The Business Lawyer Forthcoming May Meeting Daubert Standards in Calculating Damages For Shareholder Class Action Litigation

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1 The Business Lawyer Forthcoming May 2007 Meeting Daubert Standards in Calculating Damages For Shareholder Class Action Litigation Linda Allen Professor of Finance Baruch College Zicklin School of Business, CUNY February 2007 Abstract Current practice in class action litigation entails a series of arbitrary assumptions about fundamental parameters that may not meet Daubert standards of scientific evidence. This paper surveys the commonly used models used to estimate stock price inflation and the number of damaged shares for the purposes of damage calculations in shareholder class action litigation. We propose tractable methods for improving current practice using: (1) a regression approach to event studies and (2) a new model (denoted the Theoretically-grounded Microstructure Trading Model or TMTM) that incorporates the well-established literature of market microstructure and trade direction into a model that can be estimated utilizing publicly available data. Actual trading data are used to validate the assumptions of the TMTM approach and to refute the assumptions of extant trading models such as the Proportional Trading Model (PTM) and the Two Trader Model (TTM). Thanks are extended to Aron Gottesman and Oren Shmuel for extremely helpful suggestions and research assistance.

2 Meeting Daubert Standards in Calculating Damages For Shareholder Class Action Litigation Shareholder class action litigation has proliferated in recent years, with no signs of abating. However, the methodology used to calculate damages has not kept pace with developments in the field of economics and finance. The calculation of damages is not a mere arithmetic exercise since the data required to compute the number of damaged shares is typically unavailable even after discovery. Thus, damages experts must estimate economic trading models in order to impute the number of damaged shares. There are concerns that the extant approaches to the computation of damages using these economic trading models fail to meet Daubert standards. The US Supreme Court in Daubert v. Merrell Dow Pharmaceuticals, Inc. reiterated a four-part test for scientific evidence that had been described in Kumho Tire Co. v. Carmichael. 1 To be admissible, the Courts must ascertain that: (1) The theory or technique has been appropriately tested and found valid; (2) The technique or theory has been subjected to peer review and published in a respected journal or other suitable outlet; (3) The error rate is low enough so that the theory or technique is reliable; and (4) The theory or technique is generally accepted within the expert s profession. Current practice by damages experts may fail to meet these standards. This survey describes the shortcomings in current practices and proposes tractable methodologies to correct them so as to comply with the Daubert standards in calculating damages for shareholder class action suits. Two inputs are required to determine damages in fraud-on-the-market cases: (1) an estimate of the price inflation that was caused by the fraudulent disclosures; 2 and (2) an estimate of the number of damaged shares. The measure of damages is then obtained by multiplying the price inflation per share times the number of damaged shares. The first component of the damages calculation the measure of stock price inflation is typically evaluated using an event study methodology. We discuss this approach in Section 2. The second component of the damages calculation the number of damaged shares - is typically estimated using trading models that may make arbitrary assumptions U.S. 579 (1993) and 526 U.S. 137 (1999). 2 It is possible that the fraud on the market may cause share prices to decline, but most cases deal with loss causation resulting from the unwarranted inflation in share prices caused by fraudulent behavior. For concreteness, we will therefore discuss only the case of prices that were inflated by the fraudulent behavior. 2

3 about critical parameter inputs. In Section 3, we describe a new approach the Theoretically-grounded Microstructure Trading Model or TMTM 3 - that estimates retained shares using a new algorithm based on the well-established theory of market microstructure applied to objective financial data. As is well known, it is difficult or impossible to obtain actual trading data to compute the number of retained shares for use in damages calculations. Moreover, damages calculations may precede discovery and therefore must be performed by estimating a trading model that utilizes publicly available data. Extant trading models make assumptions that have not been verified in the literature. In Section 4, we utilize actual proprietary trading data in order to illustrate how the assumptions of commonly used trading models (such as the Proportional Trading Model, PTM, and the Two Trader Model, TTM) are violated, whereas the assumptions of the TMTM are corroborated. The paper concludes in Section Computing the Share Price Inflation Using Event Studies In order to calculate damages in shareholder class action litigation, the expert must perform a but-for analysis that is designed to answer the following question: What would the price of the stock have been during the Class Period, but for the fraudulent behavior? That is, a model must be formulated to evaluate the inflation in the share price that was caused by the allegedly fraudulent behavior that pumped up the stock price. Presumably, share prices return to normal shortly after the revelation of the fraud upon the market s integration of the information into securities prices. Thus, the model must derive a baseline stock return generating function, utilizing data from before the Class Period that continues the trend of stock prices from before the Class Period through the fraudulent disclosure period and then to the correction afterward. The stock price inflation on each date during the Class Period is then calculated by deducting the estimated baseline normal stock price generated by the model from the actually observed stock price. 3 The TMTM is proprietary and is the subject of a pending United States patent application. See Allen (2006). 3

