A Theory of Debt Maturity: The Long and Short of Debt Overhang

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1 A Theory of Deb Maury: The Long and Shor of Deb Overhang Douglas W. Damond and Zhguo He Ths draf: May (Frs draf: January ) Absrac Deb maury nfluences deb overhang: he reduced ncenve for hghlylevered borrowers o make real nvesmens because some value accrues o deb. Reducng maury can ncrease or decrease overhang even when shorer-erm deb s value depends less on frm value. Fuure overhang s more volale for shorer-erm deb, makng fuure nvesmen ncenves volale and nfluencng mmedae nvesmen ncenves. Wh mmedae nvesmen, shorer-erm deb ypcally mposes lower overhang; longer-erm deb can mpose less f frm value s more volale n bad mes. For fuure nvesmens, reduced correlaon beween he value of asses-n-place and profably of nvesmen ncreases he overhang of shorer-erm deb. ooh School of usness, Unversy of Chcago, and NER. The auhors graefully acknowledge research suppor from he Cener for Research n Secury Prces a Chcago ooh. Damond graefully acknowledges suppor from he Naon Scence Foundaon. We hank wo referees, semnar parcpans a MIT Sloan, OSU Fsher, Chcago ooh, Columba, Yale, Harvard, UCLA, NER Corporae Fnance meeng n Chcago, AFA n Denver, Na ergman, Hu Chen, Gusavo Manso, Gregor Mavos, Vcora Ivashna, Henr Pages, Raghu Rajan, erk Sensoy, Jeremy Sen, Rene Sulz, Sherdan Tman and especally Sewar Myers and Charles Kahn for nsghful commens.

2 . Inroducon Ths paper sudes he effecs of he deb maury on curren and fuure real nvesmen decsons of an owner of equy (or a manager who s compensaed by equy). Our analyss s based on deb overhang frs analyzed by Myers (977), who pons ou ha ousandng deb may dsor he frm s nvesmen ncenves downward. A reduced ncenve o underake profable nvesmens when decson makers seek o maxmze equy value s referred o as a problem of deb overhang, because par of he reurn from a curren new nvesmen goes o make exsng deb more valuable. Myers (977) suggess a possble soluon of shor-erm deb o he deb overhang problem. In par, hs exends he dea ha f all deb maures before he nvesmen opporuny, hen he frm whou deb n place can make he nvesmen decson as f an all-equy frm. Hence, followng hs logc, deb ha maures soon--alhough afer relevan nvesmen decsons, as opposed o before--should have reduced overhang. However, shor-erm deb s known o have several dsadvanages. For frms whou access o ousde funds o mee deb repaymens, shor-erm deb can lead o early frm closure and lqudaon (e.g., Damond (99), Gerner and Scharfsen (99)). More relevan o our paper, Gerner and Scharfsen (99) show ha, condonal on ex-pos fnancal dsress, makng a fxed promsed deb paymen due earler (.e., shorer-erm) rases he marke value of he deb and hus he frm s marke leverage, leadng o more deb overhang ex-pos. In addon, ceran drawbacks of shor-erm deb have also been suggesed by some quanave models (Tman and Tsyplakov (7), Moyen (7)) ha focus on equy holders dfferenal ables and ncenves o adjus leverage n response o new nformaon gven dfferen deb maury srucures. Our paper ams o provde a horough analyss on he effecs of deb maury on he equy ncenves o underake boh curren and fuure nvesmens, and, more mporanly, o denfy he forces ha deermne overhang. We show why he deas based on Myer s suggeson have mer, and show how and why hey can be reversed under dfferen sengs. Throughou, we frs llusrae our resuls va smple examples; hese resuls are hen generalzed n he conex of For more deals, see leraure revew a he end of he nroducon.

3 sandard models used by praconers and researchers, allowng us o esablsh her generaly and nroduce more complcaed ssues. We sress he mporance of he relave mng of nvesmen decsons, deb maury, and news abou he prospec of he frm s exsng asses (ncludng pas nvesmen). In a nushell, shorer-erm deb--because of earler repaymen and less rsk sharng usually receves less benef from an mmedae nvesmen, leadng o a lower overhang. Ths s conssen wh he deas n Myers (977). However, for frms wh fuure nvesmen opporunes, he fuure prospecs and value of exsng asses flucuae before nvesmen decsons, and he sharng of less rsk by shorer-erm deb mples more volale equy value and hence more volale deb overhang. Shorer-erm deb hus mposes sronger overhang especally n bad mes, and hs resul s relaed o Gerner and Scharfsen (99) and Tman and Tsyplakov (7). The frs seng ha we analyze s he one wh mmedae nvesmen decsons ha are made rgh afer he ssuance of he deb bu before any new nformaon s released. Exendng he logc of Myers (977), shorer-erm deb would mpose less overhang on mmedae nvesmen, and we llusrae hs resul by smple examples. There, a shorer maury makes he deb marke value less sensve o changes n he frm value, and equy holders nvesmen ncenves are dsored (downward) exacly by he ncreased value of clams oher han equy. However, hs nuve dea s ncomplee, as we furher show ha he effec of maury could reverse, dependng on he mng of nvesmen decsons and he way ha uncerany s resolved over me (.e., sae-dependen volaly). Movng from examples o sandard model frameworks, Proposon n Secon 3 proves ha, holdng he deb value and hence he frm s leverage consan, 3 shorer-erm deb mposes less overhang on mmedae nvesmen n he classc seng based on lack and Scholes (973) and Meron (974). To our knowledge, hs analycal resul s new o he leraure, and appears o be he bass for he logc exendng Myers (977). In he lack-scholes-meron seng, he asse volaly s consan. Wh covaraon of volaly and he value of asses-n-place, especally when he mng of resoluon of uncerany leads o suffcenly ncreased remanng uncerany afer bad oucomes, shorer-erm deb may have a sronger overhang even for mmedae nvesmen decsons. We show hs by an example 3 Holdng consan he borrower s leverage has no been sressed n prevous sudes abou he effec of maury on deb overhang, e.g., Gerner and Scharfsen (99). 3

