``leasing and risk``

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ELSEVIER Journal of Financial Econo mics 42 (1996)333 -364

J O U R N L O F

inancialE ONOMI S

easing and credit risk

Steven R renadier

Gradu ate School ()/ Business, Sta njo rd U niversity, Stan/brd, CA 94305-501 5, USA

Received September 1995 ; final version received M arch 1996)

A b s t r a c t

Despite empirical evidence pointing to a strong similarity between lease contracts andjunk bo nds, the theoretical m ode ling of equilibrium lease determination has been confinedprim arily to default-free leases. This pap er provid es a unified framew ork for determiningthe equilibrium credit spread on leases subject to default risk. The model is flexible enoughto be applied to a wide v ariety of real-w orld leasing structures, includ ing secu rity deposits,required up-front prepayments, embedded lease options, leases indexed to use, and leasecredit insurance contracts.

Key words: Leasing; Options; Credit riskJEL c lass i f i ca t ion :GI2; G13; G32

1 I n t r o d u c t i o n

L e a s i n g i s o ld e n p e r c e i v e d a s a su b s t it u t e f o r d e b t f o r f ir m s t h a t a re ' t o o r i s k y o r

a r e u n a b l e t o a c c e s s c o n v e n t i o n a l d e b t m a r k e t s ' ( L e a s e , M c C o n n e l l , a n d S c h a l l -h e i m , 1 9 9 0 ) . I n th e U . S ., a l m o s t o n e - t h i r d o f t o ta l b u s i n e s s i n v e s t m e n t i s l ea s e d .I n 1 9 9 3 , e q u i p m e n t l e a s i n g a lo n e w a s v a l u e d a t o v e r 1 2 5 m i l l i o n , w i t h sm a l l e rf ir m s r e p r e s e n t i n g a d i s p r o p o r t i o n a t e l y l a rg e s h a r e o f t h is a c t i v i ty. D e s p i t e e m -p i ri c a l e v i d e n c e p o i n t i n g t o a s t ro n g s i m i l a r i ty b e t w e e n l e a s e c o n t r a c t s a n d j u n kb o n d s , t h e t h e o r e t ic a l m o d e l i n g o f e q u i l i b r i u m l e a s e d e t e r m i n a t i o n h a s b e e n v i r tu -a l l y c o n f i n e d t o d e f a u l t - f r e e l e a s e s ( a n o t a b l e e x c e p t i o n i s L e w i s a n d S c h a l l h e i m ,1 9 9 2 , w h o t a k e i n to a c c o u n t d e f a u l t r i sk i n a s in g l e - p e r i o d m o d e l o f t h e e q u i -l i b ri u m r e n t a l r a t e) . G i v e n t h e l a rg e v o l u m e o f r e s e a r c h d e v o t e d t o t h e s t u d y o f

c o r p o r a t e b o n d c r e d i t r is k , i t is p u z z l i n g t h a t t h e r i s k o f le a s i n g h a s b e e n l a rg e l yi g n o r e d . A m o d e l o f l ea s e d e f a u l t r is k i s i m p o r t a n t n o t o n l y f o r t h e d e t e r m i n a t i o n

Com ments from Geert B ekaert, Michael Harrison, Patric H endershott, Craig Lew is (the rcferee), JohnLong (th e editor), Tim Riddiough, C.F. Sinn ans, and Richard Stanton are gratefully acknowledged.

0304-405X/96/ 15.00 1996 Elsevier Science S.A. All rights reservedP I I S O 3 0 4 - 4 0 5 X ( 9 6 ) 0 0 8 8 2 - 3


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334 X R. Grenadier/.lournal q[ Financial Economics 42 1996) 333 364

of equ i l ib r ium ren ta l r a tes, bu t a l so fb r an unders tand ing o f the wide v ar ie tyof lease c lauses and con ven t ions that are wri tten to deal expl ic i t ly wi th credi tr i sk . Th i s paper p rov ides a un i f i ed f ramework fo r de te rmin ing the equ i l ib r iumcredi t spread on leases subject to defaul t r i sk , and appl ies the resul ts to sev-e ra l r ea l -wor ld l eas ing a r rangements such as secur i ty depos i t s , r equ i red up- f ron tprepayments , embedded lease op t ions , l eases indexed to use , and l ease c red i tinsurance contracts .

The percept ion that credi t r i sk plays a s ignif icant ro le in leas ing arrangementsi s r e in forced by m uch of the empi r ica l l i te ra tu re on l ease y ie lds . E ar ly empi r ica lwork expressed surpr i se a t the observa t ion tha t y ie lds on l ease con t rac t s exceeded ,by a wide marg in , y ie lds on app rox ima te ly equ iva len t deb t - f inanc ing a r range-ments . Crawford , Harper, and McConne l l (1981) repor t tha t the l eases in the i rsample were wr i t t en wi th y ie lds s ign i f i can t ly h igher than re tu rns on BBB bondsdur ing the sam e per iod . T h is conf i rmed the resu l t s o f an ea r l ie r s tudy by Sorensenand Johnson (1977) . In pe rhaps the l a rges t s tudy o f l ease y ie lds , Scha l lhe im,Johnson , Lease , and M cCo nne l l (1987) , us ing a sam ple o f 363 con t rac t s , find tha tlease yie lds are se t to compensate the lessor for the defaul t potent ia l inherent inleas ing contracts . In par t icular, both the s ize and the l iquidi ty ra t io o f the lesseehad s ign i f i can t exp lana to ry power in de te rmin ing the l ease y ie ld . The smal le r thelessee and the low er i t s l iquidi ty ra t io , the high er the lease ra te w as se t. 1

A s t rong ana logy can be made be tween l ease con t rac t s and h igh-y ie ld ( junk)bonds . Al tm an (198 9) and Asqui th , Mul l ins , and W ol ff (1989 ) rep or t tha t cum u-la tive de fau l t r a tes on such bonds a re abou t 30 ov er the li f e o f the bonds . Onthose bonds tha t do defau lt , on ly abou t 40 o f the face va lue is r ecovered . As t u d y o f th e e x p o s t p e r t b r m a n c e o f l ea s e s b y L e a s e , M c C o n n e l l, a n d S c h a l lh e i m(1990) f inds that leases display s imilar defaul t proper t ies . The leases in thei rs a m p l e e x p e r ie n c e d a d e f a u lt ra te o f a p p r o x i m a t e l y 2 0 a n d a r e c o v e r y ra te o f38 re la t ive to the or iginal cost o f the asset , or 64 re la t ive to the presen t valueof the rema in ing l ease paym ents p lus es t imated sa lva ge va lue . In add i t ion , thespread be tween con t rac ted and observed y ie lds on l eases c lose ly matches tha t onjunk bonds . For the sample by Lease , McConne l l , and Scha l lhe im, the con t rac ty ie ld exceeds the rea l i zed re tu rn by 2 .62 ov er the 197 3-1982 per iod . B lum eand K eim (199 1) f ind tha t fo r lowe r-grade bonds i s sued dur ing 1977 1978 theprom ised y ie ld exceeds the rea l i zed re tu rn b y 2 .61 . Thus , the re i s ev iden ce tha tleases and high-yie ld bonds are not ent i re ly diss imilar.