4 Event studies have long been a well-established method for calculating the impact of a particular exogenous event on stock prices, dating back to Fama, Fisher, Jensen, and Roll (1969, hereinafter, FFJR). This methodology involves the specification of a market model, such as the Capital Asset Pricing Model (CAPM), that is estimated over a nonevent period so as to determine the stock price sensitivity to general market conditions. Then the model is estimated over the event period so as to calculate the normal return that would have prevailed if it were not for the event. The abnormal return is obtained by deducting the normal return from the observed actual return. Statistical measures of significance can be calculated to establish confidence intervals regarding the estimates of abnormal returns. This methodology is well suited to the problem of calculating the stock price inflation caused by the alleged fraud on the market. However, there are three major issues that must be resolved. First, the market model must be specified. The CAPM specifies a single market factor in order to estimate stock returns (in excess of the riskfree rate) as a function of market risk, as measured by beta. The CAPM has been criticized by many financial economists, not the least of which were Fama and French, who in their seminal 1993 paper introduced additional factors to the market beta. (1) The Small Minus Big factor, or SMB, measures the monthly performance of small stocks relative to big stocks, (2) The High book-to-market Minus Low book-to-market factor, or HML, measures the monthly equity returns on a portfolio of value stocks relative to growth stocks, and (3) the UMD is the momentum factor, measured as the average return on the two highest prior return portfolios minus the average return on the two lowest prior return portfolios. 4 Other specifications include an industry index, as well as a general market index. Thus, the damages expert must choose the appropriate stock return generating function in order to estimate the baseline stock return. Second, the length of the event window must be specified. Many experts utilize the announcement date only. However, often there is information leakage several days prior to the actual event, particularly in the time period before the passage of Regulation FD that mandated full and symmetric disclosure of market-moving information to all 4 The monthly Fama-French factors are available on Kenneth French s website at 4

5 market participants, not just a favored few financial institutions. 5 Moreover, the market may take a day to react to the information if the announcement is made after the market closes or very late in the trading day. Thus, researchers may utilize a 3-day event window, which starts one day prior to the event date and ends one day after the event date, specified as (-1,+1). The third, and perhaps most important consideration in designing event studies for the purposes of damages calculation is the specification of the events themselves. Typically, this is done in the context of the Complaint and is specified by the plaintiff s attorneys. However, the appropriate estimation methodology depends critically on the events that are alleged to constitute the fraud on the market. If multiple announcement events follow closely one after the other, the conditions required to perform a standard FFJR event study methodology may not be present. That is, if there are multiple event dates that are rapidly occurring, then there may be cross-sectional and intertemporal correlation in the residuals of security returns, thereby violating the econometric conditions of FFJR. Intuitively, one cannot disentangle the abnormal return associated with a single event when there are several announcements within the same event window. That is, the stock price does not have sufficient time between announcements to return to normal so as to form a baseline for the subsequent event. Brown and Warner (1985) show that the existence of serial correlation in excess returns can substantially bias event study results. This is particularly the case if there is an increase in the variance of returns over time, perhaps as uncertainty about fundamental stock value increases during the Class Period. Thus, this is potentially a significant issue for event studies used to measure the extent of stock price inflation for shareholder class action litigation. Although Brown and Warner (1985) offer some suggestions to amend the standard event study methodology, they call for further research into this issue. Schipper and Thompson (1983) offer an alternative specification that can be used with multiple, rapidly occurring event dates, and therefore may resolve the problems associated with serial correlation in stock return residuals. Their regression approach utilizes a Zellner Seemingly Unrelated Regression methodology that specifies an 5 The SEC s adoption of Regulation FD in October 2000 mandated fair disclosure of any material and forward-looking information to the market as a whole, rather than to selected institutions that used to receive the information prior to a more general public release. 5

