Inerne Appendix o Produc Marke Copeiion, Insider Trading and Sock Marke Efficiency * In his appendix, we verify ha our resuls are robus o a nuber of changes. We firs confir ha hey are insensiive o he definiion of indusries (we obain idenical resuls using hree- insead of wo-digi SIC codes o idenify indusries), he lengh of he even window (our findings are unchanged when we use a hree-day window fro -1 o +1 day), and he definiion of abnoral reurns (again, we obain idenical resuls using raw reurns and abnoral reurns wih respec o he CAPM odel). Second, he regressions in he paper cluser sandard errors by fir and year (Tables II o IV). We confir ha he Faa-McBeh (1973) procedure wih sandard errors correced for eporal dependence (using he Newey-Wes (1987) adjusen wih one lag) leads o he sae conclusions. The resuls feaured in Tables IA.I o IA.III are siilar o hose obained wih fir and year clusers, in ers of coefficiens and significance. Third, he resuls also obain using a nonparaeric analysis ha conrols for fir size. Every year, we sor firs ino size deciles and divide furher each size decile ino arke power quiniles. We copue, wihin each of he 50 groups, he equally weighed average of he share urnover, insider aciviy, and abnoral reurn around earnings announceens. The resuls are displayed in Tables IA.IV o IA.VI. In os size deciles, share urnover and insider rading increase fro he boo o he op arke power deciles, while he abnoral reurn surrounding earnings announceens decreases. These findings confir he resuls of he regressions. This paern is only reversed aong he larges firs (op decile) for he share urnover and insider urnover. This reversal is * Ciaion fora: Peress, Joel, 2010, Inerne Appendix o Produc Marke Copeiion, Insider Trading and Sock Marke Efficiency, Journal of Finance 65, 1-43, hp://www.afajof.org/ia/2010.asp. Please noe: Wiley-Blackwell is no responsible for he conen or funcionaliy of any supporing inforaion supplied by he auhors. Any queries (oher han issing aerial) should be direced o he corresponding auhor for he aricle.
consisen wih he finding ha in he urnover regressions, he coefficien on an ineraced regressor, Mk Power x Size, is negaive. Finally, we also assess he inforaiveness of sock prices using a differen approach. The paper uses he abnoral sock price reacion o earnings news as a proxy for he inforaiveness of sock prices. While his echnique has a long radiion in finance and accouning i, one ay no be enirely convinced ha i capures he inforaion conen of prices. To reinforce he evidence, we eploy a differen approach, also frequenly used in accouning (e.g.,collins e al. (1994), Durnev e al. (2003)). The idea o esiae how uch inforaion abou fuure earnings is capialized ino sock prices by regressing curren sock reurns on fuure earnings. This ehod relies on wo hypoheses. Firs, i assues ha revisions in expeced dividends are correlaed wih revisions in expeced earnings. Therefore, curren sock reurns can be expressed as a funcion of he curren unexpeced earnings and changes in expeced fuure earnings. The second assupion is ha curren unexpeced earnings can be proxied using he curren change in earnings, and ha changes in expeced fuure earnings can be proxied using changes in realized fuure earnings. I follows ha where r = a + b0 Δ E + bsδe + s + csr + s u, (IA.1) + s s r denoes he reurn on fir s sock in year, Δ E he change in is earnings in year, and E + s Δ he change in is earnings s periods ahead. In his equaion, we follow his lieraure by including fuure sock reurns, r + s fuure years of earnings changes and fuure reurns (i.e., seing s=3). ii, as conrol variables, and using hree There are wo easures of sock price inforaiveness, boh based on he significance of fuure earnings in equaion (IA.1). The firs easure is he su of he coefficiens on fuure earnings, denoed FERC (for Fuure Earnings Response Coefficien ) and defined as 3 FERC s= 1 b. s
