Workig Paper arch 27, 2007 Just Lucky? A Statistical Test for Optio Backdatig Richard E. Goldberg James A. Read, Jr. The Brattle Group Abstract The literature i fiacial ecoomics provides covicig evidece of retroactive timig of executive stock optio grats (optio backdatig ). However, the literature does ot yet cotai a methodology for detectig backdatig at idividual compaies. The preset paper seeks to fill this gap. Specifically, it describes a rigorous statistical test for backdatig based o publicly available data. I additio, it idetifies a flaw i the methodology employed by the Wall Street Joural (Forelle ad Badler, 2006) to calculate the odds that the timig of executive stock optio grats was purely radom, ad it provides experimetal evidece that this flaw could i practice distort the assessmet of odds. Richard E. Goldberg, The Brattle Group, 333 Sacrameto Street, Suite 40, Sa Fracisco, CA 94, Richard.Goldberg@brattle.com. James A. Read, Jr., The Brattle Group, 44 Brattle Street, 3rd Floor, Cambridge, A 0238, James.Read@brattle.com. 2007 The Brattle Group, Ic. Electroic copy of this paper is available at: http://ssr.com/abstract=97758
Just Lucky? A Statistical Test for Optio Backdatig Richard E. Goldberg James A. Read, Jr. The Brattle Group. Itroductio Optio backdatig the retroactive selectio of grat dates for executive stock optios i order to set favorable strike prices is curretly big ews, with over 00 compaies uder ivestigatio by the Securities ad Exchage Commissio (SEC). Regulatory scrutiy of the timig of executive stock optio grats was iitially stimulated by a lie of academic research that goes back to Yermack (997). I a sample of approximately 600 stock optio grats from 992 to 994, Yermack foud that the returs o the uderlyig stocks were ormal i the period prior to the optio grats but were abormal (positive) i the period followig optio grats. After cosiderig several possible explaatios for these results, Yermack cocluded that maagers iflueced compesatio committees to award more optios whe they aticipated improvemets i corporate performace. Lie (2005) examied a much larger sample of stock optio grats from 992 to 2002. 2 Like Yermack, he foud abormal returs i the period followig optio grats, but he also foud abormal (egative) returs i the period precedig optio grats. I additio, Lie suggested a ovel explaatio for these fidigs, amely, that grat dates might have bee selected retroactively, i light of observed (rather tha aticipated) icreases i uderlyig stock prices. Recetly, Lie ad Hero (forthcomig) report that evidece of abormal returs aroud optio grat dates declied sharply followig the Sarbaes-Oxley Act, which madated that grats be reported withi two busiess days of the grat date. Thus, the academic research provides covicig evidece that backdatig has ideed occurred. The academic literature to date has addressed optio backdatig i the aggregate but ot at the level of idividual compaies. However, earlier this year the Wall Street Joural (Forelle ad Badler, 2006) idetified several compaies for which there is a strikig patter of icreases i the prices of uderlyig stocks immediately followig optio grats. The Wall Street Joural (WSJ) also reported eye-poppig odds agaist these results havig occurred by chace. For example, they reported that the odds that the returs o shares held by Jeffrey Rich, the CEO of Affiliated Computer Services, occurred by chace were oly oe i 300 billio; the odds of returs o shares held by Louis Tomasetta, Presidet ad CEO of Vitesse Semicoductor, were oly oe i 26 billio; ad the odds of returs o shares held by Kobi Alexader, Chairma ad CEO of Comverse Techology, were oly oe i 6 billio. The WSJ computed these odds usig 2 See, e.g., the article by Peder (2006) i the Sa Fracisco Chroicle. Other papers i this lie of academic research iclude Aboody ad Kaszik (2000) ad Chauvi ad Sheoy (200). - 2 - Electroic copy of this paper is available at: http://ssr.com/abstract=97758
a upublished methodology that it devised ad implemeted with the assistace of a statisticia. (The methodology is described i the same article.) The purpose of the preset paper is threefold. First, we idetify a flaw i the statistical test the WSJ applied to detect optio backdatig. Secod, we describe a experimet we performed usig ote Carlo simulatio, which shows that this flaw could distort the assessmet of odds i practice. Fially, we develop a alterative statistical test that does ot suffer from the same flaw. To summarize briefly, the flaw i the WSJ methodology is related to the fact that it is based o aual rakigs of returs. The alterative test we propose, i cotrast, is based o returs but ot o retur raks. 2. The WSJ Calculatios The results reported by the WSJ were obtaied by applyig the followig methodology. 3 For each tradig day i which there was a stock optio grat for a particular compay, the WSJ computed the 20-day retur o the uderlyig stock. 