Portfolio Liquidation and Security Design with Private Information

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1 Porfolo Lqudaon and Secury Desgn wh Prvae Informaon Peer M. DeMarzo (Sanford Unversy) Davd M. Frankel (Melbourne Busness School and Iowa Sae Unversy) Yu Jn (Shangha Unversy of Fnance and Economcs) Ths Revson: November 5, 2017 ABSTRACT. An ssuer seeks o lqudae all or par of a porfolo of heerogeneous asses abou whch she has prvae nformaon. In conras wh he sngle-asse case, greaer opmsm may lead her o sell more of a gven asse. Bu f asses can be ranked by her nformaonal sensvy hen he ssuer frs lqudaes her leas sensve asses, whch, under weak assumpons, are her more senor asses (followng Myers s peckng order hypohess). When he ssuer can desgn new secures afer learnng her nformaon hen she has wo equvalen, opmal sraeges. She may desgn and sell, ex pos, a sngle sandard deb secury whose face value s decreasng n her nformaon. Or she may pool and ranche her asses ex ane no many prorzed deb secures and sell, ex pos, hose ranches whose senory exceeds a hreshold ha s ncreasng n her nformaon. In each case, he ssuer reans more of her cash flow when nformaon asymmeres are greaer, as has been found n he case of no-documenaon loans. DeMarzo: Sanford, CA ; pdemarzo@sanford.edu. Frankel: 200 Leceser S., Carlon, VIC 3053, Ausrala, d.frankel@mbs.edu. Jn: Shangha , Chna; jnyu@shufe.edu.cn. We are graeful o Mchael Fshman for useful commens, as well as semnar parcpans a U. Cncnna, Fudan Unversy, he Haas School of Busness, FRB-Mnneapols, and Unversy of Wesern Onaro, as well as o Ashwn Alankar for research asssance. Ths paper subsumes he resuls of a 2003 workng paper of he same name (DeMarzo 2003).

2 1. Inroducon Ths paper sudes how an ssuer should sell asses abou whch she has prvae nformaon. Examples nclude a frm ha rases capal by sellng deb, equy, and/or hybrd secures; an nvesor seekng o lqudae a porfolo; and a bank ha sells clams o he repaymens of loans ha has ssued. One can magne varous movaons for such a sale. A frm, for nsance, may need cash o nves n a worhwhle projec. A bank may seek funds n order o ssue more loans or o comply wh regulaons. We capure hese vared reasons by assumng, smply, ha he ssuer s less paen han nvesors. As hs creaes gans from rade, he ssuer would sell her enre porfolo f nformaon were symmerc. In he more realsc asymmerc nformaon case, such an offer mgh be nerpreed by nvesors as a negave sgnal. The ssuer may hus choose o sell less, n order o sgnal opmsm. In mos pror work on hs problem, he ssuer sells a sngle asse o nvesors. Examples nclude Leland and Pyle (1979), Myers and Majluf (1984), DeMarzo and Duffe (1999). In he equlbra of hese models, he ssuer sgnals opmsm by reanng a larger poron of her sngle asse. DeMarzo (2005) sudes a mulple-asse model, n whch he quany sold of each asse can be used o sgnal an ndependen dmenson of he ssuer s nformaon. He shows ha n ha seng, he porfolo lqudaon decson can be solved asse by asse as n he sngle-asse case. 1 We depar from he above papers n assumng ha he ssuer holds many asses, bu learns nformaon abou a one-dmensonal common facor such ha good news for one asse s good news for all. Ths condon holds, for example, f each asse s payou s a nondecreasng funcon of a common underlyng cash flow, and beer nformaon rases he cash flow dsrbuon n a frs-order sochasc domnance sense. Oher examples nclude a holder of a bond (or CDS) porfolo wh nformaon regardng volaly, or an ssuer of morgage-backed secures wh nformaon abou defaul or pre-paymen rsk. Whle he model s sylzed, yelds srong predcons abou opmal asse sales, lqudy, 1 He (2009) consders a wo-asse verson of Leland and Pyle (1979). The ssuer has dsnc nformaon abou each asse, bu because he ssuer s rsk-averse he cos of reanng one asse depends on reenon of he oher. The CARA-Normal framework precludes analyss of secury desgn, however, whch s a prmary focus of our paper. 1

3 and secury desgn. These predcons are conssen wh observed pracce and are confrmed n he daa. In our base model, we characerze equlbra when he ssuer has a fxed se of asses o sell. 2 We show by example ha a more opmsc ssuer may sell more of an asse a a hgher prce. 3 Ths conrass wh he sngle-asse case. Inuvely, he ssuer mos effcenly sgnals a gven ncrease n her nformaon by reanng hose asses whose expeced payous (or reurns) rse proporonally more as a resul of her roser expecaons. Bu he deny of hese more nformaonally sensve asses may alernae as he ssuer becomes progressvely more opmsc. Hence she may rean some shares of one asse o sgnal a gven level of opmsm, only o sell hs asse n s enrey and rean shares of a dfferen asse n order o sgnal ye greaer opmsm. In he remander of he paper, we focus on sengs n whch he ssuer s asses can be ordered ex ane n erms of her sensvy o he ssuer s nformaon. 4 In hs seng, as he ssuer s opmsm grows, she frs reans her mos nformaonally sensve asse, hen her second mos, and so on. Thus, an ssuer wll no sell any poron of a gven asse unless she also sells her enre holdngs of her less nformaonally-sensve asses. We nex apply hs resul o fnancal asses such as he deb, equy, and mezzanne ranches used n securzaon deals. We assume, frs, ha he payou of each of he ssuer s asses s a nondecreasng funcon of a common cash flow. 5 For nsance, he ssuer may be a frm ha sells deb and equy ha are secured by fuure operang profs, or a bank ha wshes o sell ranches of a common loan pool. We assume, moreover, ha an ncrease n he ssuer s nformaon lowers he hazard rae funcon of he underlyng cash 2 In order o oban a unque equlbrum, we assume he Inuve Creron of Cho and Kreps (1987): belefs followng an unexpeced acon mus be concenraed only on ypes ha could possbly hope o gan from he devaon. By allowng he ssuer o underprce asses and raon her allocaon, we elmnae mplausble equlbra, even hough no underprcng occurs n equlbrum. 3 Whle such behavor seems rare, s nuon plays a key role n our subsequen, more realsc resuls. 4 More precsely, we assume ha f one asse s expeced value rses proporonally more han anoher s for a gven ncrease n he ssuer s nformaon, hen does so for any such ncrease. 5 A common underlyng cash flow s no needed for our precedng resuls, whch assume only ha he expeced payou of each asse s nondecreasng n he ssuer s nformaon. For nsance, each asse may be secured by a dsnc cash flow, whose dsrbuon s ncreasng n he ssuer s nformaon. 2

