Empirical scaling rules for Value-at-Risk (VaR)

Vikentia Provizionatou a*, Sheri Markose a#, Olaf Menkens a

a CCFEA, Centre for Computational Finance and Economic Agents, #Economics Department, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United

Kingdom

April 15, 2005: Preliminary Draft (Comments Welcome, please do not quote without permission)

Abstract Value-at-Risk is undoubtedly the financial industry’s main measure of risk. Its widespread application follows with it use by regulatory authorities to calculate banks’ market risk capital requirement. The regulatory standard involves reporting the 10-day Value-at-Risk at 99 per cent confidence level on trading portfolios of banks. The current common practice is to use the daily-VaR, routinely calculated using the banks’ internal models, and scale it up to the 10-day VaR using the square-root-of-time rule. The latter, which is appropriate for Gaussian distributions, has been criticized on the grounds that asset returns data is far from Gaussian. In this paper we propose two methods for empirically and locally determined time varying scaling exponents using a recursive window framework on overlapping returns data based on order statistics. The first empirical scaling rule (Hest) assumes a ‘pseudo’ scale invariant measure of the scale exponent, which is derived by the gradient of the linear regression of the q-quantile of the returns with different holding periods in a log-log plot. The second method involves the numerical local determination of scale variant exponents (Hnum) for the q-quantile one day returns and the q-quantile of n>1 returns. We study the properties of these data determined scaling exponents and use back testing methods to compare their effectiveness for risk management with that of the square-root-of-time rule. The back testing results show that the application of the empirically determined scaling rules outperforms the square-root-of-time rule and leads to a significant amount of saving in banks’ capital. JEL classification: C00, G10, G21 Keywords: Value-at-Risk; Scaling exponent; Square-root-of-time rule; Backtesting

* Corresponding author. Tel. +44-1206-873975 Email address: vprovi@essex.ac.uk(V. Provizionatou).

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1. Introduction

Since the late 1990s Value-at-Risk (VaR) has been established as the standard method for evaluating the market risk of a portfolio of financial assets. The concept is very intuitive as VaR summarizes in one single number the maximum loss that a portfolio may incur with probability (1 – q) on a specified time scale, where q is a given confidence level. Mathematically, VaRq is the q–quantile of the portfolio return distribution, and can be calculated as )()( 1 qFRVaR kR

kq

−−= , where kR is the k–

day return on the portfolio, and kRF is the distribution function of kR and )(1 qF kR

− is the q-quantile function. Banks use VaR internally for risk management purposes, but they also have to report their portfolios’ VaR to the regulatory body. The regulators utilize VaR to calculate each bank’s market risk capital requirement, and more specifically they require banks to report their maximum loss over 10 trading days at a 99% confidence level†‡ based on data of at least 250 days length. This implies that banks have to estimate their VaR at the 99% confidence level on a 10-day horizon as accurately as possible, as this figure will determine how much of their capital has to be set aside for market risk contingencies. The need to report VaR at a 10-day horizon entails practical difficulties, which has generated significant academic research on the estimation of VaR.§ The customary method of VaR estimation is based on the order statistics of historical non-overlapping returns data. It is recognized that by this method a minimum of 10 years data is needed for VaR on 10-day returns when using a 250 day window. In a sample, for instance, of 500 historical prices (approximately 2 years of data) one can get only 50 non-overlapping observations. Using this small number of data to calculate the 99th quantile is inadequate and can involve erroneous predictions. To overcome this problem it has become the industry standard that banks calculate daily VaR and then scale it by 10 in order to get the 10-day VaR, which implies that

)(10)( 199.0

1099.0 RVaRRVaR = , where 10R is the 10-day return on the portfolio and

1R is the daily return. This is known as the square-root-of-time rule (SQRT-rule). However, it is well known that the validity of the SQRT-rule is restricted to the Guassian distribution. As the financial asset returns violates that assumption due to ‘fat tails’ and left skewness, it is increasingly being held, as we will review, that neither higher moments of distributions (such as volatility) nor their quantiles should be scaled according to the SQRT-rule. Indeed, the Basel Committee has taken on board the potential problems of the square–root–of–time rule, for in a more recent technical guidance paper (Basel Committee on Banking Supervision, 2002) it is no

† The confidence level for the risk capital requirement is set in order to ensure the system is prepared in the case of a systemic crisis. As this is regarded as a relatively unlikely event, 99% confidence level reflects the low 1% probability. The 10 days horizon VaR is selected that the Banking Supervision authorities reckon that this is the number of days banks will need in order to liquidate their assets in the case of a systemic crisis. The 99% ‡ Note the confidence level is set at 99% represents the risk capital required in the event of a systemic crisis. As this is regarded as a relatively unlikely event, there is a low probability of 1 % for this. The 10 day holding period is regarded to be the number of days banks will need in order to liquidate their assets in the event of a systemic crisis. § See Jorion (2001) or Dowd (2002) for extensive discussions of the different models for the estimation of VaR.

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longer suggested that the square–root–of–time rule be used, but that “in constructing VaR models estimating potential quarterly losses, institutions may use quarterly data or convert shorter period data to a quarterly equivalent using an analytically appropriate method supported by empirical evidence” (emphasis added). The key issue then, is to establish the alternative to this rule. This is precisely the area of contribution of our paper. In this paper we propose two methods for empirically and locally determined time varying scaling exponents using a recursive window framework on overlapping returns data based on order statistics. The first empirical scaling rule (Hest) assumes a ‘pseudo’ scale invariant measure of the scale exponent which is derived by the gradient of the linear regression of the q-quantile of the returns with different holding periods in a log-log plot. The second method involves the numerical local determination of scale variant exponents (Hnum) for the q-quantile one day returns and the q-quantile of n>1 returns. We study the properties of these data determined scaling exponents and use back testing methods to compare their effectiveness for risk management with that of the square-root-of-time rule. The back testing results show that the application of the empirically determined scaling rules outperforms the square-root-of-time rule and leads to a significant amount of saving in banks’ capital. The structure of the rest of the paper is as follows. Section 2 is a review of the available literature on the issue of scaling, and is divided in two subsections, one dealing with the theoretical background and the other dealing with the empirical results already presented in the literature. Section 3, presents our methods for calculating the scaling exponent and those used for backtesting our model. Section 4 outlines our empirical results and discusses statistical issues arising from the estimations. Then, in section 5 we perform the backtesting analysis and section 6 concludes the paper. 2. Literature review on scaling volatility and quantile measures

Even though the stylised facts of financial returns have been documented to

violate the log-normality assumption since the 1960s (Fama (1965), Mandelbrot (1967)) the square-root-of-time rule is, still, widely used due to the non-existence of a robust alternative. This is aggravated by the lack of understanding of the shortcomings and the problems that arise from the use of this rule.