4 It the raked the retur o each grat date i relatio to the 20-day returs o all other days i the same year. For example, if the 20- day retur o a grat date was the highest i the year, its rak would be, if the 20-day retur o a grat date was the secod highest i the year, the the rak would be 2, ad so o. The, uder the assumptio that each day s rak was radom, they computed the likelihood of observig a patter of retur raks as extreme as the raks observed for the grat dates. To illustrate the WSJ methodology, we will describe the result cited above for Jeffrey Rich of Affiliated Computer Services (ACS). Of the six grats received by r. Rich from 995 through 2002, the returs for two grat dates were raked highest i their respective years, the returs for two grat dates were raked secod highest, the retur for oe grat date was raked third highest, ad the retur for oe grat date was raked fourth highest (see Table ). We will represet this (uordered) outcome as {,,2,2,3,4}. If there are 252 tradig days i the year, the pickig oe day radomly i each of six years (assumig o ties) has 252 6 (approximately 256 trillio) differet possible outcomes. We ca the cout how may of these outcomes are at least as extreme as the outcome for the six ACS optio grats. There are 9 differet uordered outcomes at least as extreme but each of these outcomes ca have permutatios with the same uordered rakig. For example, i a outcome cosistig of five highest raked days ad oe secod highest raked day, the secod highest rakig day could have occurred i ay oe of six years. Table 2 lists the 9 differet outcomes at least as extreme as the {,,2,2,3,4} outcome observed for the ACS grat dates. It also reports the umber of equivalet ordered outcomes. The result is that oly 864 of the roughly 250 trillio possible outcomes are as extreme as the outcomes associated with the ACS grat dates. This is equivalet to approximately oe i 300 billio (= 256 trillio/864). 3 4 See Forelle (2006). Lie s aalysis suggests that most of the abormal returs followig stock optio grats occurs withi 20 to 30 days followig the grat date, so although the WSJ s choice of 20 days to compute returs was somewhat arbitrary, it is ot icosistet with the academic research. - 3 -
To summarize, the WSJ methodology is based o the likelihood of radomly selectig days o which the 20-day retur raked as high as the 20-day retur observed for the stock optio grats to compay executives. 3. A Problem with the WSJ Calculatios A key assumptio i the WSJ calculatios is that the selectio of stock optio grat dates is idepedet of the aual rakig of the 20-day returs. Clearly a compay s price history is kow at the time stock optios are grated. This meas that the returs are kow ad could be take ito accout i a decisio as to whe to grat optios. However, these returs also eter ito the calculatio of aual rakigs. If the rakigs are affected by iformatio that is used i selectig the grat date, the the assumptio that the grat dates are a radom samplig of the days of the year is ivalid. Is this a mior techical issue or could it be sigificat? To explore this questio, we coducted a two-part experimet usig ote Carlo simulatio. I the first part of our experimet, we drew a large umber of oe-year sample paths of daily stock prices uder the assumptio that the price P t follows a logormal radom walk, i.e., l ( Pi + Δt ) l( Pi ) = μδt + σzi Δt, where the Z i terms are draw from a stadard ormal distributio ad Δ t = tradig day = 252 year. To be specific, we assumed that μ=20%/year ad σ=50%/year /2. We the radomly selected a grat date for each sample path ad computed the 20-day grat-date retur. Fially, we repeated the simulatio 00,000 times with the followig result: The grat date had the highest raked 20- day retur i 385 of the 00,000 simulatios. This is equal to out of every 260 simulatios, close to the predicted result of out of every 252. I the secod step of our experimet we selected grat dates usig a alterative rule: Offer the grat o the first day the price had dropped each of the prior te days or, if that ever occurred, o the last day of the year. After executig aother 00,000 simulatios (with the same radom draws) usig the alterative rule, we foud that the grats were made o the highest raked day 977 times out of 00,000. This is equal to out of 02 days, about twice as ofte as predicted based o the assumptio that the grat date was selected radomly. (Although the exact umbers chage i each set of simulatios, we cosistetly foud that grat date selectio usig our simple strategy roughly doubled the chaces of the grat occurrig o the highest raked days.) If very simple strategies based o prior returs ca double the likelihood of achievig highly raked days, the strategies that compay isiders ca thik of should be able to achieve eve greater improvemets i the likelihood of highly raked grat dates. Therefore, a statistical test based o the assumptio that the rakigs of grat dates are idepedet of date selectio caot be trusted. For this reaso, we coclude that the statistical test used for screeig purposes by the WSJ is flawed. - 4 -