4 flow dsrbuon. Ths Hazard Rae Orderng (HRO) propery s sronger han frs-order sochasc domnance, bu weaker han he usual monoone lkelhood rao propery. 6 In hs seng, we show ha he ssuer s senor asses are less nformaonally sensve, and consequenly more lqud, han her junor ones. Hence as her opmsm grows, she wll rean her junor asses frs. For nsance, he ssuer wll sell senor deb before sellng junor deb, and junor deb before equy, as predced by Myers s (1984) Peckng Order Hypohess. 7 The resul uses a novel and weak noon of senory: one asse s senor o anoher f s payou grows a a slower rae han he oher s n he underlyng cash flow. Whle hs ranks he varous classes of deb by her prory n bankrupcy, also exends o oher common clams. For nsance, equy s senor o a call opon ha s wren on. The precedng resuls assume ha he ssuer s endowed wh a fxed se of asses ha she wshes o sell. We nex perm secury desgn: we le he ssuer consruc new secures ha are secured by her nal asses. In pror work on secury desgn, DeMarzo and Duffe (1999) suded he ex-ane, sngle-secury case: he ssuer desgns a sngle monoone secury, learns her nformaon, and chooses a proporon of her secury o sell. We sudy he general case. We assume he ssuer has a gven se of nal asses ha are secured by a common underlyng cash flow. She frs desgns any number of nerm secures ha are secured by her nal asses. She hen learns her nformaon and desgns any number of ex-pos secures ha are secured by her nerm secures. Ths general seng ncludes DeMarzo and Duffe (1999) as a specal case. In hs seng, he ssuer has wo equvalen, opmal sraeges She can pool her nal asses, see her nformaon, and ssue a sngle sandard deb secury whose face value s decreasng n her nformaon. 6 See Shaked and Shanhkumar (2007), p. 43, Thm. 1.C.1. 7 Whle Myers s hypohess has mxed emprcal suppor n he corporae fnance leraure (e.g. Fama and French (2002), Flannery and Rangan (2006), Opler e al (1999), and Shyam-Sunder and Myers (1999)), our conex s more sued o he sale of asse-backed secures, where reenon raes declne and lqudy mproves wh senory (see e.g. Begley and Purnanandam (2017) and Franke and Krahnen (2007)). 8 They are equvalen n he sense ha hey yeld he same ype-conngen revenue for he ssuer and promse he same aggregae fnal payou o nvesors for any cash flow realzaon. 3

5 2. She can pool her nal asses, ranche he resul no a maxmal se of prorzed deb secures, see her nformaon, and sell hose ranches whose senory exceeds a hreshold ha s ncreasng n her nformaon. 9 Ths opmal sraegy mrrors he usual srucure of loan-pool securzaon deals. 10 In he frs sraegy, he ssuer wres a sngle sandard deb secury afer seeng her nformaon. DeMarzo and Duffe (1999) assume he ssuer mus desgn her secury before seeng her nformaon. Whle hey also fnd ha deb s opmal, here s a key dfference. Wh ex-ane deb, he ssuer sgnals opmsm by sellng fewer shares, hus lowerng he payou o nvesors by a consan proporon n all saes. Wh ex-pos deb, he ssuer nsead reduces he face value, whch lowers nvesors payou n nondefaul saes whle leavng unchanged n defaul saes. Thus ex-pos deb (or, equvalenly, ex-ane ranchng) provdes a more effcen way o sgnal opmsm, as allows he ssuer o rean more of he cash flow n hose saes whose probables have rsen he mos as he resul of her greaer opmsm. 11 The above secury desgn resuls assume ha he ssuer s ype and cash flow are dscreely dsrbued. For wder applcably, we also sudy he lm as he ssuer s nformaon and her realzed cash flows become connuous. 12 In hs lm, he face value of he opmal ex-pos deb secury s gven by a smple dfferenal equaon ha has a unque soluon. Moreover, hs equaon descrbes an equlbrum of he connuous model, and he ssuer's expeced profs n he dscree model converge unformly o her expeced profs n hs equlbrum. 13 Hence, he connuous model s a good approxmaon o he dscree model. 9 In boh sraeges, he ssuer pools her nal asses. Inuvely, any funcon of hese asses payous can be rewren as a funcon solely of he underlyng cash flow, so poolng hem does no lm he ssuer s subsequen acons n any way. 10 For emprcal examples, see Begley and Purnanandam (2017) and Franke and Krahnen (2007). We are aware of one alernave explanaon for hs srucure: Dang, Goron, and Holmsröm (2015) show ha an unnformed ssuer may rean an equy ranche o dscourage nformaon gaherng by nvesors. 11 Nachman and Noe (1994) also show ha deb s he opmal ex pos secury desgn when an ssuer mus rase a fxed amoun of cash. The all-or-nohng naure of he fnancng problem leads o a poolng equlbrum n whch each ype of ssuer sells sandard deb wh he same face value. A poolng equlbrum wh deb also emerges when an nformed ssuer sells o a sngle large nvesor (Bas and Maro 2005); here, he ssuer pools n order o sheld her nformaonal rens from he nvesor. 12 Applcaons of hese connuum resuls nclude secon 4.2 of DeMarzo (2005) and secon 7.2 of Frankel and Jn (2015). DeMarzo (2005) ces an earler verson of hs paper, DeMarzo (2003), for hese resuls. 13 Manell (1996, 1997) sudes a general sequence of fne sgnalng games (whch have fne ype and message spaces) ha converges o a connuous sgnalng game ha, lke ours, has compac ype and message 4

6 The connuous soluon can be descrbed heurscally as follows. We normalze he nvesors dscoun facor o one and le 0,1 denoe he ssuer s dscoun facor. The ssuer s opmsm s ndexed by her ype 0. On seeng, he ssuer sells a sngle ex pos deb secury, secured by her cash flow, whose face value D s deermned as follows. The mos pessmsc ssuer (whose ype s zero) sells everyhng: he face value D 0 of her deb secury equals he hghes possble value of her underlyng cash flow. As her opmsm grows, she lowers he face value a he rae dd d v2 D, 11 D, 0, (1) where vd (, ) s he condonal expeced payou of a sandard deb secury wh face value D, ( D, ) s he condonal defaul probably of hs secury, and subscrps denoe paral dervaves. 14 As he equlbrum s separang, compeon among nvesors drves he secury s prce o s condonal expeced payou vd (, ). In equaon (1), he rae a whch a more opmsc ssuer lowers her face value s equal o he benef-cos rao: he nformaonal benef she receves from he decrease, dvded by he socal cos of reanng more of he cash flow. 15 Ths equaon serves as a buldng block n he appled models of DeMarzo (2005) and Frankel and Jn (2015). Is opmaly was frs derved heurscally n an earler verson of hs paper, DeMarzo (2003), n a seng n whch he ssuer desgns a sngle secury afer seeng her nformaon. In he spaces. Manell (1996) shows ha any sequence of equlbra of he fne games has a convergen subsequence ha converges o some equlbrum of he connuous game. Smlarly, Manell (1997) shows ha f a sequence of equlbra of he fne games, each of whch sasfes he Never a Weak Bes Response creron of Kohlberg and Merens (1986), converges o some equlbrum of he connuous game, hen hs lmng equlbrum sasfes he same creron. However, because Manell s fne games have fne message space whle ours has an nfne message space - he se of monoone secures - hs resuls canno be drecly appled o our seng. 14 More precsely, vd, Emn D, Y and D, Pr Y D where Y s he ssuer s underlyng cash flow, and v2 D, vd,. 15 In parcular, he numeraor capures he ssuer s benef of convncng nvesors ha she s more opmsc, whle he denomnaor equals los socal surplus from a un decrease n he face value: he expeced declne 1 n he payou o nvesors, mulpled by he ssuer s holdng cos per un reducon n he payou, 1. When he nformaonal benef of lowerng he face value s large relave o he cos, maon by lower ypes s more profable and hus, o be deerred, requres a larger declne n he face value. 5