4 n Secon.5, more formally by addng sae dependen volaly o he lack-scholes-meron model n Secon 3.3. These resuls produce new mplcaons for he effec of deb maury on mmedae nvesmen ncenves. In he second seng ha we analyze, frms have many nvesmen opporunes n he presen and he fuure. Wh fuure nvesmen opporunes, he dsrbuon of deb overhang n he fuure s relevan because he ncenve o nves depends on he overhang prevalng a he me of fuure nvesmen. In addon, deb overhang nfluences he equy s decson o defaul on deb, 4 and defaul mples ha nvesmen opporunes hereafer are no aken. In general, exacly because shorer-erm deb s less sensve o changes n frm value, leads o more volale fuure equy value and hence more volale fuure overhang: Equy has weak nvesmen ncenves and s more lkely o defaul afer poor performance of he frm s asses-nplace, and has srong nvesmen ncenves afer good performance. Examples n Secon.4 llusrae hs logc and some of s mplcaons. For frms wh nvesmen opporunes n he fuure, a balance of mananng nvesmen ncenves n fuure good and bad saes leads o an opmal neror maury srucure. Ths mechansm s formally llusraed n Secon 4 where we sudy a dynamc model ha generalzes Leland (994b, 998) o nclude a seres of fuure nvesmen opporunes. We furher show ha when nvesmen opporunes are posvely correlaed wh he frm s asses-n-place, s benefcal o have a shorer deb maury, whch mposes lower (hgher) deb overhang n mes wh hgh (low) values of asses-n-place and hence beer (worse) nvesmen opporunes. Ths resul offers a new perspecve on he emprcal predcons regardng growh frms and deb maury. For growh frms wh unceran nvesmen opporunes where exsng nvesmen projecs have realzaons ha are posvely correlaed wh he value of new nvesmen opporunes, shorer-erm deb s preferred. However, for a growh frm wh known fuure opporunes or he one where realzed asse reurns are no very nformave abou fuure opporunes, nvesmen ncenves are more effcen wh longer-erm deb. On he oher exreme, maure frms ha requre mely manenance o replace unexpecedly hgh deprecaon n mes of low cash flows should choose even longer-erm deb. Ths perspecve s dfferen 4 The nerpreaon of endogenous defaul gven deb burden as undernvesmen due o deb overhang, s menoned n, for example, Lambrech and Myers (8) and He (). 4

5 from he exsng dea ha frms wh subsanal fuure nvesmen opporunes should choose shorer-erm deb. In our dynamc seng wh mulple nvesmen opporunes, oday s nvesmen benef s posvely relaed o fuure nvesmen polces. ecause shorer-erm deb rggers earler defaul whch elmnaes fuure growh opporunes, hs negave force may feed back o oday and undermne curren nvesmen ncenves. Inuvely, f fuure growh exendng oday s nvesmen wll no occur, makes curren nvesmen less aracve. Ineresngly, n conras o a sac seng where rskless deb canno mpose any overhang, Secon 4.4 shows ha n our dynamc seng a polcy of (almos) rskless ulra-shor deb may cause srong overhang on curren nvesmen. Our framework focuses on decsons durng he perod before deb refnancng and hus fs well wh he emprcal leraure n whch shor-erm deb s ofen classfed as ha wh maury whn hree years (e.g., Johnson (3)). Indeed, emprcal work on deb maury based on he hypohess of reduced overhang of shorer-erm deb, whch mples he use of more shor-erm deb by growh frms wh large nvesmen opporunes, has had mxed success for reasons relaed o our fndngs n hs paper. 5 Shor-erm overhang s relaed o, bu dsnc from, he dea ha shorer-erm deb maury ncreases he conrol rghs of lenders o dscplne managemen, e.g., enmelech (6), Calomrs and Kahn (99), Damond (99), Damond and Rajan (a), Flannery (994), and Leland (998). 6 Shor-erm deb provdes dscplne n par because shor-erm lenders do no s dly whle a borrower msbehaves--hey demand paymen on maury. Ths s closely relaed o our pon ha shor-erm deb can have severe overhang when frm value declnes afer he deb was ssued. From hs perspecve, our paper exends he resuls n Gerner and Scharfsen (99) ha for a gven oal promsed (bu rsky) paymen o deb holders, ex-pos deb overhang s made worse f more of he fxed amoun s due sooner. Makng a fxed paymen due earler rases he marke value of he deb and hus he frm s marke leverage, ncreasng ex-pos overhang. Our paper emphaszes he mng of nvesmen, and examnes ex-ane effecs of maury on deb 5 For nsance, arclay and Smh (995) and Guedes and Opler (996) documen a negave relaon beween maury and growh opporunes, whle Sohs and Mauer (996) and Johnson (3) fnd a posve relaon once conrollng for frm leverage. 6 I also dffers from heores of maury srucure where shor-maury deb s aracve o borrowers wh prvae nformaon ha her fuure cred rang may mprove, e.g., Flannery (986) and Damond (99), or he dea ha shor-erm deb ences runs due o coordnaon ssues, e.g., Damond and Dybvg (983), He and ong (). 5

6 overhang for a gven nal leverage o solae from he pure leverage effec. 7 As a resul, our paper reconcles he resul n Gerner and Scharfsen (99) ha shorer-erm deb has more ex-pos overhang, wh he suggeson n Myers (977) ha shorer-erm deb has less ex-ane overhang. We descrbe several mporan effecs of deb maury ha have no receved much formal analyss n he leraure. We have new resuls on he effec of deb maury on deb overhang wh an mmedae nvesmen n he lack-scholes-meron seng, holdng leverage consan. Our resul on he effecs of reduced volaly when frm value s low has no been suggesed prevously. We argue n he concluson ha hs resul apples especally well o fnancal nsuons such as banks. Our analycal sudy for he case of dynamc nvesmen opporunes n he Leland seng s relaed o exsng quanave sudes on he effec of deb overhang. These sudes focus on leverage adjusmens o rade off he ax sheld agans physcal coss of defaul, and each adds some oher frcons. oh of hem sugges ha shor-erm deb s no a perfec soluon o overhang. Tman and Tsyplakov (7) use a model based on Leland (998) bu wh coss of adjusng leverage, and fnd ha shor-erm deb mproves nvesmen ncenves bu rggers earler defaul. Relave o Tman and Tsyplakov, our paper wh a smpler seng provdes analycal resuls, and furher pons ou ha wh ner-emporally lnked nvesmen ncenves shor-erm deb may hur curren nvesmen decsons due o earler defaul n he fuure. In anoher closely relaed paper, Moyen (7) sudes he effec of deb maury on overhang drecly bu focuses on an assumed asymmery n leverage adjusmen,.e., he leverage canno be adjused f here s long-erm deb, bu can be adjused every perod f shor-erm deb s ssued. Moyen fnds ha, compared o long-erm deb, a frm wh shor-erm deb has hgher (lower) leverage n good (bad) mes, bu he overall overhang effec s smlar across boh maury srucures. In conras, we follow Leland (994b, 998) where he deb burden s fxed o focus only on maury. Our analyss of dynamc nvesmen opporunes s based on Leland (994b, 998) so ha he deb refnancng rae (whch s nversely relaed o he frm s deb maury srucure) 7 We sudy he effec of maury on deb overhang by fxng he leverage exogenously for an addonal reason. Tax based heores sugges leverage ncreases when curren and fuure nvesmen opporunes become more profable, whle conrol and peckng order heores sugges he oppose. There s no an agreemen n emprcal sudes abou hese deermnans of dynamc adjusmens o frm leverage (see e.g., Fama and French ()). 6

