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The behavior of extreme values in Germany's stock index futures: An application to intradaily margin setting
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH
ELSEVIER
European Journal of Operational Research 104 (1998) 393-402
Theory and Methodology
The behavior of extreme values in Germany's stock index futures: An application to intradaily margin setting John Paul Broussard a, G. Geoffrey Booth b,, a College of Business Administration, Fairleigh Dickinson University, 285 Madison Avenue, Madison, NJ 07940, USA b Department of Finance, Louisiana State University, 2163 CEBA Building, Baton Rouge, LA 70803, USA
Received 1 November 1995; accepted 1 October 1996
Abstract Recent defaults and large financial losses attributed to derivative security investing point to an area of finance not often researched, i.e. the probabilities of observing extreme occurrences. This paper examines this behavior for German stock index futures (FDAX) contracts. Its empirical results indicate that large FDAX intradaily price changes follow a Frrchet extreme value distribution and the extreme value distribution probabilities may be confidently used to help set intradaily margin levels. © 1998 Elsevier Science B.V. Keywords: Distributions; Frrchet; Extreme values; Futures; Margin; Germany
1. Introduction Research on the distributional form of speculative asset returns has a long history and is fundamental to normative investment decision-making criteria and asset pricing models. The preponderance o f research assumes normally distributed processes and concentrates on making central tendency inference statements such as " o n average, the...." Numerous studies, however, have documented that these distributions often have tails that are thicker than those o f a normal distribution. S k e w e d distributions have also been observed. This suggests that relying only on the mean and variance to describe the data generating
* Corresponding author. Email: [email protected].
process may be inappropriate, i Whatever the thrust of the research, however, most of the statistical investigations generally use the full spectrum of available observations. The purpose of this paper is to examine the distributional form of speculative asset prices by exploring the statistical nature of the largest intraday price changes (returns), regardless o f sign, o f the German stock index futures contract, the F D A X . The primary contribution o f this research is that it offers insights into the statistical properties o f outlying observations that are often thrown out o f time-series
I The list of studies is endless. For a recent study concerning the German DAX, the stock index underlying the FDAX, see Booth et al. (1995); see also Funke (1994). For earlier German stock studies see Akgiray et al. (1989a,b) and the citations therein.
0377-2217/98/$19.00 © 1998 Elsevier Science B.V. All fights reserved. Pll S0377-22 ! 7(97)00014-3
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and cross-sectional speculative asset pricing research. These insights are provided by applying extreme value theory to estimate the probability of these 'outliers' occurring. F r o m the statistical analysis, decision-making implications concerning intradaily margin setting on G e r m a n y ' s stock index futures market are given. 2 The plan of this paper is as follows. Section 2 discusses extreme value theory and the method of analysis for the particular application examined herein. Section 3 describes the data used and presents results of the extreme value statistical tests. Section 4 examines the institutional environment surrounding F D A X contracts and offers some suggestions on using the extreme value results for the intraday margin setting decision. A summary is provided in Section 5.
2. Extreme value theory Fisher and Tippett (1928) derived three asymptotic distributions that describe extreme value behavior. 3 These distributions provide solutions to determining this behavior for all data generation processes and are useful because the distributional form of extreme values is the same as the tail behavior o f the parent distribution. Some general concepts related to the three distributional forms follow. Consider a sequence o f stationary i.i.d, random variables X I, X 2 . . . . . X. with some probability density function F. To determine the probability that the maximum
M, = max ( X t , X 2 . . . . . X,)
(1)
value of the first n variables is below a certain level
2 Statistical applications to margin setting typically assume that the distribution of futures price changes is Gaussian (e.g. see Figlewski, 1984 and Gay et al., 1986). Although Warsharsky (1989) points out that the normality assumption is inappropriate, very few efforts have been made to seek other alternatives. Examples of these efforts, which use extreme value theory, include Kofman and de Vries (1989), Kofman (1993) and Longin (1995). 3 The work of Fisher and Tippett (1928) followed the contribution of Bortkiewicz (1922) to the problem of quantifying extreme value probabilities and independently replicated the asymptotic distribution of Fr~chet (1927).
x, the cumulative probability relationship may be stated as:
Pr{ M,
(2)
Extreme value theory concentrates on studying the asymptotic distributional properties o f the scaled order statistic M,. Its objective is to find an appropriate situation where the following is satisfied: P r { a , ( M, - O,) < x} -~ G ( x ) ,
(3)
or in terms o f F : W
F"( x / a , + b,) ~ G( x ) . G(x) is one of Fisher and Tippett's three distributional forms (Fisher and Tippett, 1928); a , measures dispersion; b, is a location parameter; and w denotes ' t o weakly converge'. If there is a situation that satisfies Eq. (3), then F belongs to the domain o f attraction o f G, where G takes one of the following three forms: Typel: Type If:
G(x) G(x)
and Type 1II: G(x)
= = =
e x p ( - e -x) 0 e x p ( - x -~)
for for for
-oo0;
= =
e x p ( - ( - x) y) 1
for
x < 0,
for x ~ 0.
(4) The Type I (Gumbel) specification represents cases where the tail behavior o f the parent distribution approaches its asymptote exponentially, e.g. the Gaussian distribution. The Type II (Fr~chet) distribution denotes cases where the tail o f the underlying distribution does not decay rapidly, i.e. thick-tailed. The Type III (Weibull) limiting result describes extreme value behavior of truncated data generation processes. The underlying assumption in most financial research employing extreme value theory is that extreme asset returns exhibit Type II limiting behavior (see Koedijk et al., 1992, Jansen and de Vries, 1991, Kofman and de Vries, 1989, and Kofman, 1993). 4
4 Some useful features of extreme value theory for financial economic applications is that the limiting distributions are robust to the presence of serial correlation (Berman, 1963) and ARCH effects (de Haan et al., 1989). Also, the limiting distributions hold in a mixture of distributions environment (Leadbetter et al., 1983).
J.P. Broussard, G.G. Booth/European Journal of Operational Research 104 (1998) 393-402
The linchpin of this paper's analysis is the determination of which of the three distributional forms corresponds to FDAX extremal intraday price changes. A solution to this problem was first proposed by Jenkinson (1955) and reparameterized to a more general framework by Maritz and Munro (1967). Longin (1995, 1996) was the first to apply the generalized extreme value technique to financial time series data by examining the extremal behavior of indexes of the most actively traded U.S. stocks. According to Maritz and Munro (1967), Eq. (4) can be operationalized to estimate empirically the cumulative probability distributions for the negative ( F rain) and positive ( F max) extreme values by (notation not theirs): f min~-- 1 - exp [ - ( 1 + Tminx) l/~rmi"]
(5)
and Fmax = exp [ - ( 1 -
"rmaxx)l/rm'x],
(6)
where x represents an extreme observation and may be defined as ( r - / 3 ) / a , where r in this analysis denotes the minimal or maximal price changes in a specified time period and where a and /3 designate the corresponding dispersion and location parameters, respectively. Thus, in this form x = ( r - / 3 ) / a may be thought of as a deviation of r from its location standardized by its dispersion. The tail index r determines the type of distribution from which extreme values are drawn, and it may be either less than, equal to, or greater than zero. When ~-= 0, the limiting distribution corresponds to the Type I distribution. When r < 0, the limiting distribution signifies a Type II distribution and when r > 0, the limiting distribution is Type III. As Gumbel (1958) suggests, a straightforward way to obtain estimates of a, /3 and ~- is to estimate the following equations using standard non-linear least squares regression: --'n [--'n (N---~) ] -
1 zm--q-;/n a "~n 1
~.-~'ffnf n [ a ~n
_
T min
X(/3"~"-rmin)] +u.