7 curren verson, s shown o be opmal even f he ssuer can desgn any number of secures, boh before and afer she becomes nformed. 16 In equlbrum, a hgher face value sgnals ha he ssuer s pessmsc and hus leads nvesors o lower her valuaon of her cash flow. For hs reason, f an nvesor of a gven ype chooses a hgher face value, her securzaon revenue rses by less han he rue ncrease n value of her secury. Ths llqudy nduces hgh-value ssuers o rean more equy. Begley and Purnanandam (2017) confrm hs resul emprcally by showng ha when he equy (reaned) ranche of a pool of resdenal morgage-backed secures (RMBS s) makes up a larger proporon of he pool s face value, he loans n he pool have lower subsequen delnquency raes condonal on observables and he secures ha are sold fech hgher prces condonal on her cred rangs. Ths s mpled by (1): he ssuer s face value D s lower, and so her equy ranche s larger, when her ype s hgher whch, n he case of a loan pool, s naurally nerpreed as a lower expeced defaul rae. Begley and Purnanandam (2017) also fnd ha ssuers rean larger proporons of he face value of RMBS pools ha conan a hgher proporon of no-documenaon ( no-doc ) loans, conrollng for oher loan observables. Equaon (1) mples hs as well f we assume ha no-doc loans are more fragle: ha boh he defaul rsk D, sensvy v 2, and he nformaonal D of he ssuer s secury are greaer for no-doc loans. On hs assumpon, equaon (1) mples a more rapd declne n he face value D n he case of no-doc loans and hus a larger equy ranche for any Inuvely, as he value of her secury s more sensve o her nformaon, a no-doc ssuer gans more from convncng nvesors of her greaer opmsm. In order o deer maon by more pessmsc ypes, she mus herefore send a cosler sgnal as her opmsm grows. She does so by ncreasng he sze of her equy ranche more rapdly. 16 Tha s, an ssuer n hs more general seng sll prefers o ssue a sngle ex-pos deb secury wh face value gven by (1). 17 Ths reles on he fac ha he lowes-ype ssuer ( 0 ) always chooses he mnmum equy ranche, regardless of he proporon of no-doc loans n he pool. Snce her equy ranche rses faser n n he case of no-doc loans, s larger for any 0. 6

8 The res of he paper s organzed as follows. The base model wh a fxed se of asses s suded n secon 2. The general secury desgn game s analyzed n secon 3. We relax a monooncy assumpon n secon 4 and conclude n secon The Asse Sale Game Secon 2.1 ses ou a base model (he Asse Sale game ) n whch a prvaely nformed ssuer s endowed wh a fxed porfolo of asses. Sellng all her asses s effcen as he ssuer s relavely mpaen; for nsance, she may have aracve alernave nvesmens or face lqudy or capal requremens. However, because here s asymmerc nformaon, she may rean some of her asses n order o sgnal ha she s opmsc and hus o oban hgher prces. Sgnalng games ofen have mulple equlbra even wh one-dmensonal sgnals. The problem s poenally worse n our seng as he sgnal s muldmensonal: he quany of each asse o offer for sale. Neverheless, we show n secon 2.2 ha he game has a unque equlbrum ha sasfes he Inuve Creron. 18 In he Asse Sale game, he mos pessmsc ssuer sells her enre porfolo; as her nformaon rses, she reans hose asses ha she przes he mos relave o lower ypes, as hs s he mos effcen way o dsngush herself from hese ypes. Snce hese mosprzed asses may alernae, an ssuer of nermedae opmsm may rean an asse ha a more opmsc ssuer sells n s enrey. Ths nonmonoonc behavor, whch canno occur n he sngle-asse case, appears n wo compued examples n secon 2.3. In secon 2.4, we add a mld assumpon ha rules ou such nonmonooncy: ha he ssuer s asses can be ranked globally n erms of her nformaonal sensvy. Tha s, f one asse s expeced value rses proporonally more han anoher s for a gven ncrease n he ssuer s ype, hen does so for any such ncrease. We show ha as he ssuer s nformaon rses, she frs reans her mos nformaonally sensve asse, hen her second mos, and so on. Thus, an ssuer wll no sell any poron of an asse unless she also sells all of her less nformaonally sensve asses n her enrey. 18 The unqueness argumen wll rely on an assumpon ha he ssuer can se a prce cap for each asse. 7

9 Secon 2.5 hen relaes hs predcon o commonly observed fnancal asses such as deb, equy, and he prorzed ranches of securzed loan pools. We show ha under a mld dsrbuonal assumpon he hazard rae orderng propery senor asses are less nformaonally sensve han junor ones. Thus, by he precedng resul, she wll sell her senor asses frs. Imporanly, hs resul reles on a novel and weak noon of senory: one asse s senor o anoher f s payou rses more slowly n he underlyng cash flow.e., f s relave clam o he cash flow s sronger when he cash flow s low. Ths weak noon of senory les us exend Myers s (1984) peckng-order hypohess o a much larger class of asses. For nsance, we fnd ha a sock s senor o a call opon wren on he sock and hus wll be sold before he opon n order o rase cash The Base Model The parcpans are a sngle ssuer and a connuum of nvesors. All parcpans are rskneural and fully raonal. The ssuer s endowed wh a porfolo of n asses represened by he vecor a, where a 0 s he number of shares she owns of asse. A holder of one share of asse 1,, n n s enled o he random fuure payou F. The ssuer has prvae nformaon abou hese payous, whch s summarzed by her ype 0,, T. Condonal on he ssuer s ype, asse has an expeced payou per share of d f EF. 19 Le n n le,, n F F1 F denoe he vecor of random asse payous and f E F denoe he vecor of expeced payous condonal on. We refer o ( a, f ) as he ssuer s endowmen. We assume hroughou ha (a) he ssuer s nformaon can be ordered so ha hgher ypes are more opmsc abou he expeced payou of each asse; (b) an asse s expeced condonal payou s never negave (e,g., because of lmed lably); and (c) even he mos pessmsc ssuer hnks ha her porfolo has a posve expeced value: More precsely, he payou of asse s a funcon F of an exogenous, unknown random sae. Then f s smply he expecaon of F wh respec o he condonal dsrbuon of gven. 20 In he case of 2 ypes and 2 asses, propery (a) s no needed; see secon 4. 8

