the effectiveness of margin setting with historical, implied, or

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					The Effectiveness of Margin-Setting with Historical, Implied, or Realized Volatility
by Chor-yiu SIN (CY), WISE, Xiamen University (cysin@xmu.edu.cn) Kin LAM, Hong Kong Baptist University December 21, 2006 International Workshop on Forecasting and Risk Management Center for Forecasting Science, Chinese Academy of Sciences (CAS)

OUTLINE OF THE TALK
1 Margin-Setting as a Problem for the Practitioners: A Practical Issue 2 Information Contents of Various Volatility Forecasts: An Academic Issue 3 Using Various Volatility Forecasts to Set the Margin 4 Assessing the Effectiveness of Margin-Setting 5 Empirical Study: Data 6 Empirical Study: Results 7 Conclusions

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Margin-Setting as a Problem of the Practitioners: What is the Issue?
• Suppose an investor buys a futures (or an options) contract. • The contract may worth $500,000 but she/he only needs to pay a

margin, say, $10,000.
• The clearinghouse faces no market risk but default risk. • In the case of Hang Seng Index Futures (HSIF), 1 index point =

HK$50.

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Margin-Setting as a Problem of the Practitioners: What is the Issue?
• Suppose in Period t-1, you long/buy. • In Period t, if Hang Seng Index (HSI, the underlying asset of HSIF)

drops by 300 points, then your loss (L) is HK$15,000.
• On the other hand, suppose in Period t-1, another investor shorts/sells. • In Period t, if HSI increases by 300 points, then her/his loss (L) is

HK$15,000.
• So, in either scenario, if either investor defaults, the clearinghouse will

lose HK$15,000. In view of that, margin is necessary.

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Margin-Setting as a Problem of the Practitioners: What is the Issue?
• The dilemma is:

• If the margin is too low, the clearinghouse has a high default risk. (We say that the margin requirement is not prudent.) • If the margin is too high, the investors are not happy. (We say that the margin requirement imposes a high opportunity cost on the investors.)
• The issue is to strike a balance between prudentiality and

opportunity cost.

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Margin-Setting as a Problem of the Practitioners: Background of the Study
• A “clearing members” model:

• Baer, France and Moser (1994) • Pooling the opportunity cost of margin money and the clearing members’ loss (when there is a loss) • Margin requirement should be set to minimize the expected pooled cost.

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Margin-Setting as a Problem of the Practitioners: Background of the Study
• A “listed company” model:

• In markets such as Australia, Hong Kong and Singapore, the clearinghouse is a (part of) listed company.

• The objective of a listed company is simply maximizing profit.

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Margin-Setting as a Problem of the Practitioners: Background of the Study
• However, usually a clearinghouse takes two roles, even though it is a

listed company.
• Take Hong Kong market as an example. • The Hong Kong Exchanges and Clearing Limited (HKEx) has been a

company listed in the stock exchange since Year 2000.
• At the same time, an ordinance requires the HKEx to maintain an

orderly and fair market and to manage risks prudently.

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Margin-Setting as a Problem of the Practitioners: Background of the Study
• A “mixed” model: The clearinghouse takes two roles:

• Private role: minimizing the opportunity cost as a way to maximize profit. • Public role: maximizing the prudentiality.

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Margin-Setting as a Problem of the Practitioners: Summary
• A clearinghouse for futures contracts collects margin money from its clearing members with non-zero open position in the contract. • To maintain their positions, investors have to meet the maintenance margin requirement in the sense that the trader must replenish the account balance up to the maintenance margin level. • The purpose of this margin system is to protect the clearinghouse from members' defaults resulting from big losses due to adverse movements of futures prices.

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Margin-Setting as a Problem of the Practitioners: Summary
• Once the default risk is judged to be prudential enough, the clearinghouse’s remaining concern is the opportunity cost of the investors. • While a margin committee of a clearinghouse would not follow a mechanical formula in setting the margin, they do use some benchmark formula that makes use of some econometric measures.

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Information Contents of Various Volatility Forecasts
• As a forecast of the subsequent market volatility, implied volatility (IV) is widely believed to be informationally superior to other alternatives, such as (various versions of) historical volatility (HV). • Recently, another type of alternatives, realized volatility (RV) which makes use of the intra-daily data, is found to be superior to HV. • However the assessment is by and large based on the traditional method such as goodness-of-fit (R2 in the regression analysis).

