A Comparison of Trading Models Used for Calculating ...

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A COMPARISON OF TRADING MODELSUSED FOR CALCULATING AGGREGATEDAMAGES IN SECURITIES LITIGATION

MICHAEL BARCLAY* AND FRANK C. TORCHIO**

I

INTRODUCTION

For approximately two decades, the General Trading Model (“GTM”) hasbeen used in securities litigation to estimate the number of shares damaged byalleged fraudulent misrepresentations by defendants. The GTM estimates thefraction of in-and-out trading volume and the fraction of retained volume. “In-and-out volume” refers to shares bought and sold within the class period; “re-tained volume” refers to shares purchased and held through the final disclosurethat reveals the fraud. This is typically the last day of the class period. Esti-mates of the number of damaged shares from the GTM have been used in con-junction with a theory of true value (or conversely, artificial inflation) for thesecurity to estimate aggregate monetary damages.1

Over the years, variations of the GTM predicated on different assumptionsand/or parameters have been developed. The variations include single-tradermodels, such as the proportional and accelerated trading models, and multi-trader models.2 This article compares the results of these models and criticallyevaluates the conclusions reached in previously published research.

This article demonstrates that results from the proportional single-tradermodel, GTM (1x), are consistent with the results of multi-trader GTMs whenappropriate assumptions and parameters are used. No evidence was found toreject the GTM (1x) as a scientific method to estimate the number of damagedshares in securities litigation.

Copyright © 2001 by Michael Barclay and Frank C. TorchioThis article is also available at http://www.law.duke.edu/journals/64LCPBarclay.* Alumni Distinguished Professor of Finance and Finance Area Coordinator, William E. Simon

Graduate School of Business Administration, University of Rochester.** President, Forensic Economics, Inc., a firm that consults in securities litigation cases and com-

plex business disputes.1. For example, Forensic Economics used the GTM in conjunction with a theory of true value to

provide damage estimates of $8.5 billion in the Cendant securities litigation, the largest class action se-curities litigation settlement in history. See generally In re Cendant Corp. Sec. Litig., 109 F. Supp. 2d285 (D.N.J. 2000).

2. Virtually every securities litigation case in which aggregate damages are estimated relies on aversion of the GTM for the number of shares damaged and the damage theory for the amount of artifi-cial inflation. See id.


106 LAW AND CONTEMPORARY PROBLEMS [Vol. 64: Nos. 2 & 3

II

BACKGROUND

In securities litigation, damages arise when defendants make false or mis-leading statements that artificially inflate the stock price.3 If an investor pur-chases the stock at this artificially inflated price, and the price later declineswhen the fraud is revealed, the investor will suffer damages from paying toomuch for the stock. In general, damages per share are calculated as the artificialinflation when the shares were purchased minus the artificial inflation when theshares were sold. For example, shares purchased when the stock price was arti-ficially inflated and held through a disclosure that reveals the fraud typically areconsidered to be damaged. Shares purchased and then sold before any revela-tion of the fraud, however, are typically not considered to be damaged becausethese shares were passed on before any deflation in value.

Experts on damages in securities class actions generally do not have accessto the trading records of individual class members. Consequently, the numberof damaged shares is commonly estimated from a security’s reported dailytrading volume. Although the reported trading volume is quite reliable, thenumber of damaged shares is generally less than reported volume for severalreasons.

First, reported volume may overstate the trading volume by the plaintiffclass because it includes trades by specialists on the New York Stock Exchange(“NYSE”) or market makers on the National Association of Securities DealersAutomated Quotation system (“NASDAQ”) who buy from one investor andsell to another. One must adjust the reported volume to remove these double-counted trades. Recently published research suggests that a suitable correctionis obtained by reducing NYSE reported volume by approximately ten percentand reducing NASDAQ volume by approximately fifty-eight percent.4

A second adjustment to volume is necessary to eliminate shares that werepurchased during the class period and sold before the revelation of the allegedfraud. In many cases, these in-and-out shares have no associated damages be-cause they were purchased and sold at prices with the same artificial inflation.5

Historically, it has been common practice among economic experts for bothplaintiffs and defendants to adjust volume for non-damaged, in-and-out volumeusing a statistical trading model.6 The trading model is a mathematical model

3. Artificial inflation is the difference between the actual stock price and what the true value ofthe stock would have been but for the false or misleading information.

4. See Harindra de Silva et al., Securities Act Violations: Estimation of Damages, in LITIGATIONSERVICES HANDBOOK: ROLE OF THE ACCOUNTANT AS EXPERT WITNESS 44-31 (Roman L. Weil etal. eds., 2d ed., 1995); John F. Gould & Allan W. Kleidon, Market Maker Activity on NASDAQ: Impli-cations for Trading Volume, STAN. J.L. BUS. & FIN., Fall 1994, at 11, 14, 21.

