Analyzing Cost of Debt and Credit Spreads Using a Two Factor Model with Multiple Default Thresholds and Varying Covenant Protection.

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1 Analyzing Cos of Deb an Cei Speas Using a wo Faco Moel wih Muliple Defaul heshols an Vaying Covenan Poecion by S. Lakshmivaahan 1, Shengguang Qian 1 an Duane Sock June 16, 9 1) School of Compue Science, Univesiy of Oklahoma, Noman, OK 7319 ) Division of Finance, Michael F. Pice College of Business, Univesiy of Oklahoma, Noman, OK Conac Auho: Duane Sock, [email protected], phone ( ) Absac: he cos of eb capial fo copoaions epens on cei speas. Of couse, he amaically geae cei speas of 8 gealy incease he cos of eb fo he majoiy of bon issues. We analyze he shape of cei spea em sucues paying special aenion o he humps ha have been obseve by a numbe of eseaches. he shape of cei speas epens upon he shape of fis passage efaul. Impoanly, ou wok allows sepaaion of efaul pobabiliy ue o beach of baie vesus efaul pobabiliy ue o asses being less han face value a mauiy. We noe ha in some cases, fis passage efaul has a hump bu no in ohes. I is useful o see when an how fis passage efaul humps conibue o a humpe cei spea. he impac of ecenly popula weak covenans (covenan lie) is shown o play a majo ole in he shape of cei speas. he implicaions of ou suy ae impoan o such opics as measuing he iskiness of he banking sysem epenen upon cei spea slopes. he auhos appeciae commens mae in pesenaions a Geog-Augus Univesiy of Göingen, Souhwes Finance Symposium (ulsa), Univesiy of Oklahoma, Mulinaional Finance Sociey (Olano), Euopean FMA (uin), an Goehe Univesiy am Main (Fankfu). We ae gaeful fo commens of Robe J. Ellio (Mahemaics, Univesiy of Calgay) fo help in eveloping Appenix A. We ae also gaeful o Kay Giesecke (Sanfo Univesiy) fo avice on eveloping he moel allowing boh baie an mauiy efaul. We also appeciae commens fom Lay Johnson (Univesiy of ulsa), Olaf Kon (Geog-Augus Univesiy of Göingen), Michael Jacobs (Compolle of he Cuency), Chisian Schlag (Goehe Univesiy) an Ralph Rogalla (Goehe Univesiy). JEL classificaion: G13( Bon Inees Raes), G ( Geneal Financial Economics) Keywos: cei spea, em sucue, efaul, hump

2 Inoucion Picing cei isk an efaul poenial have always been impoan eseach opics an he cei cisis of 7 an 8 has geneae even geae inees. Numeous empiical suies of cei speas have been pefome. Fo example, Kishnan, Richken an homson (6) use cei spea slopes of bank eb o peic bank isk. As ohes, hey fin he shape of he cei spea can be posiive o negaive an is ofen humpe. Fuhemoe, hey fin ha ove ime he cei spea can change in iffeen ways fo iniviual banks. Ou pupose is o analyze he shape of cei spea em sucues. Moe specifically, we analyze he em sucue of fis passage efaul an, in un, is impac upon em sucues of efaul isky yiels an cei speas. We analyze how hese em sucues epen on such hings as covenans, he volailiy of inees aes, volailiy of fim value, coelaion of fim value wih inees aes, an ohe moel paamees. Humps occu when he em sucue slope changes fom posiive o negaive. A hump in fis passage efaul can encouage a hump in cei spea bu is no necessay fo a hump o occu. Coniions fo a hump in em sucues ae paiculaly ineesing an useful o explain. Fuhemoe, he impac of (weak) covenans upon hese em sucues is paiculaly imely given he ecen cei cisis. As we show, i is impoan o sepaae efaul ue o beach of baie vesus efaul ue o asses being less han face value a mauiy. Ceain inusies an fims may have a hump in em sucue of efaul pobabiliy ue o sengh of covenans (baies) whee ohes o no. o ou knowlege, we ae he fis o apply an analyze Giesecke s (4) sepaaion of efaul 1

3 pobabiliies. Banking egulaos shoul fin anyhing which helps peic efaul isk an he healh of he banking sysem vey useful in eveloping egulaion policies. Even hough cei spea moels have been foun useful, a numbe of empiical suies have noe a lack of explanaoy powe. Collin-Dufesne, Golsein an Main (1) fin ha some vaiables ha shoul explain cei spea changes have only limie explanaoy powe. 1 In an effo o incease explanaoy powe, Diessen (5), fo example, focuses on aing an even isk pemium bu fins such a pemium canno be esimae vey well. Chisensen (8) efes o he inabiliy of empiical moels o explain cei speas as he copoae bon cei spea puzzle. he limiaions of hese empiical finings using exan moels sugges a nee fo impove heoeical moels ha uilize fim specific o inusy specific facos such as covenans an efaul baies an, also, a nee o segegae efaul pobabiliy ino ha ue o baie (covenans) vesus classic Meon (1974) mauiy efaul. he fis sucual moel of cei speas was evelope by Meon (1974) whee he assume a consan sho em inees ae. 3 Lelan (1994) an Lelan an of (1996) also assume a consan inees ae. I is unappealing o no allow inees aes o change in a moel of bon valuaion. hus, in conas, Longsaff an Schwaz (1995), Collin-Dufesne an 1 Schaefe an Sebulaev (8) sugges ha alhough sucual moels may no peic speas an pices vey well, hey povie goo heging infomaion. Impove empiical moeling coul also impove empiical esuls. 3 hee ae wo boa classes of heoeical moels: sucual an euce fom. Fo a compehensive analysis of alenaive efaul moels, see Duffie an Singleon (3). Jaow an Poe (4) compae euce fom an sucual moels using an infomaion base pespecive. Jaow, Lano,unbull (1997) povie euce fom moel. We focus upon impovemens o sucual moels.

4 Golsein (1) an Achaya an Capene () moels inclue boh a value of he fim ( V ) pocess an, also, an inees ae ( ) pocess. Eom, Helwege, an Huang (4) empiically es alenaive moels an povie an excellen chaaceizaion of single faco vesus wo faco moels in an appenix. wo faco moels woul ceainly seem o be vey appealing when, fo example, inees aes ae moe volaile han aveage. Longsaff an Schwaz (1995) evelop a wo faco moel involving V an whee is base on he Vasicek (1977) moel. Using he sana famewok fo picing a coningen claim, Longsaff an Schwaz (1995) efine he pobabiliy of efaul as a soluion of a paial iffeenial equaion. Also, using ye anohe known esul, hey hen chaaceize he efaul ensiy as soluion of an inegal equaion. By invoking he sana isceizaion scheme, his lae equaion is euce o a sysem of linea equaions which is hen solve fo he efaul pobabiliy. Achaya an Capene () also consie he wo faco moel involving V an whee, in conas, evolves accoing o he Cox, Ingesoll, Ross (1985) moel. Using he classical (mauiy) efiniion of efaul, hey compue he efaul pobabiliy wih numeical mehos base upon a binomial ee appoximaion. Ou V an moels ae simila o Longsaff an Schwaz (1995) an we make impoan enhancemens o hei moel. We noe ha Lakshmivaahan, Qian, an Sock (8) quanify he isibuion of V an show ha i has he lognomal fom wih ime vaying mean an vaiance. We use he efiniion of efaul ue o Giesecke (4) which, moe 3

5 compehensively, has wo heshols insea of one. By combining he lognomal isibuion of V in Lakshmivaahan, Qian, an Sock (8) an he newe, moe ealisic, efiniion of efaul of Giesecke (4), we hen explicily expess he efaul pobabiliy as he sum of wo quaniies. One quaniy epesens he classical (mauiy) efaul of Meon (1974) an he ohe quaniy epesens baie efaul which, ciically, epens on covenans. Given wok by Alman, Bay, Resi an Sioni (5), we ecognize ha ecovey in even of efaul is likely bes moele as epenen upon pobabiliy of efaul insea being assume consan. Fuhemoe, pobabiliy of efaul may be convenienly expesse as a funcion of expece value of V an vaiance of V. In effec, as mauiy inceases, hee is a ace beween inceasing E ( V ) an inceasing ( ) Va V o eemine fis passage efaul pobabiliy an he shapes of he esuling em sucues of efaul pobabiliy, efaul isky aes, an cei speas. 4 hus, he impac of mauiy upon cei spea is complex. Ineesingly, pobabiliy of efaul may o may no incease wih mauiy. If he incease in E(V ) ue o geae mauiy is songe han he incease in Va (V ), he pobabiliy of efaul may ecline along wih he cei spea. his may be especially ue when efaul baies ae low ue o (ecen) weak covenan poecion (covenan lie). Such analysis helps explain humpe cei speas so fequenly foun by, among ohes, Meon (1974), Longsaff an Schwaz (1995) an Kishnan, Richken an homson (6). Also, humps in cei 4 Defaul may mean enegoiaion wih lenes o esucue o, alenaively, foeclose. he esucue o foeclose ecision is beyon he scope o ou eseach an we efe ineese eaes o Bown, Ciochei, an Riough (6) fo moe on his ecision. 4