The time scaling of volatilities for instance implicitly depends on returns being iid (identically and independently distributed), an assumption Engle (1982) argues is incorrect due to the presence of volatility clusters. When applied to quantiles, it is well- known that the square–root–of–time rule requires returns not only to be iid but also normal. It has been known at least since Mandelbrot (1963) and Fama (1965) that returns exhibit excess kurtosis, i.e. they are “fat tailed.” In general, the presence of fat tails introduces an additional bias in applications of the square–root–of–time rule to quantile forecasts.

The academic response on the issue of scaling financial time series is fairly recent. Initially, the focus was on scaling volatility into multiple-day horizons. Within this spectrum we encounter the work of Diebold and Hickman (1997), which was followed by Christoffersen and Diebold (1998), and then by Barndorff-Nielsen (1998).

With the establishment of VaR as the universal risk measure in the financial sector, the issue of scaling VaR also emerged. Hence, the work of McNeil and Frey

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(2000), Kaufmann and Patie (2003), Danielsson and Zigrand (2004) and Menkens (2004), all of them attempts to deal with the issue of scaling short-term horizon VaR to longer-term horizon VaR. 2.1 Volatility scaling

Early work on scaling has focused on volatility scaling and one such attempt is the work by Diebold et.al.(1997). The authors deal with the issue of converting 1-day volatility to h-day volatility and discuss in detail the shortcomings of the square-root-of-time rule. Even though their concern is volatility, there are many interesting ideas for our work. First of all, their main focus is in the failure of the iid assumption in financial data. They show that the square-root-of-time rule works in iid environments but fails otherwise. Their overall conclusion is clear and very motivating for our work. They support the view, that if the GARCH (1, 1) model specifies correctly the 1-day returns, then the Drost-Nijman formula should be used for scaling the k-day returns. However, if the models covered by Drost-Nijman (1993) do not correctly specify the daily returns, then their results do not apply, and there are no known analytic methods for computing the k-day volatilities from the 1-day volatilities. The only realistic alternative they find is to use a k-day model in the first place.

The issue of obtaining long-horizon volatilities from short-horizon volatilities is also the topic of Christoffersen, Diebold, and Schuermann (1998). They recognise that “First, temporal aggregation formulae are presently available only for restrictive classes of models; the literature has progressed little since Drost and Nijman. Second, the aggregation formulae assume the truth of the fitted model, when in fact the fitted model is simply an approximation, and the best approximation to h–day volatility dynamics is not likely to be what one gets by aggregating the best approximation (let alone a mediocre approximation) to one-day dynamics.”

As a result, they propose a model-free procedure to test the forecastability across different horizons. This procedure is mainly based on constructing an interval of volatilities and then testing across the volatilities range to see whether they belong in this interval. If a sequence of them does not belong, then they take this as evidence of a volatility cluster. They perform their exercise for many horizons, and conclude that clusters existence diminishes as the horizon increases. Their result is intuitive and simple to implement, but does not assist greatly our work which focuses on converting 1-day quantiles to k-day quantile.

The empirical scaling law has appeared in a volatility context in Barndorff-Nielsen (1998). His main focus is on high frequency data and the need to analyse volatility on different time scales. He points out that volatility of high frequency data is sometimes (e.g. in Mandelbrot 1963, Muller et al. 1990, and Guillaume et al. 1997) defined not as a standard deviation but as the average of absolute logarithmic price changes. Hence, he claims that considering the absolute logarithmic price changes of FX rates one often finds a scaling power law which relates the volatility over a given time interval over the size of this interval as a power. He then proceeds by defining this power scaling law as the ratio of the log of the volatility over the log of time interval.

His approach is based on the Inverse Gaussian Law as a potentially realistic distributional assumption of intraday FX returns. Even though this distribution does not capture all the stylised facts of financial returns, he finds that the empirical scaling law and the approximate scaling laws of the NIG Levy process are quite similar. Even

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though his work is again on volatility, his use of the empirical scaling law gives scope to using this approach in VaR scaling as well.

2.2 Scaling quantiles

The first paper to discuss the scaling of quantiles is the one by McNeil and Frey (2000). They generate data using a GARCH approximation and a GPD tail using the algorithm developed by Danielsson and de Vries (1997). They then observe that a power law can approximate the ratio of the k-day quantile over the 1-day quantile, i.e.

tkxkx tq

tq

λ≈/)( , where tλ depends on the initial volatility level. They find evidence of their conjecture, and propose their conditional distribution of asset returns against the current volatility background as a better VaR estimation method.

More recently Kaufmann and Patie (2003) provide an overview on methodologies that are proposed to model the evolution of risk factors over a long horizon. They study the problem of converting the 1-day quantiles to 1-year quantiles. They use four parametric approaches which are: Random Walk, AR(p), GARCH (1,1) and Heavy Tailed Power Law (Hill Estimator). They also use the empirical power law in order to compare their distributional assumptions. Their aim is to identify which model does better at modelling yearly risks and they conclude that, that is the random walk with normal innovations.

Another approach is by Danielsson and Zigrand (2004) where they find that by applying the square-root-of-time rule to the time scaling of quantiles of return distributions, risk is underestimated at an increasing rate as the extrapolation horizon is extended, and as the confidence level is raised. In particular, they postulate, that as the scaling horizon increases, the bias introduced by the square-root-of-time rule grows at a rate faster than time. They propose the use of a particular distributional assumption that can overcome the shortcomings of normality. According to them, this should be the jump diffusion process. They support the view that jump diffusions are an appropriate methodology for capturing the risks of systemic events, which while rare are large in magnitude.

Their empirical results indicate that an application of the square–root–of–time rule to the forecast of quantile-based risk estimates (such as Value–at–Risk) when the underlying data follows a jump–diffusion process is bound to provide downward biased risk estimates. Furthermore, the bias increases at an increasing rate with longer holding periods, larger jump intensities or lower quantile probabilities. The reason is that the scaling by the square root of time does not sufficiently scale the jump risk, i.e. the systemic part of the market risk.

Finally, the most recent contribution and starting point for this research paper is the work by Menkens (2004). The author deals with the issue of scaling VaR from 1-day to n-day horizon, using the principle of self-similarity. He argues that the square-root-of-time rule in valid only for normal and iid data and a more general approach would be to consider self-similar processes instead which yields the following scaling rule:

)()( 1PVaRnPVaR qHn

q ⋅=

where H is the Hurst coefficient and Pn and P1 are the n-day and 1-day return series respectively. The above equation is then used in a log form, and the Hurst exponent is derived from the gradient of a linear regression in a log-log plot. For self-similar

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processes, the empirical estimated scaling law has to be the same over all quantiles. The author finds that this is not the case for the considered financial time series. Nevertheless the author is right by saying that it is still possible to use the empirical estimated scaling rule for each quantile separately. Doing so these scaling laws are obviously no longer an estimated Hurst exponent of some self-similar process but only an approximation for a scaling rule in a certain quantile.