7 s fxed a a consan. The dynamc adjusmen of deb maury s beyond our paper, and we provde some dscusson n Secon 4.6 The res of he paper s organzed as follows. We gve a seres of examples n Secon o llusrae he key deas of hs paper. Secon 3 provdes a model wh a sngle nvesmen decson based on he lack-scholes-meron seng, and Secon 4 provdes a dynamc Lelandype model wh many fuure nvesmen decsons. In Secon 5 we conclude.. Deb Overhang: Assumpons and Examples We frs descrbe deb overhang and relaed assumpons, and hen provde numercal examples o llusrae he man nsghs delvered by hs paper.. Deb Overhang and Key Assumpons Deb overhang, frs formalzed by Myers (977), capures he nsgh ha nvesmen ofen leads o exernal benefs ha accrue o he frm s deb clams. These exernal benefs consequenly lead equy holders (or equvalenly managers who are pad n equy) who make nvesmen decsons o nernalze only par of nvesmen benefs, and hence o undernves relave o he level ha maxmzes he oal value of he frm. To sudy deb overhang, we wll make he followng assumpons hroughou.. We examne sandard deb conracs wh wo characerscs: promsed face value and maury.. We assume ha a me he frm has o rase a ceran amoun of fnancng hrough deb. Our analyss fxes he nal marke value of he deb, because we sudy deb maury for a gven amoun of leverage. However, we do no specfy he parcular reasons for why frms use deb. Ths s parly because all reasons for usng deb mus ake accoun of he poenal effec on nvesmen ncenves, and parly because here s no emprcal consensus on he relave mers of varous reasons (e.g., ax or manageral ncenves for decsons oher han nvesmen; see foonoe 7). 3. We assume ha s equy holders who conrol he frm and who carry ou nvesmen. Ths capures he dea ha corporae decsons are delegaed o hose n conrol, raher han decded by a consensus of ousde nvesors. Invesmen opporunes are los durng bankrupcy, and we mpose no exogenous bankrupcy cos oherwse. 7

8 / Sae G /6 / /3 4 / Sae /6 /3 / = = = Fgure : Tme lne for numercal examples, wh condonal probables denoed on each pah. As shown, he condonal dsrbuon gven sae G s {/,/3,/6}. We wll consder an alernave condonal dsrbuon of {/3,/3,} gven sae G. 4. We assume ha deb canno be renegoaed o brbe managers o make alernave decsons. Ths assumpon s especally relevan o deb wh many holders, as opposed o a sngle bank or ndvdual We focus on nvesmen projecs ha are subjec o deb overhang only,.e., projecs ha weakly ncrease or leave unchanged he value of each of s deb and equy clams. We do no consder rsk shfng where a large ncrease n he rsk profle of exsng asses may cause a redsrbuon of value across equy and deb clams, as descrbed n Jensen and Mecklng (976). Throughou hs paper we focus on ncremenal nvesmens whch have less chance of nroducng he possbly of rsk shfng. 6. To focus on maury only, debs wh dfferen maures are assumed o have same senory durng bankrupcy. 9. Example Seng We begn by showng our resuls va numercal examples. We laer wll show smlar resuls based on sandard lack-scholes-meron and Leland models. Asses-n-place. As n Fgure, he frm has asses-n-place whch brng fnal cash flows a dae, wh hree poenal oucomes {4,, } each occurrng wh probably /3 from he perspecve of dae. There are no cash flows on oher daes. The dscoun rae s zero. 8 Gerner and Scharfsen (99) show ha deb ha canno be renegoaed (especally f shor-erm) can mpede renegoaon of oher deb. In ha sense, even renegoaon s subjec o overhang. 9 Ths s relaed o he dluon effec when frms refnance her maurng deb n a dynamc model. We rule ou dluon by adopng he Leland (994b, 998) seng n Secon 4. For dluon ssues, see Damond (993), runnermeer and Oehmke (), and Hackbarh and Mauer (), and relaed dscusson n Secon

9 Informaon. A dae some publc nformaon arrves. A sae whch occurs wh probably /, he news s bad, and he condonal probables o reach he fnal oucomes become {/6,/3,/}. Symmercally, good news arrves a sae G wh a probably of /, and he condonal probables for hree oucomes become {/,/3,/6}. Deb face values and maures. Suppose ha he frm needs o rase 8.5 a dae. The deb can be eher long-erm (repad a dae ) or shor-erm (repad a dae ), and hey have face values of F L =.75 and F S = 8.5, respecvely, o lead o he arge dae- marke value of 8.5: = 8.5 = The lef hand sde descrbes he payoffs o long-erm deb, whch s only pad n full (.e.,.75) wh probably /3 a he oucome of 4. The rgh hand sde descrbes he shor-erm deb: wh probably ½, he frm n sae G pays deb holders he full face value 8.5; whle wh probably ½ he frm n sae defauls, and shor-erm deb holders recover he asses-n-place wh a value of 8 = 4( / 6) + ( / 3). Invesmen opporunes. For ease of llusraon we only consder nfnesmal nvesmen whch mproves he fnal payoff of he asses-n-place by ε>. We do no specfy he nvesmen cos because deb overhang can be measured by he nvesmen benef ha s capured by deb. The nvesmen decson wll be made only f s ne presen value exceeds he deb overhang. Invesmen mngs. We consder wo dfferen mngs of nvesmen. The frs s ha he frm nvess a dae before he realzaon of sae G or ; and he second s ha he frm nvess only a dae afer he realzaon of he news abou he asses-n-place (sae G or ) bu before he shor-erm deb maures. We beleve boh mng assumpons are emprcally relevan..3 Dae- Invesmen before Asses-n-place News: A enchmark Resul We frs consder he case of sngle nvesmen a dae, mmedaely afer rasng he deb. Many of he exng deas based on he dscusson n Myers (977) consder he effec of maury on deb overhang n hs parcular seng. We calculae he overhang as he expeced benef from he new nvesmen ha s capured by he deb wh gven maures. ecause he long-erm deb face value F L =.75 exceeds he nermedae oucome bu below he hghes oucome 4, he overhang occurs n boh he mddle and low saes and hus s /3ε (equy ges /3ε). For shor-erm deb, n sae he frm 9