5. However, in-and-out shares may be damaged if more than one corrective disclosure is involved.6. See generally de Silva et al., supra note 4, at 44-21 to 44-32; Dean Furbush & Jeffrey W. Smith,

Estimating the Number of Damaged Shares in Securities Fraud Litigation: An Introduction to StockTrading Models, 49 BUS. LAW. 527, 531 (1994); Jon Koslow, Estimating Aggregate Damages in Class-Action Litigation Under Rule 10b-5 for Purposes of Settlement, 59 FORDHAM L. REV. 811 (1991); Craig


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that estimates, on each day of the class period, the fraction of volume that is in-and-out volume and the fraction that is retained volume.7

Trading models generally require a calculation of the daily ratio of adjustedvolume to float, where float is defined as the number of shares that could havebeen traded on a given day, and adjusted volume is defined as reported volumeadjusted to eliminate intra-day market maker or specialist trades, and to elimi-nate insider trades.8 This “adjusted volume-to-float” ratio is an important fac-tor in virtually all trading models.9 In particular, as the number of shares in thefloat declines, the number of retained shares estimated from a trading modelalso declines, all else being the same.10

It is commonly assumed that each share purchased during the class periodhas the same chance of being sold on a subsequent day of the class period asany other share in the float. This special case of the GTM is commonly knownas the proportional trading (or proportional decay) model because of the pro-portionality assumption about trading propensities.11 The trading propensity as-sumption is sometimes referred to as the acceleration factor.12 The proportion-ality assumption treats the acceleration factor as equal to one.

If one adopts the proportional trading assumption, then shares purchased onday one of the class period have, on average, a probability of being sold on daytwo equal to the adjusted volume-to-float ratio on day two. If one assumes thatshares purchased during the class period were, on average, more likely to havebeen sold later in the class period than other shares in the float, then a multiplethat is greater than one would be applied to the adjusted volume-to-float ratio.Conversely, if one assumes that shares purchased in the class period were lesslikely to have been sold later in the class period than other shares in the float,then a multiple that is less than one would be applied to the adjusted volume-to-float ratio. When it is assumed that shares traded during the class period aremore likely to be sold later in the class period than other shares in the float, one

McCann et al., Demystifying Stock Trading Models in Securities Class Action Lawsuits (KPMG, Econ.Consulting Serv., N.Y., N.Y.), Aug. 1997, at 1.

7. When the full extent of the fraud is revealed through several partial disclosures, it is usuallyimportant to determine the number of shares that were purchased during the class period and heldthrough each partial disclosure. For example, if two distinct stock price drops can be attributed to thefraud, then in-and-out shares purchased before the first price drop, but sold before the second pricedrop, would normally be damaged in-and-out shares.

8. Float is generally determined by subtracting any shares known not to have traded over a por-tion of the class period from total shares outstanding. For example, insider trading records and quar-terly institutional holdings are used to estimate the number of shares owned by insiders and institutionsthat were not traded. The float also is adjusted for share offerings or buybacks and for short interestpositions. See de Silva et al., supra note 4, at 44-22 to 44-24.

9. Mathematically, the computation of retained shares is a function of the volume-to-float ratio.Specifically, using a GTM (1x) model, the number of retained shares from purchases on day one of athree-day class period equals the number of shares purchased on day one multiplied by the product ofthe quantity (one minus the volume-to-float ratio) on day two and the quantity (one minus the volume-to-float ratio) on day three. See id. at 44-25 to 44-32.

10. This mathematical property follows from the algebra of the GTMs.11. See Furbush & Smith, supra note 6, at 534.12. See McCann et al., supra note 6, at 4.



is applying what is called an accelerated trading model.13 Although the terms“proportional trading model” and “accelerated trading model” connote sepa-rate models, they are both GTMs with different assumptions about trading pro-pensities during the class period. Therefore, this article shall refer to a model asthe GTM (1x) when one assumes an acceleration factor of one (proportional as-sumption), and GTM (1.1x) or GTM (.9x) for acceleration factors of 1.1 and 0.9,respectively.

III

A COMPARISON OF SINGLE-TRADER AND MULTI-TRADER MODELS

Recently the proportionality assumption in the GTM (1x) has been criti-cized.14 Theoretical criticisms of the single-trader GTM (1x) center around amathematical proposition asserted by William Beaver, James Malernee, andMichael Keeley that the number of damaged shares estimated from a two-trader model is substantially less than the number estimated from a propor-tional trading model.15 The two-trader model divides a company’s shares intotwo groups: those held by active traders and those held by passive traders.Daily trading volume for each group is determined by the relative propensitiesto trade that are assumed for the active-trader and for the passive-tradergroups. Given the assumptions about the fraction of shares held by each tradergroup and the fraction of trading volume attributed to each trader group, esti-mates of retained shares and in-and-out shares are computed using a GTM (1x)for each group.