Our approach is using this exact view to estimate the scaling rule for different quantiles with an application to the FTSE-100. For this reason, we distinguish from the self-similarity theory by refusing to use the Hurst exponent terminology and instead calling the exponent simply as ‘scaling exponent’. 3. Methods

3.1 Scaling and Self-similarity

The empirical scaling behaviour of (discrete) time series data for a random variable X(t) examines the statistical properties of the relationship of X(t) ( t= v, 2v,…., kv, …..,T ) for different time increments, τ, τ >v. It is convenient to assume v = 1 as we are concerned about the scaling rule that relates returns over different holding periods and the 1-day return.

Theoretically precise scaling laws were first derived in the context of stable distributions, viz. Guassian and Levy Stable distributions that are stable under convolutions such that

X1 + X2 + ……..+ Xk ]2,0(1/1 ∈= µµ someforXk

d.

In the Guassian case, we have µ = 2 which results in the well-known square

root rule of scaling. However, with µ < 2, we have the Levy Stable distribution which implies infinite second moments. Since this does not appear to correspond with stylized facts about financial returns - we need a generic statement for the scaling rule. This arises in the following strong form of self-similarity or scale invariance. Self-similarity of the process refers to the property that the increments of X at scale τ =kv has the same distribution as any other increment τ under appropriate rescaling. The precise definition for this is given by Embrechts and Maejima (2000) is as follows: A stochastic process { }0),( ≥ttX is self-similar, if there exists 0≥H such that for any 0>k ,

{ } { }.)()( vXkkvX Hd= (1)

H is referred to as the scaling exponent, though for historical reasons it is also called the Hurst coefficient. **

** Some studies have confirmed the presence of self-similarity and scale invariance properties in

various markets: the Milan Stock Exchange by Mantegna (1991), the SP500 by Mantegna (1995), the CAC40 by Zajdenweber (1994), and foreign exchange markets by Pictet (1995), as well as individual French stocks by Belcacem (1996). However, as this remains controversial, and indeed our belief is that financial asset returns do not satisfy conditions of self-simlarity, we resort to the notion of empirically and locally determined scaling rules.

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Thus, in terms of the distribution function:

F(X(kv)) ))(())(( vXFkvXkF HHd

== . (2) Note as VaRq is the q–quantile of the portfolio return distribution: )()( 1 qFRVaR kR

kq

−−= ,

)(1 qF kR− is the generalized inverse function:

)(1 qF kR

− = inf {Rk: F(Rk) ≥ q} for 0 <q <1. From this it follows that the scaling law that applies to the distribution of returns F(Rk) also applies to the q-quantile. Thus , )()( 1RVaRkRVaR q

Hkq ⋅= , (3)

setting v=1 , here.

Then solve for the scaling exponent and get

1* )(loglog −

⎟⎟⎠

⎞⎜⎜⎝

⎛= k

xxH k

k

, (4)

where x is the quantile (VaR) and k is the holding period (5, 10, 15, 20, 25, 30). For example for the scaling law of the 10-day VaR we will need to calculate H as:

1)10(log)1()10(log −

⎟⎟⎠

⎞⎜⎜⎝

⎛=

VaRVaRH (5)

3.2.1 Estimation of the scaling exponent (Hest) In the estimation of the scaling exponent we follow the approach introduced in

Menkens (2004). The author proposes the use of quantiles in the estimation of the scaling exponent, and derives it from the gradient of a linear regression in a log-log-plot. To overcome the issue of limited data, the author uses the overlapping technique, which we also adopt in this paper. The estimation of Hest is performed for a given quantile VaRq(k) (for k = 1, 5,…,30). The seven points of the given quantile are plotted in a log-log diagram against the log(k). A linear regression in this log-log-plot gives the estimated scaling exponent.

3.2.2 Numerical derivation of the scaling rule (Hnum)

The proposed numerical derivation of the scaling rule is a straightforward and

easy to implement method. The first step is to derive the return series from the price

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series. Returns are sampled on a 1, 5, 10, 15, 20, 25, and 30-day horizon and calculated continuously as

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−kt

ttk P

PR ln (6)

VaR numbers are obtained using the returns series obtained above. The aim is to use the whole data set of size n; we start by picking the first 250 returns and calculate VaR for the different values of k (k = 1, 5, 10, 15, 20, 25, 30). Then, using (k = 1) as the reference horizon to get the scaling exponent using (4). In sequence, we move the time window (250 days) and repeat the derivation procedure m = n –250 + 1 times. Therefore, we calculate the Hnum using the overlapping sampling data method. The estimation is performed locally, i.e. for every VaR value we get an individual scaling exponent. In addition, the procedure is performed for q quantile sizes (q = 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 0.99). Given that the reference horizon is the 1-day returns the average scaling exponent for fixed k and q is given as:

∑=

=m

i

qq iHm

Hk

num

k1

)(11,1,

(7)

Therefore, the numerically derived value of the scaling exponent can be

exactly estimated for each VaR quantile of the k-day returns, as the scaling rule for converting 1-day VaR to k-day VaR. The significance of this imputed scale exponent is that this number represents the value of H for which the k-day VaR equals exactly their observed values. In other words, kHVaR(1) is exactly what the observed VaR(k) is.

3.3 Backtesting: Model Performance Techniques

The Basle Committee on Banking Supervision has incorporated backtesting into the internal models approach to market risk capital requirements. The essence of all backtesting efforts is the comparison of actual trading results with model-generated risk measures. If this comparison is close enough, the backtest raises no issues regarding the quality of the risk measurement model. In some cases, however, the comparison uncovers sufficient differences that problems almost certainly must exist, either with the model or with the assumptions of the backtest. In between these two cases is a grey area where the test results are, on their own, inconclusive.

The backtesting framework developed by the Committee is based on that adopted by many of the banks that use internal market risk measurement models. These backtesting programs typically consist of a periodic comparison of the bank’s daily value-at-risk measures with the subsequent daily profit or loss (“trading outcome”). The value-at-risk measures are intended to be larger than all but a certain fraction of the trading outcomes, where that fraction is determined by the confidence level of the value-at-risk measure.