10 value 8 s below he face value of shor-erm deb F S = 8.5. Shor-erm deb mposes an overhang of /ε as capures all of he gan a sae from he dae- nvesmen. If he nvesmen cos a dae s beween /3ε and /ε, hen hs nvesmen wll be aken f and only f he frm uses shor-erm deb. In hs example, long-erm deb mposes more overhang han shor-erm deb does, conssen wh he dscusson of Myers (977). We wll formally show hs resul n Secon 3 usng a model based on lack-scholes (973) and Meron (974). Ths nuve resul reles upon wo assumpons, whch we wll sudy n he followng subsecons. The frs s abou nvesmen mng; we show ha f equy holders make nvesmen decson a dae afer he news abou asses-n-place, hen he opmal maury wll depend on he deals of he nvesmen opporunes. The second s abou he cash-flow dsrbuon, and we show ha even for dae nvesmen, shor-erm deb may mpose sronger overhang n a dsrbuon feaurng hgher volaly followng bad news..4 Fuure Invesmen: Dae- Invesmen afer News abou Exsng Asses Now suppose ha nvesmen opporunes are avalable only a dae, so ha equy holders make nvesmen decsons afer he realzaon of he nerm sae bu before he shor-erm deb maures. Consder he case of long-erm deb frs. A sae G, he benef from an nfnesmal nvesmen ha goes o deb holders s /ε (equy also ges /ε); pu dfferenly, equy recovers he benef from nvesmen only a he oucome 4 whch occurs wh a condonal probably of ½. A smlar argumen mples ha a sae he long-erm deb overhang s 5/6ε (equy ges /6ε). Hence, long-erm deb mposes some overhang n boh saes, bu s never so severe ha equy holders recover nohng from new nvesmen. If he cos of nvesmen s less han /6ε, for example, hen here wll be nvesmen n boh saes. In conras, shor-erm deb s a hard conrac ha does no share as much rsk wh equy due o s requremen of full paymen whenever possble on s shor maury. As a resul, n sae G shor-erm deb mposes no overhang; bu n sae mposes he mos exreme overhang, whch s ε so ha shor-erm deb holders capure he enre benef of nvesmen. To see hs, he shor-erm deb becomes rskless a sae G (he frm value 6 exceeds he deb face value 8.5) and herefore wll no capure any gan from new nvesmen. However, a sae, he deerorang asses-n-place wh a value of 8 fall below he face value 8.5, so ha equy holders wll defaul a sae. There, he deb overhang s he enre nvesmen benef ε, because f equy holders were o nves rgh before he shor-erm deb maures, deb holders would

11 receve every dollar ha new nvesmen generaes. If he cos of nvesmen s less han /6ε, for example, hen here wll be nvesmen only n sae G whle here would be nvesmen n boh saes wh long-erm deb. There are wo lessons ha we learn from hs example wh dae nvesmen. Frs, shows ha when he frm s asses-n-place flucuaes, a shorer-erm deb generaes a more counercyclcal overhang (hgher overhang for low values of he frm, or weak nvesmen ncenves n fuure bad mes). Hence, when he frm s nvesmen opporunes are presen n he fuure, he opmal deb maury ha mnmzes overall overhang wll depend on he deals of fuure nvesmen opporunes n dfferen saes. Ths dea wll be formally analyzed n he dynamc model n Secon 4. Second, we can relae he sae-dependen condonal overhang n hs example back o he average dae- overhang calculaed n Secon.3. Indeed, he dae long-erm deb overhang /3ε, s jus he average of he overhangs condonal on sae G (5/6ε) and on sae (/ε). Smlarly, he dae- shor-erm deb overhang /ε s he average of he condonal overhang n sae G () and n sae (ε). Shor-erm deb mposes more volale condonal overhang, bu urns ou ha gven he parcular dsrbuon n hs example, once akng he average a dae, shor-erm deb has a lower average overhang han long-erm deb. The nex example shows ha s possble o reverse he relave orderng of dae- overhang by wsng he cash-flow dsrbuon, based on he dea ha condonal volales can affec condonal overhang a dfferen saes..5 Dae- Invesmen wh Condonal Volaly Consder he dae nvesmen seng as n Secon.3, bu modfy he dsrbuon o reduce he condonal varance of cash flows n sae G. Le he new condonal probables gven G be {/3, /3, }; he old condonal dsrbuon of {/,/3,/6} n Fgure s a mean-preservng spread over he new condonal dsrbuon. Due o symmery beween saes G and n he old dsrbuon, he new dsrbuon feaures a lower varance condonal on sae G han sae. Gven he new cash-flow dsrbuon whch mples he dae- probables over fnal cash flows o be {/4,/,/4}, we calculae he new face values needed o rase 8.5,.e., he longerm deb face value s reduced o F L = whle he shor-erm face value s unchanged a F = 8.5. S

12 In hs new example, he shor-erm overhang remans a /ε, because he shor-erm deb value 8.5 s unaffeced, and he frm value 8 n sae sll leads o defaul as n he prevous benchmark example n Secon.3. In fac, s no surprsng o have shor-erm deb overhang unchanged. To see hs, noe ha condonal varances prevalng on dae govern he dsrbuon of dae cash flows condonal on dae nformaon, and as a resul canno affec he payoff of shor-erm deb a all. In conras, he long-erm overhang s reduced o /4ε because he new long-erm deb face value F L = s below he nermedae oucome. Here, alhough shor-erm deb shares less rsk from he jump down n value a of dae, he shorerm deb ends up akng more of he fuure reurn from dae- nvesmen, and hence a more severe overhang. Wha s he reason? As shown, he dae- overhang s he average of fuure levels of overhang a dfferen saes. We observe from Secon.4 ha he relave severy beween long- and shor-erm overhang s sae dependen, n ha long- (shor-) erm deb mposes sronger (weaker) overhang n good saes. Thus, he average overhang a dae depends on he magnude of overhang condonal on fuure saes. Adjusng volales condonal on fuure saes affecs he relave magnude of condonal overhang for deb wh dfferen maures. As menoned above, because shor-erm deb ges repad a dae, he shor-erm overhang s no affeced by dae condonal volales. However, for long-erm deb, eher a lower volaly n sae G or a hgher volaly n sae reduces overhang n he correspondng sae. A lower volaly n he good sae mples ha here s lle chance of defaul for a long perod aferwards; n our new example, he frm wh long-erm deb never defauls n sae G (here s zero probably o have he oucome ), mplyng zero overhang. A hgher volaly n sae mples ha, despe a low curren value, he asses-n-place are more lkely o ncrease suffcenly o repay credors before long-erm deb maures, whch reduces overhang as equy can recover benefs from nvesmen n hese For nsance, he ncreased volaly condonal on sae does no affec overhang for a small ncremenal nvesmen. As long as he value of asses-n-place (afer nvesmen) s below he face value of shor-erm deb due before furher resoluon of uncerany, all nvesmen benef followng sae goes o shor-erm deb holders. The marke value of shor-erm deb s 8.5 (8) n sae G (), whle he marke value of long-erm deb s (5.5) n sae G (). Hence he shor-erm deb sll shares less rsk han long-erm deb. One canno consruc such an example wh sronger shor-erm overhang under he old symmerc dsrbuon.

