PATENT DOCUMENT

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. A new theoretically-grounded microstructure trading model (TMTM) method and arrangement determines retained shares for use in damages calculations for securities fraud cases and complies with Daubert standards. In particular, the TMTM method classifies trades as “buys” or “sells”, as well as estimates trading intensity using the bid-ask spread. The TMTM method has been found to yield more accurate estimates of retained shares than heretofore.

Full Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims priority to U.S. provisional application Ser. No. 60/815, 356, filed Jun. 21, 2006. 
    
    
     DESCRIPTION OF THE RELATED ART 
     Financial markets function effectively only if they are fed a steady diet consisting of full and accurate information disclosure. Recent scandals regarding failure to disclose material information, or the misleading and deceptive disclosures of information, have led to a proliferation of fraud-on-the-market litigation. Damages assessed in these cases act as the economic penalties that, ex ante, inhibit behavior that undermines the integrity of global financial markets. In order to act as an effective deterrent on bad behavior, damages must be properly determined. 
     There are two inputs required to accurately determine damages in fraud-on-the-market cases. The first is an estimate of the price inflation that was caused by the fraudulent disclosures. 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. For concreteness, only the case of prices that were inflated by the fraudulent behavior is discussed herein. The second is an estimate of the number of damaged shares. The measure of damages is then determined by multiplying the price inflation per share times the number of damaged shares. 
     However, this is easier said than done. Whereas an event study based on a theoretical market method can be used to estimate the extent of the share price inflation using daily share prices and the public record of relevant events, there is no such theoretical basis currently used to calculate the number of damaged shares. Moreover, detailed shareholding and transaction data are not available to implement a purely empirical approach to an estimation of the number of damaged shares. Current approaches rely on arbitrary assumptions that invariably result in dueling experts, because they cannot be validated by either theory or empirical investigation. 
     The problem of accurately determining damages is complicated by the fact that damage estimates may precede discovery and, therefore, must be performed using publicly available data. However, even if damages are calculated with the benefit of the discovery process, it is virtually impossible to obtain on discovery the detailed transaction data required to exactly count the number of damaged shares. This is because damages do not apply to any shares that are bought before and subsequently sold after the class period (Class Period), i.e., the time frame during which the alleged fraud or other securities law violations artificially inflated the price of the shares at issue, or to any shares that are both bought and sold within the Class Period (the “in-and-out shares”). Indeed, it is the number of shares that were bought at some date during the Class Period and only sold after the end of the Class Period that must be accurately determined, thereby focusing on only those damaged shares that were bought at inflated prices during the Class Period and only sold after the share price declined upon revelation of the fraud. Trading methods have been employed to eliminate the in-and-out shares so as to arrive at an estimate of the number of damaged shares to be used in the damage determination. 
     For a trading method to be useful, it must be tractable, i.e., can be estimated using readily publicly available data; it must not be perceived as arbitrary with respect to critical parameter values and assumptions; and it must be based on generally accepted financial theory and validated using empirical data. Unfortunately, each of the currently known prior art methods used in practice lacks one or more of these fundamental attributes. Furthermore, the U.S. Supreme Court in  Daubert v. Merrell Dow Pharmaceuticals, Inc.,  509 U.S. 579 (1993) reiterated a four-part test for scientific evidence that had been described in  Kumho Tire Co. v. Carmichael,  526 U.S. 137 (1999). The so-called Daubert standard states that to be admissible, the courts must ascertain that the theory or technique has been appropriately tested and found valid; the technique or theory has been subjected to peer review and published in a respected journal or other suitable outlet; the error rate is low enough so that the theory or technique is reliable; and the theory or technique is “generally accepted” within the expert&#39;s profession. 
     The most commonly used known method of determining the number of retained, i.e., damaged, shares for the purpose of determining damages in litigation involving U.S. Securities and Exchange Commission Rule 10b-5, which rule prohibits any act or omission resulting in fraud or deceit in connection with the purchase or sale of any security, including insider trading, is a proportional trading model (PTM) that assumes that all traders (with the exception of institutions and market makers) have the same propensity to trade. That is, the daily ratio of “adjusted volume” to “float” is determined 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 less 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 ratio of the adjusted volume to the 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 method) and that all shares have the same probability of trading on any date within the Class Period. 
     However, the restrictive and unrealistic assumptions of the PTM method, have been criticized by experts and courts alike. For example, the court in  Kaufman v. Motorola, Inc ., No. 95-C1069, 2000 WL 1506892 at 2 (N.D. Ill. Sep. 21, 2000) 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.” Finnerty and Pushner (2003) survey the literature and cite the many articles that refute the PTM method. See, for example, Beaver and Malernee (1990); Beaver, Malernee and Kealey (1993); Cone and Laurence (1994); and Mayer (2000), Bassin (2000). 
     The single, homogenous trader assumption of the PTM method, while having the benefit of tractability, is clearly arbitrary. In an effort to inject more flexibility into the basic PTM method, an accelerated trading model (ATM) has been proposed, and this method assumes a trade propensity for the single representative trader that is either proportional or accelerated based on the date of share purchase. That is, accelerated variants of the PTM method assess a greater (or lesser) probability of sale for shares more recently purchased. However, the acceleration (or deceleration) factors are arbitrary. They are not 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 methods. 
     Also known is a two trader model (TTM) which does not fare much better. Rather than assuming a single, homogenous trader, the TTM method arbitrarily posits 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 method, in which the proportion of high and low intensity traders changes each day, the TTM method assumes that there is a fixed distribution of traders and investors that does not change over time. Actually, the ATM method asymptotically converges to the TTM method over time, as the proportion of high and low intensity traders stabilizes. See Finnerty and Pushner (2003). 
     While sacrificing some tractability, the TTM method appears to be more “realistic,” but there is no way of verifying that since there is often no database available to calibrate the TTM method&#39;s assumptions. Trading propensities for each type of trader are assumed and differences of opinions cannot be resolved objectively, because there is no theoretical underpinning to the TTM method. That is, the TTM method assumes either a fixed trading propensity for investors, or for traders. For example, Bassin (2000) assumes a fixed propensity for traders (e.g., traders are more than twenty 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. This is often motivated by appealing to general market characteristics, but not related to the microstructure of the particular share&#39;s trading patterns. For example, Cone and Laurence (1994) use claims data to assert that the TTM method outperforms the PTM method or the ATM method. However, Barclay and Torchio (2001) find that the claims data are unreliable and find that the PTM method, if properly estimated, can yield virtually the same results as a more sophisticated four-trader method. 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. See, for example, Cohen, Maier, Schwartz and Whitcomb (1979) for an early survey of the literature. 
     SUMMARY OF THE INVENTION 
     One feature of this invention, briefly stated, resides in an arrangement for, and a method of, more accurately determining damages in shareholder class action litigation by offering a theoretically-grounded microstructure trading model (TMTM) that satisfies the Daubert standard and is parameterized using publicly available data. 
     In essence, two parameters of the TMTM method are estimated. First, directional trading methods, such as the modified quote rule and the tick rule (see, for example, Lee and Ready (1991); Finucane (2000); and Ellis, Michaely and O&#39;Hara (2000)) are used to distinguish between buys and sells. Whereas the known PTM, ATM and TTM methods all use aggregate trade volume as the basis of their calculations, the new TMTM method utilizes a categorization of daily share volume into the number of shares bought and sold by retail customers. 
     The direction of a trade can be determined by comparing the transaction price to the quoted “bid-ask spread”, i.e., the difference between the “ask” quote and the “bid” quote, and the price of the preceding trade. If the trade is executed at the ask quote, then it is categorized as a purchase by a retail 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 trades are classified as sells (buys) if the last price was lower (higher) than the transaction price. 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. That is, if the transaction price reflects an uptick (an increase over the last transaction price), then the trade must have been initiated by a retail buyer. Thus, a “buy” (“sell”) occurs if the retail trader initiates a purchase from (sale to) a market maker or specialist. 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 retail seller. 
     Using a readily publicly available database, such as the trade and quote (TAQ) database, which is a collection of intraday trades and quotes for all securities listed on the New York Stock Exchange (NYSE), American Stock Exchange (Amex), Nasdaq National Market System (Nasdaq) and SmallCap issues, each day&#39;s total trading volume can be divided into the total number of sales and the total number of purchases. The modified quote/tick rule classification of buys and sells satisfies the Daubert standards of reliability. Ellis, Michaely and O&#39;Hara (2000) show that the Ellis, Michaely and O&#39;Hara (2000) and the Lee and Ready (1991) algorithms correctly classify more than 75% of the trades. This is particularly true for shares that trade on the NYSE. For Nasdaq shares, that may trade within the bid-ask spread, Ellis, Michaely and O&#39;Hara (2000) proposed an algorithm to improve the trade classification accuracy, above 90% for some subsamples. 
     Thus, directional trading methods 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. The Ellis, Michaely and O&#39;Hara (2000) and the Lee and Ready (1991) quote/tick methods are incorporated herein by reference thereto. Other methods that are generally accepted and used in the market microstructure literature include Lee and Ready&#39;s (1991) algorithm used in studies of price formation and informed trading (e.g., Brennan and Subrahmanyam (1995, 1998); Easley, Kiefer and O&#39;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 parameter input into the TMTM method is an estimate of trading propensity. A 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. 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 the counterparty and the lower the dealer&#39;s inventory cost. Thus, the propensity to trade can be determined as a function of the bid-ask spread. Using readily publicly available data on the size of the daily average bid-ask spread, the trade propensity for shares bought on each date of the Class Period is determinable. 
     The novel features which are considered as characteristic of the invention are set forth in particular in the appended claims. The invention itself, however, both as to its construction and its method of operation, together with additional objects and advantages thereof, will be best understood from the following description of specific embodiments when read in connection with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a depiction of a first step of the method in accordance with this invention for determining a maximum holding period share turnover; 
         FIG. 2  is a flow chart depiction of a second step of the method in accordance with one embodiment of this invention, in which daily transaction volume is separated into daily sales and daily purchases; 
         FIG. 3  is a flow chart depiction analogous to  FIG. 2  of another embodiment of this invention, in which daily transaction volume is separated into daily sales and daily purchases; 
         FIG. 4  is a flow chart depiction of a third step of the method in accordance with one embodiment of this invention, in which the daily average sales propensity is determined as a function of the daily bid-ask spread on the date of purchase; 
         FIG. 5  is a flow chart depiction analogous to  FIG. 4  of another embodiment of this invention, in which the daily average sales propensity is determined as a function of the daily bid-ask spread on the date of purchase; 
         FIG. 6  is a flow chart depiction of a fourth step in accordance with the method of this invention for determining the total number of damaged shares over the class period; 
         FIG. 7  is a flow chart depiction of a fifth step in accordance with the method of this invention, in which a confidence interval is obtained around an expected value of the damaged shares; and 
         FIG. 8  illustrates a general purpose computer configured to execute the method of this invention, including a plurality of inputs for entry into the computer. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     An arrangement for, and a method of, more accurately determining damages in shareholder class action litigation propose a theoretically-grounded microstructure trading model (TMTM) that is parameterized using publicly available data. The number of retained or damaged shares is determined using the TMTM method in accordance with this invention, and is then compared to the number determined from the known PTM and the TTM methods, all illustrated using publicly available data for Enron Corporation, by way of a non-limiting example. 
     All trading methods utilize the float, i.e., the number of shares that are available to trade. It is common practice to calculate the float by deducting from the total number of shares outstanding the number of shares held by insiders and institutions. The float on date t is denoted as F t . To calculate the number of days it takes for the float to change hands, the ratio F t /V t  is computed where V t  is equal to the total trading volume (less dealer and specialist trades) on date t. N denotes the number of days in the maximum investor holding period, i.e., the number of days required to turn over the total volume of traded shares outstanding over the Class Period. As shown in method step  1  of  FIG. 1 , N is calculated as follows:
 