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Information Contents of Various Volatility Forecasts

• More importantly, the literature documented equivocal results, when different frequencies of data and/or different definitions of future volatility are used. See, for instance:

• Day and Lewis (1992) JoE, Canina and Figlewski (1993) RFS, Lamoureux and Lastrapes (1993) RFS, Amin and Ng (1997) RFS, Fleming (1998) JFE, Christensen and Prabhala (1998) JFE and Szakmary, Ors, Kim and Davidson (2003) JBF.
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Information Contents of Various Volatility Forecasts

• In a prequel of this paper, Lam, Sin and Leung (2004) JFM introduced another dimension of margin effectiveness. • We argued that while margin determination should be prudential, an increase in margin level drives up the investors’ cost. They then formulated a measure of opportunity cost and compare volatility forecasts in terms of such opportunity costs.

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Using Various Volatility Forecasts to Set the Margin: An Overview
• In this paper, the volatility forecast models include: • Historical volatility (HV) • Implied volatility (IV) • Realized volatility (RV) • Various versions of HV and IV are often used by the practitioners as well as the academics.

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Using Various Volatility Forecasts to Set the Margin: HV
• In particular, the HV model we consider is GARCH-GJR. Throughout we denote it as HV-GARCH. • GARCH is a volatility forecast first proposed by Bollerslev (1986) JoE. And it is modified by Glosten, Jagannathan and Runkle (1993) JF. • For each trading day t-1, we use 400 past daily returns to estimate the GARCH-GJR parameters. • This generalization from the usual GARCH-GJR estimation allows the more efficient use of updated information.

• Normality is used in the estimation but our empirical results are not sensitive to this
assumption.

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Using Various Volatility Forecasts to Set the Margin: IV
• The IV we use is calculated with the formula in Black (1976) JFE. • More precisely, the IV volatility forecast is simply the square of implied volatility calculated at the close of trading day t-1. • For sake of comparison, it is the daily variance of the return, in percentage. • This is denoted as IV-LAG. • In addition, we also consider the simple averages and its exponentially weighted moving averages, denoted as IV-SMA and IV-EWMA respectively. More precisely, RV-SMA is the 60-day moving averages, and IV-EWMA = 0.06 IV-LAG + 0.94 IV-EWMA-LAG

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Using Various Volatility Forecasts to Set the Margin: RV • The RV is the volatility forecast based on the intra-daily data. • In particular, we use the 5-minute returns. • For each trading day t-1, the average of square of the 5-minute returns
is our volatility forecast for next trading day t. This is denoted as RVLAG.

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Using Various Volatility Forecasts to Set the Margin: RV
• This approach is first suggested by Müller et al. (1990) JIMF and discussed and used by Andersen et al. (2003) Ec. • In addition, we also consider the simple averages and its exponentially weighted moving averages, denoted as RV-SMA and RV-EWMA respectively. More precisely, RV-SMA is the 60-day moving averages, and

RV-EWMA = 0.06 RV-LAG + 0.94 RV-EWMA-LAG

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Assessing the Effectiveness of Margin-Setting
• A benchmark formula for the margin (denoted as M) of a futures contract: • Pt-1|µt-1+k h t −1 |. • µt-1 and ht-1 are respectively the forecast of mean return and volatility. • Pt-1 is the price on the trading day t-1. • If the return is normally distributed, for 95% coverage probability (to be defined later), k = 1.96. Usually we will find a k larger than 1.96. • Usually one may assume that µt-1 = 0. Why?

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Assessing the Effectiveness of Margin-Setting
• The dilemma of choosing M: • If M is too low (for instance, we set the constant k very small), the default risk is high. • In order to quantify the default risk, we define two measures of the socalled prudentiality, namely coverage probability and expected shortfall. Both measures will be defined below.

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Assessing the Effectiveness of Margin-Setting
• The dilemma of choosing M: • On the other hand, if M is too high, opportunity cost to the investors may be too high. • This may defeat the functions of a stock exchange. • We will consider two measures of opportunity cost, namely, the margin amount itself and the overcharge. The latter is simply the excess margin.

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Assessing the Effectiveness of Margin-Setting
• A good volatility forecast strikes a balance between default risk and opportunity cost. • In other words, keeping the same level of default risk (coverage probability or expected shortfall), a good volatility forecast attains a lower opportunity cost (margin or overcharge). • See the following figures.