10 ASSUMPTION A (MONOTONE EXPECTED PAYOUTS). For s, f fs. In addon, f0 0 and af0 0. Ths nformaon srucure s naural when he ssuer s nformed abou a common facor ha affecs all of her asses. For nsance, he ssuer mgh own a porfolo of secures backed by a common pool of asses, such as he deb and equy of a sngle frm or he prorzed deb ranches of a loan pool, and have prvae nformaon abou he fuure value of he underlyng pool. 21 On seeng her nformaon, he ssuer chooses a quany q 0, a for sale. Le 0 q a of each asse o offer,, n q q1 qn denoe he row vecor of chosen quanes, where. 22 The ssuer may also se a prce cap p 0, for each asse. 23 If he marke clearng prce for asse exceeds p, she charges p and raons he asse. 24,, n p p1 pn denoe he vecor of prce caps. Le The ssuer s less paen han he nvesors: she dscouns fuure cash flows a some rae 0,1, whle nvesors dscoun facor s normalzed o one. 25 For nsance, he ssuer 21 The case of asse-backed secures s furher developed n secon 2.5. Oher naural examples are a loan porfolo where he ssuer has prvae nformaon abou marke volaly or prepaymen rsk, and a porfolo of eques whn an ndusry when he ssuer s prvaely nformed abou fuure ndusry profs. 22 For vecors x and y, he noaon x y means ha for all, x y. 23 There s a superfcal smlary beween he prce caps n our model and he prce-quany menu chosen n Bas and Maro (2005). In he laer paper, a sngle nvesor chooses a prce-quany par from a menu ha he ssuer specfes. Hence he ssuer chooses one of he prces ha he nvesor pays. In our model, n conras, he prce caps never bnd. Raher, her role s merely o rule ou mplausble equlbra. Inuvely, a ype ssuer who devaes could choose prce caps low enough ha any lower ype mus lose from he devaon. By he Inuve Creron, nvesors mus hen hnk her ype s or more so hey canno bd less han her valuaon of he asses. Ths can make he devaon profable, hus rulng ou he equlbrum. 24 We could also allow he ssuer o se reserve or mnmum prces for he secures. However, snce nvesors would refuse o buy overprced secures, exendng he sraegy space n hs way would play no role n equlbrum. (In oher aucon envronmens, a reserve prce s useful o exrac addonal surplus from buyers. Our model dffers n ha nvesors are homogeneous and unnformed. Hence hey earn no surplus even absen a reserve prce.) ˆ 0,1 and 25 More precsely, le he unnormalzed dscoun facors of he nvesors and he ssuer be 0, ˆ, respecvely, and le ˆf be he undscouned expeced payoffs of he asses. Then our model corresponds o f ˆ fˆ and ˆ ; ha s, we nerpre f as he condonal expeced presen value of he asse usng he nvesors dscoun rae, and as he ssuer s relave dscoun facor. 9

11 may face capal requremens or need cash o nves n worhwhle projecs. Ths dfference n dscoun facors s he source of gans from rade n he model. We assume he ssuer has no oher collaeral whch can be used o rase funds apar from her porfolo ( a, f ). We also assume, for now, ha she s resrced o sellng only her exsng secures n Secon 3 we wll relax hs assumpon and allow her o desgn new secures (such as secured deb) usng her exsng secures as collaeral. The nvesors have a common, posve pror over he dfferen possble realzaons of he ssuer s ype. On seeng he ssuer s sale decson qp,, hey form poseror belefs ( q, p). Invesors are rsk-neural, behave compevely, and have deep pockes, so her demand for asse s perfecly elasc a he prce f () ( q, p). Gven he vecor p of prce caps, he realzed prces of he asses are hus gven by he vecor p qp, p f ( ) ( qp, ), (2) where x y denoes he componenwse mnmum of vecors x and y. Suppose an ssuer of ype sells he quanes q of her asses a prces,, n p p. 1 pn She receves revenue qp from he asse sale, plus he dscouned expeced payou a q f of her reaned asses. 26 Her oal payoff he sum of hese erms can be wren as U, q, p af q p f. Our equlbrum concep s as follows. ASSET SALE EQUILIBRIUM. A perfec Bayesan equlbrum for he Asse Sale game s an ssuance sraegy q (), p () for he ssuer, a prce response funcon pqp (, ), and a poseror belef funcon ( q, p) for he nvesors, such ha he followng condons hold The concaenaon of wo vecors s always o be nerpreed as a do produc; e.g., pq p q. 27 The argumens of hese funcons are ncluded for clary. For nsance, q () should be nerpreed as a n funcon q: 0,, T ha specfes he quany vecor chosen by each ype of ssuer. 10

12 1. Payoff Maxmzaon: for any ype, he ssuer s sale decson ( q ( ), p ( )) solves max U ', ' (, q q p, p ( q, p )) subjec o 0 q a. 2. Compeve Prcng: for any sale decson ( qp,, ) he prce vecor pqp (, ) sasfes equaon (2). 3. Raonal Updang: nvesors poseror belefs ( q, p) are gven by Bayes rule f qp, s chosen by some ype of ssuer n equlbrum. Subsung he equlbrum prce funcon p p q, p U, q, p and omng he fxed erm funcon u q p f d for he prce vecor p n af, we oban he ssuer s equlbrum payoff, whch we refer o as he oucome of he equlbrum.28 I equals he addonal surplus ha a ype- ssuer receves from sellng asses o nvesors Unqueness and Compuaon Lke mos sgnalng games, he above Asse Sale game has mulple equlbra. In order o oban a unque predcon, we mus use a sgnalng game refnemen. The weakes such refnemen s he Inuve Creron of Cho and Kreps (1987). In our conex, hs creron saes ha f nvesors see an ou-of-equlbrum sale decson ( qp, ), her belefs pu wegh only on hose ypes who could possbly expec o gan from he devaon. More precsely, say a ype- ssuer devaes o ( qp., ) A suffcen condon for her o lose from hs devaon s ha her equlbrum payoff u() exceeds her maxmum payoff qp f from he devaon or, equvalenly, ha29 qp u qf (3). 28 The oucome capures all payoff-relevan feaures of he equlbrum snce compeon drves nvesors payoffs o zero. However, wo equlbra may nvolve dfferen acons bu he same oucome; e.g., f he ssuer s endowed wh wo dencal asses 1 and 2, hen he oucome depends only on he sum q1 q2 of quanes sold for each ype and no on he ndvdual quanes q 1 and q If ype ssues q for he prces p n equlbrum, we can rewre (3) as qp q p ( q q ) f, where he rgh hand sde s he opporuny cos of devang o ( qp:, ) he sum of ype s orgnal revenue and her 11