6 efaul swap em sucues have been foun by Bajlum an Lasen (7) an Lano an Moensen (5) an ohes. We ae he fis o analyze fis passage pobabiliies fo boh mauiy an baie efaul wih a wo faco moel. An impoan obsevaion is ha geae passage of ime suggess geae expece fim value bu, a he same ime, geae vaiance of V so ha he impac of mauiy on efaul pobabiliy is unclea. Recen evens in he cei cisis make ou analysis of baie efaul especially useful. ha is, a majo eeminan of baie efauls ae covenans which, if violae, may consiue efaul o sugges imminen efaul. 5 he pupose of such covenans is o help conol agency coss of eb. 6 ha is, bon buyes ae concene abou he poenial fo sock holes o ake acions ha ae o he isavanage of bon holes a some ime afe issuance. Moe specifically, he fim may uneinves in pojecs ha benefi bonholes bu have lile o no benefi fo sock holes. Pobably moe elevan o ou eseach is fim oveinvesmen in high isk pojecs ha en o pouce benefis fo sock holes bu, a he same ime, make he bon hole posiion moe isky. As an example, Paino an Weisbach (1999) fin ha fims may even inves in negaive pesen value pojecs ha en o have high volailiy. Bille, King an Maue (7) fin song evience ha bon holes equie moe covenan poecion fo fims 5 As given by Lano an Moensen (5), efaul may be viewe as iggee by covenans bu can also be viewe as inabiliy o cove equie coupons ue o liquiiy consains o inabiliy o aise new equiy capial 6 Chava an Robes (8) es he impoance of covenan violaion. hey fin ha capial invesmen eclines shaply following financial covenan violaion as ceios use he hea of acceleaing he loan o inevene in managemen. 5

7 wih high gowh oppouniies, longe mauiy eb, an geae leveage. Consisen wih hese finings, lowe ae eb ens o have moe covenan poecion. Cuiously, hee has been a ecen en in some eb makes o have much lowe covenan poecion han wha agency heoy an he Bille, King an Maue (7) finings woul sugges. Such lack of covenan poecion, eme covenan lie, has le o misleaingly low efaul aes even as he cei cisis of 7 an 8 coninue an he economy weakene. Many fea ha highe efaul aes ae meely being elaye an hese high elaye efauls will lae be vey amaging o a financial sysem ha many hough ha fully ecovee fom he cei cisis. One of ou main conibuions is explici analysis of how efaul pobabiliy is a vaying funcion of baies eemine by covenans. Secion 1 escibes he pobabiliy ensiy of efaul as epenen upon a wo faco valuaion pocess assuming he Vasicek (1977) pocess fo he inees ae. Defaul occus when fis passage of fim value his a baie befoe mauiy, o, a mauiy if value of he fim is hen less han face value of eb. In Secion we eive expessions fo efaul isky spo aes an cei speas. Pobabiliy of efaul epens upon such facos as leveage, baies, 7 an he volailiy of fim value. Secion 3 epos he compuaion fo he em sucues of isky spo aes, cei speas an efaul pobabiliies. hen, in Secion 4, we iscuss he convesion of he isk neual efaul pobabiliies o physical efaul pobabiliies. I uns ou ha he em 7 hese baies escibe levels of V whee eb holes lose paience wih he fim o whee covenans ae violae. 6

8 sucues of hese physical efaul pobabiliies ae vey simila. Secion 5 povies expessions fo how cei spea slope is explicily epenen upon efaul pobabiliy slope an suggess efinemens in peicing fuue bank isk. Concluing obsevaions ae povie in he las secion. 1. he Disibuion of V an Expession fo Defaul Pobabiliy Le he value pocess V evolve accoing o he sochasic iffeenial equaion (SDE) V V = + σ W (1.1) v v, whee he sho-em inees ae evolves accoing o he naow sense linea moel, ue o Vasicek (1977), given by he SDE 8 θ = c + σ W c,. (1.) he inees ae pocess has long un mean θ/c an eves o he mean a he ae c. he iving innovaions ae coelae an E W W = ρ v,, (1.3) whee ρ 1. Hee ρ is assume consan. he moel paamees σ, v σ, θ, c, ρ an he iniial coniions V an ae specifie. 9 8 See Anol (1974) fo chaaceizaion of pocesses as naow sense linea, geneal linea, an nonlinea. 9 he effec of incluing cash ouflows, such as iviens an inees paymens, as he faco γ is 7

9 his couple sysem of SDEs can be solve explicily an he soluion is given by whee V g e V = an g g ( e) g ( an ) = + (1.4) θ g m e c c ( ) c e = + ( 1 ), θ 1 m = σ c v (1.5) an ( ) = ( ) + ( ) ( ) + ( ) ( ). (1.6) g an v s s v s W v s W s 1 1, s 3 (Refe o Lakshmivaahan, Qian, an Sock (8) fo eails). Hee W1 ( ) an ( ) wo inepenen Wiene pocesses an ( s) ( ), ( ), ( ) 1 v s = ρσ v s = σ e v s = ρ σ. (1.7) 1 v 3 v W ae Since each of he Io inegals on he RHS of (1.6) ae maingales, hei sum (Kaazas an Sheve (1991) an Shiyaev (1999) ) can be equivalenly epesene by he ime change of a sana Wiene pocess B() as whee ( ) ( ) σ ( ) g an = B (1.8) ( ) = ( ) + ( ) ( ) + ( ) σ v s s v s s v s s 1 3 V V = ( γ ) + σ ( ) W. v v his can be aken ino accoun by eplacing 1 1 σv wih ( γ + σv ) in all he expessions ha follow. 8

10 σ c ρσ vσ c = σ v + 1 e 3 ( c c 1) 1 e ( c 1) 4c c. (1.9) µ = g, Consequenly, wih ( ) ( e) ( ) ( ) ( ) g = µ + B σ (1.1) an V V has a lognomal isibuion given by V P V 1 V ln µ V ( ) exp = V σ ( ) πσ ( ) V. (1.11) Defaul is efine by he occuence of he even whee he pobabiliy of efaul can be given as (, ) { min s } B K D = V < K o V < D (1.1) s ( ) ( ) P = P ob B K, D 1 P ob V K an minvs D = > > s (1.13) As in Giescke (4), Bockman an ule (3), Reisz an Pelich (7) an ohes, efaul occus in wo ways. Fis, in he classical case of Meon (1974), efaul occus when he value of he fim falls below he face value of he eb ( K ) a ime. We assume K = 1. Aiionally, efaul occus befoe mauiy, <, when he value of he fim falls below a baie level, D. ha is, fequenly ceios have a igh o pull he plug on he fim uing financial isess. (Noe ha he Longsaff an Schwaz (1995) efaul is 9

11 pemauiy only.) Vaious covenans in bank loans, bons, an ohe eb may conain a igh o effecively pull he plug. Similaly, bank egulaos may close a financial insiuion when equiy is below a ceain level of asses. See Coe an Schan (1999) fo analysis of his ype of covenan. Fo ou puposes, we assume, as given in Giescke (4) an assume by Bockman an ule (3), ha D is below K. hus, P is he pobabiliy he fis passage is befoe o a. Moe fomally, fis passage o efaul is given as τ = min ( τ 1, τ ) whee τ is fis passage o baie D an 1 τ is a mauiy,, if V 1, as given in Giesecke (4), fo illusaion. 1 < K. Please see Figue ( ) Refeing o he fis pa of Appenix A, i uns ou ha he close fom expession fo P is available only fo he special case when ( ) g = a + bb (1.14) whee a an b ae consans an B ( ) is he sana Wiene pocess. See Ellio an Kopp (1999) an Giesecke (4). Since (1.1) is no of he fom (1.14), in he following we seek goo linea appoximaions o µ ( ) an ( ) σ. Refeing o he secon pa of Appenix A, i can be veifie ha ( ) ( ) = m1 an σ ( ) σ ( ) µ µ = m (1.15) 1 Black an Cox (1976) wee he fis o sugges baies bu hey assume was consan. 1

12 Whee m 1 = m1 ( α ) an m = ( ) Subsiuing (1.15) in (1.1) we ge m α ae consans ha epen on he eal paamee α. 1 ( ) g = m + m B (1.16) Fo his case, following Ellio an Kopp (1999) an Giesecke (4) we eaily obain he expession fo he efaul pobabiliy. P ( ) whee K D ln m1 m1 ln + m1 V m D KV = Φ + Φ m V m (1.17) ( x) 1 x z exp π Φ = z (1.18) he fis em on he igh han sie is classical efaul ue o asses being less han K (face value) a mauiy an he secon is efaul ue o beach of baie. A sample plo of he vaiaion P ( ) vesus is given in Figue. Appenix B gives expessions (eivaives) fo he sensiiviy of efaul o vaious paamees. he fla baie we use is simila o Longsaff an Schwaz (1995). As hey o, we noe ha he baie can alenaively be given as ime vaying wih D equal o K a mauiy. Alenaive esuls using a ime vaying baie (no shown hee) ae qualiaively unchange fom hose epoe below. 11

