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Setting margin levels in futures markets: An extreme value method
Nonlinear Analysis: Real World Applications 11 (2010) 1704–1713
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Setting margin levels in futures markets: An extreme value method Tzu-chuan Kao a,1 , Chu-hsiung Lin b,∗ a
Kun Shan University, Department of Finance and Banking, 949, Da Wan Road, Yung-Kang City, Tainan Hsien, 710, Taiwan
b
National Kaohsiung First University of Science and Technology, Department of Risk Management and Insurance, 2, Juoyue Road, Nantz District, Kaohsiung, Taiwan
article
info
Article history: Received 15 February 2007 Accepted 23 March 2009 Keywords: Extreme value theory Margin setting Stock index futures Value at Risk
abstract There are of course different types of margin requirements in futures clearinghouses, and this study focuses on setting initial and maintenance margin levels. This study provides an approach, the VaR-x method that incorporates a modification of the Hill estimator based on extreme value theory (EVT) into a Student-t distribution, for setting the unconditional and conditional margin levels (i.e. initial and maintenance margin levels). Empirical applications are based on daily data for three stock index futures returns: the FTSE100, Nasdaq100 and Nikkei225. The empirical results demonstrate that given lower probabilities of margin violation, the VaR-x approach to setting unconditional margin levels is more accurate than either the normal approach or the Hill non-parametric approach proposed by Cotter [J. Cotter, Margin exceedences for European Stock Index Futures using extreme value theory, Journal of Banking and Finance 25 (2001) 1475–1502]. Additionally, this study demonstrates that using the conditional VaR-x approach to setting margin levels can better capture extreme events, thus ensuring adequate prudence, something that is particularly crucial in periods of strong fluctuation. These empirical findings suggest that the proposed approach is very useful to setting the initial and maintenance margin levels. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Margin committees face the difficult task of appropriate margin levels to balance the costs of trader default against the benefits of increased market liquidity [1]. The key problem margin committee face is that setting the margin level too high will lead to high transaction costs and thus reduced trading volume. Meanwhile, if the margin level is set too low, it may not adequately cover extreme price changes, thus incurring default risk. For guarding against default, this paper focuses on setting prudent margins designed to protect futures positions from extreme price movements. The academic literature has implemented two approaches to setting appropriate margin levels. First, the application of economic models that assume margin levels are endogenously determined for minimizing broker costs. For example, Brennan [2] and Fenn and Kupiec [3] used the concept of efficient contract design to identify economic factors determining the optimal margin level. Second is the application of statistical approaches, which is used to set margin levels that represent an amount not to be exceeded by a price change over a specified time period at an acceptable probability level. These statistical approaches include parametric and non-parametric methods that rely on Gaussian and non-normal distribution assumptions for the underlying distribution of futures price changes. Figlewski [4] and Gay et al. [5] are two examples of
∗
Corresponding author. Tel.: +886 7 6011000; fax: +886 7 6011046. E-mail addresses: [email protected] (T.-c. Kao), [email protected] (C.-h. Lin).
1 Tel.: +886 6 2727175; fax: +886 6 2050622. 1468-1218/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2009.03.025
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studies that assume futures returns follow a normal distribution. Warshawsky [6] used a non-parametric analysis procedure, which does not rely on normal distributional assumptions, and shows that it may be inappropriate to assume normally distributed futures returns. Kholodnyi [7] developed a new approach, which models price changes with spikes as a nonMarkovian stochastic process that allows for modeling spikes directly as self-reversing jumps. Recent studies on margin setting have increasingly studied the fat-tailed characteristics of returns distribution, and in doing so, applied extreme value theory (EVT) to setting margin levels. As extreme movements are central to margin setting problem, EVT directly models the tails of the returns distribution and thus could potentially yield better estimates and forecasts of default risk. Kofman [8], Longin [9,10], Booth et al. [1], Broussard and Booth [30] and Broussard [11] document that using EVT with a parametric method is appropriate for measuring the optimal margin levels. Furthermore, their studies also show that assuming normally distributed data underestimates the probabilities of margin violation because the empirical distribution of observed large price movements is too leptokurtic to be normal. Meanwhile, in EVT the limiting distribution of extreme values has non-parametric underlying assumption, thus Dewachter and Gielens [12] and Cotter [13] propose applying the Hill non-parametric method to generate measures of margin levels, showing that the margin levels estimated by the Hill non-parametric method are sufficient for futures contracts. Consequently, a consensus exists in the previous empirical research that using EVT with parametric and Hill non-parametric methods to determine margin levels can fully protect against default resulting from extreme price movements, but that for minimizing model risk the Hill non-parametric method is superior to parametric procedures which require assumptions regarding the exact distribution type of extreme price changes. However, the Hill non-parametric method still suffers from two major limitations [14]. First, the Hill estimator is biased when applied to small samples. Second, the Hill estimator used in the estimation procedure must depend on the number of order statistics [15,16]. Traditionally, the number of order statistics is determined by simulating the mean squared error minimizing number, but generally this procedure produces biased estimates [12]. Recently, Jansen and de Vries [17], Koedijk and Kool [18] and Beirlant et al. [19] respectively proposed highly effective methods of resolving this problem. However, these methods still base their tail estimates on a specific number of tail observations [14]. To correct the Hill non-parametric method proposed by Cotter [13], this study specifically provides an approach to setting margin levels, known as VaR-x that incorporates a modified version of the Hill estimator into a Student-t distribution. In line with the works of Huisman et al. [20,14] and Pownall and Koedijk [21], this study is the first to use the modified Hill estimator, which is to correct the small sample bias in tail index estimates, and does not condition its tail observations as do the Hill [22], Beirlant et al. [19], and other tail index estimators [14], to estimate the tail index of returns distribution. The estimated tail index then can be used to parameterise the number of degrees of freedom of the Student-t distribution for estimating the margin level. The fact that the number of degrees of freedom reflects the degree of tail fatness enables the capture of extreme events, thus ensuring adequate margin level, something that is especially crucial during times of high fluctuation. Additionally, studies on margin setting can consider two different returns distributions: the conditional and unconditional distributions. The unconditional analysis attempts to incorporate extreme events occurring over a long time period, therefore it is suited to the setting of initial margin levels (i.e. unconditional margin levels). While the conditional analysis reflects the change in market conditions over time, therefore it is suited to the setting of daily or maintenance margin levels (i.e. conditional margin levels) [23]. This study considers the adoption of both unconditional and conditional analyses to provide different but complementary information for the setting of initial and maintenance margin levels. In analyzing conditional margin setting, most previous studies focused on the GARCH model. For example, Cotter [13], Knott and Polenghi [24] and Cotter and Longin [25]2 considered a conditional process by applying several variants of the GARCH model to address issues relating to the time-varying behavior of futures price changes. Knott and Polenghi [24] showed that employing a GARCH model with innovations based on Student-t distribution could provide a timely and informative measure of conditional margin levels. Though the GARCH model focuses on accounting for volatility clustering, whereas extreme values do not actually cluster [26,13], setting margin levels can specifically reflect common volatility levels, but cannot deal with very low probability levels where the extreme values are located. As the problem of setting conditional margin levels is still related to the tails of the distribution of futures price changes, this study uses a conditional approach of VaR-x which simultaneously describe the tail fatness and time-varying behavior of conditional distribution to capture some of the additional upside and downside risk faced during more volatile periods. The empirical investigation applies the proposed unconditional and conditional VaR-x approach to estimate the margin levels for three different daily stock index futures series: the US Nasdaq100, the British FTSE100 and the Japanese Nikkei225. Moreover, margin levels may differ between long and short positions owing to the behavior of left and right tail price movements. Therefore, this empirical investigation respectively deals with the estimation of the margin levels for long and short positions. The proposed VaR-x approach is compared with two different methods, which are the normal method [4,10] and the Hill non-parametric method [13]. The remainder of this paper is organized as follows. Section 2 presents the theoretical foundations used to estimate unconditional and conditional margin levels. Section 3 then details the stylized fact of the empirical data. Subsequently, Section 4 illustrates the empirical results. Finally, Section 5 presents conclusions.
2 Cotter and Longin [25] extended the work of Cotter [13], applying modified Hill estimator based on EVT [14] to set daily margin levels by considering the intraday dynamics of market prices. They found that the modified Hill estimator could easily extend the multi-period margin estimation.
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2. Method 2.1. Determining unconditional margin levels The unconditional VaR-x approach proposed by Huisman et al. [20,14] is developed from EVT, which investigates the distribution of tail observations in large samples. In the limit, the tail shape follows a Pareto law for a general class of fattailed distributions. The tail fatness of the distribution is characterized by the tail index. Notably, the tail index can be used as a parameter for the number of degrees of freedom to parameterise the Student-t distribution [20,14]. Consequently, the link to the Student-t distribution, a fatter-tailed distribution, is used to measure the margin level. The unconditional VaR-x approach for measuring margin level is presented below. In futures markets, large positive price movements create default risk for short positions, while large negative price movements create default risk for long positions. Thus, the measure of the probability of margin violation resulting from abrupt price changes should be derived from extreme positive and negative price movements. Under the theoretical framework of EVT, let the limiting distribution of extreme minimum (the negative) and maximum (the positive) price changes observed over the following n trading days defined by FMin(n) and FMax(n) , where FMin(n) (FMax(n) ) describes the left (right) tail behavior of futures returns. Considering a period of n trading days, the probabilities of margin violation for long and short positions are given by: qlong = P (Min(R1 , R2 , . . . , Rn ) < −MLlong ) = FMin(n) (−MLlong ) short
q
= P (Max(R1 , R2 , . . . , Rn ) > ML
short
) = 1 − FMax(n) (ML
short
(1)
)
(2)
where R represents the futures return (or futures price change rate), and MLlong and MLshort denote the margin levels of long and short positions.3 The above two equations show that the measure of margin levels for long and short positions given a specific probability of margin violation is related to modeling the left and right tails of the returns distribution and thus to estimating the tail index of the left and right tails. To obtain an unbiased tail index estimate, a modification of the Hill estimator is used to correct the small sample bias in tail index estimates without requiring a priori selection of the number of tail observations to be included. The process used to estimate the tail index of the futures returns distribution is presented below. For simplicity, this study discusses estimation and inference for the left tail of the returns distribution.4 Let R(i) denote the ith increasing order statistic of the absolute returns such that R(i) > R(i−1) for i = 2, . . . , n. Specifying k as the number of tail observations, the tail estimator (or tail index) α(k) proposed by Hill [22] is obtained as follows:
α(k) =
k 1X
k j=1
! −1 ln R(n−j+1) − ln R(n−k)
.
