Assessing the Fairness of a Firm s Allocation of Shares in Initial Public Offerings: Adapting a Measure from Biostatistics


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1 Assessng the Farness of a Frm s Allocaton of Shares n Intal Publc Offerngs: Adaptng a Measure from Bostatstcs by Efstatha Bura and Joseph L. Gastwrth Department of Statstcs The George Washngton Unversty Introducton and Outlne In 1980, Professor Gastwrth was a vstng professor at MIT and gave a semnar on statstcs n law. Two Harvard law students attended the course. One of them went on to become a partner n a premer New York law frm specalzng n securtes law. Realzng that the case he was workng on nvolved a sgnfcant statstcal component, he contacted hs former Professor to be hs statstcal expert. Durng the hghtech boom of the late 90 s, ndvduals or frms who receved shares at the ntal offerng prce were often makng substantal proft by sellng ther shares shortly after they bought them. Some nsttutons, e.g. hedge funds, tred to ncrease the number of shares they were allocated by sharng ther profts wth ther brokers or nvestment frms. Ths practce s prohbted by law as the Wall Street frms should serve members of the publc farly. Indeed several major frms, e.g. Credt Susse Frst Boston, pad substantal penaltes to the Securtes and Exchange Commsson (SEC). In ths case, the clent was a relatvely small nvestment frm that was accused of sharng profts wth some customers, prmarly because these customers pad socalled nflated or excessve commssons on other trades made through the frm on or near the days they receved IPO (Intal Publc Offerng) shares. The case nvolved both legal and statstcal ssues. Pror to the complant, a commsson was deemed to be excessve f t exceeded fve percent. Although the publshed rule does state that commssons less than 5% mght also be judged excessve, no alternatve gudelne s provded. Snce almost all the so called nflated or excessve commssons were less than 5%, ndeed, less than 3%, the proprety of the regulators changng the crtera for commssons to be deemed excessve wthout conductng the standard rulemakng procedures was a major legal ssue. A key statstcal ssue concerned the farness of the IPO allocaton process used at the frm. If there were a set of proft sharng customers, one would expect that they would be favored n the allocaton process. Thus, one would expect that ther IPO share allocatons would be greater than ther expected or far share relatve to those of other customers whch would usually be less. Ths problem has analoges n 1
2 epdemology and equal employment. In the former one studes whether the members of a populaton that were exposed to a partcular health rsk dffers sgnfcantly from the unexposed populaton. In dscrmnaton cases, the fracton of jobs or promotons a mnorty group receved s compared to ther percentage of qualfed applcants. In both contexts, the CochranMantelHaenszel (CMH) test s a wdely used measure of dsparty between two classes of subjects. In our applcaton, the two groups were the alleged proft sharers and other smlar customers who requested IPO shares. The CMH measure was adapted n order to provde an nterpretaton of the dfference between the two groups n terms of a rato that could be easly conveyed to the hearng panel. A major ssue we faced was the defnton of farness n the allocaton of IPO shares. Ths was needed, as our CMHbased measure requres a determnaton of the expected share of customers under far allocaton practces. Arthur Levtt, the Charman of the SEC from 1993 to 2001, has stated that t s reasonable for a frm to allocate shares to ts best customers, especally those who had gven t substantal busness pror to the market and IPO boom. A measure of the farness of the allocatons of a partcular IPO needs to ncorporate the prevous busness of the customers who requested shares n that IPO n order to determne whch ones receved less than ther approprate share and whch customers were favored. The measure of the farness of the allocatons of IPO shares we proposed compares the actual number of shares the alleged proft sharers receved to the number they would have been expected to receve based on ther fracton of the busness all requestng customers gave the frm durng the prevous twelve months. The ssue of a group of subjects beng favored over another s central n dscrmnaton law cases. We expressed our measure n terms of the selecton rato used to evaluate an employment practce for possble dsparate mpact. It wll be seen that the group of alleged proft sharers were not favored. Indeed, the selecton rato measure ndcates that the proft sharers were actually dsadvantaged n the allocaton process. Background: Adaptng the CochranMantelHaenszel (CMH) Test Statstc No specfc SEC rules govern the process of allocatng shares n a securtes offerng. NASD (Natonal Assocaton of Securtes Dealers) Conduct Rule 2110 requres member frms to observe just and equtable prncples of trade. SEC or NASD rules pror to ths case precluded