6 indicator variable that takes on the value of one during any event window. Therefore, the joint effect of all events can be estimated, as well as the abnormal return associated with each event individually. Which methodology should be used depends on the specifics of each case. 3. Calculating Retained Shares The second component of the damages calculation is the computation of the number of damaged shares. The objective for the damages expert is to determine the number of retained shares that were bought at some date during the Class Period and only sold after the end of the Class Period, thereby focusing on those shares that were bought at inflated prices during the Class Period and only sold after the stock price decline upon revelation of the fraud. That is, damages do not apply to any shares that are bought and subsequently sold outside of the Class Period, or to shares that are both bought and sold within the Class Period (the so-called in and out shares ). 6 Because detailed trading data are generally not available, a damages expert must specify a trading model that can be used to eliminate the in and out shares, so as to arrive at an estimate of retained shares to be incorporated into the damage calculation. The trading model must make assumptions about trading behavior in order to estimate the number of retained shares that are eligible for damages. For a trading model to be useful for the purposes of damages calculation, it must have three major attributes: (1) It must be tractable i.e., can by estimated using readily available data; (2) It must not be perceived as arbitrary with respect to critical parameter values and assumptions; and (3) It must be based on generally accepted financial theory and validated using empirical data. Unfortunately, each of the current models used in practice lack one or more of these fundamental attributes. The most commonly used method for calculating the number of retained shares for the purposes of damages in 10B-5 cases is the Proportional Trading Model (PTM) that assumes that all traders (with the exception of institutions and market makers) have 6 For simplicity, I assume throughout this paper that the alleged fraud is completely undetected during the entire Class Period. That is, the alleged fraud coincides with the start of the Class Period and is revealed on the last date of the Class Period. If instead there are multiple, partial revelations of the fraudulent behavior periodically during the Class Period, thereby impacting the amount of share price inflation during the Class Period, then in and out shares may be damaged. The Theoretically-grounded Microstructure Trading Model (TMTM) can be applied to this more complex information structure. 6

7 the same propensity to trade. That is, the daily ratio of adjusted volume to float is calculated and used as an estimate of the factor by which the number of retained shares declines on each date during the Class Period. (See, for example, Furbush and Smith (1994)). Adjusted volume is calculated as the daily trading volume minus dealer and specialist transactions. The float is calculated as the number of shares outstanding minus institutional and insider holdings that either do not trade or are ineligible for damages. The proportion of adjusted volume to float is used as an estimate of the probability that shares purchased during the Class Period are subsequently sold within the Class Period, thereby rendering them ineligible for damages. This assumes that all traders have an equal propensity to trade (the single trader model) and that all shares have the same probability of trading on any date within the Class Period. These restrictive and unrealistic assumptions of the PTM have been criticized by experts and courts alike. For example, the 2000 federal district court ruling in Kaufman v. Motorola, Inc. stated that the proportional trading model has never been tested against reality [and] has never been accepted by professional economists finding it to be a theory developed more for securities litigation than anything else. 7 Finnerty and Pushner (2003) survey the literature and cite the many articles that refute the PTM. 8 The single, homogenous trader assumption of the PTM, while having the benefit of tractability, is clearly without scientific foundation. In an effort to inject more flexibility into the basic PTM, accelerated trading models (ATM) assume a trade propensity for the single representative trader posited in the model that is either proportional or accelerated based on the date of share purchase. That is, accelerated variants of the PTM assess a greater (or lesser) probability of sale for shares more recently purchased. However, the acceleration (or deceleration) factors are arbitrary and simply imposed on the model by assumption. These critical parameter values are not 7 Kaufman v. Motorola, Inc. No. 95-C1069, 2000 WL at 2 (N.D. Ill. Sept. 21, 2000). We test the PTM against reality in Section 4 of this paper by utilizing proprietary trading data in order to assess (and reject) the accuracy of the assumptions of the PTM. 8 See, for example, Beaver and Malernee (1990), Beaver, Malernee and Kealey (1993), Cone and Laurence (1994), Mayer (2000), Bassin (2000). 7