 N =Public Share Float/Adjusted Daily Volume=(Number of Shares Outstanding Less Institutional and Insider Holdings)/(Average Number of Shares Traded Less Market Maker and Specialist Trades)
 
     Next, on any given date, the total trading volume (denoted V t ) can be divided into the number of buys (denoted V Bt ) and the number of sells (denoted V St ) using directional trading methods, such as the Lee and Ready (1991) or the Ellis, Michaely and O&#39;Hara (2000) modified quote/tick rules, as shown in method step  2  ( FIG. 2 ), or alternative method step  2 ′ ( FIG. 3 ), of the drawings, such that V St +V Bt =V t . 
     The Ellis, Michaely and O&#39;Hara (2000) rule categorizes all trades that are executed at the ask quote as buys and all trades executed at the bid quote as sells. The Lee and Ready (1991) rule is similar, except that it classifies trades as buys (sells) if they transact at a price above (below) the midpoint of the bid-ask spread, rather than at the ask quote (bid quote). All other trades are categorized by the tick rule, such that buys (sells) take place on an uptick or a zero-uptick (a downtick or a zero-downtick). A trade is a zero-uptick (zero-downtick) if there was no change in the transaction price as compared to the preceding trade, but that the last price change was an uptick (downtick). 
     The designation of trades as either buys or sells allows the number of shares purchased on each date of the Class Period to be determined. However, the determination of the retained damaged shares must reduce the number of damaged shares by the number of in-and-out shares purchased during the Class Period that are subsequently sold during the Class Period. Thus, the buyers&#39; propensity to sell must be determined in order to determine whether the sold shares were originally bought during the Class Period or before. 
     As a simple first approximation, a microstructure-driven proportional trading rule is assumed. As described below, this will be replaced by a theoretically derived parameter value that allows for a continuum of trading intensities for any type of market participant. However, the assumption that all shares have the same probability of trade is a useful simplification for expositional purposes only. If all shares are equally likely to be traded, then the shares sold can be obtained from shareholders who bought their shares on any one of the last F t /V t  days, i.e., during the maximum holding period N. 
     For the purposes of the damage determination required herein, the days in the Class Period must be distinguished with those days before and after the Class Period. Let the Class Period extend from t=0, 1, T, where there are T days in the Class Period. Thus, days before the Class Period are denoted as t&lt;0, days during the Class Period are denoted as 0≦t≦T, and days after the Class Period are denoted as t&gt;T. Using the proportional assumption as a first approximation, the number of retained shares (denoted RS T ) is determined as follows: 
     
       
         
           
             
               
                 
