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1 Journal of Banking & Finance xxx (2010) xxx xxx Contents lists available at ScienceDirect Journal of Banking & Finance journal homeage: 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 Deartment of Finance, Minghsin University of Science and Technology, Taiwan b Deartment of Finance, National Central University, Taiwan c Deartment of Finance, Vanung University, Taiwan d Deartment of Finance, Yuan Ze University, Taiwan article info abstract Article history: Received 9 March 2010 Acceted 18 Setember 2010 Available online xxxx JEL classification: G18 G20 C20 Keywords: Value at risk Square-root-of-time rule Jum diffusion Serial deendence Heavy-tail Volatility clustering Subsamling-based test The square-root-of-time rule (SRTR) is oular in assessing multi-eriod VaR; however, it makes several unrealistic assumtions. We examine and reconcile different stylized factors in returns that contribute to the SRTR scaling distortions. In comlementing the use of the variance ratio test, we roose a new intuitive subsamling-based test for the overall validity of the SRTR. The results indicate that serial deendence and heavy-tailedness may severely bias the alicability of SRTR, while jums or volatility clustering may be less relevant. To mitigate the first-order effect from time deendence, we suggest a simle modified-srtr for scaling tail risks. By examining 47 markets globally, we find the SRTR to be lenient, in that it generally yields downward-biased 10-day and 30-day VaRs, articularly in Eastern Euroe, Central-South America, and the Asia Pacific. Nevertheless, accommodating the deendence correction is a notable imrovement over the traditional SRTR. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Following several serious financial crises in little more than a decade, including the Asian Financial Crisis of 1997, the Dot-Com Bubble of 2000, and the Global Financial Tsunami of 2008, risk management, articularly in relation to tail risks, has recently increased considerably in imortance in numerous subfields of finance. Value at Risk (VaR), defined as a worst case scenario in terms of losses on a tyical day, is a oular measure of tail risk management that is not only recommended by banking suervisors (BCBS, 1996a), but is also widely used throughout the financial industry, including by banks and investment funds, see Pérignon and Smith (2010a,b). It is even used by nonfinancial cororations in suervising in-house financial risks following the success of the J.P. Morgan RiskMetrics system. Oerationally, tail risk such as VaR is generally assessed using a 1-day horizon, and short-horizon risk measures are converted to Corresonding author. Tel.: x1890; fax: x addresses: jnwang@must.edu.tw (J.-N. Wang), jhyeh@ncu.edu.tw (J.-H. Yeh), nickcheng@mail.vnu.edu.tw (N.Y.-P. Cheng). longer horizons. A common rule of thumb, borrowed from the time scaling of volatility, is the square-root-of-time rule (hereafter the SRTR), according to which the time-aggregated financial risk is scaled by the square root of the length of the time interval, just as in the Black Scholes formula where the T-eriod volatility is given by r ffiffiffi T. Regulators also advocate the routine use of the SRTR. For examle, to avoid dulication of risk measurement systems, financial institutions are allowed to derive their two-week VaR measure by scaling u the daily VaR by SRTR; see, for examle, BCBS (1996b). In fact, horizons of u to a year are not uncommon; many banks link trading volatility measurement to internal caital allocation and risk-adjusted erformance measurement schemes, which ffiffiffiffiffiffiffiffi rely on annual volatility estimates by scaling 1-day volatility by 252. If the SRTR is to serve as a good aroximation of all quantiles and horizons, it not only requires the iid roerty of zero-mean returns, but also that of the Normality of the returns. These reassumtions are far from being realized in real world financial asset returns, rovided the numerous documented stylized facts that are conflict with these roerties. Accordingly, numerous studies have attemted to identify how these different effects give rise to bias in SRTR aroximation. The first attemt is based on the /$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.

2 2 J.-N. Wang et al. / Journal of Banking & Finance xxx (2010) xxx xxx fact that asset returns may be weakly deendent, both in levels and higher moments. As illustrated in Jorion (2001), the SRTR tends to understate long-term tail risk when the return follows a ersistent attern, but tends to overstate the tail risk of temorally-aggregated returns if it dislays mean-reverting behavior. Similarly, the resence of volatility clustering, as well-documented in the case of most financial assets since Engle (1982), Bollerslev et al. (1992), Bollerslev et al. (1994), under the dynamic setu, has been demonstrated using detailed examles of how the common ractice of converting 1-day volatility estimates to h-day estimates by SRTR scaling is inaroriate and yields overestimates of the variability of long-horizon volatility. On this, see Diebold et al. (1997) and Müller et al. (1990). Numerous extant studies have demonstrated that asset returns exhibit heavy-tails (Fama, 1965; Jansen and de Vries, 1991; Pagan, 1996). Although allowing for dynamic