Optiml Eecution of Open-Mrket Stock Repurchse Progrms Jcob Oded This Drft: December 15, 005 Abstrct We provide theoreticl investigtion of the eecution of open-mrket stock repurchse progrms. Our model suggests tht the eecution depends on vilbility of free csh nd informtion symmetry. The results highlight importnt fetures of open-mrket stock repurchse progrms: they leve the firm the option to void pyout when csh is needed for opertions, yet they lso disburse free csh s long s the stock is not severely overpriced. Becuse they re preformed t mngement discretion, however, repurchse progrms lso re-distribute welth mong shreholders. The model genertes predictions bout the completion rte of the progrms nd bout the bid-sk spred during the repurchse period tht might eplin inconsistencies mong erlier empiricl studies. I would like to thnk seminr prticipnts t Boston University. Deprtment of Finnce nd Economics, School of Mngement, Boston University, 595 Commonwelth Avenue, Boston, MA 015. Jcob Oded: (617)353 679, oded@bu.edu. 1
1 Introduction Over the lst two decdes, nnouncements of open-mrket stock repurchse progrms (henceforth, open-mrket progrms ) hve become common prctice. 1 Yet, empiricl evidence points to gret vribility ssocited with their eecution. First, there is gret vribility documented bout ctul completion rtes. In the US, Stephens nd Weisbch (1998) nd Jgnnthn, Stephens, nd Weisbch (000) document tht verge ctul repurchse rtes re only between 70 80%. Frequently only smll frction of the quntity of stock nnounced is ctully repurchsed, nd mny nnouncing firms do not repurchse t ll. Actul repurchse rtes re even lower outside the US. Ikenberry, Lkonishok, nd Vermelen (000) find verge ctul repurchse rtes to be s low s 8% of the quntity nnounced in Cnd, nd Ru nd Vermelen (00) find verge ctul repurchse rtes of only 37% in the UK. In ddition, there is gret vribility in the timing of the repurchse trde. Some firms repurchse the full quntity nnounced immeditely fter they nnounce. Others repurchse grdully, or wit for long period nd then repurchse ll or prt of the quntity nnounced (Stephens nd Weisbch (1998), Cook Krigmn nd Lech (004)). There is lso disgreement mong empiricl studies bout the ffect of the open-mrket progrm nnouncements on liquidity, mesured by the bid sk spred. In the US, Brcly nd Smith (1988) find widening of the spred. However, Miller nd McConnell (1995) find no widening of the bid sk spred, wheres Wiggins (1994), Frnz, Ru, nd Tripthy (1995), nd Cook, Krigmn, nd Lech (004) ctully find nrrowing of the bid sk spred during periods of ctul repurchses. Outside the US, Brockmn nd Chung (001) nd Ginglinger nd Hmon (003) find widening of the bid sk spred during ctul repurchse periods in Hong Kong nd Frnce, respectively. Why is there vribility in ctul repurchse nd in the bid-sk spred during the repurchse period cross firms nd cross countries? Wht is the optiml wy to eecute n open-mrket repurchse progrm? Are there ny implictions for regultory bodies? The purpose of this pper is to develop theoreticl frmework with which to nswer these questions. Unlike most erlier theoreticl investigtions of open-mrket progrms, we build on the motivtion to distribute free csh in order to void its wste. Incresing empiricl evidence suggests tht the vilbility of free csh, nd the need to void its wste, ply n importnt role in decisions to 1 See, for emple, Stephens nd Weisbch (1998), Jgnnthn, Stephens, nd Weisbch (000), nd Grullon nd Michely (00). Most of the bove studies lso hve findings on mrket depth consistent with their findings bout bid sk spred. Tht is, studies tht find nrrowing of the bid sk spred lso find n increse in mrket depth mesured by price impct on order imblnces, nd studies tht find widening of the spred lso find decrese in mrket depth.