                   
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     Equation (1) sums up for all days in the Class Period from t=0, . . . T, the total number of share purchases on each date in the Class Period (V Bt ) less the fraction of the shares that were sold (V St ) by traders who bought their shares during the Class Period, where T is the number of days in the Class Period. N denotes the number of days in the holding period; i.e., the maximum number of days required to turn over the total volume of traded shares outstanding over the Class Period. 
     Despite the restrictive assumption of proportional trading, the existing PTM, ATM and TTM methods have been improved by incorporating the theoretical underpinnings associated with methods that categorize trade direction. That is, buys and sells are distinguished so as to denote which shares are purchased during the Class Period and which are sold. This allows a more precise breakdown of total trading volume and therefore a more exact estimation of the number of retained shares. 
     However, it is also known to relate the propensity to trade (trading intensity) to the bid-ask spread. Financial intermediaries (FIs) that make markets, in particular securities, receive payment for their liquidity services in the form of the bid-ask spread. The bid-ask spread is wider, the higher the FIs&#39; order processing costs, the greater the FIs&#39; inventory holding costs, or the greater the adverse information costs, that is, the more uncertainty about the security&#39;s fundamental value. Thus, traders with a high propensity to trade will be less likely to purchase shares in illiquid markets with high bid-ask spreads. Indeed, the expected propensity to trade for a particular security can be estimated on a given date by examining the bid-ask spread in the market on the date in which the trader purchased the shares. This will provide a market measure of the security buyers&#39; likelihood to subsequently sell their shares. Share purchases of less liquid shares signal a lower expected probability of a future trade (sale). 
     Rather than arbitrarily assuming a trading intensity for one or more classes of traders, this invention proposes, as shown in method step  3  of  FIG. 4 , or in alternative method step  3 ′ of  FIG. 5 , of the drawings, the determination of an average daily propensity to trade that is consistent with a continuum of trader types. 
     Garbade (1978) asserts that “the average rate of flow of purchase and sale orders from the public to the dealer community at any point in time is a function of average dealer quotations at that time and an exogenously specified equilibrium price.” (p. 484). Equation (1), as well as the known PTM and TTM methods, assume that there is an equal probability of selling shares that were bought yesterday as compared to shares that were bought N days ago. Although the ATM method does not make this assumption, it instead makes arbitrary assumptions about the changes in the rate of trading intensity over time. Whether this is an accurate assumption can be determined using the method of trading propensity as a function of the bid-ask spread. Following Garbade (1978), the mean order arrival time of a sell order (by a retail trader) can be expressed as:
 
μ st   =a   1   +a   2 (   B −P* ) if    B &gt;P *−( a   1   /a   2 ) and 0 otherwise  (2)
 
where  B  is the average daily bid price and P* is the transaction price (assumed to be the midpoint of the bid and ask prices). Thus, if μ st  equals twenty trades per hour, that implies three minutes between trades. If there are seven hours in a trading day, then this stock has an average propensity to generate 140 sales per day. At a round lot of one-hundred shares, this implies an average sales propensity of 14,000 shares on date t.
 
     Substituting the midpoint of the bid-ask quote for the transaction price P* yields: 
     
       
         
           
             
               
                 
                   
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                         otherwise 
                       
                     
                   
                 
               
               
                 
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     where a 1 &gt;0 and a 2 &gt;0 and  B −Ā&lt;0. 
     Equation (3) implies that the wider the bid-ask spread, the lower the probability of sales μ st . This is consistent with microstructure theory that suggests that trading intensity is inversely related to liquidity, i.e., the lower the bid-ask spread, the more liquid the stock, and the higher the propensity to trade ceterisparibus. 
     Determining the sales propensity on any given purchase date t, the proportion of sales volume in equation (1) can be weighted by μ st  in order to solve for an endogenously determined acceleration factor, as shown in method step  4  of  FIG. 6  of the drawings. Therefore, equation (1) can be rewritten as follows: 
                     RS   T     =       ∑     t   =   0     T     ⁢     {       V   Bt     -       V   St     ⁢         ∑     j   =   1     t     ⁢       μ     t   -   j       ⁢     V     B   ,     t   -   j                 ∑     n   =   1     N     ⁢       μ     t   -   n       ⁢     V     B   ,     t   -   n                   }                 (   1   ’     )               
where μ st  is the daily sales probability. The value of μ st  is an endogenously determined random variable. That is, shares purchased on a given date also have an endogenously determined average propensity to be sold.
 
     Using the mean of the trade probability distribution, a measure of share trade acceleration that is a function of the daily bid-ask spread can be derived and not simply arbitrarily assumed. Using the standard deviation of the probability distribution, as shown in method step  5  of  FIG. 7  of the drawings, a confidence interval is defined around the mean, so as to arrive at the confidence interval of the range of retained shares calculations. The elapsed time between trades has an exponential probability density function. See Snyder (1975). Thus, the novel TMTM method meets the Daubert standards. 
     Publicly available data on Enron is employed herein in order to compare the results of the TMTM method to the PTM and the TTM methods. For the sake of this hypothetical example, let the Class Period extend from Dec. 29, 2000 to May 10, 2001 (90 days) and let the holdover turnover period be 90 days. Of course, the methods can be estimated for any Class Period and for any length of holdover turnover period. The periods chosen herein are for illustrative purposes only and do not correspond to the periods in actual class action litigation. Three different methods are now used to estimate the number of retained shares using the PTM, the TTM and the TMTM methods, and to compare their results. 
     The PTM method of Furbush and Smith (1994) is now estimated. The equation for the retained shares as of date t is shown as: 
     
       
         
           
             
               
                 
                   
                     RS 
                     t 
                   
                   = 
                   
                     
                       RS 
                       
                         t 
                         - 
                         1 
                       
                     
                     + 
                     
                       
                         V 
                         t 
                       
                       ( 
                       
                         1 
                         - 
                         
                           
                             RS 
                             
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                               1 
                             