(3)
Based on the method of Huisman et al. [14], a modified version of the Hill estimator can be used to correct the bias in small samples. The bias of the inverse of Hill estimator γ (k) = 1/α(k) results from that the bias is a function of the sample size. A bias corrected tail index is thus obtained by observing the bias of the γ (k) as the number of tail observations increases until κ , where κ equals half of the sample size (i.e. κ = n/2):
γ (k) = β0 + β1 k + ε(k) k = 1, 2, . . . , κ
(4)
where ε(k) denotes the error term and the inverse of the estimated βˆ 0 is an estimator of α . Since the estimate of the tail index ˆ1 is equal to the number of degrees of freedom of the Student-t distribution [20,14] we can assume that the empirical β0
distribution of futures return approximates to a Student-t distribution with ˆ1 degrees of freedom. Via the relationship of β0 the critical value and cumulative probability, the estimate of margin level ML can be derived. The procedure for obtaining the unconditional margin level estimates of long and short positions MLlong , MLshort is summarized as follows: Step 1: Use Eqs. (3) and (4) to obtain estimates of the left and right tail index ˆ1− , ˆ1+ . Furthermore, it is also necessary to β0 β0 estimate the unconditional mean µ and the unconditional variance σ 2 of the underlying futures returns distribution. Step 2: Set the left tail index estimate ˆ1− to equal the number of degrees of freedom νˆ − of the Student-t distribution. β0
Similarly, set the right tail index estimate ˆ1+ to equal the number of degrees of freedom νˆ + of the Student-t β0 distribution. 3 Notably, in the proposed approach the estimated margin level is expressed as a percentage. However, in practice, cleaning houses tend to state the margin level in terms of currency units per contract. To avoid confusion, the relationship between ‘‘percentage’’ and ‘‘currency unit’’ can be explained as follows: if the current future contract price is $100, a currency margin of $5 for a long position corresponds to a percentage margin rate of −5.13% (=100ln(95/100)), and a currency margin of $5 for a short position corresponds to a percentage margin rate of 4.88% (100ln(105/100)). 4 An estimate for right tail index is captured using a sequence of increasing order statistic of the positive returns. An estimate for the common tail index is captured using a sequence of increasing order statistic, which encompasses both sets of positive returns and the absolute value of the negative returns.
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Step 3: With a confidence level 1 − q, read the critical value S − (S + ) of the Student-t distribution with νˆ − (νˆ + ) degrees of freedom. Step 4: Transform S − into the corresponding critical value of the real futures returns distribution,5 in which case the estimate of unconditional margin level for a long position given the probability of margin violation q% is given by:
ˆ long = −S − ∗ θ + µ ML ˆ
where θ = p
σˆ νˆ − /ˆν −
−2
.
(5)
Similarly, the estimated unconditional margin level for a short position given probability of margin violation q% is given by:
ˆ short = S + ∗ θ + µ ML ˆ
where θ = p
σˆ νˆ + /ˆν +
−2
.
(6)
2.2. Testing for tail stability This study employs a test developed by Loretan and Phillips [27] to examine whether any difference exists between left and right tail behavior. The null hypothesis is given by H0 : α + = α − . The test statistics are as follows: Vˆ (α + − α − ) =
α+ − α−
(α +2 /m+ ) + (α −2 /m− )
− →
1/2
d
N (0, 1)
(7)
where α + (α − ) denotes the estimate of the right (left) tail index, and m+ (m− ) represents the number of largest order statistics used to estimate the tail index. If the null hypothesis is rejected, the optimal margin level for long and short positions should be treated separately. Meanwhile, if the null hypothesis is supported, long and short positions should have a common margin level. 2.3. Determining conditional margin levels This section presents a conditional VaR-x approach for capturing the time-varying margin levels to provide clearing houses additional information on the setting of daily or maintenance margin levels.The proposed conditional VaR-x approach assumes that the underlying distribution of futures returns is a time-varying Student-t distribution and uses a conditional modified Hill estimator to parameterise the time-varying Student-t distribution. Based on Eqs. (5) and (6), the long M Lˆ t and M Lˆ short using the conditional VaR-x approach are given by: t long
M Lˆ t
σˆ t −1 = −St−−1 ∗ θt −1 + µ ˆ t −1 where θt −1 = q − νˆ t −1 /ˆνt−−1 − 2
M Lˆ short = St+−1 ∗ θt −1 + µ ˆ t −1 t
where θt −1 = q
σˆ t −1
νˆ t+−1 /ˆνt+−1
(8)
(9)
−2
long
) denotes the conditional margin level at time t for long (short) position. To implement Eqs. (8) and (9), where M Lˆ t (M Lˆ short t we use the rolling window approach, which can respond to the changes on the volatility levels around the current period t, to compute the conditional marginal levels. Furthermore, The EWMA model, known as an integrated GARCH or IGARCH model, is used to estimate the conditional volatility σt since it relies on one parameter only and thus facilitates estimation. The EWMA estimator is given by:
σt2 = λσt2−1 + (1 − λ)R2t −1
(10)
where the parameter λ is called the decay factor and must be less than unity, and Rt denotes the rate of return at time t. This study sets λ = 0.94 used by RiskMetrics to estimate daily volatility for the purpose of calculating Value at Risk. 3. Data description To verify the performance of the proposed approach, this empirical investigation applies the proposed approach to three different stock index futures contracts: FTSE100, Nasdaq100 and Nikkei225.6 This empirical investigation examines daily prices (trading days) from their issue dates to May 20, 2005. During the selected periods, 5382 observations are obtained 5 This transformation is due to the fact that the Student-t distribution with νˆ degrees of freedom has a predetermined mean equal to zero and a variance equal to ν/ν − 2. (see [20]). 6 The FTSE100 future contract traded on LIFFE, the Nasdaq100 future contract traded on CME and the Nikkei225 future contracts traded on SGX-DT.
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Table 1 Summary statistics for the stock index futures contracts’ returns. Statistic
FTSE100
Nasdaq100
Nikkei225
Meana Std.Dev.a Maximuma Minimuma Skewness Kurtosis Jarque–Berab Ljung–Box Q (12)c Ljung–Box Q 2 (12)d ADFe PPe ARCH LM testf : χ62
0.03 1.14 8.09 −16.75 −0.7419** 15.6987*** 36 655.47*** 37.94*** 1653.43*** −34.35*** −72.95*** 778.65***
0.04 2.32 15.44 −12.23 0.0468 6.4515*** 1142.46*** 21.67** 894.45*** −23.32*** −49.44*** 313.86***
−0.01 1.60 22.97 −32.58 −1.2112*** 52.0546*** 459 227.2*** 67.34*** 680.94*** −32.37*** −70.91*** 754.65***
**
Significant at 5% level. Significant at 1% level. Statistics are expressed in percentages. b Jarque–Bera is a test statistic for testing whether the series is normally distributed. c Q (8) indicates the Ljung–Box Q -statistics at lag 8 by log return series, it is a test statistic for the null hypothesis that there is no autocorrelation up to order 8. d Q 2 (8) indicates the Ljung–Box Q -statistic at lag 8 by squared log return series, it is a test statistic for the null hypothesis that there is no autocorrelation up to order 8. e ADF and PP indicate respectively augmented Dickey and Fuller [31,32] and Phillips and Perron [33] unit root tests for whether the series is stationary. f ARCH LM test is a Lagrange multiplier test for autoregressive conditional heteroskedasticity in the residuals [34]. χ62 indicates the Engle LM test statistic at lag 6 by residuals of the AR(1), it is a test statistic for the null hypothesis that there is no ARCH up to order 6 in the residuals. *** a
Table 2 Test of tail stability for stock index futures returns. Contract
m−
α−
m+
α+
V (α + − α − )
FTSE100 Nasdaq100 Nikkei225
1248 516 1051
3.5083 4.9017 3.8652
1335 591 1085
4.1138 3.7235 3.7747
−5.1093*** −0.6326
4.8210***
Note: Modified Hill estimates, α , are calculated for the left and right tails for each futures contracts. The symbols −, + represent the left and right tails, respectively. Tail stability is calculated in the last column. *** Significant at 1% level.
for the FTSE100, as well as 2300 for the Nasdaq100 and 4569 for the Nikkei225. The time series of futures prices is created based on the close price of the nearest contract to maturity and up to the last trading day for the period before the delivery month, and the dataset is downloaded from Datastream. The futures returns are measured by the first difference of the natural logarithm of the close prices. Table 1 lists preliminary statistics of the returns for FTSE100, Nasdaq100 and Nikkei225 futures contracts. The mean returns for the entire period are nearly zero. The unconditional distributions of the returns are fat-tailed or leptokurtic, as demonstrated by high kurtosis and highly significant Jargue–Bera statistics. The returns distribution of the FTSE100 and Nikkei225 display a negative skewness (Nasdaq100 contract excepted). These findings indicate that the unconditional distributions of FTSE100, Nasdaq100 and Nikkei225 returns can be characterized by fat-tailed distributions and that the left and right tails should be treated, respectively. Additionally, this investigation also reports the ADF and PP statistics, which indicate that the return series are stationary. Ljung–Box statistics for the returns themselves and for the squared returns and LM test are also presented, and these statistics confirm that the empirical return series contains autocorrelation and ARCH effects, suggesting that conditional modeling of short run returns is beneficial. The dynamics are introduced in the conditional VaR-x approach. 4. Empirical results 4.1. Unconditional margin level analysis To implement Eqs. (3) and (4), this investigation uses the full samples of the FTSE100, Nasdaq100 and Nikkei225 futures returns to estimate tail indexes. Table 2 lists the number of tail observations and the modified Hill index estimates for the left and right tails, which correspond to the shape parameter used to calculate the long and short margin levels. All tail index estimates range between three and five, which indicate three returns distributions have fat-tailed characteristics. Additionally, the test statistics V (α + −α − ) present that the left and right tail estimates significantly differ, with the exception of the Nikkei225 contract. However, the left and right tail index estimates of the Nikkei225 futures returns distribution still differ slightly. According to the above result, different tail index estimates are confirmed to have different margin levels.