nnounce nd eecute open-mrket progrms. For emple, Grullon nd Michely (004) find tht the progrm nnouncement return is higher for firms tht re more likely to overinvest, nd Stephens nd Weisbch (1998), nd Oswld nd Young (004) find tht ctul repurchses depend on the vilbility of free csh. 3 For most of the nlysis, we tke the progrm nnouncement s given in order to focus on the eecution. Assuming uncertinty nd symmetric informtion bout the firm vlue, we show tht the eecution is the solution to n optimiztion problem over wste-prevention benefits from pying out free csh nd gins (or losses) from the informed trde of the firm. Specificlly, if the firm lerns tht it does not hve free csh it refrins from eecuting the progrm so s not to hurt investment. If, insted, the firm lerns tht it does hve free csh, it will lwys eecute the repurchse when the stock is undervlued, becuse in this cse it benefits from preventing the wste of free csh nd it lso ccrues trding gins from the (informed) repurchse trde. However, when the stock is overvlued, the firm fces trdeoff between wste prevention gins nd trding losses. Hence, the firm is less likely to eecute the repurchse when the stock is overvlued; the higher the overvlution, the less likely is the eecution. Thus, open-mrket progrms enhnce vlue to shreholders by distributing free csh but lso result in welth trnsfers mong shreholders becuse of the informed/strtegic trde of the firm. The model provides predictions bout ctul repurchse rtes nd the bid sk spred tht might eplin the discrepncy mong the empiricl studies cited bove: When uncertinty bout the firm vlue is reltively high, ctul repurchses re driven by the motivtion to tke dvntge of informtion through strtegic trding, nd, hence, open-mrket progrms re chrcterized with low completion rtes nd high bid sk spreds. In this cse, epected welth trnsfers mong shreholders (epropritions) becuse of the firm s informed trde re more significnt thn epected vlue enhncement through the disbursement of free csh. In contrst, when uncertinty bout the firm vlue is reltively low, ctul repurchses re driven by the motivtion to distribute free csh in order to void its wste, nd open-mrket progrms re thus chrcterized with higher completion rtes nd lower bid sk spreds. In this cse, epected vlue enhncement through the disbursement of free csh is more significnt thn epected welth epropritions. These results nturlly generte testble predictions bout how the eecution will depend on firm chrcteristics, such s vlue vs. growth, lrge vs. smll, etc. Becuse the model suggests tht open-mrket progrms enhnce epected firm vlue but t thesmetimeresultinwelthepropritions,the model lso hs importnt regultory impli- 3 On the gency costs of free csh flow see, for emple, Jensen (1986) Stultz (1990) nd Btes (005). 3
ctions: Whenever welth epropritions re more significnt thn the enhncement of vlue, open-mrket progrms nd their eecution should be regulted or even forbidden. In contrst, when the sitution is reversed, regultory bodies should encourge open-mrket progrms nd void regulting the eecution, s regultion could discourge eecutions nd thereby ecerbte the wste of free csh nd welth epropritions mong shreholders groups. We show tht this impliction is brodly consistent with the cross-country evidence (see Section 4). The model highlights two importnt fetures of open-mrket progrms tht hve been lrgely ignored in the theoreticl literture nd tht might eplin their incresing populrity. First, the model suggests tht open-mrket progrms provide the firm with finncil fleibility, i.e., the firm retins the option not to eventully repurchse, should the vilbility of free csh chnge. 4 Most firms hve considerble mounts of csh on their blnce sheet t the time they nnounce repurchse progrm. Our thrust is tht, t the time they mke the nnouncement, whether this csh is free or not is yet to be determined. Second, in most of the eisting literture, the trding gins ssocited with the (informed) repurchse trde re generlly viewed s negtive property of open-mrket progrms (e.g., Brcly nd Smith (1988)) wheres the model here suggests tht these trding gins do not represent zero sum gme. Specificlly, if free csh disbursement is vlue enhncing, then when the firm eecutes the repurchse under uncertinty nd symmetric informtion, it is privtely informed not only bout the true vlue of the stock, but lso bout the vlue enhncement through the repurchse trde. This increses the motivtion to eecute the repurchse even when the stock is overvlued. Most erlier theoreticl investigtions of open-mrket repurchse progrms, focus on signling undervlution motivtion. Signling motivtion, however, seems inconsistent with the noncommitting nture of open-mrket progrms confirmed with low ctul repurchse rtes nd does not eplin the mied results on the bid sk spred. Further, even mong the signling ppers, very few consider the optionlity of the progrms (i.e., distinguish between nnouncement nd ctul repurchse). The lter group includes Ikenberry nd Vermelen (1996), Bhttchry nd Dittmr (003), nd Oded (005). All three ppers build signling stories bsed on the optionlity of open-mrket progrms, but bstrct from the disbursement of free csh. Brennn nd Thkor (1990) lso consider the optionlity of open-mrket progrms, but their focus is on welth trnsfers ssocited with the eecution rther thn on signling undervlution. Interestingly, the gency costs of free csh re lrgely ignored in theoreticl work bout repurchses in generl nd for open-mrket repurchse progrms in prticulr. To our knowledge, the only theoreticl ppers tht consider free csh distribution s motivtion in 4 Supporting evidence on the fleibility of open-mrket progrms is provided in Jgnnthn, Stephens, nd Weisbch (000), Guy nd Hrford (000), nd Brv et l (005). 4