13 The Inuve Creron saes ha on seeng he devaon ( q, p ), nvesors mus pu zero probably on ype f hs ype s no wllng o choose ( q, p ) bu some oher ype mgh be: f condon (3) holds for bu fals for some oher ype s. Tha s: THE INTUITIVE CRITERION. A perfec Bayesan equlbrum of he asse sale game wh poseror belef funcon (,) and oucome u s nuve f, on seeng any quany vecor 0 q a and prce cap vecor p, nvesors poseror probably ( q, p) s zero for any ype ha sasfes (3) as long as here s some ype s for whch he nequaly s reversed: for whch qp u() s qf () s. An equlbrum ha sasfes he Inuve Creron wll be called nuve, as wll nvesors belefs n ha equlbrum. If an oucome equlbrum, we wll call he oucome nuve as well. u s suppored by an nuve Clearly, a smple way ensure nuve belefs s for nvesors o respond o a devaon ( q, p ) by pung all of her wegh on a ype for whch he opporuny cos u () qf() of he gven devaon s a a mnmum. Snce no ype has a lower opporuny cos of devang o ( q, p ) han hs ype, here s no prce vecor p p ha makes some ype s wllng o devae o ( q, p ) f does no also make ype wllng o choose hs devaon: belefs are nuve. Snce echncally more han one ype may mnmze he opporuny cos u () qf() of devang, we can break es by pung all of nvesors wegh on he lowes such ype. Summarzng, a suffcen condon for he belef funcon o be nuve s ha, followng a devaon ( q, p ), sasfes where q q, p 1 (4) qmn argmn u qf s he lowes ype ha mnmzes u qf. (5) dscouned cos of ransferrng q uns of each asse o nvesors raher han q. Type wll no devae o (, ) qp f hs opporuny cos exceeds her maxmum revenue qp from dong so,.e. f (3) holds. 12

14 We wll show ha any nuve equlbrum of he Asse Sale game sasfes he followng propery. FAIR PRICING. An equlbrum s farly prced f, for all and, q 0 mples p ( q ( ), p ( )) f( ). Tha s, he prce assgned o any asse ha s sold n equlbrum equals he condonal expeced payou of hs asse. Ths propery s relaed, bu no dencal, o he noon of a separang equlbrum. Unlke separaon, far prcng mples ha he ssuer s prce caps never bnd and ha nvesors payoffs are dencally zero, so socal welfare s gven by he ssuer s payoff funcon u(). And unlke far prcng, separaon mples ha nvesors can nfer he ssuer's acual nformaon n equlbrum, whle far prcng mples only ha hey can nfer he condonal expeced payou of any asse ha he ssuer chooses o sell whch s all hey need n order o compue he asse s far-marke value. We now show ha here exss a unque nuve oucome of he Asse Sale game, and ha hs oucome s suppored by a farly prced equlbrum. Moreover, hs equlbrum maxmzes he payoff of an ssuer of each ype whn he se of farly prced equlbra and hence s effcen whn hs se (snce, n a farly prced equlbrum, nvesors mus break even on average). To consruc he equlbrum, we recursvely calculae he maxmum surplus u () achevable by each ype subjec o far prcng and o he ncenve consran ha no lower ype would choose o mae he gven ype: RECURSIVE LINEAR PROGRAM (RLP). For ype 0 af, le q 0 a and le u denoe he gans from sellng all of he ssuer s endowmen. For any hgher ype 0, defne 30 u max 1 qf( ) s.. f f( s 0qa q ( ) ) u ( s) for all s, 30 Any ype s ha chooses he same q as ype wll earn revenue of qf(), and have cos qf(s), for oal surplus q(f()f(s)), whch mus no exceed u (s) n order o manan ncenve compably. 13

15 and le q denoe any quany vecor ha solves he gven maxmzaon problem and yelds he payoff u. We wll now show ha he unque nuve oucome s he soluon For each, le ype s quany vecor cap vecor be any soluon o p f u o he RLP. 31 q () be any soluon o RLP and le ype s prce (whch wll mply ha he prce caps do no bnd f he equlbrum s farly prced). Le he poseror belefs q, p be gven by Bayes s Rule f q, p s chosen by some ype n equlbrum and by (4) and (5) f no (wh u replaced by u n (5)). Fnally, le he marke prce funcon p, q p be he resul of subsung for n equaon (2). Snce belefs are nuve by consrucon, e sasfes he Inuve Creron f s an equlbrum. The man resul of hs secon s ha e s an equlbrum, s farly prced, yelds he oucome u, and s effcen (and opmal for he ssuer) whn he se of farly prced equlbra. Moreover, any nuve equlbrum of he Asse Sale game has he same oucome u as e. Fnally, n e, a hgher quany sold of any asse canno rase he prce of any asse. PROPOSITION 1. The RECURSIVE LINEAR PROGRAM has a unque soluon u, whch s srcly posve and nonncreasng n he ssuer s ype, and s he unque nuve oucome. The profle Moreover, for each ype, e defned above s an nuve equlbrum wh oucome u. u s boh he hghes aanable ssuer s payoff and he maxmum socal welfare n any farly prced equlbrum. Fnally, for any quany vecors q q (wheher or no chosen n equlbrum), no asse prce s hgher under q han q: p q, p p q, p PROOF: Appendx.. 31 Wheher u corresponds o an equlbrum s no ye obvous; we have no, for example, checked wheher he ncenve consrans for hgh ypes no o mmc low ypes are sasfed. We verfy hs below. 14

16 2.3. Nonmonoonc Issuance In he sngle-asse case, a more opmsc ssuer reans more of her asse n order o dsngush herself from more pessmsc ypes (e.g., Leland and Pyle 1979, DeMarzo and Duffe 1999). Wh mulple asses, hs monooncy propery can fal o hold. We show hs va wo compued examples. The frs s a smple example wh hree ypes of ssuer, whose soluon can be easly calculaed by hand. EXAMPLE 1. Assume he ssuer owns one share each of wo asses: a1 a2 1. The ssuer has hree possble ypes 0,1,2 and her dscoun facor s 12. The condonal expeced payous of he asses are f 0 1,1, 1 2,3 f, and f 2 5,4. Tha s, asse 2 (resp., 1) responds relavely more when he ssuer s ype rses from 0 o 1 (resp., from 1 o 2). One can hen verfy ha n he soluon q o RLP, he ssuer sells her enre porfolo when her ype s 0, 2/3 shares of asse 1 and no shares of asse 2 when her ype s 1, and no shares of asse 1 and 4/15 shares of asse 2 when her ype s 2. Inuvely, for each ncrease n her ype, he ssuer mos effcenly dsngushes herself from he nex lower ype by reanng more of he asse whose valuaon s rased proporonally more by he gven ype ncrease. In parcular, he ncrease n her ype from 1 o 2 leads her o sell more of asse 2, n conras wh he sngle-asse case where opmsc ssuers always sell less. The nex example llusraes he same prncple n a more realsc seng wh many ssuer ypes. EXAMPLE 2. Suppose he ssuer holds 1 share of each of 2 asses, wh expeced payous of each (condonal on = 0,1,, 200) gven n Fgure Whle he overall sensvy o nformaon for each asse s smlar (a 10% ncrease n value over he range of ), he rgh panel shows ha asse 2 s reurn s more sensve o ncreases n for 50 and asse 1 s more sensve when Parameer assumpons are avalable by reques. 15