The under ly ing ap proach o f th i s paper is in the t rad i tion o f Mi l l e r and U pton(1976 ) and G renad ie r (1995) , wi th the focus on the econom ic aspec t s o f l easing .Leas ing i s s imp ly a me chan ism for se l ling the use o f an asse t fo r a spec if i ed

1However. two other variables that could proxy for default risk, the lessee 's return on assets andleverage ratio, did not have significant explanatory power. Schallheim (1994) states that 'perhaps thelessees size and liquidity ratios capture the default potential in the sample to such a degree that theother two ratios do not help explain lease yields'.


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per iod o f time , wi thou t necess i t a t ing a t r ans fe r o f ownersh ip . T hus , l eas ing p ro-v ides a separa t ion o f ownersh ip f rom use , wi th the l e ssee rece iv ing the benef i t so f use and the l e ssor rece iv ing the l ease pay m ent f low p lus the residua l va lueof the asset . The eco nom ic f ram ew ork o f the mod e l fac i li t a tes an op t ion-pr ic ingapproach to equ i l ib r ium lease va lua t ion . Th is approach has been fo l lowed by sev-e ra l au thors . Smi th (1979) p rov ides an op t ion- l ike charac te r i za t ion fo r a l ease : aT-year l ease i s economica l ly equ iva len t to a por t fo l io tha t s imul taneous ly pur-chases the under ly ing asse t and wr i t es a European ca l l op t ion on the asse t wi thexp i ra t ion da te T and ze ro exerc i se p r ice . McConne l l and Scha l lhe im (1983) andS c h a l lh e i m a n d M c C o n n e l l ( 1 9 8 5 ) u s e a n o p t i o n - p r ic i n g f r a m e w o r k to p r o v i d ea thorough ana lys i s o f a wide v ar ie ty o f l ease op t ions and insurance con t rac t s.Grenad ie r (1995) combines an op t ion-pr ic ing approach wi th a compet i t ive indus -t ry equ i l ib r ium to p rov ide a un i f i ed f ramework fo r va lu ing l eases .

W hi le th i s approach is in tu i tive and fac i l it a tes t ack l ing som e of the more com -p l ica ted aspec t s o f r ea l -wor ld l easing , i t a l so neg lec t s som e very impo r tan t l eas ingconside rat ions . In par ticular, there is a vas t l i tera ture on the im po rtanc e o f taxesin the lease- vers us-b uy decis ion (e .g . , Schall , 1974; M yers , D i ll , and Bau t is ta ,1976; B rea ley and You ng , 1980; Lew is and Scha l lhe im, 1992). W hi le the p resen tana lys i s focuses on the no n tax a spec t o f l easing , the inc lus ion o f t ax con s idera -t ions would add g rea te r r ea l i sm to the ana lys i s . Unfor tuna te ly, incorpora t ing t axcons idera t ions in to the p resen t mode l would vas t ly inc rease i t s complex i ty.

Th is pape r beg ins w i th a mode l o f the equ i l ib r ium de te rmina t ion o f l easera tes when the payments a re sub jec t to de fau l t r i sk . In equ i l ib r ium, the p romisedlease paym ents m us t com pensa te the l e ssor fo r bo th the fo rgo ne use o f the asse tand the po ten t ia l con sequenc es o f de fau l t . In the m ode l , the u nder ly ing a sse t sva lue ev o lves s tochas t ica l ly. T he ev en t o f de fau l t i s de te rmined when the l e ssee ss tochast ic cash f low (or asset value) fa l ls to a lower defaul t barr ier. Upon defaul t ,the l e ssor rece ives on ly a f rac t ion o f the l ease s va lue due to the cos t s o f de fau l t.However, in equi l ibr ium, the contracted lease ra te i s se t such that the lessor i sindifferent betw een lessees o f d i fferent credi t qual i ty. Th e lease cred i t spread isde te rmined such tha t any sa le o f the u se o f the asse t ove r a g iven per iod o f t imemus t have the same equ i l ib r ium va lua t ion .

Fo l lowing an ana lys i s o f l ease c red i t r isk in genera l , I m ov e on to ana lyzesom e o f the mo s t com m on leas ing co nven t ions tha t dea l d i rec t ly wi th c red i t ri sk .To m ot iva te the app l ica tions , cons ider the fo l low ing exa m ples o f rea l is t ic l eas ingar rangements :

1. Yo u r c o m p a n y i s n e g o t ia t in g t o l e a se c o m p u t e r e q u ip m e n t . T h e m a n u f a c t u r e ri s wi l l ing to l ease to you , p rov ided you pu t a secur i ty depos i t in to esc row.

H ow should thc inclus ion of a secur i ty depo si t affect the equi l ibr ium renta lra te?

2. Y ou are leas ing office spac e to a tenant o f que st ionable credi t qual i ty. W hileyou se t a r en ta l r a te tha t is nea r the l eve l o f yo ur sa fe r t enan ts , you dem and


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336 s . R . G r e n a d i e r / J o u r n a l ()] F i n a n c ia l E c o n o m i c s 4 2 ( 1 9 9 6 ) 3 3 3 ~ 6 4

that some of the future rent payments be made up-front. How many months'prepaid rent can you demand in equilibrium?

3. As an issuer of asset-backed securities, you bundle lease contracts. You areseeking an insurer to provide credit enhancement so that the securities canobtain a AAA rating. How much should you pay for such insurance?

The first contract provis ion 1 analyze is the security deposit requirement.Security deposits are very common on leases of capital equipment. As statedin Schallheim (1994), 'third-party guarantees and the pledging of other assetscan be the difference between approval to lease and no credit approval'. Securitydeposits lead to a lessening (although not necessarily a complete removal) ofcredit risk. Of course, in return the contracted lease payment must be lower.

The second contract provision I analyze is the requirement of prepaid rent.For example+ it is quite common in the rental of residential property for the firstand last months' rent to be paid in advance. In fact, in many leasing contractssubstantial prepayment of rent is required in order to lessen the potential lossesof default. In return, the rental rate on the lease is lowered.

The third application of the model is the valuation of lease credit insuranceand guarantees. Such insurance provides payments to the lessor in the event ofdefault. For example, in the leasing of commercial real estate space, the land-lord may require the purchase of a third-party guarantee of the promised rentalpayments for the full term of the lease. Such guarantees are sought for small ornewly established tenants. In addition, such insurance exists in the form of creditenhancement on lease-backed securities. Similar to other asset-backed securities,lease-backed securities can involve full or partial insurance against credit loss inthe form of credit enhancement provisions.

The fourth application of the model deals with valuing options to purchasethe asset under conditions of default risk. Many leases contain an option for thelessee to purchase the underlying asset for a fixed price at the end of the leaseterm. Such options clearly have value, and of course affect the equilibrium rentalrate. McConnell and Schallheim (1983) provide a detailed analysis of the optionto purchase. The option to purchase helps align the incentives of the lessee andlessor. The option provides an incentive to the lessee to maintain the asset andto make the required rental payments. Should default occur, the lessee loses allrights to a potentially valuable option. The model is used to value this option topurchase in an cnvironment of credit risk.

The final application is an analysis of percentage leases, the prevailing formof shopping center lease. In the U.S., shopping centers contain over 4.5 billionsquare feet and account for over 700 billion of retail sales (Shopping CenterDirections, 1992). Percentage leases are indexed to a store's sales, so that thelandlord shares a portion of the tenant's success. These sharing arrangements areespecially pronounced for the smaller mall tenants. Thus, the impact o f creditrisk is likely to be sizable.