The second easure is he increenal power of fuure earnings in explaining curren reurns, denoed FINC (for Fuure earnings INCreenal explanaory power ). To copue i, we run he sae regression as equaion IA.1 (again on an annual frequency) excluding he forward looking ers and copue is R². Then we subrac i fro he R² of equaion IA.1. Thus, FINC is defined as FINC R² 3 3 a + b0 ΔE + b E c r u s 1 s Δ s + s 1 s s + = + = + R² a + b0 ΔE + u. Hence, FERC and FINC capure how uch inforaion abou fuure earnings is conained in curren reurns: he higher heir values, he greaer he inforaion conen of sock reurns. To assess he ipac of firs arke power on heir FERC and FINC, we group firs ino arke power groups and easure FERC and FINC wihin each group. To allow for ie variaion in coefficiens, we rebalance groups every year. To iigae he iporan influence of size, we firs sor firs (every year) ino size deciles, which we divide furher ino wo arke power groups. In his way, we obain 20 groups of firs in every year. We denoe by i, j FERC and i, j FINC he response coefficien and increenal explanaory power of fuure earnings in size group i (i=1 o 10) and arke power group j (j=1 o 2). If arke power speeds up he incorporaion of inforaion abou fuure earnings ino curren reurns, hen FERC and FINC should increase fro he low (j=1) o he high (j=2) arke power group. Defining Mk power duy, a duy variable ha equals zero (one) in he low (high) arke power group, he coefficiens on Mk power duy should be significanly posiive in he following regressions: i, j FERC = cons an + β * Mk _ power _ duy + γ * i + δ * Conrol + error, i, j i, j FINC = cons an + β * Mk _ power _ duy + γ * i + δ * Conrol + error. i, j
In hese regressions, we conrol for size by including as a regressor he index i, which easures he fir s size group, iii and for oher facors by including as regressors he average value of a conrol variable wihin size decile i and arke power group j in year, i, j denoed Conrol. We use he sae conrols as in he paper: reurn on asses, leverage, and he arke-o-book raio. Because we do no have Aihud s illiquidiy easure over he full saple period (1969 o 2002), we use insead share urnover. We use annual daa fro Copusa covering he 1969 o 2002 period. The saple ends in 2002 because we need earnings and reurns daa up o 2005. In each of hese 34 years, I for 10 x 2 groups, resuling in 680 observaions. The esiaion resuls are presened in Table IA.VII. In all specificaions, he coefficien on Mk power duy is saisically significan and posiive. Including fuure earnings as an explanaory variable in equaion (IA.1) s leads o a higher loading on fuure earnings and raises is R² ore aong high arke power firs han aong low arke power firs. The effec is also econoically significan. For exaple, a coefficien on Mk power duy of 1 (0.04) in he FERC (FINC) regressions iplies ha FERC (FINC) is larger by 1 (0.04) aong high arke power firs relaive o low arke power firs. This aouns o a relaive increase of 107% (30%) given ha he average FERC (FINC) is 0.93 (0.13). These resuls confir hose obained fro easuring abnoral sock reurns around earnings announceens (Table III). Overall, we conclude ha here is convincing evidence ha produc arke power enhances rading, including ha by insiders, and he inforaiveness of sock prices, in line wih he odel.
References Abarbanell, J., W. Lanen and R. Verrecchia, 1995, Analyss forecass as proxies for invesor beliefs in epirical research, Journal of Accouning and Econoics 20, 31 60. Aiase, R. and L. Baber, 1994, Trading volue reacions o annual accouning earnings announceens: he increenal role of predisclosure inforaion asyery, Journal of Accouning and Econoics 17, 309 329. Baber, L. and Y. Cheon, 1995, Differenial price and volue reacions o accouning earnings announceens, Accouning Review 70, 417 441. Barron, Orie, and Paela Suerke, 1998, Dispersion in analys s earnings forecass as a easure of uncerainy, Journal of Accouning, Audiing, and Finance 13, 243 268. Collins, D.W., S. P. Kohari, J. Shanken and R. Sloan, 1994, Lack of Tieliness and Noise as Explanaions for he Low Coneporaneous Reurn-Earnings Associaion. Journal of Accouning and Econoics 18, 289 324. Durnev, A, R. Mork, B. Yeung and P. Zarowin, 2003, Does Greaer Fir-Specific Reurn Variaion Mean More or Less Infored Sock Pricing?, Journal of Accouning Research, 41(5), 797-836. Faa Eugene, and Jaes MacBeh, Risk, 1973, Reurn and equilibriu: Epirical ess, Journal of Poliical Econoy 81, 607-636. Harris, Milon and Ar Raviv, 1993, Differences of opinion ake a horse race, Review of Financial Sudies 6, 473 506. Kandel, E. and N. Pearson, 1995, Differenial inerpreaion of public signals and rade in speculaive arkes, Journal of Poliical Econoy 103, 831 872 Ki, Jeong-Bon, Izhak Krinsky and Jason Lee, 1997, Insiuional holding and rading volue reacions o quarerly earnings announceens, Journal of Accouning, Audiing and Finance 12, 1 14.