Comparing the risk measures with the trading outcomes simply means that the bank counts the number of times that the risk measures were larger than the trading outcome. The fraction actually covered can then be compared with the intended level of coverage to gauge the performance of the bank’s risk model. In some cases, this

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last step is relatively informal, although there are a number of statistical tests that may also be applied.

It is with the statistical limitations of backtesting in mind that the Basle Committee has introduced a framework for the supervisory interpretation of backtesting results that encompasses a range of possible responses, depending on the strength of the signal generated from the backtest. These responses are classified into three zones, distinguished by colours into a hierarchy of responses. The green zone corresponds to backtesting results that do not themselves suggest a problem with the quality or accuracy of a bank’s model, and involves up to 4 violations for the 0.99 VaR at a 250-days backtest. The yellow zone encompasses results that do raise questions in this regard, but where such a conclusion is not definitive, and involves 5 to 9 violations. The red zone indicates a backtesting result that almost certainly indicates a problem with a bank’s risk model, and involves more than 10 violations.

Now, we look in more detail the methods used in this paper for backtesting the proposed empirical scaling law. The window size for the VaR calculations is 250 trading days, hence the number of violations expected for the different quantile values are given on the following table.

Table 1: Violations for each quantile used on the 250-day window

Quantile 0.70 0.75 0.80 0.85 0.90 0.95 0.99

Window = 250 75 62.5 50 37.5 25 12.5 2.5

The violations are calculated as:

{ }∑=

−<=

n

iVaRR

violp

ittin

V251

250 1 (8)

A good estimation of VaR will lead to a value for Vviol, which is close to the numbers given in the above table.

In addition, we will need to compare the models’ performance and decide which one is over- and which one is under-estimating VaR. To do so, we will report the difference in violations given by each rule and the benchmark violations range. We will therefore calculate the values as

Performance = Rule-Vviol - Benchmark-Vviol

So that, when Performance is positive, the rule in question has under-estimated risk, i.e. the quantile value is higher than it should have been (to the right of the k-day VaR). Similarly, when Performance is negative the rule in question has over-estimated risk, i.e. the quantile value is smaller than it should have been (to the left of the k-day VaR).

Finally, we will be reporting in what percentage is the assumed scaling rules outperform the SQRT- rule. Hence we are calculating the number of times, within our sample of results, each scaling rule does better in forecasting VaR values than the SQRT- rule.

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4. Empirical Results

4.1 Data Description The data set studied is the FTSE–100 Stock Index price series from January 2,

1986 to November 11, 2003. The data set is obtained from DataStream (University of Essex, Albert Sloman Library). The series is the daily closing prices and the total number of observations is 4516. The series is then divided in four sampling windows, with the first one being the AllData (whole time series), the second being the 2500 (December 20, 1993 - November 11, 2003), the third being the 1500 (December 3, 1997 - November 11, 2003) and the last one being the 500 (November 20, 2001 - November 11, 2003).

The descriptive statistics of the series analysed are given in Table 2. Two significant patterns emerge from the observation of the moments across the different holding periods. First, one can see that the standard deviation (volatility) increases as the holding period increases. This is in accordance with the idea that daily returns are not as volatile as for example monthly (k = 20) returns, since the magnitude of the change in price from the beginning of the month to the last day of the month will be much higher than that of the change in price from one day to the next. This can also be seen from the increasing size of the minimum and maximum returns across the different holding periods. Secondly, another pattern can be detected in the excess kurtosis range. It is very clear that the size of the excess kurtosis diminishes as the holding period increases. Again, this pragmatic observation follows the common intuition that as the data holding period becomes larger, the data is approximating the Normal Distribution and as a result the excess kurtosis is decreasing with a tendency to reach zero.

These two patterns can be observed graphically in the following figures. Figure 1 is a graphical representation of the FTSE-100 (AllData Sample) for different (k) holding periods. It is obvious that as the holding period increases from 1 to 30, the series is becoming much more volatile. In figure 2, we show the Q-Q plots of the same series again ranging from k = 1 to k = 30. Clearly, the series fits the Normal Plot much better for the longer horizon, i.e. the series is more closely approximated by the Normal Distribution. It should be noted, that the large deviation on the left tail that is persistent, and only slightly decreasing, in all cases is due to the inclusion of the 1987 Crash in our sample.

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Table 2: Descriptive Statistics for the four samples of the FTSE-100 Index at the k-day holding periods.

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Figure 1: FTSE-100 returns (AllData sample) for different (k) holding periods

Figure 2: FTSE-100 Q-Q Plots (AllData Sample) for different (k) holding periods

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4.2.1 Scaling exponent (Hest) estimation In this section we report the estimation results of the scaling exponent based

on the pseudo assumption of self-similarity. The scaling exponents have been estimated using the log-log plot method, based on seven holding periods given by k = (1, 5, 10, 15, 20, 25, 30). As a result, the scaling estimated exponent is stable across holding periods, but changes daily as the moving window alters. Also, Hest is different across quantiles. Figure 3 reports the Hest values for the sample period of 1500 days and the 0.99 to 0.70 quantiles. This figure shows how volatile the Hest is across time, as the method of overlapping data reveals.

Figure 4 outlines the estimation process, which entails a log-log scale regression of the VaR(k) quantiles versus the k-holding period. The estimated scaling law Hest is found as the gradient of each VaR line. The intercept is an estimate for VaR(1) at each given quantile. It is obvious that under this assumption the scaling exponent is stable across holding periods; however it is different across quantiles.

Figure 3: Estimated scaling exponents for the FTSE-100 time series at the 1500-days sample (1998-2003). Results of the regression analysis are presented for the

left tail quantiles ranging from a size of 0.99 to 0.70.

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Figure 4: VaR(k) quantiles as a function of k in a log-log scale for the FTSE-100 time series in the 1500-days sample period (1998-2003). Each curve corresponds

to a different quantile ranging from 0.99 to 0.70.

4.2.2 Scaling exponent numerical derivation

In this section we report the results for the numerically derived scaling

exponent Hnum computed for k = (5, 10, 15, 20, 25, 30). These exponents for the AllData sample of the FTSE-100 index are reported in figures 5 and 6 for the left and right tail respectively.

In figure 5, we have the scaling law estimates for the 0.99 VaR (left tail) for k = (5, 10, 20, 30). The exponents are computed locally for every recursive window of 250 days using equation (4). The first observation is that the scaling exponent H for k = 5, is closer to H = 0.5, than it is for the other 3 horizons. Actually, for k = 20, and k = 30 H appears to be systematically below the square-root-rule-of-time.

On the other hand, when we move on to Figure 6, we observe that the right tail has a different behaviour. From approximately 1989 to 1997, for k = 5, k = 10, and k=20, the scaling exponent is systematically higher than H = 0.5. From then onwards, it goes systematically below H = 0.5. What is also very interesting is that for k = 30, the scaling exponent does not follow this pattern and is almost throughout the whole period studied below H = 0.5.