The two-trader model has two key assumptions that are not present in thesingle-trader model: (1) The percentage of total shares outstanding held by ac-tive and passive traders are different; and (2) The active and passive tradershave different trading propensities (for example, the active trader may be as-

13. See Furbush & Smith, supra note 6, at 533.14. The criticism generally has emanated from individuals associated with Lexecon or Cornerstone

Research, two firms that provide economic consulting and expert testimony in securities class actioncases primarily for defendants. See generally Janet Cooper Alexander, The Value of Bad News in Secu-rities Class Actions, 41 UCLA L. REV. 1421, 1461-62 (1994); William H. Beaver & James K. Malernee,Estimating Damages in Securities Fraud Cases Slides, in HOW TO PREPARE FOR AND SUCCESSFULLYTRY A SECURITIES CLASS ACTION IN THE POST-REFORM ERA 631 (PLI Corp. Law & Practice CourseHandbook Series No. 627, 1990); William H. Beaver et al., Potential Damages Facing Auditors in Secu-rities Fraud Cases, in ACCOUNTANTS’ LIABILITY: THE NEED FOR FAIRNESS 113 (John T. Behrendt etal., eds., 1994); Kenneth R. Cone & James E. Laurence, How Accurate are Estimates of AggregateDamages in Securities Fraud Cases?, 49 BUS. LAW. 505 (1994); Edward J. Yodowitz & Steven J. Kol-leeny, Changing the Standard of Joint and Several Liability to a Proportional Liability Rule, inSECURITIES CLASS ACTIONS: ABUSES AND REMEDIES 31 (National Legal Ctr. for the Public Interested., 1994).

15. For example, Alexander offers no independent analysis but merely asserts that “the use of theproportional trading assumption in calculating aggregate class damages accordingly may inflate the to-tal class damages by 100%.” Alexander, supra note 14, at 18, citing Beaver et al., supra note 14. Skud-der, having cited Alexander and also the empirical findings of Cone and Laurence as his bases, assertsthat “because the proportional decay model overstates the retention shares traded during the class pe-riod, damages will be overstated.” Michael Y. Scudder, The Implications of Market-Based DamagesCaps in Securities Class Actions, 92 NW. U.L. REV. 435, 451 (1997).



sumed to be two times, five times, ten times, or even twenty times more likelyto trade than the passive trader).16

Beaver et al. simulate a case in which they assume that the class period is100 days, the number of shares outstanding is 10 million, daily volume is 100,000shares, twenty percent of shares are held by active traders, and active tradersare twenty times more likely to trade than passive traders.17 They report thattheir two-trader model yields 3,477,350 retained shares.18 They compute thenumber of retained shares using a GTM (1x) to be 6,339,680.19 Because the two-trader model yields only 54.9% (3,447,350 / 6,339,680) of the retained shares ob-tained from their GTM (1x), they conclude that the GTM (1x) overstates re-tained shares and, therefore, overstates aggregate damages.20

To illustrate this mathematical proposition, Beaver et al. compared the re-sults of each model to actual depository records that they obtained whileworking on an unnamed securities case.21 The authors state that the class periodwas 128 days and that the average trading volume was 2.2% of the shares out-standing.22 Their published work claims that their two-trader model matchedthe actual depository data, which showed that 49.5% of total shares outstandingwere retained and submitted for damage claims.23 The results of their GTM(1x), however, yielded a number of retained shares that was 94.1% of sharesoutstanding.24

A. Beaver, Malernee, and Keeley’s Results from Their GTM (1x) Are FlawedBecause They Do Not Compute Float and They Do Not Adjust ReportedVolume

In their comparison to the depository records, Beaver et al. mischaracter-ized how the single-trader GTM is used in current practice.25 First, they used to-tal shares outstanding instead of float in their volume-to-float ratio to computeretained shares.26 As discussed above, it is standard practice—and critical to theanalysis—that the float used to compute retained shares exclude all sharesknown not to have traded over the class period.27 Beaver et al. make no attemptto do so in their GTM (1x). The amount of shares that would typically be ex-

16. The additional assumptions about relative trading propensities and the distribution of tradingpropensities are viewed by some researchers as a disadvantage. See Edward A. Dyl, Estimating Eco-nomic Damages in Class Action Securities Fraud Litigation, 12 J. FORENSIC ECON. 10, 11 (1999).

17. Beaver et al., supra note 14, at 125.18. See id. at 126.19. See id.20. See id. at 126-27.21. See id. at 128-30.22. See id. at 128.23. See id.24. See id.25. Also, Beaver et al. do not explain whether the depository records they obtained include all

trading volume over the class period. See id. at 128-30.26. See id. at 128.27. See de Silva et al., supra note 4, at 44-21 to 44-37.



cluded from the float is substantial. For example, it is common for float to beless than 50% of shares outstanding for a class period of only 128 days.28