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Table 3 Unconditional margin levels for a long position. Contract
Method
Probability of margin violation 0.05
0.04
0.03
0.02
0.01
FTSE
Historical Normal Hill(1) Hill(2) VaR-x
1.6769 1.8488 2.1794 2.2284 1.6335*
1.8289 1.9695 2.3470 2.4127 1.7987*
2.0751 2.1179* 2.5823 2.6728 2.0215
2.3819 2.3152 2.9545 3.0879 2.3574*
3.1475 2.6262 3.7192 3.9520 3.0049*
0.005 3.6031 2.9107 4.6818 5.0581 3.7689*
0.004 3.8805 2.9977 5.0419 5.4763 4.0447*
0.003 4.3700 3.1068 5.5473 6.0668 4.4244*
0.002 4.8533 3.2556 6.3468 7.0089 5.0110*
0.001 5.6865 3.4975 7.9896 8.9704 6.1741*
Nasdaq100
Historical Normal Hill(1) Hill(2) VaR-x
3.6987 3.7716* 5.0361 4.7634 3.5687
4.0178 4.0168* 5.3305 5.1280 3.8857
4.4333 4.3183* 5.7354 5.6396 4.3018
5.0428 4.7191 6.3590 6.4485 4.9071*
6.1070 5.3507 7.5860 8.1090 6.0099*
7.6239 5.9288 9.0498 10.1971 7.2236*
7.9056 6.1055 9.5787 10.9777 7.6425*
8.3477 6.3272 10.3064 12.0729 8.2055*
9.0410 6.6293 11.4270 13.8045 9.0464*
9.8113 7.1208 13.6319 17.3592 10.6290*
Nikkei225
Historical Normal Hill(1) Hill(2) VaR-x
2.4353 2.6434 3.1838 3.1701 2.4051*
2.6809 2.8128 3.4025 3.4135 2.6338*
2.9567 3.0209 3.7067 3.7550 2.9395*
3.3260 3.2977* 4.1822 4.2951 3.3952
4.0847 3.7338 5.1406 5.4044 4.2577*
5.2265 4.1330 6.3186 6.8001 5.2529*
5.5266 4.2550 6.7526 7.3221 5.6071*
5.8579 4.4081 7.3563 8.0547 6.0910*
6.2413 4.6167 8.2999 9.2132 6.8306*
7.1607 4.9561 10.2019 11.5927 8.2729*
Note: The optimal number of tail observations in Hill (1), m, are calculated following the method proposed by Phillips et al. [28]. The optimal number of tail observations in Hill (2), m, are calculated as 10% of sample size [29]. The optimal number of tail observations in the VaR-x, m, are calculated following the method proposed by Huisman et al. [14]. This table gives the optimal margin level (as percentage) at given probabilities of margin violation for a long position. Five different methods are used to compute the margin levels. * Indicates the margin level estimates is close to empirical margin levels.
Sequentially, this investigation implements Eqs. (5) and (6) to obtain the estimates of unconditional margin levels for long and short positions given different probabilities of margin violation, which range from 0.1% to 5%. The performance of the proposed approach is assessed by comparing three different methods of computing the margin level. The first method is the normal approach [4,10], which assumes that returns approximate a normal distribution. The second and third methods rely on the Hill estimator proposed by Cotter [13]. In the second method (named Hill(1)), for selecting m largest order statistics, we completely follow the work of Cotter [13] using the method proposed by Phillips et al. [28]. To simplify the dynamic process for estimating the margin level proposed by Cotter [13], the third method (named Hill(2)) as proposed by Quintos et al. [29] adopts 10% of the sample size for selecting m. These methods derived from statistical models can be compared with the historical approach as done by Longin [10]. Empirical results of unconditional margin levels in percentage form for the long and short positions are respectively presented in Tables 3 and 4. For the long and short positions in FTSE100, Nasdaq100 and Nikkei225 contracts, the unconditional VaR-x approach is more appropriate since it generates theoretical margin levels that closely mirror the empirical distribution when the probabilities of margin violation are below 1%. Furthermore, for lower margin violation probabilities from 0.1% to 1%, underestimation bias is found in margin levels calculated based on the assumption of normality, whereas overestimation bias is found in margin levels that are calculated following the work of Cotter [13], which assumes that the returns approximate a extreme value distribution. Taking the long position of FTSE100 contracts as an example, the estimated margin level equals 3.6979% under the historical approach for 0.5% margin violation probability compared with 3.8709% (2.9802%, 4.7637%, 5.2052%) under the VaR-x (Normal, Hill(1), Hill(2)) approach. Summarizing the above findings, this investigation demonstrates that given lower probabilities of margin violation, the unconditional VaR-x approach to setting margin levels is more accurate than the normal approach and the Hill nonparametric approach proposed by Cotter [13]. 4.2. Conditional margin level analysis In analyzing conditional margin levels, this investigation used the conditional VaR-x approach to incorporate dynamics into margin setting, thus reflecting the current volatility levels in the trading environment. This empirical investigation uses a moving window of 500 observations to dynamically estimate the margin levels for long and short positions by the conditional VaR-x approach, the conditional normal approach and the conditional Hill non-parametric approach. To assess the predictive performance of different margin setting methods, this investigation uses the ‘‘backtesting’’ procedure. Namely, using each of the approaches, given different probabilities of margin violation,7 this investigation forecasts daily margin levels for long and short positions. These margin level forecasts are then compared with the actual returns on that particular day, and the number of days when the actual returns exceed the forecast margin level are counted. The
7 This empirical study shows that the unconditional VaR-x approach is more appropriate for computing optimal margin requirements given lower probabilities of margin violation. Thus in empirical analysis of conditional margin requirements, this study merely estimates the conditional margin requirement given lower probabilities of margin violation, which ranges from 0.5% to 0.1%.
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Table 4 Unconditional margin levels for a short position. Contract
Method
Probability of margin violation 0.05
0.04
0.03
0.02
0.01
FTSE
Historical Normal Hill(1) Hill(2) VaR-x
1.7451 1.9035 2.3433 2.1960 1.7567*
1.9038 2.0243 2.4840 2.3636 1.9180*
2.0810 2.1727 2.6780 2.5987 2.1325*
2.4310 2.3699 2.9773 2.9703 2.4501*
3.0220 2.6809 3.5685 3.7327 3.0447*
3.6458 2.9655 4.2772 4.6908 3.7217*
3.8135 3.0524 4.5340 5.0488 3.9605*
4.0176 3.1616 4.8880 5.5510 4.2853*
4.5326 3.3103 5.4343 6.3447 4.7786*
4.9010 3.5522 6.5135 7.9732 5.7309*
Nasdaq100
Historical Normal Hill(1) Hill(2) VaR-x
3.4286 3.8511 4.6220 4.4262 3.4735*
3.8648 4.0963 4.9480 4.8171 3.8067*
4.2812 4.3978 5.4025 5.3724 4.2533*
5.0205 4.7985 6.1149 6.2655 4.9220*
6.1404 5.4302 7.5570 8.1495 6.1965*
7.7705 6.0083 9.3393 10.5999 7.6794*
8.7508 6.1850 9.9981 11.5360 8.2099*
9.1633 6.4067 10.9165 12.8659 8.9370*
9.8190 6.7088 12.3560 15.0047 10.0526*
10.5728 7.2002 15.2699 19.5164 12.2419*
Nikkei225
Historical Normal Hill(1) Hill(2) VaR-x
2.3319 2.6201 3.1505 3.0533 2.3678*
2.5739 2.7894 3.3589 3.3113 2.5974*
2.9056 2.9976 3.6481 3.6764 2.9049*
3.3887 3.2743 4.0984 4.2603 3.3644*
4.1766 3.7105 5.0008 5.4812 4.2380*
5.0572 4.1097 6.1018 7.0521 5.2513*
5.2404 4.2317 6.5055 7.6481 5.6131*
5.9525 4.3848 7.0656 8.4913 6.1085*
6.2168 4.5934 7.9379 9.8400 6.8673*
7.1602 4.9327 9.6856 12.6600 8.3532*
0.005
0.004
0.003
0.002
0.001
Note: The optimal number of tail observations in Hill(1), m, are calculated following the method proposed by Phillips et al. [28]. The optimal number of tail observations in Hill(2), m, are calculated as 10% of sample size [29]. The optimal number of tail observations in the VaR-x, m, are calculated following the method proposed by Huisman et al. [14]. This table gives the optimal margin level (as percentage) at given probabilities of margin violation for a short position. Five different methods are used to compute the margin levels. * Indicates the margin level estimates is close to empirical margin levels.