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Figure A

A Opportunity cost index B

P Q

prudentiality index

PI0

Figure B
cost

A B

Opportunity

index

prudentiality index

PI0

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Assessing the Effectiveness of Margin-Setting
• All in all, two measures of opportunity cost: • margin amount itself. • overcharge OC, which is defined as: OC = M - L OC = 0 if M ≥ L, if M < L,

where L is the actual loss of the investor's position (short or long).

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Assessing the Effectiveness of Margin-Setting
• All in all, two measures of prudentiality index: • coverage probability, which simply means the probability that the margin level is large enough to cover the loss, that is Probability(M ≥ L) • shortfall SF, which is defined as: SF = 0 SF = L - M if M ≥ L, if M < L,

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Empirical Study: Data
• From July 31, 1996 to September 5, 2003. • Altogether 1710 daily observations, after deleting the missing data. • Daily close of HSIF: Datastream. • Daily implied volatility: HKEx (Hong Kong Exchanges and Clearing Limited). • Tick-by-tick HSIF: HKEx. • See Tables 2.1 - 2.3 for the summary statistics. It should be noted that the RV's reported in Table 2.3 is normalized such that its mean is the same as that of HVGARCH.

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Table 2.1 Summary statistics of the original daily data
Lower Upper Mean Median SD Kurtosis Skewness Min. Quartile Quartile Max. IV a 29.086 27.000 11.580 6.412 1.680 9.500 20.600 34.800 139.000 HSIF b 12121 11593 2571 -0.779 0.352 6610 10040 13890 18390 HSIF Return c 0.023 -0.027 2.234 17.087 0.400 -22.509 -1.087 1.105 24.799 a The implied volatility is in percentage. It runs from July 31, 1996 to September 5, 2003. That amounts to 1710 observations. b The HSIF is the daily close, in index point. It also runs from July 31, 1996 to September 5, 2003. That amounts to 1710 observations. c The HSIF Return is the close-to-close daily return, in percentage. Computed from the HSIF series, it runs from August 1, 1996 to September 5, 2003. That amounts to 1709 observations.

Table 2.2 Summary statistics of the five-minute data
5-minute return 5-minute return Squared Nobs. 81619 81619 Mean 0.001 0.037 Median 0.002 0.008 SD 0.191 0.136 Min. -3.412 0.000 Lower Quartile -0.087 0.001 Upper Quartile 0.089 0.030 Max. 3.482 12.122

Table 2.3 Summary statistics of daily volatility forecast a
Mean Median HV-GARCH 3.792 1.510 IV-LAG 3.891 2.893 IV-SMA 3.879 2.990 IV-EWMA 3.885 3.043 RV-LAG 3.792 2.292 RV-SMA 3.776 2.628 RV-EWMA 3.782 2.699 All the data run from August 1, observations. SD 8.834 3.784 2.849 2.967 6.872 3.490 3.771 1996 to Lower Upper Kurtosis Skewness Min. Quartile Quartile Max. 156.491 10.737 0.966 1.166 2.904 173.255 85.084 5.959 0.358 1.684 4.806 76.671 1.662 1.472 0.602 1.746 4.695 14.541 2.947 1.726 0.620 1.769 4.942 16.195 390.139 15.633 0.167 1.201 4.270 197.864 5.211 2.186 0.543 1.493 4.631 19.378 12.251 3.001 0.485 1.430 4.566 27.793 September 5, 2003, in daily percentage. That amounts to 1708

a

Empirical Study: Results

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Table 4.1(a) Empirical coverage probability (ECP) versus average overcharge (AOC)
ECP (%) 95 (i) HV-GARCH K AOC 2.645 (ii) IV-LAG K AOC (iii) IV-SMA k AOC 2.177 2.273 2.449 2.859 3.159 3.891 5.669 312.55 (4.49) 332.97 (4.65) 370.77 (4.95) 459.71 (5.63) 525.44 (6.13) 687.01 (7.37) 1080.96 (10.56) (iv) IV-EWMA K AOC 2.107 2.203 2.409 2.725 3.108 3.621 5.215 295.49 (4.28) 315.85 (4.44) 359.97 (4.77) 428.33 (5.28) 511.97 (5.90) 624.82 (6.75) 976.81 (9.54) (v) RV-LAG k AOC 2.516 2.689 2.897 3.166 3.815 4.485 5.484 341.37 (5.95) 375.67 (6.36) 417.22 (6.85) 471.33 (7.49) 603.03 (9.04) 739.47 (10.69) 943.73 (13.18) (vi) RV-SMA K AOC 2.307 2.438 2.600 2.901 3.517 3.771 5.268 322.86 (4.96) 349.82 (5.21) 383.48 (5.52) 446.55 (6.10) 576.96 (7.29) 631.04 (7.78) 951.27 (10.78) (vii) RV-EWMA k AOC 2.214 2.355 2.557 2.886 3.269 3.813 4.624 300.94 (4.76) 329.80 (5.03) 371.52 (5.42) 440.16 (6.06) 520.87 (6.80) 636.24 (7.86) 808.83 (9.50)