deendence in the conditional variance artially contributes to the letokurtic nature, the GARCH effect alone does not exlain the excess kurtosis in financial asset returns. On the one hand, this motivates studies to emloy their emirical GARCH modeling with student-t or generalized error distributions to account for heavier tails. On the other hand, researchers have turned to models that generate rice discontinuities to resolve the emirical regularity. Researchers have long realized that financial time series exhibit certain unusual and extreme violent movements, known as jums and modeled using jum diffusions develoed by Merton (1976) that create discontinuous samle aths. See Andersen et al. (2002), Pan (2002), Eraker et al. (2003), Becker et al. (2009), Câmara (2009) for recent evidence on the revailing henomena of jums in rice rocesses. Nonetheless, how the underlying jums influence the SRTR aroximation of longer-term tail risks remained unclear until the work of Danielsson and Zigrand (2006). They intuitively and clearly show that SRTR tends to underestimate the time-aggregated VaR and the downward bias deteriorates with the time horizon owing to the existence of negative jums. However, it remains unseen if in general rice jums are not confined to downside extreme losses only, would the SRTR-induced downward-bias move in the other direction instead or become negligible? Although we sound different alarms from distinct ersectives by disclosing SRTR scaling as being inaroriate and misleading, with documented uward biases for some effects and downward biases for others, it is unclear after all whether the overall validity of the SRTR is aroriate or not for ractical risk imlementation given that all these effects coexist in a given asset. However, this aer is not merely concerned with individual effects, such as a weak deendence of returns, volatility scaling, rice discontinuities or letokurticity, as is the case for the literature on the time scaling erformance of the SRTR. Instead, we are interested in the interactions among these stylized facts on the scaling of tail risks via the alication of the SRTR. To our knowledge, no revious investigation has reconciled the quality of aroximation in time-aggregated tail risks using the SRTR under various confounding factors. This study fills this void by first devising a general framework for disentangling and searately estimating the sensitivity toward each systematic risk factor. To examine the overall erformance of the SRTR aroximation and characterize the otential bias, we define a bias function using a benchmark VaR based on averaging a set of subsamled non-overlaing temoral aggregated VaRs. Based on Monte Carlo exeriments, this investigation demonstrates that deendence at the return level is the dominant bias factor. The SRTR leads to a systematic underestimation (overestimation) of risk when the return follows a ersistent (mean-reverting) rocess, and can do so by a substantial margin. Moreover, the magnitude of downward (uward) bias increases with the time horizon. However, volatility clustering tends to drive the timeaggregated VaR to slightly underestimate its true value. Alternatively, the heavy-tailed nature of the underlying return overstates the time-aggregated VaR via the SRTR. Perhas surrisingly, unlike the solely unilateral downside jums secified by Danielsson and Zigrand (2006) that indicate a severe underestimation bias, the Monte Carlo allowing for both sided jums with Poisson arrival erformed in this study suggests that there is a slight overestimation when scaling with the SRTR. In view of these results, roer tests for a reliminary verification of the alicability of the SRTR in ractice are required. This study first recommends a new informal but informative subsamling-based test, comlementing the variance ratio test develoed by Lo and MacKinlay (1988), 1 for emirical studies. Moreover, it also contributes to the literature by suggesting a simle modified-srtr that is robust to the time deendence-induced biases. By utilizing 47 markets included in the MSCI index, including both develoed and emerging markets, this study demonstrates that the SRTR underestimates 10-day and 30-day VaRs by an average of aroximately 5.7% and 13%, resectively. We also observe that the severity of downward bias is greater for emerging markets in Eastern Euroe, Central and South America, and the Asia Pacific. For some develoed markets, even when the model assumtions are violated, the SRTR scaling yields results that are correct on average, as shown in the global investigation. This occurs because the underestimation resulting from the dynamic deendence structure is counterbalanced by the overestimation resulting from the excess kurtosis and jums. Hence SRTR scaling can be aroriate in some cases. Although its widesread use as a tool for aroximate horizon conversion is understandable, caution is, however, necessary. We believe that the use of certain retests as we roosed beforehand is imortant and may illuminate the alicability of SRTR in the ractical aroximation of tail risks. Our newly-roosed modified-srtr aroach is shown to be effective in alleviating the bias attributable to the firstorder effect from time deendence and the deendence correction is a notable imrovement over the traditional unadjusted raw SRTR. The remainder of this aer is organized as follows. Section 2 formally defines the time-aggregated VaR and SRTR scaling. Section 3 then erforms algebraic analysis, in conjunction with Monte Carlo simulations, to disentangle each isolated different stylized effect on the SRTR. This section also briefly reviews the variance ratio test devised by Lo and MacKinlay (1988). Section 4 introduces the suggested variance ratio test and a newly-develoed subsamlebased test for retesting the alicability of the SRTR. More imortantly, we introduce a new tail risk scaling rule the Modified- SRTR. Section 5 subsequently summarizes the global emirical study based on data from 47 develoed and emerging markets included in the MSCI index. Finally, Section 6 resents the conclusions. 