17 f f 1 /f f Fgure 1: Expeced Payoffs of Asses for EXAMPLE 2. Asse 1 s reurns are less sensve o nformaon han asse 2 s reurns for low, bu are more sensve for hgh. Fgure 2 depcs he equlbrum sraeges for 0.9 (gven by he sold lne wh crcles for = 0, 10, 200). The lowes ype sells all shares of boh asses. Inally, she sgnals ype ncreases by sellng fewer shares of asse 2, whch s more sensve han asse 1 s o her nformaon when he ssuer s ype s low. Her approach changes when her ype rses above 50: asse 1 now becomes more nformaon-sensve, so she sgnals beer nformaon by sellng fewer shares of asse 1 and more shares of 2. Fnally, when her ype s so hgh ha asse 1 s reaned n s enrey, he ssuer has no choce bu o sell less of asse 2 as a sgnal (even hough asse 2 s less nformaonally sensve han asse 1 n hs range). Also depced n Fgure 2 s he equlbrum prce funcon, as a conour plo showng he ype nvesors nfer gven any quany q; darker shades ndcae lower ypes, whle whe lnes ndcae he so-prce conours for = 0, 10, 200. On each such conour, nvesors poseror belefs and hus he prces of boh asses are consan. As he conours are downward slopng, nvesors become more pessmsc as he ssuer sells more shares of eher asse. Moreover, for ypes beween 50 and 120, belefs drop dsconnuously for a small ncrease n eher quany. For example, sarng from q (60), a small ncrease n he quany sold of eher secury wll cause belefs o drop dsconnuously from 60 o below 20; smlarly, a small ncrease from q (90) would cause belefs o drop o 0. 16

18 Dsconnues occur when he bndng ncenve consran n RLP s non-local; for example, he ype wh he greaes ncenve o mmc ype 60 s ype 18. The possbly ha non-local consrans may bnd dsngushes our seng from sandard sgnalng models n whch a sngle crossng propery holds. =200 q 2 () =120 1 = q 1 () Fgure 2: Equlbrum for EXAMPLE 2. Crcles show equlbrum quanes along he sold curve; shadng ndcaes so-prce conours gven he belefs n (4). Noe he dsconnuy n prces along he upper rgh boundary of he equlbrum curve Informaonally Ordered Asses We saw n secon 2.3 ha greaer opmsm may somemes lead an ssuer o sell more shares of a gven asse. Ths nonmonooncy occurs because asses can alernae n her nformaonal sensvy as he ssuer s opmsm grows. In he remander of he paper, we focus on sengs n whch asses sensvy o he common facor can be uncondonally ranked: f one asse s more sensve han anoher for a gven ncrease n opmsm, hen s more sensve for any such ncrease. 17

19 In hs seng, he Asse Sale game has a smple oucome. The lowes ype ssuer sells all her asses as before. As her ype rses, she reans her more nformaonally sensve asses frs. Inuvely, hese asses are always more effcen sgnals of greaer opmsm. As a resul, he model predcs ha a frm wll never sell an asse unless also sells all of s less sensve asses n her enrey, as predced by Myers s (1984) Peckng Order hypohess Informaon Sensvy We wll say ha one asse s more nformaonally sensve han anoher f s expeced value changes proporonally more for a gven change n he ssuer s ype. To ensure ha proporonal changes are well-defned, we assume he condonal expeced payoff of each asse s always posve: ASSUMPTION B (POSITIVE EXPECTED PAYOFFS). For all, f The formal defnon s as follows. INFORMATIONAL SENSITIVITY. Asse s more nformaonally sensve han asse j a ype f, for any lower ype s, f f s f f s j j. If hs holds for each ype, hen asse s more nformaonally sensve han asse j. 34 The asses 1,,n dsplay ncreasng nformaon sensvy (IIS) f each asse s more nformaonally sensve han any lower-ndexed asse j. 35 In hs seng, he ssuer prefers o rean her more nformaonally sensve asses frs as her opmsm grows. Formally: PROPOSITION 2. Suppose asse s more nformaonally sensve a han asse j. Then n any soluon q o he RECURSIVE LINEAR PROGRAM, f q () a hen j j 33 Under hs requremen, a secury can have zero payous n some saes as long as, for any ype, here are saes n whch he secury s payou s srcly posve. 34 Equvalenly, s more nformaonally sensve han j f f ()/ f () s ncreasng n. 35 IIS s equvalen o he funcons f () beng log-supermodular n (, ). j 18

20 q () 0: he ssuer wll no rean any shares of asse j unless she also reans all shares of asse. PROOF: Appendx Hurdle Class Sraeges If we furher assume IIS hen, by PROPOSITION 2, he ssuer wll choose o sell all of her less nformaonally sensve asses and rean all of her more nformaonally sensve ones, wh he exac cuoff, or hurdle class, deermned by her ype: PROPOSITION 3. Le he ssuer s asses dsplay Increasng Informaon Sensvy. Then an ssuer of each ype wll choose a hurdle class quany vecor: a vecor of he form q a,, a, q,0,,0 for some hurdle class c1,, n1 1 c1 and some hurdle class quany q 0, a q q s c c. 36 Moreover, for any ypes s,, whence chooses a weakly lower hurdle class han s does. PROOF: Appendx. Inuvely, a low ype has a sronger ncenve o sell addonal asses by choosng a hgher hurdle class han a hgh ype because, whle he mpac on revenue s he same, he opporuny cos of parng wh he addonal shares s smaller for he low ype. Hence, a low ype canno choose a lower hurdle class han a hgh ype does n equlbrum. Any hurdle class vecor a,, a, q,0,,0 for q 0, a 1 c1 c c c s unquely denfed by c he oal number q c1 c a 0 c of shares ha he ssuer sells (of all asses combned). Ths number can be hough of as a one-dmensonal sgnal. Thus, under IIS he ssuer s sgnalng problem reduces o he famlar and racable one-dmensonal case. One remanng complcaon s ha for each ype 0, RLP ncludes 1 ncenve compably (IC) consrans: one for each lower ype s. The nex resul shows ha, n fac, he IC consran for he nex lower ype 1 s bndng and, moreover, denfes a unque hurdle class vecor for ype : 36 When c equals n + 1, q equals a: he ssuer sells her enre porfolo. 19