repurchse policy re Chowdhry nd nd (1994), nd Lucs nd McDonld (1998). However, these studies do not distinguish between nnouncement nd ctul repurchse, nd thus pply more to tender offers thn to open-mrket progrms. The rest of this pper is orgnized s follows. The ssumptions re set up nd discussed in Section. Section lso demonstrtes the min ide using numericl emple. A generl formultion nd solution is given in Section 3. Implictions tht were briefly discussed bove nd possible etensions re discussed in Section 4. Section 5 concludes. Assumptions nd Emple There re three dtes indeed by t =0, 1,. All gents re risk neutrl, the interest rte is zero, nd there re no tes or trnsction costs. Consider n equity-finnced firm. At t =0, the firm owns project nd some csh, where it is uncler wht portion of the csh will be needed to finnce the project nd wht portion of the csh is free csh. At t =1thefirm genertes ssets in plce with vlue of à {A, A + X} with equl probbility, where 0 <X, nd relizes free csh C {0,C} with equl probbility, where 0 <C,ndwhereà nd C re independent. 5 Thus, there re four eqully likely outcomes for the firm vlue V t t =1: V 1 {A, A + C, A + X, A + X + C}. We will generlly omit the time inde for t =1,smost of the ction hppens on this dte. At t = thefirm is sold/dismntled, nd investors get the vlue of their shres. The firm is run by mnger who mimizes the terminl vlue per shre. 6 Informtion is symmetric t t =0. However,tt = 1 only the mnger observes the reliztion of à nd C, wheres ll other gents know only the distribution of these vribles. The prcticl interprettion here is tht the mnger observes the relized vlue of the firm s projects nd in ddition observes wht portion of the firm s csh is ctully needed for the projects, wheres the rest becomes free csh. The shreholders nd the mrket do not observe this informtion yet. At t =, ll informtion is publicly known. There re shres outstnding t t = 0, nd we normlize the vlues of A, X, C, V to be vlues per shre using lowercse letters,, c, v, respectively. At t =0thefirm cn nnounce repurchse progrm tht it my eecute t t =1. 7 The firm cn buy bck shres t t =1 5 See lso Section 4 on the ssumption tht à nd C re independent 6 It will be shown in Section 3.3. tht, in our set up, this objective function is equivlent to mimizing the epected welth of the originl shreholders. Whose vlue the firm is mimizing is still n open question in corporte finnce. See, for emple, Myers nd Mjluf (1984). 7 Since our focus is optiml eecution of the progrm, we will lter tke it s given tht the firm hs progrm it cn eecute t t =1. In the model, for the firm, nnouncing lwys domintes not nnouncing, nd, since t t = 0 ll informtion is symmetric, the nnouncement hs no signling content. We will consider the cse 5
only with free csh (otherwise the vlue of ssets in plce is severely dmged). If the firm does hve free csh but does not distribute it with repurchse t t = 1, portion (1 δ) of the free csh is lost, where δ (0, 1]. 8 Without loss of generlity, we ssume tht the firm will repurchse whenever indifferent. Like most pyout policy models, we ssume tht borrowing is not llowed. At t = 1 there is mrket for the stock. Liquidity trders plce quntity bids A < nd 3 B < they wnt to buy nd sell respectively. The mrket mker sets prices p 3 A,p B in the buy nd sell mrkets, respectively, before investors plce their quntity bids (nticipting the possibility of informed trde from the firm side) to ern zero epected profit. 9 The following emple demonstrtes how uncertinty in the vlue of ssets in plce nd uncertinty of free csh interct to determine the progrm eecution. Emple 1: Consider cse with high uncertinty of the vlue of ssets in plce nd reltively low uncertinty of free csh (shown in Figure 1). At t =0firm F hs =10 shres. At t = 1, the vlue of ssets in plce is relized to be either 7 or 1 with equl probbility, nd free csh is relized to be or 0 with equl probbility. Thus, =0.7, =0.5, c =0., nd there re four possible sttes s described in Figure 1. Assume tht t t =1 liquidity buyers plce quntity orders A = B = 3, nd suppose further tht if the free csh is not distributed t t =1thenδ =0.8,i.e. thewsterteis1 δ =0.. Suppose tht the firm does not nnounce repurchse progrm. Then the epected firm vlue t t =is 0.5(7 + 1 + 0.8 ) = 10.3. Since there is no informed trde t t = 1, this is the price the mrket mker will sell nd buy for t t = 1. Tht is, without repurchse progrm p A = p B =1.03. If, insted, the firm does nnounce progrm t t =0,tt = 1 it will buy shres only in the upper stte in Figure 1. without progrm only for comprison. We lso tke it s given tht firms must mke their progrms publicly known (nnounce) beforehnd. The only country in which firms re not required to nnounce their progrms beforehnd is the US, nd even there, nnouncing is the norm. 8 We tke the gency problem s given, s we wnt to focus on the eecution itself. Thus, we refrin from modeling the resons for the wste nd do not model ny benefits for the mnger. Models tht ssume the mnger does not benefit from the wste of free csh include Chowdhry nd nd (1994). See lso Section 4 on this ssumption. We eclude δ = 0 to simplify the nlysis. Our results would hold lso for δ =0. 9 The mrket mechnism we use is stndrd nd is empolyed, for emple, in oe (00). We focus on t =1 becuse this is where the repurchse tkes plce, but it could be ssumed tht the mrket opens lso t t =0 nd t =. The restriction on liquidity trde is without loss of generlity in order to limit the discussion to the fesiblerngeoftheresults. 6
