15.19 4.26 2.26 30.00 28.47 7.71 2.76

Panel B: EuropeAustria 15.00 15.63 �4.00 0.45 28.88 34.04 �15.17b 1.37Belgium 11.56 12.55 �7.92 1.26 20.81 23.31 �10.72 1.21Denmark 11.88 12.05 �1.44 0.11 21.53 23.44 �8.16 0.59Finland 17.21 18.40 �6.50 �0.80 31.42 31.04 1.23 0.29France 12.29 13.85 �11.25 �2.20b 21.21 22.62 �6.24 �1.26Germany 14.88 15.46 �3.71 �1.16 25.96 26.84 �3.27 �0.57Greece 18.44 17.02 8.30 2.99a 34.26 28.78 19.05b 3.05a

Ireland 14.29 16.53 �13.54b 0.93 26.37 30.27 �12.88 1.26Italy 13.06 14.14 �7.64 0.81 23.82 24.83 �4.08 1.15Luxembourg 16.30 16.16 0.89 2.40b 31.21 36.37 �14.21 3.29a

Netherlands 13.28 14.66 �9.39 �0.14 24.86 29.19 �14.83b 0.84Norway 15.63 15.45 1.17 �0.31 29.00 31.56 �8.12 0.55Portugal 11.65 11.34 2.76 2.38b 22.55 22.00 2.51 3.04a

Spain 12.29 13.29 �7.53 �0.99 22.64 22.74 �0.45 0.14Sweden 12.72 13.27 �4.14 �2.17b 22.70 24.02 �5.53 �0.87Switzerland 11.31 12.20 �7.33 �0.39 20.45 21.16 �3.40 0.25UK 10.09 10.99 �8.13 �1.64 17.33 19.56 �11.40 �1.00

Average 13.64 14.29 �4.67 0.09 25.00 26.58 �5.63 0.79

Panel C: Eastern EuropeCzech R. 14.11 15.70 �10.11 1.06 26.78 31.81 �15.81b 1.63Hungary 18.03 18.59 �2.97 0.24 33.98 44.36 �23.39a 1.16Poland 14.57 14.59 �0.14 1.80 27.01 29.14 �7.28 2.23b

Russia 29.56 31.16 �5.14 0.80 51.19 55.44 �7.67 0.47

Average 19.07 20.01 �4.59 0.97 34.74 40.19 �13.54 1.38

Panel D: Central and South AmericaArgentina 22.94 21.72 5.61 1.50 42.20 43.24 �2.39 1.82c

Brazil 18.62 22.18 �16.04b �1.28 34.19 38.19 �10.46 �0.16Chile 11.41 11.70 �2.48 4.28a 20.27 23.49 �13.71 2.89a

Mexico 14.55 14.29 1.80 0.60 25.82 27.01 �4.38 0.69Peru 18.51 18.27 1.32 2.86a 35.57 38.48 �7.55 3.15a

Venezuela 18.63 22.65 �17.72b 0.63 34.37 47.28 �27.30a 1.31

Average 17.44 18.47 �4.59 1.43 32.07 36.28 �10.97 1.62

Panel E: North AmericaCanada 10.53 11.10 �5.13 �0.87 19.08 22.26 �14.29 �0.06US 9.72 10.98 �11.42 �2.40a 16.84 21.38 �21.23b �1.32

Average 10.13 11.04 �8.28 �1.64 17.96 21.82 �17.76 �0.69

Panel F: Asia PacificAustralia 8.92 9.22 �3.16 �0.72 15.78 17.75 �11.13 �0.22Hong Kong 15.51 16.38 �5.32 �0.53 28.08 31.27 �10.21 0.13India 15.75 15.77 �0.10 0.42 28.60 28.16 1.57 0.95Israel 11.06 10.86 1.83 0.81 20.91 18.90 10.64 1.88c

Japan 12.48 11.30 10.44 �1.71c 21.50 24.88 �13.59 �1.03Jordan 11.99 11.11 7.87 0.44 22.59 20.81 8.59 1.24Korea 18.67 16.65 12.14 �0.21 34.27 31.31 9.46 0.71Malaysia 13.02 15.90 �18.11c 0.44 24.55 28.58 �14.10 0.97New Zealand 7.37 8.49 �13.25 0.45 13.52 15.49 �12.69 1.01Pakistan 18.99 22.56 �15.80b 3.82a 35.67 44.73 �20.25b 3.82a

