IMES DISCUSSION PAPER SERIES

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1 IES ISCUSSION PAPER SERIES ynac odel of Ced Rsk n Relaonshp endng: A Gae-heoec Real Opons Appoach Takash Shaa and Tesuya Yaada scusson Pape No. 009-E-7 INSTITUTE OR ONETARY AN ECONOIC STUIES BANK O JAPAN -1-1 NIHONBASHI-HONGOKUCHO CHUO-KU TOKYO JAPAN You can donload hs and ohe papes a he IES We se: hp://.es.o.o.p o no epn o epoduce hou pesson.

2 NOTE: IES scusson Pape Sees s cculaed n ode o sulae dscusson and coens. Ves epessed n scusson Pape Sees ae hose of auhos and do no necessaly eflec hose of he Bank of Japan o he Insue fo oneay and Econoc Sudes.

3 IES scusson Pape Sees 009-E-7 ach 009 ynac odel of Ced Rsk n Relaonshp endng: A Gae-heoec Real Opons Appoach Takash Shaa and Tesuya Yaada Asac We develop a dynac ced sk odel fo he case ha anks copee o collec he loans fo a f fallng n dange of ankupcy. We apply a gae-heoec eal opons appoach o nvesgae ank s opal saeges. Ou odel eveals ha he ank h he lage loan aoun naely he an ank povdes an addonal loan o suppo he deeoang f hen he ohe ank collecs s loan. Ths suggess ha hee ess aonal foeaance lendng y he an ank. Copaave sacs sho ha as he lqudaon value s loe he opal e ng fo he non-an ank coes a an eale sage of usness donun and he opal lqudaon ng y he an ank s delayed fuhe. As he nees ae of he loan s loe he opal e ng fo he non-an ank coes eale. These analyses ae conssen h he foeaance lendng and eposue concenaon of an anks oseved n Japan. Keyods: Ced sk; Relaonshp lendng; Real opon; Gae heoy; Concenaon sk JE classfcaon: G1 G3 G Assocae Pofesso Tokyo eopolan Unvesy (E-al: shaa@u.ac.p) Assocae eco Insue fo oneay and Econoc Sudes Bank of Japan (E-al: esuya.yaada@o.o.p) The auhos ould lke o hank asaak Ka Hosh Osano Junch Ia pacpans of he eseach eeng a RIETI and he saff of he Insue fo oneay and Econoc Sudes (IES) he Bank of Japan fo he useful coens. Ves epessed n hs pape ae hose of he auhos and do no necessaly eflec he offcal ves of he Bank of Japan.

4 I1. Inoducon o he d 1990s Japanese anks suggled h nonpefong loan poles. The poles paly eeged fo he sk eedded n elaonshp ankng. The dsadvanages and advanages of he elaonshp have een dscussed n he ankng heoy leaue (fo deals see Boo [000] and Elyasan and Goldeg [004]). As s ell knon he fs advanage s a educon of neffcency seng fo asyec nfoaon eeen a ank and a f. The second s o faclae an plc long-e conac hough sk shang such ha a ank anans a sale loan nees ae even f he f s ced sk flucuaes. These enefs hoeve poenally un no dsadvanages. o eaple onopolsc lendng as a consequence of a longe elaonshp eeen a ank and a f gves se o he hold-up pole ha s he ank s song aganng poe gves he f an ncenve o oo fo ohe anks. The f heefoe pefes ulple ankng elaonshps despe addonal adnsave coss. Anohe dsadvanage s he sof-udge pole hch coes fo an plc long-e conac. When Japanese anks suggled h he nonpefong loan pole as ofen ced as a ypcal eaple of he pole posed y Japanese anks foeaance lendng o deeoang fs n ode o avod losses fo fs ankupcy. Banks aay polces egadng he suppo fo fs educed he dscplne fo ced sk anageen. 1 ulple ankng elaonshps n soe counes ae coned h a an ank syse. In hs syse one pacula ank ha holds he lages shae of a f s de s defned as a an ank. The an ank faces he esponsly of onong he f s condon n eun fo holdng he lages shae of lendng and povdng ohe fnancal sevces o he f. Whn he an ank syse a sof-udge pole gh e oe seous. When he ced condon of he f osens he non-an ank gh collec s ousandng loans fo he f. Unless he an ank povdes an addonal loan o fll he shoage of f s loan deand he an ank edaely suffes fo he f s ankupcy ecause of a lack of lqudy. We heefoe nae he an ank s addonal loan nsead of he non-an ank de assupon. oeaance lendng hoeve gh lead o a lage loss fo he an ank n he fuue. An aay lendng polcy n elaonshp ankng s equed fo a cean decson ule. As fo he non- 1 The sof-udge pole hghlghs he dffeence eeen e ane effcency and e pos effcency. In hs pape e ane effcency coesponds o he opal e saegy hou hough fo ohe anks lendng saeges as dscussed lae. The equlu of he gae h consdeaon of counepas saeges epesens e pos effcency. Hence aay polcy hee eans he an ank s polcy hou consdeaon of he ohe anks polces. 1