To ern zero epected profit, the mrket mker must set p A such tht 3[(p A 0.7) + (p A (0.7+0.8 0.)) + (p A (1.))] + (p A ( 7+5 10 p A ))(3 + p A )=0 whichuponsolutionimpliesp A =1.1093. The implied verge terminl stock vlue t t =is 7+5 0.5[0.7 + 0.86 + 1. +( 10 )] = 1.056. 1.1093 This is lso the price t which the mrket mker buys for t t = 1 (no dverse selection on sell mrket), tht is, p B = E [p ]=1.056. 10 In comprison to the cse where the firm does not nnounce progrm, liquidity buyers lose 0.16, originl shreholders gin totl 0.6 (i.e. 0.06 per shre regrdless of when they sell). Socil welth increses becuse of the repurchse by 0.6 0.16 = 0.1. ow, consider insted, cse with low uncertinty of the vlue of ssets in plce reltive to the uncertinty of the free csh (shown in Figure 1b). Specificlly, consider firm G, for which everything is the sme s for firm F,eceptthttt = 1 the vlue of the ssets in plce is relized to be either 9 or 10 with equl probbility (the free csh is still either or 0 with equl probbility). Thus, for firm G, = 0.9, = 0.1, c = 0.. There re four possible sttes s described in Figure 1b. If the firm does not nnounce repurchse progrm, the epected firm vlue t t =is 0.5(9 + 10 + 0.8 ) = 10.3 nd p A = p B =1.03 (sme s for firm F ). However, if firm G nnounces progrm t t =0,t t = 1 it will buy shres not only in the upper stte in Figure 1b, but rther in both sttes in which it hs free csh. To ern zero epected profit, the mrket mker must set p A such tht 9 3[(p A 0.9) + (p A 1)] + (p A ( 10 ))(3 + )+(p A ( 9+1 p A p A 10 ))(3 + )=0 p A p A whichuponsolutionimpliesp A =1.089. The implied verge terminl stock vlue t t =is 0.5[0.9+1.0+ 9 10 1.089 + 9+1 10 ]=1.0576. 1.089 10 One cn verify tht the firm will not buy in the stte with low sset vlue with free csh: if it does not repurchse, terminl vlue per shre would be 0.7 + 0.8 0. = 0.86. If it does repurchse, terminl vlue per shre would be 7/(10 (/1.1093)) = 0.8540, nd hence the firm is better off not repurchsing. 7
This is lso the price t which the mrket mker buys for t t = 1 (no dverse selection on sell mrket), tht is p B = E[p ]=1.0576. 11 In comprison to the cse where the firm does not nnounce progrm, liquidity buyers lose 0.0760, originl shreholders gin totl 0.760 (i.e. 0.076 per shre regrdless of when they sell). Socil welth increses becuse of the repurchse by 0.760 0.0760 = 0.. Tble 1 highlights the differences between Firm F nd firm G in the bove emple. Emple 1 demonstrtes tht, when uncertinty in the vlue of ssets in plce reltive to the uncertinty of free csh is low (Firm F in Tble 1), n open-mrket progrm will result in highercompletionrtendlowerbid skspredincomprisontothecseinwhichuncertinty in the vlue of ssets in plce reltive to uncertinty in the free csh is high (Firm G in Tble 1). Furthermore, when uncertinty bout the vlue of ssets in plce is low, the increse in socil welth is higher nd there is lso less welth trnsfer from liquidity/outside investors to insiders. For both firms, the inherited fleibility of open-mrket progrms leds to informed trde from the firm side. Mngers repurchse to enhnce the vlue of terminl shres. This vlue enhncement comes prtly t new shreholders epense nd prtly becuse the repurchse prevents the wste of free csh. 3 The Forml Model Becuse informed trde is possible only in the buy mrket, we focus on this mrket nd denote A, p A p. Given the ssumptions in Section, the mrket mker s zero-epected-profit condition is X [Pr{j}(p v vj,r j )( + r j p )] = 0 (1) j where j indictes the four possible outcomes (sttes) of the firm vlue V t t =1,wherer j is the number of shres the firm repurchses t t =1insttej, ndv vj,r j is the vlue of ech shre t t =dependingonv j,thevluepershreinsttej relized t t =1,ndonr j. Definition 1 Equilibrium is set ({r j },p) consisting of repurchse strtegy {r j } (0, C p ) set by the mnger given {v j },p to mimize the terminl vlue per shre v,ndpricep set 11 One cn verify tht the firm will indeed buy in stte with low sset vlue with free csh: If it does not repurchse, terminl vlue per shre would be 0.9+0.8 0. = 1.06. If it does repurchse, terminl vlue per shre would be 9/(10 (/1.089)) = 1.1039, nd hence the firm is better off repurchsing. 8
by the mrket mker, such tht condition (1) is stisfied. It is immedite to show tht if the firm does not nnounce repurchse progrm p = E[v ]= + + δc. Henceforth we tke it s given tht the firm nnounces repurchse progrm t t = 0, nd we focus on the optiml eecution (see lso footnote 7). Lemm 1 In ny equilibrium, the firm never repurchses in the sttes v =, v = + nd it lwys repurchses with ll vilble csh in the stte v = + + c. Proofs of ll Lemms nd Propositions pper in the Appendi. Accordingly, we cn write the mrket mker s zero-epected-profit condition (1) s: (p ) +(p v v=+c )( + r p,v=+c )+(p ( + )) +(p ( A + X ))( + C )=0. () C p p This condition essentilly requires tht the verge of the differences between the price tht the mrket mker is willing to sell for nd the terminl vlue of shre, weighted by the quntity he sells in ech stte, is equl to zero. The first nd the third terms correspond to the sttes with low nd high sset vlue, respectively, where the firm hs no csh nd therefore does not repurchse. The lst term corresponds to the stte with high sset vlue nd csh (v = + + c). By Lemm 1, in this stte the firm will lwys repurchse. In this stte the terminl vlue per shre is A+X nd the mrket mker sells + C shres. The second term C p p correspondstotheinterestingsttewithlowssetvluendwithcsh(v = + c) ndin which the decision