21 PROPOSITION 4. If he asses dsplay Increasng Informaon Sensvy, hen q 0 a and, for each 0, q s he unque hurdle class vecor ha sasfes he ncenve compably consran for ype 1, whch s PROOF: Appendx q f f q f 2.5. Applcaon: Prorzed Asses. (6) To apply he precedng resul, one mus know wheher one asse s more sensve han anoher o an ssuer s nformaon. In pracce, hs condon may be hard o check. In hs secon we gve a smple rule ha can be used o make hs deermnaon: under a weak dsrbuonal condon, more senor asses are less nformaonally sensve. Hence, by he pror resul, hey wll be sold frs, as predced by Myers s (1984) peckng-order hypohess. Imporanly, we develop hs resul usng a novel, generalzed noon of senory ha perms he rankng of asses ha are no ordered by src prory. A deb asse has src prory over anoher asse f s owners mus be pad s face value before he owners of he oher asse are pad anyhng. Whle our noon agrees wh src prory n hs case, also apples when neher asse has src prory. Generally speakng, senory refers o he clam of one asse versus anoher o a fuure cash flow ha secures boh asses. Hence, we assume ha he payou of each asse {1,, n} of he ssuer s gven by some nondecreasng, nonnegave funcon F Y of a common, sochasc cash flow Y. The cash flow Y mgh mgh represen a frm s fuure operang profs or he aggregae repaymens on a bank s loan porfolo. The dsrbuonal assumpon we need s as follows. HAZARD RATE ORDERING (HRO). For any ype, he poseror suppor of Y concdes wh he pror suppor. Moreover, for any ypes s, he rao 20

22 Y y s Y y Pr / Pr s decreasng n y n he suppor of Y whenever he rao s well-defned. 37 Whle sronger han frs-order sochasc domnance, HRO s weaker han he monoone lkelhood rao propery, whch s commonly assumed n sgnalng envronmens. 38 If he condonal hazard rae of Y gven s well defned, HRO saes ha hs hazard rae s decreasng n he ype ; however, HRO also apples when he hazard rae s undefned. 39 As for senory, le us say ha asse s senor o asse j f he laer asse has a relavely sronger clam on he cash flow when he cash flow s low: f he payou rao / ry F Y F Y s ncreasng n he cash flow Y. The precse defnon, whch j perms he denomnaor o be zero and he rao o be locally consan, s as follows. PRIORITIZED ASSETS. Asse s senor o asse j (and j s junor o ) f he payou Fj( Y ) o asse j can be wren as he produc of he payou FY ( ) of asse and a nonnegave, nondecreasng funcon ry ( ) of he cash flow ha akes more han one value wh posve probably on ha poron of he suppor of he cash flow where asse s payou s posve. 40 The asses F are prorzed f any asse s senor o any hgher-ndexed asse j. Src prory s mpled by he above defnon: f asse has src prory over asse j, he funcon r(y) s zero whle asse s beng repad, grows whle asse j s beng repad, and becomes consan once asse j has been repad n full. However, he defnon s more general: asse may be senor o asse j even f he payous o boh rse wh he cash flow, as long as he payou o asse j rses relavely faser. 37 The only pon y n he suppor of Y a whch he rao may be undefned s he he upper boundary of Y s suppor, where Pr Y y may equal zero. 38 See Shaked and Shanhkumar (2007), p. 43, Thm. 1.C The condonal hazard rae s h y / 1 H y where H denoes he condonal dsrbuon of he cash flow Y gven he ype, and h s he correspondng densy. Ths hazard rae s undefned f he condonal dsrbuon s dscree or has aoms, whle he condon n HRO s sll well defned n hs case. 40 I.e., here s a consan 0 Pr F Y 0 and r Y c Pr F Y 0 and r Y c are c such ha and boh posve. Ths requremen rules ou rval cases where secures dffer only on measure zero oucomes. 21

23 Some examples appear n Fgure 3. The secures are ranked n order, wh F 1 he mos senor and F 5 he mos junor, wh he only excepon ha F 4 s noncomparable wh F 2 or F 3. Thus ( F1, F4, F 5) and ( F1, F2, F3, F 5) are prorzed ses. Fgure 3: Examples of Prorzed Secures. Secures are ranked wh F 1 mos senor and F 5 mos junor, wh he excepon ha F 4 s non-comparable wh F 2 and F 3. We can now relae a secury s senory o he order n whch wll be sold: under HRO, junor secures wll be reaned frs as an ssuer becomes more opmsc. Or, equvalenly, senor secures wll be sold frs: PROPOSITION 5. Suppose HRO holds and he secures are prorzed. Then he asses sasfy Increasng Informaonal Sensvy, so her lqudaon s as se ou n PROPOSITION 3 and PROPOSITION 4. In parcular, he ssuer wll no sell any shares of a gven asse unless she also sells all more senor asses n her enrey. PROOF: Appendx. 3. Secury Desgn Havng solved he general porfolo lqudaon problem, we now apply he above ools o he problem of secury desgn. More precsely, we generalze he sngle ex-ane secury 22

24 case suded by DeMarzo and Duffe (1999) by permng an ssuer o desgn mulple secures before and/or afer she receves her nformaon. We frs show ha any nuve equlbrum of hs game s equvalen o one n whch he ssuer pools her nal asses, sees her nformaon, and hen desgns and sells a sngle secury whose payou equals he aggregae payou of he secures she sells n he orgnal equlbrum. To furher sharpen hese predcons, we hen mpose he above Hazard Rae Orderng propery. We also assume monooncy: he payou receved by nvesors, as well as ha reaned by he ssuer, s a nonnegave and nondecreasng funcon of he fnal cash flow. 41 Under hese wo assumpons, we show ha he ssuer s opmal ex-pos secury s sandard deb wh a face value ha s decreasng n her nformaon. We esablsh hs by provng a formal equvalence beween ex-pos secury desgn and a sraegy n whch he ssuer pools her nal asses, desgns a maxmal se of prorzed deb secures or ranches ha are secured by he pool, sees her nformaon, and hen chooses how much of each ranche o sell as n he asse sale game. 42 By PROPOSITION 5, he ssuer wll sell hose ranches whose senory les above a hreshold ha s ncreasng n her nformaon. Bu hs sraegy s equvalen o sellng a sngle sandard deb secury whose face value s decreasng n he ssuer s nformaon. Thus, he wo sraeges are equvalen and opmal. In each such sraegy, he ssuer begns by poolng her nal asses. In DeMarzo (2005), n conras, s somemes beer no o pool. Why? In our seng, he nal asses paron a common underlyng cash flow. Hence, each pre-exsng asse can be exacly replcaed by a new asse ha s secured by he pool of nal asses. As poolng does no resrc he ssuer, canno harm her. In DeMarzo (2005), n conras, he nal asses are backed by dsnc cash flows. Ths gves rse o wo new effecs. Frs, poolng prevens 41 Monoony s a common assumpon n he secury desgn leraure; see, e.g., DeMarzo (2005), DeMarzo and Duffe (1999), Frankel and Jn (2015), Har and Moore (1995), Mahews (2001), and Nachman and Noe (1994). I can be jusfed by supposng he ssuer has free dsposal over her cash flow Y and can also conrbue cash o nflae. Hence, f her payou were decreasng n Y, she would freely dspose of some cash n order o rase hs payou. And f, alernavely, he payou o nvesors were fallng n Y, he ssuer would conrbue cash o nflae Y, hus payng nvesors less and rasng her payou by more han he amoun conrbued. Fnally, nonnegavy s movaed by he common lmed-lably feaure of fnancal asses. 42 The wo sraeges are equvalen because any monoone secury can be mmcked by a suable porfolo of ranches and vce-versa. Equvalence means ha he porfolos sold by he ssuer n he wo sraeges rase he same revenue from nvesors and, for any realzaon of he underlyng cash flow, enal he same fnal aggregae payou o nvesors. 23