5 an ank also faces unceany egadng he e ng fo lendng o he f. The opal ng s deened y he ade-off eeen an ncease n ced sk and an oppouny o gan fuue eanngs fo he loan o he f. In hs pape e popose a heoecal odel o easue anks ced sk n a gae eeen a an ank and non-an ank concenng he e ng and he decsons aou he de assupons. s assung ha a f oos fo one ank e eane he opal e saegy fo he lendng o he f. The saegy can e developed usng eal opons heoy usng he sochasc pocess of f value. Ths appoach s developed y eland [1994] and ella-baal and Peaudn [1997]. Real opons heoy pays aenon o he ank s ang opon o collec s loan fo he deeoang f consdeng he f ay avod ankupcy. I ay e opal fo he ank o e lae. The eal opons odel gves a heshold level of f value fo he ank s decson on eng unde unceany of f value. Baa [001] developed a heoecal odel o nvesgae opal ng n ank s ng off s nonpefong loan usng eal opons appoach. Second e assue ha a f oos fo o anks: one s he an ank and he ohe s he non-an ank. Ths seup noduces a gae-heoec ve no he eal opons odel. We eend he eal opons odel of ella-baal and Peaudn [1997] o a gae-heoec eal opons odel. Ths appoach s developed y and Pndyck [1994] o eplan he opal eny ng n a ake n hch anohe playe also as fo hs/he opal eny e. They sho ha hee ess an equlu hee one of he poenal enans nvess eale han he ohe. In addon he nvesen ng of he fs-ove s eale han he noncopeve eal opons case. Genade [1996] appled hs appoach o a eal esae ake o eplan oveuldng a vaan of ovenvesen as a ae o ne enans. Weeds [00] appled hs appoach o fs R& nvesen and copaed he esuls of a coopeave gae h hose of a noncoopeave gae. These sudes eaned he eny gae h eal opons heoy hle ou sudy focuses on he e gae. Slaly o he eny gae ou gae-heoec eal opons odel has a unque equlu. The equlu analyss shos ha a dffeence n he loan aoun eeen he o anks esuls n a dffeence n he opal ng of e. The an ank akes de assupons n es of s azaon of he loan value even f he non-an ank es eale. Ou odel does no desce aonal foeaance lendng u does gve oh anks easue of he ced sk ased on he oulook of he gae. In addon e eane

6 hough copaave sacs he pacs of changes n eogenous vaales such as he lqudaon value of he f he nees ae of loan and he volaly of f value on oh anks e saeges. Each ank deenes s opal e saegy y akng no accoun he ohe ank s opal saegy and he equlu s gven n hs gaeheoec suaon. These copaave sacs eveal he follong. 1) The loe he lqudaon value of he f he eale he non-an ank es. In conas he loe he lqudaon value he lae he an ank lqudaes he f. ) The loe he nees ae of he loan he eale he non-an ank es. 3) The hghe he volaly of f value he lae oh anks e and lqudae he f. Hoeve uch hghe volaly causes an ncenve o e fo he non-an ank. These esuls ae conssen h he foeaance lendng and eposue concenaon oseved hn an anks n Japan. The eande of he pape s oganzed as follos. Secon eplans he enchak odel of onopoly lendng developed y ella-baal and Peaudn [1997]. I shos ho he eal opons appoach helps us deene he opal ng of e. Secon 3 eends he enchak odel o a gae n hch a f oos fo o anks. Secon 4 eanes he equlu of he odel desced n Secon 3. Secon 5 dscusses he copaave sacs and plcaons. Secon 6 concludes.. Benchak odel fo onopoly lendng s e eane sple onopoly lendng usng eal opons heoy follong ella-baal and Peaudn [1997]. We eploe a odel n hch a f fnances s usness h de and equy. One ank supples he loan and a epesenave equy holde conols he f. (1) odel sengs We denoe he sales of he f as X and assue ha X follos a geoec Bonan oon unde a sk-neual easue: dx = μ X d σ X dz X 0 = (1) hee μ and σ ae consan and z s a sandad Bonan oon pocess. We assue consan vaales fo he follong paaees: 3