So, the first results indicate that firstly the scaling exponent is time varying, as it is not constant throughout the period analysed. Secondly, it changes behaviour as the holding period (k) changes, and thirdly it is systematically different than H= 0.5.

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Figure 5: Scaling Law estimates for the 0.99 VaR (Left Tail) of FTSE-100 AllData sample for different (k) holding periods

Figure 6: Scaling Law estimates for the 0.99 VaR (Right Tail) of FTSE-100 AllData sample for different (k) holding periods

In sequence, the scaling properties between the VaR of k-days and 1-day are presented in table 3 and 4 (left and right tail respectively). Table 3 results present the statistical evidence of VaR scaling as the numerical scaling exponents (Hnum) derived from Equation (4) above. Using observed VaR values of the FTSE-100 for the k-day

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returns, the implied scale exponent is the value of H in Equation (4). Under the null hypothesis Ho, the value of the scale exponent should be H = 0.5. Rejection of the null hypothesis will imply that the currency series tested did not conform to a Gaussian random walk over the sample period.

Numerical results of Hnum are in general lower than H = 0.5 in the estimates of Table 3. Typically the values in Table 3 are closer to H = 0.5 when the difference between the return intervals k- and daily-return is small. That is, when the VaR of weekly returns (k = 5) are estimated by rescaling daily-VaR, implied H values are closer to H = 0.5, than when daily return standard deviations are rescaled to estimate the standard deviation of 6-week returns (k = 30). For example, the implied scaling exponent for the 0.99 VaR of the AllData sample gives H = 0.4946 for the 5-day scaling and H = 0.3553 for the 30-day scaling. Table 3: Implied Scale Exponent (H) for k-day horizons of the FTSE-100 Index

estimated at three left-tail VaR levels. Implied Scale Exponent (H)

FTSE Horizon VaR0.90 VaR0.95 VaR0.99 FTSE Horizon VaR0.90 VaR0.95 VaR0.99 Left Tail k = Left Tail k = AllData 5 0.4987 0.4864 0.4946 1500 5 0.4787 0.4456 0.4335

10 0.4673 0.4705 0.4518 10 0.4629 0.4431 0.4369 20 0.4525 0.4756 0.4296 20 0.4857 0.4592 0.4123 30 0.4181 0.3961 0.3553 30 0.3575 0.3493 0.3366

2500 5 0.4748 0.4662 0.4685 500 5 0.4854 0.4575 0.4497 10 0.4379 0.4372 0.4333 10 0.5129 0.4745 0.5040 20 0.4064 0.4266 0.3950 20 0.5271 0.4783 0.4264 30 0.4398 0.4204 0.3877 30 0.3098 0.3028 0.2812

Table 4 shows the same estimation results performed on the right tail of the

FTSE-100 time series. In this case, we cannot conclude that the short horizon scaling exponents are closer to H = 0.5, as this appears to be true in two of the analysed samples that is the AllData and the 2500. In the other two, however, the 5-day scaling exponent has much lower estimates than the 30-day scaling exponent. Also, we observe that the estimates of the right tail scaling exponent are also in general lower than H = 0.5, with a few exceptions in the AllData sample which are nevertheless only slightly larger than that. Table 4: Implied Scale Exponent (H) for k-day horizons of the FTSE-100 Index

estimated at three right-tail VaR levels. Implied Scale Exponent (H)

FTSE Horizon VaR0.90 VaR0.95 VaR0.99 FTSE Horizon VaR0.90 VaR0.95 VaR0.99 Right Tail Right Tail

AllData 5 0.5105 0.5185 0.5077 1500 5 0.3871 0.3801 0.4236 10 0.5116 0.5179 0.4906 10 0.3909 0.3869 0.4075 20 0.5158 0.5100 0.4963 20 0.3911 0.3828 0.4050 30 0.4177 0.4478 0.4258 30 0.4699 0.4522 0.4417

2500 5 0.4636 0.4644 0.4728 500 5 0.3127 0.3585 0.4452 10 0.4467 0.4479 0.4416 10 0.3201 0.3534 0.3269 20 0.4471 0.4384 0.4419 20 0.2945 0.3512 0.3411 30 0.3544 0.3845 0.3855 30 0.4800 0.4492 0.4296

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In principle, when H < 0.5 the SQRT-rule is overestimating the VaR, while

when H > 0.5 the SQRT-rule is underestimating VaR. This can be observed in Table 5, where we present the k-day VaR values for the FTSE-100 time series at different quantiles. Typically, as our results in Table 3 indicated the numerically derived scaling exponent (Hnum) was less than 0.5, and as a result the SQRT-rule has substantially overestimated VaR values. The bold-type values correspond to the daily VaR values scaled to k-day using Hnum.

Table 5: Results for the k-day Value-at-Risk of FTSE-100 Index estimated using

the numerical scaling exponent (Hnum) and the SQRT-rule.

Sample Horizon VaR 0.90 VaR 0.95 VaR 0.99 AllData (k) k^0.5 k^H k^0.5 k^H k^0.5 k^H

5 2.646% 2.660% 3.572% 3.479% 5.701% 5.891% 10 3.735% 3.539% 5.045% 4.858% 8.057% 8.066% 20 11.417% 10.288% 7.149% 7.806% 5.298% 5.073% 30 6.496% 6.580% 8.761% 9.036% 13.998% 11.487%

2500 5 2.976% 2.886% 4.014% 3.779% 6.082% 5.729% 10 4.203% 3.815% 5.672% 5.036% 8.600% 7.774% 20 5.963% 5.280% 8.043% 7.079% 12.196% 9.533% 30 7.308% 6.052% 9.861% 8.059% 14.961% 10.833%

1500 5 3.672% 3.547% 5.062% 4.631% 7.755% 6.998% 10 5.184% 4.849% 7.154% 6.391% 10.968% 9.924% 20 7.349% 7.290% 10.124% 9.140% 15.521% 12.236% 30 9.005% 8.544% 12.399% 10.877% 19.018% 14.541%

500 5 4.448% 4.361% 6.362% 5.927% 10.027% 9.220%

10 6.266% 6.544% 8.919% 8.395% 14.048% 14.197% 20 8.845% 9.814% 12.472% 11.666% 19.605% 15.709%

30 11.086% 11.313% 15.912% 13.816% 24.306% 19.458%

5. Model Performance: Backtesting Results

In this section we present the backtesting results of the SQRT-rule, the numerically derived Hnum and the estimated scaling exponent Hest. The objective of backtesting is to identify whether VaR(k) has been correctly approximated using each of the rules. For example, when k = 10, we want to see whether VaR(1)*10Hnum or VaR(1)*10Hest is better than VaR(1)*101/2 as an approximation of VaR(10). To do this, we need to backtest both VaR(1)*10Hnum and VaR(1)*10Hest in addition to VaR(1)*101/2. If the model for VaR(10) is correct then, we should find violations close to their theoretical values. For example for the 0.90-VaR(10), we should find 25 violations in our window of 250 10-day observations. This process involves the following steps:

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1. Take the first value of the VaR(1)*10Hnum and compare it with the next 10-day return. If the negative return (for the left tail) is larger than the VaR, then count this as a violation.