brokerage frms from chargng commssons n excess of 5%. Unlke many brokerage houses, ths frm dd not have a schedule for commssons for trades of varous szes. From ts ncepton the frm allowed ts customers to place ther own commssons on ther trades. As a consequence of the deregulaton of the securtes ndustry, ths practce was legal and was not unque to the frm. In the case, the regulators examned about two and half years of data and dentfed several groups of accounts as proft sharers at dfferent tmes durng the nvestgaton. The fnal set of accounts was selected by a new crteron for excessve commssons and the alleged volatons were clamed to have occurred n a sx month perod. A stock transacton was categorzed as an nflated rate commsson f the trade 2
3 nvolved 10,000 or more shares wth a commsson of 20 or more cents per share. Also, for the accounts wth such trades, trades of 1,000 or more shares wth commsson of 75 or more cents per share were also consdered nflated. These crtera were met by about 30 nsttutonal accounts of the frm. The regulator s expert decded to drop several of these because ther tradng pattern was not smlar to that of the other members of ths group. Ths fnal set of allegedly favored customers wll be referred to as the APS (Alleged Proft Sharers) group n ths paper. The CochranMantelHaenszel (CMH) test compares two groups on a bnary response, stratfyng the data nto K subgroups or strata wth smlar levels of other relevant covarates. Agrest provdes a full dscusson of the test and ts applcatons. The data are gven by a seres of K 2x2 contngency tables. Tradtonally, n each table the rows correspond to the "Treatment group" values (e.g "Placebo", "Drug A") and the columns to the "Response" values (e.g "No change," "Improvement"). The value of the test statstc s gven by CMH = K = 1 ( observed K = 1 expected ), V where V s the varance of the usual test of the equalty of proportons n the th stratum. The CMH test statstc s used to test the null hypothess that the response s condtonally ndependent of the treatment. If we consder the APS group as the treatment group and a comparable set of other customers as the control, then for every IPO a 2x2 table contanng the number of requests each group made and ther success rate can be formed. The data for each IPO defne a separate stratum. Ths analyss was carred out after we notced that the aggregate success rates of retal (nonnsttutonal) customers exceeded those of the APS customers. Ths was qute surprsng as nsttutonal customers typcally gve nvestment frms much more busness than retal customers. The CMH test for the 2x2 tables across all IPO allocatons obtaned that ther success rates were not statstcally sgnfcantly dfferent. The success rate analyss does not account for volume dfferences. That s, the fact that a customer was allocated IPO shares does not contan nformaton as to the sze of the allocaton wth respect to the total number of IPO shares the frm had avalable and the number of shares allocated to other customers. Thus, n addton to the CMH formal test, t s useful to have a measure of the magntude of the dfference observed expected ) relatve to the total expected value of the response. We use ( 3
4 K = 1 ( observed K = 1 expected ) expected to measure the dsparty between the actual and expected allocatons as a percent of the expected number of shares. The numerator of ths measure s the numerator of the CMH statstc whle the denomnator reflects ther expected total shares under a far system. Postve values of the rato ndcate an allocaton exceedng ther expected number whereas negatve values ndcate allocatons below expected. Ths measure s smlar to the attrbutable rsk type measures used n epdemology. To mplement ths measure one needs to determne the expected number of shares a customer would receve under a far allocaton system. Expectancy Analyss: Was the allocaton far? As already stated, there were no specfc crtera or rules governng the allocaton process set by the SEC, NASD or the frm. In our search for understandng allocaton practces n the street, we came across statements of two SEC commssoners that t s reasonable for a frm to consder ts busness relatonshp wth a customer n allocatng IPO shares. The problem s to translate the factors a frm could approprately take nto account n the calculaton of a far expected number of shares for each customer who requests shares n a partcular IPO. Some measurable factors that could be used n evaluatng a customer s busness relatonshp wth the frm are: (1) how long the customer has been wth the frm, (2) the amount of busness an account generated for the frm over a perod of tme, and (3) how actve the account has been. Whle the potental of a customer to generate future busness s also a reasonable factor, t s qute subjectve and brokers are usually not asked to record such predctons at the tme they allocate shares. Data on the length of tme a customer had been wth the frm was not as accurate as we frst thought. Some customers came