                           
                           F 
                         
                       
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                   4 
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     where RS t  is the cumulative retained shares as of date t, V t  is the volume of trading on date t, and F is the float (total number of shares outstanding less the shares held by insiders and institutions). 
     An estimate of Enron&#39;s float over the Class Period is estimated as 213,672,332 shares. All three methods use the same float value and thus the comparative results are not sensitive to this measure. Table 1 shows the determination of the PTM method using equation (4) and daily volume data for the days of the Class Period (obtained from TAQ) and the average float of 213,672,332 shares. As shown in Table 1, the PTM method estimates that the retained shares as of the last date of the Class Period is 176,549,400 shares. 
     The PTM method assumes that all shares have an equal probability of trading on any given date. In contrast, the TTM method assumes that some traders have a greater propensity to trade than others. These high propensity security transactors are called traders, whereas the lower propensity, buy and hold participants are called investors. The Finnerty and Pushner (2003) TTM method assumes that investors have a fixed, assumed trading propensity on each date of the Class Period. The daily propensity for traders to transact is then derived. Finnerty and Pushner (2003) assume (without verification) that 33.3% of the float is held by traders and 66.7% is held by investors, who have a constant assumed daily trading probability of 0.2%. As in the estimation of the PTM method (and subsequently in the estimate of the TMTM method), the daily volume of transactions obtained from the TAQ database is utilized, as well as the total float of 213,672,332 shares. By assumption, there is no intraday trading by investors, but 20% of traders&#39; net daily trades is assumed to be retraded by the same investors. Thus, 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. 
     Table 2 calculates the retained shares over the Class Period for the two types of traders. Investors are shown to have a total of 53,992,643 retained shares and traders a total of 23,738,767 shares. Thus, the TTM method determines that the total retained shares for damages calculation to be 77,731,410. The disparity between the PTM and the TTM methods&#39; estimates of retained shares stems from assumed differences in method specifications and parameters. Since these assumptions are not, and cannot be, verified in the context of methods without any theoretical underpinnings, the veracity of either of these estimates cannot be validated. 
     The implicit assumption of the PTM method is that all shares have an equal probability of trading on each date of the Class Period. This trading intensity is calculated as the average daily volume of trades divided by the float. Using the data for the PTM method estimates shown in Table 1, this implies an average daily trading propensity of around 1.9%. However, this is simply an ad hoc estimation that is not grounded in any theory of market microstructure or method of trading behavior. Moreover, the TTM method makes equally arbitrary assumptions about trade propensity for each of the exogenously specified trader classes. That is, the daily trading propensity for investors is assumed to be 0.2%, whereas the daily trading propensity for traders averages 4.3%. 
     By contrast, in accordance with this invention, the TMTM method utilizes microstructure theory in order to determine each day&#39;s trading propensity as a continuous function of the daily bid-ask spread. Table 3 shows the TMTM method&#39;s determination of the daily average trading intensity for each date within the Class Period and for the 90 days prior to the start of the Class Period that constitutes the maximum turnover holding period. As discussed above, the daily average trading propensity (μ st ,) is estimated using equation (3). The Garbade (1978) specification of a 1 =average time between public orders=420/(No. of Trades Per day) is used where 420 minutes are the total number of minutes in a trading day, and a 2 =0.5a 1 . From Table 3, the daily average trading propensity varies across the days in the Class Period and therefore invalidates the assumption of a constant trading propensity for all traders (in the Furbush-Smith PTM method) or for investors (in the Finnerty-Pushner TTM method). 
     To estimate the number of retained shares, equation (I′) requires the categorization of the daily volume of trades into sells and buys using the Lee and Ready (1991) and the Ellis, Michaely and O&#39;Hara (2000) algorithms. Results using the Lee and Ready (1991) algorithm are shown in Table 4 and for the Ellis, Michaely and O&#39;Hara (2000) algorithm in Table 5. All trades are classified on each date up to 90 days before the start of the Class Period until the end of the Class Period. Following Lee and Ready (1991), all trades are classified using quote data available five seconds prior to the transaction time. To classify early morning trades, the tick value of the last trades of the last trading day is used. 
     Table 6 compares the determination of the number of retained shares over the Class Period using the three methods. The PTM method of Furbush and Smith (1994) yields the highest estimate of 176,549,400 retained shares. The TTM method of Finnerty and Pushner (2003) yields the lowest estimate of 77,731,410 shares. The TMTM method yields retained shares estimates of between 95,652,351 to 113,151,273 shares. Moreover, the TMTM method 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 summary, the TMTM method uses actual publicly available market data, has well-accepted theoretical underpinnings, and is not subject to the same concerns associated with other known trading methods that rely on arbitrary assumptions to set key parameter values. Indeed, the TMTM method shows, for a specific example, that the fundamental assumption of the PTM and the TTM methods, i.e., constant daily trading intensities, is not met. Moreover, the TMTM method&#39;s determination of damages lies well within the extremes of the PTM and the TTM methods for a specific example. The novel TMTM method meets Daubert standards of reliability, general acceptance in the profession and accuracy of estimation as compared to the existing methods used to estimate retained shares for damages calculations in securities fraud litigation. The method steps 1-5 depicted in the drawings can be executed by a programmed computer under the control of a program. 
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 The Proportional Trading Model 
               
             
          
           
               
                 Days in the C.P. 
                 Daily Volume 
                 RS(t) 
               
               
                   
               
             
          
           