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In a more general equi l ibr ium set t ing, such factors as the specif ic terms ofthe l ease con t rac t and the de fau l t p ropens i ty o f the l e ssee wi ll be d e te rminedendog enous ly. Ho wev er, the va lua t ion appro ach deve lo ped in th is paper wi l l se rveas an essen t ia l com pon ent o f the assoc ia ted genera l equ i l ib r ium prob lem . Forexam ple , whi le the de fau l t bo und ary i s l ike ly to be chosen as pa r t o f a l e ssee sop t imiza t ion p rob lem, the equ i l ib r ium lease ra te wi l l be de te rmined by the p resen tm od e l s fo rm ula t ion , con t ingen t on the resu l ting op t im al de fau l t boun dary, In theone-per iod mode l o f Lewis and Scha l lhe im (1992) , bo th the l ease ra te and thedefau l t p robab i l i ty a re de tenn ined endogenous ly. More genera l ly, the type o fcon t rac t chosen shou ld be the on e tha t is mo s t e ffi c ien t in t e rm s o f min im iz ingmoni to r ing cos t s , adverse se lec t ion , mora l hazard , t r ansac t ions cos t s , and o therfac to rs re la ted to the p rob lem s of con t rac ting be tw een l essees and l essors . Sm i tha n d W a k e m a n ( 1 9 8 5 ) p r o v i d e a u n i fi ed a n a l y s is o f h o w t h e e x i s te n c e o f v a r i o u sincent ive issues exp la in the use of d is t inct contractual pro vis ion s foun d in leases .However, con t ingen t on the chosen con t rac t type , the p resen t mode l p rov ides theequi l ibr ium lease value and renta l ra te .

The pa per p roceed s as fo l lows . Sec t ion 2 deve lo ps the mo de l o f equ i l ib r iumleas ing under cond i t ions o f c red i t r i sk . Sec t ion 3 ana lyzes the equ i l ib r ium leasecredi t spread. Sect ion 4 character izes the equi l ibr ium secur i ty deposi t provis ion.Sec t ion 5 ana lyzes the p repayment c lause in r i sky l eases . Sec t ion 6 ana lyzeslease de fau l t insurance . Sec t ion 7 p rov ides an an a lys i s o f l ease purch ase op t ions .Sec t ion 8 ana lyzes pe rcen tage l eases , and Sec t ion 9 conc ludes .

2 . T h e b a s i c m o d e l

In th i s sec t ion , I p rov ide an equ i l ib r ium de te rmina t ion o f l ease ra tes when thelease payments a re sub jec t to c red i t r i sk . Under a l ease a r rangement , the l e ssorse ll s the use o f the asse t fo r a spec i fi ed pe r iod o f t ime , and the l e ssee p rom ises tom ake spec i fi ed lease paym ents ove r the l ease t e rm. In equ i l ib r ium , the p rom isedlease payment i s se t so as to compensa te the l e ssor fo r bo th the fo rgone use o fthe asse t and the po ten t i a l consequ ences o f de fau lt .

There a re two sources o f uncer tain ty. F i r st , the se rv ice f low of the l easedasse t ( and hence i t s va lue) i s s tochas t i c . Second , the t iming and consequenceso f defau l t are a lso s tochast ic . The equ i l ibr ium lease valuat ion wi l l take thesefac to rs in to account . T he so lu tion ap proac h wi l l be tha t o f op t ion-pr ic ing ana lys i s .Trad i t ional ly, in the opt ion -pr ic ing l i tera ture , pr ices of con t ingen t c la ims are basedon a rb i t rage a rguments . However, such an approach requ i res assumpt ions abou tthe l iquidi ty o f the und er lying asset . In the case of leas ing mark ets , whe re theunder ly ing asse t s such as o ff i ce bu i ld ings and heavy equ ipment a re sub jec t tosubstant ia l t ransact ions costs , indivis ibi l i ty, and the inabi l i ty to be sold shor t , sucharb i t rage a rguments a re pa r t i cu la r ly ques t ionab le . An equ i l ib r ium approach re laxesthe t r adab i l i ty assumpt ions needed fo r a rb i t rage p r ic ing , a l though an appropr ia te


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equilibrium model must be chosen. For simplicity, I assume risk neutrality, sothat all assets are priced to yield an expected rate of return equal to the risk-freerate, r. This seemingly restrictive assumption can be relaxed by adjusting thedrift rates to account for a risk premium in the manner of Cox and Ross (1976).

The underlying demand for the use of the leased asset results from the value ofits use as an input in economic activity. Let the value of the service flow from theasset be denoted by S ( t ) , where S ( t ) evolves as the following diffusion process:

dS = ~ . , . (S , t )Sd t + c r, (S , t )Sdz , . , (1)

where :~(S,t) is the instantaneous conditional expected percentage change in Sper unit time, ,.(S, t) is the instantaneous conditional standard deviation per unittime, and dz,. is the increment of a standard Wiener process. The sign of theexpected growth rate, :s(S,t), is not restricted. Thus, the service flow from anasset may appreciate or depreciate over time.

Initially, consider the simple case of a T-year lease to a riskless lessee. Thisrepresents the sale of the use of the asset for T years in return for a sure paymentflow of R ( T ) . The value of the use of the asset for T years, Y ( S , T ) ,can beexpressed as follows:

/ / )( S , T ) = E e - r r S ( t ) d t . (2)As in many models of risky bond pricing (e.g., Merton, 1974; Black and Cox,1976; Leland, 1994), I assume that a riskless asset exists that pays a constant rateof interest, r. This simplification permits us to focus on the credit risk aspects ofleasing directly. To extend the model to a stochastic interest rate environment, aterm structure model (e.g., Cox, Ingersoll, and Ross, 1985; Vasicek, 1977) mustbe appended to the asset valuation model. 2

The equilibrium riskless lease rate, R(T), will then be the payment flow' whoseannuity value equals g( T). Thns, R ( T ) must satisfy the following equilibrium

equality:

( r ) y ( s , T ) (3)R (T ) 1 e ,-r

The value of the asset is simply the present value of the service flow of theasset. The value of the underlying asset, V ( S ) , is then the perpetuity value of S(t):

V ( S ) = lim Y ( & T ) . (4)

2 A n e x t e n s i o n o f t h e p r e s e n t m o d e l t o a c c o m o d a t e s t o c h a s t i c i n t e r e s t r a t e s c o u l d b e a c c o m p l i s h e d i na m a n n e r s i m i l a r t o B r c n n a n a n d S c h w a r t z 1 9 8 0 ) a n d K i l n , R a m a s w a m y, a n d S u n d a r e s a n 1 9 9 3 ) .U s i n g n u m e r i c a l s o l u t i o n t e c h n i q u e s , t h e y f i n d t h a t t h e y i e l d s p r e a d s b e t w e e n c o r p o r a t e a n d Tr e a s u r ybon ds a re qu i te in sens i t ive to in te res t ra te uncer ta in ty.