(7)
395
and
--
1
/ n o t max T max 1
,/.max
•/ n [ amax. - "rmax z(rmax--/3max)] q-U m,
(8)
where m is the rank assigned to each return after sorting all observations from lowest to highest and N is the number of observations for either the extreme positive or negative returns. Kinnison (1985) notes that the goal of this regression is to fit the expected cumulative probability frequencies to their empirical extreme value. An alternative to this approach is to estimate the parameters of Eq. (5) and Eq. (6) using the maximum likelihood method as described by Tiago de Oliveira (1973). Both these parametric methods provide consistent estimates. Indeed, Longin (1996) reports in his analysis of the tail shape properties of the returns of NYSE stocks that the two methods provide similar results, although the asymptotic standard errors for the regression method are somewhat higher. Nevertheless, as is demonstrated in the empirical exercise that follows, the parameter p-values are generally either very small or very large, thereby negating any potential interpretational difficulties. 5
3. D a t a
and
results
Tick-by-tick data are provided by Germany's screen-based futures market, the Deutsche
s Non-parametrictechniquesto analyzeextremevalue behavior exist. Examples include those developed by Hill (1975) and Pickands (1975). These,however,only focus on tail shape. Moreover, the Hill (1975) approach, which seems to be the most frequentlyused in financialeconomics,is limited only to Fr6chet distributions. In their studies of futures margin setting Kofman and de Vries (1989) and Kofman(1993) employ the Hill estimator. Longin (1995), however, relies on non-linear least squares regression.
J.P. Broussard, G.G. Booth / European Journal of Operational Research 104 (1998) 393-402
396
Table 1 Estimated parameters for minimum intradaily price changes Contract
N
Most negative r nan (%)
~'min(tailindex)
t~m*n (dispersion)
/3min (location)
R2
Jun. 1992 Sep. 1992 Dec. 1992 Mar. 1993 Jun. 1993 Sep. 1993 Dec. 1993 Mar. 1994 Jun. 1994 Sep. 1994
115 179 186 187 174 175 174 165 176 162
- 1.527 -4.684 - 4.788 -4.687 -4.341 -2.180 -2.157 - 4.034 -3.226 -3.478
-0.206 -0.341 - 0.229 -0.210 -0.202 0.003 * 0.031 * * - 0.140 -0.097 -0.108
0.189 0.291 0.397 0.359 0.312 0.365 0.412 0.462 0.503 0.456
-0.359 -0.395 - 0.504 -0.589 -0.458 -0.488 -0.519 - 0.602 -0.610 -0.661
0.989 0.990 0.989 0.989 0.986 0.994 0.985 0.996 0.991 0.996
The most negative r nan is the minimum intradaily return observed for each contract and N is the number of observations. All t~min (/3 nan) values are significantly positive (negative) at p-values < 0.0001. Except for two instances the ~.nan values are significantly negative (p-value < 0.0001). The exceptions are highlighted with asterisks, with * denoting positive but not significant (p-value = 0.719) and * * indicating positive and significant (p-value = 0.025). In the non-linear context, R2 is used solely as a descriptive statistic.
T e r m i n b ~ r s e ( D T B ) , a n d c o v e r all c o n t r a c t s w i t h at l e a s t six m o n t h s o f t r a d i n g d a t a f r o m J a n u a r y 1992 t h r o u g h S e p t e m b e r 1994. T h i s s e l e c t i o n p r o c e s s all o w s f o r t h e a n a l y s i s o f all f u t u r e s c o n t r a c t s f r o m J u n e 1992 to S e p t e m b e r 1994. D u r i n g this t i m e p e r i o d , t h e D A X a c h i e v e d l o w s o f a r o u n d 1400 a n d highs around 2300. Toward the end of the time p e r i o d a n a l y z e d , the D A X w a s t r a d i n g at a l e v e l around 2000. W h e n a p p l y i n g e x t r e m e v a l u e t h e o r y to a n y set o f data, a n i m p o r t a n t p r a c t i c a l i s s u e to b e r e s o l v e d is the f r e q u e n c y o f m e a s u r e m e n t . T h e s e l e c t i o n m a y
a f f e c t t h e m a g n i t u d e s o f a a n d / 3 b u t n o t ~', the last b e i n g i n d e p e n d e n t o f t e m p o r a l a g g r e g a t i o n (Feller, 1971). K i n n i s o n (1985), h o w e v e r , states t h a t the m e a s u r e m e n t i n t e r v a l s s h o u l d o c c u r at e q u a l d i s t a n t p o i n t s in time. T h e s e l e c t i o n p r o c e d u r e o f c h o o s i n g d a i l y m a x i m a a n d m i n i m a r e s u l t s in a n a t u r a l a n d c o n s i s t e n t i n t e r v a l , s i n c e t h e f o c u s o f this a p p l i c a t i o n is o n i n t r a d a y m a r g i n setting. O n l y t r a d i n g d a y s w i t h more than two transactions and having both positive a n d n e g a t i v e p r i c e c h a n g e s are c o n s i d e r e d . T h e s e d a t a w e r e c l e a n e d o f m i s t r a d e s , w h i c h are t r a n s a c t i o n s w h e r e s o m e e r r o r w a s m a d e in e n t e r i n g the
Table 2 Estimated parameters for maximum intradaily price changes Contract
N
Most positive r max (%)
~'max(tailindex)
a max (dispersion)
/3 max (location)
R2
Jun. 1992 Sep. 1992 Dec. 1992 Mar. 1993 Jun. 1993 Sep. 1993 Dec. 1993 Mar. 1994 Jun. 1994 Sep. 1994
115 179 186 187 174 175 174 165 176 162
1.707 2.906 2.690 2.424 3.705 2.813 2.747 3.706 3.546 2.845
-0.202 -0.269 - 0.017t 0.035t1 -0.154 -0.141 -0.072 -0.087 -0.112 -0.068
0.212 0.277 0.446 0.414 0.299 0.315 0.367 0.455 0.387 0.421
0.365 0.375 0.496 0.607 0.472 0.448 0.494 0.607 0.619 0.647
0.983 0.990 0.986 0.995 0.977 0.995 0.996 0.997 0.996 0.996
The most positive r max is the maximum intradaily return observed for each contract and N is the number of observations. All a max and /3 max values are significantly positive at p-values < 0.0001. Except for two cases the 1"max values ale significantly negative (p-value < 0.0001). The exceptions are highlighted by daggers, with t denoting negative but not significant (p-value = 0.167) and t t indicating positive and significant (p-value ~ 0.001). In the non-linear context, R 2 is used solely as a descriptive statistic.
J.P. Broussard, G.G. Booth/European Journal of Operational Research 104 (1998) 393-402
trade. Mistrade information was provided directly by the DTB. To examine maximum and minimum intradaily price changes, all possible combinations of intradaily price changes are calculated. Price changes are obtained taking the natural logarithm of a later transaction's price level and subtracting the natural logarithm of an earlier transaction's price level (In Pt In P t - b ) and multiplying by 100. After all possible combinations of transactional price changes are calculated, a maximum and a minimum for each day are selected. 6 The basic empirical results are displayed in Tables 1 and 2. Table 1 provides the results of using Eq. (7) for the minimum intradaily returns. For eight of the 10 contracts, ~.n~n values are significantly negative (p-values <0.0001). The ~.~n values for September and December 1993 contracts are positive but the former is not significantly different from zero (p-value = 0.719). Table 2 gives the results of applying Eq. (8) to the largest positive intradaily price changes. Except for the December 1992 and March 1993 contracts, z ~x estimates are negative and significant (p-values < 0.0001). The r ~ x for the former contract is negative but not significant (p-value = 0.167). In contrast, the ~.r~x for the latter is positive and significant ( p - v a l u e < 0 . 0 0 0 1 ) . The overall goodness-of-fit is highlighted by the very high R2s for Eq. (7) and Eq. (8) for every contract. Therefore, similar to the results found in Longin (1995, 1996) and implied by Jansen and de Vries (1991), Kofman and de Vries (1989), and Kofman (1993), the extreme values resulting from both increases and decreases in FDAX contract prices generally follow a Frrchet limiting distribution. The tails of two contracts, however, follow a Gumbel limiting distribution, whereas two are described by a Weibull distribution. 7 One characteristic revealed in Tables 1 and 2 is that the tail index parameter estimates are
6 The intradaily price changes are calculated using SAS' PROC IML routine to create dynamic vector sizes. Each daily maximum and minimumis chosen from an (T(T - 1)/2)× 1 vector where T is the number of transactions occurring in that day. 7 The T-valuesfor the two Weibull distributions are nearly zero, indicating that for these contractsthe degree of truncation is small. If binding price limits were instituted, more evidence of truncation should occur.