               
                 0 
                 1,814,800 
                 1,814,800 
               
               
                 1 
                 3,230,200 
                 5,017,565 
               
               
                 2 
                 4,240,700 
                 9,158,682 
               
               
                 3 
                 9,181,100 
                 17,946,251 
               
               
                 4 
                 5,072,700 
                 22,592,897 
               
               
                 5 
                 2,460,300 
                 24,793,054 
               
               
                 6 
                 3,921,500 
                 28,259,531 
               
               
                 7 
                 3,531,100 
                 31,323,620 
               
               
                 8 
                 2,613,800 
                 33,554,246 
               
               
                 9 
                 2,456,400 
                 35,624,903 
               
               
                 10 
                 4,283,500 
                 39,194,229 
               
               
                 11 
                 5,377,700 
                 43,585,489 
               
               
                 12 
                 3,440,400 
                 46,324,107 
               
               
                 13 
                 3,092,500 
                 48,746,153 
               
               
                 14 
                 5,009,900 
                 52,613,120 
               
               
                 15 
                 4,952,800 
                 56,346,378 
               
               
                 16 
                 8,294,000 
                 62,453,212 
               
               
                 17 
                 7,135,400 
                 67,503,042 
               
               
                 18 
                 5,745,600 
                 71,433,501 
               
               
                 19 
                 3,048,800 
                 73,463,046 
               
               
                 20 
                 7,970,200 
                 78,692,998 
               
               
                 21 
                 9,804,500 
                 84,886,617 
               
               
                 22 
                 4,266,700 
                 87,458,265 
               
               
                 23 
                 3,355,800 
                 89,440,501 
               
               
                 24 
                 3,443,000 
                 91,442,306 
               
               
                 25 
                 3,507,800 
                 93,448,923 
               
               
                 26 
                 2,345,700 
                 94,768,738 
               
               
                 27 
                 3,390,500 
                 96,655,471 
               
               
                 28 
                 1,873,100 
                 97,681,267 
               
               
                 29 
                 1,936,400 
                 98,732,433 
               
               
                 30 
                 2,829,200 
                 100,254,333 
               
               
                 31 
                 2,031,600 
                 101,332,714 
               
               
                 32 
                 4,522,700 
                 103,710,553 
               
               
                 33 
                 3,300,200 
                 105,408,928 
               
               
                 34 
                 2,300,300 
                 106,574,443 
               
               
                 35 
                 3,012,600 
                 108,084,433 
               
               
                 36 
                 3,320,800 
                 109,725,433 
               
               
                 37 
                 3,083,500 
                 111,225,488 
               
               
                 38 
                 3,040,300 
                 112,683,183 
               
               
                 39 
                 2,914,500 
                 114,060,680 
               
               
                 40 
                 4,054,600 
                 115,950,889 
               
               
                 41 
                 4,168,500 
                 117,857,321 
               
               
                 42 
                 2,299,700 
                 118,888,554 
               
               
                 43 
                 2,668,200 
                 120,072,151 
               
               
                 44 
                 3,443,800 
                 121,580,724 
               
               
                 45 
                 4,161,400 
                 123,374,265 
               
               
                 46 
                 2,958,700 
                 124,624,613 
               
               
                 47 
                 2,655,400 
                 125,731,249 
               
               
                 48 
                 6,090,300 
                 128,237,832 
               
               
                 49 
                 5,884,600 
                 130,590,724 
               
               
                 50 
                 6,124,600 
                 132,972,135 
               
               
                 51 
                 8,064,000 
                 136,017,763 
               
               
                 52 
                 4,311,700 
                 137,584,756 
               
               
                 53 
                 3,426,700 
                 138,804,986 
               
               
                 54 
                 4,211,100 
                 140,280,488 
               
               
                 55 
                 9,842,500 
                 143,661,174 
               
               
                 56 
                 10,233,000 
                 147,014,085 
               
               
                 57 
                 7,729,200 
                 149,425,323 
               
               
                 58 
                 4,113,200 
                 150,662,080 
               
               
                 59 
                 3,717,400 
                 151,758,312 
               
               
                 60 
                 3,488,200 
                 152,769,058 
               
               
                 61 
                 5,721,300 
                 154,399,807 
               
               
                 62 
                 3,717,100 
                 155,430,927 
               
               
                 63 
                 3,087,400 
                 156,272,470 
               
               
                 64 
                 3,224,500 
                 157,138,684 
               
               
                 65 
                 2,883,000 
                 157,901,471 
               
               
                 66 
                 3,899,000 
                 158,919,153 
               
               
                 67 
                 4,911,800 
                 160,177,794 
               
               
                 68 
                 2,481,700 
                 160,799,107 
               
               
                 69 
                 3,689,400 
                 161,712,049 
               
               
                 70 
                 3,266,200 
                 162,506,315 
               
               
                 71 
                 2,112,700 
                 163,012,223 
               
               
                 72 
                 3,681,800 
                 163,885,150 
               
               
                 73 
                 4,758,400 
                 164,993,891 
               
               
                 74 
                 6,189,200 
                 166,403,903 
               
               
                 75 
                 3,704,300 
                 167,223,366 
               
               
                 76 
                 3,111,000 
                 167,899,648 
               
               
                 77 
                 3,045,500 
                 168,552,052 
               
               
                 78 
                 2,779,100 
                 169,138,903 
               
               
                 79 
                 2,615,200 
                 169,683,961 
               
               
                 80 
                 3,774,700 
                 170,461,052 
               
               
                 81 
                 2,217,200 
                 170,909,440 
               
               
                 82 
                 2,263,200 
                 171,362,381 
               
               
                 83 
                 1,762,900 
                 171,711,458 
               
               
                 84 
                 4,200,500 
                 172,536,350 
               
               
                 85 
                 3,971,600 
                 173,300,959 
               
               
                 86 
                 2,741,500 
                 173,818,939 
               
               
                 87 
                 3,433,500 
                 174,459,344 
               
               
                 88 
                 5,186,900 
                 175,411,240 
               
               
                 89 
                 3,959,700 
                 176,120,281 
               
               
                 90 
                 2,441,700 
                 176,549,400 
               
               
                   
               
               
                 RS(t) is calculated using the Furbush and Smith (1994) model. 
               
               
                 RS i  = RS i−1  + V i  (1−(RS i−1 /F)) 
               
             
          
         
       
     
     
       
         
               
             
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
             
             
               
                 Calculating the Retained Shares Using the Two Trader Model (Finnerty and Pushner, 2003) 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
               
               
                   
               
               
                 Trader Retained Shares 
                 238,084 
                 238,561 
                 239,039 
                 239,517 
                 239,997 
                 240,477 
                 240,959 
                 241,441 
               
               
                 Investor Retained Shares 
                 21,638 
                 36,443 
                 46,047 
                 75,038 
                 60,554 
                 37,315 
                 55,295 
                 53,467 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 16 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
                 23 
               
               
                   
               
               
                 Trader Retained Shares 
                 245,335 
                 245,826 
                 246,319 
                 246,812 
                 247,306 
                 247,801 
                 248,297 
                 248,794 
               
               
                 Investor Retained Shares 
                 93,617 
                 132,286 
                 135,047 
                 129,833 
                 89,291 
                 171,203 
                 207,091 
                 145,912 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 31 
                 32 
                 33 
                 34 
                 35 
                 36 
                 37 
                 38 
               
               
                   
               
               
                 Trader Retained Shares 
                 252,807 
                 253,313 
                 253,820 
                 254,328 
                 254,837 
                 255,348 
                 255,859 
                 256,371 
               
               
                 Investor Retained Shares 
                 138,504 
                 138,504 
                 206,406 
                 173,067 
                 179,322 
                 170,942 
                 190,491 
                 186,348 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 46 
                 47 
                 48 
                 49 
                 50 
                 51 
                 52 
                 53 
               
               
                   
               
               
                 Trader Retained Shares 
                 260,506 
                 261,027 
                 261,550 
                 262,074 
                 262,598 
                 263,124 
                 263,651 
                 264,179 
               
               
                 Investor Retained Shares 
                 306,652 
                 248,644 
                 234,746 
                 433,976 
                 455,992 
                 499,050 
                 620,116 
                 471,524 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
                 68 
               
               
                   
               
               
                 Trader Retained Shares 
                 268,440 
                 268,977 
                 269,515 
                 270,055 
                 270,596 
                 271,137 
                 271,680 
                 272,224 
               
               
                 Investor Retained Shares 
                 699,524 
                 1,013,534 
                 812,556 
                 734,767 
                 785,674 
                 744,111 
                 966,128 
                 1,179,504 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 76 
                 77 
                 78 
                 79 
                 80 
                 81 
                 82 
                 83 
               
               
                   
               
               
                 Trader Retained Shares 
                 276,615 
                 277,168 
                 277,723 
                 278,279 
                 278,837 
                 279,395 
                 279,954 
                 280,515 
               
               
                 Investor Retained Shares 
                 1,366,853 
                 1,246,398 
                 1,267,001 
                 1,215,268 
                 1,189,422 
                 1,630,473 
                 1,102,683 
                 1,148,902 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
               
                 Trader Retained Shares 
                 241,924 
                 242,409 
                 242,894 
                 243,380 
                 243,868 
                 244,356 
                 244,845 
               
               
                 Investor Retained Shares 
                 43,726 
                 42,609 
                 67,252 
                 81,760 
                 63,982 
                 61,156 
                 89,151 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
               
               
                   
               
               
                 Trader Retained Shares 
                 249,292 
                 249,791 
                 250,291 
                 250,792 
                 251,295 
                 251,798 
                 252,302 
               
               
                 Investor Retained Shares 
                 128,513 
                 135,867 
                 143,010 
                 107,012 
                 148,319 
                 92,389 
                 97,163 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
               
               
                   
               
               
                 Trader Retained Shares 
                 256,884 
                 257,399 
                 257,914 
                 258,430 
                 258,948 
                 259,466 
                 259,985 
               
               
                 Investor Retained Shares 
                 190,439 
                 190,150 
                 251,700 
                 268,320 
                 175,579 
                 204,399 
                 257,569 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 54 
                 55 
                 56 
                 57 
                 58 
                 59 
                 60 
               
               
                   
               
               
                 Trader Retained Shares 
                 264,708 
                 265,237 
                 265,768 
                 266,301 
                 266,834 
                 267,368 
                 267,903 
               
               
                 Investor Retained Shares 
                 419,156 
                 504,947 
                 852,769 
                 975,099 
                 966,360 
                 723,452 
                 704,176 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
               
               
                   
               
               
                 Trader Retained Shares 
                 272,769 
                 273,315 
                 273,862 
                 274,411 
                 274,960 
                 275,510 
                 276,062 
               
               
                 Investor Retained Shares 
                 746,065 
                 1,050,187 
                 997,464 
                 715,049 
                 1,154,990 
                 1,436,184 
                 1,779,301 
               
               
                   
               
             
          
           
               
                   
                 Bought on Day No: 
               
             
          
           
               
                   
                 84 
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
               
               
                   
               
               
                 Trader Retained Shares 
                 281,076 
                 281,639 
                 282,203 
                 282,768 
                 283,334 
                 283,901 
                 284,469 
               
               
                 Investor Retained Shares 
                 922,149 
                 1,951,953 
                 1,965,820 
                 1,544,065 
                 1,899,583 
                 2,621,424 
                 2,320,206 
               
               
                   
               
             
          
           
               
                 Total Number of Trader Retained Shares 
                 23,738,767 
                   
               
               
                 Total Number of Investor Retained Shares 
                 53,992,643 
               
               
                 Total Number of Retained Shares Using the TTM, Finnerty and Pushner (2003) 
                 77,731,410 
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 ESTIMATING THE DAILY AVERAGE TRADING PROPENSITY 
               
               
                 USING THE TMTM 
               
             
          
           
               
                 Day No. 
                 VOLUME 
                 Propensity 
               
               
                   
               
             
          
           