8 calibrated to the data, because data, by and large, do not exist. They are not grounded in theory, because there is no extant theoretical foundation for the PTM and the ATM. 9 Two trader models (TTM) do not fare much better. Rather than assuming a single, homogenous trader, TTMs arbitrarily posit the existence of two types of traders with different trading intensities: investors (who basically buy and hold the shares) and traders (who have a higher propensity to trade than do investors). In contrast to the ATM, in which the proportion of high and low intensity traders changes each day, the TTM assumes that there is a fixed distribution of traders and investors that does not change over time. 10 While sacrificing some tractability, TTMs appear to be more realistic, but we have no way of verifying that since there is often no database available to calibrate the model s assumptions. Trading propensities for each type of trader are simply assumed and differences of opinions cannot be resolved objectively because there is no theoretical underpinning to the TTM. That is, the TTM is an arbitrary model that assumes either a fixed trading propensity for investors or for traders without any scientific foundation. 11 This is often motivated by appealing to general market characteristics, but not related to the microstructure of the particular stock s trading patterns. For example, Cone and Laurence (1994) use claims data to assert that the TTM outperforms the PTM or the ATM. However, Barclay and Torchio (2001) find that the claims data are unreliable and find that the PTM, if properly calibrated, can yield virtually the same results as a more sophisticated four-trader model. Moreover, the well established academic literature on market microstructure suggests that trading propensity is not constant, but instead is a function of market conditions, such as the bid-ask spread, information flows, liquidity needs, etc. (for example, see Cohen, Maier, Schwartz and Whitcomb (1979) for an early survey of the literature). Estimates using actual 9 Due to the similarity between the PTM and the ATM, I do not estimate the ATM separately in the comparison of models conducted in this paper (see Table 1). 10 However, the ATM asymptotically converges to the TTM over time, as the proportion of high and low intensity traders stabilizes. See Finnerty and Pushner (2003). 11 For example, Bassin (2000) assumes a fixed propensity for traders (e.g., traders are more than 20 times more likely to trade than investors), whereas Finnerty and Pushner (2003) assume a fixed trading intensity for investors (such that 0.2 percent of the shares held by investors trade on any given date). There is no theoretical basis for either of these assumptions. Moreover, our analysis in Section 4 of this paper shows that both assumptions do not conform to actual trading patterns. 8

9 proprietary trading data presented in Section 4 of this paper support this theoretical proposition and refute the assumptions of the PTM, ATM and TTM. Allen (2006) presents the Theoretically-grounded Microstructure Trading Model (hereinafter, TMTM), designed to satisfy Daubert standards and fill the gap in practice by mobilizing academic literature to offer a theoretically grounded trading model that can be parameterized using publicly available data. 12 In essence, the TMTM utilizes the theory of market microstructure to estimate two fundamental parameters of the model: (1) the categorization of aggregate trading volume into buys and sells and (2) the average daily trading propensity. First, the modified quote rule and the tick rule (see, for example, Lee and Ready (1991), Finucane (2000), and Ellis, Michaely and O Hara (2000)) are used to distinguish between buys and sells. Whereas the PTM, ATM and TTM approaches all use aggregate trade volume as the basis of their calculations, the new TMTM approach utilizes a categorization of daily volume into the number of shares bought and sold by public, non-institutional customers. Thus, the TMTM calculates the net purchases of shares by public investors on each date of the Class Period. The direction of the trade can be determined by comparing the price to the quoted spread and the price of the preceding trade. If the trade is executed at the ask quote, then it must be a purchase by a public customer from a market maker or broker. If the trade is executed at the bid quote, then it is categorized as a sale by the public to a dealer. Moreover, since market makers cannot sell (buy) on a downtick (uptick), then we classify trades as sells (buys) if the last price was lower (higher) than the transaction price. 13 That is, if the transaction price reflects an uptick (an increase over the last transaction price), then the trade must have initiated by a non-institutional buyer. 14 Similarly, if the transaction price reflects a downtick (a decrease compared to the last transaction price), then the trade must have been initiated by a non-institutional seller. Using a readily available database (the Trade and Quote (TAQ) database made available by the NYSE), therefore, each day s total trading volume can be divided into the total number of sales and the total 12 See L. Allen, A New Theoretically-Grounded Microstructure Trading Model For Calculating Damages in Shareholder Class Action Litigation, Stanford Journal of Law, Business and Finance, January An uptick occurs if the last transaction price was less than or equal to the transaction price. A downtick occurs if the last transaction price was greater than or equal to the transaction price. 14 Thus, a buy ( sell ) occurs if the non-institutional trader initiates a purchase from (sale to) a market maker or specialist. 9