This same confusion over the proper use of the GTM (1x) is evident in thework of Daniel Fischel and David Ross.29 They state that an indirect test of theGTM (1x) demonstrates that “the proportional trading decay model would pre-dict that 142.6 million of Intel’s 148.7 million shares outstanding were boughtduring 1990, although the institutional holding data shows that institutions heldand did not trade 77.4 million.”30 Fischel and Ross predetermine the results oftheir indirect test by omitting an extremely important parameter of the GTM:the number of institutional shares that did not trade.31 If they had properly usedthe institutional data to estimate Intel’s float, the number of buy and holdshares predicted could not possibly have exceeded 71.3 million, the number ofshares in Intel’s float (71.3 million shares in the float is equal to 148.7 million to-tal shares outstanding less 77.4 million shares held and not traded by institu-tions).32 Therefore, Fischel and Ross put forth a straw man calculation of re-tained shares by failing to adjust the float properly in their model for theinstitutional trading data. Then they criticize the single-trader GTM becausethe calculated estimation of retained shares does not comport with the institu-tional trading data.33

Second, Beaver et al. apparently failed to adjust volume for specialist ormarket-maker trading.34 Adjusting the reported volume lowers the number ofretained shares. It is common practice to adjust reported volume for specialist(NYSE firms) or market maker (NASDAQ firms) trading.35 As previouslynoted, reported volume should be reduced by approximately twenty percent forNYSE stocks and by approximately fifty-eight percent for NASDAQ stocks.While this adjustment was not discussed in Beaver et al.’s published work, Cor-nerstone Research summarized the work in its Web page and stated that thisadjustment to volume is “an additional problem not discussed here [in their pa-per].”36 Because the authors did not adjust reported volume when they esti-mated damaged shares, their proportional trading model overstated retainedshares by omitting this important parameter.

28. For example, see Plaintiffs Expert Reports, In re Gaming Lottery Sec. Litig., No. 96 Civ. 5567,2001 U.S. Dist. LEXIS 1204, at *1 (S.D.N.Y. Feb. 8, 2001).

29. See generally Daniel R. Fischel & David J. Ross, The Use of Trading Models to Estimate Aggre-gate Damages in Securities Fraud Litigation: A Proposal for Change, in SECURITIES CLASS ACTIONS:ABUSES AND REMEDIES 131 (National Legal Ctr. for the Public Interest ed., 1994).

30. Id. at 139.31. See id.32. This analysis of float does not reflect any effect from insider holdings or short interest posi-

tions. See id.33. See id.34. See Beaver et al., supra note 14, at 128-30.35. See de Silva et al., supra note 4, at 44-21 to 44-37; see also Gould & Kleidon, supra note 4, at 13.36. William H. Beaver et al., Stock Trading Behavior and Damage Estimation in Securities Cases 8

(1993) (unpublished manuscript, available at the Cornerstone Research website) (last modified Jan. 19,2000) <http://www.cornerstone.com/fram_sea.html>.



Chart A of the Appendix shows the results from a simulation using a GTM(1x) for a class period of 128 days and for average daily volume of 2.2% of totalshares outstanding. With no adjustment to the float or reported volume, theGTM (1x) trading model yields damaged shares that are 94.2% of shares out-standing, which is similar to the 94.1% reported by Beaver et al. using actualdata. Using the same GTM (1x), but now estimating float to be fifty percent ofshares outstanding and adjusting reported volume by the NYSE factor oftwenty percent, the number of damaged shares is only 49.2% of shares out-standing (see Chart A). Therefore, with the correct adjustment to reportedvolume and plausible assumptions about float, the number of retained sharesestimated from a GTM (1x) are approximately equal to the actual number ofretained shares as reported by Beaver et al. based on depository records.37

B. The Twenty Percent Active and Eighty Percent Passive Trader AssumptionUsed by Beaver, Malernee, and Keeley Appears to Have Been Chosen toExaggerate the Differences Between the Proportional Trading Model andthe Two-Trader Model

The assumptions that Beaver et al. used to estimate their two-trader modelexaggerate the difference between the number of retained shares using the two-trader model and the proportional trading model. Beaver et al. illustrate an ex-ample in which active traders hold a constant twenty percent of the stock over a100-day class period and active traders are twenty times more likely to tradethan passive traders.38 They show that the two-trader model yields fifty-fivepercent of the total retained shares that result from a GTM (1x).39

As Chart B demonstrates, the twenty percent active-trader assumption Bea-ver et al. chose for their hypothetical example results in the greatest possibledifference between the models holding the relative propensity of active andpassive traders to trade alike. That is, if one assumes that the active tradersheld more than twenty percent, or less than twenty percent of the shares out-standing, the number of retained shares from the two-trader model increases,and the difference between retained shares estimated from the two-tradermodel and retained shares estimated from the GTM (1x) declines.

Moreover, most of the sample data of trading activity do not support thetwenty percent active and eighty percent passive distribution asserted by Beaveret al.40 To the contrary, the trading data conform more to a normal, bell-shapeddistribution.41 Kenneth Froot, Andre Perold, and Jeremy Stein present a de-composition of share turnover for nine groups of traders: Active Pension Funds,Passive Pension Funds, Foundations/Endowments, Self-directed Households,

37. Since Beaver et al. reveal neither the company nor the class period for which they are calcu-lating damaged shares, it is not possible to make an accurate assessment of the float.