7 : he opeang coss of he f : he pncpal of he ank s loan C: he lqudaon value of he f c: he ao of loan value coveed y he lqudaon (defned y C/) : he sk-fee ae.e. he dscoun ae unde he sk-neual easue and : he nees ae of he loan. We assue ha he df of he f s sales μ s less han he sk-fee ae. 3 We also assue ha he nees ae of he loan s geae han. The equy holde and he ank deene he opal saeges especvely ased on he coon knoledge of he sochasc pocess of X and he cuen value of X. The choces of he equy holde ae ehe o un he f o o go ankup a each unde oseved X. In he case of ankupcy he f s oned y he de holde.e. he ank and afe ankupcy he ank uns he f. The ank s choce s ehe o un he f o o lqudae a each afe he ankupcy. We denoe he ankupcy e as τ and he lqudaon e as τ c. We also denoe as he flaon ses of he nfoaon on X and as he se of soppng es on he nfoaon se condonal epecaon aou =. s gven y. The No e foulae he opzaon pole fo he equy holde. The equy holde decdes he opal ng of ankupcy o aze he equy value n each and he azed value s gven y: E = τ ( s) ( X ) a e ( X s ) ds τ The equy value E() equals he azed pesen value of he f s pofs efoe he f s ankupcy ( < τ ). Afe he ankupcy ( τ ) he equy holde leaves he f s oneshp o he ank. We denoe he opal τ as τ. Afe ankupcy occus he ank decdes he ng of lqudaon o aze he loan value unde he gven value of τ. The opzed ng τ he azaon pole as follos: c. () s deened y τ = τ c c ( s) ( s) X a e ds e ( X ) τ τ s ds e ( τ c ) C (3) We assue c s less han 1. 3 Ths assupon s a necessay condon fo he dscouned pesen value of he f s pof o convege o a fne value. If >μ holds hen he negal [0 ) e (X 0 e μ ) d conveges o X 0 / ( μ). Hoeve he negal dveges o nfny f μ. 4

8 hee () epesens he azed loan value a e efoe τ. The fs e of he condonal epecaon n equaon (3) epesens he pesen value of he nees ncoes efoe ankupcy occus ( < τ ). The second e epesens he pesen value of he ank s eanngs n he s peods fo he ankupcy o he lqudaon ( τ ). The eanngs of he f s one ae gven y X s dung he s peods. < < τ c The hd e epesens he pesen value of he lqudaon value of he f. () The opon values of ankupcy and lqudaon The opzaon pole fo he equy holde n equaon () can e solved analycally (see Append 1). The azed equy value E() s gven y: and: E ( ) = μ μ = 1 < ( μ ) fo (4) (5) hee: = 1/ μ / σ ( μ / σ 1/ ) / σ < 0. The fs e of he gh-hand sde of equaon (4) s he pesen value of he f s eanngs efoe he ankupcy. 4 Ths value s an nceasng funcon of he nal f s sales and he df of he f s sales μ. I s also a deceasng funcon of he opeang cos he nees ae and he loan aoun. Noe ha does no depend on he volaly σ n he sochasc pocess of X ecause he value s deved as he opzaon of he epeced value unde he sk-neual easue. The second e epesens he opon value of he equy holde onng he gh o ankup he f. We nae he ankupcy opon. The ankupcy opon depends on he volaly σ and nceases as σ nceases. 5 Ths ples ha he ankupcy opon ecoes oe valuale as he f s sales ecoe oe volale. To eane ho he opon value eanngs e sho he nepeaon of he heshold value coespondng o τ. The e ( ) n equaon (4) s he 4 The e s dscouned y μ hle he e s dscouned y. Ths s ecause X has he df μ as shon n equaon (1). 5 ffeenaon of equaon (4) h espec o σ yelds de/dσ = {( E/ ) ( / ) E/ } / = ( E/ ) ( / σ) (Noce ha E/ = 0). Thus de/dσ > 0 ecause E/ = { /( μ) ( )/} (/ ) log(/ ) > 0 and ( / σ) > 0. 5

9 poaly ha eaches he heshold value 6 ha s he poaly of ankupcy. We can easly check ha ( ) 0 as and ( ) = 1 a =. Thus he fs e n he equy value E() donaes as and eaches zeo a ankupcy =. In addon he agnal value of he equy a ankupcy E ( ) also eaches zeo. uheoe E() soohly passes zeo a =. The equy value funcon s depced y gue 1. The ankupcy opon nceases he equy value hch ples ha n un deceases he value of loan fo he ank. Once he opal ng of he ankupcy s gven y he e hen eaches he heshold value of (5) he azed loan value () and he heshold value of fo he lqudaon ae gven y: = ( ) fo < (6) hee: μ μ c c = c (7) and: The devaons of hese ae shon n Append. c = c ( μ ). 1 (8) Equaon (6) s nepeed as follos. The fs e epesens he pesen value of he nees ncoes and he second e s he negave opon value esulng fo he equy holde s opon on ankupcy. The lae deceases he loan value as declnes. s close o as eaches hle s close o he fs e n equaon (6) as. epesens he loan value afe ankupcy gven y (7). Equaon (7) shos he pesen value of he f ha he ank nhes fo he equy holde a he e τ. The fs and second es epesen he value o he ank of unnng he f afe he ankupcy. The ank oans he oal pof of he f hle he equy holde oans he f s pof afe nees payens as shon n equaon (4). The hd e s he opon value of he ank havng he gh o pospone 6 Scly speakng (/ ) s he pesen value of he poaly of ankupcy hch s gven y E(ep( τ )) = [0 ) e E( τ < d). See and Pndyck [1994]. 6