2. Move the window size one day further and repeat the above with the second VaR(1)*10Hnum.

3. Repeat the process until all the estimated VaR numbers have been examined. 4. Sum up the number of violations found, multiply by 250 and divide by the

sample size. 5. Repeat steps 1-4 for VaR(1)*101/2 and VaR(1)*10Hest. 6. Repeat steps 1-5 for the right tail, with an adjustment in step 1, where we now

have a violation if the positive return is higher than the VaR. In table 6 and 7, we present the number of violations and performance results for

the left tail. We are using the 1500-day sample of the FTSE-100 Index data, and presenting the results for the 5-, 10-, 20- and 30-day returns. We are examining the scaling properties of seven quantiles ranging from 0.70 to 0.99. In table 6, we can observe how many violations the numerically derived empirical scaling rule (Hnum) and the estimated scaling rule (Hest) generated as opposed to the square-root-of-time rule (SQRT-rule). Clearly the three rules offer quite different numbers of violations in all quantiles shown and holding periods. The benchmark violations are given on the last column of the table.

Table 6: Average Violations reported for the square-root-of-time rule (SQRT) and the scaling law exponent (H) at different holding periods (k) and VaR (left

tail) quantiles.

k=5

k=10

k=20

k=30 TheoreticalRule/VaR SQRT Hnum

Hest SQRT Hnum Hest SQRT Hnum Hest SQRT Hnum Hest Values

0.70 71.8 76.9 73.8 66.5 70.8 71.8 68.0 72.9 72.1 50.0 56.6 53.8 75 0.75 59.8 65.1 60.2 54.1 59.0 56.3 59.5 63.8 62.8 38.7 43.1 41.6 62.5 0.80 48.6 53.1 48.6 41.5 45.4 43.1 48.8 55.2 52.1 26.1 31.7 28.0 50 0.85 34.4 43.1 35.0 29.9 34.4 32.3 37.7 41.8 39.3 13.7 20.8 19.1 37.5 0.90 22.2 29.7 24.2 17.9 25.0 22.4 24.7 28.2 28.2 6.1 14.1 12.0 25 0.95 10.8 16.1 12.2 9.6 13.0 12.2 13.2 17.5 18.1 1.7 8.8 7.6 12.5 0.99 2.6 5.1 4.7 3.1 4.9 4.5 3.5 7.4 7.0 0.0 3.2 2.7 2.5

The next issue examined, is whether each rule is actually over- or under-

estimating the true VaR. This answer is given in table 7, where we calculate the Performance measure, given as the difference between each model’s violations and the benchmark violations. When the performance measure is negative, the assumed rule has over-estimated VaR, i.e. it has given higher VaR numbers than what the true VaR(10) was. Similarly, when the Performance measure is positive the assumed rule has under-estimated VaR(10); i.e. it has given lower than the true VaR estimates. In Table 7, we have highlighted the cases where performance was negative as this is the most frequent case. We observe the SQRT-rule and the Hest rule are the ones that over-estimate VaR in almost all of the cases, while the Hnum rule overestimates mostly in the 30-day return case.

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Table 7: Performance results of the SQRT rule and the scaling exponent H rule. Over-estimation or Under-estimation is reported for the SQRT-rule and the H-

rule at different holding periods (k) and VaR (left tail) quantiles.

k=5

k=10

k=20

k=30 lRule/VaR SQRT Hnum

Hest SQRT Hnum Hest SQRT Hnum Hest SQRT Hnum Hest

0.70 -3.2 1.9 -1.2 -8.5 -4.2 -3.2 -7.0 -2.1 -2.9 -25.0 -18.4 -21.2 0.75 -2.7 2.6 -2.3 -8.4 -3.5 -6.2 -3.0 1.3 0.3 -23.8 -19.4 -20.9 0.80 -1.4 3.1 -1.4 -8.5 -4.6 -6.9 -1.2 5.2 2.1 -23.9 -18.3 -22 0.85 -3.1 5.6 -2.5 -7.6 -3.1 -5.2 0.2 4.3 1.8 -23.8 -16.7 -18.4 0.90 -2.8 4.7 -0.8 -7.1 0.0 -2.6 -0.3 3.2 3.2 -18.9 -10.9 -13.5 0.95 -1.7 3.6 -0.3 -2.9 0.5 0.7 0.7 5.0 5.6 -10.8 -3.7 -4.9 0.99 0.1 2.6 2.2 0.6 2.4 2.0 1.0 4.9 4.5 -2.5 0.7 0.2

The final question is, whether the assumed scaling rule does any better in VaR performance, and if so at what percentage is it outperforming the SQRT-rule. One rule outperforms the other, if the latter is closer to the benchmark violations range. In this sample of results, the numerically derived empirical scaling rule is closer to the benchmark 60.71 per cent of the times; i.e. Hnum outperforms the SQRT-rule 17 out of the 28 (4 holding periods times 7 quantiles) cases studied here. It is crucial to note that the Hest outperforms the SQRT-rule 75 per cent of the times, i.e. 21 out of the 28 cases seen here.

In the following tables 8 and 9, we look at the results of the right tail. The analysis is the same, but here we are looking at the positive returns of the distribution. Table 8 shows in a similar manner as table 6, that the rules do not give the exact number of violations expected from the true VaR(10). In addition, we can see in Table 9 that the SQRT-rule is over-estimating the right tail as well. All three rules here systematically over-estimate VaR up to the 0.95 quantile and up to the 20-day holding period. Hence, the over-estimation in the right tail is more systematic and substantial in size than it is for the left tail.