4. A Alterative Test The basis for the alterative statistical test we propose is the fact that, while aual rakigs of returs may deped o past prices, the returs themselves should ot. For example, i our simulatios we foud that the mea 20-day log-price retur (defied as the differece i the atural logarithm of prices) followig grat dates was almost idetical for both the radom ad strategically selected grat dates:.65% for the radomly selected dates ad.58% for the dates selected strategically. (The populatio retur of 20%/year correspods to.59% over 20 days ad the stadard deviatio of estimates over 00,000 samples should be equal to.045%.) Therefore, a statistical test should be based o a compariso of returs aroud grat dates to typical returs, ot o the aual rak of grat-date returs. To follow up o this isight, we costruct a T test for the differece i mea returs over grat dates ad mea returs over typical tradig days. We assume that the daily log-price returs l( P i+ / Pi ) ( Pi + Pi )/ Pi for tradig days i the period i=,,n+w ca be modeled as idepedet ad idetically distributed radom variables of the form P X i i l + = Δt + Z P μ σ i i Δt where the Z i are idepedet stadard ormal draws. I this case, the W-day log-price returs for tradig days i=,,n ca be writte as P Yi l i+ W = P i W k = 0 X i+ k W = μ WΔt + σ WΔt Zi+ k μy + σ Y Z WΔt k = 0 Y i where the Z Y i are stadard ormal draws that are correlated whe withi W days of oe aother such that E Y Y { Z Z } i j W i = W 0 j if i - j W otherwise Our ull hypothesis is that the set of optio grat dates were selected i a way that did ot deped o the returs followig the grat dates. If that hypothesis is correct, the the mea W- day log-price retur o the grat dates m is a ubiased estimator of the populatio retur μ Y. Suppose the that we observe that m is much higher tha the estimator m N computed usig the full data set of N tradig days. We would like to develop a statistical test to see if such a observed differece is solely due to chace. A T test ca be used to test the sigificace of a differece i meas. However, it requires that the tradig data beig compared with the grat-date data satisfy certai coditios, amely, that the W-day returs for each sample of tradig data used to estimate the populatio mea ad variace be idepedet idepedet of each other ad idepedet of the returs o each grat date. This ca be accomplished by parsig the full set of tradig data ito a mesh of - 5 -
data poits that are at least W days apart from each other ad also from each grat date. (Sice the data poits are W-day mea returs, all data poits less tha W days apart will be correlated with each other.) We ca calculate the mea ad stadard deviatio of W-day returs over the mesh of parsed data poits (m ad S, respectively) ad the perform a T test to see if a observatio that m is much greater tha m is statistically sigificat. The T test statistic will have - degrees of freedom ad be equal to T S ( m m ) / + 2 + earby grats j, k W i AX 0, G ( j) i W G ( k) where m m = = j= Y i ( j) G m= Y i ( m) are the estimated meas over the grat dates ad over the parsed data poits, respectively, where i G (j) ad i (m) eumerate the tradig days correspodig to the grat dates ad to the parsed data poits, respectively; S = ( Yi m m ) ( ) m= 2 is the stadard deviatio of W-day returs estimated usig the parsed data set; ad the sum over earby grats is a sum over distict pairs of grat dates that are withi W days of each other. The observatio that returs are higher followig grat dates tha they are o other tradig days correspods to a high T test statistic. Whether or ot the test statistic is higher tha would likely result solely from radom chace ca be determied by referece to values of the oe-tailed T distributio with - degrees of freedom. If the likelihood of achievig the observed test statistic is less tha a reasoable threshold, the default hypothesis that optio grat dates are selected without forekowledge of future prices will eed to be rejected. 5. Compariso of Test Results To compare our test with the WSJ test, we computed the odds of achievig the results the WSJ reported for ACS, Vitesse Semicoductor, ad Comverse Techology (see Table 3). I each case we verified the result reported by the WSJ based o our uderstadig of their methodology ad the applied the T test described herei. Sice for 20-day returs there are 20 differet parsed data sets that our method would allow for the test (depedig o how may days from the first day i the period we start our mesh of poits), all of the results we report are from the parsed data set with the highest odds of occurrig. Grat date iformatio was obtaied from - 6 -