wth a newly hred broker so ther assocaton wth the broker was longer than would be ndcated by the frm s records. Also, several major customers had substantal other busness relatonshp wth the frm. The only hghly relable data avalable to us were the tradng data of all customers over a perod of two years from whch we could easly extract the commsson busness each customer gave the frm. Snce an nvestment frm would naturally allocate more shares to customers who had gven t more busness than to other customers, the relatonshp between the allocatons of shares n all IPO deals and the prevous year s commsson busness customers gave the frm was studed. We focused on one year s pror busness snce t would account for seasonal varatons n the tradng patterns of dfferent customers. The frm also thought that ts brokers would have a relatvely accurate pcture of the busness ther customers gave them over the last year. 4
5 Frst, we nvestgated whether aggregate hstorcal commsson busness would be a good predctor of IPO allocatons especally n comparson to the new cents per share crteron used by the regulator. We compared the fracton of IPO shares allocated to customers requestng those shares to the twelve month average of the cents per share they put on trades and the correspondng average of ther total commsson busness. As the total commsson the customers gave over the year pror to the IPO was not normally dstrbuted the Spearman correlaton coeffcent was used to assess the strength of the relatonshp between the number of shares allocated and total commsson. The average Spearman correlaton between IPO shares allocated and total commssons n all offerngs was.58 and was hghly sgnfcant (pvalue <.0001). In contrast, the Spearman correlaton coeffcent between IPO shares allocated and cents per share was only.04, a nonsgnfcant result. Ths calculaton was made because the regulator defned a commsson to be nflated n terms of cents per share. It showed that a customer s allocaton was not related to ther average commsson n that metrc. We consdered an allocaton of IPO shares to a set of requestng customers to be far f the fracton of the total number of shares a customer receved was proportonal to ther share of pror busness. In detal, for each offerng the expected number of shares a customer who expressed nterest would receve was determned by ts fracton of the total pror busness gven the frm amongst the customers who were nterested. Once ths expected fracton, F, was determned, the expected number of shares the customer should receve s just F tmes the number of shares the frm had to allocate. For example, f the frm had 20,000 shares to allocate and a customer s fracton F of pror busness gven by all requestng customers over the last year was.25, the customer would be expected to receve 5,000 shares. To assess whether the APS customers receved more than ther far share, for each offerng we summed the expected fractons (Fs) of the APS customers who requested shares to determne ther total, say FT. The expected number of shares the APS customers should receve, assumng an allocaton proportonal to pror busness, s FT tmes the number of shares the frm had to allocate. To contnue the example, f FT were.60, e.g., three APS customers wth F s of.25,.20 and.15 asked for shares, they would be expected to receve.6 x 20,000 = 12,000 shares of the offerng. By comparng the actual number of shares to the expected number of shares we can assess whether the APS customers receved more or less than ther far share. Ths s accomplshed by calculatng the dfference between the actual and expected numbers of shares for each offerng and summng over all IPO deals. The results can be summarzed by comparng the actual fracton of shares receved by the APS customers n the offerngs of nterest to ther fracton expected. The few new customers,.e. the customers who had not executed any trades n the year pror to the month of the offerng, were excluded from the computatons, as they would be expected to receve zero shares snce they had no busness n the pror twelve months. The effect of ths was qute small as about 96% of all IPO shares went to customers wth pror busness. Of course, some of these customers mght have had an account wth the frm but had not been actve n the year precedng the month of an IPO deal. 5