to repurchse depends on the model prmeters. In this term, the vlue of r (repurchse) is either C or 0, depending on whether or not the firm repurchses in this stte, p A nd v v=+c is either or + δc depending on whether or not the firm repurchses in this C p stte, respectively. An importnt feture of repurchses under symmetric informtion tht is reflected in () is the nonlinerity in vlue introduced through the firm s trde. Specificlly, when the firm does repurchse to tke dvntge of its privte informtion, the per-shre vlue increses not only becuse trding gins re dded to the vlue of the terminl shres, but lso becuse these trding gins re shred by reduced number of shres. Definition A Full Repurchse Equilibrium is n equilibrium in which the firm repurchses t t =1whenever it hs free csh, i.e. in both sttes v = + c nd v = + + c. A Prtil Repurchse Equilibrium is n equilibrium in which the firm repurchses t t =1only when it hs free csh nd the sset vlue is high, i.e. only in stte v = + + c. 9
In ny full repurchse equilibrium, condition () becomes (p ) +(p A )( + C C p p + X )+(p ( + )) +(p (A ))( + C )=0, (3) C p p wheres in ny prtil repurchse equilibrium, condition () becomes (p ) +(p ( + δc)) +(p ( + )) +(p ( A + X ))( + C )=0. (4) C p p The following Lemm presents the solution for the price p of (3) nd (4) in full repurchse equilibrium nd in prtil repurchse equilibrium, respectively. Lemm In ny full repurchse equilibrium, the price p t which the mrket mker sells t t =1is r ³ ( + c + c) + ( + c + c) + c ³( + +c) ( + ) p =. (5) In ny prtil repurchse equilibrium, the price p t which the mrket mker sells t t =1is p = r ³ + +(1+ δ c )c + + +(1+ δ c )c 4 4 4 4 + c ³( + + c) (3 + + δc). (6) The firm s decision bout whether or not to repurchse in the stte v = +c depends on the one hnd on how deep the undervlution is nd on the other hnd on how severe the wste is. Specificlly, since by ssumption the mnger mimizes the vlue of the terminl shres, she will not buy if the terminl stock vlue without repurchse is higher thn the terminl stock vlue with the repurchse, tht is, if + δc > C p, which fter rerrngement is equivlent to p> δ + c. (7) 10
Otherwise, tht is, if p + c, thefirm will lwys repurchse (recll tht without loss of δ generlity we hve ssumed tht the firm will repurchse whenever indifferent). The importnt nd nonintuitive insight, which is reflected in (7), is tht when coming to decide whether to repurchse or not, the mnger does not compre the vlue to the price, but rther compres the projected terminl vlues under ech lterntive. 1 The following Lemm combines condition (5)withtherequirementtht(7)doesnot hold to give necessry nd sufficient condition for full repurchse equilibrium; it lso combines conditions (6) nd (7) to give necessry nd sufficient condition for prtil repurchse equilibrium. Lemm 3 A necessry nd sufficient condition for full repurchse equilibrium is µ 4+ δ 4 µ 1 c ³+ δ δ ( +4) µ +1. (8) A necessry nd sufficient condition for prtil repurchse equilibrium is µ 4+ δ 4 µ µ 1 >c δ δ +4 1 (1 + )( +1). (9) Conditions (8) nd (9) re the bsis for our results. We first demonstrte the results for specil cses. 3.1 Specil Cses 3.1.1 Specil Cse 1: δ =1 Consider first the cse in which δ = 1, tht is, the cse where there is no csh wste regrdless of whether the firm repurchses or not. In this cse the sole purpose of repurchsing is trding gins bsed on symmetric informtion. The repurchse lso does not increse socil welth. Proposition 1 Suppose δ =1.Ifboth < +1 (10) 1 ote tht p>+ δc is not enough to ssure no repurchse. Tht is, to ssure no repurchse it is not enough tht the price is higher thn the terminl stock vlue without repurchse. Tht ssurnce requires the stronger restriction, reflected in (7), tht the terminl stock vlue without repurchse be higher thn the terminl stock vlue with repurchse. 11
nd ³ +1 6 c (11) hold, the outcome is full repurchse equilibrium. Otherwise, the outcome is prtil repurchse equilibrium. The intuition for Proposition 1 is s follows. When δ = 1, csh wste does not ffect repurchse policy. A full repurchse equilibrium requires tht the firm will buy in the stte with low vlue of sset in plce nd free csh (v = + c). Otherwise, prtil repurchse equilibrium previls (the firm lwys buys in the other stte with free csh v = + + c). For the firm to repurchse in the stte with low vlue of ssets in plce nd free csh, the price must not be higher thn the stock vlue in this stte. This is becuse when δ = 1 condition (7) becomes p + c. The equilibrium price, in turn, must provide the mrket mker with zero epected profit nd hence reflects the epected vlue, pushed somewht higher to reflect the level of dverse selection ssocited with the repurchse. Adverse selection, however, is positively correlted with both vribility in the vlue of ssets in plce,, nd with the level of free csh when the firm does hve csh, c (i.e. the vribility of free csh). When is significnt, the price tht the mrket mker sets to ern zero epected profit will lwys be too high for the firm to repurchse in the stte with low sset vlue nd with csh (v = + c), no mtter wht the vlue of c is. However, when is sufficiently low, its effect on the price becomeslesssignificnt so tht with enough vribility in free csh (when c is high enough), the stock vlue with free csh will be higher thn the price tht gives the mrket mker zero epected profit even if the vlue of ssets in plce is relized to be low nd prtil repurchse equilibrium cnnot hold nd full repurchse equilibrium will previl. 