number of such days is called the number of exceedences. The predictive margin violation probability then can be obtained using the number of exceedences divided by the total number of observed returns. If the predictive margin violation probability estimated by a certain method is close to the theoretical probability, then the certain method has performs well for conditional margin setting. Table 5 lists empirical results for predictive margin violation probability versus theoretical margin violation probability. For the FTSE100 and Nikkei225 contracts, the conditional VaR-x approach outperforms other methods. For the Nasdaq100 contract, the Hill non-parametric approach outperforms other methods; however, when adopting the decay factor λ = 0.895 which is smaller than λ = 0.94 used by RiskMetrics to estimate daily volatility for all series, we find that the conditional VaR-x approach provides more accurate forecasts of the true conditional margin level.8 The result implies that the decay factor may vary across series and the choice of decay factor lead to different performance. Additionally, for all contracts, the computed predictive margin violation probabilities by the normal and Hill non-parametric approaches exceed the theoretical margin violation probabilities, and the predictive margin violation probabilities calculated using the conditional VaR-x approach are less than or equal to the theoretical margin violation probabilities. Thus margin levels estimated by the normal and Hill non-parametric approaches have an underestimation bias, and those estimated by the conditional VaR-x approach have a slight overestimation bias. Consequently, using the conditional VaR-x approach to set conditional margin levels can capture the additional downside and upside risk faced during times of greater fluctuation. To clearly compare the performance of the conditional VaR-x, normal, and Hill non-parametric approaches, Figs. 1–6, given 0.5% probability of margin violation, also plot the computed conditional margin level forecasts for the long and short positions of the FTSE100, Nasdaq100, and Nikkei225 contracts and the corresponding actual daily returns observed over the same time horizon. For the long positions, the VaR-x approach can capture some of additional downside risk faced during the more volatile periods, beyond that associated with using the normal approach. This is exemplified by the fact that boundaries of VaR-x approach lie below those of the normal method. Meanwhile, for the short position, the VaR-x approach can capture some of the upside risk faced during the more volatile periods. Additionally, the charts of the conditional margin levels calculated using the Hill non-parametric approach are not very sensitive in capturing the volatility of actual returns. This is exemplified by the smooth boundaries of the Hill non-parametric approach. Therefore, for conditional margin setting, the conditional VaR-x approach provides more accurate and prudent forecasts of the conditional margin level than the normal and Hill non-parametric approaches.
8 Here is a list of the computed predictive margin violation probabilities for the Nasdaq100 contract by the conditional VaR-x approach, which set the value of decay factor to 0.895: Theoretical probability Position Predictive probability (VaR-x) *
0.01 Long 0.0100
Short *
0.005 Long
0.0067 0.0044
Short *
0.004 Long
0.0033 0.0033
Short *
0.003 Long
0.002
Short
Long *
0.0028 0.0017 0.0028
0.0011 0.0022
Indicates the predictive probability of margin violation is close to theoretical probability of margin violation.
0.001
Short
Long *
0.0011
Short *
0.0006*
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Table 5 Backtesting results: theoretical vs. predictive probability of margin violation. Contract
Method
Theoretical probability of margin violation 0.01
FTSE100
Nasdaq100
Nikkei225
Normal Hill VaR-x Normal Hill VaR-x Normal Hill VaR-x
0.005
0.004
0.003
0.002
0.001
Long
Short
Long
Short
Long
Short
Long
Short
Long
Short
Long
Short
0.0150* 0.0230 0.0168 0.0150 0.0161 0.0072* 0.0213 0.0231 0.0140*
0.0129 0.0164 0.0090* 0.0122 0.0111* 0.0067 0.0176 0.0228 0.0127*
0.0115 0.0108 0.0073* 0.0072* 0.0089 0.0022 0.0145 0.0137 0.0070*
0.0092 0.0088 0.0039* 0.0089 0.0056* 0.0028 0.0127 0.0114 0.0057*
0.0104 0.0093 0.0064* 0.0072 0.0056* 0.0022 0.0130 0.0112 0.0054*
0.0090 0.0063 0.0033* 0.0061 0.0044* 0.0028 0.0122 0.0083 0.0041*
0.0091 0.0071 0.0060* 0.0044 0.0028* 0.0011 0.0106 0.0093 0.0039*
0.0070 0.0053 0.0020* 0.0044 0.0039 0.0028* 0.0112 0.0065 0.0039*
0.0075 0.0040* 0.0040* 0.0028 0.0022* 0.0011 0.0096 0.0057 0.0036*
0.0061 0.0039 0.0014* 0.0039 0.0022* 0.0017 0.0093 0.0039 0.0021*
0.0044 0.0018* 0.0027 0.0017 0.0011* 0.0006 0.0075 0.0026 0.0013*
0.0039 0.0012 0.0010* 0.0033 0.0017 0.0006* 0.0073 0.0013* 0.0013*
Note: In conditional VaR-x and normal approaches, σ 2 are estimated by EWMA approach, which the decay factor λ is set to 0.94. ‘‘Long’’ and ‘‘short’’ indicate ‘‘long position’’ and ‘‘short position’’, respectively. * Indicates the predictive probability of margin violation is close to theoretical probability of margin violation.
Fig. 1. Actual returns vs. computed conditional margin levels of the long position for conditional VaR-x, normal, and Hill non-parametric approaches.
Fig. 2. Actual returns vs. computed conditional margin levels of the long position for conditional VaR-x, normal, and Hill non-parametric approaches.
5. Conclusions The previous empirical research agrees that using EVT to determine margin levels can fully protect against default resulting from extreme price movements. The main contribution of this study is to provide an approach, the VaR-x method based on EVT, that incorporates a modified version of the Hill estimator into a Student-t distribution, for setting the unconditional and conditional margin levels. This approach is especially appealing because the modified Hill estimator can avoid small sample bias and does not condition its tail observations as does the Hill estimator. Moreover, rather than using the Hill non-parametric approach developed by Cotter [13], this study uses the Student-t distribution, parameterised by the tail index estimated from the modified Hill estimator, to estimate the margin level. Empirical study involving three stock index futures contracts, the FTSE100, the Nasdaq100 and the Nikkei225, demonstrates the accuracy of the proposed approach.
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Fig. 3. Actual returns vs. computed conditional margin levels of the long position for conditional VaR-x, normal, and Hill non-parametric approaches.
Fig. 4. Actual returns vs. computed conditional margin levels of the short position for conditional VaR-x, normal, and Hill non-parametric approaches.
Fig. 5. Actual returns vs. computed conditional margin levels of the short position for conditional VaR-x, normal, and Hill non-parametric approaches.
The analysis of unconditional margin setting first demonstrated that behavior differs between the left and right tails of returns distribution. This suggests that different margin levels should be set for the long and short positions. Second, given the lower probabilities of margin violation, the unconditional VaR-x approach is a more appropriate means of setting margin level than the normal approach or the Hill non-parametric approach developed by Cotter [13]. Third, this study shows that the underestimation bias in margin levels that are computed assuming normality, whereas the Hill non-parametric approach has a phenomenon of overestimating margin level. In the analysis of conditional margin setting, this study evidences that conditional VaR-x approach compared with conditional normal and conditional Hill non-parametric approaches is better able to capture the nature of extreme events, particularly crucial in periods of greater fluctuation. In sum, the empirical evidence demonstrates that the unconditional and conditional VaR-x approaches offer more accurately forecast of margin levels and can capture extreme events, thus ensuring adequate prudence for the setting of initial and daily or maintenance margin levels.
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Fig. 6. Actual returns vs. computed conditional margin levels of the short position for conditional VaR-x, normal, and Hill non-parametric approaches.