Ki, O. and R. Verrecchia, 1991a, Trading volue and price reacions o public announceens, Journal of Accouning Research 29, 302 321. Ki, O. and R. Verrecchia, 1991b, Marke reacion o anicipaed announceens, Journal of Financial Econoics 30, 273 310. Ki, O. and R. Verrecchia, 1994, Marke liquidiy and volue around earnings announceens, Journal of Accouning and Econoics 17, 41 67. Newey, Whiney, and Kenneh Wes, 1987, A siple, posiive sei-definie, heeroscedasic and auocorrelaion consisen covariance arix, Econoerica 55, 703-708. Uaa, S. and W. Cready, 1997, Insiuional ownership, differenial predisclosure precision and rading volue a announceen daes, Journal of Accouning and Econoics 24, 129 150.
Table IA.I Marke Power and Turnover: Faa-Macbeh Regressions wih Newey-Wes Sandard Errors This able presens resuls of annual Faa-MacBeh regressions of urnover on arke power. Sandard errors are adjused for serial correlaion using he Newey-Wes procedure wih one lag. The absolue value of -saisics are displayed below he coefficien esiaes. The sybols ***, **, and * denoe significance a he 1%, 5%, and 10% levels, respecively, for he wo-ailed hypohesis es ha he coefficien equals zero. See he ables in he paper for variable definiions. Turnover Mk power 0.9248 0.5303 0.3613 0.3488 0.5567 2.5014 1.4955 1.7771 4.26*** 5.15*** 4.19*** 3.63*** 5.64*** 6.54*** 3.90*** 4.36*** Mk power * Size -0.3319-0.188-0.2023 7.40*** 3.76*** 3.95*** Size 0.1289 0.0998 0.1001 0.1265 0.1741 0.1259 0.1549 3.24*** 2.39** 2.41** 3.01*** 3.95*** 2.71*** 3.26*** Illiquidiy -0.0184-0.0181-0.0172-0.018-0.0168 3.27*** 3.23*** 3.11*** 3.29*** 3.11*** Reurn on asses 0.3605 0.393-0.058 0.3168-0.1128 1.89* 2.02** 0.23 1.72* 0.47 Marke-o-book 5.6956 5.2156 5.1997 1.85* 1.94* 1.93* Leverage -0.8553-0.8617 7.15*** 7.17*** Consan -0.2349-0.9293-0.7174-0.7337-0.7321-1.1849-0.8664-0.8921 3.00*** 3.75*** 2.81*** 2.93*** 2.89*** 4.30*** 3.04*** 3.09*** Observaions 25798 25791 25732 25462 25389 25791 25732 25389
Table IA.II Marke Power and Insider Trading: Faa-Macbeh Regressions wih Newey-Wes Sandard Errors This able presens resuls of annual Faa-MacBeh regressions of insider rading on arke power. In he lef panel, insider rading aciviy is easured as he log of he raio of a fir s annual oal insider rading dollar volue o he fir s arke capializaion, and i is denoed Insider urnover. In he righ panel, i is easured as he log of he raio of he fir s annual nuber of insider rades o is nuber of acive insiders, denoed Nuber of insider rades. Sandard errors are adjused for serial correlaion using he Newey-Wes procedure wih one lag. The absolue value of -saisics are displayed below he coefficien esiaes. The sybols ***, **, and * denoe significance a he 1%, 5%, and 10% levels, respecively, for he wo-ailed hypohesis es ha he coefficien equals zero. See he ables in he paper for variable definiions. Panel A: Insider urnover Mk power 0.0036 0.0055 0.003 0.0027 0.0025 0.0183 0.0124 0.0122 2.82*** 4.86*** 3.14*** 2.54** 2.03** 4.79*** 3.09*** 3.30*** Mk power * Size -0.0021-0.0015-0.0016 4.36*** 2.71*** 3.23*** Size -0.0006-0.0006-0.0006-0.0006-0.0003-0.0004-0.0004 5.45*** 5.71*** 5.69*** 4.23*** 2.57** 2.93*** 