38. See Beaver et al., supra note 14, at 126.39. See id.40. See KENNETH A. FROOT ET AL., SHAREHOLDER TRADING PRACTICES AND CORPORATE

INVESTMENT HORIZONS (National Bureau of Econ. Research Working Paper No. 3638, 1991).41. See id. at 57 tbl. 1.



Bank Trust Departments for Households, Insurance Companies, Mutual Funds,Foreign Investors, and Others. 42 Froot et al. report the percentage held by eachgroup and the average share turnover for each group.43

Using turnover statistics from Froot et al., relative trading propensities werecomputed by dividing each group’s reported turnover statistic by the turnoverstatistic for the group with the lowest turnover (Passive Pension Funds). Thetrading propensities ranged from 1.0 to 7.4. The data show that five percent ofequity holdings are Passive Pension Funds that have the lowest turnover andthus a trading propensity of one. Approximately fifty-six percent of equityholdings have a trading propensity between 1.5 and 1.9, approximately thirty-two percent of equity holdings have trading propensities between 2.9 and 3.8,and seven percent of equity holdings have trading propensities between 6.5 and7.4. Chart A shows the results of this calculation.

These statistics were used to estimate a four-trader model before comparingthe results from this four-trader model with the GTM (1x) using the same as-sumptions about daily volume and float used by Beaver et al. The number ofretained shares estimated from the four-trader model using the statistics fromFroot et al. are approximately ninety-two percent of the results from the GTM(1x).

Thus, a more sophisticated multi-trader model that makes use of actual dataabout trading propensities results in retained shares estimates that are quitesimilar to that of the GTM (1x). This result should not be surprising. TheGTM (1x), when used correctly, is essentially a two-trader model becauseshares are eliminated from the float and therefore eliminated from damages.Shares that are eliminated from the float have a trading propensity that is equalto zero, because they are known not to have traded during the class period.Thus, the model has one (often large) group of traders with a known tradingpropensity of zero during the class period. Shares that remain in the float, bydesign, have much more homogeneous trading propensities because the shareswith the lowest trading propensities have been removed from total shares out-standing.

The sensitivity of the four-trader model results was tested by maintainingthe general bell-shaped distribution for the number of traders in each tradergroup while increasing the range of relative trading propensities. That is, thestudy assumed that the lowest and highest trading-propensity groups are eachapproximately five to seven percent of shares outstanding but increased therange of trading propensities. Even assuming that the highest trading propen-sity is forty times greater than the lowest trading propensity, the results werestill eighty-nine percent of the GTM (1x) (see Chart B). Thus, if the multi-trader model is applied with a distribution of traders that conforms to the ob-served empirical data, it does not result in the exaggerated differences of nearlyfifty percent when compared with the proportional trading model as asserted by

42. See id.43. See id.



the critics. The unrealistic trading propensities assumed by Beaver et al. in theirtwo-trader model significantly understate the number of retained shares whencompared with a multi-trader model using a distribution of trading propensitiesthat is more consistent with observed trading propensities.

C. The Magnitude of the Difference Between the Two-trader Model and theGTM (1x) Is Sensitive to the Length of the Class Period

Beaver et al. use a hypothetical class period of 100 days when they comparethe results from a proportional trading model with a two-trader model.44 ChartC shows the ratio of the number of retained shares from the two-trader model(with twenty percent active shares and eighty percent passive shares) to thenumber of retained shares from the GTM (1x) for various assumptions aboutrelative trading propensities and the length of the class period. Chart C illus-trates that as the class period becomes shorter than 100 days or greater than 150days, this ratio increases using any assumption about trading propensities.

IV

EMPIRICAL EVIDENCE

Kenneth Cone and James Laurence’s work is the most frequently cited em-pirical evidence on the accuracy of trading models. They cite two cases in whichthe damage estimates from the plaintiffs’ experts, who used a proportionaltrading model, exceeded the aggregate claims filed.45 Cone and Laurence criti-cize the proportional trading assumption (referring to it as the “uniform” as-sumption) on several grounds and state that “the ultimate test of any model liesin how well it predicts the bottom line.”46 The “bottom line” for them is the dif-ference between the predicted aggregate damages and the actual aggregatedamages awarded through the claims process.47 The two cases they cite involveMidwestern and Storage Technology, NASDAQ and NYSE stocks, respectively.