13 . Risky Spo Raes an Cei Speas As Depenen Upon Fis Passage Pobabiliy of Defaul.a. Deivaion of Expessions fo Spo Raes an Cei Speas Given isibuions log, V V an V V we now evelop compac expessions fo efaul isky spo aes, R ( ), an he spo ae spea ove a case wih no efaul isk, Rf ( ) ( ) ( ) is enoe as ( ) R R f level an volailiy of. Hee S. his combines he wo pocesses fo an V whee he V epesen efaul isk. Deb wih no efaul isk an mauiy has pesen value a of PVf ( ) = PAR. Fo a bon wih efaul isk, he pesen value a is ( = ) = ( ) ( ) + ( ) ( ) ( ) PV PVf 1 P PVf P RR ( P ). (.1) Hee, P ( ) is he fis ime (fis passage) he fim value his a level, K o D, whee efaul occus. 11 RR is he ecovey ae of pincipal in case he fim efauls an we expess his as a funcion of P. 11 Ou fis passage efaul pobabiliy shoul be isinguishe fom a haza ae which is a efaul inensiy measue ypically saing a an ening a +1. Cumulaive haza aes fom a seies of noneceasing aes fom zeo o a muli-peio hoizon. See Duffie, Saia an Wang (7) fo an example. 1

14 We can now eive he efaul isky bon pice a mauiy, PV ( ) efaul-fee bon pesen value, PV ( ) spea ae eive fom he efaul-fee spo ae R ( ) a mauiy, PV ( ). ( ) f, fom he, he pa value of he bon. he isky spo ae an f an he efaul isky bon pice PV is he pesen value of he bon which may efaul a. PV ( ) PV ( ) f ( ) ( ) ( ) Rf ue o he isk of efaul. o efine he isky spo ae, R ( ) which sells a ( ) PV = PV e (.), we easily consuc a efaul isky bon, PV a pesen, an is value a PAR a mauiy. hus, ( ) ( ) R f ( ) PV e = PAR = PV. (.3) Afe subsiuing, ( ) hen, iviing boh sies by PV ( ) ( ) ( ) ( ) R R f PV e e PAR PV an simplifying we obain = =. (.4) f 1 1 R ( ) = ln + R f 1 1 RR ( P ( )) P ( ) ( ) (.5) an 13

15 1 1 S ( ) = R ( ) Rf ( ) = ln 1 1 RR P P ( ( )) ( ) (.6) As long as ( ) ( ) 1 RR P, P > an ( ) 1-[1-RR (P )] P <1 (.7) which implies ha R ( ) R ( ) >. f Alman, Bay,Resi, an Sioni (5) an Alman, Resi, an Sioni (5) have moele he ecovey ae as epenen upon efaul pobabiliy an foun a negaive elaion. he logic is ha when efaul aes ae high, he economy is ypically weak an he value of asses is elaively epesse compae o cases whee he economy is songe. We use he below funcional fom ha Alman, Resi, an Sioni (5) foun o be he bes fi. ( ( )) = ( ( )) RR P a P, (.8) RR b RR whee a RR =.1457 an b RR =.81..b. Compuaions of Risky Spo Raes an Cei Speas Given he above sysem, we may now analyze P, R ( ), an S ( ) em sucues as epenen upon vaious eemining paamees whee we sess he impac of he em sucue of P upon R ( ) an S ( ). Ou examinaion has an analyical avanage compae o 14

16 pio eseach in ha we can eaily compue expece values an vaiances fo a wo faco moel fom he above isibuion fo V. Assuming he popula Vasicek (1977) moel of sho em inees aes, we can vey easily plo E ( V ), Va ( V ), an P fo any paamees of a paicula inees ae pocess. hen we analyze how paamee values such as he efaul baie an complex ineacions icae vaious levels an shapes of P, R ( ), an S ( ). Again, E ( V ) values ha gow fase wih ime obviously en o euce efaul isk because, in he gea majoiy of cases, he value of he fim is assume o if upwa wih he passing of ime. In he above E ( V ) expessions, he gowh ae in fim value is obviously affece by he level of inees aes (highe inees aes, highe V gowh) an he pocess in he alenaive inees ae moels. Howeve, as E ( V ) gows wih ime, he vaiance of V also inceases wih ime. ha is, hee is a ace beween he gowh of expece value an vaiance o eemine he pecise impac of on P. A lage ae of incease in pobabiliy of efaul ue o ime passage ens o incease he elaive slope of R ( ) an S ( ) even hough he slopes may be genly posiive o even negaive. Howeve, we noe ha he impac of a fis passage pobabiliy ha gows wih ime is complex. ha is, fis passage efaul pobabiliy may gow wih ime bu, S ( ) an R ( ) may o may no incease wih ime because he pesen value of he expece loss is iminishe wih geae ime. Also, he geae he level of inees aes, he moe pesen value is 15

17 iminishe. Wih espec o level an sucue of inees aes, we noe ha Kishnan,Richken, an homson (8) fin ha he shape of he iskless em sucue can help peic cei speas. In fac, hey sugges ha cuen cei speas an isk fee yiels cuves impoun pacically all necessay infomaion fo peicing cei speas. I is quie ineesing o emonsae ha he pobabiliy of efaul may no necessaily incease wih ime. Le us consie how he isibuion of V explains P. Figue 3 is one example of how ou analysis pemis eaile analysis of efaul speas. Hee we assume V = 15, c =.3, θ =.18, =.6, σ =., level of inees aes. 1 his is a fla ( ) f σ =., D = 6, an no coelaion beween fim value an v R em sucue as θ / c =. Expece value of he fim ises wih mauiy whee i is 498 a a mauiy of. Va ( V ), in he secon panel, inceases o abou 37, a mauiy. P always gows wih mauiy in his case which is shown in he hi panel. he fouh panel isplays he eivaive of P wih espec o which, in his case, is always posiive alhough i consisenly eclines. he las wo panels isplay R ( ) an S ( ). Noe ha R ( ) peaks a aoun =3 which is in conas o he fla Rf ( ). If R f is fla, any R hump hee is oally ue o efaul isk an he S shape. hus, even hough P consisenly inceases (no hump) beyon =3, is impac on he spo 1 Diffeen suies have foun a wie vaiaion in esimaes of σ an ohe paamees nee fo compuaion. We use vaious paamee values fo σ, θ an c esimae an use by Piske (1998), Zeyune an Gupa (7) an Ai-Sahalia (1996). We use σ values esimae by Paino, Poeshman, an Weisbach (5). Appenices show v ou esuls ae obus o a wie ange of paamee values. 16

18 ae weakens in his case because he pesen value of he expece loss eclines wih ime. S ( ) also peaks a aoun =3 an hen eclines. Figue 4 is anohe se of gaphs epicing efaul speas whee θ is now.3. hus, θ / c is.1 an geae han (.6) such ha he R ( ) em sucue is posiive. Hee expece value of he fim gows much moe apily han above bu, a he same ime, so oes vaiance aoun expece value. Fo his paicula case, he ne effec is o geneally euce pobabiliy of efaul elaive o he pevious figue. Ineesingly, he pobabiliy of efaul peaks an hen falls an hus is eivaive becomes negaive. Fo his case, R ( ) inceases bu S ( ) peaks a lile bi ealie han ou pevious figue. he ealie S ( ) always peak an subsequen seep ecline is ue o he hump in P. We examine he funcional elaionship beween S an P in moe eail in a lae secion. he P hump is couneinuiive a fis glance bu geae E ( V ) gowh may ominae he incease in vaiance so ha efaul eclines wih. We explain he shape of P em sucues in moe eail below. I is consucive o also consie gaphs of he lognomal isibuion of V an V log. V See Figue 5, panels a, b, c, an, whee he isibuions fo iffeen mauiies ae given again assuming he iniial value ( V ) is 15 an face value of eb is 1. Pesen value of eb will be much lowe han 1 fo long mauiies. Panels a, b, an c use a ρ of zeo. Hee, we again use a fla R ( ) em sucue as θ/c equals alhough he level is now 1%. Fo he momen, we 17

19 assume only he classic (mauiy) efaul case whee he baie (D) is zeo fo his illusaion. In fac, he ole of alenaive (pe-mauiy) efaul baies canno be isplaye in gaphs of isibuions of V an V log. ha is, isibuions only aess value a ime. A ime, V if V is less han 1, efaul occus as illusae by he aea o he lef of he veical line. One may compae he classic efaul aeas fo he iffeen mauiies o eemine if classic mauiy efaul pobabiliy inceases o eceases wih he assume iscee mauiies. ha is, in Figue 5a we see ha he aea fo V less han 1 (efaul) if = is less han fo =, 5, an 1, hus yieling a hump. he maximum fo hese cases is a = 5. A shocoming of Figue 5a is ha, given he amaic asymmey, one canno see how E ( V ) behaves wih espec o. hus we give Figue 5b fo V log V whee, given a symmeic isibuion, one can see ha he expece value of he isibuion inceases wih. he incease may seem small bu ealize ha a small incease in V log V epesens a lage incease in V. Figue 5c is he humpe P cuve associae wih he 5a an b. Finally, Figue 5, using a ρ of.5, shows simila behavio. We again noe ha Figue 5 oes no consie baie efaul bu sill illusaes behavio obus o lae analysis incluing D values. he figues of his secion have use example paamee values ha convenienly buil upon ou moels fo V an an, also, en o mos easily illusae shapes an popeies we 18