331.50 2.110 288.58 (6.34) (4.24) 96 2.848 368.60 2.247 317.17 (6.82) (4.47) 97 3.092 413.62 2.438 357.41 (7.40) (4.78) 98 3.427 475.91 2.668 406.39 (8.19) (5.16) 99 3.894 563.50 2.981 473.61 (9.29) (5.68) 99.5 4.402 659.59 3.407 565.90 (10.49) (6.39) 99.8 5.376 844.52 4.124 721.95 (12.83) (7.64) Figures in brackets are the standard errors of AOC.

Table 4.1(b) Difference in average overcharge (AOC) (compared with IV-LAG)
(i) HV-GARCH (ii) IV-LAG (iii) IV-SMA (iv) IV-EWMA (v) RV-LAG (vi) RV-SMA (vii) RV-EWMA ECP Difference in Difference in Difference in Difference in Difference in Difference in Difference in (%) AOC AOC AOC AOC AOC AOC AOC 95 42.92** NA 23.97** 6.91** 52.78** 34.27** 12.35** (4.88) (2.32) (1.65) (3.90) (2.44) (1.95) 96 51.44** NA 15.80** -1.31 58.50** 32.66** 12.63** (5.28) (2.46) (1.76) (4.20) (2.59) (2.08) 97 56.20** NA 13.35** 2.56 59.81** 26.07** 14.11** (5.76) (2.68) (1.94) (4.56) (2.77) (2.27) 98 69.52** NA 53.32** 21.94** 64.93** 40.16** 33.77** (6.42) (3.08) (2.16) (5.02) (3.10) (2.60) 99 89.89** NA 51.84** 38.37** 129.43** 103.35** 47.27** (7.33) (3.43) (2.45) (6.25) (3.84) (2.98) 99.5 93.69** NA 121.11** 58.92** 173.57** 65.14** 70.34** (8.29) (4.16) (2.85) (7.47) (4.08) (3.53) 99.8 122.57** NA 359.00** 254.86** 221.78** 229.32** 86.88** (10.14) (6.18) (4.27) (9.23) (5.97) (4.32) Figures in brackets are the standard errors of differences in AOC. †, * and ** denote significance at 10%, 5% and 1% respectively. Assuming a 2-tailed z-test under the usual assumptions and estimation procedure, the critical values used are 1.645, 1.960 and 2.576 respectively.

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Table 4.2(i) Summary statistics of the overcharge of different volatility models (ECP = 98%)
Mean HV-GARCH IV-LAG IV-SMA IV-EWMA RV-LAG RV-SMA RV-EWMA 475.910 406.392 459.707 428.328 471.326 446.549 440.160 Median 397.128 379.521 428.383 399.939 421.050 412.543 403.783 SD 338.290 213.361 232.741 218.218 309.372 251.966 250.241 Kurtosis 23.075 0.269 -0.349 -0.109 12.485 0.100 0.737 Skewness 3.347 0.548 0.361 0.431 1.925 0.590 0.759 Minimum 0 0 0 0 0 0 0
Lower Quartile Upper Quartile

298.742 255.156 288.532 268.056 255.123 254.667 248.435

562.844 545.663 618.629 580.301 631.237 614.928 594.346

Maximum 4279.194 1254.231 1221.147 1143.734 4080.576 1327.625 1547.891

Table 4.2(ii) Summary statistics of the margin level of different volatility models (ECP = 98%)
Mean HV-GARCH IV-LAG IV-SMA IV-EWMA RV-LAG RV-SMA RV-EWMA 653.145 583.503 635.455 604.125 648.827 622.725 617.190 Median 567.864 569.419 636.931 609.787 594.616 608.071 605.539 SD 360.275 224.777 231.541 219.477 344.767 261.615 265.389 Kurtosis 28.048 0.814 -0.897 -0.675 16.347 -0.458 0.488 Skewness 3.962 0.690 0.290 0.357 2.395 0.503 0.750 Minimum 307.544 180.182 235.449 227.812 119.999 234.944 232.253
Lower Quartile Upper Quartile