References [1] G.G. Booth, J.P. Broussard, T.P. Martikainen, Prudent margin levels in the finnish stock index futures market, Management Science 43 (1997) 1177–1188. [2] M.J. Brennan, A theory of price limits in futures markets, Journal of Financial Economics 16 (1986) 213–233. [3] G.W. Fenn, Kupiec, Prudential margin policy in a future-style settlement system, Journal of Futures Markets 13 (1993) 389–408. [4] S. Figlewski, Margins and market integrity: Margin setting for stock index futures and options, The Journal of Futures Markets 4 (1984) 385–416. [5] G.D. Gay, W.C. Hunter, R.W. Kolb, A comparative analysis of futures contract margins, The Journal of Futures Markets 6 (1986) 307–324. [6] M.J. Warshawsky, The adequacy and consistency of margin requirements: The cash futures, and options segments of the equity markets, Review of Futures Markets 8 (1989) 420–437. [7] V.A. Kholodnyi, The non-markovian approach to the valuation and hedging of European contingent claims on power with scaling spikes, Journal of Nonlinear Analysis: Hybrid Systems 2 (2) (2008) 285–309. [8] P. Kofman, Optimizing futures margins with distribution tails, Advances in Review of Futures Markets 6 (1993) 263–278. [9] F.M. Longin, Optimal margin level in future markets: A parametric extreme-based method, in: Proceeding Ninth Chicago Board of Trade Conference on Futures and Options, Bonn, Germany, 1995. [10] F.M. Longin, Optimal margin levels in futures markets: Extreme price movements, Journal of Futures Markets 19 (1999) 127–152. [11] J.P. Broussard, Extreme-value and margin setting with and without price limits, The Quarterly Review of Economics and Finance 41 (2001) 365–385. [12] H. Dewachter, G. Gielens, Setting futures margins: The extremes approach, Applied Financial Economics 9 (1999) 173–181. [13] J. Cotter, Margin exceedences for European stock index futures using extreme value theory, Journal of Banking and Finance 25 (2001) 1475–1502. [14] R. Huisman, K.G. Koedijk, C.J.M. Kool, F. Palm, Tail-index estimates in small samples, Journal of Business and Economic Statistics 19 (1) (2001) 208–216. [15] K.G. Koedijk, M.M.A. Schafgans, C.G. de Vries, The tail index of exchange rate returns, Journal of International Economics 29 (1990) 93–108. [16] M.C.A.B. Hols, C.G. de Vries, The limiting distribution of extreme exchange rates return, Journal of Applied Econometrics 6 (1991) 287–302. [17] D. Jansen, C. de Vries, On the frequency of large stock returns: Putting booms and busts into perspective, The Review of Economics and Statistics 73 (1991) 18–24. [18] K.G. Koedijk, C. Kool, Tail estimates and the EMS target zone, Review of International Economics 2 (1994) 153–165. [19] J. Beirlant, P. Vynckier, J. Teugles, Tail-index estimation, pareto quantile plots regression diagnostics, Journal of the American Statistical Association 91 (1996) 1659–1667. [20] R. Huisman, K.G. Koedijk, R.A.J. Pownall, VaR-x: Fat tails in financial risk management, Journal of Risk 1 (1) (1998) 47–61. [21] R.A.J. Pownall, K.G. Koedijk, Capturing downside risk in financial markets: The case of Asian crisis, Journal of International Money and Finance 18 (1999) 853–870. [22] P. Hill, A simple general approach to inference about the tail of a distribution, The Annals of Statistics 3 (1975) 1163–1174. [23] J. Cotter, K. Dowd, Extreme spectral risk measures: An application to futures clearinghouse margin requirements, Journal of Banking & Finance 30 (2006) 3469–3485. [24] R. Knott, M. Polenghi, Assessing central counterparty margin coverage on futures contracts using GARCH models, Paper provided by Royal Economic Society in its series Royal Economic Society Annual Conference 2004 with number 6, 2004. [25] J. Cotter, F. Longin, Margin requirements with intraday dynamics, Working paper, 2004. [26] J. Danielsson, C.G. de Veries, Value at risk and extreme returns, Financial Markets Group Discussion Paper. No. 273, London School of Economics, London, 1997. [27] M. Loretan, P.C.B. Phillips, Testing the covariance stationary of heavy-tailed time series, Journal of Empirical Finance 1 (1994) 221–248. [28] P.C.B. Phillips, J.W. McFarland, P.C. McMahon, Robust tests of forward exchange market efficiency with empirical evidence from the 1920s, Journal of Applied Econometrics 11 (1996) 1–22. [29] C. Quintos, Z. Fan, P.C.B. Phillips, Structural change tests in tail behavior and the Asian crisis, Review of Economic Studies 68 (2001) 633–663. [30] J.P. Broussard, G.G. Booth, The behavior of extreme values in German stock index futures: an application to margin setting, Journal of Operational Research 104 (1998) 393–402. [31] D. Dickey, W.A. Fuller, Distribution of the estimates for autoregressive time series with a unit root, Journal of the American Statistical Association 74 (1979) 427–431. [32] D. Dickey, W.A. Fuller, Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica 49 (1981) 1057–1072. [33] P. Phillips, P. Perron, Testing for a unit root in time series regression, Biometrica 75 (1988) 335–346. [34] R.F. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50 (1982) 987–1007.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Setting margin levels in futures markets: An extreme value method Tzu-chuan Kao a,1 , Chu-hsiung Lin b,∗ a
Kun Shan University, Department of Finance and Banking, 949, Da Wan Road, Yung-Kang City, Tainan Hsien, 710, Taiwan
b
National Kaohsiung First University of Science and Technology, Department of Risk Management and Insurance, 2, Juoyue Road, Nantz District, Kaohsiung, Taiwan
article
info
Article history: Received 15 February 2007 Accepted 23 March 2009 Keywords: Extreme value theory Margin setting Stock index futures Value at Risk
abstract There are of course different types of margin requirements in futures clearinghouses, and this study focuses on setting initial and maintenance margin levels. This study provides an approach, the VaR-x method that incorporates a modification of the Hill estimator based on extreme value theory (EVT) into a Student-t distribution, for setting the unconditional and conditional margin levels (i.e. initial and maintenance margin levels). Empirical applications are based on daily data for three stock index futures returns: the FTSE100, Nasdaq100 and Nikkei225. The empirical results demonstrate that given lower probabilities of margin violation, the VaR-x approach to setting unconditional margin levels is more accurate than either the normal approach or the Hill non-parametric approach proposed by Cotter [J. Cotter, Margin exceedences for European Stock Index Futures using extreme value theory, Journal of Banking and Finance 25 (2001) 1475–1502]. Additionally, this study demonstrates that using the conditional VaR-x approach to setting margin levels can better capture extreme events, thus ensuring adequate prudence, something that is particularly crucial in periods of strong fluctuation. These empirical findings suggest that the proposed approach is very useful to setting the initial and maintenance margin levels. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Margin committees face the difficult task of appropriate margin levels to balance the costs of trader default against the benefits of increased market liquidity [1]. The key problem margin committee face is that setting the margin level too high will lead to high transaction costs and thus reduced trading volume. Meanwhile, if the margin level is set too low, it may not adequately cover extreme price changes, thus incurring default risk. For guarding against default, this paper focuses on setting prudent margins designed to protect futures positions from extreme price movements. The academic literature has implemented two approaches to setting appropriate margin levels. First, the application of economic models that assume margin levels are endogenously determined for minimizing broker costs. For example, Brennan [2] and Fenn and Kupiec [3] used the concept of efficient contract design to identify economic factors determining the optimal margin level. Second is the application of statistical approaches, which is used to set margin levels that represent an amount not to be exceeded by a price change over a specified time period at an acceptable probability level. These statistical approaches include parametric and non-parametric methods that rely on Gaussian and non-normal distribution assumptions for the underlying distribution of futures price changes. Figlewski [4] and Gay et al. [5] are two examples of
∗
Corresponding author. Tel.: +886 7 6011000; fax: +886 7 6011046. E-mail addresses: [email protected] (T.-c. Kao), [email protected] (C.-h. Lin).
1 Tel.: +886 6 2727175; fax: +886 6 2050622. 1468-1218/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2009.03.025
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studies that assume futures returns follow a normal distribution. Warshawsky [6] used a non-parametric analysis procedure, which does not rely on normal distributional assumptions, and shows that it may be inappropriate to assume normally distributed futures returns. Kholodnyi [7] developed a new approach, which models price changes with spikes as a nonMarkovian stochastic process that allows for modeling spikes directly as self-reversing jumps. Recent studies on margin setting have increasingly studied the fat-tailed characteristics of returns distribution, and in doing so, applied extreme value theory (EVT) to setting margin levels. As extreme movements are central to margin setting problem, EVT directly models the tails of the returns distribution and thus could potentially yield better estimates and forecasts of default risk. Kofman [8], Longin [9,10], Booth et al. [1], Broussard and Booth [30] and Broussard [11] document that using EVT with a parametric method is appropriate for measuring the optimal margin levels. Furthermore, their studies also show that assuming normally distributed data underestimates the probabilities of margin violation because the empirical distribution of observed large price movements is too leptokurtic to be normal. Meanwhile, in EVT the limiting distribution of extreme values has non-parametric underlying assumption, thus Dewachter and Gielens [12] and Cotter [13] propose applying the Hill non-parametric method to generate measures of margin levels, showing that the margin levels estimated by the Hill non-parametric method are sufficient for futures contracts. Consequently, a consensus exists in the previous empirical research that using EVT with parametric and Hill non-parametric methods to determine margin levels can fully protect against default resulting from extreme price movements, but that for minimizing model risk the Hill non-parametric method is superior to parametric procedures which require assumptions regarding the exact distribution type of extreme price changes. However, the Hill non-parametric method still suffers from two major limitations [14]. First, the Hill estimator is biased when applied to small samples. Second, the Hill estimator used in the estimation procedure must depend on the number of order statistics [15,16]. Traditionally, the number of order statistics is determined by simulating the mean squared error minimizing number, but generally this procedure produces biased estimates [12]. Recently, Jansen and de Vries [17], Koedijk and Kool [18] and Beirlant et al. [19] respectively proposed highly effective methods of resolving this problem. However, these methods still base their tail estimates on a specific number of tail observations [14]. To correct the Hill non-parametric method proposed by Cotter [13], this study specifically provides an approach to setting margin levels, known as VaR-x that incorporates a modified version of the Hill estimator into a Student-t distribution. In line with the works of Huisman et al. [20,14] and Pownall and Koedijk [21], this study is the first to use the modified Hill estimator, which is to correct the small sample bias in tail index estimates, and does not condition its tail observations as do the Hill [22], Beirlant et al. [19], and other tail index estimators [14], to estimate the tail index of returns distribution. The estimated tail index then can be used to parameterise the number of degrees of freedom of the Student-t distribution for estimating the margin level. The fact that the number of degrees of freedom reflects the degree of tail fatness enables the capture of extreme events, thus ensuring adequate margin level, something that is especially crucial during times of high fluctuation. Additionally, studies on margin setting can consider two different returns distributions: the conditional and unconditional distributions. The unconditional analysis attempts to incorporate extreme events occurring over a long time period, therefore it is suited to the setting of initial margin levels (i.e. unconditional margin levels). While the conditional analysis reflects the change in market conditions over time, therefore it is suited to the setting of daily or maintenance margin levels (i.e. conditional margin levels) [23]. This study considers the adoption of both unconditional and conditional analyses to provide different but complementary information for the setting of initial and maintenance margin levels. In analyzing conditional margin setting, most previous studies focused on the GARCH model. For example, Cotter [13], Knott and Polenghi [24] and Cotter and Longin [25]2 considered a conditional process by applying several variants of the GARCH model to address issues relating to the time-varying behavior of futures price changes. Knott and Polenghi [24] showed that employing a GARCH model with innovations based on Student-t distribution could provide a timely and informative measure of conditional margin levels. Though the GARCH model focuses on accounting for volatility clustering, whereas extreme values do not actually cluster [26,13], setting margin levels can specifically reflect common volatility levels, but cannot deal with very low probability levels where the extreme values are located. As the problem of setting conditional margin levels is still related to the tails of the distribution of futures price changes, this study uses a conditional approach of VaR-x which simultaneously describe the tail fatness and time-varying behavior of conditional distribution to capture some of the additional upside and downside risk faced during more volatile periods. The empirical investigation applies the proposed unconditional and conditional VaR-x approach to estimate the margin levels for three different daily stock index futures series: the US Nasdaq100, the British FTSE100 and the Japanese Nikkei225. Moreover, margin levels may differ between long and short positions owing to the behavior of left and right tail price movements. Therefore, this empirical investigation respectively deals with the estimation of the margin levels for long and short positions. The proposed VaR-x approach is compared with two different methods, which are the normal method [4,10] and the Hill non-parametric method [13]. The remainder of this paper is organized as follows. Section 2 presents the theoretical foundations used to estimate unconditional and conditional margin levels. Section 3 then details the stylized fact of the empirical data. Subsequently, Section 4 illustrates the empirical results. Finally, Section 5 presents conclusions.