               
                 −91 
                 1,504,500 
                 0.1283 
               
               
                 −90 
                 1,557,000 
                 0.1131 
               
               
                 −89 
                 2,703,900 
                 0.0226 
               
               
                 −88 
                 2,941,800 
                 0.0000 
               
               
                 −87 
                 1,984,800 
                 0.0647 
               
               
                 −86 
                 1,886,300 
                 0.0831 
               
               
                 −85 
                 1,379,700 
                 0.1191 
               
               
                 −84 
                 1,488,300 
                 0.1552 
               
               
                 −83 
                 2,258,100 
                 0.1823 
               
               
                 −82 
                 1,776,400 
                 0.0987 
               
               
                 −81 
                 1,718,000 
                 0.1281 
               
               
                 −80 
                 4,258,600 
                 0.0632 
               
               
                 −79 
                 2,741,400 
                 0.0805 
               
               
                 −78 
                 1,449,600 
                 0.1019 
               
               
                 −77 
                 2,647,200 
                 0.0815 
               
               
                 −76 
                 5,214,700 
                 0.1292 
               
               
                 −75 
                 1,616,100 
                 0.1312 
               
               
                 −74 
                 2,687,700 
                 0.1244 
               
               
                 −73 
                 4,092,700 
                 0.0602 
               
               
                 −72 
                 2,691,000 
                 0.0922 
               
               
                 −71 
                 3,214,300 
                 0.0000 
               
               
                 −70 
                 4,450,500 
                 0.0000 
               
               
                 −69 
                 3,127,600 
                 0.0000 
               
               
                 −68 
                 2,818,200 
                 0.0092 
               
               
                 −67 
                 2,594,700 
                 0.1008 
               
               
                 −66 
                 3,082,200 
                 0.1288 
               
               
                 −65 
                 1,901,800 
                 0.0737 
               
               
                 −64 
                 2,012,400 
                 0.1177 
               
               
                 −63 
                 2,363,500 
                 0.0039 
               
               
                 −62 
                 2,244,200 
                 0.0199 
               
               
                 −61 
                 2,240,100 
                 0.1763 
               
               
                 −60 
                 2,777,100 
                 0.0485 
               
               
                 −59 
                 2,253,900 
                 0.1483 
               
               
                 −58 
                 1,994,400 
                 0.0875 
               
               
                 −57 
                 1,609,900 
                 0.1497 
               
               
                 −56 
                 2,021,400 
                 0.1049 
               
               
                 −55 
                 1,634,800 
                 0.0852 
               
               
                 −54 
                 2,297,700 
                 0.0237 
               
               
                 −53 
                 2,045,700 
                 0.1091 
               
               
                 −52 
                 1,635,100 
                 0.1331 
               
               
                 −51 
                 2,371,800 
                 0.0219 
               
               
                 −50 
                 2,549,900 
                 0.0635 
               
               
                 −49 
                 1,933,400 
                 0.1737 
               
               
                 −48 
                 1,804,900 
                 0.1099 
               
               
                 −47 
                 2,302,700 
                 0.0507 
               
               
                 −46 
                 1,516,300 
                 0.0518 
               
               
                 −45 
                 2,169,200 
                 0.0000 
               
               
                 −44 
                 2,189,400 
                 0.0766 
               
               
                 −43 
                 1,583,200 
                 0.2016 
               
               
                 −42 
                 1,831,100 
                 0.1350 
               
               
                 −41 
                 2,153,100 
                 0.0193 
               
               
                 −40 
                 2,649,600 
                 0.1693 
               
               
                 −39 
                 2,693,700 
                 0.1056 
               
               
                 −38 
                 3,310,300 
                 0.0000 
               
               
                 −37 
                 3,147,900 
                 0.0000 
               
               
                 −36 
                 1,959,200 
                 0.0943 
               
               
                 −35 
                 1,791,200 
                 0.2835 
               
               
                 −34 
                 2,077,700 
                 0.2373 
               
               
                 −33 
                 2,331,300 
                 0.3159 
               
               
                 −32 
                 2,239,900 
                 0.0000 
               
               
                 −31 
                 2,354,200 
                 0.2248 
               
               
                 −30 
                 1,899,900 
                 0.2221 
               
               
                 −29 
                 1,310,400 
                 0.2221 
               
               
                 −28 
                 1,973,500 
                 0.1621 
               
               
                 −27 
                 1,869,300 
                 0.1769 
               
               
                 −26 
                 1,554,500 
                 0.1444 
               
               
                 −25 
                 4,046,600 
                 0.0000 
               
               
                 −24 
                 1,128,600 
                 0.2417 
               
               
                 −23 
                 1,504,100 
                 0.2222 
               
               
                 −22 
                 1,405,100 
                 0.2462 
               
               
                 −21 
                 6,545,300 
                 0.0000 
               
               
                 −20 
                 10,863,800 
                 0.0000 
               
               
                 −19 
                 5,640,100 
                 0.0000 
               
               
                 −18 
                 2,390,800 
                 0.0261 
               
               
                 −17 
                 3,609,600 
                 0.0722 
               
               
                 −16 
                 3,854,000 
                 0.0000 
               
               
                 −15 
                 2,830,500 
                 0.1318 
               
               
                 −14 
                 3,765,100 
                 0.0108 
               
               
                 −13 
                 2,835,800 
                 0.0000 
               
               
                 −12 
                 1,892,100 
                 0.1011 
               
               
                 −11 
                 2,988,100 
                 0.0000 
               
               
                 −10 
                 3,804,600 
                 0.0344 
               
               
                 −9 
                 5,272,200 
                 0.0311 
               
               
                 −8 
                 2,107,700 
                 0.0795 
               
               
                 −7 
                 2,657,100 
                 0.1649 
               
               
                 −6 
                 3,112,300 
                 0.1245 
               
               
                 −5 
                 2,890,300 
                 0.1607 
               
               
                 −4 
                 1,953,300 
                 0.0955 
               
               
                 −3 
                 3,063,6000 
                 0.0436 
               
               
                 −2 
                 3,184,800 
                 0.0000 
               
               
                 −1 
                 2,073,800 
                 0.0907 
               
               
                 0 
                 1,814,800 
                 0.0891 
               
               
                 1 
                 3,230,200 
                 0.0000 
               
               
                 2 
                 4,240,700 
                 0.0000 
               
               
                 3 
                 9,181,100 
                 0.0141 
               
               
                 4 
                 5,072,700 
                 0.0798 
               
               
                 5 
                 2,460,300 
                 0.1120 
               
               
                 6 
                 3,921,500 
                 0.0000 
               
               
                 7 
                 3,531,100 
                 0.0460 
               
               
                 8 
                 2,613,800 
                 0.0778 
               
               
                 9 
                 2,456,400 
                 0.0915 
               
               
                 10 
                 4,283,500 
                 0.0521 
               
               
                 11 
                 5,377,700 
                 0.0000 
               
               
                 12 
                 3,440,400 
                 0.1185 
               
               
                 13 
                 3,092,500 
                 0.0817 
               
               
                 14 
                 5,009,900 
                 0.0389 
               
               
                 15 
                 4,952,800 
                 0.0720 
               
               
                 16 
                 8,294,000 
                 0.0021 
               
               
                 17 
                 7,135,400 
                 0.0310 
               
               
                 18 
                 5,745,600 
                 0.1020 
               
               
                 19 
                 3,048,800 
                 0.1447 
               
               
                 20 
                 7,970,200 
                 0.0890 
               
               
                 21 
                 9,804,500 
                 0.1087 
               
               
                 22 
                 4,266,700 
                 0.0959 
               
               
                 23 
                 3,355,800 
                 0.1051 
               
               
                 24 
                 3,443,000 
                 0.0806 
               
               
                 25 
                 3,507,800 
                 0.1404 
               
               
                 26 
                 2,345,700 
                 0.1398 
               
               
                 27 
                 3,390,500 
                 0.1278 
               
               
                 28 
                 1,873,100 
                 0.1552 
               
               
                 29 
                 1,936,400 
                 0.1613 
               
               
                 30 
                 2,829,200 
                 0.0717 
               
               
                 31 
                 2,031,600 
                 0.1605 
               
               
                 32 
                 4,522,700 
                 0.0000 
               
               
                 33 
                 3,300,200 
                 0.0378 
               
               
                 34 