6 To further llustrate how the calculatons are carred out, we present the followng hypothetcal example. Suppose the frm had a total of 1,000 shares of an IPO to dstrbute to the 10 customers who requested shares. If we assume that two of these customers were new to the frm and were allocated a total of 100 shares, there reman 900 shares to allocate to the remanng 8 customers. Also assume that among the 8 customers, the alleged proft sharers fracton of the pror twelvemonth busness s.60 and they were allocated 500 shares. Table 1 below summarzes the calculatons. Table 1: Hypothetcal Example to Illustrate the Calculaton of Expected Number of Shares Allocated to APS Customers and Comparson wth Actual Number Number of Receved Fracton of Aggregate Expected Number of IPO Dfference (Receved IPO Pror Year Shares Expected) Shares Busness APS *0.6= All Other *0.4= Accounts Total Table 2 reports the number of shares n all IPO deals the APS customers receved and the number of shares they would have been expected to receve on the bass of ther fracton of the commssons receved by the frm n the prevous 12 months. Table 2: Comparson of Actual and Expected Number of Shares Allocated to the APS Customers Actual Expected Dfference Number Number 57 IPOs 1,279,775 1,806, ,811 As the total number of IPO shares allocated to all customers wth pror busness was 3,118,025, the actual and expected percentages of these offerngs allocated to these accounts are gven n a barplot n Fgure 1. The APS customers receved 41.04% of all IPO shares but on the bass of ther pror commssons they would have been expected to receve 57.94% so they only were allocated 41.04/57.94=.7083 or 70.83% of ther expected number. Thus, n relatve terms, the APS receved 70.83% of the IPO shares they would have been expected to receve had shares been allocated n proporton to a customer s commssons durng the prevous twelve months. 6
7 Fgure 1. Comparson of Actual and Expected Percent of Shares Allocated to APS Customers To confrm that these results reflectng the allocaton process of IPO shares were not unque to the sxmonth perod specfc to the law case, a smlar expectancy analyss was conducted for the subsequent sx months. In that perod the APS receved about 66.15% of ther expected number of shares or equvalently, 33.85% fewer shares than would be expected on the bass of ther busness durng the prevous twelve months. A Senstvty Analyss: Demonstratng the Robustness of the Results When presentng the results of a statstcal analyss n court t s useful to provde a senstvty analyss showng that the man nference remans unaffected by the nevtable devatons of the data from the theoretcal deal. Several such analyses are descrbed n our related paper lsted n the references. Here we summarze an analyss answerng an nterestng nonstatstcal queston posed by the lawyers. 7
8 For purposes of the regulators analyss, an nflated rate transacton was a stock trade nvolvng: () commssons of $.20 per share or more on 10,000 shares or more, or () commssons of $.75 or more on 1,000 shares or more. What effect would these two crtera have on the number of shares the APS customers would be expected to receve had these crtera been n exstence at the tme and enforced by the frm? To answer ths queston we adjusted the pror busness for all customers who had executed such trades n the twelvemonth perod pror to the revew perod as well as durng the revew perod as follows. All trades satsfyng crteron () were assgned commssons calculated by settng the cents per share commsson rate to $.19. All trades satsfyng crteron () were assgned commssons by settng cents per share to $.74. All other commssons were left ntact. Then the expected number of shares for each IPO was calculated from these adjusted data, whch removed the alleged excessve component of commssons gven by the APS customers from the analyss. Note that the pror busness of the APS customers s reduced and, consequently, ther expected fracton of IPO shares becomes smaller. Table 3 reports the number of IPO shares the APS accounts receved and the number of shares they would have been expected to receve on the bass of ther fracton of the adjusted commssons they would have had gven the frm n the prevous twelve months. For each offerng the expected number of shares for the APS customers s determned from ther fracton of the adjusted total commsson busness durng the prevous twelve months of all customers requestng that specfc offerng as descrbed n the man expectancy analyss. Table 3: Comparson of Actual and Adjusted Expected Number of Shares Allocated to the APS Customers Actual Number Expected Number Dfference 57 IPOs 1,279,775 1,542, , Fgure 2 dsplays a barplot wth the percentages of actual and expected number of shares for the APS group. The actual remans the same as n Fgure 1. The new expected value, after the adjustment, s now 49.47%. In relatve terms, even after subtractng from ther pror busness the allegedly excessve component of ther commsson, the APS customers receved 17.04% fewer IPO shares they would have been expected to receve had shares been allocated n proporton to a customer s adjusted commssons durng the prevous twelve months. 8