397
achieving smaller magnitudes, which is indicative of tail behavior tending towards thinner tails.
4. Implications for intradaily margin setting The difficulty of deciding on an appropriate margin level is highlighted by Telser (1981a) when he states "[t]here is no formula that can determine the optimal size of the margin." As pointed out by Baer et al. (1994), among others, the appropriate margin depends upon many factors and trade-offs are involved. Some of these factors may be related to the risk aversion level of the participants (e.g. see Telser, 1981b and Hunter, 1986) or the volatility of the underlying asset of the futures contract (e.g. see Hull, 1993 and Tucker, 1991). Additionally, Gay et al. (1986) and Rutz (1988) suggest that price levels, volume, and levels of speculation and hedging positions may be potential factors affecting the appropriate margin. Since arriving at an appropriate margin is a multi-faceted process, Brenner (1981) states that margins are optimally set by a designated margin committee similar to the one found at the DTB. Since one underlying objective of setting a margin is to protect the exchanges integrity from trader default due to adverse market movements, an important ingredient into the decision-making process is the probability that a specified margin will be violated. To illustrate the use of the information provided by the above statistical analysis to measuring this probability, this section applies the analysis to setting intradaily margin levels for FDAX contracts. Before doing so, however, it is helpful to briefly describe the institutional environment. The central organization for trading and clearing both options and futures transactions in Germany is the DTB. All DTB trading members fit into one of three categories: (1) General Clearing Member (GCM), (2) Direct Clearing Member (DCM), and (3) Non-Clearing Member (NCM). GCMs are traders who can clear their own trades, client trades, and transactions from NCMs. DCMs can settle transactions only for their own or clients' accounts. NCMs must settle their transactions through a GCM. GCMs and DCMs are directly responsible to the DTB for all obligations they have agreed to clear. NCMs and
398
J.P. Broussard, G.G. Booth~European Journal of Operational Research 104 (1998) 393-402
clients o f m e m b e r firms, however, have no direct responsibility to the DTB. To b e c o m e a D T B trading m e m b e r , or exchange participant, and gain access to the D T B trading system, there are several r e q u i r e m e n t s that need to be met. Generally, the m e m b e r firm m u s t be an operating b u s i n e s s e s t a b l i s h m e n t with m e a n i n g f u l e x c h a n g e business c o n d u c t e d on the D T B . There are restrictions and basic training qualifications that m u s t be m e t by the participants' personnel. Since the D T B is a totally electronic e x c h a n g e , participants m u s t equip themselves with D T B authorized hardware and software. D u r i n g the time period a n a l y z e d in this paper, the D T B e m p l o y e d daily m a r g i n settlement procedures that arrived at the appropriate m a r g i n r e q u i r e m e n t by netting positions within 21 m a r g i n classes. These classes are m a i n t a i n e d to lessen the potential impacts o f extreme price changes in i n d i v i d u a l assets. The D A X m a r g i n class c o n t a i n s the F D A X contracts, the O D A X (Options on the D A X contracts), and the
O F D X ( O p t i o n s on the F D A X contracts). The initial daily m a r g i n for a single F D A X contract (long or short) is 130 index points, which is equivalent to D M 13,000 (130 multiplied by 100, the F D A X contract size). T h u s d u r i n g the period o f analysis, the margin, in percentage terms (130 d i v i d e d by the D A X value and multiplied by 100), r a n g e d from 5.6% to 9.2%. At the e n d o f the period the effective m a r g i n was a p p r o x i m a t e l y 6.5%. It should be noted that the D T B recently l o w e r e d the daily m a r g i n r e q u i r e m e n t for F D A X contracts to 100 index points ( D M 10,000) per contract. A t a D A X level of 2000, this n e w level is e q u i v a l e n t to a 5% initial margin. As stated in the April 1995 D T B Reporter, the m a i n reason for the reduction in m a r g i n requirements was a believed decrease in volatility o f the F D A X contract prices. G i v e n the i m p o r t a n c e of the D T B in the G e r m a n (and E u r o p e a n ) capital markets, it is appropriate to consider setting levels on an intradaily rather than the current daily basis. Intradaily m a r g i n levels m a y be beneficial d u r i n g periods of high market volatil-
Table 3 lntradaily price changes given margin violation probabilities for the June 1993 contract Margin violation probabilities
Intradaily price changes (%) Short futures position
Long futures position 0.100 0.050 0.040 0.030 0.020 0.010 0.005 0.004 0.003 0.002 0.001 0.0005 0.0004 0.0003 0.0002 0.0001
Estimated distribution
Empirical distribution
Estimated distribution
Empirical distribution
-
-
1.277 1.599 1.709 1.855 2.072 2.474 2.920 3.074 3.280 3.586
1.215 1.498 !.559 1.571 1.599 2.008 3.705 3.705 3.705 na na na na na na na
1.349 1.730 1.863 2.043 2.314 2.830 3.422 3.631 3.914 4.342
- 5.160
-
6.100 6.432 6.883 7.564 8.867
1.320 1.610 1.641 1.799 2.062 2.147 4.341 4.341 4.341 na na na na na na na
4.156
4.789 5.007 5.300 5.736 6.547
The margin violation probabilities are F rain and 1 - F max for the long and short positions, respectively. Various probabilities are provided. Using the estimated extreme value distribution parameters, the intraday price change associated with each margin violation probability for each position is calculated. Where possible, the corresponding price changes derived from the empirical distribution are also provided. Note that, according to Table 1 and Table 2, the most negative price change is -4.341% and the most positive is 3.705%, respectively.
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ity. Having procedures in place to institute intradaily margin calls may also have the effect of lowering the daily margin requirement because of the ability to shore-up futures positions within the day. The decreased margin requirement therefore could increase trader participation and hence the liquidity of the market. The results of Section 3 provide useful input into setting a desired intraday margin level. Consider again Tables 1 and 2. The values for the most negative price changes range from - 1 . 5 2 7 % to - 4 . 7 8 8 % with a mean of - 3 . 5 1 0 % . The values of the most positive price changes are more clustered with a range of 1.707% to 3.706% and a mean of 2.909%. These percentages indicate that the daily initial margin target of 6.5% (and the 5% margin level just recently set) is quite conservative in an intradaily context. Apparently, no futures position was subject to a margin call, at least not on the first day of the transaction. Additional insights into the issue may be gained by directly examining the probability of a margin violation for long and short futures positions. A long (short) position's risk of margin violation is associated with negative (positive) returns. Thus, for a long position, the probability that a specified margin is violated is F ran, which is defined by Eq. (5). In contrast and as a result of being concerned with the other tail of the distribution, the margin violation probability for a short position is 1 - F m~x, where F m~ is given by Eq. (6). In either case, margin violation probability increases as ct increases, 1/31 increases and ~- becomes more negative. Table 3 contains the results for margin violation probability analyses for long and short futures positions. Only one contract (June 1993) is presented for brevity. Interpretation of Table 3 is as follows. The first column contains various margin violation probabilities. The second (fourth) column presents the return for the long (short) position calculated using a specific margin violation probability and the appropriate estimated values for a , fl and ~-. The absolute value of the return is equivalent to the margin requirement. The long (short) position return derived from the empirical distribution is provided in the third (fifth) column for benchmark purposes. A comparison of the second and third columns and fourth and fifth columns indicates that the data points de-
399
Table 4 Margin violation probabilities for various margin levels associated with the June 1993 contract's estimated distribution Margin (%) Marginviolation probabilities
10.00 5.00 4.00 3.00 2.00 1.00 0.50 0.40 0.30 0.20 0.10 0.05 0.04 0.03 0.02 0.01
Long futures position
Shortfutures position
Estimated distribution
Estimated distribution
0.000 0.001 0.003 0.008 0.032 0.202 0.584 0.702 0.819 0.916 0.975 0.990 0.992 0.993 0.995 0.996
0.000 0.000 0.00 I 0.004 0.023 0.189 0.598 0.721 0.838 0.930 0.981 0.992 0.994 0.995 0.996 0.997
The margin violation probabilities are F mjn and 1 - F max for the long and short positions, respectively. Various margins (i.e. the absolute value of the intraday price changes) are provided. Using the estimated extreme value distribution parameters, the margin violation probability associated with each price change for each position is calculated.
rived from the estimated distributions closely track those of the corresponding empirical distributions. 8 Two examples serve to illustrate the content of Table 3. For a long futures position associated with a margin violation probability of 0.001, the intraday price change is - 5 . 1 6 0 % , assuming the estimated Frrchet distribution. In other words, for every one out of 1000 trading days (about once in four years), an intradaily price decline of at least 5.160% is expected. Turning to a short futures example, the intradaily price increase associated with the same margin violation probability is 4.156%. The probability of a 5% price increase is approximately 0.0004, less than half that of price decrease of equal magni-
s The empirical distribution for minimal (maximal) price changes cannot be used to assess the probability of violating a margin that is greater than 4.341% (3.705%) for a long (short) position. This is because the requisite empirical observations have not been experienced.