10 number of purchases. 15 The modified quote/tick rule classification of buys and sells satisfies the Daubert standards of reliability. Ellis, Michaely and O Hara (2000) show that the algorithms proposed by Ellis, Michaely and O Hara (2000) and Lee and Ready (1991) correctly classify more than 75% of the trades. 16 Moreover, the models are generally accepted and used in the market microstructure literature. For example, Lee and Ready s (1991) algorithm is used in studies of price formation and informed trading (e.g., Brennan and Subrahmanyam (1995, 1998), Easley, Kiefer and O Hara (1995), Harris and Schultz (1997), and Chakravarty and McConnell (1999)) and in studies measuring trading costs using effective spreads (e.g., Bessembinder (1997), Madhavan and Cheng (1997) and Kumar, Sarin and Shastri (1998)). The second input into the new Theoretically-grounded Microstructure Trading Model (denoted, TMTM) is an estimate of trading propensity. Trade involves the search for a counterparty willing to accept a given transaction price. The likelihood that a trade will take place depends on the probability that the search for a counterparty will be successful. The bid-ask spread is a measure of the search costs, as well as the cost to the dealer of holding an inventory of shares. 17 All else equal, the narrower the bid-ask spread, the greater the likelihood of a trade (see, for example, Garbade (1978)). That is, the greater the probability of success in the search for a counterparty and the lower the dealer s inventory cost. Thus, the TMTM solves for the propensity to trade as a function of the bid-ask spread. Using readily available data on the size of the daily average bidask spread, the TMTM derives the trade propensity for shares bought on each date of the Class Period. The TMTM puts together these two features to formulate a novel approach to estimating retained shares for damages calculations. First, it utilizes the classification of buys (denoted V Bt ) and sells (denoted V St ) to define net purchases of shares (V Bt - V St ) by 15 Another advantage of using the TAQ database is that the total volume data include regional exchanges and trades on electronic communication networks (ECNs), whereas other commonly used databases, such as CRSP, are limited to trades on NYSE, AMEX and Nasdaq only. 16 This is particularly true for stocks that trade on NYSE. For Nasdaq stocks, that may trade within the bidask spread, Ellis, Michaely and O Hara (2000) propose an algorithm to improve the trade classification accuracy, above 90% for some subsamples. Thus, academic models classifying trade direction have a long, well-established history, dating back to Holthausen, Leftwich and Mayers (1987) and Hasbrouck (1988), and are generally accepted by academics, practitioners and regulators. 17 The bid-ask spread is the difference between the ask price minus the bid price. 10

11 public buyers for each date of the Class Period. Summing up over all the dates of the Class Period gives an aggregate (unadjusted) initial estimate of retained shares. However, this unadjusted initial estimate understates the number of shares eligible for damages since it assumes that all shares sold on any date of the Class Period were also purchased during the Class Period. That is, all share sales are assumed to be in and out shares. In actuality, many of the share sales may have been purchased prior to the Class Period, and thus, these sales do not reflect in and out shares, and do not have to be deducted from the retained shares calculation. To add back in the number of shares sold during the Class Period that were purchased prior to the start of the Class Period, the TMTM utilizes a theoretically-grounded trading propensity model to calculate an adjustment factor. The parameters that determine the adjustment factor are not arbitrarily assumed, but rather are estimated in the context of a microstructure model using objective financial data on bid-ask spreads and the number of trades per day. The TMTM arrives at the total amount of retained shares eligible for damages by aggregating the daily net buys plus the adjustment factor for each date of the Class Period. Table 1 compares the TMTM to the PTM and TTM approaches using publicly available data for Enron. For the sake of this hypothetical example, let the Class Period extend from 12/29/2000 to 5/10/2001 (90 days) and let the average holding turnover period (i.e., the number of days required to turn over the total volume of traded shares outstanding over the Class Period) be 90 days. 18 The PTM specification of Furbush and Smith (1994) estimates that the retained shares as of the last date of the Class Period is 176,549,400 shares. 19 The Finnerty and Pushner (2003) TTM approach assumes two types of transactors: (1) investors that are assumed to have a fixed, assumed trading propensity of 0.2% on each date of the Class Period, and (2) traders, whose daily propensity for traders to transact is then derived. 20 By assumption, there is no intraday trading by investors, but 20% of traders net daily trades are assumed to be retraded by the same investors. Thus, 18 Of course, the models can be estimated for any Class Period and for any length of holdover turnover period. These periods are chosen for illustrative purposes only and do not correspond to the periods in actual class action litigation. 19 For more detail regarding these model estimations, see Allen (2006). 20 Finnerty and Pushner (2003) assume (without verification) that 33.3% of the float is held by traders and 66.7% held by investors. 11