                 2,300,300 
                 0.1042 
               
               
                 35 
                 3,012,600 
                 0.0635 
               
               
                 36 
                 3,320,800 
                 0.0670 
               
               
                 37 
                 3,083,500 
                 0.0602 
               
               
                 38 
                 3,040,300 
                 0.0675 
               
               
                 39 
                 2,914,500 
                 0.0988 
               
               
                 40 
                 4,054,600 
                 0.1011 
               
               
                 41 
                 4,168,500 
                 0.0991 
               
               
                 42 
                 2,299,700 
                 0.1028 
               
               
                 43 
                 2,668,200 
                 0.1548 
               
               
                 44 
                 3,443,800 
                 0.0762 
               
               
                 45 
                 4,161,400 
                 0.1232 
               
               
                 46 
                 2,958,700 
                 0.1579 
               
               
                 47 
                 2,655,400 
                 0.0561 
               
               
                 48 
                 6,090,300 
                 0.0000 
               
               
                 49 
                 5,884,600 
                 0.0658 
               
               
                 50 
                 6,124,600 
                 0.0117 
               
               
                 51 
                 8,064,000 
                 0.0072 
               
               
                 52 
                 4,311,700 
                 0.0000 
               
               
                 53 
                 3,426,700 
                 0.1044 
               
               
                 54 
                 4,211,100 
                 0.1342 
               
               
                 55 
                 9,842,500 
                 0.0000 
               
               
                 56 
                 10,233,000 
                 0.0003 
               
               
                 57 
                 7,729,200 
                 0.0000 
               
               
                 58 
                 4,113,200 
                 0.0875 
               
               
                 59 
                 3,717,400 
                 0.0681 
               
               
                 60 
                 3,488,200 
                 0.0497 
               
               
                 61 
                 5,721,300 
                 0.0023 
               
               
                 62 
                 3,717,100 
                 0.0146 
               
               
                 63 
                 3,087,400 
                 0.0236 
               
               
                 64 
                 3,224,500 
                 0.0209 
               
               
                 65 
                 2,883,000 
                 0.0926 
               
               
                 66 
                 3,899,000 
                 0.0742 
               
               
                 67 
                 4,911,800 
                 0.0086 
               
               
                 68 
                 2,481,700 
                 0.0547 
               
               
                 69 
                 3,689,400 
                 0.0280 
               
               
                 70 
                 3,266,200 
                 0.1242 
               
               
                 71 
                 2,112,700 
                 0.1449 
               
               
                 72 
                 3,681,800 
                 0.0621 
               
               
                 73 
                 4,758,400 
                 0.0915 
               
               
                 74 
                 6,189,200 
                 0.0452 
               
               
                 75 
                 3,704,300 
                 0.1144 
               
               
                 76 
                 3,111,000 
                 0.0826 
               
               
                 77 
                 3,045,500 
                 0.1108 
               
               
                 78 
                 2,779,100 
                 0.1317 
               
               
                 79 
                 2,615,200 
                 0.1073 
               
               
                 80 
                 3,774,700 
                 0.1597 
               
               
                 81 
                 2,217,200 
                 0.1566 
               
               
                 82 
                 2,263,200 
                 0.0734 
               
               
                 83 
                 1,762,900 
                 0.1010 
               
               
                 84 
                 4,200,500 
                 0.0032 
               
               
                 85 
                 3,971,600 
                 0.0306 
               
               
                 86 
                 2,741,500 
                 0.1221 
               
               
                 87 
                 3,433,500 
                 0.0679 
               
               
                 88 
                 5,186,900 
                 0.0674 
               
               
                 89 
                 3,959,700 
                 0.0201 
               
               
                 90 
                 2,441,700 
                 0.0939 
               
             
          
           
               
                   
                 AVERAGE 
                 0.0712 
               
               
                   
                   
               
               
                   
                 Calculation of the daily average sales propensity using Garbade (1978). μ st  = a 1  + a 2  (  B  − P*) where a1 = 420/No. of Trades Per Day and a2 = 5a1 
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 Retained Shares Using the TMTM 
               
               
                 Lee and Ready (1991) Algorithm 
               
             
          
           
               
                   
                 Day No. 
                 Retained Shares 
               
               
                   
                   
               
             
          
           
               
                   
                  0 
                 971,800 
               
               
                   
                  1 
                 1,615,977 
               
               
                   
                  2 
                 2,028,629 
               
               
                   
                  3 
                 3,904,019 
               
               
                   
                  4 
                 2,633,944 
               
               
                   
                  5 
                 1,094,176 
               
               
                   
                  6 
                 1,490,290 
               
               
                   
                  7 
                 1,398,318 
               
               
                   
                  8 
                 1,330,041 
               
               
                   
                  9 
                 1,350,455 
               
               
                   
                 10 
                 1,107,305 
               
               
                   
                 11 
                 2,584,345 
               
               
                   
                 12 
                 2,004,084 
               
               
                   
                 13 
                 1,373,094 
               
               
                   
                 14 
                 2,696,785 
               
               
                   
                 15 
                 3,053,542 
               
               
                   
                 16 
                 3,729,331 
               
               
                   
                 17 
                 3,352,513 
               
               
                   
                 18 
                 3,220,591 
               
               
                   
                 19 
                 1,219,177 
               
               
                   
                 20 
                 4,379,624 
               
               
                   
                 21 
                 5,977,782 
               
               
                   
                 22 
                 1,079,452 
               
               
                   
                 23 
                 1,429,234 
               
               
                   
                 24 
                 1,754,075 
               
               
                   
                 25 
                 1,152,802 
               
               
                   
                 26 
                 504,174 
               
               
                   
                 27 
                 1,197,239 
               
               
                   
                 28 
                 714,742 
               
               
                   
                 29 
                 733,245 
               
               
                   
                 30 
                 1,275,783 
               
               
                   
                 31 
                 759,092 
               
               
                   
                 32 
                 1,011,778 
               
               
                   
                 33 
                 456,528 
               
               
                   
                 34 
                 333,690 
               
               
                   
                 35 
                 487,694 
               
               
                   
                 36 
                 682,032 
               
               
                   
                 37 
                 854,814 
               
               
                   
                 38 
                 665,987 
               
               
                   
                 39 
                 588,379 
               
               
                   
                 40 
                 1,373,567 
               
               
                   
                 41 
                 789,750 
               
               
                   
                 42 
                 879,842 
               
               
                   
                 43 
                 119,496 
               
               
                   
                 44 
                 0 
               
               
                   
                 45 
                 176,468 
               
               
                   
                 46 
                 349,000 
               
               
                   
                 47 
                 0 
               
               
                   
                 48 
                 0 
               
               
                   
                 49 
                 943,427 
               
               
                   
                 50 
                 1,078,740 
               
               
                   
                 51 
                 3,267,046 
               
               
                   
                 52 
                 968,434 
               
               
                   
                 53 
                 464,216 
               
               
                   
                 54 
                 0 
               
               
                   
                 55 
                 642,065 
               
               
                   
                 56 
                 1,459,798 
               
               
                   
                 57 
                 1,708,040 
               
               
                   
                 58 
                 722,244 
               
               
                   
                 59 
                 631,783 
               
               
                   
                 60 
                 0 
               
               
                   
                 61 
                 0 
               
               
                   
                 62 
                 156,344 
               
               
                   
                 63 
                 49,266 
               
               
                   
                 64 
                 0 
               
               
                   