13 Figure demonstrtes the results in Proposition 1 by mens of grph. The figure illustrtes how the decision of whether or not to repurchse in the stte v = + c depends on the vribility in the vlue of ssets in plce,, nd the vribility in free csh, c. Theverticl dshed line indictes where condition (10) holds with equlity. To the right of this line the vribility in the firm vlue, introduced through the vribility in ssets in plce, is too high so tht it pushes the stock price too high for full repurchse equilibrium to eist, nd therefore prtil repurchse equilibrium previls. To the left of the dshed line, equilibrium type depends on the level of free csh. Specificlly, the solid curved line indictes where condition (11) holds with equlity. Below this curved line, vribility in the firm vlue due to vribility in csh 13 For simplicity we hve ssumed csh distribution of {0,C}. If, insted, we chose two positive vlues of csh (s is for ssets in plce), the qulittive results should not chnge. However, this will significntly complicte the nlysis. 1
is too low (c is too low), so tht in the stte with low sset vlue nd csh the firm will not repurchse nd hence prtil repurchse equilibrium previls. Above this line (c is sufficiently high), full repurchse equilibrium previls. ote tht deeper mrket (lrger )menstht the dshed line gets pushed to the right nd the solid curved line gets pushed down, so tht the region in which full equilibrium previls widens nd the region in which prtil equilibrium previls nrrows. Also, in the region where < +1, the smller is, the lower the required level of c for full repurchse equilibrium to eist. 3.1. Specil Cse : =0 Suppose there is no vribility in the vlue of ssets in plce i.e., =0. Proposition When =0 full repurchse equilibrium lwys eists nd prtil repurchse equilibrium never eits. Intuitively, when there is no vribility in the vlue of ssets in plce, only vribility in free csh determines the vribility in the firm vlue. Becuse the mrket mker sets price to ern zero epected profit, the firm will be undervlued whenever it hs free csh. Thus, regrdless of the wste rte, the firm will lwys repurchse when it hs free csh. When δ < 1, however, with = 0 the repurchse completely prevents the wste of free csh. 3. The generl cse In the generl cse, full repurchse equilibrium nd prtil repurchse equilibrium re not mutully eclusive. Thus, we hve to nlyze them seprtely. We first investigte eistence of full repurchse equilibrium bsed on condition (8) nd then investigte eistence of prtil repurchse equilibrium bsed on condition (9). 3..1 Full repurchse equilibrium Proposition 3: (Eistence of full repurchse equilibrium) A full repurchse equilibrium lwys eists if δ + (1) nd never eists if ( ³ +) +1 δ. (13) + 13
In the rnge + < δ < full repurchse equilibrium eits if c c F where nd does not eist otherwise. µ c F,, δ, ³ 1 δ ( ³ +) +1 (14) + δ 4+ 4 δ. (15) ( + )( +1) +4 Proposition 3 suggests tht eistence of full repurchse equilibrium depends primrily on the reltion between δ nd. Only when δ nd meet prticulr joint conditions does the reltion between these vribles nd c lso mtter. Specificlly, when both δ nd re low, full repurchse equilibrium lwys holds, nd when both δ nd re high, full repurchse equilibrium never holds. Otherwise, if both re neither too smll nor too high, eistence will depend on the vribility of free csh c, where there is some level of c, c F,bovewhich full repurchse equilibrium eists nd below which it does not eist. Thus, in this region, for sufficiently high c full repurchse equilibrium will eist. Figure 3 demonstrtes the results of Proposition 3 by mens of grph. The figure illustrtes how eistence of full repurchse equilibrium depends on the vribility in the vlue of ssets in plce,, nd the wste rte cptured by δ. The solid (curved) line indictes where condition (1) holds with equlity. The dotted (curved) line indictes where condition (13) holds with equlity. In the re bove the dotted line full repurchse equilibrium never eists. In the re below the solid line full repurchse equilibrium lwys eists. In the re cptured between the lines (where (14) holds), full repurchse equilibrium eits if c>c F,wherec F is defined in (15). The intuition for Proposition 3 is s follows. In the re bove the dotted line, δ is high (free csh wste is not significnt) nd hence the intuition in Proposition 1 for the cse with high still goes though. Tht is, if δ nd re high (up nd to the right of the dotted line in the figure), the wste of free csh is not mteril; wheres the dverse selection introduced though the vribility in vlue of ssets in plce is strong, so tht the sk price tht the mrket mker sets is very high. Consequently, in the stte in which the vlue of ssets in plce is relized to be low the firm is better off not repurchsing. Although in this cse some free csh is lost, thereby reducing shreholders vlue, the lterntive of pying too much for the shres would hurt shre vlue even more. Between the curved lines, the effect of free csh wste becomes significnt, so tht the intuition of Proposition 1 no longer holds. Specificlly, in this region, vribility in 14