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Table 5 Margin violation probabilities for a 3% margin for all contracts
Contract
Jun. 1992 Sep. 1992 Dec. 1992 Mar. 1993 Jun. 1993 Sep. 1993 Dec. 1993 Mar. 1994 Jun. 1994 Sep. 1994
Margin violation probabilities using the estimated distributions Long futures position
Short futures position
0.001 0.016 0.020 0.015 0.008 0.001 0.001 0.015 0.020 0.017
0.002 0.009 0.005 0.002 0.004 0.005 0.004 0.001 0.009 0.009
The margin violation probabilities are F nan and 1 - F mx for the long and short positions, respectively.
tude. Expressed in terms of time. 0.0004 is equivalent to once in every 10 years. Table 4 recasts Table 3 to be more 'user friendly' to the decision maker. In this table the margins are specified and the margin violation probability is calculated. An example for a long futures position shows that for a 3% margin (a - 3 % price change) there is a 0.008 probability of a margin violation. The corresponding probability for a short position is 0.004. In terms of margin violation frequency, these probabilities convert to eight violations per 1000 trading days (twice a year) and four violations per 1000 trading days (once a year) for the long and short positions, respectively. To extend this example, Table 5 provides the margin violation probabilities associated with a 3% margin for all contracts. Two points are noteworthy. First, the probabilities for both long and short positions vary considerably among contracts. For the long (short) position the range is from 0.001 (0.001) to 0.020 (0.009). Second, the margin violation probabilities appear to be asymmetric, with those associated with a long position often being greater than those with a short position. For instance, as pointed out above, for the June 1993 contract, the probability of violating a 3% margin is twice as likely for a long position than for a short position. There are, however, cases (e.g. the September 1993 contract) where the reverse obtains. To incorporate the notion of margin violation
probability into the margin setting process, the DTB must address several related issues. First, it must determine the amount of risk that it is willing to bear, with the time until a margin requirement is violated being an intuitive way of expressing this risk tolerance. Second, the DTB must express its margin requirement in percentage terms rather than in customary points to permit the margin to correspond to price changes. 9 Third, it must decide whether to impose different margins for short and long positions, since the extant analysis shows that the risk is asymmetric. Finally, the DTB must recognize that the extremal risk characteristics vary among contracts. One possible scenario that integrates the above issues is the following. The DTB concludes that is willing to accept the possibility of a margin violation occurring once every 50 trading days (approximately two months) and it never wants to bear more risk. Further, the DTB decides that different margins for long and short positions may be unnecessarily administratively cumbersome, thereby opting for a single margin requirement. Finally, they adopt a percentage margin scheme. Under these conditions, according to the figures given in Table 5, the DTB should set the margin to be 3% for short and long positions. For many contracts and positions this will provide the DTB with greater protection than desired but it will never (according to the historical record) provide less.
5. Summary The purpose of this paper was to describe the statistical properties of FDAX extreme intradaily price movements and to examine the ability of extreme value theory to provide reliable information on the incidence of intradaily margin violation probabilities. The statistical results of this study indicate that the intradaily extremes in Germany's FDAX con-
9 Expressing the margin in percentage ratber than points has the additional benefit of keeping the margin risk constant even though the value of the FDAX changes. For instance, in a rising market, all other things being equal, the possibility of a margin violation increases if the point convention is used.
J.P. Broussard. G. G. Booth~European Journal of Operational Research 104 (1998) 393-402
tracts generally follow a Type II (Frrchet) limiting extreme value distribution, thereby confirming the results of previous research concerning the distribution of futures' price changes. Moreover, the results indicate that extreme value theory should provide useful information to the D T B ' s margin setting committee. In particular, the extreme value technique generates probabilities of incurring abrupt intradaily price changes and should be included in the decision-making calculus used to set margin requirements.
Acknowledgements This paper was inspired by Francois Longin and the other participants at the Conference on Multivariate Extreme Value Estimation with Applications to Economics and Finance, Erasmus University, Rotterdam, May 2 6 - 2 8 , 1994. The authors thank Otto Loistl of Wirtschaftsuniverist~it Wien and formerly of the Deutsche B~irse, JiSrg Franke, Kerstin Miiller, Dirk Schermoly and Anselm Jumpertz of the DTB for providing data and vital information related to DTB operations. The authors also thank two anonymous reviewers for their helpful comments. All errors belong to the authors.
References Akgiray, V., Booth, G.G., and Loistl, O. (1989a), "German stock market's resiliency to world-wide panics", Zeitschrifl fdr Betriebswirtschafl 59, 968-978. Akgiray, V., Booth, G.G., and Loistl, O. (1989b), "Statistical models of German stock returns", Journal of Economics l, 17-33. Baer, H.L., France, V.G., and Moser, J.T. (1994), "Opportunity cost and prudentiality: An analysis of futures clearinghouse behavior", Policy Research Working Paper 1340, The World Bank Policy Research Department. Berman, S.M. (1963), "Limiting theorems for the maximumterm in stationary sequences", Annals of Mathematical Statistics 35, 502-516. Booth, G.G., Chowdhury, M., Harem, J.J., and Loistl, O. (1995), "The nature of predictability of German stock returns", in: D.K. Ghosh (ed.), New Advances in Financial Economics, Pergamon Press, Oxford. Bortkiewicz, L. yon. (1922), "Variationshreite mad mittlerer Fehler", Sitzungsberichte den Berliner Mathematik Gesellschaft 21, 3- i 1.
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Brenner, T.W. (1981), "Margin authority: No reason for a change", The Journal of Futures Markets 1,487-490. de Haan, U, Resnick, I.S., Rootz~nH., and de Vries, C.G. (1989), "Extremal behavior of solutions to a stochastic difference equation with applications to ARCH process~', Stochastic Processes and their Applications 32, 213-224. Feller, W. (1971), An Introduction to Probability Theory and Applications 2, Wiley, New York. Figlewski, S. (1984), "Margins and market integrity: Margin setting for stock index futures and options", The Journal of Futures Markets 4, 385-416. Fisher, R.A., and TippeR, L.H.C. (1928), "Limiting forms of the frequency distributionof the largest and smallest member of a sample", Proceedings, Cambridge Philosophy Society 24/2, 180-190. Frrcbet, M. (1927), "Sur la loi de probabilit6 de l'rcart maximim", Annuals de la Soci~t~ polonaise de Math~matiques 6, 93-116. Fmake, M. (1994), "Testing for nonlinearity in daily German stock returns", Allegemeines Statistisches Archiv 78, 281292. Gay, G.D., Hunter,W.C., and Kolb, R.W. (1986) "A comparative analysis of futures contract margins", The Journal of Futures Markets 6, 307-324. Gumbel, E.J. (1958), Statistics of Extremes, Columbia University Press, New York. Hill, B.M. (1975), "A simple general approach to inferenceabout the tail of a distribution", Annals of Statistics 3, 1163-1173. Hull, J.C. (1993), Options, Futures and Other Derivative Securities, Prentice Hall, EnglewoodCliffs. Hunter, W.C. (1986), "Rational margins on futures contracts: Initial margins", Review of Research in Futures Markets 5, 160-173. Jansen, D.W., and de Vries, C.G. (1991), "On the frequency of large stock returns: Putting booms and busts into perspective", The Review of Economics and Statistics 73, 18-24. Jenkinson,A.F. (1955), "The frequencydistributionof the annual maximum (or minimum)values of meteorological elements", Quarterly Journal of the Royal Meteorology Society 7, 145158. Kinnison, R.R. (1985), Applied Extreme Value Statistics, Battelle Press, Columbus. Koedijk, K.G., Stork, P.A., and de Vries, C. G, (1992), "Differences between foreign exchange rate regimes: The view from the tails", Journal of International Money and Finance 11,462-473. Kofman, P., and de Vries, C.G. (1989) "Potato futures returns: A tail investigation", Review of Futures Markets 8, 244-258. Kofman, P. (1993), "Optimizing futures margins with distribution tails", Advances in Review of Futures Markets 6, 263-278. I_,eadbetter, M.R., Lindgren, G., and Rootz~n, H. (1983), Extremes and Related Properties of Random Sequences and Processes, SpringerVerlag, New York.