2.38** Illiquidiy 0 0 0 0 0 1.33 1.29 1.34 1.15 1.19 Reurn on asses 0.0109 0.0119 0.0121 0.0105 0.0117 6.04*** 6.43*** 6.63*** 6.07*** 6.78*** Marke-o-book 0.0234 0.025 0.0248 1.32 1.36 1.35 Leverage 0.0005 0.0004 0.35 0.3 Consan 0.0062 0.0094 0.0093 0.0092 0.0092 0.0077 0.008 0.008 19.44*** 15.57*** 15.17*** 15.19*** 15.06*** 11.85*** 9.34*** 9.71*** Observaions 26040 26031 25970 25676 25597 26031 25970 25597 Panel B: Nuber of insider rades Mk Power 0.2149 0.215 0.1591 0.1485 0.1687 0.5368 0.3615 0.3808 7.10*** 6.76*** 6.85*** 7.14*** 7.47*** 5.08*** 6.64*** 7.02*** Mk Power * Size -0.0526-0.0329-0.0344 4.47*** 4.70*** 5.03*** Size -0.0003-0.0018-0.0017 0.0008 0.0071 0.0029 0.0058 0.32 1.48 1.33 0.66 4.68*** 2.15** 4.59*** Illiquidiy -0.0022-0.0022-0.0021-0.0021-0.002 1.6 1.55 1.49 1.59 1.47 Reurn on asses 0.2327 0.2686 0.2235 0.2252 0.2135 2.11** 2.81*** 2.16** 2.07** 2.09** Marke-o-book 1.3271 1.2818 1.2814 2.22** 2.30** 2.29** Leverage -0.0838-0.0855 4.01*** 4.08*** Consan 0.2176 0.2193 0.2281 0.2236 0.2243 0.1762 0.2006 0.1957 9.86*** 9.07*** 9.93*** 10.17*** 10.29*** 9.41*** 10.05*** 9.46*** Observaions 23091 23080 23037 22743 22674 23080 23037 22674
Table IA.III Marke Power and Sock Price Inforaiveness: Faa-Macbeh Regressions wih Newey-Wes Sandard Errors This able presens resuls of annual Faa-MacBeh regressions of he absolue abnoral reurn surrounding an earnings announceen on arke power. Sandard errors are adjused for serial correlaion using he Newey-Wes procedure wih one lag. The absolue value of -saisics are displayed below he coefficien esiaes. The sybols ***, **, and * denoe significance a he 1%, 5%, and 10% levels, respecively, for he woailed hypohesis es ha he coefficien equals zero. See he ables in he paper for variable definiions. Sock price inforaiveness Mk power -0.0995-0.048-0.0207-0.0197-0.0196-0.0369-0.189-0.0753-0.0753-0.1076 16.03*** 6.68*** 3.79*** 3.51*** 2.75*** 4.47*** 11.03*** 8.41*** 8.89*** 9.18*** Mk power * Size 0.0242 0.0094 0.0096 0.0119 6.77*** 3.71*** 3.90*** 11.61*** Size -0.0172-0.0154-0.0157-0.0158-0.0194-0.0206-0.0167-0.0171-0.0211 10.33*** 10.05*** 10.53*** 10.66*** 28.55*** 10.03*** 9.28*** 9.69*** 27.81*** Illiquidiy 0.001 0.001 0.001 0.0014 0.001 0.0009 0.0014 4.77*** 4.75*** 4.62*** 5.21*** 4.77*** 4.59*** 5.17*** Reurn on asses -0.0885-0.0948-0.0948-0.0857-0.0869-0.0921-0.0823 6.11*** 7.58*** 6.51*** 7.68*** 6.03*** 6.28*** 7.26*** Marke-o-book 0.2484 0.2358-0.0051 0.2341-0.0062 1.66* 1.67* 0.22 1.67* 0.27 Leverage 0.0018 0.0292 0.0021 0.0296 0.25 5.90*** 0.3 5.84*** Turnover 0.0267 0.0268 12.40*** 12.58*** Consan 0.1595 0.2486 0.2349 0.2357 0.236 0.2558 0.2669 0.2423 0.2434 0.2652 21.37*** 24.51*** 30.03*** 31.24*** 30.63*** 44.69*** 21.27*** 25.43*** 26.83*** 50.49*** Observaions 23432 23417 23415 23113 23040 22718 23417 23415 23040 22718