12 the retention rate for investors is fixed at 99.8%, whereas the retention rate for traders fluctuates each day with the volume of transactions, averaging 95.7% over the Class Period. Using the hypothetical Enron example, see Allen (2006), investors are shown to have a total of 53,992,643 retained shares and traders a total of 23,738,767 shares. Thus, Table 1 shows that the TTM estimates total retained shares for damages calculation to be 77,731,410. Finally, Allen (2006) estimates the new Theoretically-grounded Microstructure Trading Model (TMTM) for the hypothetical Enron case and obtains retained shares estimates of between 95,652,351 to 113,151,273 shares. Table 1 shows that the TMTM approach yields an estimate of damaged shares that lies between the extremes of the PTM and the TTM approaches. The new TMTM approach is a clear improvement over the extant models currently used for damages calculations because it uses objective market data in order to estimate the parameters of a theoretically-grounded trading model based on the academic literature. The TMTM approach generates critical parameter values using objective data applied to theories of the microstructure of security markets that are generally accepted by academics, regulators and practitioners. In particular, those well-established models do not support the crucial assumption of PTM, ATM and TTM algorithms in use for damages calculations that assume that the likelihood of any trader to sell their shares (i.e., trade propensity) is a fixed, assumed amount. In the next section, we use actual trading data to show that the assumption of constant trading propensities is violated in practice. Moreover, we show that the daily trading propensity is a function of the bid-ask spread, thereby supporting the specifications underlying the TMTM approach. 4. Validating the Assumptions of the TMTM As noted earlier, all extant trading models used for the purposes of damages calculations make assumptions about daily trading propensity. That is, the likelihood that any trader will sell his or her shares on any given date is often an arbitrarily specified, assumed parameter. In none of the extant models (e.g., the PTM, the ATM and the TTM) is this crucial parameter estimated as a function of objective market data. Indeed, none 12

13 of the extant approaches specify a theoretical model of the microstructure of financial markets that would determine the likelihood of any trader to trade on any given date. In contrast, the TMTM is formulated on the basis of a theoretical microstructure trading model. The likelihood of any trader to trade on any given date is a function of market conditions. In particular, theoretical models show that the bid-ask spread is an important measure of market conditions that impacts trading propensity. That is, all else equal, a trader is more likely to trade if the bid-ask spread is relatively narrow, thereby reducing transaction costs and making it more likely that the buyer will find a seller and vice versa. In this section, we utilize proprietary trading data for a large publicly-traded company over the period extending from 1998 to 2004 in order to calculate the holding periods for the customers of several large financial institutions. We do not have sufficient data to calculate the number of retained shares, but we can utilize this proprietary database in order to assess the validity of the assumptions of the various trading models. That is, we determine whether the daily trading propensity is constant, as assumed in the PTM and TTM approaches, or whether it is more closely approximated by a function of the bid-ask spread, as assumed in the TMTM approach in accordance with the market microstructure academic literature. By hand, we match buys and sells for each individual customer (identified by customer code) and calculate the number of days that each position was held, denoted H; i.e., H is the number of days in a roundtrip transaction. We utilize those transactions for which we can match the number of shares bought and sold for each individual customer. However, if a purchase had no subsequent sale, we consider that the position was held to the end of the sample period, and calculate H as the number of days between the buy date and the last record for that broker/dealer. 21 If the sell date precedes the buy date, we consider that transaction to be a short sale and denote it using a negative value for H. If the buy and the sell occur on the same date, then we consider it to be an intraday transaction and denote the holding period H as 0. At the beginning (end) of each customer record, we drop sells (buys) with no earlier (later) buys (sells). We obtained 21 We also perform the analysis without these observations by dropping all holding periods over 500 days. See Model 3 results presented in Table 3. 13

14 proprietary data on 3,045 roundtrip transactions. 22 Table 2 provides the sample statistics. The average holding period is 389 days. However, when we drop all short sales (negative holding periods as shown in Panel B of Table 2) and intraday trades, the average holding period is 535 days (see Table 2). We utilize the daily spread (the highest ask price minus the lowest bid price on each trading date, as obtained from CRSP) on the buy date as an independent variable in an empirical regression model designed to determine whether the holding period H increases (trading propensity decreases) as the spread increases, as posited by both the TMTM and academic studies of market microstructure. Essentially, the model examines whether the bid-ask spread can be used to explain the length of the holding period. A positive coefficient on the spread variable in the model implies that, all else equal, the higher the bid-ask spread, the longer the holding period and the lower the trade propensity. This is consistent with the assumptions of the TMTM, but violates the constant trade propensity assumptions of the other trading models. Table 3 provides the regression results for three different model specifications. In all models, the coefficients on the spread variable are positive, consistent with the TMTM. Model 1 utilizes all observations, including short sales and intraday trades. Model 2 eliminates short sales and intraday trades. In Model 3, we eliminate short sales and intraday trades, as well as all trades with holding periods exceeding 500 days, so as to eliminate any trades that were mistakenly attributed as buy and hold through the end of the observation period, i.e., buy trades that had no subsequent reported sell date in the database. As shown in Table 3, the coefficient on the spread variable is positive for all three models, and is statistically significant (at the 5% level or better) for Models 2 and 3. That is, the highly statistically significant coefficient of in Model 3 implies that, all else equal, a one point increase in spread is consistent with a day increase in the investor s holding period. These results are consistent with the theoretical foundations 22 Although we characterize the trades as roundtrip transactions, we include in the sample buy and hold investors that do not sell prior to the end of the data period. That is, by construction, we consider all buys with no subsequent trades as buy and hold trades and compute the number of days in the holding period as the number of days from the purchase to the end of the data period. To get an estimate of the proportion of buy and hold investors in our sample, Table 3, model 3, shows that the number of long trades with less than 500 days (excluding intraday trades) is 1,468. There are 667 intraday trades in our sample. Thus, approximately 910 transactions out of a total of 3,045 (=3,045-1, ) is comprised of buy and hold investors. 14