Philippines 15.99 14.70 8.82 2.44b 30.61 28.84 6.13 3.28a

Shanghai 16.84 15.45 8.98 0.52 31.42 24.88 26.29b 1.53Singapore 13.52 16.94 �20.20b 2.10b 25.01 29.44 �15.05c 2.33b

Thailand 17.45 19.36 �9.87 2.04b 33.35 35.59 �6.27 2.93a

Taiwan 14.29 14.09 1.39 1.12 26.61 25.67 3.68 1.93c

Average 14.12 14.59 �2.29 0.76 26.17 27.09 �2.46 1.43

Total average 14.75 15.32 �3.48 0.62 27.12 29.05 �5.64 1.21

a Those test statistics that are statistically significant at the confidence levels of 99%.b Those test statistics that are statistically significant at the confidence levels of 95%.c Those test statistics that are statistically significant at the confidence levels of 90%.

J.-N. Wang et al. / Journal of Banking & Finance 35 (2011) 1158–1169 1167

0 10 20 30 40 50 60−40

−30

−20

−10

0

10

20

30

40

VaR (h) %

ξ(h)

%

h=10h=30

Fig. 6. Scatter Plot of VaR (h) with the corresponding bias f(h) from MVaR. Based on47 markets, we sketch two scatter plots of VaR (10) and VaR (30) with theircorresponding f(10) and f(30) produced under our newly-proposed modifiedscaling MVaR (h) as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih� VRðhÞ

pVaRð1Þ � VaRðhÞ respectively, from January 2, 1997

to December 31, 2009.

1168 J.-N. Wang et al. / Journal of Banking & Finance 35 (2011) 1158–1169

It is evident that calculating VaR over a short horizon, followedby SRTR scaling to convert to longer-term tail risks, is likely to beinappropriate and misleading, particularly for markets in EasternEurope, Central and South America, or the Asia Pacific. Caution isnecessary in applying the SRTR. This investigation offers a new ap-proach to verifying and recommending the practical applicabilityof the SRTR, considering the potential presence of a mixture of con-founding dynamic dependence, distributional properties andjumps. This study proposes performing the simple subsample-based test described here, to complement to a typical variance ra-tio test in verifying the existence of a synthetic bias based on theinteraction of these underlying effects. The variance ratio test,w*(h), helps identify the first-order effect causing the bias, i.e., itjustifies the existing serial dependence. Restated, a positive (nega-tive) w*(h) indicates positive (negative) series dependence. To illus-trate the practical usefulness of the new procedures, this study firstconsiders the 10-day case.

Eleven markets had significantly positive w*(10) and four hadsignificantly negative w*(10). By focusing on the markets exhibitingpersistent returns (a positive w*(10)), their average f(10) was�13.59% which is over twice the average for all markets. However,the average f(10) was 7.39% among the four mean-reverting mar-kets (a negative w*(10)). For the 30-day case, 15 markets exhibitedsignificantly positive w*(30), and their average f(10) was �18.7%.Therefore, the empirical results reveal that positive serial depen-dence indeed causes the SRTR to severely understate the time-aggregated VaR.

As discussed in Section 3.3, volatility clustering and jump com-ponents both only slightly distort the approximation performanceof SRTR. Nonetheless, a heavy-tail exerts a greater influence thanthe above two factors. The average kurtosis of 1-day, 10-day, and30-day returns for all markets are 13.98, 6.80, and 5.44, respec-tively. Fig. 5 shows the scatter plots of the kurtosis of daily returnsagainst the corresponding kurtosis of 10-day and 30-day returnsand indicates that almost all kurt[rt] are larger than the corre-sponding kurt[Rk

t ðhÞ]. Tail heaviness is decreasing through tempo-ral aggregating daily returns into 10-day or 30-day returns. Asargued previously, VaR (10) or VaR (30) is overestimated owing tothe smoothing of the heavy-tails.

5.3. New findings after serial dependence correction

To illustrate the practical usefulness of the new procedures, themain results after implementing the modified-SRTR are summa-rized in Table 4. By comparing the robust modified-SRTR withthe traditional raw SRTR, the cross-sectional average of the overallbias, f(10), from the 47 market indices improved from �5.7% to�3.48% and that for the longer horizon, f(30), improved from�13% to �5.64%. Therefore, this robust scaling rule via MVaR in-duces less bias when estimating the multi-period VaR. In particu-lar, more evident results indicating the robustness of MVaR couldbe found in Table 4. When estimating the 10-day VaR using theSRTR, there are 10 markets that are significantly biased; however,using the modified-SRTR, only five markets remain significantlybiased. Similar results are found in the cases for the 30-day VaR.The number of significant biased markets is reduced from 12 to5. In other words, most strong upward or downward biases dis-closed as significant in Table 3 are largely due to the failure toaccommodate the serial time dependence of return in the VaR scal-ing using the raw SRTR. Once such dependence is properly ad-justed, the bias from the first-order effect is largely mitigated.