432.166 395.523 438.598 408.405 396.309 412.403 397.187

720.221 725.888 799.982 753.385 828.775 784.279 768.456

Maximum 4903.194 2034.784 1231.151 1153.738 4720.576 1374.584 1702.170

Table 4.2(iii) Summary statistics of the overcharge of different volatility models (ASF = 2.0)
Mean HV-GARCH IV-LAG IV-SMA IV-EWMA RV-LAG RV-SMA RV-EWMA 536.793 469.073 793.876 687.200 541.443 610.786 527.860 Median 447.837 439.678 739.454 641.115 484.266 567.793 489.163 SD 370.060 233.379 339.636 298.783 343.442 313.930 283.453 Kurtosis 23.961 0.274 -0.513 -0.214 13.196 0.020 0.738 Skewness 3.442 0.565 0.404 0.475 1.993 0.600 0.781 Minimum 0 0 0 0 0 0 0
Lower Quartile Upper Quartile

340.655 298.822 519.414 441.670 301.972 362.213 308.018

626.804 619.178 1045.512 901.668 717.186 824.874 700.693

Maximum 4744.164 1396.958 1873.497 1642.876 4596.438 1690.297 1778.197

Table 4.2(iv) Summary statistics of the margin level of different volatility models (ASF = 2.0)
Mean HV-GARCH IV-LAG IV-SMA IV-EWMA RV-LAG RV-SMA RV-EWMA 715.083 647.361 972.163 865.487 719.730 789.075 706.149 Median 621.715 631.735 974.422 873.599 659.596 770.506 692.818 SD 394.439 249.377 354.227 314.430 382.443 331.501 303.641 Kurtosis 28.048 0.814 -0.897 -0.675 16.347 -0.458 0.488 Skewness 3.962 0.690 0.290 0.357 2.395 0.503 0.750 Minimum 336.708 199.900 360.206 326.370 133.113 297.705 265.728 Lower Quartile 473.148 438.808 670.997 585.093 439.617 522.569 454.435 Upper Quartile 788.519 805.328 1223.868 1079.322 919.343 993.785 879.218 Maximum 5368.164 2257.465 1883.502 1652.880 5236.438 1741.779 1947.511

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Figure 4.1 Average Overcharge vs Empirical Coverage Probability
1600

1400

Average Overcharge (Index Point per Day)

1200

1000 HV-GARCH IV-LAG RV-EWMA 800

600

400

200 95 95.2 95.4 95.6 95.8 96 96.2 96.4 96.6 96.8 97 97.2 97.4 97.6 97.8 98 98.2 98.4 98.6 98.8 99 99.2 99.4 99.6 99.8 100 Empirical Coverage Probability (%)

27

Figure 4.2 Recursive Average Overcharge of Different Models (ECP=98%)
700

600 Recursive Average Overcharge (Index Point per Day)

500

400 HV-GARCH IV-LAG RV-EWMA 300

200

100

0
8/1/1996 5/28/1997 3/24/1998 1/18/1999 11/14/1999 9/9/2000 7/6/2001 5/2/2002 2/26/2003

Date

28

Figure 4.3 Average Overcharge vs Empirical Coverage Probability
1600

1400

Average Overcharge (Index Point per Day)

1200

1000 IV-SMA IV-LAG IV-EWMA 800

600

400

200 95 95.2 95.4 95.6 95.8 96 96.2 96.4 96.6 96.8 97 97.2 97.4 97.6 97.8 98 98.2 98.4 98.6 98.8 99 99.2 99.4 99.6 99.8 100 Empirical Coverage Probability (%)

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Figure 4.4 Recursive Average Overcharge of Different Models (ECP=98%)
700

600 Recursive Average Overcharge (Index Point per Day)

500

400 IV-SMA IV-LAG IV-EWMA 300

200

100

0
8/1/1996 5/28/1997 3/24/1998 1/18/1999 11/14/1999 9/9/2000 7/6/2001 5/2/2002 2/26/2003

Date

30

Conclusions
• Implied volatility (IV) performs best, compared with the GARCH-GJR and the recently developed realized volatility. • More precisely, IV strikes a good balance between prudentiality and opportunity cost, as far as the margin-setting of the futures contract is concerned. • IV even beats its own smoothing versions, namely simple moving averages and exponentially weighted moving averages. • This is confined to the case of Hang Sang Index Futures though. • It remains to see if our conclusions carry over to other markets and/or other products, or more ambitiously, other respects of risk management.

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