2 Cotter and Longin [25] extended the work of Cotter [13], applying modified Hill estimator based on EVT [14] to set daily margin levels by considering the intraday dynamics of market prices. They found that the modified Hill estimator could easily extend the multi-period margin estimation.
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2. Method 2.1. Determining unconditional margin levels The unconditional VaR-x approach proposed by Huisman et al. [20,14] is developed from EVT, which investigates the distribution of tail observations in large samples. In the limit, the tail shape follows a Pareto law for a general class of fattailed distributions. The tail fatness of the distribution is characterized by the tail index. Notably, the tail index can be used as a parameter for the number of degrees of freedom to parameterise the Student-t distribution [20,14]. Consequently, the link to the Student-t distribution, a fatter-tailed distribution, is used to measure the margin level. The unconditional VaR-x approach for measuring margin level is presented below. In futures markets, large positive price movements create default risk for short positions, while large negative price movements create default risk for long positions. Thus, the measure of the probability of margin violation resulting from abrupt price changes should be derived from extreme positive and negative price movements. Under the theoretical framework of EVT, let the limiting distribution of extreme minimum (the negative) and maximum (the positive) price changes observed over the following n trading days defined by FMin(n) and FMax(n) , where FMin(n) (FMax(n) ) describes the left (right) tail behavior of futures returns. Considering a period of n trading days, the probabilities of margin violation for long and short positions are given by: qlong = P (Min(R1 , R2 , . . . , Rn ) < −MLlong ) = FMin(n) (−MLlong ) short
q
= P (Max(R1 , R2 , . . . , Rn ) > ML
short
) = 1 − FMax(n) (ML
short
(1)
)
(2)
where R represents the futures return (or futures price change rate), and MLlong and MLshort denote the margin levels of long and short positions.3 The above two equations show that the measure of margin levels for long and short positions given a specific probability of margin violation is related to modeling the left and right tails of the returns distribution and thus to estimating the tail index of the left and right tails. To obtain an unbiased tail index estimate, a modification of the Hill estimator is used to correct the small sample bias in tail index estimates without requiring a priori selection of the number of tail observations to be included. The process used to estimate the tail index of the futures returns distribution is presented below. For simplicity, this study discusses estimation and inference for the left tail of the returns distribution.4 Let R(i) denote the ith increasing order statistic of the absolute returns such that R(i) > R(i−1) for i = 2, . . . , n. Specifying k as the number of tail observations, the tail estimator (or tail index) α(k) proposed by Hill [22] is obtained as follows:
α(k) =
k 1X
k j=1
! −1 ln R(n−j+1) − ln R(n−k)
.
(3)
Based on the method of Huisman et al. [14], a modified version of the Hill estimator can be used to correct the bias in small samples. The bias of the inverse of Hill estimator γ (k) = 1/α(k) results from that the bias is a function of the sample size. A bias corrected tail index is thus obtained by observing the bias of the γ (k) as the number of tail observations increases until κ , where κ equals half of the sample size (i.e. κ = n/2):
γ (k) = β0 + β1 k + ε(k) k = 1, 2, . . . , κ
(4)
where ε(k) denotes the error term and the inverse of the estimated βˆ 0 is an estimator of α . Since the estimate of the tail index ˆ1 is equal to the number of degrees of freedom of the Student-t distribution [20,14] we can assume that the empirical β0
distribution of futures return approximates to a Student-t distribution with ˆ1 degrees of freedom. Via the relationship of β0 the critical value and cumulative probability, the estimate of margin level ML can be derived. The procedure for obtaining the unconditional margin level estimates of long and short positions MLlong , MLshort is summarized as follows: Step 1: Use Eqs. (3) and (4) to obtain estimates of the left and right tail index ˆ1− , ˆ1+ . Furthermore, it is also necessary to β0 β0 estimate the unconditional mean µ and the unconditional variance σ 2 of the underlying futures returns distribution. Step 2: Set the left tail index estimate ˆ1− to equal the number of degrees of freedom νˆ − of the Student-t distribution. β0
Similarly, set the right tail index estimate ˆ1+ to equal the number of degrees of freedom νˆ + of the Student-t β0 distribution. 3 Notably, in the proposed approach the estimated margin level is expressed as a percentage. However, in practice, cleaning houses tend to state the margin level in terms of currency units per contract. To avoid confusion, the relationship between ‘‘percentage’’ and ‘‘currency unit’’ can be explained as follows: if the current future contract price is $100, a currency margin of $5 for a long position corresponds to a percentage margin rate of −5.13% (=100ln(95/100)), and a currency margin of $5 for a short position corresponds to a percentage margin rate of 4.88% (100ln(105/100)). 4 An estimate for right tail index is captured using a sequence of increasing order statistic of the positive returns. An estimate for the common tail index is captured using a sequence of increasing order statistic, which encompasses both sets of positive returns and the absolute value of the negative returns.
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Step 3: With a confidence level 1 − q, read the critical value S − (S + ) of the Student-t distribution with νˆ − (νˆ + ) degrees of freedom. Step 4: Transform S − into the corresponding critical value of the real futures returns distribution,5 in which case the estimate of unconditional margin level for a long position given the probability of margin violation q% is given by:
ˆ long = −S − ∗ θ + µ ML ˆ
where θ = p
σˆ νˆ − /ˆν −
−2
.
(5)
Similarly, the estimated unconditional margin level for a short position given probability of margin violation q% is given by:
ˆ short = S + ∗ θ + µ ML ˆ
where θ = p
σˆ νˆ + /ˆν +
−2
.
(6)
2.2. Testing for tail stability This study employs a test developed by Loretan and Phillips [27] to examine whether any difference exists between left and right tail behavior. The null hypothesis is given by H0 : α + = α − . The test statistics are as follows: Vˆ (α + − α − ) =
α+ − α−
(α +2 /m+ ) + (α −2 /m− )
− →
1/2
d
N (0, 1)
(7)
where α + (α − ) denotes the estimate of the right (left) tail index, and m+ (m− ) represents the number of largest order statistics used to estimate the tail index. If the null hypothesis is rejected, the optimal margin level for long and short positions should be treated separately. Meanwhile, if the null hypothesis is supported, long and short positions should have a common margin level. 2.3. Determining conditional margin levels This section presents a conditional VaR-x approach for capturing the time-varying margin levels to provide clearing houses additional information on the setting of daily or maintenance margin levels.The proposed conditional VaR-x approach assumes that the underlying distribution of futures returns is a time-varying Student-t distribution and uses a conditional modified Hill estimator to parameterise the time-varying Student-t distribution. Based on Eqs. (5) and (6), the long M Lˆ t and M Lˆ short using the conditional VaR-x approach are given by: t long
M Lˆ t
σˆ t −1 = −St−−1 ∗ θt −1 + µ ˆ t −1 where θt −1 = q − νˆ t −1 /ˆνt−−1 − 2
M Lˆ short = St+−1 ∗ θt −1 + µ ˆ t −1 t
where θt −1 = q
σˆ t −1
νˆ t+−1 /ˆνt+−1
(8)
(9)
−2
long
) denotes the conditional margin level at time t for long (short) position. To implement Eqs. (8) and (9), where M Lˆ t (M Lˆ short t we use the rolling window approach, which can respond to the changes on the volatility levels around the current period t, to compute the conditional marginal levels. Furthermore, The EWMA model, known as an integrated GARCH or IGARCH model, is used to estimate the conditional volatility σt since it relies on one parameter only and thus facilitates estimation. The EWMA estimator is given by:
σt2 = λσt2−1 + (1 − λ)R2t −1
(10)
where the parameter λ is called the decay factor and must be less than unity, and Rt denotes the rate of return at time t. This study sets λ = 0.94 used by RiskMetrics to estimate daily volatility for the purpose of calculating Value at Risk. 3. Data description To verify the performance of the proposed approach, this empirical investigation applies the proposed approach to three different stock index futures contracts: FTSE100, Nasdaq100 and Nikkei225.6 This empirical investigation examines daily prices (trading days) from their issue dates to May 20, 2005. During the selected periods, 5382 observations are obtained 5 This transformation is due to the fact that the Student-t distribution with νˆ degrees of freedom has a predetermined mean equal to zero and a variance equal to ν/ν − 2. (see [20]). 6 The FTSE100 future contract traded on LIFFE, the Nasdaq100 future contract traded on CME and the Nikkei225 future contracts traded on SGX-DT.