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S . R , G r e n a d i e r / J o u r n a l o F i n a n c ia l E c o n o m i c s 4 2 ( 1 9 9 6 ) 3 3 3 ~ 6 4 9

N o w c o n s i d e r a T- y e a r l e a s e w h e n t h e l e s s e e is s u b j e c t t o th e r i s k o f d e f a u lt .W h i l e t h e l e s s e e m a y p a y t h e p r o m i s e d r e n t a l r a t eP T ) o v e r t h e e n t i r e l e a s ep e r i o d , t h e r e i s a l so t h e p o s s i b i l i t y t h at t h e p r o m i s e d p a y m e n t i s n o t r e a l i z e d .L e a s e d e f a u l t i s c o n t r a c t u a l l y d e fi n e d , a n d m a y c o n s i s t o f l a te p a y m e n t s , f a i l u r et o m a i n t a i n t h e a s s e t , o r f a i l u r e to f u l f i ll t h e c o v e n a n t s o f o t h e r f i n a n c i a l c o n -t r a c t s. T h e c a u s e s o f d e f a u l t a r e a l s o v a r i e d . B u s i n e s s f a i l u r e , l i q u i d i t y c r i s e s , a n dd i s s a t i s f a c ti o n w i t h t h e p e r f o r m a n c e o f th e l e a s e d a s s e t m a y u n d e r l i e t h e f a i lu r et o u p h o l d t h e t e r m s o f th e l e as e .

L e a s e d e f a u l t m a y b e b r o k e n d o w n i n to tw o p a r t s: t h e o c c u r r e n c e o f d e f a u l ta n d t h e c o n s e q u e n c e s o f d ef a u l t. To m o d e l t h e o c c u r r e n c e o f d e f a u l t, I f o l l o wB l a c k a n d C o x ( 1 9 7 6 ) , L e l a n d ( 1 9 9 4 ) , a n d L o n g s t a f f a n d S c h w a r t z ( 1 9 9 5 ) b ya s s u m i n g t h a t d e f a u l t o c c u r s w h e n a f in a n c i a l s t at e v a r i a b l e o f t h e l e s s e e , su c ha s c a s h f l o w o r a s s e t v a l u e , f a l l s t o a l o w e r t h r e s h o l d K . T h i s d e f i n i t i o n o fd e f a u l t i s c o n s i s t e n t w i t h b o t h t h e c a s e i n w h i c h t h e l e s s e e i s i n s o l v e n t b e c a u s ei t d o e s n o t g e n e r a t e s u f f i c ie n t c a s h f l o w to m e e t c u r r e n t o b l i g a t i o n s , a s w e l l a s t h ec a s e i n w h i c h t h e l e s s e e s a ss e t s v i o l a t e m i n i m u m n e t w o r t h o r w o r k i n g - c a p i t a lr e q u i r e m e n t s . S u c h a d e f a u l t c o n d i t i o n c a n t h u s a c c o u n t f o r b o th f l o w - b a s e d a n ds t o c k - b a s e d i n s o lv e n c y, a s d i s c u s se d in Wr u c k ( 1 9 9 0 ) a n d K i m , R a m a s w a m y ,a n d S u n d a r e s a n ( 1 9 9 3 ) . O t h e r s i m i l a r a p p r o a c h e s t o m o d e l i n g th e o c c u r r e n c e o fd e f a u l t ar e d i s p l a y e d in M e r t o n ( 1 9 7 4 ) , D u f f le an d S i n g l e t o n ( 1 9 9 5 ) , a n d J a r r o wa n d Tu m b u l l ( 1 9 9 5 ) .

L e t X t ) d e n o t e a s t a t e v a r i a b l e o f t h e l e s s e e , s u c h a s c a s h f l o w o r a s s e t v a l u e .I f X t ) f a l l s t o t h e b o u n d a r y K , t h e l e s s e e d e f a u l t s . T h i s c o u l d r e p r e s e n t l a t e o rm i s s e d p a y m e n t s , o r o u t r i g h t a b r o g a t i o n o f t h e l e as e . T h u s , d e f a u l t o c c u r s o n aT- y e a r l e a s e i f t h e f i rs t p a s s a g e t i m e o fX t ) t o th e b o u n d a r y K i s l e s s t h a nT. M a t h e m a t i c a l l y, d e fa u l t o c cu r s a t t i m e t * w h e r e t* = i n f [ t


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S.R. G renadier/JourmH of Financial Economics 42 1996) 33 33 64 3 4 1

i n i s

dF = c~ S , t )S Fss +p S ,X, t)c~ . S , t) c ~ X, t )SXFsx

1 2 ]+ 5c ;~ X, t )X2Fxx + ~ S , t )SFs + ex X, t )XFx + F t d t

+ ~s S , t )SFsd z, + ax X, t )X Fx dzx . 6 )

I n a d d i t i o n t o t h e c a p i t a l g a i n r e c e i v e d o n t h i s l e a s e , t h e l e s s o r r e c e i v e s a c a s hi n f l o w d u e t o th e r e n ts o f P d t . T h e r e f o r e , t h e to t a l e x p e c t e d r e t u r n o n F p e r u n i tt i m e , / ~ f , i s d e f i n e d a s

I~ F dt - E [d F +FPdt ] . 7 )

S e t t in g t h e e x p e c t e d r e t u rn e q u a l t o t h e e q u i l i b r iu m r e t u rn r a n d s i m p l i f y i n gy i e l d s t h e f o l l o w i n g e q u i l i b r i u m p a r t i a l d i f f e r e n t i a l e q u a t i o n :

1 2 1 2a , S , t )SZFss + p S ,X, t )a , S , t)ax X, t )SXF sx + ~ax X, t)X2F~c

+ ~ , S , t )SFs + ex X, t )XF y + F~ + P - rF . 8 )

T h i s p a r t ia l d i f fe r e n ti a l e q u a t i o n i s s o l v e d s u b j e c t to b o u n d a r y c o n d i t i o n s d e -f in e d b y A l t e rn a t i v e 2 . A t th e m o m e n t d e f a u lt o c c u r s, w h e nX t ) ~ K ,t h e l e s s o ro b t a i n s a f r a c t i o n o f t h e v a l u e o f t h e r e m a i n i n g l e a s e , 1 - c o ) -Y [S t), T - t].T h i s d e f a u l t b o u n d a r y c o n d i t i o n c a n b e w r i t t e n a s

F S , K , t ; P, T ) - - 1 - c o)- Y[S t ) , T t ] . 9 )

A l t e r n a t i v e l y, i f n o d e f a u l t o c c u r s o v e r t h e l if e o f t h e l e a s e , th e t e r m i n a l c o n -d i t io n e n s u r e s t h a t t h e v a l u e o f t h e r e m a i n i n g r e n t a l p a y m e n t s e q u a l s z e r o a tm a t u r i t y :

F S , X , T; P, T ) ~ O . 1 0 )

N o w, b y s p e c i f y in g t h e p a r a m e t e r s o f t h e s t o c h as ti c p r o c e s s e s f o rS t ) a n dX t) , [es S , t ) ,c~x X, t ),p S ,X, t ) ] ,w e c a n s o l v e p a r t i a l d i f f e re n t i a l e q u a t i o n 8 )s u b j e c t t o b o u n d a r y c o n d i t i o n s 9 ) a n d 1 0 ) . I n g e n e r a l , c l o s e d - f o r m s o l u t i o n sw i l l n o t b e a v a i l a b l e . H o w e v e r , s u c h e q u a t i o n s c a n b e s o l v e d u s i n g n u m e r i c a lt e c h n i q u e s .