9 Fgure 2. Comparson of Actual and Adjusted Expected Percent of Shares Allocated to the APS Customers Drawng on an Analog n Equal Employment: Selecton Rato Analyss To further apprecate the magntude and mportance of the shortfall n the number of IPO shares receved by the APS customers dentfed as proft sharers by the regulator, ths shortfall can be nterpreted as a selecton rato used n equal employment cases. It s the rato of the pass rate of a mnorty group of job applcants to the pass rate of job applcants from the majorty group and s used to assess whether a job requrement (e.g., attanng a certan score on a test) has a dsparate mpact on mnorty applcants. In that applcaton, the government gudelne s that selecton ratos less than.80 or fourffths (e.g., mnorty applcants receve passng scores on a test at a rate less than 80%, or fourffths, the rate of Caucasans) ndcate a dsparate mpact so that the specfed job requrement needs to be shown to be jobrelated. Although the selecton rato s defned for the comparson of two success rates, Karys et al. and Gastwrth and Greenhouse have translated t to our stuaton. Suppose the fracton π of job applcants s mnorty and they receve the fracton p (<π) of the jobs. 9
10 Assume there s a total of N applcants of whch n are hred. Then the mnorty success rate s np/πn and the majorty success rate s n(1p)/(1π)n. The selecton rato then s gven by np πn n(1 p) (1 π ) N p 1 p = π 1 π Notce that the selecton rato s the rato of the odds a hre s mnorty to the odds an applcant s mnorty. In our applcaton the fracton of shares the APS receved s p and ther fracton of pror busness s π. The selecton rato for the APS customers calculated from the data n Fgure 1 s.5053 well below the value of the government gudelne,.80. Ths ndcates that on the bass of ther pror commsson busness the APS customers were dsadvantaged relatve to other customers n the IPO allocaton process used at the frm. The selecton rato for the data n Fgure 2, where the alleged excessve or nflated commssons were adjusted downward to comply wth the new crteron, s.711, stll below the.80 threshold. Thus nterpretng the results of the expectancy analyss n terms of the gudelnes used n equal employment cases shows that the selecton rato of the APS group ndcates they were dsfavored rather than favored by the frm. A Fnal Note Whle the IPO frenzy durng the late 1990 s was unusual, there remans a substantal IPO market as new companes grow and decde to go publc. The methodology descrbed n ths paper should be useful both to regulators and ndvdual frms who desre to check the farness of ther IPO allocatons. In partcular, a frm can readly montor ts ndvdual brokers to ensure that none of them s favorng a few customers by gvng them notceably more shares than ther pror busness merts. If a customer has an unusually hgh success rate or s often allocated more shares than expected, the frm could then examne the pattern of commsson busness the customer gves to see whether t s concentrated near the tme the customer receved shares n an IPO and s related to the proft potental of the IPO. The pattern that the regulatory body n the case motvatng ths research used really does not capture proft sharng actvty around the days of an IPO. Recall that only trades of 10,000 shares or more wth commssons of 20 cents or more per share or trades of at least 1,000 shares wth commssons of 75 cents per share or more were consdered nflated or excessve. Thus, a customer who traded 5,000 shares and placed a commsson of 50 cents per share would go undetected even though he/she gave a commsson of $2,500, exceedng a $2,000 commsson on a trade of 10,000 at 20 cents per share. It would seem more approprate to consder the total commsson busness near the recept of an IPO and examne the rato of these commssons to ther projected proft from the IPO shares they were allocated. If there had been a formal or nformal qud pro 10
11 quo arrangement or understandng these ratos would be expected to concentrate around the understood share. As far as the authors know, ths s the frst use of concepts orgnally developed n bostatstcs n a securtes law case. The factors used to determne the expected number of shares a customer would receve under a far system rely on practces consdered approprate n the ndustry. In another case, other factors mght be ncorporated provded relable data on them are avalable. References and Further Readng Agrest, A. (1990). Categorcal Data Analyss. New York: Wley. Gel, Y. Mao, W. and Gastwrth, J. L. (2005). The Importance of Checkng the Assumptons Underlyng Statstcal Analyss: Graphcal Methods for Assessng Normalty. Jurmetrcs Journal, 46, Gastwrth, J. L. (1988). Statstcal Reasonng n Law and Publc Polcy. Volume 1 Statstcal Concepts and Issues of Farness. San Dego: Academc Press. Gastwrth,, J. L., Bura, E. and Modarres, R. (2005). Statstcal Methods for Assessng the Farness of the Allocaton of Shares n Intal Publc Offerngs. Law, Probablty and Rsk. 4, Gastwrth, J. L., Modarres, R. and Bura, E. (2005). The Use of the Lorenz Curve, Gn Index and Related Measures of Relatve Inequalty and Unformty n Securtes Law. Metron. In press. Gastwrth, J. L. and Greenhouse, S. W. (1987). Estmatng a common relatve rsk: Applcaton n equal employment. Journal of the Amercan Statstcal Assocaton, 82, Karys, D., Kadane, J. B., and Lehoczky, J. P. (1997). Jury Representatveness: A Mandate for Multple Source Pools. Calforna Law Revew, 65,
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