2. Time-aggregated value at risk The 1-day VaR, defined as VaR (1), measures the maximum ossible loss over one trading day under a given confidence level 100 (1 c). Suosing that the initial investment of the asset is$1 and R is the random rate of return, then, the asset value at the 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 rš > cg: 1 Finding that using SRTR to estimate Share Ratios causes bias when returns exhibit serial deendence, Lo (2002) suggests using the variance ratio test as a retest. Other related works include Huang (1985) and Ayadi and Pyun (1994), among many others. ð1þ

3 J.-N. Wang et al. / Journal of Banking & Finance xxx (2010) xxx xxx 3 Following the concern of 718 (Lxxvi) in the Basel II Accord, we denote the confidence level as 99% in this aer. The VaR (1) of an asset can simly be estimated through the quantile function of the historical returns. Suosing that a sequence of T daily log rices of an asset f t g T t¼0 is available, then its daily returns are fr t g T t¼1, where r t = t t 1. By letting q() denote the quantile function, given P(r t 6 q(0.01)) = 0.01, the value of VaR (1) is defined as q(0.01). However, in ractice, it is usually hard to estimate the regulatory h-day VaR, VaR (h), since the time horizon needed for the VaR (h) is quite long, esecially when h is large. For examle, 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 er year). Therefore, the Basel Committee on Banking Suervision suggests that banks scale VaR (1) u to 10 days using SRTR. More generally, it says ffiffiffi that VaR (1) could be converted to VaR (h) by multilying by h. ffiffiffi Under the I.I.D. and zero mean assumtions, without doubt, h VaR (1) is equal to VaR (h). The urose of this aer is to investigate the validity of the SRTR in time-aggregated VaR. Before roceeding with our algebraic analysis, we need a true VaR (h) as a benchmark for the comarison. In ractice it is usually difficult if not ossible to estimate the regulatory benchmark VaR. In this study, we recommend finding the benchmark h-eriod VaR, VaR (h), through a subsamling scheme on the return series, and then use it for further characterization of the biases in our Monte Carlo exeriments as well as in emirical studies. This quantity is also emloyed to develo an informative retest for examining the overall alicability of the SRTR in reality in a later section. Before considering VaR (h), we need to generate the daily rices where the samle size may not be too short and construct an h-day return series from the original data. By leaving the first h rices as seeds, one may begin by subsamling the rice series with a fixed length of h days with one of the seeds as the starting oints. In this regard, we confront a total of h 1 different subsamles of h-horizon return series and the kth subsamle time-aggregated return from a non-overlaing interval, denoted by fr k s ðhþgbðt=hþ 1c s¼1, is defined as hþk fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} k 1 ; 2hþk hþk 1 ;...; fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} shþk ðs 1Þhþk ;...; fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} R k 1 ðhþ R k 2 ðhþ R k s ðhþ where k =1,2,...,h 1. Given PðR k s ðhþ 6 Q kð0:01þþ ¼ 0:01, a secific h-day VaR denoted by VaR k (h) can be comuted intuitively by Q k (0.01) for the kth series. For each k, there will be a corresonding VaR k (h). As these fvar k ðhþg h 1 are obtained from different rices without overlaing return eriods, a benchmark naturally arises to be defined as VaRðhÞ ¼ 1 h 1 X h 1 VaR k ðhþ: Suosing the SRTR is correct, the scaled ffiffiffi h VaR (1) shall be equal to VaR (h). To examine whether the SRTR is tenable to serve as a good aroximation for the multi-horizon VaR, given our subsamled and averaged benchmark VaR (h), we define a bias function f(h) to measure the aroximation error of the SRTR in scaling tail risks by, 2 ffiffiffi h VaRð1Þ fðhþ ¼ 1: ð3þ VaRðhÞ ffiffiffi 2 Danielsson and Zigrand (2006) calculate VaR (h)/( h VaR (1)) as an indicator of bias. ð2þ If f(h) is ositive (negative), using SRTR roduces an overestimated (underestimated) time-aggregated VaR. In the next section, we construct the Monte Carlo exlorations for different non-i.i.d. returns features to investigate the influences of serial deendence, heavytailed distributions, jums, and volatility clustering on the timeaggregated VaR, resectively. 