If there remains any significant bias after adjusting for the timedependence using the new MVaR, it might be contributed by otherbias factors apart from the time dependence. For instance, if theremaining bias is negative, this may suggest the existence of astrong GARCH effect, a non-negligible negative mean or a combina-

tion of both. Similarly, a significant upward bias may be attribut-able to the strong inherent infrequent jump components orheavy-tail phenomenon.

Fig. 6 characterizes the overall bias for 10-day and 30-day VaRcomputed from the newly-proposed modified-SRTR, MVaR, acrossall markets in the sample. Interestingly, the dependence correctionthat makes a notable improvement over the traditional unadjustedraw SRTR is visually evident in Fig. 6. Obviously the scatters of thebiases are now more centered around zero with bias magnitudesmuch smaller than that presented in Fig. 4.

6. Conclusions

Scaling with the SRTR is simple and has been widely employedin practice, and, even in some instances is required by regulation,as a tool for approximating longer horizon tail risks in the financialindustry. The ugly facts based on the real world asset returns makethe optimistic pre-assumptions on which SRTR scaling is built farfrom credible, and thus the performance of SRTR scaling is doubt-ful. This study examines and reconciles different potential bias fac-tors in financial return series from the literature to clarify howbiased the SRTR may be by considering alternative return charac-teristics: including serial dependence, volatility clustering, heavy-tails, and jumps. By complementing the variance ratio test, thisstudy proposes a new test that is both intuitive and simple of theoverall validity of the SRTR based on subsampling.

This study finds that serial dependence severely biases theapplicability of SRTR, and that the heavy-tail results in an upwardsbias for the SRTR; By contrast, a non-zero mean in the daily levelgives rise to only mild bias, so that the effect of overlooking jumpsor volatility clustering is less relevant in scaling time-aggregatedVaR. The empirical evidence presented in this study, covering 47developed and emerging markets included in the MSCI index,shows that the raw SRTR is a lenient rule, which on average under-estimates 10-day and 30-day VaR. This implies that financial insti-tutions using SRTR will be insufficiently prudent and will fail toresolve their inadequate capital requirements.

On the one hand, for some developed markets, even when thepre-assumptions are violated, the SRTR scaling yields results thatare generally correct, as is shown in the global investigation. This sit-uation occurs because the underestimation arising from the dy-namic dependence structure is counterbalanced by the

J.-N. Wang et al. / Journal of Banking & Finance 35 (2011) 1158–1169 1169

overestimation arising from the excess kurtosis and jumps. Hencethe SRTR scaling may serve in its place in assessing the multi-periodVaR. On the other hand, the SRTR scaling of tail risk is likely to bevery inappropriate and misleading, particularly for markets in East-ern Europe, Central and South America, and the Asia Pacific.Although its widespread use as a tool for approximating horizonconversion is understandable, caution is necessary.

As we have shown that the dynamic serial dependence serves asthe first-order effect that biases the validity of the SRTR among theother stylized features, the present paper fills this void by develop-ing a simple and robust scaling rule utilizing the estimated vari-ance ratio, the modified-SRTR (MVaR (h)), for estimating andscaling the multi-horizon tail risks. Interestingly, it turns out thatthe dependence correction is a notable improvement over the tra-ditional unadjusted raw SRTR. This study concludes that the use ofcertain pretests, as proposed above, is an important step and mayilluminate the applicability of the original SRTR in practical tail riskapproximation. Given the demonstrated performance and theirempirical simplicity, the newly-proposed test as well as the mod-ified-SRTR approach are likely to appeal to researchers and practi-tioners alike when estimating longer horizon VaRs.

Acknowledgements

The authors are very grateful to an anonymous referee for his/her helpful comments and suggestions that lead to substantialimprovements of the paper. Wang’s work on this paper was partlyfunded by the National Science Council in Taiwan(NSC 98-2410-H-159-010) and Yeh thank the National Science Council in Taiwan forfinancial support via Grant No. NSC 96-2415-H-008-009. The usualdisclaimer applies.

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