13 saes. In hs new example, he overhang n sae s /ε. 3 condonal overhang, he dae long-erm overhang s /4ε. Averagng ou hese levels of In sum, volaly ha s hgher n he bad sae or lower n he good sae reduces long-erm deb overhang for boh saes a dae, whch helps average ou o a lower long-erm deb overhang a dae. In conras, he change n volaly afer shor-erm deb maures has no effec on s overhang. Ths resul and assocaed nuon wll be furher llusraed n Secon 3.3 n he lack-scholes-meron framework..6 Plan of he Res of he Paper So far we have used smple numercal examples o llusrae our man resuls. The res of he paper formalzes hese resuls usng models ha are commonly used by researchers and praconers n corporae fnance. In Secon 3 we use he lack-scholes-meron model o sudy he maury effec of deb overhang on mmedae nvesmen, whch corresponds o he resuls shown n Secon.3 and Secon.5. We hen adop he Leland framework o sudy he role of deb maury on overhang when he frm wh flucuang values of asses-n-place has access o nvesmen opporunes over me. Ths analyss corresponds o he seng n Secon.4 where a frm chooses nvesmen afer he realzaon of news abou s asses-n-place. 3 Immedae Invesmen n he lack-scholes-meron Model Much of he nuon ha shorer-erm deb enhances he ncenve for nvesmen decsons, such as Myers (977), comes from he lack and Scholes (973) model and he sudy of rsky corporae deb n Meron (974) where equy s a European call opon wh a srke prce equal o he deb face o be repad on s maury dae. Ths secon analyzes he effec of maury on deb overhang n a lack-scholes-meron seng. Alhough many have dscussed he effec of maury on deb overhang based on he dscusson n Myers (977), we are unaware of any exsng formal analyss n he lack-scholes-meron seng. 3. The lack-scholes-meron Seng The frm has some exsng asses n place, wh curren marke value denoed by V. The asse value follows a log-normal dffuson, and s value a any fuure me > s exp σ V, = V + σ Z 3 In hs example here s anoher ndrec effec of a lower long-erm deb face value under he new dsrbuon. 3

14 where Z N(, ), and he volaly σ s a consan. Laer we nroduce sae dependen volales. Whou loss of generaly we se he rsk-free rae o be zero. Followng Meron (974), he frm has a zero-coupon deb ssue ha maures a me wh a face value F, and hs s he frm s only deb. A me, f he frm valuev s below F, deb holders ake he defauled frm o obanv ; oherwse, deb holders are repad n full by F. ecausev follows a marngale, a shorer maury of deb can equvalenly be vewed as a deb o be repad afer a smaller amoun of resoluon of uncerany abou he frm s asses. Fnally, recall ha we rule ou physcal bankrupcy coss (.e., he asse can be lqudaed any me for s valuev ) and renegoaon of he deb n reurn for a changed nvesmen decson. As n our numercal examples before, we consder a sngle nvesmen opporuny a dae modeled as a small scale expanson of exsng asses. 4 We examne he effec of deb maury on deb overhang by varyng, he me horzon o a sngle deb maury. To focus on maury only, our analyss conrols for he frm s nal leverage. More specfcally, we adjus he face value F o hold consan he me- marke value of deb when we vary. 3. Sronger Shor-Term Deb Overhang wh Consan Volaly We frs esablsh a benchmark resul under consan volaly: Longer-erm deb mposes sronger overhang on he me- nvesmen for a gven marke leverage. I s well-known n hs seng ha he payoff o equy holders wll be reduced by deb overhang, whch can be measured by he ncrease n he value of exsng deb as a resul of he scale expanson ofv. How does deb maury affec he amoun of overhang? m Consder shor-erm (long-erm) deb wh face value F ( F ), wh maury m ( m ), where > m. The sandard lack-scholes calculaon gves he correspondng dae- deb value as ( ) ( ) ( ) ( σ ) ( ) + σ ln V F.5 m D V F m = V N d + FN d m d = ;,, where,,. σ m 4 In he lack-scholes-meron seng wh me- nvesmen only, he frm s refnancng polcy a me-,.e., wheher he frm refnances exsng deb wh newly ssued equy or newly ssued deb, s rrelevan. Recall ha he deb n consderaon s he only deb ha he frm has, whch mples ha he frm wll refnance hs deb a maury dae whou exsng clams (oher han equy). ecause he ne presen value of he dae- nvesmen underaken wll be known on dae, and fuure nvesors break even, equy holders wll recover any gan from he nvesmen, excep hose gong o deb holders exsng a dae. 4

15 Deb overhang s measured by ( ;, ) D DV Fm V, whch capures he mpac of a change n V frm value on he value of exsng deb. We sudy he wedge beween wo deb overhangs: ( ) ( ) D D V ; F, m D V ; F, m, () V V V where face values F > F are chosen o hold consan he nal frm leverage: Proposon formally saes ha ( ) DV ( F m) DV; Fm, ;, =. DV n Eq. () s negave. Inuvely, gven he same dae- marke deb value, shorer-erm deb always gans less from any margnal ncrease of he me- asses-n-placev, resulng n beer equy holders nvesmen ncenves. Proposon. Under he lack-scholes-meron seng, we have D ( V ; F, m ) < D ( V ; F, m ) whenever DV ( ; Fm, ) DV ( ; F, m) V V =. Ths mples ha for a gven nal deb marke value, long-erm deb mposes sronger overhang han shor-erm deb. 5 oh he example n Secon.3 and he lack-scholes-meron seng n Proposon have sae-ndependen volaly,.e., uncerany resolves a he same rae n good and bad saes. The nex subsecon relaxes hs assumpon. 3.3 Sronger Shor-Term Overhang wh Sae-Dependen Volaly Recall ha n he example n Secon.5, even wh a sngle nal nvesmen, condonal volales ha ncrease n bad saes can reverse he resul ha shorer-erm deb mposes less overhang. Followng hs dea, we now show ha a sae-dependen volaly (more specfcally, hgher volaly gven low asses-n-place sae) n he lack-scholes-meron seng can lead o a sronger shor-erm deb overhang. Consder he followng smple modfcaon of he lack-scholes-meron model, where ( ) shor-erm deb (long-erm deb) wll maure a m = ( m = ). Suppose ha he frm s asses- n-place value a he end of perod s V = V exp z.5σ + z.5σ, where z and z have zero mean and follow he normal dsrbuon wh varancesσ and σ, respecvely. Thus, he. asses-n-place value on dae s V = Vexp( z.5σ ) 5 All proofs are n he appendx. 5