3. Characterizing biases: algebraic analysis with Monte Carlos Different biases arise from different data generating rocesses, excet for the re-assumed I.I.D. Gaussian case with zero mean. To accommodate a wide sectrum of stylized facts documented in the literature, we consider a fairly general data generation rocess (DGP) for daily return r t that follows a non-zero mean ARMA (1,1)-GARCH (1,1) model with Poisson jums as r t ¼ l þ / 1 r t 1 þ a t þ h 1 a t 1 þ J t ; a t ¼ r t t ; r 2 t ¼ a 0 þ b 1 r 2 t 1 þ a 1a 2 t 1 ; where t =1,2,...,T and J t is a comound Poisson rocess with jum size distributed as Nð0; r 2 j Þ and constant jum intensity k. We allow GARCH (1, 1) to govern the evolution of the conditional variance of a t over time. { t } is a sequence of I.I.D. N (0,1), a 0 >0,a 1 P 0, b 1 P 0, and a 1 + b 1 < 1. Assuming there are 250 trading days er year, we let a 0 ¼ 0:15 ð1 a 1 b 1 Þ ; ð5þ 250 simly to control the annualized volatility to be roughly about 15%. Through the Monte Carlo simulation, a sequence of 5000 daily returns, r 1,r 2,...,r 5000, is constructed, which amounts to a samling eriod of about 20 years. Then we subsamle (h 1) sequences of h-day temoral aggregated returns from the above daily returns. The VaR (1) of the daily returns is defined as the 1%-quantile for this simulation. The time-aggregated VaR (h), h = 10 or 30, is comuted through the average of the subsamled quantities. With 2000 relications, we denote the true VaR (1) and VaR (h) as the means of the 1-day and h-day VaRs, resectively. We examine the following secific DGPs by restricting certain arameters to isolate the different effects that might have an imact on the time-aggregated VaR. We will come back and reconcile all these imacts on the SRTR scaling of a tail risk later Non-zero mean As the validity of SRTR scaling for quantiles hinges on a series of assumtions, this subsection resents the bias arising from the rimary factor of a non-zero mean for the underlying return. By letting / 1 = h 1 = a 1 = b 1 =0,r t = r and assume a zero jum (r j = 0), a non-zero mean model is r t ¼ l þ a t ; where a t = r t. Assuming Normality for simlicity, the daily VaR (1) is VaRð1Þ ¼ l ru 1 ð0:01þ; where U 1 () is the inverse normal cumulative distribution function. Straightforwardly, we can find the VaR under a longer holding eriod h to be VaRðhÞ ¼ h l ffiffiffi h ru 1 ð0:01þ: ð8þ Therefore, the bias indicator f is l þ ru 1 ð0:01þ fðhþ ¼ffiffiffi 1: ð9þ h l þ ru 1 ð0:01þ We let l = 0.08%, 0.04%, 0.02%, 0.02%, 0.04%, and 0.08%, which imly that the means of their annualized returns are 20%, 10%, 5%, ð4þ ð6þ ð7þ

4 4 J.-N. Wang et al. / Journal of Banking & Finance xxx (2010) xxx xxx Table 1 Temorally-Aggregated VaR under Different Scenarios. We consider different DGPs of returns as shown in Eq. (4) where t =1,2,...,5000. ffiffiffiffiffiffi For each simulation, ffiffiffiffiffiffi we generate the 10- day and 30-day VaRs through the SRTR and the subsamling aroach. We reeat 2000 times and reort the means of 10 VaR (1), VaR (10), f(10), 30 VaR (1), VaR (30), and f(30). ffiffiffiffiffiffi 10 VaR (1) VaR (10) f(10)% ffiffiffiffiffiffi 30 VaR (1) VaR (30) f(30)% Panel A: Non-zero mean models with different l% Panel B: AR (1) models with different / Panel C: MA (1) models with different h Panel D: GARCH (1,1) models with different (a 1,b 1 ) (0.130,0.820) (0.150,0.800) (0.130,0.840) (0.150,0.820) Panel E: Student-t models with different l Panel F: Jum models with different (k,r j ) (0.058,0.020) (0.058,0.030) (0.082,0.020) (0.082,0.030) %, 10%, and 20%, resectively. 3 The simulation results are reorted in Panel A of Table 1. Interestingly, SRTR tends to underestimate (overestimate) VaR (10) or VaR (30) due to the effect of a negative (ositive) l. However, as the readers will see in the following subsection, we show that this non-zero-mean-induced bias is slight and only of second-order imortance in the resence of serial deendence Series deendence By fixing l = a 1 = b 1 =0, r t = r, and assuming a zero jum (r j = 0), a weakly stationary ARMA (1,1) model for r t is r t ¼ / 1 r t 1 þ a t þ h 1 a t 1 ; where a t = r t. The k-order autocorrelation of r t is ð10þ q 1 ¼ / 1 þ h 1r 2 var½r t Š ; q k ¼ / 1q k 1 ; for k > 1; ð11þ where the variance of r t is var½r t Š¼ ð1 þ 2/ 1h 1 þ h 2 1 Þr2 : ð12þ 1 / The average annual return of the examed 47 markets is 6.61%. Only two among them, namely, Portugal and Turkey, have higher average annual returns that exceed 20%. Then the variance of R k s ðhþ is written as ( ) var½r k Xh 1 X h 1 t ðhþš ¼ Covðr t i ; r t j Þ¼var½r t Š h þ 2 Xh 1 ðh kþq k i¼0 j¼0 (!!) ¼ var½r t Š h þ 2 h 1 /h 1 1 / 1 ð1 / 1 Þ 2 / 1 þ h 1ð1 / 2 1 Þ 1 þ 2/ 1 h 1 þ h 2 1 : ð13þ Therefore, based on the above result it is straightforward to show qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VaRðhÞ ¼ var½r k t ðhþš U 1 ð0:01þ; ð14þ where U 1 () is the quantile function from the standard Normal distribution. Then we can find the bias f through (3), (12) (14), i.e., fðhþ¼ ffiffiffi (!!) h h h þ 2 1 /h 1 / 1 / 1 ð1 / 1 Þ 2 1 þ h 1ð1 / 2 1 Þ 1=2 1: 1 þ 2/ 1 h 1 þ h 2 1 ð15þ We let h 1 = 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 h 1 = 0.7, 0.5, 0.2, 0.2, 0.5, 0.7 are also examined. The results in anel B and anel CofTable 1 show that the SRTR yields severe overestimation (in the case of negative serial correlation) or severe underestimation results (in the case of ositive serial correlation) for VaR (10) and VaR (30). According to (15), the atterns of bias function across horizons, f(10), f(30), and f(90), are lotted in Figs. 1 and 2.