Longin, F. (1995), "Optimal margins in futures markets: A parametric extreme-based approach", Proceedings, Ninth Chicago Board of Trade Conference on Futures and Options,
Bonn.
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Longin, F. (1996), "The asymptotic distribution of extreme stock market returns", Journal of Business 69, 383-408. Maritz, J.S., and Munro, A.H. (1967), "'On the use of the generalized extreme-value distribution in estimating extreme percentiles", Biometrics 23, 79-103. Pickands, J. (1975), "'Statistical inference using extreme order statistics", Annals of Statistics 3, 119-131. Rutz, R.D. (1988), "Clearance, payment, and settlement systems in the futures, options and stock markets", The Review of Futures Markets 7, 346-370. Telser, L.G. (1981a), "Margins and futures contracts", The Journal of Futures Markets 1/2, 225-253.
Telser, L.G. (1981b), "Why there are organized futures markets", The Journal of Law and Economics, 1-22. Tiago de Oliveira, J. (1973), Statistical Extremes - A Survey, Center of Applied Mathematics, Faculty of Sciences, Lisbon. Tucker, A.L. (1991), Financial Futures, Options, and Swaps, West Publishing, New York. Warsharsky, M.J. (1989), "The adequacy and consistency of margin requirements: The cash, futures, and options segments of the equity markets", The Review of Futures Markets 8, 420-437.
ELSEVIER
European Journal of Operational Research 104 (1998) 393-402
Theory and Methodology
The behavior of extreme values in Germany's stock index futures: An application to intradaily margin setting John Paul Broussard a, G. Geoffrey Booth b,, a College of Business Administration, Fairleigh Dickinson University, 285 Madison Avenue, Madison, NJ 07940, USA b Department of Finance, Louisiana State University, 2163 CEBA Building, Baton Rouge, LA 70803, USA
Received 1 November 1995; accepted 1 October 1996
Abstract Recent defaults and large financial losses attributed to derivative security investing point to an area of finance not often researched, i.e. the probabilities of observing extreme occurrences. This paper examines this behavior for German stock index futures (FDAX) contracts. Its empirical results indicate that large FDAX intradaily price changes follow a Frrchet extreme value distribution and the extreme value distribution probabilities may be confidently used to help set intradaily margin levels. © 1998 Elsevier Science B.V. Keywords: Distributions; Frrchet; Extreme values; Futures; Margin; Germany
1. Introduction Research on the distributional form of speculative asset returns has a long history and is fundamental to normative investment decision-making criteria and asset pricing models. The preponderance o f research assumes normally distributed processes and concentrates on making central tendency inference statements such as " o n average, the...." Numerous studies, however, have documented that these distributions often have tails that are thicker than those o f a normal distribution. S k e w e d distributions have also been observed. This suggests that relying only on the mean and variance to describe the data generating
* Corresponding author. Email: [email protected].
process may be inappropriate, i Whatever the thrust of the research, however, most of the statistical investigations generally use the full spectrum of available observations. The purpose of this paper is to examine the distributional form of speculative asset prices by exploring the statistical nature of the largest intraday price changes (returns), regardless o f sign, o f the German stock index futures contract, the F D A X . The primary contribution o f this research is that it offers insights into the statistical properties o f outlying observations that are often thrown out o f time-series
I The list of studies is endless. For a recent study concerning the German DAX, the stock index underlying the FDAX, see Booth et al. (1995); see also Funke (1994). For earlier German stock studies see Akgiray et al. (1989a,b) and the citations therein.
0377-2217/98/$19.00 © 1998 Elsevier Science B.V. All fights reserved. Pll S0377-22 ! 7(97)00014-3
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and cross-sectional speculative asset pricing research. These insights are provided by applying extreme value theory to estimate the probability of these 'outliers' occurring. F r o m the statistical analysis, decision-making implications concerning intradaily margin setting on G e r m a n y ' s stock index futures market are given. 2 The plan of this paper is as follows. Section 2 discusses extreme value theory and the method of analysis for the particular application examined herein. Section 3 describes the data used and presents results of the extreme value statistical tests. Section 4 examines the institutional environment surrounding F D A X contracts and offers some suggestions on using the extreme value results for the intraday margin setting decision. A summary is provided in Section 5.
2. Extreme value theory Fisher and Tippett (1928) derived three asymptotic distributions that describe extreme value behavior. 3 These distributions provide solutions to determining this behavior for all data generation processes and are useful because the distributional form of extreme values is the same as the tail behavior o f the parent distribution. Some general concepts related to the three distributional forms follow. Consider a sequence o f stationary i.i.d, random variables X I, X 2 . . . . . X. with some probability density function F. To determine the probability that the maximum
M, = max ( X t , X 2 . . . . . X,)
(1)
value of the first n variables is below a certain level
2 Statistical applications to margin setting typically assume that the distribution of futures price changes is Gaussian (e.g. see Figlewski, 1984 and Gay et al., 1986). Although Warsharsky (1989) points out that the normality assumption is inappropriate, very few efforts have been made to seek other alternatives. Examples of these efforts, which use extreme value theory, include Kofman and de Vries (1989), Kofman (1993) and Longin (1995). 3 The work of Fisher and Tippett (1928) followed the contribution of Bortkiewicz (1922) to the problem of quantifying extreme value probabilities and independently replicated the asymptotic distribution of Fr~chet (1927).
x, the cumulative probability relationship may be stated as:
Pr{ M,
(2)
Extreme value theory concentrates on studying the asymptotic distributional properties o f the scaled order statistic M,. Its objective is to find an appropriate situation where the following is satisfied: P r { a , ( M, - O,) < x} -~ G ( x ) ,
(3)
or in terms o f F : W
F"( x / a , + b,) ~ G( x ) . G(x) is one of Fisher and Tippett's three distributional forms (Fisher and Tippett, 1928); a , measures dispersion; b, is a location parameter; and w denotes ' t o weakly converge'. If there is a situation that satisfies Eq. (3), then F belongs to the domain o f attraction o f G, where G takes one of the following three forms: Typel: Type If:
G(x) G(x)
and Type 1II: G(x)
= = =
e x p ( - e -x) 0 e x p ( - x -~)
for for for
-oo
= =
e x p ( - ( - x) y) 1
for
x < 0,
for x ~ 0.
(4) The Type I (Gumbel) specification represents cases where the tail behavior o f the parent distribution approaches its asymptote exponentially, e.g. the Gaussian distribution. The Type II (Fr~chet) distribution denotes cases where the tail o f the underlying distribution does not decay rapidly, i.e. thick-tailed. The Type III (Weibull) limiting result describes extreme value behavior of truncated data generation processes. The underlying assumption in most financial research employing extreme value theory is that extreme asset returns exhibit Type II limiting behavior (see Koedijk et al., 1992, Jansen and de Vries, 1991, Kofman and de Vries, 1989, and Kofman, 1993). 4
4 Some useful features of extreme value theory for financial economic applications is that the limiting distributions are robust to the presence of serial correlation (Berman, 1963) and ARCH effects (de Haan et al., 1989). Also, the limiting distributions hold in a mixture of distributions environment (Leadbetter et al., 1983).