20 wish o sess. We have foun hese qualiaive esuls obus o alenaive paamees whee Appenix C gives some sample alenaive paamee esuls. 3. Paamee Values ha Deemine Shape of P he above figues fo V isibuions an P ae a sa a unesaning he geneal heoy of P shape. We now moe sysemaically analyze how paamees ineac o ceae a hump as i may appea ineesing an couneinuiive o many. Ou analysis shows ha he hump can appea o isappea by, fo example, vaying D, he slope of he em sucue ( R f ), σ v, an leveage. Fo moe analyics on he sensiiviy of P o iffeen paamees, please see Appenix B. Sensiiviy o Baies (D) Baie values, unlike mos ohe paamees, can be viewe as a ecision vaiable ha boowes an lenes negoiae a issuance. Bockman an ule (3) sugges he mos common baies ae covenans such as eb/equiy an imes inees eane equiemens. Fuhemoe, hey conen ha beaching any baie can igge eb ecall, efaul, o bankupcy whee moe han one can occu simulaneously. If lenes ae incline o have lile paience fo a fim wih low cei qualiy, hey may negoiae a high value fo D wheeas ohe lenes may be much moe olean an hus allow a lowe D. Lenes may equie a elaively low D, if fo example, asses o be claime in efaul ae angible an en o be elaively liqui. 19

21 In he yeas pio o he cei cisis of 7-8, a en owa weake covenan poecion was one example of boowe fienly, libeal lening ems ha may have le o an oveleveage financial sysem. One of he ousaning examples of libeal lening ems was he enency fo many leveage loans o be covenan lie whee he eb ha weak o no covenans. One explanaion is ha bank eman o synicae loans was so high hey wee willing o agee o any covenan sucue. 13 As anohe example of libeal lening ems, oggle high yiel bons became popula whee, along wih weak covenan poecion, he issue coul choose o pay an inees in he fom of newly issue bons. he effec of issuing eb wih weak covenan poecion ealie in he cenuy has le o angeously misleaing low efaul aes ealy in he cei cisis. Moe specifically, weak covenan poecion means ha efaul baies ae vey low hus pemiing boowes o coninue opeaing even if aios such as imes inees eane ae vey low. Fich has epoe ha efauls in 6, 7 an ealy 8 wee hus vey low even hough cei qualiy may have gealy eeioae. Because low efauls (misleaingly) sugges low isk an oleance fo isk, moe aggessive eb issuance is encouage. In he long em, weak covenan poecion may well no euce efaul bu meely shif (vey high) efaul o lae aes. Defaul aes ae even moe misleaing low if, as is ofen suspece, weak covenans euce he abiliy of 13 See A. Deniz, Sponsos Will No Allow he Feae Raf of Loan Defauls, Financial imes, July 16, 8.

22 lenes o esic managemen fom uneaking iskie pojecs (oveinvesmen) eflece in a shif owa geae σ v a couple yeas afe eb issuance. 14 Delaye efauls coul be ba fo analyss an invesos who may have peice he wos of he cei cisis was ove afe iniial cei shocks woke hough he sysem wihin a ypical peio of ime. 15 Raings agencies such as Fich an Sana an Poo s now ecognize such covenan isk an ajus hei aings fo he egee of covenan poecion. Bockman an ule (3) esimae implie D (as oppose o moe naowly efine explici covenan D) values fo numeous inusies an fin a vey wie vaiaion. Reisz an Pelich (7) sugges impovemens o B baie esimaes. We accep he concep ha hee ae boh explici (covenans) an implici baies a which lenes effecively pull he plug on boowes an we hus assume vaious D values esimae fom pevious suies See Fich Raings, Cei Make Reseach U.S. Leveage Loan Covenan Decline Acceleaing in 7, an Dealbeake he Mysey of Low Defauls fo Leveage Loans, Januay 1, In one conas o loans wih weak covenans, Reues ( Golman Sees Fase Pickup in US Junk Bon Defauls, June 3, 8, noes ha many bank loans ha easonably song covenans he effecively foce home builes o liquiae lan an invenoy o foce covenan compliance. 16 Of couse, if D is beache an echnical efaul occus, hen lenes may pemi subsequen eoganizaion of he fim o, alenaively, liquiaion may occu. his choice is beyon he scope of ou eseach. See, fo example, Davyenko an Sebulaev (7) fo analysis of how enegoiaion poenial can affec cei speas. Also, see Bown, Ciochei, an Riough (6) fo how financial isess may be esolve. Uhig-Hombug (5) escibes coniions fo bankupcy as i vaies acoss iffeen counies. ha is, in he Unie Saes, filing fo bankupcy is pemie une boae coniions (insolvency no equie) han in Gemany an Canaa. Davyenko an Fanks (8) fin iffeences in ceio ighs acoss iffeen counies whee such iffeences coul affec baie values. 1

23 Ou above figues fo canno fully illusae fis passage efaul a V isibuions wee useful o illusae some basic issues bu < when a ealisic D is impose. he opeaive saemen fo efaul is: Wha is he pobabiliy ha V will no ecline o D befoe an, also, finish above boh D an K a? If gowh in V ominaes gowh in Va ( V ), hen P may ecline if he ominaion is song enough. 17 Of couse a highe D inceases P bu we pefe o focus on he shape of efaul em sucue insea of he level of P. Figue 6 helps analyze he siuaion by iviing P ino efaul pobabiliy ue o he sum of 1.) beaching baie D befoe mauiy an.) V < K a mauiy. We call hese, especively, baie efaul an (classic) mauiy efaul. o bes visually illusae he impoan elaionships, we now assume base paamees of =.6, θ =.3, c =.3, σ =., σ =., ρ =, V = 15, K = 1 whee we noe ha his suggess an upwa v if in inees aes an posiive ( ) f R as θ/c is.1 an =.6. In some cases, a lowe D ha is less likely o be beache shinks baie efaul o zeo an simply makes P he classic Meon (1974) case. A geae D aises boh baie an oal efaul pobabiliy. D values in he fou iffeen panels ae 5, 6, 7 an 9. As inceases, baie efaul always inceases as moe cumulaive poenial beaching evens occu. Figue 6, panel a, D = 5, shows no significan poenial fo beaching he baie unil abou = 1. In conas, panel (D = 9) shows consieable baie efaul poenial much ealie, a =. As D inceases, baie 17 Hee we again assume K > D.

24 efaul ens o supass classic mauiy efaul a some mauiy. his figue also shows ha as D inceases, he hump of classic mauiy efaul ens o be neualize by baie efaul such ha he (oal) P hump isappeas in he las wo panels. Fo moe eails on he inees ae sysem fo assume paamees, he las panels of his figue inclue spo aes, consoliae P plos, an P eivaives wih espec o an D. A hump in cei spea occus fo all D values in he las panel. he impac of vaying D values can be sensiive o he assume paamees. In Figue 7 we change θ o.18 such ha hee is no expece if in aes ( R ( ) f is fla) because θ/c is.6 an equal o. Now he (oal) P hump is gone alhough he classic mauiy hump emains. Also noe ha P is much highe han he pevious se of figues whee his is likely because he if em in V is lowe. Unlike he pevious figue, baie efaul excees classic mauiy efaul only fo long mauiies in panel. Fo moe eails on he inees ae sysem fo he assume paamees, he las panels of his figue inclue spo aes, consoliae P plos, an he P eivaives wih espec o an D. Sensiiviy of P o σ v, Leveage, an σ Nex we analyze he impac of σ on he shape of P v fo a posiive em sucue as use in Figue 6. In Figue 8 we incease σ o.5 (fom.) an again use inceasing D v values in he fis fou panels. Now he (oal) P hump isappeas. In each panel, P is 3

25 highe han in he ealie coesponing figue. Relae o hump isappeaance, baie efaul becomes geae han classic mauiy efaul a = in panel b an a even ealie mauiies in lae panels. he las hee panels povie moe eails abou inees aes fo he assume paamees. Figue 9 shows how he hump vaies fo iffeen levels of leveage as given by V. If V is only 11, he P cuve is fla fo a sho while an hen uns seeply negaive. As V inceases up o 18, he hump occus bu ens o become moe genle fo geae V. In Figue 1 we noe ha, given ha he if of he V pocess is epenen upon, he basic elaion is ha geae volailiy, σ, in inees aes will incease volailiy of V an hus incease he pobabiliy of efaul, speas an R values. Howeve, we noe ha non-zeo coelaion in V an pocesses can eihe enhance o evese his effec. he moe posiive ρ is, he moe volailiy in he V pocess an he geae P. Howeve, if ρ is negaive, he elaion coul be evese. Figue 1 epesens he ρ = -.5, an +.5 cases an is consisen wih he above. In hese cases, all P cuves ae humpe. When ρ is zeo, geae σ values aise P bu he elaion is evese when ρ is negaive. Fuhemoe, he P spea beween iffeen σ values is much lage fo posiive ρ compae o ρ=. Appenix B shows he eivaive of P wih espec o σ. 4

26 Sensiiviy of P o ρ he sign an sengh of he coelaion, ρ, beween he an V pocesses can have a vey lage impac on ceain fims an hei P shapes. Pime examples ae fims in he banking, eal esae, an consucion inusies. Low inees aes fom (appoximaely) 1-6 fosee vey song gowh an pefomance in hese inusies bu subsequen changes in inees aes, along wih he ecen cei cisis, have ofen le o moe ecen isasous pefomance fo some fims in hese inusies. 18 Highe ρ values geneae geae levels of P because he vaiance of V epens on he coelaion beween euns on V an changes. If he coelaion is posiive (negaive), he coelaion inceases (eceases) he vaiance of V an inceases (eceases) he level of P. 19 We can escibe he elaion in moe eail wih he below. Le W 1, an W, be wo inepenen sana Wiene incemen pocesses. he wo coelae pocesses W v, an W, can be expesse as linea funcions of W 1, an W, given by W, = W1,, W = ρ W + 1 ρ W. If ρ is -1, he sochasic pa of he V pocess (befoe scaling v, 1,, fo iffeing volailiies) is oally offse by he sochasic pa of he pocess. On ohe han, 18 One ousaning example of a lage negaive ρ is Nohen Rock of he UK whee he bank faile o hege agains inees ae inceases. As hei cos of boowing ose, he value of he bank plummee. See he Weck of Nohen Rock, Bloombeg Makes, May 8 by Richa omlinson an Ben Livesey. 19 Longsaff an Schwaz (1995) escibe spea behavio similaly. 5