10 he lqudaon of he f n ode o e on he f s ecovey. The opon value s popoonal o he poaly of lqudaon ( ) c and he loss (o pof) a lqudaon hch s gven y he dffeence eeen he lqudaon value C(= c) and he value of unnng he f. s an nceasng funcon of. I eaches he lqudaon value C; as appoaches c he heshold value of he lqudaon gven y equaon (8). The soohness condon ( c ) = 0 s assued n he azaon of equaon (3) o oan equaons (6) (7) and (8) hch ae equed fo he opaly of he lqudaon heshold c. gue 1 shos he value funcons of he equy and he loan ha coespond o equaons () and (3). The opal ngs of he ankupcy and he lqudaon ae gven y he pons hee he value funcon cuves ae soohly pasng o zeo and C especvely. 7 The loan value cuve has an uppe ound gven y he pesen value of nees ncoes. The ound coesponds o he value n he case of.e. ankupcy poaly 0. As long as C s less han.e. c s less han a un he equy holde acceps ankupcy eale han he ank acceps lqudaon. gue 1. The value of equy and loan n onopoly lendng loan value / equy value loan value ( ) c pncpal aoun lqudaon value equy value E ( ) f's sales 0 lqudaon ankupcy 7 See and Pndyck [1994]. 7

11 3. Eended odel fo duopoly lendng In hs secon e eend he enchak odel o he case hee a f oos fo o anks. In he pevous secon one ank povded a loan o he f and he ank s collecng he loan ean ha he f s lqudaed y he ank. In he duopoly lendng case he equy holde uns he f as long as he oal aoun of he loan s ananed. In he duopoly case he opal ankupcy ng s eale han hen anks collec he loans n he onopoly case. We heefoe nvesgae an e gae eeen o anks hee one of he anks gh ake a de assupon hen he ohe ank collecs s loan. The decson of he de assupon depends on he opzaon pole fo he ank ha faces he fs acon y he ohe ank. In hs secon e eane he opzaon pole fo he fs-ove and fo he ohe assgnng a ank o he ole of ehe he fs-ove o he ohe. (1) odel sengs Nehe ank deenes s saegy coopeavely. We have o anks: ank A and ank B. Bank es eale han ank ( { A B} ) and hus e efe o as he leade and as he folloe. The odel n hs secon yelds he opal saegy gven he counepa s saegy pepang fo he noncoopeave gae n he ne secon. All paaees ecep he loan aoun ae he sae fo anks A and B and denoed as n Secon. Each loan aoun fo anks A and B s denoed as A and B especvely. We also defne A and B as each ank s shae of oal loan ha s = { A B} hee A B =. If ank es eale han ank ( { A B} ) hen he folloe ank gh ake a de assupon of he aoun of he loan ha he leade ank collecs fo he f. When he folloe ank lqudaes he f afe he de assupon ank oans he lqudaon value C hee C <. We do no assue hehe ank A o B ecoes he leade a hs sage u e ll sho ha he dffeence eeen A and B deenes he leade lae on. The opzaon pole fo he equy holde s he sae as n Secon. uheoe he opal ankupcy ng τ s gven as n Secon. Gven he ankupcy ng gven y: τ he opzaon pole fo he leade ank s 8

12 τ ( s) ( s) X = a e ds e ( X ) τ τ τ e s ( τ ) hee s he leade s loan value and τ s he e ng fo he leade ank. The fs e of he gh-hand sde n equaon (9) epesens he pesen value of he leade s nees ncoes efoe he ankupcy occus ( < τ ). The second e epesens he pesen value of he leade s eanngs n he peods fo he ankupcy o he e ( τ < τ ). 8 Because e assue ha oh anks anan he eng loan shaes afe he ankupcy he second e s coposed of he f s pof dvded y s lendng shae afe he ankupcy. The hd e epesens he pesen value of he leade s collecon of s loan pncpal a =τ. τ s deved fo he opzaon n equaon (9). The folloe ank deenes τ c he opal ng of he lqudaon afe akes a de assupon a τ. τ c gh e close o τ u e assue τ c us e lae han. We eclude he possly of sulaneous es of oh anks ecause he connuous pocess assues anconcdence eeen and fo heeogeneous anks h espec o loan aoun. The opzaon pole fo he folloe ank s gven y: τ τ ( s) ( s) X = a e ( ) ds e X s ds τ c τ (10) τ c ( τ ) ( s) ( τ c ) e e ( X ) ds e C s τ. τ The fs e n equaon (10) epesens he pesen value of he folloe s nees ncoe efoe he ankupcy ( < τ ). The second e epesens he pesen value of he folloe s eanngs afe he ankupcy ( τ < τ ). The hd e epesens he pesen value of he folloe s ne loan nsead of he leade s collecon a =. The fouh e epesens he pesen value of he folloe s eanngs n he peod fo o. The las e epesens he pesen value of lqudaon a = τ c. τ c ds τ (9) τ 8 If he leade es efoe he ankupcy he negal neval n he fs e has o e changed fo [ τ ] o [ τ ] and he second e s no needed. 9