16 To nroduce sae-dependen volaly, we allow he volaly σ o be dependen on dae asses-n-place z. Parcularly, for some consan Q we se σ when z > Q Q L = σ H when z σ whereσ L σ H. Ths formulaon mples ha he asse volaly s hgher n low value saes (or, a negavely skewed dsrbuon). In fac, hs paern can be generaed by he exsence of volaly ha s no scaled wh he asse value. 6 I s also a naural resul when he borrower s asses are deb conracs, for example a bank, where volaly falls n good saes when deb asses become defaul free. We se he long-erm deb face value F Vexp( Q.5σ ) =, so ha he conngen volaly s lower (hgher) for regons ofv beng above (below) F. We have he followng proposon. Proposon. We adjus suffcenly small. F such ha DV ( ; F, ) DV ( ; F,) =, and suppose ha ε > s Example. If σ = σ = ε >,.e., whou conngen volaly, long-erm deb mposes L sronger overhang han shor-erm deb; H Example. If σ = ε > = σ,.e., wh conngen volaly, shor-erm deb mposes H sronger overhang han long-erm deb. L Wh wo conrasng examples, Proposon shows ha he condonal volaly could lead o sronger shor-erm deb overhang for dae- nvesmen. In example, he asse dsplays sae-conngen z volaly, a paern ha s n sharp conras o example wh consan z volaly (whch s a specal case of Proposon ). The nuon s smlar o ha whch we provde n Secon.5. For shor-erm deb ha s refnanced a dae, wheher he volaly of z s conngen or no does no affec s overhang. In conras, for long-erm deb, he volaly of z maers. To see hs, afer he bad realzaon z = Q η, he rsk of z reduces overhang (as equy holders can recover some nvesmen benefs), and hs force s presen n boh cases wh conngen and consan volales (he same volaly σ H = ε ). However, afer he good 6 For nsance, consder randomness n he fxed cos; hen a fxed absolue volaly becomes a larger percenage of volaly when asse values are decreased. 6

17 realzaon z = Q+ η, he case of conngen volaly has a lower long-erm deb overhang. I s because wh conngen volaly σ L =, he dae- frm value V says consan a = exp( +.5 ) whch s above F Vexp( Q.5σ ) V V Q η σ whou fuure defaul; bu wh consan volalyσ =, hence zero overhang L = ε he frm value may deerorae a dae, leadng o poenal overhang. These comparsons resul n a sronger shor-erm deb overhang n Proposon. Proposon llusraes how sae-dependen volaly (hgher volaly n worse saes) n he lack-scholes-meron seng could lead o sronger shorer-erm deb overhang. However, he exsence of sae-dependen volaly s no suffcen for sronger shorer-erm overhang. Wha s general and shown n he proof of Proposon s ha hs sae-dependen volaly reduces he dfference beween long-erm and shor-erm overhang. Moreover, Proposon demonsraes ha hs effec can be suffcenly srong o overurn he posve long-shor overhang wedge esablshed n Proposon. 3.4 Deb Maury, Sae-Conngen Overhang, and Condonal Volaly In hs subsecon we offer anoher nsghful way o undersand he role of condonal volaly. As suggesed by he numercal example n Secon.4, he relave severy of long- and shor-erm overhang depends on he fuure sae of frm value, and longer-erm deb mposes more (less) overhang n good (bad) saes. The average of hese fuure sae-dependen overhang severes deermnes he me- nvesmen ncenves, and Proposon shows ha n he lack-scholes-meron consan volaly seng, once conrollng he me- marke deb value, he average long-erm deb overhang always exceeds he average shor-erm overhang. Condonal volaly s a way o ws he sae-dependen overhang o poenally delver a greaer dae- average shor-erm overhang, hus reversng Proposon. In he lack-scholes- Meron seng he effecve deb maury s nversely relaed o he speed of resoluon of uncerany, whch s also asse volaly. From hs perspecve, he condonal volaly allows us o ws he effecve deb maures gven dfferen saes. In Example wh conngen volaly n Proposon, a dae- good saes, he zero dae- volaly mples no dfference beween long-erm deb (ha maures a dae-) and shor-erm deb (ha maures a dae-). Ths mnmzes he (posve) wedge: he excess of long-erm over shor-erm overhang. In conras, a dae- bad saes, shor- and long-erm debs dffer gven he posve dae- volaly, 7

18 whch preserves he negave wedge beween long- and shor-erm overhang. In sum, hgher condonal volales a lower asses-n-place saes can reduce he posve excess of long-erm overhang over shor-erm overhang n good mes whle preservng he negave dfference n bad mes. For he sngle nal nvesmen, hs ncreases dae- average overhang of shor-erm deb compared o long-erm. 4 Deb Overhang wh Dynamc Invesmen To examne he long horzon effecs of deb maury, we need a racable framework wh dynamc nvesmen opporunes ha goes beyond he lack-scholes-meron model. We have wo goals for hs dynamc analyss. Frs, we would lke o sudy deb overhang for a frm wh sochasc values of asses-n-places and wh access o fuure nvesmen opporunes. Ths wll generalze he examples n Secon.4 where nvesmen s made afer he realzaon of news abou asse values. Second, all of our prevous examples and lack-scholes-meron models have had refnancng (f any) occurrng a a me where here s no oher exsng deb ousandng. When deb s refnanced, he ncenves o refnance or defaul are nfluenced by he maury of exsng ousandng deb on ha dae, whch s anoher form of deb overhang. As we show, hese wo crucal feaures (whch are mssng from sac models) lead o neresng mplcaons abou shor-erm deb overhang. 4. The Seng and Valuaons Models wh mulple deb ssues and dynamcs n he value of asses are dffcul o analyze, and he mos racable exsng framework s based on Leland (994b, 998) whch akes as fxed parameers boh he frequency of refnancng and he oal amoun of promsed repaymens of deb. There, equy holders always have access o funds o cover he nvesmen coss or losses a refnancng. Defaul hen occurs only when her ncenve o njec more fundng s nsuffcen. Ths allows us o elmnae ssues of lmed lqudy (e.g., Damond, 99) and focus nsead on deb overhang by examnng he equy holders ncenve o njec funds no he frm. 4.. Frm asses Consder a frm ha generaes cash flows a a rae of whch evolve as follows:. We nerpre as asses-n-place d = d +σ dz. () 8