27 if ρ is 1, he sochasic pas of he wo pocesses ae equal an any volailiy of one pocess is enhance by he ohe. Appenix B povies moe eails on he elaion beween ρ an P. Figue 11 isplays cases whee ρ vaies fom songly negaive o songly posiive fo iffeen levels of σ v. In panel a, ρ = -.5, he level of P is lowe han panels b, ρ =, an c, ρ= +.5. In all hee panels, noe ha he P shape is posiive fo he highes σ v (.3), essenially fla fo he secon highes σ v (.5) if mauiy excees 5, an humpe fo all lowe levels of σ v. hus, ρ has a song effec on boh level an shape of P he figues of his secion have use example paamee values ha convenienly buil upon ou moels fo V an an, also, en o bes illusae shapes an popeies we wish o sess. We have foun he esuls obus o alenaive paamees whee Appenix D gives some sample alenaive paamee esuls. 4. Physical pobabiliies We have shown analysis of he em sucue of fis passage efaul in a wo faco moel whee we sugges he mos ineesing aspecs ae he sensiiviy of efaul o baies an he hump in he P em sucue. Reseaches moeling efaul shoul fin hese esuls useful. Also, financial analyss eveloping invesmen saegies shoul noe ou finings. Howeve, some may sugges ha he usefulness is limie in ha ou P values ae isk- 6

28 neual pobabiliies, no physical pobabiliies. We espon in wo ways. Fis, Chou an Wang (6) mainain ha isk neual pobabiliies sill povie a ceible anking of fims accoing o suscepibiliy o efaul. Secon, we ansfom ou isk-neual pobabiliies o physical as given below. he pocess o obain physical pobabiliy of efaul is o muliply make pice of isk by σ an a his pouc o he if em. Reisz an Pelich (7) use.15 fo he make pice v of isk ( λ ). Fo compleeness, we also use lowe an highe values an compae o he isk neual (zeo λ ) case. he esuls ae in Figue 1 whee panel a is he isk-neual case an panels b, c, epesen λ s of.75,.15 an.3, especively. In each panel we plo cuves fo iffeen values of σ an show ha geae v σ values aise P v. As expece, P values ecline wih geae λ bu he shape of P is obus o iffeen λ. ha is, if a hump occue in he isk neual case, i also occus in he physical pobabiliy. 5. Cei spea slope as a funcion of P slope he above analysis of efaul pobabiliy can be use o evelop invesmen an heging saegies. Fuhemoe, egulaos can use ou analysis o evelop policy. In his conex we noe ha many economiss an egulaos favo manaoy bank issuance of Reisz an Pelich (7) ge hei esimae (λ ) fom Huang an Huang (3). he compuaion is fim gowh ae in excess of isk fee ae ivie by volailiy of asse value. 7

29 suboinae eb as a way o enhance make iscipline of banks. In fac, he Gamm- Leach-Bliley Ac of 1999 equies lage banks o have a leas one suboinae eb issue ousaning a all imes. Kishnan,Richken, an homson (6) fin ha ha he shape of cei spea em sucues may vey well help peic bank isk whee he basic iea is ha cei spea slopes can peic fowa speas an isk. A song posiive slope ens o successfully peic geae fowa cei speas an geae isk. Cei spea slopes can compue fo iffeen iscee inevals whee, fo example, one may use he hee yea spea less he one yea o esimae slope. Alenaively, as in Kishnan,Richken,an homson (6), one coul use he seven yea spea less he hee yea spea. If a hump in cei spea occus beween he mauiies use fo compuaion, say a =, he esimae slope beween one an hee coul be fla bu his may be a misleaing epesenaion of fuue isk. In ealiy, fo he hump case, he slope woul be posiive befoe = an negaive afe. Regulaos may inepe a zeo slope as suggesing no nee fo acion (no nee fo aggessive U. S. easuy puchase of 8

30 low gae bank asses a high pices) wheeas he ealiy is ha acion may be neee wihin a one yea hoizon. We now explicily expess cei spea slope as a funcion of efaul pobabiliy slope. Ou ealie expession fo cei spea may be ewien as whee 1 S = R ( ) Rf ( ) = S 1 (5.1) ( ( ) ) ( ) S1 = ln 1 1 RR P P. hus he spea is S 1 (always posiive) scale by 1/. Because S 1 is epenen upon hough P, S 1 S1 P ( ) = P ( ) (5.). Also, (RR) (1 RR) P S1 P = P (1 RR) P (5.3) 9

31 whee RR = RR(P ) fo simpliciy. Since RR given in (.8) is less han one an is also negaively elae o P, he eivaive of S 1 wih espec o P is clealy always posiive. his also hols fo a consan RR. Fuhemoe, he slope of he cei spea em sucue is S 1 S 1 1 = S 1 (5.4). Subsiuing, we obain an alenaive expession fo cei spea slope as epenen upon he slope of P. S 1 P ( ) S1 1 = P ( ) S 1 (5.5) P can be foun in Appenix B. As he eivaive of S 1 wih espec o P is always posiive, he only sign inefinie em is P /. he las em, (1/ ) (S 1 ), always euces he slope. Consie he impac of P /. If his eivaive is posiive an an song, as ofen seen in ou figues fo sho mauiies, he slope of S ens o be posiive. As inceases, P / ens o eihe flaen o even change sign. If i meely flaens, he 3

32 slope of S may o may no emain posiive epening he magniue of changes elaive o (1/ )S 1. If P / becomes negaive, he slope of S will clealy become negaive an may well become songly negaive. We may apply he above o numeous figues. Fis, consie Figue 3 whee hee is no hump in pobabiliy. Noneheless, he spea has a hump ue o he magniue of (1/ ) S 1 evenually becoming ominan. Figue 6 conains P cuves ha have a hump, say *, an ohes ha o no. he hump in cei speas fo siuaions associae wih humpe P cuves is ue o boh P humps an (1/ ) S 1. Given ha locaion of he S hump can be impoan, we analyze how he locaion of he S hump may change wih paamee values. ha is, assuming eveyhing else fixe, oes he mauiy a which he spea hump occus, **, incease o ecease wih vaious paamees? Figue 13 shows ha ** vaies wih D. Fo D values beween 5 an 7, ** genly inceases an hen inceases much moe apily unil D is abou 9 a which poin i eclines. Ohe figues, no shown, eveal ha ** is negaively elae o σ, σ v, an ρ bu posiively elae o V o. 31

33 6. Conclusion Song gowh in cei eivaives an he ecen cei cises have boh incease he eman fo impove moels of cei isk. We noe ha popula sucual moels commonly uilize simulaneous pocesses fo V an sho em inees aes,, whee changes in V ae epenen upon he paicula pocess assume. Having he isibuion fo V pemis us o moe easily analyze fis passage efaul pobabiliies an em sucues of cei isk in eail. o ou knowlege, no one else has analyze fis passage pobabiliies fo boh mauiy an baie efaul wih a wo faco moel. Geae passage of ime suggess geae expece fim value bu, a he same ime, geae vaiance of V so ha he impac of mauiy on efaul pobabiliy is complex. he behavio of efaul pobabiliy wih passage of ime has a clea impac on em sucue of cei speas whee a song gowh in pobabiliy of efaul inceases he slope of he em sucue of cei speas. I is ineesing o noe ha pobabiliy of efaul may isplay a hump wih espec o mauiy. his esul is hols fo many combinaions of paamees escibing he simulaneous V an pocesses. We fin ha he hump can be especially clea fo smalle baie values, smalle volailiy of inees aes, lage volailiy in V, an geae leveage. he impac of smalle baies is especially ineesing ue o he ecen en owa weake covenan poecion in many ecen eb issues. Weake covenan poecion euces he nea em pobabiliy of efaul bu many fea ha weake covenans meely elay efauls. he applicabiliy of he eseach is enhance by he fac ha he hump appeas fo boh isk neual 3

34 pobabiliies an convesions o physical pobabiliies. We povie explici expessions fo how cei spea slope is a funcion of efaul pobabiliy slope an sugges ha, given ou esuls, egulaos can evelop impove peicions of cei isk. 33

35 Appenix A his appenix has wo pas. In he fis pa we show ha fo he pocess g of he ype (1.1), he sana meho fo compuing he efaul pobabiliy oes no lea o a close fom soluion. Base on his, in pa wo we eive he bes linea appoximaion o µ ( ) an σ ( ) using a close fom expession fo he efaul pobabiliy is obaine. he analysis in he main boy of his pape is base on he lae close fom soluion. Nee fo appoximaion Since V g = e, fom (1.13) i follows ha ( ) P 1 P log min log = ob g > K an g s > D (A.1) s Clealy, compuing he secon em on he igh han sie of (A.1) involves compuing he join ensiy of g an min g s s whee g is given by (1.1). By way of simplifying noaion, le η 1 ( ) = σ ( ). Since η ( ) is an inceasing funcion, seing η ( ) = τ, we ge η ( τ ) ( ) τ = η when. Hence (1.1) becomes g 1 1 ( ( )) ( ) ( ) ( ) B ( ) = whee η τ = µ η τ + τ (A.) ( ) 1 1 Defining X ( τ ) = g η ( τ ) an λ ( τ ) µ η ( τ ) =, (A.) becomes, fo all τ, 34