J.P. Broussard, G.G. Booth/European Journal of Operational Research 104 (1998) 393-402
The linchpin of this paper's analysis is the determination of which of the three distributional forms corresponds to FDAX extremal intraday price changes. A solution to this problem was first proposed by Jenkinson (1955) and reparameterized to a more general framework by Maritz and Munro (1967). Longin (1995, 1996) was the first to apply the generalized extreme value technique to financial time series data by examining the extremal behavior of indexes of the most actively traded U.S. stocks. According to Maritz and Munro (1967), Eq. (4) can be operationalized to estimate empirically the cumulative probability distributions for the negative ( F rain) and positive ( F max) extreme values by (notation not theirs): f min~-- 1 - exp [ - ( 1 + Tminx) l/~rmi"]
(5)
and Fmax = exp [ - ( 1 -
"rmaxx)l/rm'x],
(6)
where x represents an extreme observation and may be defined as ( r - / 3 ) / a , where r in this analysis denotes the minimal or maximal price changes in a specified time period and where a and /3 designate the corresponding dispersion and location parameters, respectively. Thus, in this form x = ( r - / 3 ) / a may be thought of as a deviation of r from its location standardized by its dispersion. The tail index r determines the type of distribution from which extreme values are drawn, and it may be either less than, equal to, or greater than zero. When ~-= 0, the limiting distribution corresponds to the Type I distribution. When r < 0, the limiting distribution signifies a Type II distribution and when r > 0, the limiting distribution is Type III. As Gumbel (1958) suggests, a straightforward way to obtain estimates of a, /3 and ~- is to estimate the following equations using standard non-linear least squares regression: --'n [--'n (N---~) ] -
1 zm--q-;/n a "~n 1
~.-~'ffnf n [ a ~n
_
T min
X(/3"~"-rmin)] +u.
(7)
395
and
--
1
/ n o t max T max 1
,/.max
•/ n [ amax. - "rmax z(rmax--/3max)] q-U m,
(8)
where m is the rank assigned to each return after sorting all observations from lowest to highest and N is the number of observations for either the extreme positive or negative returns. Kinnison (1985) notes that the goal of this regression is to fit the expected cumulative probability frequencies to their empirical extreme value. An alternative to this approach is to estimate the parameters of Eq. (5) and Eq. (6) using the maximum likelihood method as described by Tiago de Oliveira (1973). Both these parametric methods provide consistent estimates. Indeed, Longin (1996) reports in his analysis of the tail shape properties of the returns of NYSE stocks that the two methods provide similar results, although the asymptotic standard errors for the regression method are somewhat higher. Nevertheless, as is demonstrated in the empirical exercise that follows, the parameter p-values are generally either very small or very large, thereby negating any potential interpretational difficulties. 5
3. D a t a
and
results
Tick-by-tick data are provided by Germany's screen-based futures market, the Deutsche
s Non-parametrictechniquesto analyzeextremevalue behavior exist. Examples include those developed by Hill (1975) and Pickands (1975). These,however,only focus on tail shape. Moreover, the Hill (1975) approach, which seems to be the most frequentlyused in financialeconomics,is limited only to Fr6chet distributions. In their studies of futures margin setting Kofman and de Vries (1989) and Kofman(1993) employ the Hill estimator. Longin (1995), however, relies on non-linear least squares regression.
J.P. Broussard, G.G. Booth / European Journal of Operational Research 104 (1998) 393-402
396
Table 1 Estimated parameters for minimum intradaily price changes Contract
N
Most negative r nan (%)
~'min(tailindex)
t~m*n (dispersion)
/3min (location)
R2
Jun. 1992 Sep. 1992 Dec. 1992 Mar. 1993 Jun. 1993 Sep. 1993 Dec. 1993 Mar. 1994 Jun. 1994 Sep. 1994
115 179 186 187 174 175 174 165 176 162
- 1.527 -4.684 - 4.788 -4.687 -4.341 -2.180 -2.157 - 4.034 -3.226 -3.478
-0.206 -0.341 - 0.229 -0.210 -0.202 0.003 * 0.031 * * - 0.140 -0.097 -0.108
0.189 0.291 0.397 0.359 0.312 0.365 0.412 0.462 0.503 0.456
-0.359 -0.395 - 0.504 -0.589 -0.458 -0.488 -0.519 - 0.602 -0.610 -0.661
0.989 0.990 0.989 0.989 0.986 0.994 0.985 0.996 0.991 0.996
The most negative r nan is the minimum intradaily return observed for each contract and N is the number of observations. All t~min (/3 nan) values are significantly positive (negative) at p-values < 0.0001. Except for two instances the ~.nan values are significantly negative (p-value < 0.0001). The exceptions are highlighted with asterisks, with * denoting positive but not significant (p-value = 0.719) and * * indicating positive and significant (p-value = 0.025). In the non-linear context, R2 is used solely as a descriptive statistic.
T e r m i n b ~ r s e ( D T B ) , a n d c o v e r all c o n t r a c t s w i t h at l e a s t six m o n t h s o f t r a d i n g d a t a f r o m J a n u a r y 1992 t h r o u g h S e p t e m b e r 1994. T h i s s e l e c t i o n p r o c e s s all o w s f o r t h e a n a l y s i s o f all f u t u r e s c o n t r a c t s f r o m J u n e 1992 to S e p t e m b e r 1994. D u r i n g this t i m e p e r i o d , t h e D A X a c h i e v e d l o w s o f a r o u n d 1400 a n d highs around 2300. Toward the end of the time p e r i o d a n a l y z e d , the D A X w a s t r a d i n g at a l e v e l around 2000. W h e n a p p l y i n g e x t r e m e v a l u e t h e o r y to a n y set o f data, a n i m p o r t a n t p r a c t i c a l i s s u e to b e r e s o l v e d is the f r e q u e n c y o f m e a s u r e m e n t . T h e s e l e c t i o n m a y
a f f e c t t h e m a g n i t u d e s o f a a n d / 3 b u t n o t ~', the last b e i n g i n d e p e n d e n t o f t e m p o r a l a g g r e g a t i o n (Feller, 1971). K i n n i s o n (1985), h o w e v e r , states t h a t the m e a s u r e m e n t i n t e r v a l s s h o u l d o c c u r at e q u a l d i s t a n t p o i n t s in time. T h e s e l e c t i o n p r o c e d u r e o f c h o o s i n g d a i l y m a x i m a a n d m i n i m a r e s u l t s in a n a t u r a l a n d c o n s i s t e n t i n t e r v a l , s i n c e t h e f o c u s o f this a p p l i c a t i o n is o n i n t r a d a y m a r g i n setting. O n l y t r a d i n g d a y s w i t h more than two transactions and having both positive a n d n e g a t i v e p r i c e c h a n g e s are c o n s i d e r e d . T h e s e d a t a w e r e c l e a n e d o f m i s t r a d e s , w h i c h are t r a n s a c t i o n s w h e r e s o m e e r r o r w a s m a d e in e n t e r i n g the
Table 2 Estimated parameters for maximum intradaily price changes Contract
N
Most positive r max (%)
~'max(tailindex)
a max (dispersion)
/3 max (location)
R2
Jun. 1992 Sep. 1992 Dec. 1992 Mar. 1993 Jun. 1993 Sep. 1993 Dec. 1993 Mar. 1994 Jun. 1994 Sep. 1994
115 179 186 187 174 175 174 165 176 162
1.707 2.906 2.690 2.424 3.705 2.813 2.747 3.706 3.546 2.845
-0.202 -0.269 - 0.017t 0.035t1 -0.154 -0.141 -0.072 -0.087 -0.112 -0.068
0.212 0.277 0.446 0.414 0.299 0.315 0.367 0.455 0.387 0.421
0.365 0.375 0.496 0.607 0.472 0.448 0.494 0.607 0.619 0.647
0.983 0.990 0.986 0.995 0.977 0.995 0.996 0.997 0.996 0.996
The most positive r max is the maximum intradaily return observed for each contract and N is the number of observations. All a max and /3 max values are significantly positive at p-values < 0.0001. Except for two cases the 1"max values ale significantly negative (p-value < 0.0001). The exceptions are highlighted by daggers, with t denoting negative but not significant (p-value = 0.167) and t t indicating positive and significant (p-value ~ 0.001). In the non-linear context, R 2 is used solely as a descriptive statistic.