F i n a l l y, to d e r i v e t h e e q u i l i b r iu m r i s k y r e n t o n a T - y e a r l e a se , I u s e t h e e q u i l i b -r i u m c o n d i t i o n t h a t t h e v a l u e s o f A l t e r n a t i v e s 1 a n d 2 m u s t b e e q u a l. T h e r e f o r e ,t h e e q u i l i b r i u m r e n t o n a T- y e a r l e a s e s u b j e c t t o d e f a u l t r i s k ,P T ) , i s s e t a t t imez e r o s u c h t h a t t h e f o l l o w i n g e q u a l i t y i s s a t i s f i e d :

F S , X , O ; P T ) , T ) = Y S , T ) . 1 1)

I w i l l n o w a s s u m e t h a tS t ) a n d X t ) f o l l o w c o r r e l a t e d g e o m e t r i c B r o w n i a nm o t i o n s j o in t ly l o g - n o r m a l ) . T h u s~s S, t) , ccx X,t) ,a n d p S , X , t ) a r e e q u a l t o


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34 S.R. Grenadier/Journal o/ Financial Economics 42 1996) 333 364

the constants ,~,., ~ , and p , respect ive ly. Such an ass um ption is ve ry s tandard inthe f inancia l and real -opt ions l i tera ture . This wi l l permit a c losed-form solut ionto be obta ined.

Under the cor re la ted geomet r ic Brownian mot ion assumpt ion , the so lu t ion toEq . (8 ) sub jec t to boundary cond i t ions (9 ) and (10) i s

F S X . t ; P ,T ) cP l P,~ ) b2 X,z) P. - + c b 3 y , r ) . 1 o ) Sr r fZs

( 1 2 )

wh ere z - T t , and the funct ions q51(P,r ) ,q O 2 ( X , 2 ) ,and 4~3(X,r) are presentedin the Appendix.

The va lue o f Al te rna t ive 2 ,F S ,X , t;P, T) ,whi le complex , i s qu i t e eas i ly in -terpreted. The f i rs t term , @ l(P,z ) , i s equal to the valu e o f the renta l f low i fthe lease is r i sk-f ree . The second term, q~2(X,r) -P/r) , i s the va lue o f thepotent ia l loss o f al l con tracted renta ls i f the lessee defaul ts . Th e th i rd term ,q~3(X,r) . (1 v)) S// - ~ s ) ) , i s t he v a l u e o f t he r e n t th a t is r e c o v e r e d s h o u ldthe lessee defaul t . Therefore , the to ta l value to leas ing the asset according toAltern at ive 2 is equal to a por t fol io con sis t ing o f ( i ) a r iskless lease , ( i i) a shor tposi t ion in a contract which pays out the credi t loss under a lease , and ( i i i ) along pos i t ion in a con t rac t which pays the amount recovered f rom a l ease de fau l t .

By app ly ing the so lu t ion to the log-normal case de r ived in Eq . (12) and in theAppendix to the equ i l ib r ium ren t equa t ion , Eq . (11) , the equ i l ib r ium ren t on alease subject to credit r isk is

r . S - e r ~ , )r , 0 ) ]p ( T ) = ( ~ ) [ 1 - @ 3 ( X , T ) . ( I]- 7 7 - 7Y 7 ~-2(~-~k ~ ] 13)

3 . A n a l y s i s o f th e l e a s e c r e d it r i s k s p r e a d

Les sors o f assets wi l l adjus t the rents cha rged to a r isky lessee to ensureindifference re la t ive to a lessee wi th no credi t r i sk . The credi t r i sk spread, D(T) ,is def ined as P T ) - R T ).In th is sect ion, I analyze the factors that inf luencethe s i ze o f the cush ion dem ande d on r i sky l eases .

In essence, th is analys is i s ana logo us to the s tudy o f the r isk s t ructure o fin te res t ra tes . In M er ton (1974) , the com para t ive s t at ic s o f the d i ffe rence be tw eenthe y ie ld on a de fau l t ab le bond and a r i sk less bond a re de r ived . Unl ike the caseof the bond , which i s e ssen t ia lly the sa le o f a lump sum o f cash in re tu rn fo ra ser ies of fu ture cash f lows, a lease is the sa le of the ec on om ic ben ef i ts ofa specif ic asset . Thus , the lease credi t r i sk spread wi l l be re la ted to var iablesunder ly ing the p rocesses o f bo th the l e ssee s f i rm va lue and the l eased asse t , a swel l a s numerous con t rac tua l p rov i s ions .


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S.R. Grenadier/Journal o1 Financial Economics 42 (1996) 333 364 343

W h e n S a n d X a re c o r r el a te d g e o m e t r i c B r o w n i a n m o t io n s , a c lo s e d - f o r msolu tion fo r the equ i l ib r ium c red i t sp read can be de r ived . Su b t rac ting Eq . (13)f rom the so lu t ion to Eq . (3 ) p rov ided in the Appendix , the c red i t r i sk sp read canbe wri t ten as

D T) = ~ 1 - e - r - qO2(X, T ) i - ~ - e r ~ - J

14)

The impa c t o f va r ious p a ram ete r changes on the c red i t r i sk sp read a re i l lust ratedin Figs . la throu gh l f . 3

F ig . l a p lo t s the impac t o f l ease matur i ty (T) on the sp read , In essence , th i s

is a p lot of the term s t ructure o f lease credi t spreads . T he ter m s t ructure o f leasec red i t sp reads i s upward-s lop ing . Tha t i s , fo r longer- t e rm leases , the p remium tha ta r isky tenant must pay re la t ive to a r iskless tenant increases . The in tui t ion iss imple: the long er the term, the grea ter the l ikel ihood o f defaul t . Since the lessorrecognizes th i s a t the ou t se t , the ren t mus t compensa te fo r the h igher expec tedlosses . Con s ider the emp i r ica l im pl ica t ions o f th is r esu l t fo r the l eas ing o f rea les ta te . F or the leas ing o f res ident ia l space, sh or t - term leases are s tandard wi th thevas t major i ty o f lease t e rms in the range o f one mo nth to one year. Ho wev er, fo rthe leas ing of off ice space, long er- term leases are s tandard, typica l ly in the range

of th ree to ten years (DiPasqua le an d W hea ton , 1996). There fo re , the mo de lwould p red ic t tha t r i sky o ff i ce t enan t s wi l l be charged a g rea te r r en t p remiumthan r i sky apar tm ent t enan ts . In add it ion , the co ncav i ty o f the t e rm s t ruc tu re o flease c red i t sp reads sugges t s tha t fo r long- te rm leases , the p remium requ i red toc o m p e n s a t e f o r r i s k m o d e r a t e s .