15 for the TMTM that suggest that the wider the bid-ask spread, the longer the holding period and the lower the trading propensity. The results presented in Table 3 are not consistent with the assumptions of constant trading propensity inherent in the PTM and TTM approaches. 5. Conclusions We apply academic standards to the topic of calculating damages for the purposes of shareholder class action litigation. Such calculations require the computation of: (1) the extent of the stock price inflation from the allegedly fraudulent announcements on each date of the Class Period, and (2) the number of damaged shares that were purchased at inflated prices during the Class Period, but not resold subsequently during the Class Period. We offer suggestions to improve the methodology currently used by damages experts to estimate each of these components, so as to meet Daubert standards of scientific verifiability and objectivity. We describe an alternative regression approach to the event study methodology currently used to estimate stock price inflation during the Class Period. We show that the conditions of the standard event study methodology may be violated if event dates are bunched together so as to create possible serial correlation in the residuals of the estimated stock returns. In order to estimate the number of damaged shares, we offer a new Theoreticallygrounded Microstructure Trading Model (TMTM) that utilizes publicly available data in order to estimate a model that is founded on the principles of market microstructure. Thus, we distinguish between buys and sells on each date of the Class Period. Moreover, we estimate the daily trading propensity (in order to adjust daily net purchases to reflect in and out shares) as a function of the bid-ask spread. Analysis performed using proprietary trading data supports the assumptions of the TMTM, and is inconsistent with the assumptions of commonly used trading models (the Proportional Trading Model, PTM, the Accelerated Trading Model, ATM, and the Two Trader Model, TTM). 15

16 References Allen, L., A New Theoretically-Grounded Microstructure Trading Model For Calculating Damages in Shareholder Class Action Litigation, Stanford Journal of Law, Business and Finance, January Barclay, M. and F. C. Torchio, A Comparison of Trading Models Used for Calculating Aggregate Damages in Securities Litigation, Law and Contemporary Problems, vol. 64, nos. 2&3, Spring/Summer 2001, pp Bassin, W.M., A Two Trader Population Share Retention Model for Estimating Damages in Shareholder Class Action Litigations, 6 Stanford Journal of Law, Business and Finance, 49, W.H. Beaver and J.K. Malernee, Estimating Damages in Securities Fraud Cases, Cornerstone Research, W.H. Beaver, J.K. Malernee, and Keeley, Potential Damages Facing Auditors in Securities Fraud Cases, in Accountants Liability: The Need for Fairness 113 (John T. Behrendt t al., eds., 1994). Brennan, M.J. and A. Subrahmanyam, Investment Analysis and Price Formation in Securities Markets, Journal of Financial Economics, vol. 38, 1995, pp Brown, S.J., and J.B. Warner, 1985, Using Daily Stock Returns: The Case of Event Studies, Journal of Financial Economics 14, Chakravarty, S. and J.J. McConnell, Does Insider Trading Really Move Stock Prices? Journal of Financial and Quantitative Analysis, vol. 34, 1999, pp Cohen, K.J., S.F. Maier, R.A. Schwartz, D.K. Whitcomb, Market Makers and the Market Spread: A Review of Recent Literature, Journal of Financial and Quantitative Analysis, November 1979, pp Cone, K.R., and J.E. Laurence, How Accurate are Aggregate Damages in Securities Fraud Cases? 49 Business Law 505, Easley, D., N.M. Kiefer, and M. O Hara, Cream-Skimming or Profit-Sharing? The Curious Role of Purchased Order Flow, Journal of Finance, vol. 51, 1996, pp Ellis, K., R. Michaely, M. O Hara, The Accuracy of Trade Classification Rules: Evidence from Nasdaq, Journal of Financial and Quantitative Analysis, vol. 35, no. 4, December 2000, pp Fama, Eugene, L. Fisher, Michael Jensen, and Richard Roll. The Adjustment of Stock Prices to New Information. International Economic Review 10 (February 1969),