Table 8: Average Violations reported for the square-root-of-time rule (SQRT) and the scaling law exponent (H) at different holding periods (k) and VaR

(right tail) quantiles.

k=5

k=10

k=20

k=30 TheoreticalRule/VaR SQRT Hnum

Hest SQRT Hnum Hest SQRT Hnum Hest SQRT Hnum Hest Values

0.70 62.9 70.6 73.0 57.2 71.2 70.2 55.7 71.7 71.1 70.4 80.1 88.3 75 0.75 47.7 58.1 59.1 46.4 59.6 58.2 39.1 59.5 58.9 58.5 66.4 75.9 62.5 0.80 34.0 45.5 43.3 34.2 47.8 45.2 29.4 46.7 48.1 48.1 49.6 62.2 50 0.85 22.2 34.0 32.4 21.2 35.4 34.8 18.5 33.4 34.7 34.7 38.9 52.1 37.5 0.90 14.0 23.1 20.6 11.8 21.6 22.0 9.3 22.7 24.0 25.7 30.3 41.0 25 0.95 4.9 13.1 9.5 4.9 12.0 11.8 2.9 10.9 13.5 12.2 19.6 29.6 12.5 0.99 1.6 3.6 3.6 1.2 5.3 4.5 0.2 3.6 5.9 4.6 10.1 17.9 2.5

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Table 9: Performance results of the SQRT rule and the scaling exponent H rule. Over-estimation or Under-estimation is reported for the SQRT-rule and the H-

rule at different holding periods (k) and VaR (right tail) quantiles.

k=5

k=10

k=20

k=30 lRule/VaR SQRT Hnum

Hest SQRT Hnum Hest SQRT Hnum Hest SQRT Hnum Hest

0.70 -12.1 -4.4 -2 -17.8 -3.8 -4.8 -19.3 -3.3 -3.9 -4.6 5.1 13.3 0.75 -14.8 -4.4 -3.4 -16.1 -2.9 -4.3 -23.4 -3.0 -4.0 -4.0 3.9 13.4 0.80 -16.0 -4.5 -6.7 -15.8 -2.2 -4.8 -20.6 -3.3 -1.9 -1.9 -0.4 12.2 0.85 -15.3 -3.5 -5.1 -16.3 -2.1 -2.7 -19.0 -4.1 -2.8 -2.8 1.4 14.6 0.90 -11.0 -1.9 -4.4 -13.2 -3.4 -3.0 -15.7 -2.3 0.7 0.7 5.3 16.0 0.95 -7.6 0.6 -3.0 -7.6 -0.5 -0.7 -9.6 -1.6 1.0 -0.3 7.1 17.1 0.99 -0.9 1.1 2.1 -1.3 2.8 2.0 -2.3 1.1 4.5 2.1 7.6 15.4

Finally, we look at how often the Hnum and Hest rules outperform the SQRT-rule. In this case we have 23 out of the 28 cases where Hnum rule outperforms the SQRT-rule; i.e. 82.14 per cent of the times. The Hest rule outperforms the SQRT-rule in 18 out of the 28 cases, which corresponds to 64.28 per cent. Therefore, so far we have established that the numerically derived scaling exponent appears to do better at the right tail, while the estimated scaling exponent performs better at the left tail.

Looking more closely at these results, one can observe that both the Hnum and Hest scaling rules do not perform well in the 0.99 quantile, which is undoubtedly the most important one as far as practitioners are concerned. To investigate why this is happening and whether it is indeed a bad result, we use backtesting charts for the three rules. These charts graph the time series to be backtested, in our case the 10-day returns of the FTSE-100 in the 1500 days sample. As the first 250 are used for the VaR window, the backtesting starts from day 251, which in this sample is 31-Dec-98. For each 10-day return the VaR number is also graphed, and the cases where the return is higher than the VaR are the cases of violations. The lower bound in the graphs corresponds to the left tail of the series and is thus the negative (losses) VaR. The upper bound corresponds to the positive returns and is therefore the positive (gains) VaR. Figure 7 is the backtesting chart for the numerically derived empirical scaling rule, figure 8 is the same for the estimated scaling rule and figure 9 is for the SQRT-rule.

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Figure 7: Backtesting chart for FTSE 100. VaR is estimated using the numerically derived empirical scaling rule on a 10-day horizon. The time series is the 10-day returns on the FTSE ranging from 31-12-1998 to 11-11-2003. Lower VaR corresponds to the left tail (losses) and upper VaR corresponds to the right tail (gains).

Figure 8: Backtesting chart for FTSE 100. VaR is estimated using the estimated scaling rule (Hest) on a 10-day horizon. The time series is the 10-day returns on the FTSE ranging from 31-12-1998 to 11-11-2003. Lower VaR corresponds to the left tail (losses) and upper VaR corresponds to the right tail (gains).

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Figure 9: Backtesting chart for FTSE 100. VaR is estimated using the SQRT-rule on a 10-day horizon. The time series is the 10-day returns on the FTSE ranging from 31-12-1998 to 11-11-2003. Lower VaR corresponds to the left tail (losses) and upper VaR corresponds to the right tail (gains).

The first observation on these graphs is that VaR violations appear in clusters. This is strong evidence of non-stationary data. What is happening in effect is that when a large loss occurs it is followed by a sequence of large losses, which results in a sequence of violations of VaR. For instance, this is really evident in March and September 2001, as well as in September 2002. The second observation is that the VaR numbers obtained with the numerically derived scaling rule (Hnum) and the estimated scaling rule (Hest) are much closer to the actual 10-day returns as the bounds for these scaling rules are much tighter than the ones of the SQRT-rule. Crucially, these results are inter-related, as the fact that the Hnum and the Hest scaling rules move closely with the returns implies that when a large loss occurs it will inevitably lead to a violation. In other words, as these rules offer tighter bounds they are more likely to lead to a violation when a big loss occurs. As these large losses appear in clusters, the scaling rules appear to have more violations than the SQRT-rule at the 99th quantile. We deal with this issue by observing where these clusters do occur (i.e. more than 3 violations in a sequence) and count 1 violation for each cluster. Performing this modification in the example studied here, we get average violations for the empirical scaling law (0.99 10-day VaR) is 2.03, while for the SQRT-rule it is 3.1. Clearly, the numerically derived scaling rule now is closer to the theoretical value of 2.5. Returning to the second observation of tighter bounds, we identify that the importance of this result is extremely high in particular from a practitioner’s point of view. What is crucial here is that if a bank had used the either of the scaling rules to scale the 1-day VaR to the 10-day VaR then, it would have saved a significant amount of its capital. Since, the 10-day VaR is used to calculate each bank’s capital requirement, by having a lower VaR, which does not lead to a punishment from the