5 J.-N. Wang et al. / Journal of Banking & Finance xxx (2010) xxx xxx Tail behavior If r t is normally distributed, R k s 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 not hard. We can then calculate VaR (1) or VaR (h) using (14). Under the Normal case, it is easy to find that the bias function is only related to variances, i.e., fðhþ ¼ h var½r 1=2 tš : ð16þ var½r t ðhþš Nevertheless, without the Normal assumtion, not only the variance, but also the tail behavior before/after temoral aggregation, may affect f(h). While the work of Dacorogna et al. (2001) has demonstrated that, excet for the boundary case of Normality, any other heavytailed distribution under a stable law leads the SRTR to underestimate the VaR, to our knowledge, there is no general theoretical ζ h=10 h=30 h= φ 1 Fig. 1. Bias functions of aggregated VaR with AR (1) models. Consider the model from (10) given / 1 = 0. We lot different time horizons of bias functions, f(10), f(30), and f(90), which are determined by (15). ζ h=10 h=30 h= θ 1 Fig. 2. Bias functions of aggregated VaR with MA (1) models. Consider the model from (10) given h 1 = 0. We lot the different time horizons of bias functions, f(10), f(30), and f(90), which are determined by (15). model that can rovide a good interretation of the relationshi between the time-aggregated VaR and tail behavior. In articular, the question is a little involved from the distributional ersective since there are some heavy-tailed distributions that may not be closed under temoral aggregation, even if they are iid generated. For instance, a student-t with 2 degrees of freedom would scale like the SRTR in the tails; however, the fat-tail may no longer exist after aggregation and thus this case gives rise to another source of SRTR aroximation error beyond the discussions in the literature. In addition, while the letokurticity of the observed returns may be attributable to its underlying distributional roerty, it can also be the consequence of a higher moment deendence such as volatility clustering, as well as some occasional rice discontinuities or jums. We will therefore investigate these issues through the numerical analysis of three oular economic henomena that have contributed to the excess kurtosis in the stylized facts: volatility clustering, heavy-tailed distributions and rice jums Volatility clustering By letting l = / 1 = h 1 = 0 and assuming no jum, we consider the following GARCH (1, 1) model r t ¼ r t t ; r 2 t ¼ a 0 þ b 1 r 2 t 1 þ a 1r 2 t 1 : ð17þ Drost and Nijman (1993) derive the temoral aggregation of the GARCH rocesses and show, under regularity conditions, that the corresonding samle ath of R t (h) follows a similar GARCH (1,1) rocess with different arameters. The results have been suggested to convert short-run volatility into long-run volatility in Christoffersen and Diebold (1997) and Diebold et al. (1998). They do, however, oint out that using SRTR is inaroriate and roduces overestimates of the variability of long-horizon volatility. While these works highlight the dangers of SRTR in the scaling of time-varying volatilities into longer horizons, we take a different route. We conduct a series of Monte Carlo exeriments to exlore the robustness of the SRTR in scaling VaR in the resence of GARCH effects in the underlying return series. We entertain airs of (a 1,b 1 ) with a 1 + b 1 95 or 97 for the GARCH (1, 1) models. Panel D of Table 1 shows that the SRTR tends to yield only a slightly underestimated VaR (10) or VaR (30) in the resence of volatility clustering in terms of the negative fs ranging from 1.43% to 2.93%. These downward biases are intuitively reasonable for overlooking the time-varying risks Heavy-tailed distribution To demonstrate the effect of different distributional considerations from the literature, we let the return rocess be r t ¼ rx t ; ð18þ where {x t } is a sequence of indeendent and identical student-t distributions with m degrees of freedom. On the variance of the aggregated entity, since {x t } is indeendent, var [R t (h)] is equal to var [r t ] multilied by h. Nevertheless, it is intuitive that if we add h daily returns to a h-day return, the long-tailedness aearing in the daily return shortens as h increases. To see this, Fig. 3 shows the robability density functions of daily and 10-day returns with m = 5 and it is rather obvious that the tail art has been diluted after aggregation. By setting m = 3, 5, 7, and 9 to obtain different heavy-tailed return distributions from (18), we find that an overestimated aggregated VaR based on the SRTR is due to the heavier tail in the daily return distribution. To robe further into what may have gone wrong with the SRTR scaling aroximation, we reort both the variance and kurtosis of the daily, 10-day and 30-day returns under different DGPs in Table 2. By cross insecting across the

6 6 J.-N. Wang et al. / Journal of Banking & Finance xxx (2010) xxx xxx Standardized 10 day return density Standardized 1 day return density jum diffusion rocess 4 by letting l = / 0 = / 1 = h 1 = a 1 = b 1 = 0 and r t = r to isolate the effect from jums r t ¼ r t þ J t ; ð19þ where J t is a comound Poisson rocess with constant jum intensity k and random jum size distributed as Nð0; r 2 j Þ. The aggregated variance of r t can be written as " # var½r t ðhþš ¼ var Xh 1 ¼ h var½r t Š: ð20þ i¼0 r t i þ Xh 1 i¼0 J t i Fig. 3. Probability densities of 1-day and 10-day returns. We generate the daily returns which follow model (18) with m = 5. Then, we deict the robability densities of standardized 1-day and 10-day returns. Panel E in Table 1 and anel D in Table 2, it is readily seen that the uward bias comes solely from the decrease in kurtosis, and not from the scaling of volatility, since var[r k t ð10þ] 10 var[r t] whereas kurt[r k t ð10þ] < kurt[r t]. Similar atterns can be found in R k s ð30þ. Therefore, the SRTR overestimates VaR (10) and VaR (30) by about 5 11% with different m for failing to take into account the change in tail behavior among the original high frequency returns and the