Table IA.IV Marke Power and Turnover: Double-sor Table This able presens he average urnover for firs sored every year ino size deciles and arke power quiniles. See he ables in he paper for variable definiions. Size Marke Power Low 2 3 4 High High - Low -sa for High - Low Sall -0.587-0.594-0.482-0.454-0.451 0.135 1.75 2-0.633-0.603-0.366-0.208-0.322 0.311 4.873 3-0.532-0.519-0.204-0.091-0.078 0.454 6.473 4-0.335-0.417-0.312 0.015-0.044 0.291 3.993 5-0.282-0.16-0.217 0.133 0.134 0.416 5.827 6-0.205-0.066 0.007 0.126 0.211 0.416 6.748 7-0.002-0.042-0.089 0.023 0.198 0.199 3.11 8 0.119 0.043 0.094 0.06 0.283 0.164 2.756 9 0.217 0.09 0.141 0.161 0.171-0.046-0.678 Big 0.267 0.188 0.15-0.049-0.064-0.331-5.951
Table IA.V Marke Power and Insider Trading: Double-sor Table This able presens he average insider urnover and he nuber of insider rades for firs sored every year ino size deciles and arke power quiniles. See he ables in he paper for he variable definiions. Panel A: Insider urnover Size Marke Power Low 2 3 4 High High - Low -sa for High - Low Sall 0.006 0.008 0.008 0.008 0.006 0-0.119 2 0.006 0.007 0.008 0.009 0.01 0.003 1.984 3 0.005 0.006 0.006 0.008 0.008 0.003 2.402 4 0.006 0.007 0.01 0.009 0.012 0.005 2.238 5 0.005 0.007 0.009 0.011 0.008 0.004 3.272 6 0.008 0.007 0.007 0.009 0.011 0.003 1.43 7 0.006 0.005 0.007 0.006 0.008 0.002 1.654 8 0.005 0.006 0.007 0.006 0.007 0.002 1.389 9 0.008 0.006 0.005 0.006 0.005-0.003-1.922 Big 0.005 0.003 0.004 0.003 0.005 0-0.116 Panel B: Nuber of insider rades Size Marke Power Low 2 3 4 High High - Low -sa for High - Low Sall 0.177 0.176 0.226 0.24 0.279 0.102 4.23 2 0.179 0.21 0.223 0.322 0.271 0.092 4.428 3 0.204 0.225 0.281 0.307 0.33 0.126 5.254 4 0.222 0.232 0.248 0.311 0.304 0.082 3.636 5 0.229 0.258 0.25 0.299 0.328 0.099 4.242 6 0.192 0.253 0.267 0.302 0.315 0.123 6.084 7 0.197 0.269 0.254 0.269 0.279 0.081 3.543 8 0.231 0.248 0.232 0.261 0.295 0.064 2.828 9 0.205 0.238 0.205 0.249 0.299 0.094 3.367 Big 0.206 0.197 0.217 0.23 0.23 0.024 1.192
Table IA.VI Marke Power and Sock Price Inforaiveness: Double-sor Table This able presens he average absolue abnoral reurn surrounding an earnings announceen for firs sored every year ino size deciles and arke power quiniles. See he ables in he paper for he variable definiions. Size Marke Power Low 2 3 4 High High - Low -sa for High - Low Sall 0.223 0.218 0.21 0.203 0.178-0.045-6.668 2 0.193 0.19 0.177 0.177 0.166-0.026-5.137 3 0.19 0.162 0.166 0.16 0.157-0.032-6.295 4 0.168 0.155 0.144 0.152 0.152-0.016-3.246 5 0.156 0.143 0.137 0.142 0.144-0.012-2.698 6 0.15 0.138 0.126 0.133 0.137-0.013-3.193 7 0.142 0.128 0.123 0.124 0.127-0.015-3.465 8 0.139 0.123 0.119 0.115 0.119-0.02-4.965 9 0.134 0.118 0.107 0.107 0.115-0.02-4.694 Big 0.115 0.102 0.096 0.089 0.097-0.018-4.568
Table IA.VII Marke Power and Sock Price Inforaiveness: Measuring Sock Price Inforaiveness by he Fuure Earnings Response Coefficien, FERC, and he Increenal Power of Fuure Earnings in Explaining Curren Reurns, FINC This able presens resuls of regressions of FERC and FINC on arke power and oher fir characerisics over he 1969-2002 period. Every year, firs are sored firs ino size deciles and hen ino wo arke power groups. The following regression is run in each year and wihin each size and arke power group: 3 3 r = a + b0 Δ E + bsδe + s + csr + s + u, (IA.1) s= 1 where r denoes he reurn on fir s sock in year, Δ E he change in is earnings in year, and Δ E + s he change in is earnings s periods ahead. FERC and FINC are esiaed