A. Testing Trading Models with Anecdotal Claims Data Can Be GrosslyMisleading

1. Storage Technology. For the Storage Technology case, Cone andLaurence report that the proportional trading model overestimates the numberof buy-and-hold shares that entered the class compared with the buy-and-holdclaims submitted.48 They report that for Storage Technology, only 9.3 million

44. See Beaver et al., supra note 14, at 125.45. The two cases cited are Biben v. Card, 789 F. Supp. 1001 (W.D. Mo. 1992) and Levit v. Aweida,

630 F. Supp. 1072 (D. Colo. 1986). See Cone & Laurence, supra note 14, at 507.46. Id. at 522, 530.47. See id. at 530.48. See id. at 537.



buy-and-hold shares were submitted for a claim, which was about one-third ofthe estimated buy-and-hold shares from their proportional trading model.49

A comprehensive analysis of share ownership in the Storage Technologycase, however, demonstrates that use of claims data as a benchmark for thenumber of Storage Technology shares that were bought and held is misleadingand inappropriate. In reaching this conclusion, institutional holdings of StorageTechnology over the class period as reported by Thomson Financial SecuritiesData (“Thomson”) were reviewed. Thomson records the shares of a stock heldby each institution at the end of each quarter from the 13-F Securities and Ex-change Commission (“SEC”) filings that institutions are required to file. Ac-cording to Thomson, there were 135 institutions that held 15.6 million shares(approximately fifty percent of shares outstanding) of Storage Technology atthe end of the quarter before the beginning of Storage Technology’s class pe-riod. The data show that approximately 15.4 million of the 15.6 million sharesowned by institutions were sold during the class period.50 Thus, the 9.3 millionshares submitted for claims reflect only sixty percent of the buy-and-hold sharesoriginally held by institutions and none of the buy-and-hold shares originallyheld by individual investors (who owned approximately fifty percent of the totalshares outstanding). The 9.3 million buy-and-hold shares submitted for claimsare a misleading and inappropriate benchmark for assessing the efficacy of anytrading model.

Just how poor is the use of claims data to assess the trading model for Stor-age Technology? Although we do not have data on individual shareholders aswe do for institutions, certain facts about this case may provide some insight.The class period is over two years long, Storage Technology lost seventy-fivepercent of its market value of equity over the same time period in which themarket increased by sixty-five percent, and institutions turned over nearly theirentire holdings of Storage Technology stock during the class period.51 Reportedvolume over the class period was over 125 million shares, which, with 33 millionshares outstanding, reflects a share turnover of about 380%.52 It would be re-markable to find that that many individuals who held Storage Technology stockbefore the class period did not sell during the class period. Thus, the number ofbuy-and-hold shares from claims submissions vastly underestimates the actualnumber of shares of Storage Technology that were bought and held.

Contrary to the conclusion of Cone and Laurence, the proportional tradingmodel may well provide a very good estimate of buy-and-hold shares in thiscase. It is clear, however, that Storage Technology’s claims data are of little use

49. See id.50. For each institution, the lowest balance of shares it held during the class period was determined

before subtracting that balance from the number of shares it held before the beginning of the class pe-riod.

51. See S&P 500 Index, Standard and Poors (visited Mar. 8, 2001) <http://www.spglobal.comindexmain500_data.html>; Center for Research in Securities Prices (visited Mar. 7, 2001) <http://www.gsb.uchicago.edu/research/crsp/index/html>.

52. See Center for Research in Securities Prices, supra note 51.



in assessing whether the number of buy-and-hold shares predicted by the GTM(1x) is accurate (see Chart C). This is because the Storage Technology case is anold case dating back to 1982, and the case was decided over six years after theend of the class period at issue.53 According to deposition testimony, the claimsadministration for this case was poorly conducted by today’s standards.54 Evi-dently, the proof of claims form was difficult to understand as the form saidnothing about damages and was required to be returned by registered mail.55

2. Midwestern. For Midwestern, Cone and Laurence report in Table 6 oftheir report that the buy-and-hold shares submitted for claims equaled eighty-seven of the shares predicted from a GTM (1x) when a proper adjustment toNASDAQ reported volume was used.56 Considering the potential for claimsdata to vastly underestimate actual buy-and-hold shares, as evidenced by theStorage Technology case, this difference of 230,000 shares (13% of the 1.78million share prediction) would appear to support the GTM (1x) result, ratherthan to provide evidence against its use (See Table D). It is not contended herethat the Midwestern claims data supports the use of the proportional tradingmodel, but rather that the claims data fail to support the critics’ contentions thatthe proportional trading model dramatically overstates retained shares.

B. Testing Trading Models with Anecdotal Claims Data Can Be Inappropriate

Cone and Laurence simultaneously test the efficacy of the proportional as-sumption and the reasonableness of plaintiffs’ damages theory. In most of theiranalysis, Cone and Laurence focus on aggregate dollar damages rather than onthe number of damaged shares.57 Because they compare the aggregate dollardamages estimated with a proportional trading model with the aggregate dam-age claims, their analyses involve a simultaneous testing of the results of theproportional trading model (damaged shares) and the plaintiff’s damage theory(artificial inflation per share). Unless the per share damages used in claims ad-ministration are exactly the same as the per share damages used by the damagesexpert, estimated aggregate dollar damages will always deviate from the aggre-gate dollar damages that result from claims. This deviation will occur even ifthe number of damaged shares estimated by the expert equals the number ofdamaged shares derived from claims.