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Table 1 Summary statistics for the stock index futures contracts’ returns. Statistic
FTSE100
Nasdaq100
Nikkei225
Meana Std.Dev.a Maximuma Minimuma Skewness Kurtosis Jarque–Berab Ljung–Box Q (12)c Ljung–Box Q 2 (12)d ADFe PPe ARCH LM testf : χ62
0.03 1.14 8.09 −16.75 −0.7419** 15.6987*** 36 655.47*** 37.94*** 1653.43*** −34.35*** −72.95*** 778.65***
0.04 2.32 15.44 −12.23 0.0468 6.4515*** 1142.46*** 21.67** 894.45*** −23.32*** −49.44*** 313.86***
−0.01 1.60 22.97 −32.58 −1.2112*** 52.0546*** 459 227.2*** 67.34*** 680.94*** −32.37*** −70.91*** 754.65***
**
Significant at 5% level. Significant at 1% level. Statistics are expressed in percentages. b Jarque–Bera is a test statistic for testing whether the series is normally distributed. c Q (8) indicates the Ljung–Box Q -statistics at lag 8 by log return series, it is a test statistic for the null hypothesis that there is no autocorrelation up to order 8. d Q 2 (8) indicates the Ljung–Box Q -statistic at lag 8 by squared log return series, it is a test statistic for the null hypothesis that there is no autocorrelation up to order 8. e ADF and PP indicate respectively augmented Dickey and Fuller [31,32] and Phillips and Perron [33] unit root tests for whether the series is stationary. f ARCH LM test is a Lagrange multiplier test for autoregressive conditional heteroskedasticity in the residuals [34]. χ62 indicates the Engle LM test statistic at lag 6 by residuals of the AR(1), it is a test statistic for the null hypothesis that there is no ARCH up to order 6 in the residuals. *** a
Table 2 Test of tail stability for stock index futures returns. Contract
m−
α−
m+
α+
V (α + − α − )
FTSE100 Nasdaq100 Nikkei225
1248 516 1051
3.5083 4.9017 3.8652
1335 591 1085
4.1138 3.7235 3.7747
−5.1093*** −0.6326
4.8210***
Note: Modified Hill estimates, α , are calculated for the left and right tails for each futures contracts. The symbols −, + represent the left and right tails, respectively. Tail stability is calculated in the last column. *** Significant at 1% level.
for the FTSE100, as well as 2300 for the Nasdaq100 and 4569 for the Nikkei225. The time series of futures prices is created based on the close price of the nearest contract to maturity and up to the last trading day for the period before the delivery month, and the dataset is downloaded from Datastream. The futures returns are measured by the first difference of the natural logarithm of the close prices. Table 1 lists preliminary statistics of the returns for FTSE100, Nasdaq100 and Nikkei225 futures contracts. The mean returns for the entire period are nearly zero. The unconditional distributions of the returns are fat-tailed or leptokurtic, as demonstrated by high kurtosis and highly significant Jargue–Bera statistics. The returns distribution of the FTSE100 and Nikkei225 display a negative skewness (Nasdaq100 contract excepted). These findings indicate that the unconditional distributions of FTSE100, Nasdaq100 and Nikkei225 returns can be characterized by fat-tailed distributions and that the left and right tails should be treated, respectively. Additionally, this investigation also reports the ADF and PP statistics, which indicate that the return series are stationary. Ljung–Box statistics for the returns themselves and for the squared returns and LM test are also presented, and these statistics confirm that the empirical return series contains autocorrelation and ARCH effects, suggesting that conditional modeling of short run returns is beneficial. The dynamics are introduced in the conditional VaR-x approach. 4. Empirical results 4.1. Unconditional margin level analysis To implement Eqs. (3) and (4), this investigation uses the full samples of the FTSE100, Nasdaq100 and Nikkei225 futures returns to estimate tail indexes. Table 2 lists the number of tail observations and the modified Hill index estimates for the left and right tails, which correspond to the shape parameter used to calculate the long and short margin levels. All tail index estimates range between three and five, which indicate three returns distributions have fat-tailed characteristics. Additionally, the test statistics V (α + −α − ) present that the left and right tail estimates significantly differ, with the exception of the Nikkei225 contract. However, the left and right tail index estimates of the Nikkei225 futures returns distribution still differ slightly. According to the above result, different tail index estimates are confirmed to have different margin levels.
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Table 3 Unconditional margin levels for a long position. Contract
Method
Probability of margin violation 0.05
0.04
0.03
0.02
0.01
FTSE
Historical Normal Hill(1) Hill(2) VaR-x
1.6769 1.8488 2.1794 2.2284 1.6335*
1.8289 1.9695 2.3470 2.4127 1.7987*
2.0751 2.1179* 2.5823 2.6728 2.0215
2.3819 2.3152 2.9545 3.0879 2.3574*
3.1475 2.6262 3.7192 3.9520 3.0049*
0.005 3.6031 2.9107 4.6818 5.0581 3.7689*
0.004 3.8805 2.9977 5.0419 5.4763 4.0447*
0.003 4.3700 3.1068 5.5473 6.0668 4.4244*
0.002 4.8533 3.2556 6.3468 7.0089 5.0110*
0.001 5.6865 3.4975 7.9896 8.9704 6.1741*
Nasdaq100
Historical Normal Hill(1) Hill(2) VaR-x
3.6987 3.7716* 5.0361 4.7634 3.5687
4.0178 4.0168* 5.3305 5.1280 3.8857
4.4333 4.3183* 5.7354 5.6396 4.3018
5.0428 4.7191 6.3590 6.4485 4.9071*
6.1070 5.3507 7.5860 8.1090 6.0099*
7.6239 5.9288 9.0498 10.1971 7.2236*
7.9056 6.1055 9.5787 10.9777 7.6425*
8.3477 6.3272 10.3064 12.0729 8.2055*
9.0410 6.6293 11.4270 13.8045 9.0464*
9.8113 7.1208 13.6319 17.3592 10.6290*
Nikkei225
Historical Normal Hill(1) Hill(2) VaR-x
2.4353 2.6434 3.1838 3.1701 2.4051*
2.6809 2.8128 3.4025 3.4135 2.6338*
2.9567 3.0209 3.7067 3.7550 2.9395*
3.3260 3.2977* 4.1822 4.2951 3.3952
4.0847 3.7338 5.1406 5.4044 4.2577*
5.2265 4.1330 6.3186 6.8001 5.2529*
5.5266 4.2550 6.7526 7.3221 5.6071*
5.8579 4.4081 7.3563 8.0547 6.0910*
6.2413 4.6167 8.2999 9.2132 6.8306*
7.1607 4.9561 10.2019 11.5927 8.2729*
Note: The optimal number of tail observations in Hill (1), m, are calculated following the method proposed by Phillips et al. [28]. The optimal number of tail observations in Hill (2), m, are calculated as 10% of sample size [29]. The optimal number of tail observations in the VaR-x, m, are calculated following the method proposed by Huisman et al. [14]. This table gives the optimal margin level (as percentage) at given probabilities of margin violation for a long position. Five different methods are used to compute the margin levels. * Indicates the margin level estimates is close to empirical margin levels.
Sequentially, this investigation implements Eqs. (5) and (6) to obtain the estimates of unconditional margin levels for long and short positions given different probabilities of margin violation, which range from 0.1% to 5%. The performance of the proposed approach is assessed by comparing three different methods of computing the margin level. The first method is the normal approach [4,10], which assumes that returns approximate a normal distribution. The second and third methods rely on the Hill estimator proposed by Cotter [13]. In the second method (named Hill(1)), for selecting m largest order statistics, we completely follow the work of Cotter [13] using the method proposed by Phillips et al. [28]. To simplify the dynamic process for estimating the margin level proposed by Cotter [13], the third method (named Hill(2)) as proposed by Quintos et al. [29] adopts 10% of the sample size for selecting m. These methods derived from statistical models can be compared with the historical approach as done by Longin [10]. Empirical results of unconditional margin levels in percentage form for the long and short positions are respectively presented in Tables 3 and 4. For the long and short positions in FTSE100, Nasdaq100 and Nikkei225 contracts, the unconditional VaR-x approach is more appropriate since it generates theoretical margin levels that closely mirror the empirical distribution when the probabilities of margin violation are below 1%. Furthermore, for lower margin violation probabilities from 0.1% to 1%, underestimation bias is found in margin levels calculated based on the assumption of normality, whereas overestimation bias is found in margin levels that are calculated following the work of Cotter [13], which assumes that the returns approximate a extreme value distribution. Taking the long position of FTSE100 contracts as an example, the estimated margin level equals 3.6979% under the historical approach for 0.5% margin violation probability compared with 3.8709% (2.9802%, 4.7637%, 5.2052%) under the VaR-x (Normal, Hill(1), Hill(2)) approach. Summarizing the above findings, this investigation demonstrates that given lower probabilities of margin violation, the unconditional VaR-x approach to setting margin levels is more accurate than the normal approach and the Hill nonparametric approach proposed by Cotter [13]. 4.2. Conditional margin level analysis In analyzing conditional margin levels, this investigation used the conditional VaR-x approach to incorporate dynamics into margin setting, thus reflecting the current volatility levels in the trading environment. This empirical investigation uses a moving window of 500 observations to dynamically estimate the margin levels for long and short positions by the conditional VaR-x approach, the conditional normal approach and the conditional Hill non-parametric approach. To assess the predictive performance of different margin setting methods, this investigation uses the ‘‘backtesting’’ procedure. Namely, using each of the approaches, given different probabilities of margin violation,7 this investigation forecasts daily margin levels for long and short positions. These margin level forecasts are then compared with the actual returns on that particular day, and the number of days when the actual returns exceed the forecast margin level are counted. The
7 This empirical study shows that the unconditional VaR-x approach is more appropriate for computing optimal margin requirements given lower probabilities of margin violation. Thus in empirical analysis of conditional margin requirements, this study merely estimates the conditional margin requirement given lower probabilities of margin violation, which ranges from 0.5% to 0.1%.