13 () The valuaon fo each loan value The azed leade s loan value can e oaned analycally as follos: ( ) fo < = (11) hee: = μ μ (1) and: = 1 ( μ ). (13) See Append 3 fo he devaon. Equaon (11) epesens he leade s loan value efoe he ankupcy. The fs e epesens he pesen value of he nees ncoes and he second e coesponds o he negave opon value seng fo he equy holde s eecuon of he ankupcy opon. These foulas ae sla o hose n he enchak odel (6) hle he loan value afe he ankupcy ncludes nsead of c n he enchak odel (7). The second e epesens he posve opon value of he leade eng ale o e eale han he folloe. The posve opon value s deened y ) he poaly ha he leade collecs s loan a.e. ( ) and ) he loan pncpal nus he loan value easued y he pesen value of he pofs n he case of =. The lae epesens he leade s ne gan hen he leade collecs s loan a. The ulple of he gan and he e poaly yelds he opon value of ealy e fo lendng. Noe ha he heshold value of he leade s e shon n equaon (13) s ndependen of s loan aoun. Ths s ecause he leade deenes he e ng y he gap eeen he oal loan aoun and he value of oal loan a. n he second e of he equaon (1) opeaes only as a ulple. The opon value nceases as fo aove and eaches he au a = hee he leade eecses s e opon. A hs pon he leade s loan value s equal o s loan pncpal. We assue ha he loan value pases soohly o he loan pncpal y he consan ( ) = 0 a he azaon of equaon (9). The ankupcy heshold n (5) s lage han he leade s e heshold n (13) 10

14 ecause e assue he loan nees ae s geae han he sk-fee ae. The leade heefoe alays es afe he ankupcy. gue depcs he leade s loan value funcon gven y equaon (11) noalzng he value y s loan pncpal. The value s alays lage han a un fo he assupon ha he leade can collec he loan pncpal a he e y he folloe s de assupon. The value pases soohly o he noalzed loan pncpal value.e. a un. As he f s sales nceases he poaly of ankupcy n equaon (11) deceases and he loe poaly educes he negave opon value n asolue es hch nceases he loan value. The loan value conveges o he pesen value of he nees ncoe he fs e of equaon (11) as. The leade s loan value s lage han he loan value of he onopoly lendng ecause he leade holds he opon o e eale hch he onopoly ank does no. As shon n gue a declne of nceases he dffeence eeen noalzed and n he onopoly case. gue. The value of he leade s loan ( 注 ) loan value leade's loan value ( ) 1 loan pncpal loan value (h onopoly lendng) ( ) c lqudaon value equy value E ( ) f's sales 0 lqudaon leade's eng ankupcy 11

15 Ne e consde he folloe s loan value gven ankupcy heshold and e heshold as follos:. The aal value s oaned ( ) fo < = hee ( ) (14) μ μ = (15) and: c μ μ = c. (16) c See Append 3 fo devaon. The folloe s loan value n equaon (14) s sla o he leade s value n equaon coespondng o he folloe s loan (11) u s dffeen y he e value afe he ankupcy. The second e of n equaon (15) s he negave opon value fo he folloe. The leade s opon o e yelds a negave opon fo he folloe. 9 s defned y equaon (16). The fs e of equaon (16) s he pesen value of he f s pofs nus he leade s loan pncpal hch s dencal o he aoun of he de assupon fo he folloe. The second e s he opon value fo he folloe of lqudang he f. The gap eeen he lqudaon value c and he f s dscouned pofs easued y = c epesens he gan fo he lqudaon. The aoun of he gan and he poaly of lqudaon ( ) c yeld he lqudaon opon value. The elaon c holds y he assupon C <. Ths ples ha he opal lqudaon ng s lae han he leade s e. I s aonal fo he folloe o ake he de assupon n hs odel and o un he f y self unl eaches c. Ths s ecause ha he folloe canno ecove s loan pncpal edaely y he assupon C < and hus has he ncenve o ake a de assupon o a fo he f s ecovey. gue 3 depcs he folloe s loan value funcon y a sold old lne. I s alays less han he onopoly loan value ecause of he negave opon value. Naually anks A and B oh an o avod eng he folloe. In he ne secon e nvesgae 9 The negave opon s defned y usng he gap eeen he loan value afe he leade s e and he loan value easued y = efoe he de assupon. The lae s geae han he foe fo he foe efoe he de assupon fo >. 1

16 he equlu n he gae eeen he o anks. gue 3. The value of he folloe s loan loan value ( 注 ) pncpal of lendng leade's loan ( ) ( ) onopoly lendng 1 c c lqudaon value lqudaon value - leade's loan aoun folloe's loan ( ) E ( ) equy value f's sales 0 lqudaon ankupcy 4. The equlu In hs secon e eane he equlu of he odel desced n Secon 3. Whou loss of genealy e assue ha ank A has a lage loan han ank B. We egad ank A as he an ank and ank B as he non-an ank. The only dffeence eeen he an and non-an ank s he loan aoun. s e eane he folloe s loan value assung each ank s n he poson of he folloe. gue 4 depcs he folloe s value fo he an ank A y a sold hn lne unde he assupon ha fo he non-an ank B y sold old lne unde he assupon ha A = 80% and B = 0%. The folloe s value fo he nonan ank s less han ha fo he an ank ecause he uden of he de assupon on he non-an ank s lage han ha on he an ank n he case ha he f s fnally lqudaed. The lage s n equaon (16) he lage he negave opon value n equaon (15) n asolue value. The lage negave opon value fo he non- 13