25 he ssuer from sgnalng ha one cash flow s hgh whle anoher s low, whch s harmful o her. Second, by dversfyng away he resdual rsk n he cash flows, poolng also allows he ssuer o ssue deb wh a lower defaul rsk and hus o reap more of he gans from rade wh nvesors. 43 Wheher poolng s opmal depends on he radeoff beween hese wo forces, as shown n Theorem 5 n DeMarzo (2005). 44 As hey are somewha echncal, he proofs of hs secon s resuls appear n our onlne appendx (DeMarzo, Frankel, and Jn 2017) The General Secury Desgn Game In he Asse Sale game nroduced n secon 2, he ssuer chooses how many shares o sell of each of a gven se of asses. Ths lms her; for nsance, she canno borrow money usng some of her asses as collaeral. Dong so s equvalen o sellng a deb secury ha s secured by her asses. In hs secon we dscard hs resrcon: he ssuer can desgn general secures (no only deb) ha are secured by her asses. She can do so boh before and afer she sees her nformaon. Thus, he model s a generalzaon of boh DeMarzo and Duffe (1999), n whch he ssuer desgns a sngle secury before seeng her nformaon, and an earler verson of hs paper (DeMarzo 2003) n whch she desgns a sngle secury ex pos. As n secon 2.5, we assume he ssuer s endowed wh a se of nal asses ha are secured by a common, sochasc cash flow Y. 45 Before learnng her ype, he ssuer desgns a se of nerm secures whose payous are funcons of he payous of he nal asses. Afer she sees her ype, she hen desgns a se of ex-pos secures whose payous are funcons of he payous of he nerm secures The second effec can also be seen n he presen model, as a benef o he ssuer from a reducon n he condonal volaly of her underlyng cash flow; see EXAMPLE 5 below. 44 The proof of Theorem 5 n DeMarzo (2005) reles on an earler verson of he resuls of hs secon whch appeared n he workng-paper verson of our paper, DeMarzo (2003). 45 Ths means ha he sum of nal asse payous equals he realzaon Y of he cash flow. Ths assumpon s equvalen o assumng ha, n addon o her nal asses, he ssuer can also use any resdual cash flow as collaeral for her secures. Hence, for all nens and purposes, hs resdual cash flow s an nal asse. 46 Requrng he ssuer o ssue nerm secures s no resrcve: any of hem can be a pass-hrough secury whose payou s dencally equal o he payou of some nal asse. However, hs flexbly s needed n order o reduce he general secury desgn problem o he asse lqudaon problem of our base model. 24

26 In hs secon, we show ha any nuve equlbrum of hs game s equvalen o one n whch he ssuer pools her nal asses and sells a sngle ex-pos secury whose payou s a funcon of he pool s fnal value. In subsequen secons we show ha under addonal mld assumpons, hs secury s sandard deb. Formally, he general secury desgn game s as follows. THE GENERAL SECURITY DESIGN (GSD) GAME. The ssuer s endowed wh a n 1 fne se FY F Y n common cash flow: F Y of nal asses whose payou funcons paron he 1 Y. Tmng s as follows. k 1. The ssuer desgns a fne se I I k 1 k each such secury k s some funcon k whch we wre, for brevy, as I. K k lably: I Y0, Y k1 K of nerm secures; he payou of I F Y of her nal asse payous Y. 47 The nerm asses sasfy lmed P P of j 2. The ssuer sees her ype and offers nvesors a fne vecor j 1 some number of ex-pos secures, where he payou of ex-pos secury j s j a funcon P I Y of her nerm asses payous. These payous also J j sasfy lmed lably: K k P I Y 0, I Y j1. A he same k1 me, he ssuer comms o a global revenue cap : f her ssuance revenue exceeds he cap, she keeps and dscards he res. 48 J 3. Invesors assgn a prce j p o each ex-pos secury j. Le I P J j j1 W Y P I Y (7) 47 Formally, we defne he new funcon ˆk I sasfyng Iˆk Y I k FY and hen rename o 48 Lke he prce caps n he Asse Sale model, we use he revenue cap n order o rule ou mplausble equlbra. The cap never bnds n equlbrum. k I. 25

27 denoe he aggregae payou promsed o nvesors, gven he nerm secury vecor I and he ex-pos secury vecor P. By lmed lably, hs payou les n 0,Y. If a ype- ssuer chooses he nerm secury vecor I, he ex-pos secury vecor P, and he revenue J j I cap, her expeced surplus from ssuance equals W p E Y : her j1 P ssuance revenue less he dscouned expeced value of he secures ssued. The followng wo examples llusrae he generaly and flexbly of he GSD game. We begn wh a wo-secury generalzaon of DeMarzo and Duffe (1999). EXAMPLE 3. The ssuer may desgn nerm asses conssng of wo prorzed secures: a senor ranche I 1 Y mn D, Y 2 1 ranche I Y mn D, Y I Y wh face value D and a junor wh face value D. Afer dscoverng her ype, she may hen sell a quany ex-pos secures are hen gven by z of each ranche 1,2. Formally, her wo P I Y z I Y for 1,2. The followng s a wo-secury generalzaon of he one-secury case consdered by DeMarzo (2005) and Frankel and Jn (2015). EXAMPLE 4. The ssuer may sell senor and junor deb ex pos wh ypeconngen face values D and D respecvely. Formally, she desgns a sngle exane pass-hrough secury gven by 1 1 I y y, whle her wo ex-pos secures are hen 2 1 mn, and P Iy mn D, y P Iy. P I y D y We wll refer o he acon P, ha he ssuer chooses afer learnng her ype as her expos acon. A sraegy for he ssuer s a rple I, P,, whch specfes her nerm asse vecor I and, for any ype and nerm asse vecor I (whch equals I unless he ssuer devaed), her ex-pos acon PI, I p j s a vecor j 1 J j j p where p p I, P,. A prcng funcon for nvesors s he prce of ex-pos secury j when he ssuer chooses he acon I, P,. The ssuer s condonal expeced payoff from hs acon, gven her ype and he nvesors prcng funcon p, s hus 26