temorally-aggregated ones in anel E of Table Jums While fat-tailed distributions may be more suited for daily internal risk management, they may not be suited for the modeling of uncommon and unexected systemic events. We consider a general Table 2 Variance and kurtosis of returns. We generate r t from the DGPs as (18) and (19). Thus R t (10) can be calculated by r t. To characterize the otential biases attributable to the scaling of variance or aggregation bias due to changing tail behavior, for each DGP we reort the corresonding variance and kurtosis of r t, R t (10), and R t (30). Variance (%) Kurtosis 1-day 10-day 30-day 1-day 10-day 30-day Panel A: AR (1) models with different / Panel B: MA (1) models with different h Panel C: GARCH (1, 1) models with different (a 1,b 1 ) (0.130,0.820) (0.150,0.800) (0.130,0.840) (0.150,0.820) Panel D: Student-t models with different l Panel E: Jum models with different (k,r j ) (0.058,0.020) (0.058,0.030) (0.082,0.020) (0.082,0.030) We let k = 0.058, and r j = 2%, 3% to allow for a variety of combinations of jum intensities and sizes. 5 by cross insecting across the Panel F in Table 1 and anel E in Table 2, readers may find the results are similar to revious heavy-tailed cases: the roblem with the SRTR in jum models is not due to their variance scaling but to their changing tail behavior. It is intuitive that a return distribution with jums has heavier tails. Furthermore, summing over h daily returns smoothes out the infrequent jum effects. Thus, the SRTR rovides an overestimated aggregated VaR. We also find var[r k t ð10þ] 10 var[r t] while kurt[r k t ð10þ]< kurt[r t ]. These simulation results shown in anel F of Table 1 indicate that jums indeed let the SRTR roduce an overestimated time-aggregated VaR. Nevertheless, under reasonable k and r j settings, we also find that the size of the systematic overestimation bias from the SRTR only has a slight imact, as they are less than 1.6% in aroximating VaR (10) and less than 2.3% in aroximating VaR (30). Our result is largely different from that of Danielsson and Zigrand (2006) who allow for downside jums only in their setu and it is reasonable to document downward bias in their case. Instead, we not only find uward bias via the SRTR scaling in the resence of Poisson jums, but, we document that the biases may not be that much even after we consider some sizable jum intensities and jum sizes A summary The receding analysis yields the following findings. First, the weak deendence in returns dominates among all the confounding factors considered in this study when the SRTR is used to estimate VaR (h). Positive serial deendence leads to a severe underestimation in the SRTR s aroximation of VaR (h), while severe overestimation occurs in the case of negative serial deendence. Given these results, this study rooses using the variance ratio test, tyically emloyed to test for market efficiency, to examine the synthetic underlying serial deendence in emirical studies as a retest of the alicability of VaR scaling using the SRTR. Second, using the SRTR may roduce an overestimate or underestimate of VaR (h) because of the changes in tails. In cases of overestimates of the student-t distribution and jums, the heavy-tailed nature is smoothed out by aggregating daily returns. However, volatility clustering may lead to the SRTR resulting in a slight downward bias owing to neglecting the time-varying nature. To summarize, this study carefully uses the SRTR to estimate the time-aggregated VaR, when the real data exhibits serial deendence, volatility clustering, a heavy-tailed distribution, or jums. 4 Danielsson and Zigrand (2006) demonstrate that the square-root-of-time rule leads to systematic underestimation of risks, and their setu allows for downside jums that reresent losses only. However, there is no a riori theoretical reason to restrict, let alone exect, the rices to jum down only and therefore we entertain our jum comonent to jum symmetrically and our arameters to be in line with the jum diffusion literature. 5 Andersen et al. (2002) show that the jum intensity is about 0.014, that is, 14 times every thousand trading days on average, for the daily S&P 500 cash index. They also estimate the jum size arameter r j to be at 1.5%. In the simulation conducted by Huang and Tauchen (2005), their r j varies from 0 to 2.5%.

7 J.-N. Wang et al. / Journal of Banking & Finance xxx (2010) xxx xxx 7 4. Pretesting SRTR alicability and a modified-srtr 4.1. Variance ratio test From the viewoint of emirical exloration, Lo and MacKinlay (1988) roose a statistic to test the hyothesis of a random walk. The general h-eriod variance ratio statistic VR (h) is denoted as VRðhÞ var½r tðhþš h var½r t Š ¼ 1 þ 2 Xh 1 1 k q h k ; ð21þ where q k is the kth order autocorrelation coefficient of {r t }. When VR (h) = 1, this means that {r t } follows the random walk hyothesis; when VR (R) 1, {r t } exhibits serial deendence. Moreover, we regard VR (h) as an indicator which measures the synthetical effects on different degrees of serial deendence. If VR (h) is significantly larger (smaller) than one, we say this series is characterized by a synthetically ositive (negative) serial deendence of {r t }. It is intuitive that the ositive (negative) serial deendence causes the SRTR to be underestimated (overestimated). Lo and MacKinlay (1988) define the following statistic to estimate the VR (h) of Eq. (21), VRðhÞ :¼ 1 þ 2 Xh 1 1 k ^q k ; q ð22þ where ^q k denotes the autocorrelation coefficient estimators. Under the random walk hyothesis, VRðhÞ still aroaches one. For the standard inferences, it is necessary to comute its asymtotic variance. First denote a heteroskedasticity-consistent estimator of the asymtotic variance of q k, " #" # 2 ^d k ¼ T XT X T ðr j ^lþ 2 ðr j k ^lþ 2 ðr j ^lþ 2 ð23þ j¼kþ1 where ^l 1 T P T r k. Then, the following is a heteroskedasticity-consistent estimator of the asymtotic variance of VRðhÞ, ^# h 4 Xh 1 1 k 2^dk : ð24þ h Regardless of the resence of general heteroskedasticity, the standardized statistic w*(h) can