17 an ank akes he folloe s value cuve fo he ank loe han ha fo he an ank. When he non-an ank collecs s loan pncpal he ank aandons he fuue pofs gven y. n gue 4 he coss pon of he cuve and he B pncpal oh noalzed y B gves he heshold value of fo he opal e ng fo he non-an ank. Hoeve s no an opal heshold n he equlu of he e gae. If he an ank s a folloe he sae dscusson holds. P A lendng ) a = P A P B P P coesponds o he heshold fo he an ank. Noe ha fo A < < B he nonan ank deduces ha he an ank does no e eale and s opal o connue P P he lendng. In sho he non-an ank should e a =. 10 P o < < oh anks have an ncenve o anan he lendng. If he non-an ank es a ha pon s opal fo he an ank o connue lendng y akng a de assupon ecause he lqudaon value coespondng o c n gue 4 s he ne copaave value fo he an ank o decde s saegy ehe o un he f o o lqudae. The de assupon 3) fo < he an ank uns he f and 4) a = he c < an ank lqudaes he f. We denoe he equlu e heshold as P. P B dscusson aove gves he gae equlu 1) fo > P A A B A oh anks anan he he non-an ank collecs s loan and he an ank akes a P A P A B c 10 In he sc sense he e heshold s slghly lage han A P. A A P he non-an ank s ndffeen eeen eng and connung opeaons. 14

18 gue 4. The equlu ( 注 ) loan value loan pncpal A A = B B A A A A = B B = 1 c P A P B f's sales B B 5. Copaave sacs and plcaons In hs secon e consde ho eogenous condons affec he (non-) an ank e saegy. We change he f s eogenous paaees such as 1) he lqudaon value of he f ) he nees ae of he loan 3) he volaly of he f s sales. The enchak paaees ae gven as follos. f of he pocess X :μ 0% Volaly of X :σ 10% s sales: Opeang cos: 1 Inees ae 5% Rsk-fee ae % Pncpal 10 Pncpal of an ank A 7 qudaon value C 6 Recovey ao c (= C/) 60% 15

19 (1) Copaave sacs on he f s lqudaon value s e eane ho he anks e saeges vay h he lqudaon value. In gue 5 he vecal as shos he ecovey ae of he loan pncpal a he lqudaon nsead of he lqudaon value and he hozonal as epesens cuen f s sales.e. he nal value fo fuue developen of he pocess. The heshold pons P and c ae shon fo he case of he enchak. The an esuls ae as follos. ) The e heshold fo he non-an ank P nceases as he ecovey ae c deceases. Ths ples ha he non-an ank ends o e eale hen he ecovey ae s loe. 11 ) The lqudaon heshold fo he an ank c deceases as he ecovey ae c deceases. Ths ples ha he an ank ends o hesae egadng lqudaon hen he lqudaon value deceases. Because he opon value of delayng lqudang he f s hghe fo he loe ecovey ae s aonal fo he an ank o a fo he ecovey of he f s sales. ) The ankupcy heshold fo he equy holde s ndependen of he lqudaon value. Because he lqudaon value C does no eceed he loan aoun hs condon povdes he equy holde h less ncenve o e on he sales ecovey han he an ank dscussed n Secon. In gue 5 e can also oseve ha he ankupcy heshold gh e lage han he non-an ank s e heshold a a hgh ecovey ae such as oe han 80%. In hs case oh of he anks connue lendng afe he ankupcy. 11 In he gae-heoec suaon he non-an ank deenes he e saegy copang he folloe s value h he loan pncpal (he value of eng as a leade) and hus he folloe s value plays an poan ole heeas he non-an ank alays es as a leade a he equlu. 16

20 ecovey ae c 100% gue 5. The anks e saeges and he f s lqudaon value 90% 80% lqudaon 70% 60% ankupcy non-an ank s e P enchak paaees 50% 40% 30% 0% 10% 0% f s sales () Copaave sacs on he nees ae Ne e eane ho he anks e saeges vay h he nees ae. The an esuls shon n gue 6 ae as follos. ) The e heshold of he non-an ank P nceases as he loan nees ae deceases. The ank ends o e eale hen he loan ae s loe. The loe he ncoes fo loan he loe he value of oh anks loan. 1 The lqudaon heshold c s ndependen of he nees ae (he c cuve s a vecal lne n gue 6). Ths s ecause he f s pof elongs o he anks afe he ankupcy and he decson of he lqudaon s ndependen of he nees ae. ) The ankupcy heshold nceases as he loan ae nceases. The equy holde of he f ends o ankup he f eale ecause he hghe nees payen deceases he equy value. o loan aes of oe han 6% n gue 6 he ankupcy occus efoe he non-an ank s e. o hs eason he non-an 1 The declnes n he loan values hasen he non-an ank o collec s loan hle he ng of lqudaon y he an ank does no change. Once he an ank akes s de assupon all pofs elong o he an ank and heefoe he heshold c s ndependen of he lendng ae. 17