36 X ( τ ) λ( τ ) B( τ ) = + (A.3) Define whee λ ( τ ) ( ) τ 1 τ Λ τ = exp λ ( s) B ( s) λ ( s) s (A.4) λ τ =. Define a new measue P λ as τ P λ = Λ P (A.5) τ hen, by Gisanov heoem, X ( τ ) is a sana Wiene pocess an he join isibuion of X an X s min s is given by F b c E I X b X c s λ (,, ) = Λ { <, min s < } (A.6) whee b an c an eal consans, I {A} is he sana inicao funcion, an E λ is he expecaion wih espec o he new measue P λ. Explici compuaion of (A.6) involves he knowlege of he join isibuion of λ ( s) B ( s), X an X s. o ou knowlege his min s isibuion is no known. 35

37 If λ ( s) = a is a consan howeve, hen λ ( s ) B ( s ) isibuion is known (Giesecke (4), Elliol an Kopp (1999)). _ = ab an in his case he join Founaely, fo he case when he an η ( ) σ ( ) pocess follows he Vasicek moel, he ( ) µ in (1.5) = in (1.9) ae he sum of a linea an a small nonlinea em. So, we coul effecively appoximae µ ( ) an ( ) σ by linea funcion. Bes linea appoximaions o µ ( ) an ( ) σ : In his secon pa we seek bes (in he sense of leas squaes) linea appoximaions o µ ( ) an σ ( ). o his en, using (1.4) (1.1), ewie µ ( ) an ( ) ( ) m e ( ) µ σ as µ = + (A.7) an σ ( ) σ e ( ) v = + (A.8) σ whee θ 1 m = σ c v, 1 θ eµ e c c c ( ) = ( 1 ) (A.9) 36

38 an σ ρσ σ 3 c v c ( ) = 1 ( + + 1) + 1 ( + 1) e e c c e c σ 4c c. (A.1) Fom (A.7) (A.1) i eaily follow ha boh µ ( ) an ( ) linea em an a nonlinea em. While boh µ ( ) an ( ) ( ) ( ) σ ae expesse as a sum of a σ ae efine fo all wih µ = = σ, fo puposes of applicaion in he main boy of his pape, we seek he bes linea appoximaions µ ( ) an σ ( ) fo µ ( ) an ( ) σ especively, fo in a finie ange, say α <. ypically, he value of α is icae by he maximum mauiy of bons, say α = 3. minimize Fo he bes linea appoximaion o µ ( ), le ( ) 1 µ = m. hen we seek m 1 ha α ( ) µ ( ) ( ) f1 m1 = m 1 (A.11) By ouine compuaions i can be veifie ha he minimizing m 1 is given by 3 θ 1 cα m1 ( α ) = m c α + e ( cα + 1) 1 c α c (A.1) whee m ( ) 1 α m as α. 37

39 A plo of m1 ( α ) if given in Figue A.1 an an illusaion of he bes linea appoximaion of µ ( ) by µ ( ) ( α) = m fo is given in Figue A m 1 (α) α Figue A.1 Plo of m1 ( α ) vs. α. =.6, θ =.3,.3 c =, σ =., σ =., ρ =.5 v µ() an µ() µ() µ() Figue A. An illusaion of he bes linea appoximaion µ ( ) ( α) m1 ( α ) = fo α = 3. =.6, θ =.3,.3 = m whee c =, σ =., σ =., ρ =.5 v 1 38

40 minimizes Fo he bes linea appoximaion o σ ( ), le ( ) σ = m. We seek m ha ( ) α ( ) σ ( ) f m = m (A.13) I can be veifie ha he minimizing m is given by 3σ ( ) cα m ( 3 3 α = σ v + c α e c α 5c α 6cα ) 3 8c α ρσ σ v 1 cα + c α e 4 3 ( c α 3cα 3) 3 c α (A.14) m α σ an ( ) v as α. A plo of m ( α ) vs. α is given in Figue A.3. An illusaion of he bes appoximaion of σ ( ) by m ( ) α fo α 3 is given in Figue A.4. 39

41 m (α) α Figue A.3 Plo of m ( α ) vs. α. =.6, θ =.3,.3 c =, σ =., σ =., ρ =.5 v σ () an σ () σ () σ () σ = whee Figue A.4 An illusaion of he bes linea appoximaion ( ) m ( ) m ( α ) =.4 fo α = 3. =.6, θ =.3,.3 c =, σ =., σ =., ρ =.5 v 4

42 APPENDIX B Sensiiviy of Defaul Pobabiliy o Paamees In his appenix we eive explici expessions fo he sensiiviy of he efaul pobabiliy P ( ) in (1.17) wih espec o vaious paamees. o simplify ou analysis efine an K ln m1 V f1 = f1 ( K, V, m1, m, σ v, ) =, (B.1) m m1 f = f ( m1, m, σ v ) =, (B.) m D ln + m1 KV 3 = 3 (,,, 1,, σ v, ) = (B.3) f f K V D m m whee m1 = m1 ( α ) an m m ( α ) hen, P ( ) in (1.17) akes he fom m = ae given by (A.6) an (A.8) in Appenix A, especively. f D P K D f f (,, ) = Φ ( ) + Φ ( ) 1 3 V (B.4) whee, ecall fom (1.18) ha, f ( f ) φ ( ) Φ = z z (B.5) an φ ( z) is he sana nomal ensiy. Hence, if f is a funcion of x, hen ( ) Φ f = φ ( f ) f x x. (B.6) Fom (B.4) an (B.6), we eaily obain ha 41

43 ( ) f f P f D f f D φ = + + Φ x x V x x V 3 ( f ) 1 φ ( f ) ( f ) log ( f ) (B.7) he paial eivaives of f 1, f, an f 3 in (B.1) - (B.3) wih espec o m 1, m,, D, V an K ae given in able B.1. By combining (B.7) wih he appopiae enies in his able, we can obain sensiiviy of P wih espec o six paamees of inees. x m 1 m able B.1 Deivaive of f 1, f, f 3 in (B.1) - (B.3) f 1 x m K log V m 1 ( m ) 3 K log m 1 m1 V 3 m ( m ) D V K 1 V m K 1 m f x f 3 x m m m m 1 D log KV m ( m ) + m1 log 3 D 1 1 KV 3 ( m ) m D m V 1 m 1 K m + m Recall fom (A.6) an (A.8) in Appenix A ha m 1 an m in un epen on he paamees -, c, θ, σ v, σ an ρ, of he moel. In able B., we povie expessions fo 4

44 he eivaive of m 1 an m wih espec o hese six paamees. By combining (B.7) wih enies in able B.1 an B., we can eive expession fo he sensiiviy of P ( ) wih espec o hese paamees. hus, P P m1 P m = + y m y m y 1 (B.8) whee y { c θ σ σ ρ},,,,, v able B. Deivaive of m 1, m y y m c α cα c α e cα c c α + + cα c α e ( cα ) θ ( ) c σ v σ ρ θ 3θ 1 cα + c α e 5 3 ( cα 1) 1 c c α θ + cα 3 3 cα cα e c α c 9 θ 1 cα 4 3 c α + e ( cα + 1) 1 c α c σv y m 15σ cα 3 3 c α + e 6 3 ( c α + 5c α + 6cα + 3) 3 8c α 3σ cα cα e 5 3 ( 4c α + 4c α + cα ) 8c α 4ρσ σ v 1 cα c α e 5 3 ( c α 3cα 3) 3 c α ρσ σ v cα 3 + cα e 4 3 ( cα + c α ) c α 6ρσ 1 ( ) c α 3σ cα 3 3 c α e 5 3 ( c α 5c α 6cα 3) 3 8c α ρσ 1 ( 3 3) 3 c α 6σ σ v 1 cα c α e 4 3 ( c α 3cα 3) 3 c α cα σ v + c α + e c α + 3cα v cα + c α + e c α + cα

45 Appenix C We pefome exensive esing of sensiiviy of behavio o iffeen paamee values an foun he esuls pesene in he ex ae obus. Fo example, he esuls ae obus o mean evesion, c, being.5 insea of.3. Please see examples below. Counepa o Figue 4c, c =.5 Counepa o Figue 5a, c= = P P obabiliy ensiy of V = 5.. = Mauiy = Value of fim V Counepa o Figue 5c, c= P Mauiy 44

46 Appenix D We pefome exensive esing of sensiiviy of behavio o iffeen paamee values an foun he esuls pesene in he ex ae obus. Fo example, he esuls ae obus o mean evesion, c, being.5 insea of.3. Also, esuls ae obus o iffeen σ values. Please see examples below. Counepa o Figue 6a, c=.5 Counepa o Figue 6, c=.5 a mauiy befoe mauiy oal a mauiy befoe mauiy oal.1.1 P P Counepa o Figue 7a, c=.5 Counepa o Figue 7, c=.5.5 a mauiy befoe mauiy oal.5 a mauiy befoe mauiy oal.. P.15 P