22

Basle Committee, the bank will have to lock away less funds. In other words, the bank has more money for its other business purposes. Both of the scaling rules lead to a saving equal to the difference between how much the capital requirement is, provided it is calculated with the SQRT-rule, and how much it could have been using the scaling rules. This level of saving is shown in figures 10 to 13, where the left as well as the right tail are analysed for completeness. The time series is the percentage capital saving a bank could have, had it used each of the scaling rules instead of the SQRT-rule. It is calculated as the difference between the SQRT-VaR and H-VaR. Positive values indicate saving, as the SQRT-VaR was larger than the H-VaR, while negative values indicate loss. We observe in figures 10 and 11 the capital saving made with the numerically derived scaling exponents Hnum and the estimated scaling exponents Hest respectively. Both rules lead to a significant amount of saving as opposed to the SQRT-rule which is widely used in the industry. The average percentage of capital saving a bank trading the FTSE-100 could have had is 1.051% with the Hnum rule, and 1.701% with the Hest rule. Indeed, the empirical analysis of this paper indicates that given the choice of data and sample period the rule that would actually save banks more capital is the Hest at the left tail VaR of 0.99 confidence level, which is the exact quantile that the Basel Accord uses to derive capital requirements. We have also analysed the right tail of the returns distribution, and present the results in figures 12 and 13, for the Hnum and the Hest respectively. Even though the right tail represents the profits of trading and is not used in practice for capital requirements, we still look at it on the basis of statistical completeness. So, we have that both rules outperform the SQRT-rule and offer significant amount of “saving” (we do realise that on the right tail the out-performance is not exactly saving). Here, on average we have, a bank trading the FTSE-100 could have had is 2.408% with the Hnum rule, and 2.271% with the Hest rule. Hence, in the right tail we find that the numerically derived scaling exponent Hnum is actually performing better.

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Figure 10: Capital savings (%) of the Hnum rule relative to the SQRT-rule. Savings for the left tail are calculated as )1(10 VaR⋅ )1(10 VaR

numH ⋅− .

Figure 11: Capital savings (%) of the Hest rule relative to the SQRT-rule. Savings for the left tail are calculated as )1(10 VaR⋅ )1(10 VaRHest ⋅− .

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Figure 12: Capital savings (%) of the Hnum rule relative to the SQRT-rule. Savings for the right tail are calculated as )1(10 VaR⋅ )1(10 VaR

numH ⋅− .

Figure 13: Capital savings (%) of the Hest rule relative to the SQRT-rule. Savings for the right tail are calculated as )1(10 VaR⋅ )1(10 VaRHest ⋅−

25

6. Conclusion This paper has dealt with the issue of scaling the daily VaR to a k-day horizon. This issue has been crucial in the risk management sector, as all banks are required to report the 10-day Value-at-Risk. So far, the rule used to scale 1-day VaR to k-day is the SQRT-rule. Clearly, this rule is problematic as it only holds when the data in question is normally distributed and iid. What we propose is the application of an empirically derived scaling process. The scaling exponent should be estimated locally for the data series in question and should be time-dependent as well as data-dependent. In other words, the scaling exponent should be implied from the individual data series. We call this scaling exponent the numerical exponent, Hnum, and compare it to the self-similarity scaling exponent, Hest, which is estimated as the gradient of a log-log-plot on each quantile and the set of holding periods. This paper was divided in six sections, where at first we presented the scaling theory and the background already existent in the literature for the scaling law. Then we presented our method of calculating the numerically derived scaling exponent (Hnum) and the estimated scaling exponent (Hest) and proceeded into the empirical application on the FTSE-100 Index. To make the analysis thorough and our conclusions robust we analysed the data series in six different time scales, and looked at several quantiles of four different samples taken from the FTSE-100 Index. Our results can be summarised as follows. Firstly, we discovered that the numerically derived scaling exponent is time varying, and at the same time its behaviour changes as the holding period (k) alters. Also, it is systematically different than H = 0.5. Secondly, we found that the implied results of the scale exponent (Hnum) are in general lower than H = 0.5. Also, we observed that in the left tail of the data series the Hnum values are closer to H = 0.5 when the difference between the return intervals k and daily return is small. That is, when the VaR of weekly returns (k = 5) is estimated by rescaling daily-VaR, the implied Hnum values are closer to H = 0.5, than when daily return standard deviations are rescaled to estimate the standard deviation of 6-week returns (k = 30). However, when we looked at the right tail we could not make the same conclusions. Finally, we proceeded into the model performance section where we did a backtesting analysis on the k-day VaR values obtained with both of the scaling exponents and the ones obtained with the SQRT-rule. We performed the analysis following the rules set out by the Basle Accord (1996). We presented the number of violations observed within our sample and found that both the SQRT-rule and the scaling exponent rules offer values different to the benchmark. In order to decide which one of the three performs best, we looked at the difference between each model’s violations and the benchmark. Using this approach, we can easily see which model under-predicts and which one over-predicts the VaR at the examined samples. The first pattern that emerged was that in the left tail, the SQRT-rule over-estimated VaR much more often than the scaling exponent rules did. In the right tail, all of them appeared to over-estimate the true VaR. However, looking at the actual sizes of the performances we can calculate how many times each rule was closer to the benchmark. We then had, that the numerical scaling exponent was closer to the benchmark 64 per cent of the times in the left tail, and 78 per cent in the right tail. The estimated scaling exponent was closer to the benchmark 75 per cent of the times in the left tail, and 64 per cent in the right tail. This result indicates that the estimated scaling exponent performed better in the left tail (where most of the concern is), while the numerical scaling exponent performed better in the right tail. However, what is

26

crucial to note here is that both scaling rules outperform the SQRT-rule which is currently the industry standard. Finally we made a more thorough backtesting analysis by looking at the backtesting charts for all three rules. There, we made the crucial observation that the k-day VaRs calculated using the scaling exponent rules were actually following more closely the movements of the market and thus contributing to a capital saving to the banks who used this VaR to calculate their capital requirement. We also show the size of this saving by graphing the difference between the SQRT-VaR and our proposed H-VaR. The latter is significantly smaller in our samples thus making a saving to banks capital, who had they used the SQRT-rule would have had to maintain a higher capital requirement. The findings follow the violations analysis, where we confirm again that the estimated scaling rule performed better in the left tail, while the numerical scaling rule did so in the right tail. Therefore, what we have shown is that the rationale for the use of the SQRT-rule is undoubtedly flawed when used with financial series. In addition, we have proved that its basic principle that all quantiles and all time horizons should be scaled at the power of 0.5 is completely unrealistic. Our approach in scaling shows that the scaling exponent is not only time varying but also changes as the holding period increases or decreases, and also as the quantile size alters. We have also presented positive results where the VaR numbers estimated using the scaling exponent on a numerical derivation approach, and on the estimated approach, are actually saving banks money, as they are closer to the actual movements of the market. We consequently propose that the scaling law should instead be implied from the data series in question and not be taken as granted that it is a constant number equal to 0.5.

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