21 ank s e heshold P s ndependen of he loan ae.e. he P cuve s vecal. Because he pofs of he fs ae shaed y oh anks n popoon o he loan aoun he loan ae does no ae. These esuls sugges ha lo loan aes end o ncease he loan eposue concenaon o he an ank and povde a song ncenve fo he equy holde o loe he f s sales. nees ae 10% gue 6. The anks e saeges and he loan nees ae 9% 8% 7% 6% 5% lqudaon ankupcy P non-an ank s e enchak paaees 4% 3% % f s sales (3) Copaave sacs on he volaly of f sales nally e eane ho he anks e saeges vay h he volaly of he f s sales σ. The an esuls shon n gue 7 ae as follos. ) The non-an ank ends o e lae as sales volaly nceases ecause he opon value o e on he ecovey nceases. Hoeve n he case of hghe volaly such as ha geae han 5% he e heshold P ends ackad. We povde easons fo hs elo. ) As he volaly nceases he equy holde ends o educe f sales. Ths s a 18

22 sla ncenve o he an ank hch has an ncenve o delay he lqudaon of he f. Ths s ecause he ang opon value o ankup and lqudae he f nceases as he volaly nceases. volaly σ 40% gue 7. The anks e saeges and he volaly of he f s sales 36% 3% 8% 4% 0% 16% 1% 8% lqudaon ankupcy P non-an ank s e enchak paaees 4% f s sales In he sandad eal opons odel he opal heshold s a onooncally nceasng funcon of sales volaly as shon y he ankupcy cuve n gue 7. On he ohe hand n he gae-heoec eal opons odel he opal heshold s no alays a onooncally nceasng funcon of sales volaly as shon n Ka and Shaa [005]. The eason s as follos. Noe ha he folloe s value plays an poan ole n deenng he e saegy. As eplaned n (15) (17) he folloe s value s coposed of he o negave opon values and one posve opon value such ha: The folloe s opon value = negave opon value suffeng fo he equy s opon o ankup he f negave opon value suffeng fo he leade ank s opon o e ealy 19

23 posve opon value geneaed y he folloe ank s opon o lqudae he f. The elave szes of he posve and negave opon values ake he e heshold cuve ackad endng. The ncease n he volaly heghens all hee opons values fo he an ank.e. he folloe on an asolue value ass. o loe volaly he ncease n he posve opon value eceeds he ncease n he negave one hch akes he e heshold fo he folloe loe. Ths n un loes he opal e heshold fo he leade. o hghe volaly n conas he ncease n he posve opon value s less han he ncease n he negave one. The e heshold heefoe ses as he volaly nceases. 6. Concluson Ths pape developed a dynac odel fo ced sk n elaonshp lendng. I consdeed he case n hch he an ank and he non-an ank play a gae of e fo he deeoang lendng and eaned he opal e saeges y applyng a gae-heoec eal opons appoach. Ou odel shoed ha each ank deenes he opal e saegy y akng no accoun he ohe ank s opal saegy and ha he equlu of he gae depends on he dffeence n he loan aoun eeen he o anks. The an ank h a lage loan aoun akes a de assupon aonally n a sense of s azaon of he loan value. The pape also used copaave sacs o eane he effec of eogenous vaales such as he lqudaon value of he f and he loan nees ae on he anks saeges. s a lo lqudaon value akes he non-an ank e eale heeas enhances he an ank s ncenve o delay he lqudaon hle ang fo he f s ecovey. Second a lo loan ae leads o he ealy e of he non-an ank. These echanss acceleae he concenaon of he an ank s eposue o he deeoang f. nally e llusaed ho ou odel can e fuhe developed. s ould e neesng o nvesgae asyec nfoaon aou he f eeen he o anks. The an anks ay have dffeen nfoaon on he sochasc pocess of nal sales. Second s possle fo he an ank o evalze he f once he ank ons he f. The ank ay educe he f s opeang cos and pove he goh ae of he f s sales. I ould e neesng o nvesgae hese eensons o ou odel. 0