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Table 4 Unconditional margin levels for a short position. Contract
Method
Probability of margin violation 0.05
0.04
0.03
0.02
0.01
FTSE
Historical Normal Hill(1) Hill(2) VaR-x
1.7451 1.9035 2.3433 2.1960 1.7567*
1.9038 2.0243 2.4840 2.3636 1.9180*
2.0810 2.1727 2.6780 2.5987 2.1325*
2.4310 2.3699 2.9773 2.9703 2.4501*
3.0220 2.6809 3.5685 3.7327 3.0447*
3.6458 2.9655 4.2772 4.6908 3.7217*
3.8135 3.0524 4.5340 5.0488 3.9605*
4.0176 3.1616 4.8880 5.5510 4.2853*
4.5326 3.3103 5.4343 6.3447 4.7786*
4.9010 3.5522 6.5135 7.9732 5.7309*
Nasdaq100
Historical Normal Hill(1) Hill(2) VaR-x
3.4286 3.8511 4.6220 4.4262 3.4735*
3.8648 4.0963 4.9480 4.8171 3.8067*
4.2812 4.3978 5.4025 5.3724 4.2533*
5.0205 4.7985 6.1149 6.2655 4.9220*
6.1404 5.4302 7.5570 8.1495 6.1965*
7.7705 6.0083 9.3393 10.5999 7.6794*
8.7508 6.1850 9.9981 11.5360 8.2099*
9.1633 6.4067 10.9165 12.8659 8.9370*
9.8190 6.7088 12.3560 15.0047 10.0526*
10.5728 7.2002 15.2699 19.5164 12.2419*
Nikkei225
Historical Normal Hill(1) Hill(2) VaR-x
2.3319 2.6201 3.1505 3.0533 2.3678*
2.5739 2.7894 3.3589 3.3113 2.5974*
2.9056 2.9976 3.6481 3.6764 2.9049*
3.3887 3.2743 4.0984 4.2603 3.3644*
4.1766 3.7105 5.0008 5.4812 4.2380*
5.0572 4.1097 6.1018 7.0521 5.2513*
5.2404 4.2317 6.5055 7.6481 5.6131*
5.9525 4.3848 7.0656 8.4913 6.1085*
6.2168 4.5934 7.9379 9.8400 6.8673*
7.1602 4.9327 9.6856 12.6600 8.3532*
0.005
0.004
0.003
0.002
0.001
Note: The optimal number of tail observations in Hill(1), m, are calculated following the method proposed by Phillips et al. [28]. The optimal number of tail observations in Hill(2), m, are calculated as 10% of sample size [29]. The optimal number of tail observations in the VaR-x, m, are calculated following the method proposed by Huisman et al. [14]. This table gives the optimal margin level (as percentage) at given probabilities of margin violation for a short position. Five different methods are used to compute the margin levels. * Indicates the margin level estimates is close to empirical margin levels.
number of such days is called the number of exceedences. The predictive margin violation probability then can be obtained using the number of exceedences divided by the total number of observed returns. If the predictive margin violation probability estimated by a certain method is close to the theoretical probability, then the certain method has performs well for conditional margin setting. Table 5 lists empirical results for predictive margin violation probability versus theoretical margin violation probability. For the FTSE100 and Nikkei225 contracts, the conditional VaR-x approach outperforms other methods. For the Nasdaq100 contract, the Hill non-parametric approach outperforms other methods; however, when adopting the decay factor λ = 0.895 which is smaller than λ = 0.94 used by RiskMetrics to estimate daily volatility for all series, we find that the conditional VaR-x approach provides more accurate forecasts of the true conditional margin level.8 The result implies that the decay factor may vary across series and the choice of decay factor lead to different performance. Additionally, for all contracts, the computed predictive margin violation probabilities by the normal and Hill non-parametric approaches exceed the theoretical margin violation probabilities, and the predictive margin violation probabilities calculated using the conditional VaR-x approach are less than or equal to the theoretical margin violation probabilities. Thus margin levels estimated by the normal and Hill non-parametric approaches have an underestimation bias, and those estimated by the conditional VaR-x approach have a slight overestimation bias. Consequently, using the conditional VaR-x approach to set conditional margin levels can capture the additional downside and upside risk faced during times of greater fluctuation. To clearly compare the performance of the conditional VaR-x, normal, and Hill non-parametric approaches, Figs. 1–6, given 0.5% probability of margin violation, also plot the computed conditional margin level forecasts for the long and short positions of the FTSE100, Nasdaq100, and Nikkei225 contracts and the corresponding actual daily returns observed over the same time horizon. For the long positions, the VaR-x approach can capture some of additional downside risk faced during the more volatile periods, beyond that associated with using the normal approach. This is exemplified by the fact that boundaries of VaR-x approach lie below those of the normal method. Meanwhile, for the short position, the VaR-x approach can capture some of the upside risk faced during the more volatile periods. Additionally, the charts of the conditional margin levels calculated using the Hill non-parametric approach are not very sensitive in capturing the volatility of actual returns. This is exemplified by the smooth boundaries of the Hill non-parametric approach. Therefore, for conditional margin setting, the conditional VaR-x approach provides more accurate and prudent forecasts of the conditional margin level than the normal and Hill non-parametric approaches.
8 Here is a list of the computed predictive margin violation probabilities for the Nasdaq100 contract by the conditional VaR-x approach, which set the value of decay factor to 0.895: Theoretical probability Position Predictive probability (VaR-x) *
0.01 Long 0.0100
Short *
0.005 Long
0.0067 0.0044
Short *
0.004 Long
0.0033 0.0033
Short *
0.003 Long
0.002
Short
Long *
0.0028 0.0017 0.0028
0.0011 0.0022
Indicates the predictive probability of margin violation is close to theoretical probability of margin violation.
0.001
Short
Long *
0.0011
Short *
0.0006*
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Table 5 Backtesting results: theoretical vs. predictive probability of margin violation. Contract
Method
Theoretical probability of margin violation 0.01
FTSE100
Nasdaq100
Nikkei225
Normal Hill VaR-x Normal Hill VaR-x Normal Hill VaR-x
0.005
0.004
0.003
0.002
0.001
Long
Short
Long
Short
Long
Short
Long
Short
Long
Short
Long
Short
0.0150* 0.0230 0.0168 0.0150 0.0161 0.0072* 0.0213 0.0231 0.0140*
0.0129 0.0164 0.0090* 0.0122 0.0111* 0.0067 0.0176 0.0228 0.0127*
0.0115 0.0108 0.0073* 0.0072* 0.0089 0.0022 0.0145 0.0137 0.0070*
0.0092 0.0088 0.0039* 0.0089 0.0056* 0.0028 0.0127 0.0114 0.0057*
0.0104 0.0093 0.0064* 0.0072 0.0056* 0.0022 0.0130 0.0112 0.0054*
0.0090 0.0063 0.0033* 0.0061 0.0044* 0.0028 0.0122 0.0083 0.0041*
0.0091 0.0071 0.0060* 0.0044 0.0028* 0.0011 0.0106 0.0093 0.0039*
0.0070 0.0053 0.0020* 0.0044 0.0039 0.0028* 0.0112 0.0065 0.0039*
0.0075 0.0040* 0.0040* 0.0028 0.0022* 0.0011 0.0096 0.0057 0.0036*
0.0061 0.0039 0.0014* 0.0039 0.0022* 0.0017 0.0093 0.0039 0.0021*
0.0044 0.0018* 0.0027 0.0017 0.0011* 0.0006 0.0075 0.0026 0.0013*
0.0039 0.0012 0.0010* 0.0033 0.0017 0.0006* 0.0073 0.0013* 0.0013*
Note: In conditional VaR-x and normal approaches, σ 2 are estimated by EWMA approach, which the decay factor λ is set to 0.94. ‘‘Long’’ and ‘‘short’’ indicate ‘‘long position’’ and ‘‘short position’’, respectively. * Indicates the predictive probability of margin violation is close to theoretical probability of margin violation.
Fig. 1. Actual returns vs. computed conditional margin levels of the long position for conditional VaR-x, normal, and Hill non-parametric approaches.
Fig. 2. Actual returns vs. computed conditional margin levels of the long position for conditional VaR-x, normal, and Hill non-parametric approaches.
5. Conclusions The previous empirical research agrees that using EVT to determine margin levels can fully protect against default resulting from extreme price movements. The main contribution of this study is to provide an approach, the VaR-x method based on EVT, that incorporates a modified version of the Hill estimator into a Student-t distribution, for setting the unconditional and conditional margin levels. This approach is especially appealing because the modified Hill estimator can avoid small sample bias and does not condition its tail observations as does the Hill estimator. Moreover, rather than using the Hill non-parametric approach developed by Cotter [13], this study uses the Student-t distribution, parameterised by the tail index estimated from the modified Hill estimator, to estimate the margin level. Empirical study involving three stock index futures contracts, the FTSE100, the Nasdaq100 and the Nikkei225, demonstrates the accuracy of the proposed approach.
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Fig. 3. Actual returns vs. computed conditional margin levels of the long position for conditional VaR-x, normal, and Hill non-parametric approaches.
Fig. 4. Actual returns vs. computed conditional margin levels of the short position for conditional VaR-x, normal, and Hill non-parametric approaches.
Fig. 5. Actual returns vs. computed conditional margin levels of the short position for conditional VaR-x, normal, and Hill non-parametric approaches.
The analysis of unconditional margin setting first demonstrated that behavior differs between the left and right tails of returns distribution. This suggests that different margin levels should be set for the long and short positions. Second, given the lower probabilities of margin violation, the unconditional VaR-x approach is a more appropriate means of setting margin level than the normal approach or the Hill non-parametric approach developed by Cotter [13]. Third, this study shows that the underestimation bias in margin levels that are computed assuming normality, whereas the Hill non-parametric approach has a phenomenon of overestimating margin level. In the analysis of conditional margin setting, this study evidences that conditional VaR-x approach compared with conditional normal and conditional Hill non-parametric approaches is better able to capture the nature of extreme events, particularly crucial in periods of greater fluctuation. In sum, the empirical evidence demonstrates that the unconditional and conditional VaR-x approaches offer more accurately forecast of margin levels and can capture extreme events, thus ensuring adequate prudence for the setting of initial and daily or maintenance margin levels.
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Fig. 6. Actual returns vs. computed conditional margin levels of the short position for conditional VaR-x, normal, and Hill non-parametric approaches.
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