Therefore, the comparison of predicted and actual aggregate dollar damagesis of limited usefulness when testing the accuracy of a particular trading modelbecause the comparison confounds the effects of the damage theory (value line)and the effects of the trading model itself. Although there is certainly a strong

53. See Deposition Testimony of John Torkelsen, Ziemack v. Centel, No. 92-C-3551, 1997 U.S.Dist LEXIS 2198 (N.D. Ill. Feb. 26, 1997).

54. See id.55. See id.56. Cone and Laurence reduce the reported volume by 67%, which they indicate is within the plau-

sible range for the NASDAQ reporting bias. See Cone & Laurence, supra note 14, at 515.57. See generally Cone & Laurence, supra note 14.



connection between aggregate damages and the number of damaged shares,conclusions about the accuracy of trading models should not be dependentupon a particular damage theory.

Moreover, there is an important economic distinction between the numberof damaged shares and the number of shares submitted for claims in class actionsecurities litigation. In any securities litigation, it is unlikely that all sharehold-ers who were damaged would file claims. Each shareholder faces an economicdecision about whether the time and effort required to retrieve trading recordsand complete the proof of claims forms is worth the expected damage award.But this individual decision about whether to file a claim does not alter the eco-nomic fact that a given share was damaged.

Understanding the relation between trading models and claims data canprovide important insights about both the trading model and the claims process.But, given the fundamental economic distinction between damaged shares andclaims filed, Cone and Laurence are making a legal conclusion when they assertthat the test of a trading model is its ability to predict claims. This conclusion iserroneous, and the logic behind it is flawed in a way that may result in bad pub-lic policy, as explained below.

It is generally accepted that the number of claims filed in a given case is afunction of the amount of the damage award. The greater the damage award,the more likely shareholders are to spend their time and resources retrievingtrading records and completing the proof of claims forms. The damage awardis, in turn, a function of the predicted aggregate damages that are based in parton the results of a trading model. Therefore, it follows logically that the num-ber of expected claims filed is a function of the predicted damages. Thus, tryingto estimate the number of claims filed in the damage calculation is circular.That is, the lower the damage estimate, the lower the number of expectedclaims will be, but the lower the number of expected claims, the lower the dam-age estimate will be, and so on. This circular logic is in itself a compelling rea-son to keep the damages estimates separate from any estimates of claims filed.

C. More Recent Evidence from Claims Data

The two cases cited by Cone and Laurence provide only anecdotal evidenceregarding the ability of trading models to predict claims data. Conclusions froma sample of two observations will not have a high degree of statistical signifi-cance, particularly when the results of the analyses differ dramatically betweenthe two cases.

In addition, the two cases examined by Cone and Laurence are now quiteold. There are other more recent cases in which the claims data do not supportthe critics’ contentions that the proportional trading model overestimates thenumber of damaged shares. For example, in the In re Health Management, Inc.Securities Litigation case, plaintiffs’ expert estimated that 5.631 million shares



were damaged based on a GTM (1x).58 The claims data showed that 5.014 mil-lion shares, or eighty-nine percent of the estimated number of damages shares,were submitted for claim.59 Furthermore, had the plaintiffs’ expert reduced re-ported NASDAQ volume by sixty-seven percent, as advocated by Cone andLaurence, instead of the fifty percent reduction actually used, the damagedshares from claims submissions as a percent of the GTM (1x) estimated shareswould be greater than ninety percent. Therefore, it is more reasonable to con-clude that the GTM(1x) is supported by the claims data in this case.

V

SUMMARY

Single-trader models (including the proportional and accelerated tradingmodels) and multi-trader models are all GTMs with different assumptionsand/or parameters. This article compared the results of these models and criti-cally evaluated the conclusions reached in previously published research.

Beaver, Malernee, and Keeley’s comparison between their two-tradermodel and the proportional trading model is flawed because they do not com-pute float and they do not adjust reported volume.60 Furthermore, the twentypercent active to eighty percent passive trader assumption used by Beaver et al.appears to have been chosen to exaggerate the differences between the propor-tional trading model and the two-trader model. The unrealistic trading propen-sities assumed in their two-trader model significantly understates the number ofretained shares when compared with a multi-trader GTM using a distribution oftrading propensities that is more consistent with actual observed trading pro-pensities.

The most often cited empirical evidence on the accuracy of the tradingmodels is of highly questionable use. First, testing trading models with claimsdata can be misleading and inappropriate. Second, the individual decision ofwhether to file a claim does not alter the economic fact that a given share wasdamaged. Moreover, given the fundamental economic distinction betweendamaged shares and claims filed, the logic of using claims data as a benchmarkis flawed in a way that may result in bad public policy.