the vlue of ssets in plce still motivtes no repurchse in the stte with low sset vlue (with free csh) but the potentil benefit from preventing free csh loss now significntly motivtes repurchse. Consequently, in this region, eistence of full repurchse equilibrium depends on the level of free csh c when the firm does hve csh (vribility in free csh). Higher vribility in free csh mgnifies the benefit from wste prevention more thn it mgnifies the loss from pying higher price set by the mrket mker to compenste for higher dverse selection. This is not only becuse benefits from wste prevention re higher, but lso becuse these benefits re shred by reduced number of shres. Thus, in this region, there is some level of c given in (15), bove which full repurchse equilibrium eists nd below which it does not eist. Below the solid line, full repurchse equilibrium lwys eists. Intuitively, in this region the wste rte is so high tht the firm is willing to buy bck shres even if the price is very high. This is becuse the lterntive is losing most of the csh nd severely dmging firm vlue. Lst, note tht higher does not ffect the solid line but does push the dotted line down, decresing the prevlence of full repurchse equilibrium. This is becuse lower liquidity increses the effect of dverse selection on price (pushes it up) so tht, other things equl, when eistence of full repurchse equilibrium does depend on dverse selection (i.e., when the wste does not dominte), full repurchse equilibrium is less likely to previl. 3.. Prtil repurchse equilibrium Proposition 4: (Eistence of prtil repurchse equilibrium) 4A. (Cse 1) Suppose < 4 then ( + < +) ³ +1. + In this cse prtil repurchse equilibrium never eits if nd lwys eists if δ + ³ + ³ +1 + (16) δ. (17) 15
In the rnge + < δ < prtil repurchse equilibrium eits if c<c P where nd does not eist otherwise. 4.B (Cse ) Suppose then µ c P,, δ, 1 δ ( +) ³ +1 + ³ (18) δ 4+ 4 δ ³ +1 (19) +4 1 1+ > 4 ( +) ³ +1 + In this cse prtil repurchse equilibrium never eits if < +. δ ( +) ³ +1 + (0) nd lwys eists if In the rnge + ( +) ³ +1 + prtil repurchse equilibrium eits if c>c P where c P otherwise. 4.C (Cse 3) Suppose = 4 δ. (1) < δ < + () is defined in (19) nd does not eist 16
then ( +) ³ +1 + In this cse prtil repurchse equilibrium never eits if nd lwys eists otherwise. δ + = +. Proposition 4 is rther long nd seems comple becuse the reltion between the restrictions on δ chnges with, nd three seprte cses must thus be considered. However, the sme results re obtined in ll three cses: eistence of prtil repurchse equilibrium depends primrily on the reltion between δ nd. Only in certin region does the reltion between these vribles nd c lso mtter. Specificlly, when both δ nd re low prtil repurchse equilibrium never holds, nd when both δ nd re high prtil repurchse equilibrium lwys holds. Otherwise, eistence of prtil repurchse equilibrium depends on the vribility in free csh c s follows. If δ is high but is low, there is some level of c below which prtil repurchse equilibrium eists nd bove which it does not eist, nd if is high but δ is low prtil repurchse equilibrium eists bove tht sme level of c nd does not eist below tht level. Figure 4 demonstrtes the results of Proposition 4 by mens of grph. The figure illustrtes how the eistence of prtil repurchse equilibrium depends on the vribility in the vlue of ssets in plce,, nd the wste rte cptured by δ (high wste rte is low δ). As in Figure 3, the solid (curved) line indictes where condition (16) holds with equlity. The dshed (curved) line indictes where condition (17) holds with equlity. In the re bove both lines prtil repurchse equilibrium lwys eists. In the re below both lines prtil repurchse equilibrium never eists. In the re cptured between the lines, to the left of their crossing point, prtil repurchse equilibrium eists if c<c P,wherec P is definedin(19),wheresin the re cptured between these lines to the right of their crossing point prtil repurchse equilibrium eits if c>c P. (A prtil repurchse equilibrium does not eist t the crossing point.) The intuition for Proposition 4 is s follows. In the re bove the solid line, δ is reltively high (the wste is not significnt), so tht the intuition of Proposition 1 still goes through. Tht is, if nd δ re high (to the right of the solid line in the figure), the effect of the vribility in vlue of ssets in plce on the price is strong enough to deter the firm from repurchsing if the vlue of ssets in plce is relized to be low, regrdless of the relized level of free csh. Above 17
the solid line nd to the left of the dshed line (i.e., between the curved lines to the left of their crossing point), the vribility of ssets in plce is low enough to render the vribility in the level of csh importnt. If vribility in free csh is low (when c is sufficiently low), prtil equilibrium will eist. Otherwise, if vribility in free csh is high (when c is sufficiently high), prtil repurchse equilibrium cnnot eist, becuse in the sttes in which the firm does hve free csh, given the price set by the mrket mker to ern zero epected profit, repurchsing results in higher terminl stock vlue even if the vlue of ssets is plce is relized to be low. The sitution is different thn in tht of Proposition 1 when δ becomes significntly low, i.e., belowthesolidline,becuseinthisregionthewsteoffreecshbecomessignificnt so tht the firm will repurchse regrdless of the price nd therefore prtil repurchse equilibrium never holds. This is in turn becuse the unppeling lterntive is to wtch the free csh dispper without contributing to the firms vlue. However, prtil repurchse equilibrium my still be vible even if the wste rte is high (δ low) if vribility in the vlue of ssets in plce is sufficiently high (between the curved lines nd to the right of their crossing point). In this region, becuse vribility in the vlue of ssets in plce is high, higher vribility in free csh mgnifies welth epropritions through dverse selection more thn it mgnifies benefits from wste prevention. As result, the price tht ssures zero epected profit tothemrketmker ssuming tht the firm buys only in the high stte increses very quickly in c. Consequently,in this region, there is some level of c, givenin(19), belowwhichprtil repurchse equilibrium cnnot eist but bove which it cn. Lst, note tht higher (lower liquidity) does not ffect the solid line but does push the dshed line down nd thereby increses the prevlence of prtil repurchse equilibrium. This is becuse lower liquidity increses the effect of dverse selection on price (pushes it up), so tht other things equl prtil repurchse equilibrium is more likely to previl. 