47 Refeences Achaya, V.V. an J.N. Capene () Copoae Bon Valuaion an Heging wih Sochasic Inees Rae an Enogenous Bankupcy. he Review of Financial Suies, Vol 15, pp Ai-Sahalia, Yacine (1996) esing Coninuous-ime Moels of he Spo Inees Rae he Review of Financial Suies, Vol. 9, No. (Summe, 1996), pp Alman, E.I. ; B. Bay; A. Resi; an A. Sioni (5), he Link beween Defaul an Recovey Raes: heoy,empiical Evience,an Implicaions Jounal of Business, Vol. 78, 3-7. Alman, E.I.;A. Resi; an A. Sioni, (5) Defaul Recovey Raes in Cei Risk Moeling: A Reviw of he Lieaue an Recen Evience, Jounal of Financial Lieaue, (Wine), Anol, L. (1974) Sochasic Diffeenial Equaions: heoy an Applicaions, John Wiley & Sons, New Yok. Bajlum, C. an P.. Lasen (7) "Accouning anspaency an he em Sucue of Cei Defaul Swap Speas", woking pape, Copenhagen Business School an Univesiy of Aahus. Bille, M..;.H.D. King; D.C. Maue (7) Gowh Oppouniies an he Choice of Leveage, Deb Mauiy an Covenans, Jounal of Finance, Vol.6 (Apil 7), pp Black, F. an J. Cox (1976) Valuing Copoae Secuiies: Some Effecs of Bon Inenue Povisions Jounal of Finance, Vol.31, pp Bockman, P., an H. ule (3) "A Baie Opion Famewok fo Copoae Secuiy Valuaion," Jounal of Financial Economics, Vol. 67, Bown, D. R. ; B. A. Ciochei, an. J. Riiough (6) heoy an Evience on he Resoluion of Financial Disess, Review of Financial Suies, 19(4): Chava, S. an M.R. Robes (8) How oes Financing Impac Invesmen? he Role of Deb Covenans Jounal of Finance, fohcoming. 46

48 Chou, H.C. an D. Wang (6) Pefomance of Defaul Risk Moels wih Baie Opion Famewok an Maximum Likelihoo Esimaion: Evience fom aiwan, woking pape, Ming Chuan Univesiy an Chung Yuan Chisian Univesiy. Chisensen, J. (8) he Copoae Bon Cei Spea Puzzle, Feeal Reseve Bank of San Fancisco Economic Lee, Numbe 8-1, Mach 14. Collin-Dufesne, P. an R. S. Golsein (1), Do Cei Speas Reflec Saionay Leveage Raios?, Jounal of Finance, Vol. 56, Collin-Dufesne, Piee, Robe S. Golsein, an J. Spence Main, (1) "he Deeminans of Cei Spea Changes", Jounal of Finance, Vol. 56, Coe, J. an C. Schan (1999) he effec of Accouning-base eb covenans on Equiy Valuaion Jounal of Accouning an Economics, Cox, J., J.Ingesoll an S.Ross (1985) A heoy of he em Sucue of Inees Raes, Economeica, Vol. 53, Davyenko, S. A. an I. A. Sebulaev (7) Saegic Acions an Cei Speas: An Empiical Invesigaion, Jounal of Finance, vol. 6, Davyenko, S. A. an J. R. Fanks (8) Do Bankupcy Coes Mae? A Suy of Defauls in Fance, Gemany, an he U.K., Jounal of Finance, vol. 63, Dealbeake (8) he Mysey of Low Defauls fo Leveage Loans, Januay 1, Deniz, A. (8) Sponsos Will No Allow he Feae Raf of Loan Defauls, Financial imes, July 16. Diessen,J. (5) Is Defaul Risk Pice in Copoae Bons?, he Review of Financial Suies, 5, Vol. 18, No. 1. Duffie, D., L. Saia, an K. Wang (7), Muli-peio copoae efaul peicion wih sochasic covaiaes, Jounal of Financial Economics, Duffie, D. an K. Singleon (3) Cei Risk: Picing, Measuemen, an Managemen, Pinceon Univesiy Pess, Pinceon, New Jesey. 47

49 Ellio, R.J. an P.E. Kopp (1999), Mahemaics of Financial Makes, Spinge-Velag, New Yok. Eom, Y.H.; J. Helwege; an J. Huang (4) Sucual Moels of Copoae Bon Picing: An Empiical Analysis, Review of Financial Suies, Vol. 17, Fich Raings, Cei Make Reseach U.S. Leveage Loan Covenan Decline Acceleaing in 7, www. fichaings.com. Giesecke, K. (4) Cei Risk Moeling an Valuaion: An Inoucion Chape 16 in Cei Risk: Moels an Managemen by D. Shimko, Risk Books, Lonon. Huang, J. Z. an M. Huang (3) How much of he Copoae-easuy Spea is Due o Cei Risk?, Sanfo Univesiy woking pape. Kaazas, I. an S.E. Sheve (1991) Bownian Moion an Sochasic Calculus, Spinge- Velag, New Yok. Kishnan, CNV, PH Richken, an JB homson, (6) On Cei Spea Slopes an Peicing Bank Risk, Jounal of Money, Cei, an Banking, Vol. 38, Kishnan, CNV, PH Richken, an JB homson, (8) Peicing Cei Speas, woking pape, Case Wesen Reseve Univesiy. Lakshmivaahan,S., S. Qian, an D. Sock (8) he Disibuion of he Value of he Fim an Sochasic Inees Raes, woking pape, Univesiy of Oklahoma. Lano, D. an A. Moensen (5) Revisiing he Slope of he Cei Cuve, Jounal of Invesmen Managemen, Vol.3, 6-3. Jaow, R. A. ; D. Lano; an S.M. unbull (1997) A Makov Moel fo he em Sucue of Cei Risk Speas, Review of Financial Suies, Vol. 1, Jaow, R. A. an P. Poe (4) Sucual Vesus Reuce Fom Moels: A New Infomaion Base Pespecive Jounal of Invesmen Managemen, Vol., 1-1. Lelan, H. E. (1994) Copoae Deb Value, Bon Covenans an Opimal Capial Sucue, Jounal of Finance, Vol. 49, Lelan, H. E. an K. B. of (1996) Opimal Capial Sucue, Enogenous Bankupcy, an he em Sucue of Cei Speas, Jounal of Finance, Vol. 51,

50 Longsaff, F. A. an E. S. Schwaz (1995) A Simple Appoach o Valuing Risky Fixe an Floaing Rae Deb, Jounal of Finance, Vol. 5, Meon, R. C. (1974) On he Picing of Copoae Deb: he Risk Sucue of Inees Raes, Jounal of Finance, Vol. 9, Paino, R., an M. S. Weisbach, (1999) Measuing Invesmen Disoions Aising fom Sockhole-Bonhole Conflics, Jounal of Financial Economics, Vol. 53, 3-4. Paino, R., A. M. Poeshman, an M. S. Weisbach, (5) Measuing Invesmen Disoions When Risk-Avese Manages Decie Whehe o Uneake Risky Pojecs Financial Managemen, Vol. 34, Issue 1 (Sping 5) pp Piske, M. (1998) Nonpaameic ensiy esimaion an ess of coninuous ime inees ae moels Review of Financial Suies, Vol. 11, Reisz, A. S. an C. Pelich (7), A Make Base Famewok fo Bankupcy Peicion, Jounal of Financial Sabiliy, Vol. 3, Schaefe, Sephen an I.A. Sebulaev, (8) Sucual Moels of Cei Risk ae Useful: Evience fom Hege Raios on Copoae Bons, Jounal of Financial Economics, fohcoming. Shiyaev, A.N. (1999) Essenials of Sochasic Finance: Facs, Moels, heoy, Wol Scienific, New Yok. omlinson, Richa, an B. Livesey, (8) he Weck of Nohen Rock, Bloombeg Makes, May, Vol 8, Uhig-Hombug, M. (5), Cash-flow Shoage as an Enogenous Bankupcy Reason, Jounal of Banking an Finance, Vol. 9, Vasicek, O. (1977) An equilibium chaaceizaion of he em sucue, Jounal of Financial Economics, Vol. 37, Zeyun, S. an A. Gupa (7) A Compaaive Suy of he Vasicek an CIR Moels of he Sho Rae, Beiche es Faunhofe, Numbe

51 Figue 1 Fis-passage Defaul Appoach: Baie o Classic Mauiy Defaul is efine by he occuence of he even whee he pobabiliy of efaul can be given as (, ) = { < min s < } B K D V K o V D s ( ) ( ) P = P ob B K, D 1 P ob V K an minvs D = > > s Defaul occus in wo ways. Fis, in he classical case of Meon (1974), efaul occus when he value of he fim falls below he face value of he eb ( K ) a ime. Aiionally, efaul occus befoe mauiy, <, when he value of he fim falls below a baie level, D. We use a fla baie as in Longsaff an Schwaz (1995). he baie can alenaively be shown as ime vaying wih D equal o K a mauiy. Ou esuls ae qualiaively unchange by using an alenaive baie shape. V Souce: Giesecke (4) 5

52 Figue he vaiaion in P fo α of 3 he pobabiliy of efaul is shown o vay wih α. A peak occus a abou a mauiy of six. =.6, θ =.3, c =.3, σ =., σ =., ρ =.5, V = 15, K = 1, D = 6, λ = v.5.45 α= P