J.P. Broussard, G.G. Booth/European Journal of Operational Research 104 (1998) 393-402
trade. Mistrade information was provided directly by the DTB. To examine maximum and minimum intradaily price changes, all possible combinations of intradaily price changes are calculated. Price changes are obtained taking the natural logarithm of a later transaction's price level and subtracting the natural logarithm of an earlier transaction's price level (In Pt In P t - b ) and multiplying by 100. After all possible combinations of transactional price changes are calculated, a maximum and a minimum for each day are selected. 6 The basic empirical results are displayed in Tables 1 and 2. Table 1 provides the results of using Eq. (7) for the minimum intradaily returns. For eight of the 10 contracts, ~.n~n values are significantly negative (p-values <0.0001). The ~.~n values for September and December 1993 contracts are positive but the former is not significantly different from zero (p-value = 0.719). Table 2 gives the results of applying Eq. (8) to the largest positive intradaily price changes. Except for the December 1992 and March 1993 contracts, z ~x estimates are negative and significant (p-values < 0.0001). The r ~ x for the former contract is negative but not significant (p-value = 0.167). In contrast, the ~.r~x for the latter is positive and significant ( p - v a l u e < 0 . 0 0 0 1 ) . The overall goodness-of-fit is highlighted by the very high R2s for Eq. (7) and Eq. (8) for every contract. Therefore, similar to the results found in Longin (1995, 1996) and implied by Jansen and de Vries (1991), Kofman and de Vries (1989), and Kofman (1993), the extreme values resulting from both increases and decreases in FDAX contract prices generally follow a Frrchet limiting distribution. The tails of two contracts, however, follow a Gumbel limiting distribution, whereas two are described by a Weibull distribution. 7 One characteristic revealed in Tables 1 and 2 is that the tail index parameter estimates are
6 The intradaily price changes are calculated using SAS' PROC IML routine to create dynamic vector sizes. Each daily maximum and minimumis chosen from an (T(T - 1)/2)× 1 vector where T is the number of transactions occurring in that day. 7 The T-valuesfor the two Weibull distributions are nearly zero, indicating that for these contractsthe degree of truncation is small. If binding price limits were instituted, more evidence of truncation should occur.
397
achieving smaller magnitudes, which is indicative of tail behavior tending towards thinner tails.
4. Implications for intradaily margin setting The difficulty of deciding on an appropriate margin level is highlighted by Telser (1981a) when he states "[t]here is no formula that can determine the optimal size of the margin." As pointed out by Baer et al. (1994), among others, the appropriate margin depends upon many factors and trade-offs are involved. Some of these factors may be related to the risk aversion level of the participants (e.g. see Telser, 1981b and Hunter, 1986) or the volatility of the underlying asset of the futures contract (e.g. see Hull, 1993 and Tucker, 1991). Additionally, Gay et al. (1986) and Rutz (1988) suggest that price levels, volume, and levels of speculation and hedging positions may be potential factors affecting the appropriate margin. Since arriving at an appropriate margin is a multi-faceted process, Brenner (1981) states that margins are optimally set by a designated margin committee similar to the one found at the DTB. Since one underlying objective of setting a margin is to protect the exchanges integrity from trader default due to adverse market movements, an important ingredient into the decision-making process is the probability that a specified margin will be violated. To illustrate the use of the information provided by the above statistical analysis to measuring this probability, this section applies the analysis to setting intradaily margin levels for FDAX contracts. Before doing so, however, it is helpful to briefly describe the institutional environment. The central organization for trading and clearing both options and futures transactions in Germany is the DTB. All DTB trading members fit into one of three categories: (1) General Clearing Member (GCM), (2) Direct Clearing Member (DCM), and (3) Non-Clearing Member (NCM). GCMs are traders who can clear their own trades, client trades, and transactions from NCMs. DCMs can settle transactions only for their own or clients' accounts. NCMs must settle their transactions through a GCM. GCMs and DCMs are directly responsible to the DTB for all obligations they have agreed to clear. NCMs and
398
J.P. Broussard, G.G. Booth~European Journal of Operational Research 104 (1998) 393-402
clients o f m e m b e r firms, however, have no direct responsibility to the DTB. To b e c o m e a D T B trading m e m b e r , or exchange participant, and gain access to the D T B trading system, there are several r e q u i r e m e n t s that need to be met. Generally, the m e m b e r firm m u s t be an operating b u s i n e s s e s t a b l i s h m e n t with m e a n i n g f u l e x c h a n g e business c o n d u c t e d on the D T B . There are restrictions and basic training qualifications that m u s t be m e t by the participants' personnel. Since the D T B is a totally electronic e x c h a n g e , participants m u s t equip themselves with D T B authorized hardware and software. D u r i n g the time period a n a l y z e d in this paper, the D T B e m p l o y e d daily m a r g i n settlement procedures that arrived at the appropriate m a r g i n r e q u i r e m e n t by netting positions within 21 m a r g i n classes. These classes are m a i n t a i n e d to lessen the potential impacts o f extreme price changes in i n d i v i d u a l assets. The D A X m a r g i n class c o n t a i n s the F D A X contracts, the O D A X (Options on the D A X contracts), and the
O F D X ( O p t i o n s on the F D A X contracts). The initial daily m a r g i n for a single F D A X contract (long or short) is 130 index points, which is equivalent to D M 13,000 (130 multiplied by 100, the F D A X contract size). T h u s d u r i n g the period o f analysis, the margin, in percentage terms (130 d i v i d e d by the D A X value and multiplied by 100), r a n g e d from 5.6% to 9.2%. At the e n d o f the period the effective m a r g i n was a p p r o x i m a t e l y 6.5%. It should be noted that the D T B recently l o w e r e d the daily m a r g i n r e q u i r e m e n t for F D A X contracts to 100 index points ( D M 10,000) per contract. A t a D A X level of 2000, this n e w level is e q u i v a l e n t to a 5% initial margin. As stated in the April 1995 D T B Reporter, the m a i n reason for the reduction in m a r g i n requirements was a believed decrease in volatility o f the F D A X contract prices. G i v e n the i m p o r t a n c e of the D T B in the G e r m a n (and E u r o p e a n ) capital markets, it is appropriate to consider setting levels on an intradaily rather than the current daily basis. Intradaily m a r g i n levels m a y be beneficial d u r i n g periods of high market volatil-
Table 3 lntradaily price changes given margin violation probabilities for the June 1993 contract Margin violation probabilities
Intradaily price changes (%) Short futures position
Long futures position 0.100 0.050 0.040 0.030 0.020 0.010 0.005 0.004 0.003 0.002 0.001 0.0005 0.0004 0.0003 0.0002 0.0001
Estimated distribution
Empirical distribution
Estimated distribution
Empirical distribution
-
-
1.277 1.599 1.709 1.855 2.072 2.474 2.920 3.074 3.280 3.586
1.215 1.498 !.559 1.571 1.599 2.008 3.705 3.705 3.705 na na na na na na na
1.349 1.730 1.863 2.043 2.314 2.830 3.422 3.631 3.914 4.342
- 5.160
-
6.100 6.432 6.883 7.564 8.867
1.320 1.610 1.641 1.799 2.062 2.147 4.341 4.341 4.341 na na na na na na na
4.156
4.789 5.007 5.300 5.736 6.547
The margin violation probabilities are F rain and 1 - F max for the long and short positions, respectively. Various probabilities are provided. Using the estimated extreme value distribution parameters, the intraday price change associated with each margin violation probability for each position is calculated. Where possible, the corresponding price changes derived from the empirical distribution are also provided. Note that, according to Table 1 and Table 2, the most negative price change is -4.341% and the most positive is 3.705%, respectively.