proxy statemets filed by the compaies. Stock prices were obtaied from Bloomberg. The WSJ calculatios addressed six CEO grats over the period 995 to 2002 for ACS, ie CEO grats over the period 994 to 200 for Vitesse Semicoductor, ad eight CEO grats over the period 994 to 200 for Comverse Techology. We used the same grats over the same time periods whe we carried out the T tests show i Table 3. Lookig i detail at the ACS calculatios, the mea 20-day log-price retur we calculated for the six ACS grat dates was equal to 32.7% while the mea over all tradig days i the period from 995 to 2002 was oly 2.2%. For the mesh of 88 idepedet data poits startig o the 8 th tradig date (/9/95) we estimated that the mea ad stadard deviatio of 20-day log-price returs were.4% ad.0%, respectively, resultig i a test statistic of 6.736. Comparig to the T distributio with 87 degrees of freedom, we fid that the likelihood of a test statistic this large or greater is oly about i.2 billio roughly a factor of 250 greater tha the likelihood computed by the WSJ. Although the T tests for all three compaies show much lower odds tha the umbers estimated by the WSJ, we too fid that it is highly ulikely that the grat dates were chose without iformatio about subsequet price chages. I all three cases, that hypothesis ca be rejected with 99.99% cofidece. 5 The mai reaso that the T test gives odds so much lower tha those reported by the WSJ is the limited umber of grat dates. For example, i the case of Vitesse Semicoductor, the mea 20-day log-price retur o grat dates is 36% ad the stadard deviatio (computed over the mesh startig o the first day of the period) is 23%; for ie grat dates this gives a stadard error of almost 8%. Hece, the sample mea is less tha five stadard errors large ad, after takig ito accout the estimated populatio mea (over the mesh) ad the limited umber of mesh data poits (83), the test statistic is equal to oly 4.36. It is a testamet to the size of the effect that, with so few data poits, the ull hypothesis ca be excluded with such a high degree of cofidece. 6. Coclusios The coclusios of our T test did ot differ from the coclusios of the WSJ test for the three cases we compared. Had these cases bee less extreme, however, the two tests could have led to differet coclusios. Furthermore, ulike the WSJ test, our test is ot subject to the objectio that it depeds o how the compay made its grat date decisio. The critical poit is that the returs followig grat dates should ot be sigificatly higher tha returs followig comparable o-grat tradig days. Our test makes this compariso i a statistically soud maer, yet it ca be carried out easily usig publicly available data. 5 Rejectig a hypothesis at the 99.99% cofidece level implies that the likelihood of the hypothesis beig true is less tha or equal to 0.0% which correspods to odds of less tha i 0,000. - 7 -
Refereces Aboody, D., Kaszik, R, 2000. CEO stock optio awards ad the timig of corporate volutary disclosures. Joural of Accoutig Ecoomics 29, 73 00. Chauvi, K. W., Sheoy, C., 200. Stock price decreases prior to executive stock optio grats. Joural of Corporate Fiace 7, 53 76. Forelle, C., 2006. How the joural aalyzed stock-optio grats, Wall Street Joural (arch 8), A5. Forelle, C., Badler, J., 2006. The perfect payday. Wall Street Joural (arch 8), A. Hero, R. A., Lie, E., forthcomig. Does backdatig explai the stock price patter aroud executive stock optio grats? Joural of Fiacial Ecoomics. Lie, E., 2005. O the timig of CEO stock optio awards. aagemet Sciece 5, 802-82. Peder, K., 2006. Backdatig scadal has more tha 00 compaies i SEC's sights. Sa Fracisco Chroicle (September 7). Yermack, D., 997. Good timig: CEO stock optio awards ad compay ews aoucemets. Joural of Fiace 52, 449 476. - 8 -
Year Date Price 20 Day Retur Aual Rak 995 9-ar-95 $22.50 34% 996 8-ar-96 $33.75 3% 2 997 7-Apr-97 $2.2 30% 2 998 8-Oct-98 $23.06 60% 2000 -Jul-00 $32.88 43% 4 2002 23-Jul-02 $35.75 35% 3 Table : Stock optio grats made to Jeffrey Rich of ACS durig 995-2002 Sources: Grat iformatio from ACS proxy statemets. Stock prices from Bloomberg. Calculatios by the authors. - 9 -
Sorted Outcome Rakigs for Six Grats Worst Secod Worst Third Worst Third Best Secod Best Best Number of Equivalet Outcomes 4 3 2 2 80 4 3 2 20 4 3 30 4 2 2 2 60 4 2 2 60 4 2 30 4 6 3 3 2 2 90 3 3 2 60 3 3 5 3 2 2 2 60 3 2 2 60 3 2 30 3 6 2 2 2 2 5 2 2 2 20 2 2 5 2 6 Total Extreme Outcomes 864 Total Possible Outcomes 256,096,265,048,064 Probability of Extreme Outcome 3.4E-2 Odds: i 296,407,74,76 Table 2: The likelihood of choosig six days at radom with returs raked as high as the returs o the ACS grat dates - 0 -
Compay Reported WSJ Odds ( i X) Computed WSJ Odds ( i X) Odds From T Test ( i X) Affiliated Computer Services 300,000,000,000 296,407,74,76 726,485,63 Vitesse Semicoductor 26,000,000,000 26,38,745,280 26,449 Comverse Techology 6,000,000,000 6,225,89,90 26,37 Table 3: A compariso of odds computed usig the WSJ test ad our proposed T test - -