53 Figue 3 he Behavio of Defaul Risky Spo Raes an Speas Depenen upon he Disibuion of Pobabiliy of Defaul Inceasing wih Mauiy (Fla R f em Sucue) Speas ae epenen upon he isibuion of V an ae compue assuming he below paamees..6 θ =, c =.3, σ =., σ =., ρ =, V = 15, K = 1, 6 =,.18 v V : D =, λ = Panel a Panel b 1 1 x Expecaion of V Vaiance of V Mauiy Panel c Mauiy Panel P Deivaive of P w Mauiy Mauiy Panel e Panel f R f R Spo Rae S ( ) = R ( ) - R f ( ) Mauiy Mauiy 5

54 Figue 4 he Behavio of Defaul Risky Spo Raes an Speas Depenen upon he Disibuion of V : Pobabiliy of Defaul Deceasing wih Mauiy (Posiive R f em Sucue) Speas ae epenen upon he isibuion of V an ae compue assuming he below paamees. =.6, θ =.3, c =.3, σ =.,. σ =, ρ =, V = 15, K = 1, D = 6, v λ = Panel a Panel b 1 1 x Expecaion of V Vaiance of V Mauiy Mauiy Panel c Panel P Deivaive of P w Mauiy Mauiy Panel e Panel f.1 16 x 1-3 Spo Rae S = R - R f R f R Mauiy Mauiy 53

55 Figue 5a Pobabiliy Densiy of V When ρ = Assuming he Vasicek (1977) moel, we compue he pobabiliy ensiy of he value of he fim assuming an iniial value of 15 an ohe given paamees. =.1, θ =.3, c =.3, V = 15, σ =., ρ =, σ =., v = Pobabiliy ensiy of V = 5 = 1 = Value of fim V 54

56 Figue 5b Pobabiliy Densiy of V log V When ρ = =.1, θ =.3, c =.3, σ =., ρ =, σ =., V = 15 v 1. = Pobabiliy ensiy of log(v /V ) = 5 = 1 = log(v /V ) 55

57 Figue 5c Fis Passage Defaul Pobabiliy his pesens he pobabiliy ensiy cuve fo ealie pas of his figue. =.1, θ =.3, c =.3, σ =., ρ =, σ =., V = 15, K = 1, D = 5 v P Mauiy 56

58 Figue 5 Pobabiliy Densiy of V When ρ =.5 Assuming he Vasicek (1977) moel, we compue he pobabiliy ensiy of he value of he fim assuming an iniial value of 15 an ohe given paamees. ρ is assume o be posiive.5 insea of zeo. =.1, θ =.3, c =.3, σ =., σ =., V = 15 v = Pobabiliy ensiy of V = 1 = 5 = Value of fim V 57

59 Figue 6 Fis Passage Defaul as Depenen Upon Alenaive Baies (D) (Posiive Risk- Fee em Sucue) Assuming paamees given below, we analyze he sensiiviy of efaul an cei speas o alenaive baie values. he shape of efaul pobabiliy vaies wiely. =.6, θ =.3, c =.3, σ =., ρ =, σ =., V = 15, K = 1, λ = v Panel a Panel b D = 5 D = a mauiy a mauiy.14 befoe mauiy oal.14 befoe mauiy oal P.8 P Panel c Panel D=7 D = a mauiy befoe mauiy oal.14 a mauiy befoe mauiy oal P.8 P

60 Figue 6 (coninue) Panel e Panel f D=5 D=6 D=7 D=8 D=9 Spo Rae.9.8 R f R (D=5) R.7 (D=6) R (D=7) R (D=8).6 R (D=9) Mauiy oal P Mauiy Panel g Panel h Deivaive of P w D=5 D=6 D=7 D=8 D=9 Deivaive of P w... D D=5 D=6 D=7 D=8 D= Mauiy Mauiy Panel i S = R - R f D=5 D=6 D=7 D=8 D= Mauiy 59

61 Figue 7 Fis Passage Defaul as Depenen Upon Alenaive Baies (D) (Fla Risk Fee em Sucue) Assuming paamees given below, we analyze he sensiiviy of efaul an cei speas o alenaive baie values given, in conas o a pevious figue, a fla isk fee em sucue. =.6, θ =.18, c =.3, σ =., ρ =, σ =., V = 15, K = 1, λ = v Panel a Panel b D= 5 D=6.3.3 a mauiy befoe mauiy oal a mauiy befoe mauiy oal P.15 P Panel c Panel D = 7 D = a mauiy befoe mauiy oal.5 a mauiy befoe mauiy oal.. P.15 P

62 Figue 7 (coninue) Panel e Panel f Spo Rae R f R (D=5) R (D=6) R (D=7) R (D=8) R (D=9) oal P D=5 D=6 D=7 D=8 D= Mauiy Mauiy Panel g Panel h Deivaive of P w D=5 D=6 D=7 D=8 D=9 Deivaive of P w... D D=5 D=6 D=7 D=8 D= Mauiy Mauiy Panel i S = R - R f D=5 D=6 D=7 D=8 D= Mauiy 61

63 Figue 8 Fis Passage Defaul as Depenen Upon Baies wih Incease Volailiy of Fim Value (σ v ), Posiive Risk-Fee em Sucue Assuming an incease volailiy of fim value an paamees given below, we analyze he sensiiviy of efaul an cei speas o alenaive baie values. =.6, θ =.3, c =.3, σ =., ρ =, σ =.5, V = 15, K = 1, λ = v Panel a Panel b D=5 D= a mauiy a mauiy.3 befoe mauiy oal.3 befoe mauiy oal P P Panel c Panel D= 7 D = a mauiy befoe mauiy oal.3 a mauiy befoe mauiy oal P P

64 Figue 8 (coninue) Panel e Panel f D=5 D=6 D=7 D=8 D=9 Spo Rae R f R (D=5) R (D=6) R (D=7) oal P R (D=8) R (D=9) Mauiy Mauiy Panel g Panel h Deivaive of P w D=5 D=6 D=7 D=8 D=9 Deivaive of P w... D D=5 D=6 D=7 D=8 D= Mauiy Mauiy Panel i S = R - R f D=5 D=6 D=7 D=8 D= Mauiy 63

65 Figue 9 Impac of Vaiaion in Leveage Upon Pobabiliy Densiy (Posiive Risk-Fee em Sucue) he shape of he P em sucue can vay wih leveage. he em sucue can be can be pacically negaive houghou o have a quie ponounce hump. =.6, θ =.3, c =.3, σ =., ρ =, σ =., V = 15, K = 1, D = 6, λ = v.5. V =11 V =13 V =15 V =16 V =18.15 P Mauiy 64

66 Figue 1 Impac of Vaiaion in Inees Rae Volailiy Upon Pobabiliy Densiy Using vaious inees ae volailiies an ohe paamees given below, em sucues of efaul ae compue.he hump is quie obus o he iffeen volailiies. =.6, θ =.3, c =.3, σ =., V = 15, K = 1, D = 5 Panel a : ρ = -.5 v.7 σ =.1 σ =.3.6 σ =.5 σ = P Mauiy Panel b: ρ=.7 σ =.1 σ =.3.6 σ =.5 σ = P Mauiy Panel c: ρ = P.3. σ =.1 σ =.3.1 σ =.5 σ = Mauiy 65

67 Figue 11 Impac of Vaiaion in ρ Upon Pobabiliy Densiy he coelaion beween he inees ae pocess an fim value pocess can have a lage impac on he shape of em sucue of efaul pobabiliy. A hump may o may no occu. =.6, θ =.3, c =.3, σ =., ρ =, V = 15, K = 1, D = 5, λ = Panel a: ρ =.5.3 σ v =.1 σ v =.15.5 σ v =. σ v =.5 σ v =.3. P Mauiy Panel b: ρ =.3 σ v =.1 σ v =.15.5 σ v =. σ v =.5 σ v =.3. P Mauiy Panel c: ρ =.5.3 σ v =.1.5 σ v =.15 σ v =. σ v =.5. σ v =.3 P Mauiy 66

68 Figue 1 Convesion of Risk Neual o Physical Pobabiliies he level of physical pobabiliy of efaul epens upon he make pice of isk (λ) assume. Noneheless, he shape of he cuves is obus o all assume λ. =.6, θ =.3, c =.3, σ =., ρ =, σ =., V = 15, K = 1, D = 6 v Panel a Panel b λ = λ = σ v =.1 σ v =.1 σ v =.15.5 σ v =.15 σ v =..5 σ v =. σ v =.5 σ v =.5 σ v =.3. σ v =.3. P.15 P Mauiy Mauiy Panel c Panel λ =.15 λ = σ v =.1 σ v =.15 σ v =. σ v =.5 σ v = σ v =.1 σ v =.15 σ v =. σ v =.5 σ v =.3.. P.15 P Mauiy Mauiy 67

69 Figue 13 Sensiiviy of S Hump Locaion o Baie Value (D) he mauiy a which he hump in cei spea occus is sensiive o he paamees assume o compue efaul pobabiliy an cei spea. Hee he mauiy a which he S hump occus, **, is shown o be elae o baies (D) in a complex way. σ =., σ =., ρ =.5, D = 1, V = 15, =.6, θ =.3, c =.3 v ** D 68