19 Here, σ s he consan volaly, and { Z : } < s he sandard rownan moon. Dfferng from sandard Leland sengs, n Eq. () he growh rae s he endogenous nvesmen decson conrolled by equy holders. For smplcy, we assume ha {, } akes a bnary value,.e., equy holders can decde o nves or no. The nvesmen cos s modeled as λ d because he nvesmen benef scales wh as well. We assume a consan neres (dscoun) rae r > n hs nfne horzon model. If equy holders always nves, hen he presen value of he frm, gven he curren value of asses-nplace, s rs ( ) λ e ( s λ s) ds =. (3) r Comparng hs value o he value / rwhou nvesmen a all, we assume ha λ r < whch ensures ha nvesmen a every nsan would maxmze he oal value of he frm. Denoe he nvesmen polcy by ( ) whch depends on curren asses-n-place. In Proposon 3 we wll show ha n equlbrum equy holders use a smple hreshold polcy,.e., nves whenever he value of asses-n-place exceeds a crcal level ( ) : =. (4) < As menoned n Secon., nvesmen can be only aken by equy holders, and fuure nvesmen opporunes are los when deb holders ake over he frm from bankrupcy. Ths leads o an endogenous cos of fnancal dsress. Unlke Leland s models, we mpose no oher exogenous coss of fnancal dsress. 4.. Saonary deb srucure The frm has one un of deb wh a consan aggregae prncpal face value of P. As n Leland (994b, 998), we ake a smple refnancng polcy whch governs he frm s maury srucure. Under hs framework wh refnancng frequency f, a each nsan a consan fracon of deb, fd, becomes due and mus be refnanced o keep he amoun of oal deb ousandng 9

20 consan. Ths solaes he effec of maury from changes n he amoun of deb. 7 Ths saonary deb srucure descrbes a frm whch smoohes ou neres and prncpal paymens o avod spkes n refnancng acvy. One mmedae applcaon of he consan refnancng rae s analyss of borrowers who for some exogenous reason have a parcular deb maury. For example, banks ssue shor-erm deposs and have a very shor deb maury. More generally, he saonary deb srucure s assumed for racably, bu s a sensble place o sar. For dealed dscusson abou hs refnancng polcy, see Secon 4.6. One can show ha he average deb maury s m f. The hgher he rollover frequency f, he shorer he deb maury. To he exreme, f f goes o nfny (so m goes o zero), hen he deb represens zero maury demandable deb ha maures mmedaely afer he ssuance. The advanage of hs seng s ha, because each bond s rered exponenally, a any pon of me he frm s exsng bonds--ncludng hose jus newly ssued--are dencal. esdes racably, we adop hs framework because he overall refnancng rae s he mos relevan varable o characerze a frm s deb maury srucure, and we rea hs refnancng rae as a parameer. For undersandng overhang, hs s a reasonable reamen because he refnancng rae s essenally he frequency of reprcng, and reprcng o reflec he benefs of new nvesmen s cenral o he equy holders ncenves o nves. Thus, hs framework preserves he key dfference beween long- and shor-erm deb n regard o overhang due o wealh ransfer o deb holders. Ineresngly, n addon o he usual posve force of reprcng o reduce overhang on nvesmen, we wll see anoher offseng effec where shorer-erm deb leads equy o defaul earler n bad mes, whch exacerbaes overhang. The laer effec s closely relaed o rollng over deb, o whch we urn nex Rollng over deb The marke value of he frm s deb s denoed by D( ). In refnancng, he frm ssues ( ) ( m) d uns of new bonds o receve oal proceeds of D ( ) m d, payng ( ) P m d o rere maurng bonds. The marke prce of newly ssued bonds flucuaes wh asses-n-place leadng o ne paymens o bond holders whch we refer o as rollover gans/losses of, 7 Deb reremen n hs fashon s smlar o a snkng fund ha connuously buys back deb a par wh a consan rae of repaymen.

21 D m ( ) P d. 8 Equy holders are he resdual clamans of he rollover gans or losses: any gan wll be mmedaely pad ou o equy holders and any loss wll be pad off by ssung more equy a s marke prce. Thus, he ne cash flow o equy holders s d λ d + D ( ) P d. m The frs erm s he frm s cash flows, he second erm s he nvesmen cos, and he hrd erm s he rollover loss. As emphaszed by He and ong (), when he asses-n-place deerorae n value, equy holders absorb he rollover loss by ssung addonal equy o preven bankrupcy, and hs loss s amplfed by he rollover frequency f = m. Equy holders are wllng o njec cash o repay he maurng deb holders as long as he opon value of keepng he frm alve (and hence choosng o defaul laer) jusfes he expeced rollover losses. Ths leads o defaul when he equy value drops o zero, whch occurs when he frm s assesn-place drops o an endogenously deermned hreshold Valuaons and opmal polces The deb value sasfes he followng equaon: σ rd( ) = ( ) D ( ) + D ( ) + ( P D( ) ). (5) m The lef-hand sde s he requred reurn for he deb, whch equals he expeced ncremen n he deb value on he rgh-hand sde. The frs wo erms capure he flucuaon n n Eq (). The hrd erm s he change of he deb value due o reremen: a ( m) d fracon of deb maures, wh he valuaon change beng he prncpal paymen P mnus he bond value before rerng. We need boundary condons o solve Eq.(5). Frms wh exremely profable asses-nplace = never defaul and he defaul-free deb value s p P ( mr) +. From now on we rea he defaul-free deb value p as he prmve parameer (nsead of he saed prncpal value P ). On he oher hand, equy defauls when =, and deb holders receve he frm wh a 8 ecause of zero-coupon deb, dscounng mples he frm always ncurs rollover losses. Rollover gans could occur f we nsead assumed a bond ssued a par by seng a coupon rae hgher han r. Wheher rollover gans are possble or no s no essenal o our analyss. As shown n He and ong (), he key s ha ncreased rollover losses for lower values of he asses-n-place ncrease equy holders ncenve o defaul.

22 value of ( ) D = r whou fuure nvesmen (here s no exogenous bankrupcy cos). One can formally show ha p> r,.e., on he dae of defaul here s a loss o deb holders. 9 Equy holders value E( ) sasfes he followng equaon: re( ) = max + E ( ) + σ E ( ) λ ( P D( ) ) (6) {, } m We have omed he opmal defaul polcy here; equy holders defaul a some endogenous level and receve zero. The opmzaon n Eq. (6) wh respec o leads o an nvesmen polcy n Eq. (4). The nex proposon verfes he opmaly of he hreshold nvesmen sraegy, and gves equy and deb values as soluons o Eq. (5) and Eq. (6), respecvely. Proposon 3. There exss a unque gven by Eq. (4). Gven wh ( ) and he equy value s E = λ so ha he opmal nvesmen polcy s and ( ) E ( λ) γ 3 γ p+ A = r γ4 δ4 γ δ p+ + 3 A A 3 < < r, ( ) D γ = p+ A f γ δ p + A + A 3 f < < where consans γ, γ, γ3, γ4, δ, δ4, A, A, A3,,, and 3are gven n he Appendx. The expresson for equy value s nuve. When he frm nvess, he equy value s he frm value ha would preval f he frm always nvesed, ( λ) ( r ) n Eq. (3), mnus he defaul-free deb value p, wh he adjusmen for poenal fuure defaul and soppng nvesmen (a leas emporarly). Ousde he nvesmen regon < <, he equy value s he frm value whou nvesmen rmnus he defaul-free deb value p, akng no accoun boh poenal fuure defaul and comng back o he nvesmen regon. 9 Ths can be seen by a sandard real opon argumen. Suppose ha rp ; hen D( ) rskless. Wh he opon o defaul, equy holders mus ncur srcly negave cash flows a equy holders can se =, a he cash flow for equy s a leas ( P p) / m rp = pand he deb s. From Eq. (6), snce = >, conradcon.