be used to test the hyothesis of a random walk, i.e., ffiffiffi w T ðvrðhþ 1Þ ðhþ ¼ qffiffiffiffiffi a Nð0; 1Þ; ð25þ ^# h where a denotes for asymtotically distributed as A new subsamle-based test for overall SRTR alicability While the variance ratio test is in sirit in line with the SRTR, it offers only a artial icture since it is informative in detecting only secifically the deendence structure of the return series. As the deendence structure is of first-order imortance, this aer offers a new and simly comlementary aroach to test the overall validity of alying the SRTR to scale VaR to a secific asset by utilizing the subsamles we used to construct our benchmark VaR (h). Since the way fvar k ðhþg h 1 is constructed is based on fr k s ðhþgbðt=hþ 1c s¼1, the h-eriod return from the non-overlaing subsamling of the original rices, the subsamled fvar k ðhþg h 1 is emloyed to comute the benchmark VaR (h) by taking the subsamle average. Before we arrive at a formal test for the SRTR s ffiffiffi validity, a roerly comuted standard error of the bias term, h VaRð1Þ VaRðhÞ, is needed. Nonetheless, the well-documented time deendence j¼1 may carry over to the time-aggregated returns and thus the simle variance estimator as defined by " # 1=2 1 X h 1 R 0 ðhþ ¼ ðvar k ðhþ VaRðhÞÞ 2 ; h 2 may not be sufficient to accommodate the generality of the return rocess. Moreover, the R 0 (h) defined above is also subject to the small samle bias since h is commonly limited to 10 or 30. To accommodate both of these concerns, we choose to rely on the block bootstraed samles to roduce a reliable standard error for a formal test. This is done by taking into account the otential time deendence by retaining the dynamic structure of the underlying returns by randomly drawing subsamles using blocks of consecutive returns, thereby alleviating the small samle bias roblem and imroving the testing erformance. The imlementation rocedure is as follows. We choose a block length of 10 and these blocks could be overlaing. Secifically, the data are divided into T 9 blocks, with the first block being {r 1,r 2,...,r 10 }, the second block being {r 2,r 3,...,r 11 },...,..., and the last block being {r T 9,r T 8,...,r T }. We then randomly resamle T/10 blocks to construct a new bootstraed samle of T days of returns. For each bootstraed resamle, we calculate and save the ffiffiffi values of VaR (1), the subsamled and averaged VaR (h), and h VaRð1Þ VaRðhÞ. By reeating the above rocedure for 5000 relications, ffiffiffi we obtain a bootstraed samling distribution of the bias, h VaRð1Þ VaRðhÞ, based on the 5000 bootstraed resamles. ffiffiffi Therefore, the bootstraed standard deviation of h VaRð1Þ VaRðhÞ, defined as R(h), can now be calculated. Intuitively, under certain regularity conditions, it is easy to argue from the Central Limit Theorem 6 that ffiffiffi h VaRð1Þ VaRðhÞ Nð0; 1Þ: ð26þ RðhÞ This statistic serves as our benchmark retest for the overall validity of the SRTR in our subsequent analysis, after considering different confounding dynamic and distributional roerties that revail in real asset returns Scaling tail risk with a new modified-srtr As we have shown, the dynamic serial deendence in the return rocess, among the other stylized features, serves as the first-order effect that biases the validity of the SRTR in scaling quantiles. In view of this, the subsection moves one ste further to roose a simle and robust correction to the existing SRTR. It is a well-acceted fact that a variance ratio greater than 1 suggests the existence of a ositive deendence in the underlying return series, and the oosite situation holds true for the case of a VR (h) of less than 1. This simle correction thus mainly makes use of the estimated variance ratio as indicated in Eq. (22) to adjust the raw SRTR by taking the time deendence structure into consideration. 7 Accordingly, we formally define an estimator which estimates VaR (h) through this robust rule as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi MVaRðhÞ ¼ h VRðhÞ VaRð1Þ: ð27þ We thus refer to it as the modified-srtr (MVaR). Note that if a time series is serially uncorrelated, the variance ffiffiffi ratio is 1 and therefore MVaR (h) will simly reduce to h VaR (1), which is essentially 6 The Central Limit Theorem holds for weakly deendent data as long it is aroriately standardized by a standard error that accounts for the underlying deendence. 7 We thank the anonymous referee for ointing this suggestion out to us.

8 8 J.-N. Wang et al. / Journal of Banking & Finance xxx (2010) xxx xxx Table 3 Global Evidence on SRTR Scaling of Tail Risk. We consider 47 markets globally with both develoed markets and emerging markets listed in the MSCI. The samle eriod extends from January 2, 1997 to December 31, 2009 (3391 trading days). For each market, the time-aggregated VaR (10 days and 30 days) comuted through both the simle SRTR and subsamling aroach are reorted. We also reort the overall bias, f, along with its significance based on the block-bootstraed standard error, as well as the statistic of the variance ratio test, w*. ffiffiffiffiffiffi 10 VaR (1) VaR (10) f(10)% w*(10) ffiffiffiffiffiffi 30 VaR (1) VaR (30) f(30)% w*(30) Panel A: Africa Morocco a a South Africa a 1.90 c Turkey Average Panel B: Euroe Austria a 1.37 Belgium c b 1.21 Denmark Finland France b Germany Greece a a Ireland a a 1.26 Italy Luxembourg b a 3.29 a Netherlands a 0.84 Norway Portugal b c 3.04 a Sain Sweden b Switzerland UK Average Panel C: Eastern Euroe Czech R b a 1.63 Hungary a 1.16 Poland b 2.23 b Russia Average Panel D: Central and South America Argentina b 1.82 c Brazil Chile b 4.28 a a 2.89 a Mexico Peru c 2.86 a a 3.15 a Venezuela a a 1.31 Average Panel E: North America Canada US a Average Panel F: Asia Pacific Australia Hong Kong India Israel c Jaan b 1.71 c Jordan Korea c Malaysia a a 0.97 New Zealand Pakistan a 3.82 a a 3.82 a Philiines b a Shanghai Singaore a 2.10 b a 2.33 b Thailand a 2.04 b a 2.93 a Taiwan c Average Total average 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%.