The differences between actual damaged shares and shares reported asdamaged from claims data are significant and appear to vary greatly from caseto case. In the Storage Technology case, the 9.3 million shares submitted forclaims reflect only sixty percent of the buy-and-hold shares originally held byinstitutions that owned approximately fifty percent of the total shares out-standing before the class period and subsequently sold during the class period.

58. Order Approving Distribution of Settlement Fund, In re Health Management, Inc. Sec. Litig.,180 F.R.D. 40 (E.D.N.Y. 1999) (No. CV 96-889 (ADS)(ARL)).

59. See Rule 26 Report of J.B. Torkelsen, In re Health Management, Inc. Sec. Litig., 180 F.R.D. 40(E.D.N.Y. 1999) (No. CV 96-889 (ADS)(ARL)); Deposition Testimony of J.B. Torkelsen, Oct. 20,1999, at 1352-53, Health Management (No. CV 96-889).

60. See Beaver et al., supra note 14, at 128-30.



Thus, the 9.3 million buy-and-hold shares submitted for claims are a misleadingand inappropriate benchmark, because that figure is well short of the actualdamaged shares for institutional holders alone.

Conclusions from a sample of two observations will not have a high degreeof statistical significance, particularly when the results of the analyses differdramatically between the two cases as they do here. Furthermore, the two casesexamined by Cone and Laurence are now dated, and there are other more re-cent cases in which the claims data do not support the critics’ contention thatthe proportional trading model overestimates the number of damaged shares.

The results of the single-trader GTM (1x) are consistent with the results ofmulti-trader GTMs when appropriate assumptions and parameters are used.The estimated damaged shares from a properly specified four-trader model inwhich the trading propensities are generated from the observed empirical dis-tribution yields about ninety-two percent of the estimated damaged shares froma GTM (1x). Thus, there is no evidence that would reject the GTM (1x) as ascientific method to estimate the number of damaged shares in securities litiga-tion.

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TABLE B

Total Float = shares Outstanding 10,000,000

Daily Volume 100,000

Days in Class period 100

GTM (1x) Total

Acceleration Factor 1.0

Retained Shares 6,339,677

Four Trader Model Based on Data from Froot et al.

Total 1 2 3 4

Fraction of Float 100% 5% 56% 32% 7%

Propensity to Trade NA 1.0 1.6 3.6 6.5

Daily Volume 100,000 1,966 35,114 45,025 17,895

0.39% 0.63% 1.44% 2.62%Retained Shares 5,838,155 162,848 2,611,896 2,415,942 647,469

Percent of GTM (1x) 92.1%

Four Trader Based on Sensitivity Analysis on Trading Propensities

Total 1 2 3 4

Fraction of Float 100% 5% 56% 32% 7%

Propensity to Trade NA 1.0 10.0 30.0 40.0

Daily Volume 100,000 278 31,052 53,088 15,582

0.06% 0.56% 1.70% 2.28%Retained Shares 5,629,189 27,071 2,386,391 2,589,426 626,300

Percent of GTM (1x) 88.8%

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12,4

79,5

179,

730,

506

7,52

2,20

76,

011,

026

5,02

4,91

24,

745,

633

5,69

3,87

85,

733,

258

5,67

9,39

35,

793,

542

815,

285

8,79

7,38

6

Sour

ce: T

hom

son

Fin

anci

al S

ecur

itie

s D

ata



TABLE DRATIO OF RETAINED SHARES FROM CLAIMS SUBMISSIONS TO THE RETAINED

SHARES FROM THE GTM (1X) FOR MIDWESTERN

Retained Shares Estimated from GTM (1x) Corrected forProper Adjustment of 60% of Reported Volume 1,880,000

Retained Shares resulting from Claims Submissions(source: Cone and Laurence) 1,550,000

Ratio of Retained Shares from Claims Submission toRetained Shares from GTM (1x) 82.4%



DATA FOR B AND C

Total Float 10,000,000

Daily Volume 100,000

Days in Class period 25

One Trader Total 1

Acceleration Factor 1 99.00%

1.010%

Damaged Shares 2,221,786 2,221,786

Two Trader Total 1 2

Fraction of Float 100% 80% 20%

Propensity to Trade 1 2

Daily Volume 66,667 33,333

0.84% 1.69%

Damaged Shares 2,196,317 1,510,176.73 686,140.19





0.72% 2.19%

Damaged Shares 2,148,866 1,312,572.27 836,293.98





0.63% 2.56%

Damaged Shares 2,098,541 1,160,591.74 937,948.96







0.56% 2.86%

Damaged Shares 2,051,161 1,040,097.85 1,011,063.09





0.36% 3.70%

Damaged Shares 1,878,796 684,495.44 1,194,300.09





0.26% 4.11%

Damaged Shares 1,779,288 510,025.82 1,269,262.00





0.21% 4.35%

Damaged Shares 1,716,262 406,414.49 1,309,847.84

A Comparison of Trading Models Used for Calculating ...

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