17 Fama, E. F., and K. French, Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33, Finnerty, J. and G. Pushner, An Improved Two-Trader Model for Measuring Damages in Securities Fraud Class Actions, 8 Stanford Journal of Law, Business and Finance, Spring 2003, pp Finucane, T.J., A Direct Test of Methods for Inferring Trade Direction from Intra-Day Data, Journal of Financial and Quantitative Analysis, vol. 35, no. 4, December 2000, pp Furbush, D and J.W. Smith, Estimating the Number of Damaged Shares in Securities Fraud Litigation: An Introduction to Stock Trading Models, The Business Lawyer, 49 Business Law 527, February 1994, pp Garbade, K.D., The Effect of Interdealer Brokerage on the Transactional Characteristic of Dealer Markets, Journal of Business, vol. 51, no. 3, July 1978, pp Harris, J.H. and P.H. Schultz, The Importance of Firm Quotes and Rapid Executions: Evidence from the January 1994 SOES Rules Change, Journal of Financial Economics, vol. 45, 1997, pp Hasbrouck, J., Trades, Quotes, Inventories and Information, Journal of Financial Economics, vol. 22, 1988, pp Houlthausen, R.W., R.W. Leftwich and D. Mayers, The Effect of Large Block Transactions on Security Prices: A Cross-Sectional Analysis, Journal of Financial Economics, vol. 19, 1987, pp Kumar, R., A. Sarin, and K. Shastri, The Impact of Options Trading on the Market Quality of the Underlying Security: An Empirical Analysis, Journal of Finance, vol. 53, 1998, pp Lee, C.M. and M.J. Ready, Inferring Trade Direction from Intraday Data, Journal of Finance, vol. 46, no. 2, June 1991, pp Madhavan, A. and M. Cheng, In Search of Liquidity: Block Trades in the Upstairs and Downstairs Markets, Review of Financial Studies, vol. 10, 1997, pp Schipper, K. and R. Thompson, 1983, The Impact of Merger Related Regulations on Shareholders of Acquiring Firms, Journal of Accounting Research 21, Snyder, D.L., Random Point Processes, 1975, New York: Wiley-Interscience. 17

18 Table 1 Summary of Methods Estimated Retained Shares PTM Furbush Smith (1994) 176,549,400 TTM Finnerty Pushner (2003) 77,731,410 TMTM Using Lee Ready (1991) 95,652,351 Using Ellis, Michaely & O'Hara (2000) 113,151,273 Table 2 Descriptive Statistics of the Sample of Proprietary Trading Data Panel A: All Observations Variable No. of Average Standard Minimum Maximum Observations Deviation Holding 3, ,064 Period, H Price, P 3, Volume, V 3,045 22, , ,000,000 Spread, BA 3, Panel B: Only Positive Holding Periods (No Short Sales or Intraday Trades): Variable No. of Average Standard Minimum Maximum Observations Deviation Holding 2, ,064 Period, H Price, P 2, Volume, V 2,224 14,293 40, ,000 Spread, BA 2, Notes: The holding period, H, is defined as the number of days in a roundtrip (buy and sell) transaction obtained from proprietary individual customer records reported over the period from 1998 to The price, P, is the closing price from the CRSP database on each trading date. The volume, V, is the daily volume from the TAQ database. The spread is the difference between the highest ask price minus the lowest bid price on each trading date. 18

19 Table 3 Regression Results Dependent Variable: Holding Period, H Variable Model 1 All Observations Model 2 + Holding Periods Model 3 + Holding <500 days Intercept *** (34.67) *** (47.24) (10.20) Price, P -6.76*** (1.43) -9.84*** (1.92) 2.40*** (0.41) Volume, V ( ) *** (0.0004) ** (0.0001) Spread, BA (11.14) 36.50** (14.53) 10.69*** (3.32) No. of Obs. 3,045 2,224 1,468 R % 2.22% 5.37% Notes: Standard errors in parentheses. ***, **, * denote significance at the 1%, 5%, 10% levels, respectively. Model 1 utilizes all observations. Model 2 uses only positive holding period observations, eliminating all intraday trades and short sales. Model 3 uses only positive holding periods of less than 500 days, eliminating all intraday trades, short sales and all questionably long holding periods. 19