Fig . lb p lots the im pac t of the f ract ion o f the lease value los t in defau l t ((~J) onthe spread. The spread is increas ing and l inear in {o, as i s readi ly apparent f romEq. (14) . Th i s is a s expec ted , s ince the im pac t o f de fau l t i s m ore de t r imenta lthe greater i s (u . Th ere are sev eral empir ical imp l icat ions . For ex am ple , i f thelessor has g rea te r nego t ia t ing power than the l e ssee ( say, due to f i rm s ize o rresources) , then ( , ) (or s imi lar ly the expected value of ~o) is l ikely to be lower.Th is wo uld im ply that , al l e lse equal , larger lessors can charge lowe r credi tspreads than sm al ler lessors . In addi t ion, (fo is l ikely to depe nd on the n um be r ofother c redi tors o f the lessee . I f the lessee defaul ts o n the lease , i t i s a lso l ikely to

3Wh i le the r e su l t s he re a re de r ived und er a base case se t o f in i ti a l pa ram ete r va lues , t hey ho ld und era wide va r i e ty o f r ea li s t ic p a ra ine te r va lues . H ow eve r, i t i s impo r tan t to no te tha t fo r s ign i f i can t lyn e g a t i v e v a l u e s o f p , a n d r e la t iv e l y s a l e l e a s es h i g hX/K), t h e c r e d i t s p r e a d c a n a c t u a l l y b e c o m e

nega t ive . T ha t i s , one mig h t theore t i ca l ly c ha rg e a low er r en t fo r a l e s see sub jec t to de fau l t than fo ra l e s see wi th no c red i t r i sk . The key to th i s su rp r i s ing re su l t i s in the nega t ive co r re l a t ion . Defau l ti s m o s t l i k e l y t o o c c u r w h e n t h e a s s e t c a n b e r e - l e a s e d a t u n e x p e c t e d l y h i g h v a l u e s , p o t e n t i a l l y e v e ngrea t eno ugh to com pen sa te fo r the f r ac t ion o f the se rv ice f low, {o , lo s t in de fau l t . The cond i t ionsunder which th i s r e su l t ob ta ins a re un l ike ly to be obse rved in ac tua l l eas ing marke t s .


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3 4 4 S . R . G r e n a d i e r 1J o u r n a l q / F i n a n c i a l E c o n o m i c s 42 ( 1 9 9 6 ) 3 3 3 3 6 4

SpreadF i g l a

1o

1

Spread

2

0 I I I I I0.05 095

S p r e a d 40

3O

20

io

o

Spread2

05

Fig lcSpread 2

I1.5

X/K

i5

1

05

(I

Spread10

J

/ ~ p =0

= . 2

I I I I 0

05(5

I I I Io I

P

o5

F i g . I . A n a l y s i s o f t h e l e a s e c r e d i t s p r e a d .

A l e a s e c r e d i t s p r e a d i s d e f i n e d a s t h e d i f f e r e n c e b e t w e e n t h e e q u i l i b r i u m r e n t p a y m e n t o n a

7 ' - y e a r l e a s e s u b j e c t t o c r e d i t ri s k a n d t h a t c h a r g e d o n a r i s k le s s T- y e a r l e a s e . T h e l e a se c r e d i ts p r e a d i s d e r i v e d i n E q . ( 1 4 ) . T h e s ix g r a p h s i n th e l i g u r c s h o w t h e e ff e c t o f c h a u g e s i n t h e u n d e r -

l y i n g p a r a m e t e r v a l u e s o n t h e e q u i l i b r i u m c r e d i t s p r e a d . F i g . l a s h o w s t h e e ff e c t o f i n c r e a s i n g T,t h e t e r m o f t h e l e a s e . F i g . l b s h o w s t h e e f f e c t o f i n c r e a s i n g ~o, t h e p r o p o r t i o n o f t h e l e a s e v a l u e

l o s t in th e e v e n t o f d e fa u l t . F i g . l c s h o w s t h e e f f e ct o f i n c r e a s i n g t h e r a t i o o f t h e l e s s e e ' s a s s e tv a l u e t o t h e d e f a u l t t r i g g e r ,X/ K: t il e c l o s e r t h e r a t i o i s t o o n e , t h e g r e a t e r th e l i k e l i h o o d o f d e f a u l t .F i g . l d s h o w s t h e e ff e c t o f t h e c o r r e l a t i o n , p , b e t w e e n t h e l e a s e d a s s e t v a l u e a n d t h e l e s s e e ' s a s s e t

v a l u e . F i g . l e s h o w s t h e e l t; ec t o f t h e v o l a t i l i t y o f th e l e a s e d a s s e t , o -~ , f o r d i f f e r e n t v a l u e s o f t h ec o r r e l a t i o n c o e f f i c ie n t . F i g . I f s h o w s t h e e f f e c t o f t h e v o l a t i l it y o f t h e l e s s e e ' s a s s e t v a l u e , cry. T h e

de fau l t pa r a m e te r va lue s a r e : q O , ~v - 0 . 02 , o-~ - 0 . 15 , o2, 0 . 15 , p - 0 . 5 , r 0 . 05 , u ) 0 . 2 ,

T - 5, S 10, an d X,, 'K - 1.2.


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s[R. Gretuutier Journal o Financial Econom ics 42 (1996) 33 3~ 64 345

defaul t on i ts o ther f inancia l obl igat ions . Thu s , the pr ior i ty o f the c la ims, and there la tive ba rga in ing po we r o f the assoc ia ted c la imants , wi ll de te rmine the ex pec tedm agn i tude of c ,) . Al l e lse equal , the lease ra te sh ould be high er the g reater thenum ber o f fixed ob l iga t ions o f the l e ssee .

F ig. l c p lo t s the impac t o f the ra tio o f the l e ssee s a sse t va lue (o r cash f low)to de fau l t t r igger( X / K ) . Not ice f rom Eq . (1 4) tha t the va r iab les X and K a ffec tD ( T ) only th rough the ra t ioX / K . Thus, th is ra t io , combined wi th the s tochast icp r o p e r t i e s o fX ( t ) , se rves to de f ine the l ike lihood o f de fau l t. A s one w ould ex-pect , the credi t spread is decreas ing in the ra t ioX/ 'K. The higher the ra t io , thelower the l ike l ihood of de fau l t dur ing the l ease te rm. A sX / K increases , the credi tspread fa l ls to zero and the r isky lease ra te approaches that of a r i sk-f ree lease . Inaddi t ion, the spread is convex in the ra t io~,"K. This suggests that the spread in-c reases rap id ly as the l e ssee s de fau l t p robab i l i ty inc reases. T he impl ica t ion i s tha twhi le the re may be l i t t l e d i ffe rence in the ren t p remium charged to AAA lesseesand AA lessees , the re wi ll be a s ign if i can t p rem ium charged to jun k t enan ts .

F ig . ld p lo t s the im pac t o f the cor re la t ion (p ) be tween the l e ssee s f inn va lueand the under lying asset on the spread. The spread is increas ing in the corre la t ion.The g rea te r the cor re la t ion , the w orse the an t i c ipa ted imp ac t o f de fau l t. A h ighercor re la tion m eans tha t w hen the f i rm s asse ts f a l l enough in va lue to t r iggerdefaul t , i t i s m ore l ikely that the un der lyin g asset (an d the lea se s valu e) w i llbe low. S ince h igher co r re la tion m agni f i es the im pac t o f de fau l t, the sp read iswidened. This resul t has important empir ical impl icat ions . I t suggests that thespread charged to l e ssees wil l depend on how m ov em ents in the l e sse e s cashf lows are cor re la ted w i th m ov em ents in the under ly ing l eased asse t s va lue . Th iscou ld me an that the indu st ry of the lessee wi l l mat ter in se t t ing the lease term .For example , cons ider two lessees f rom d i ffe ren t indus t r i es who s ign l eases tb rthe same asset , under the same terms, and wi th ident ical defaul t probabi l i t ies . Thef irm who sc fo r tunes a rc m os t cor re la ted wi th the va lue o f the und cr ly ing asse twi l l pay a higher rent . This would imply, for ins tance, that the rent charged to ana i r l ine fo r l eas ing an a i rc ra f t would be h igher than tha t charged fo r a comparab lyr isky operat ing company ( in an indust ry unrela ted to the a i rcraf t indust ry) that i sleas ing the a i rcraf t for i t s corporate f lee t . Similar ly, the rents charged to tenantsin a sh opping mal l wi l l va ry acco rd ing to the cor re la t ion o f the sa les o f anindividual s tore wi th the sa les o f the m al l as a whole . A l l e lse equal , a mal lshop w hose sa les possess low or nega t ive cor re la t ion w i th the sa les o f the o thers tores in the mal l should pay a lower rent .