in each year and wihin each size-arke power group. FERC is defined as FERC 3 s= 1bs, and represens he fuure earnings response coefficien. FINC is defined as FINC R² 3 3 R² a + b E b E c r u a b0 E u 0Δ + s 1 s s + s 1 s s + + + Δ Δ = + = + and capures he increenal power of fuure earnings in explaining curren reurns. Higher FINC and FERC correspond o sock prices ha conain ore inforaion abou fuure earnings. The righ-hand-side variable in he regressions displayed in he able below is FERC in Panel A and FINC in Panel B. The ain regressor is Mk power duy, a duy variable ha equals zero (one) in he low (high) arke power group. The conrol variables are he size group and he average value of he reurn on asses, leverage, arke-o-book, and share urnover, in each year and wihin each size-arke power group. Sandard errors are clusered by year. The absolue value of -saisics are displayed below he coefficien esiaes. The sybols ***, **, and * denoe significance a he 1%, 5%, and 10% levels, respecively, for he wo-ailed hypohesis es ha he coefficien equals zero. See he ables in he paper for he variable definiions. s= 1
Panel A: FINC Mk power duy 0.0129 0.0129 0.0398 0.0381 0.0422 0.0422 1.73* 1.73* 2.40** 2.25** 2.53** 2.53** Size group 0.0024 0.0026 0.0027 0.0037 0.0037 1.71* 1.82* 1.93* 1.88* 1.88* Reurn on asses -0.5102-0.5026-0.5721-0.5721 1.62 1.58 1.87* 1.87* Marke-o-book 0.0014 0.0016 0.0016 1.34 1.48 1.48 Leverage -0.0387-0.0387 0.27 0.27 Turnover -0.0064-0.0064 0.82 0.82 Consan 0.1271 0.1141 0.1242 0.1209 0.1184 0.1184 24.35*** 13.72*** 11.46*** 10.44*** 4.58*** 4.58*** Observaions 680 680 680 680 680 680 R² 0.0058 0.0121 0.0233 0.0259 0.0275 0.0275 Panel B: FERC Mk power duy 1.1277 1.1277 1.04 1.0112 0.8299 0.8299 4.71*** 4.70*** 3.05*** 3.02*** 2.18** 2.18** Size group 0.001 0.0003 0.0024 0.1653 0.1653 0.02 0.01 0.05 2.20** 2.20** Reurn on asses 1.6645 1.7952 2.9714 2.9714 0.23 0.25 0.36 0.36 Marke-o-book 0.0243 0.0126 0.0126 0.69 0.38 0.38 Leverage -12.7295-12.7295 2.05** 2.05** Turnover 0.307 0.307 1.31 1.31 Consan 0.366 0.3603 0.3273 0.27 2.1537 2.1537 2.07** 0.89 0.65 0.55 1.86* 1.86* Observaions 680 680 680 680 680 680 R² 0.032 0.032 0.0321 0.0326 0.0447 0.0447
i Many analyical and epirical sudies show ha price reacions o earnings announceens depend on he conen of he public announceen relaive o he qualiy of he pre-announceen inforaion. Theoreical sudies include Ki and Verrecchia (1991a, 1991b, 1994), Harris and Raviv (1993), and Kandel and Pearson (1995). Epirical sudies include Aiase and Baber (1994), Baber and Cheon (1995), Abarbanell, Lanen and Verrecchia (1995), Uaa and Cready (1997), Ki, Krinsky and Lee (1997), and Barron and Suerke (1998). ii Collins e al. (1994) argue ha using he acual fuure earnings inroduces an error-in variables proble in equaion (IA.1) because he heoreically correc regressor is he unobservable change in expeced fuure earnings. Fuure reurns are included as conrol variables o correc for his bias. However, our findings are unchanged when we drop fuure reurns fro equaion (IA.1). iii The resuls are unchanged if we conrol for size using he average size wihin each size decile raher han he size decile index i.