24 Append 1. The opzaon pole fo he equy holde In hs append e sho he soluon of he opzaon pole fo he equy holde n equaon (). The equaon s equvalen o he follong HJB (Halon Jaco Bellan) equaon: E d ( X ) e a ( X ) d [ E( X )] = d 0 (A-1) hee E ( X ) epesens he value of he f fo he equy holde. Applyng Io s lea o he HJB equaon (A-1) e oan he follong odnay dffeenal equaons of E : 1 = E μ E E σ (A-) h ounday condons: E E E ( ) ( μ ) ( ) ( ) = 0 ( ) = 0. (A-3) The fs condon n (A-3) eques ha E conveges o he pesen value of he f s pofs as. The condon ecludes a ule condon. The second and hd condons eque ha E pases soohly o zeo a. These condons ae called he value achng condon and sooh pasng condon especvely. Equaon (A-) s an Eule dffeenal equaon and can e solved analycally h he condons (A-3) hch deene he value of he f fo he equy holde and he ankupcy heshold y: E = μ μ 0 fo fo < (A-4) = ( μ ) 1 (A-5) hee s he negave oo of he chaacesc equaon σ ( 1) μ = defned y = 1 μ σ ( μ σ 1 ) σ.. s 1

25 Append. The opzaon pole fo he onopoly ank Ths append shos he soluon of he opzaon pole fo he ank n he case of onopoly lendng gven he ankupcy heshold deved n Append 1. Equaon (3) s equvalen o he HJB equaon: ( X ) hee ( X ) lea o = e e d d a d [ ( X )] d ( X ) d [ ( X )] d C fo < τ fo τ (A-6) epesens he value of loan fo he onopols ank. Applyng Io s (A-6) e oan he dffeenal equaon of 1 : = μ fo σ < (A-7) h ounday condons: and: ( ) 0 ( ) = ( ) (A-8) 1 = ( ) μ fo σ (A-9) h ounday condons: ( c ) = C ( c ) = 0. ( μ ) (A-10) Equaons (A-7) and (A-9) ae Eule dffeenal equaons and can e solved analycally. Equaon (A-9) s solved fs and he soluon s appled fo equaon (A-7). The value of he loan and lqudaon heshold c ae deved y he ounday condons (A-10) as follos: = μ c ( ) c C μ C c = c ( μ ). 1 fo fo c < fo < c (A-11) (A-1)

26 Append 3. The opzaon pole fo he leade ank Ths append shos he soluon of he opzaon pole fo he leade ank n he case of a duopoly. Equaon (9) s equvalen o he HJB equaon: [ ] [ ] < = X d X e X d e X d d d d fo a fo τ τ (A-13) hee epesens he value of loan fo he leade ank. Applyng Io s lea o ( X ) (A-13) e oan he dffeenal equaon fo X : < = fo 1 μ σ (A-14) h ounday condons: = 0 (A-15) and: = fo 1 μ σ (A-16) h ounday condons: = =. 0 μ (A-17) Equaons (A-14) and (A-16) can e solved analycally and he value of loan and he e heshold fo leade ae deved y he ounday condons (A-17) as: < < = < fo fo fo μ μ (A-18). 1 μ = (A-19) 3

27 Append 4. The opzaon pole fo he folloe ank Ths append shos he soluon of he opzaon pole fo he folloe ank. Equaon (10) s equvalen o he HJB equaon: hee ( X ) lea o = e ( X ) d e e d d a d [ ( X )] d [ ] ( X ) d ( X ) d [ ] ( X ) d ( X ) C d } fo fo fo < τ τ < τ τ (A-0) epesens he value of loan fo he folloe ank. Applyng Io s (A-0) e oan he dffeenal equaon of ( X ) 1 μ : = σ fo (A-1) < h ounday condons: ( ) 0 ( ) = ( ) (A-) 1 σ (A-3) = ( ) μ fo < h ounday condons: ( ) ( ( μ ) ) ( ) = ( ) (A-4) and: 1 σ (A-5) = ( ) μ fo h ounday condons: ( ) μ ( c ) = c ( c ) = 0. (A-6) The aove equaons can e solved ackad analycally fo (A-5) o (A-3) and 4

28 hen o (A-1). The lqudaon heshold fo folloe s deved y he ounday condons c (A-6) as follos: < < < = < < fo fo fo fo c c c c c c c μ μ μ μ (A-7). 1 μ = c c (A-8) 5

29 Refeences Baa N. Opal ng n anks e-off decsons unde he possle pleenaon of a susdy schee: A eal opon appoach oneay and Econoc Sudes 19 (3) pp Boo A. Relaonshp ankng: Wha o We Kno Jounal of nancal Ineedaon 9 pp A. and R. Pndyck Invesen unde Unceany Pnceon Unvesy Pess Elyasan E. and. Goldeg Relaonshp lendng: a suvey of he leaue Jounal of Econoc and Busness 59 pp Genade R. The Saegc Eecse of Opons: evelopen Cascades and Oveuldng n Real Esae akes Jounal of nance 51 pp Ka. and T. Shaa Real Opons n An Olgopoly ake Kyoo Econoc Reve 74 pp eland H. Copoae e Value Bond Covenans and Opal Capal Sucue Jounal of nance 49 pp ella-baal P. and W. Peaudn Saegc e Sevces Jounal of nance 5 pp Weeds H. Saegc elay n a Real Opons odel of R& Copeon Reve of Econoc Sudes 69 pp