Fig . le p lots the im pac t of the vola t i l i ty of the unde r lying asse t (or,) on thespread. S imu lat ions rcv cal that the effect dep end s on the s ign of p . I f p i s posi t ive ,the spre ad is increasin g in r~,.. If p is ne gativ e, the spre ad is dec reas ing in rT,..If /) is zero , the sp read is co nsta nt in r ;, .. Th e in tuit ion is as follows . I f p isposi t ive , the even t of del ;aul t i s l ikely to occur w hen the un der lyin g asset valueis depressed. I f cr~ is h igh, there is a greater l ikel ihood o f unu sual ly low re co ve ryva lues . Thus , the sp read m us t be h igh. C onv erse ly, i f p i s nega t ive , the even t o f


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346 s. R. Grenadier/Journal ~[ Financial Economies 42 1996) 333 364

default is likely to occur when the underlying asset value is unexpectedly high.With high a~, there is a greater likelihood of unusually high recovery values.Thus, the spread will be low. When p is zero, the event of default is uncorrelatedwith recovery values; the increased volatility is not priced.

Fig. I f plots the impact of the volatility o f the lessee s financial state variable(~x) on the spread. The spread is increasing in ax. For any initial level of X,the expected time of default is nondecreasing in ax. Because the lessor prefers toavoid the consequences of lease default, the spread must be set high enough tocompensate for such risk. This would imply that lessees with lower cash flow orasset volatility will be charged lower rents, all else held constant. For example,firms with divisions in diversified lines of businesses may pay lower rents thanmore concentrated firms.

4 Th e equi l ibr ium secur i ty depo s i t

Many real-world lease contracts contain clauses that provide some form ofprotection to the lessor against lessee default. A very common clause in leasesis for the payment of a security deposit. A security deposit can take a varietyof forms. The lease could require the lessee to place funds in escrow as securityagainst default or require a letter of credit from a third party. Alternatively,the lessee may be required to pledge personal assets as collateral in the eventof default. All of these methods lead to a lessening (although not necessarilya complete removal) of credit risk. Of course, in return the contracted leasepayment must be lower.

The model permits a determination of the equilibrium lease rate on a lease witha security deposit clause. Consider the following lease arrangement. A lessee andlessor agree to a T-year lease with a security deposit of SM. In addition, theyagree on a rental rate of PJ T,M) . The goal of this section is determining thelevel of rent, Y T, M ) , that is consistent with the security deposit arrangement.

It is important to emphasize that the inclusion of the security deposit clausewill provide an incentive for the lessee not to default. Therefore, the defaultboundary K is likely to be lower for a firm faced with a security deposit clause.Thus, K should be considered as endogenous to the contractual provisions. Whilethe present model does not explicitly model the effect on the firm s default incen-tives, the model holds true provided we interpret K as the outcome of a lessee soptimization problem. A similar story can be told with regard to the parameterat, the volatility of the lessee s assets or cash flow. With a security deposit pro-vision, the firm has an incentive to lessen the volatility of its other assets, thusmaking default less likely. Once again, one should interpret ~x as the outcomeof a lessee s optimization problem.

Consider the following simple security deposit clause. The lease stipulates thata fixed amount, M, must be deposited in escrow as a security deposit. The


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348 X R. Grenadier~Journal oj Financial Economics 42 1996) 333 364

E q u i l i b r i u m R e n t

12

11

100 2 4 6 8

D e p o s i t

Fig. 2. The effect of security deposit on equilibrium rent.

This graph shows thc equilibriuln rent on a risky five-year lease as the amount of the security depositincreases. A security deposit represents f un ds placed into escrow by the lessee to cove r potcntial lossesincurred in the event of default. A security deposit serves to m itigate the impact o f credit risk . [f therisky Icase requires no security deposit, then the equilibrium rent is 11.50. If a 5 security deposit isrequired, thcn the equilibrium rent falls to 10.60. The def:ault parameter v alues are ~, = 0, ~, - 0.02,a., 0 .1 5 , a,. - 0.15, p - 0 .5, r 0 .0 5 , ~o - 0.2, T 5, S 10, andX/K 1.2.

I n t h e A p p e n d i x , a s o l u t i o n f o rW S , X , o ; p D , T, M ) i s d e r i v e d f o r t h e c a s e i nw h i c h S a n d X e v o l v e as co r r el a te d g e o m e t r i c B r o w n i a n m o t i o n s. C o m b i n e dw i t h t h e s o l u t i o n f o rF S , X , t ; P, T ) , t h e e q u i l i b r i u m s o l u t io n f o r P Z ) ( T , M ) c a nb e d e r i v e d .

F i g . 2 p l o ts t h e e q u i l i b r i u m l e a s e ra t e a s a f u n c t i o n o f t h e s e c u r i t y d e p o s i t . O na f i v e - y e a r l e a s e , t h e r e n t f a l ls f r o m 11 . 5 0 w i t h n o s e c u r i t y d e p o s i t t o 1 0 . 6 0w i t h a 5 s e c u r i t y d e p o s i t . I n e q u i l i b r i u m , t h e r e w i l l b e a n i n f in i t e n u m b e r o f

c o m b i n a t i o n s o f r e n t a n d s e c u r i t y d e p o s i t t h a t l e a v e th e p a r t i e s i n d if f e re n t . O fc o u r s e , t h e p r e c i s e c o m b i n a t i o n c h o s e n s h o u l d b e ef f i c ie n t i n t h a t it m i n i m i z e st h e c o s t s o f f a c to r s s u c h a s a d v e r s e s e l e c t i o n , m o r a l h a z a r d , c r e d i t v e r i f i c a t i o n ,a n d m o n i t o r i n g .

5 The equilibrium prepaym ent clause in risky leas es

A n o t h e r f o r m o f p r o t e c t i o n a g a i n s t c r e d i t r is k is a p r e p a y m e n t c l a u se . A v e r yc o m m o n c l a u s e in l e a se s r e q u i re s t h e l e s s e e t o m a k e o n e o r m o r e p a y m e n t s i na d v a n c e . A s n o t ed b y S c h a l l h e i m ( 1 9 9 4 ) , a h n o s t a l l l e a s es d e m a n d s o m e f o r mo f p r e p a y m e n t , b u t th e a m o u n t w i l l v a r y a c c o r d i n g t o t h e d e g r e e o f c r e d i t ri sk .P r e p a y m e n t s in th e r a n g e o f o n e t o s ix m o n t h s i n a d v a n c e a r e q u i t e c o m m o n .


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S.R . Grenadier Journal of Financial Economics 42 (1996) 333-364 349

Acco rd ing to the con t rac t , i f the l e ssee pay s n m on ths ' r en t in adva nce , the f ina ln m onths o f the l ease a re ren t - f ree .