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ity. Having procedures in place to institute intradaily margin calls may also have the effect of lowering the daily margin requirement because of the ability to shore-up futures positions within the day. The decreased margin requirement therefore could increase trader participation and hence the liquidity of the market. The results of Section 3 provide useful input into setting a desired intraday margin level. Consider again Tables 1 and 2. The values for the most negative price changes range from - 1 . 5 2 7 % to - 4 . 7 8 8 % with a mean of - 3 . 5 1 0 % . The values of the most positive price changes are more clustered with a range of 1.707% to 3.706% and a mean of 2.909%. These percentages indicate that the daily initial margin target of 6.5% (and the 5% margin level just recently set) is quite conservative in an intradaily context. Apparently, no futures position was subject to a margin call, at least not on the first day of the transaction. Additional insights into the issue may be gained by directly examining the probability of a margin violation for long and short futures positions. A long (short) position's risk of margin violation is associated with negative (positive) returns. Thus, for a long position, the probability that a specified margin is violated is F ran, which is defined by Eq. (5). In contrast and as a result of being concerned with the other tail of the distribution, the margin violation probability for a short position is 1 - F m~x, where F m~ is given by Eq. (6). In either case, margin violation probability increases as ct increases, 1/31 increases and ~- becomes more negative. Table 3 contains the results for margin violation probability analyses for long and short futures positions. Only one contract (June 1993) is presented for brevity. Interpretation of Table 3 is as follows. The first column contains various margin violation probabilities. The second (fourth) column presents the return for the long (short) position calculated using a specific margin violation probability and the appropriate estimated values for a , fl and ~-. The absolute value of the return is equivalent to the margin requirement. The long (short) position return derived from the empirical distribution is provided in the third (fifth) column for benchmark purposes. A comparison of the second and third columns and fourth and fifth columns indicates that the data points de-
399
Table 4 Margin violation probabilities for various margin levels associated with the June 1993 contract's estimated distribution Margin (%) Marginviolation probabilities
10.00 5.00 4.00 3.00 2.00 1.00 0.50 0.40 0.30 0.20 0.10 0.05 0.04 0.03 0.02 0.01
Long futures position
Shortfutures position
Estimated distribution
Estimated distribution
0.000 0.001 0.003 0.008 0.032 0.202 0.584 0.702 0.819 0.916 0.975 0.990 0.992 0.993 0.995 0.996
0.000 0.000 0.00 I 0.004 0.023 0.189 0.598 0.721 0.838 0.930 0.981 0.992 0.994 0.995 0.996 0.997
The margin violation probabilities are F mjn and 1 - F max for the long and short positions, respectively. Various margins (i.e. the absolute value of the intraday price changes) are provided. Using the estimated extreme value distribution parameters, the margin violation probability associated with each price change for each position is calculated.
rived from the estimated distributions closely track those of the corresponding empirical distributions. 8 Two examples serve to illustrate the content of Table 3. For a long futures position associated with a margin violation probability of 0.001, the intraday price change is - 5 . 1 6 0 % , assuming the estimated Frrchet distribution. In other words, for every one out of 1000 trading days (about once in four years), an intradaily price decline of at least 5.160% is expected. Turning to a short futures example, the intradaily price increase associated with the same margin violation probability is 4.156%. The probability of a 5% price increase is approximately 0.0004, less than half that of price decrease of equal magni-
s The empirical distribution for minimal (maximal) price changes cannot be used to assess the probability of violating a margin that is greater than 4.341% (3.705%) for a long (short) position. This is because the requisite empirical observations have not been experienced.
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Table 5 Margin violation probabilities for a 3% margin for all contracts
Contract
Jun. 1992 Sep. 1992 Dec. 1992 Mar. 1993 Jun. 1993 Sep. 1993 Dec. 1993 Mar. 1994 Jun. 1994 Sep. 1994
Margin violation probabilities using the estimated distributions Long futures position
Short futures position
0.001 0.016 0.020 0.015 0.008 0.001 0.001 0.015 0.020 0.017
0.002 0.009 0.005 0.002 0.004 0.005 0.004 0.001 0.009 0.009
The margin violation probabilities are F nan and 1 - F mx for the long and short positions, respectively.
tude. Expressed in terms of time. 0.0004 is equivalent to once in every 10 years. Table 4 recasts Table 3 to be more 'user friendly' to the decision maker. In this table the margins are specified and the margin violation probability is calculated. An example for a long futures position shows that for a 3% margin (a - 3 % price change) there is a 0.008 probability of a margin violation. The corresponding probability for a short position is 0.004. In terms of margin violation frequency, these probabilities convert to eight violations per 1000 trading days (twice a year) and four violations per 1000 trading days (once a year) for the long and short positions, respectively. To extend this example, Table 5 provides the margin violation probabilities associated with a 3% margin for all contracts. Two points are noteworthy. First, the probabilities for both long and short positions vary considerably among contracts. For the long (short) position the range is from 0.001 (0.001) to 0.020 (0.009). Second, the margin violation probabilities appear to be asymmetric, with those associated with a long position often being greater than those with a short position. For instance, as pointed out above, for the June 1993 contract, the probability of violating a 3% margin is twice as likely for a long position than for a short position. There are, however, cases (e.g. the September 1993 contract) where the reverse obtains. To incorporate the notion of margin violation
probability into the margin setting process, the DTB must address several related issues. First, it must determine the amount of risk that it is willing to bear, with the time until a margin requirement is violated being an intuitive way of expressing this risk tolerance. Second, the DTB must express its margin requirement in percentage terms rather than in customary points to permit the margin to correspond to price changes. 9 Third, it must decide whether to impose different margins for short and long positions, since the extant analysis shows that the risk is asymmetric. Finally, the DTB must recognize that the extremal risk characteristics vary among contracts. One possible scenario that integrates the above issues is the following. The DTB concludes that is willing to accept the possibility of a margin violation occurring once every 50 trading days (approximately two months) and it never wants to bear more risk. Further, the DTB decides that different margins for long and short positions may be unnecessarily administratively cumbersome, thereby opting for a single margin requirement. Finally, they adopt a percentage margin scheme. Under these conditions, according to the figures given in Table 5, the DTB should set the margin to be 3% for short and long positions. For many contracts and positions this will provide the DTB with greater protection than desired but it will never (according to the historical record) provide less.
5. Summary The purpose of this paper was to describe the statistical properties of FDAX extreme intradaily price movements and to examine the ability of extreme value theory to provide reliable information on the incidence of intradaily margin violation probabilities. The statistical results of this study indicate that the intradaily extremes in Germany's FDAX con-
9 Expressing the margin in percentage ratber than points has the additional benefit of keeping the margin risk constant even though the value of the FDAX changes. For instance, in a rising market, all other things being equal, the possibility of a margin violation increases if the point convention is used.
J.P. Broussard. G. G. Booth~European Journal of Operational Research 104 (1998) 393-402
tracts generally follow a Type II (Frrchet) limiting extreme value distribution, thereby confirming the results of previous research concerning the distribution of futures' price changes. Moreover, the results indicate that extreme value theory should provide useful information to the D T B ' s margin setting committee. In particular, the extreme value technique generates probabilities of incurring abrupt intradaily price changes and should be included in the decision-making calculus used to set margin requirements.
Acknowledgements This paper was inspired by Francois Longin and the other participants at the Conference on Multivariate Extreme Value Estimation with Applications to Economics and Finance, Erasmus University, Rotterdam, May 2 6 - 2 8 , 1994. The authors thank Otto Loistl of Wirtschaftsuniverist~it Wien and formerly of the Deutsche B~irse, JiSrg Franke, Kerstin Miiller, Dirk Schermoly and Anselm Jumpertz of the DTB for providing data and vital information related to DTB operations. The authors also thank two anonymous reviewers for their helpful comments. All errors belong to the authors.
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