23 Fnally, he endogenous nvesmen hreshold defaul boundary sasfes ( ) sasfes he smooh-pasng condon ( ) E = λ, and he endogenous E =. The dealed equaons and seps n solvng for hese wo endogenous varables are gven n he Appendx. 4. Opmal Deb Maury Recall ha he example n Secon.4 llusraed he followng dea. For fuure nvesmen opporunes, shor-erm deb hurs he frm s ncenves o nves especally n bad mes. However, long-erm deb mposes a less sae-conngen overhang, and reduces he frm s.9.9 Opmal Defaul and Invesmen Polces.95 Frm Value Defaul oundary.85 Opmal maury m*=4.8.8 Invesmen oundary Maury m=/f Maury m=/f Fgure : Opmal defaul polcy (sold lne) and nvesmen polcy (dashed lne) and frm values for dfferen deb maures. The parameers are r=%, σ=5%, =7%, λ=9, =, and D =. nvesmen ncenves especally n good mes (relave o shor-erm deb). As we llusrae now, hs rade-off generally leads o an neror opmal maury choce. We choose r = %, σ = 5%, = 7%, and λ = 9. We normalze he dae- asses-n-place o =, and se he arge dae- deb value D =. The lef panel of Fgure graphs he opmal nvesmen and defaul polces, and he rgh panel graphs he dae- frm value; boh are ploed agans deb maury m. As before, o fx dae- deb value D, when varyng maury m we search for he defaul-free deb value p so ha D = always. In hs model, because shorer-erm deb requres equy holders o absorb greaer rollover losses (ncurrng hgher fnancng coss) once he frm s asses-n-place deerorae, equy holders defaul earler as hey refuse o subsdze deb holders, a sympom of deb overhang. Ths Though no repored, n our example p s ncreasng n deb maury o compensae for he greaer defaul rsk assocaed wh longer-erm deb. 3

24 resul can also be seen by observng ha, because shorer-erm deb does no share as much rsk, leads o more volale equy value and hence equy holders defaul opon falls no he money more ofen. Graphcally, n he lef panel of Fgure we observe ha defaul boundary rses (hence earler defaul) for shorer deb maury m. As defaul desroys fuure nvesmen opporunes, earler defaul caused by shor-erm overhang hurs he frm value. Now focus on nvesmen polcy. In he lef panel of Fgure, we fnd ha he nvesmen hreshold frs decreases wh deb maury for m ha s below abou, hen ncreases wh deb maury aferwards. We wll devoe Secon 4.4 o dscuss he range of very shor maury deb (.e., m below abou ) where shorer maury reduces nvesmen ncenves. In hs secon, we only focus on he ncreasng regon where shorer deb maury mproves nvesmen ncenves. For deb maury m beng above abou, equy holders are more relucan o nves (a hgher hreshold ) facng deb wh longer maury (a greaer m ). Relave o shor-erm deb, alhough he long-erm deb holders--due o less frequen reprcng--share more losses wh equy holders when he asses-n-place deerorae, hey also share more gans gven good news. Consequenly, as more nvesmen benef goes o deb holders (ncreased overhang), equy holders wll se a hgher nvesmen hreshold wh longer-erm deb. The combnaon of hese wo forces (one on defaul polcy, and he oher on nvesmen * polcy) leads o an neror opmal maury choce ( m = 4 ) ha maxmzes he nal frm value, as shown n he rgh panel of Fgure. To furher llusrae he mechansm, Fgure 3 plos he margnal mpac of he frm s asses-n-place on equy value,.e., E ( ) for m = 5 (hn sold lne), m = 4 (hck sold lne) and m = 3 (hn dashed lne). We can drecly compare he equy holders nvesmen ncenve E ( ) shorer deb maury 5 λ, bu also defaul early when E ( ) ncenve E ( ) o he nvesmen cos λ (fla doed lne). Frms wh m = have he seepes E ( ) curve: hey nves early once E ( ) crosses hs zero. As shown, he flaer equy holders nvesmen under a longer deb maury m = 3 gves he oppose effec: he frm nvess lae and defauls lae. The curve m = 4 balances hese wo forces and delvers he hghes frm value. 4

25 E'(): Impac of Asses-n-place on Equy Value m=5, seepes opmal maury m=4 m=3, flaes λ=9, nvesmen hreshold Fgure 3: The mpac of asses-n-place on equy value Underlyng he paern of he shorer he deb maury, he seeper he equy holders nvesmen ncenves s he nsensvy of shor-erm deb value wh respec o he frm value, due o more frequen reprcng. Ths ranslaes o a greaer volaly of deb overhang of shorerm deb, a resul conssen wh he example n Secon.4. As shown n Fgure 3, equy holders wh 5 m = have lower E ( ) earler, bu also have hgher E ( ) Asses-n-place for low values of asses-n-place and herefore defaul for hgh values of asses-n-place whch fosers effcen nvesmen. Hence, alhough no sharng gans makes shor-erm deb beer a preservng equy s ncenve o nves n good mes, no sharng losses n bad mes pushes equy holders o defaul, elmnang fuure nvesmen opporunes. In conras, equy holders wh m = 3 have worse nvesmen ncenves n good mes, bu hey are also wllng o hold on longer o rean fuure nvesmen opporunes, snce long-erm deb shares more losses n bad mes. efore we move on o he nex subsecon, recall ha n he lef panel of Fgure, for very shor maury (below abou years), boh he nvesmen and defaul hresholds decrease wh deb maury. In hs range, an even shorer-erm deb presens a double evl--frms are no only more lkely o defaul bu also less lkely o nves. Ths resul s relaed o he assumpon of ner-emporally lnked nvesmen, a opc ha we dscuss n Secon 4.4. for deb maures m=5 (hn sold lne), m=4 (hck sold lne), and m=3 (hn dashed lne). The nvesmen cos s also ploed (fla doed lne) so ha he nvesmen hreshold sasfes. The parameers are r=%, σ=5%, =7%, λ=9, =, and D =. 5