9 J.-N. Wang et al. / Journal of Banking & Finance xxx (2010) xxx xxx 9 the case for the tyical raw SRTR. However, it should be noted that the new scaling rule corrects for serial deendence only and thus there remain other otential bias factors that might distort the scaling of multi-horizon tail risks. 5. Emirical evidence 5.1. Data ζ(h) % h=10 h= VaR (h) % Fig. 4. Scatter Plot of VaR (h) with the corresonding f(h). Based on 47 markets, we sketch two scatter lots of VaR (10) and VaR (30) with their corresonding f(10) and f(30), resectively, from January 2, 1997 to December 31, To examine whether the SRTR is an aroriate method for obtaining the time-aggregated VaR, this study emloys daily stock market index returns from 47 markets listed in the Morgan Stanley Caital International (MSCI) index from Datastream, which includes most develoed and emerging markets. The samle eriod ranges from January 2, 1997 to December 31, 2009 (3391 trading days). The markets are divided into six regions, Africa, Euroe, Eastern Euroe, Central and South America, North America, and the Asia Pacific. For each market, VaR (1) is obtained from the 1%-quantile of daily returns and the corresonding VaR (10) and VaR (30) are calculated by (2). This study also reorts the comuted biases from (3). Because the degree of series deendence dominates in the SRTR aroximation, we also examine its existence via the variance ratio tests for each market. Furthermore, this study alies the newly-roosed test statistic in (26) to test whether the overall SRTR induced biases are significantly overestimated or underestimated. The test statistics that are statistically significant at confidence levels of 99%, 95% and 90% are marked with ***,** and *, resectively in Table Main findings Table 3 lists the emirical results for each market. The means of VaR (10) and VaR (30) are 15.32% and 29.05%, resectively. In resenting this table in visual form, f(10) denotes the aroximation bias obtained using the SRTR to estimate VaR (10), which generally yields downward bias of about 5.7%. Usually, severe downward biases are associated with ositive serial deendence, as indicated by a ositive and significant w* statistic. Meanwhile, the bias grows raidly with an increasing horizon. For instance, when the time horizon is increased to 30 days, the averaged understated bias grows to 13%. Fig. 4 illustrates the results of Table 3 grahically, and sketches the scatter lot of the benchmarks VaR (10) and VaR (30) with their corresonding biases, f(10) and f(30), in ercentage terms for each market. Excet for Jaan and Korea for the 10-day horizon, the uward biases f are generally below 10%, and are insignificant among the other nine markets exeriencing SRTR overestimation. However, the markets exeriencing SRTR underestimation of over 10% include 13 markets for VaR (10) and 26 markets for VaR (30). To be secific, according to the subsamle-based test statistic resented in (26), while the SRTR-scaled 10-day VaR significantly underestimates the benchmark VaR (10) in 10 markets, usually due to ersistent returns, two markets (namely, Jaan and Korea) exerience significant SRTR overestimation that may be attributable to their different mean-reverting behavior in terms of return deendence. When considering a 30-day horizon, 18 markets are significantly underestimated, and none are significantly overestimated. As a whole, the above reliminary results suggest that SRTR is a lenient rule for scaling longer-term tail risks, corresonding to a situation of insufficient rudence and financial institutions facing a combination of extreme risk and inadequate caital requirements. Most notably, the results differ among the six geograhical areas surveyed. Interestingly, North America dislays the lowest average tail risks. Meanwhile, while Eastern Euroe and Central and South America have larger VaRs than the other areas, their average bias is roughly double that of the other areas. Although the average SRTR bias, f(10), in the Asia Pacific is just 4.8%, removing the overestimated outliers of Jaan and Korea turns the average f(10) in the Asia Pacific into 8.22%, equaling Eastern Euroe and Central and South America. Within the Asia Pacific, there are five markets, namely, Malaysia, New Zealand, Pakistan, Singaore, and Thailand, with the VaRs being underestimated by over 15% (10)] 60 (30)] 60 kurt[r k t 40 kurt[r k t kurt [r t ] kurt [r t ] Fig. 5. Scatter Plot of kurt[r t ] with its corresonding scaled kurt[r k t ðhþ]. Using 47 markets, we sketch two scatter lots of kurt[r t] with their corresonding kurt[r k t ð10þ] and kurt[r k t ð30þ] from January 2, 1997 to December 31, 2009.

10 10 J.-N. Wang et al. / Journal of Banking & Finance xxx (2010) xxx xxx Table 4 Global Evidence of Modified-SRTR Scaling. We recomute the time-aggregated VaR (10 days and 30 days) using the newly-roosed modified-srtr scaling rule, MVaR (h), for the 47 countries regions and the overall bias f, along with its significance based on the block-bootstraed 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: Africa Morocco a a South Africa c 1.90 c Turkey Average Panel B: Euroe Austria b 1.37 Belgium Denmark Finland France b Germany Greece a b 3.05 a Ireland b Italy Luxembourg b a Netherlands b 0.84 Norway Portugal b a Sain Sweden b Switzerland UK Average Panel C: Eastern Euroe Czech R b 1.63 Hungary a 1.16 Poland b Russia Average Panel D: Central and South America Argentina c Brazil b Chile a a Mexico Peru a a Venezuela b a 1.31 Average Panel E: North America Canada US a b 1.32 Average Panel F: Asia Pacific Australia Hong Kong India Israel c Jaan c Jordan Korea Malaysia c New Zealand Pakistan b 3.82 a b 3.82 a Philiines b a Shanghai b 1.53 Singaore b 2.10 b c 2.33 b Thailand b a Taiwan c Average Total average 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%.