3..3 Coeistence of full nd prtil repurchse equilibrium Figure 5 combines Figures 3 nd 4 to demonstrte the rnges of eistence for both prtil nd full repurchse equilibri, depending on nd δ. In the re below both the solid nd the dshed lines (i.e., to the left of their crossing point nd below the solid line) both nd δ re low, hence only full repurchse equilibrium eists. However, in the re below the solid line bove the dshed line (i.e., to the right of their crossing point between the lines) δ is low but is now reltively high, so prtil repurchse equilibrium cn lso eist if c<c P.Inthere bove the dotted line both δ nd re high, hence only prtil repurchse equilibrium eists. However, in the re below the dotted line but bove both the dshed nd solid lines is high but δ is reltively not s high so tht full repurchse equilibrium cn lso eist if c>c F. In the re bove the solid line but below the dshed line (to the left of their crossing point) 18
eistence of both equilibri depends on the level of c. Ifc is sufficiently low to render c<c P, only prtil repurchse equilibrium eists, nd if c is sufficiently high to render c>c F,only full repurchse equilibrium eists. If c F <c<c P, i.e., if ³ 1 δ δ 4+ 4 δ ³ 1 <c< ( + ) +4 1 δ ³ +4 δ 4+ 4 δ (3) 1 (1 + )( +1) both prtil nd full repurchse equilibri cn eist. Consider condition (3) on c. If δ = 1 both limits on c re identicl, nd the rnge is thus empty. In this cse, s we hve seen erlier (specil cse δ = 1), full repurchse equilibrium nd prtil repurchse equilibrium re mutully eclusive. For ny δ < 1wegetrngeofvluesforc in which both equilibri eist. Within the discussed re (the tringle shped re in Figure 5), this rnge on c indicted in (3) widens with the decrese in δ. Proposition 5:When eistence of full repurchse equilibrium depends on c, the vlue of c F increses in nd δ. When eistence of prtil repurchse equilibrium depends on c, thevlue of c P increses in nd δ for < 4 nd decreses with nd δ otherwise. The following emple demonstrtes coeistence of prtil nd full repurchse equilibri. Emple : Suppose A = 7,X = 1,C = 5, = 10, A = 1, nd δ = 0.55. Then there eists prtil repurchse equilibrium for p = 1.808. There lso eists full repurchse equilibrium for p = 1.67. In Figure 5, these equilibri re between the dshed nd the solid lines to the right of their mutul crossing point. To understnd the coeistence result, suppose the mrket mker sets the price to p =1.808. Condition (4) becomes (1.808 0.7)+(1.808 (0.7+0.55 0.5))+(1.808 (0.7+0.1))+(1.808 7+1 10 5 )(1+ 5 1.808 )=0. 1.808 The mrket mker gins in the sttes, {, + } nd loses in the sttes { + c, + + c}. He mkes zero epected profit. The firm does not repurchse in the stte +c nd will not devite; 7 if it did, the terminl vlue per shre would be =0.968, which is lower thn 0.975 (the 10 5 1.808 terminl vlue without repurchse in tht stte). Thus, prtil repurchse equilibrium 19
eists. ow suppose the mrket mker sets price p = 1.67. Condition (3) becomes (1.67 0.7)+(1.67 7 10 5 1.67 )(1+ 5 1.67 7+1 )+(1.67 (0.7+0.1))+(1.67 10 5 )(1+ 5 1.67 )=0. 1.67 The mrket mker gins in the sttes, {, + c} nd loses in the stte { +, + + c}. He mkes zero epected profit. The firm repurchses whenever it hs free csh. The firm does repurchse in the stte + c nd will not devite; if it did, the terminl vlue per shre would 7 be 0.975, which is lower thn = 1.014 (the terminl vlue with repurchse in tht 10 5 1.67 stte). The intuition for the coeistence here is s follows. Going from the prtil repurchse equilibrium to the full repurchse equilibrium, the mrket mker reduces the price nd therefore hisgininthesttes{, + } is reduced nd his loss in the stte v = + + c is incresed. However, with tht lower price, it now pys the firm to repurchse in the stte v = + c. The mrket mker now mkes money in the stte v = + c, not only t the liquidity buyers epense, but lso t the firm s epense. This dditionl gin in the stte v = +c compenstes the mrket mker for lower gins in the other sttes, nd gin he ends up with zero epected profit. The firm pys 1.67 per shre to relize only 1.0147 on the terminl dte. However, it is hppy to do so; if it does not, 45% of the csh will be lost, which is more thn its trding loss 1 1.014 =37.6%. Emple uses etreme wste rte (δ =0.55) in order to demonstrte 1.67 the intuition. In the other coeistence rnges, emples with lower losses to the firm (or higher gins) cn be given. 3.3 The good equilibrium nd the bd equilibrium Once we hve estblished the res of eistence, the question is whether we cn stte tht the full repurchse equilibrium is better thn the prtil repurchse equilibrium s Emples 1 nd suggest. This is our gol in this subsection. More specificlly, we will demonstrte tht, in full repurchse equilibrium, completion rte nd socil welth re higher nd bid sk spred nd welth epropritions re lower. We first consider completion rte nd socil welth nd then turn to consider bid sk spred nd socil welth. When performing the nlysis, we must mke sure we do not compre pples to ornges. Foremple,itwillnotbecorrecttocompre the socil welth improvement in full repurchse equilibrium with high δ to the socil welth improvement in prtil repurchse equilibrium with low δ (reltive to no repurchse). 0