                 65 
                 0 
               
               
                   
                 66 
                 401,386 
               
               
                   
                 67 
                 0 
               
               
                   
                 68 
                 673,857 
               
               
                   
                 69 
                 1,414,320 
               
               
                   
                 70 
                 318,117 
               
               
                   
                 71 
                 0 
               
               
                   
                 72 
                 607,789 
               
               
                   
                 73 
                 0 
               
               
                   
                 74 
                 1,340,618 
               
               
                   
                 75 
                 54,601 
               
               
                   
                 76 
                 542,486 
               
               
                   
                 77 
                 978,065 
               
               
                   
                 78 
                 510,689 
               
               
                   
                 79 
                 1,172,280 
               
               
                   
                 80 
                 1,538,071 
               
               
                   
                 81 
                 302,098 
               
               
                   
                 82 
                 344,117 
               
               
                   
                 83 
                 140,796 
               
               
                   
                 84 
                 0 
               
               
                   
                 85 
                 0 
               
               
                   
                 86 
                 52,274 
               
               
                   
                 87 
                 0 
               
               
                   
                 88 
                 0 
               
               
                   
                 89 
                 1,284,698 
               
               
                   
                 90 
                 0 
               
               
                   
                 TOTAL 
                 95,652,351 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 5 
               
             
             
               
                   
               
               
                 RETAINED SHARES USING THE TMTM 
               
               
                 Ellis, Michaely &amp; O&#39;Hara (2000) Algorithm 
               
             
          
           
               
                   
                 Day No. 
                 Retained Shares 
               
               
                   
                   
               
             
          
           
               
                   
                  0 
                 951,600 
               
               
                   
                  1 
                 1,507,466 
               
               
                   
                  2 
                 1,948,255 
               
               
                   
                  3 
                 4,192,201 
               
               
                   
                  4 
                 2,539,562 
               
               
                   
                  5 
                 1,079,224 
               
               
                   
                  6 
                 1,616,831 
               
               
                   
                  7 
                 1,567,190 
               
               
                   
                  8 
                 1,252,708 
               
               
                   
                  9 
                 1,375,092 
               
               
                   
                 10 
                 1,735,584 
               
               
                   
                 11 
                 2,423,795 
               
               
                   
                 12 
                 1,835,099 
               
               
                   
                 13 
                 1,144,920 
               
               
                   
                 14 
                 2,726,891 
               
               
                   
                 15 
                 2,774,508 
               
               
                   
                 16 
                 4,353,766 
               
               
                   
                 17 
                 3,308,778 
               
               
                   
                 18 
                 2,970,011 
               
               
                   
                 19 
                 1,312,309 
               
               
                   
                 20 
                 5,011,160 
               
               
                   
                 21 
                 6,304,371 
               
               
                   
                 22 
                 1,680,682 
               
               
                   
                 23 
                 1,430,038 
               
               
                   
                 24 
                 1,771,823 
               
               
                   
                 25 
                 1,853,115 
               
               
                   
                 26 
                 917,558 
               
               
                   
                 27 
                 1,362,455 
               
               
                   
                 28 
                 836,086 
               
               
                   
                 29 
                 746,311 
               
               
                   
                 30 
                 1,396,109 
               
               
                   
                 31 
                 761,741 
               
               
                   
                 32 
                 1,441,245 
               
               
                   
                 33 
                 820,552 
               
               
                   
                 34 
                 744,105 
               
               
                   
                 35 
                 990,477 
               
               
                   
                 36 
                 1,307,348 
               
               
                   
                 37 
                 652,305 
               
               
                   
                 38 
                 1,056,649 
               
               
                   
                 39 
                 874,917 
               
               
                   
                 40 
                 1,597,450 
               
               
                   
                 41 
                 1,239,205 
               
               
                   
                 42 
                 916,641 
               
               
                   
                 43 
                 734,795 
               
               
                   
                 44 
                 597,319 
               
               
                   
                 45 
                 475,923 
               
               
                   
                 46 
                 552,773 
               
               
                   
                 47 
                 268,203 
               
               
                   
                 48 
                 298,132 
               
               
                   
                 49 
                 1,854,274 
               
               
                   
                 50 
                 2,153,912 
               
               
                   
                 51 
                 3,129,437 
               
               
                   
                 52 
                 891,444 
               
               
                   
                 53 
                 684,194 
               
               
                   
                 54 
                 973,566 
               
               
                   
                 55 
                 1,322,693 
               
               
                   
                 56 
                 2,224,765 
               
               
                   
                 57 
                 2,252,667 
               
               
                   
                 58 
                 1,062,970 
               
               
                   
                 59 
                 615,360 
               
               
                   
                 60 
                 206,759 
               
               
                   
                 61 
                 441,664 
               
               
                   
                 62 
                 726,524 
               
               
                   
                 63 
                 533,601 
               
               
                   
                 64 
                 226,188 
               
               
                   
                 65 
                 610,278 
               
               
                   
                 66 
                 426,945 
               
               
                   
                 67 
                 416,955 
               
               
                   
                 68 
                 924,703 
               
               
                   
                 69 
                 1,037,265 
               
               
                   
                 70 
                 133,781 
               
               
                   
                 71 
                 421,547 
               
               
                   
                 72 
                 798,777 
               
               
                   
                 73 
                 819,659 
               
               
                   
                 74 
                 1,269,999 
               
               
                   
                 75 
                 0 
               
               
                   
                 76 
                 637,523 
               
               
                   
                 77 
                 1,002,831 
               
               
                   
                 78 
                 437,275 
               
               
                   
                 79 
                 637,704 
               
               
                   
                 80 
                 1,056,581 
               
               
                   
                 81 
                 401,360 
               
               
                   
                 82 
                 47,616 
               
               
                   
                 83 
                 290,594 
               
               
                   
                 84 
                 385,199 
               
               
                   
                 85 
                 0 
               
               
                   
                 86 
                 95,233 
               
               
                   
                 87 
                 0 
               
               
                   
                 88 
                 0 
               
               
                   
                 89 
                 743,295 
               
               
                   
                 90 
                 856 
               
               
                   
                 TOTAL 
                 113,151,273 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
             
               
               
               
             
           
               
                 TABLE 6 
               
             
             
               
                   
               
               
                 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 &amp; O&#39;Hara (2000) 
                 113,151,273 
               
               
                   
               
             
          
         
       
     
       FIG. 8  illustrates a general purpose computer  20  constructed to implement the present invention. The computer  20  includes a central processing unit (CPU)  22  which communicates with a set of input/output (I/O) devices  24  over a bus  26 . The I/O devices  24  may include a keyboard, mouse, video monitor, printer, etc. The CPU  22  also communicates with a memory  28  over the bus  26 . The interaction between the CPU  22 , the devices  24 , the bus  26  and the memory  28  are well known in the art. The instant invention is directed towards the operation of these elements with respect to a set of data and programs stored in the memory  28 . 
     It will be understood that each of the elements described above, or two or more together, also may find a useful application in other types of embodiments differing from the types described above. 
     While the invention has been illustrated and described as embodied in an arrangement for and a method of determining damages in shareholder class action litigation, it is not intended to be limited to the details shown, since various modifications and structural changes may be made without departing in any way from the spirit of the present invention. 
     Without further analysis, the foregoing will so fully reveal the gist of the present invention that others can, by applying current knowledge, readily adapt it for various applications without omitting features that, from the standpoint of prior art, fairly constitute essential characteristics of the generic or specific aspects of this invention and, therefore, such adaptations should and are intended to be comprehended within the meaning and range of equivalence of the following claims. 
     What is claimed as new and desired to be protected by Letters Patent is set forth in the appended claims.