Prepayments p rov ide p ro tec t ion to the l e ssor, and in re tu rn the l e ssee mus trece ive a lower l ease ra te . For any g iven con t rac t r en t , the mode l can be usedto de te rmine the equ i l ib r ium f rac t ion o f the l ease tha t shou ld be p repa id . F orexam ple , cons ider a T-ye ar r i sky l ease . Su ppose tha t the l e ssor charges the r i sk -f ree renta l ra te ofR ( T ) or, s imi la r ly, any ren t payment l e ss than the equ i l ib r iumrent P ( T ) . In add i t ion , a p repayment o f 6 years i s r equ i red in advance . Thus ,t h e l e s s o r r e c e i v e s a n u p - f r o n t p a y m e n t o f6 R(T ) . T h e q u e s t i o n i s , h o w m u c hs h o u l d t h e p r e p a y m e n t p e r i o d 6 b e ?

To d e t e r m i n e t he e q u i l ib r i u m p r e p a y m e n t p e ri o d , I o n c e a g a i n u se t h e a rg u m e n ttha t any two methods o f se l l ing the se rv ice f low of an asse t fo r T years shou ldhave the same va lue . The f i r s t a l t e rna t ive i s to s imply l ease the asse t fo r T yearsto a r iskless tenant . By col lect ing a rent f low ofR ( T ) , th is a l ternat ive has a valueo f Y(S, T) .The second a l t e rna t ive i s to l ease to a r i sky t enan t fo r T years , chargethe rent R ( T ) , and co l l ec t an up- f ron t secur i ty depos i t o f 6 years ren t ,6 R( T) .I f the le ssee m ake s a ll paym ents up to t ime T - 6 , then the l e ssee re ta ins useo f t he a s s e t fo r th e r e m a i n d e r o f th e t e r m a n d m a k e s n o r e n t p a y m e n t s . I f th elessee de fau l t s a t any t ime t < T - 6 , then the l e ssor re - l eases the equ ipm ent(a f te r su ffe r ing a de fau l t loss o f the f rac t ion w of the rem ain ing l ease va lue) toa r i sk less t enan t fo r the rem ainder o f the t e rm.

S ince bo th l eas ing a l t e rna t ives se l l the asse t ' s use fo r p rec i se ly T years , eachmu s t have the same va lue . Th e va lue o f the r isk less lease i s s imp lyY(S, T) .T h eva lue o f the secon d a l t e rna t ive i s on ly a s l igh t tr ans form at ion o f Al te rna t ive 2 inSec t ion 2 o f the paper. C ons ider a ( T + 7~)-year r i sk ly lease wi th a ren ta l paym entf low of P for the f i rs t T years , and no required rent for the f inal i? years . D eno tethe va lue o f th is l ease by15(S,X, t ;P, T, T) . t~( S,X ,t ;P , T, T)wil l so lve the samedifferent ia l equat ion asF (S ,X , t ; P, T) ,with the excep t ion tha t the r igh t -hand s ideof bou ndary cond i t ion (9 ) wi ll be (1 - o ) ) .Y[ S( t), T + T - t].Since the seconda l te rna t ive i s equ iva len t to an up- f ron t payment o f6 . R ( T ) plus a r isky leasewi th a t e rm o f T years wi th l ease pay m ents o f R (T ) pa id fo r the fi rs t T 6 yearsonly, i t s in i t ia l value can be wri t ten as6 R (T ) + F (S , X ,0; R (T ), T - 6 , 6) .

Using th i s equ i l ib r ium re lat ion cond i t ion , the equ i l ib r ium prepa ym ent , 6 , i sse t such that the value o f the a l ternat ives are equal . Th is valu e of 6 sa t is f ies thefo l lowing equa l i ty :

Y ( S ,T ) 6 * . R ( T ) + F ( S , X , O ; R ( T ) , T 6 ~ ,6 ) . ( 1 9 )

Under the assumpt ion tha t S and X fo l low cor re la ted geomet r ic Brownianmot ions , a c losed- fo rm so lu t ion fo rF ( S , X , t ; P, T, T ) i s obta ined. F wi l l have aso lu t ion iden tica l to tha t o f F in Eq . (32 ) o f the App endix , w i th the so le excep -t ion that bs(X.z) wi l l now equal(X/K)C 2G (X,z, a 4 ) - e-(r-~-') (T+T)G(X,z, a3).

Fig. 3 p lo t s the e ffect o f inc reas ing c red i t ri sk on the equ i l ib r ium pre pay m entper iod . In th i s f igure , I a l low for the l e ssee to become decreas ing ly r i sky by


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350 S. R. G renadier~Jou rnal o1 Financial Econ om ics 42 (1 996) 3 3 3 3 6 4

EquilibriumP r e p a y m e n t P e r i o d

3 yrs .

2 yrs.

1yrs .

0 yrs.

~....... ' %

' , - .

% - . .

: '. .. .. o-x = .25 . . .: ..~ =.15 .... ...

'..c~ = .05 ' - .. . . . . . . I . I . . . . . . . . . . . . . . . . . I . . . . . . . . . . . . . . . . . 1

1.25 1.5 1.75 2

X / K

F i g. 3 . T h e e f fe c t o f c re d i t r is k o n t h e e q u i l i b r i u m p r e p a y m e n t p e r i o d .

T h i s g r a p h s h o w s t h e e q u i l i b r i u m p e r i o d o f r e n t w h i c h m u s t b e p r e p a i d o n a r i sk y f i v e - y e a r le a s e ,a s the l e s s ee s in i ti a l c red i t pos i t ion v a r i e s . On th i s l ease , a r i sk les s r en t i s cha rged , p lus the l e s so rd e m a n d s a p e r i o d o f p r e p a i d r e n t . A s t h e r a ti o o f th e l e s s e e s a s s e t v a l u e t o t h e d e f a u l t t r i g g e r,X/K r i s e s, t h e l e s s e e s l ik e l i h o o d o f d e f a u l t f al ls : i f X / K - l , t h e l e s s e e w i l l d e f au l t i m m e d i a t e l y ; i fX/K --~ oc t he l ease i s r i sk les s . For l e s sees on the b r in k o f de fau l t , a s ign i f i can t por t ion o f the l easem u s t b e p r e p a id . F o r l e s s e e s fa r f r o m th e d e f a u l t b o u n d a r y, a l m o s t n o p r e p a y m e n t i s r e q u i r ed . T h i se ffec t i s dem ons t ra t ed fo r th ree d i ff e ren t l eve l s o f the l e s se e s vo la t i l it y, ~ r~ . The g rea te r the u nce r t a in tyo f th e l e s s e e s a b i li t y t o m a k e t h e f u t u r e l e as e p a y m e n t s , t h e g r e a t e r t h e r e q u i r e d p r e p a y m e n t p e r i o d .Th e defa ul t pa ram ete r va lue s are ~s. - 0 , ~x - 0 .02, ~ . - 0 .15, p - 0 .5, r 0 .05, co 0 .2 , T = 5 ,and S 10.

r a i s i n g t h e r a t i oX / K f r o m j u s t a b o v e u n i ty ( at w h i c h d e f a u l t o c c u r s i m m e d i -a t e l y ) to l e v e l s at w h i c h d e f a u l t b e c o m e s v i r t u a l l y i m p o s s i b l e . F o r le a s e s o n th et h r e s h o l d o f d e f a u l t , a s i g n i fi c a n t p o r t i o n o f t h e p a y m e n t s o n a f i v e - y e a r l e a s ea r e re q u i r e d u p - f r o n t i n o r d e r t o j u s t i f y c h a rg i n g t h e r i sk l e s s r e n t. F o r l e s s e e sf a r f r o m t h e d e f a u l t b o u n d a r y, a l m o s t n o p r e p a y m e n t is r e q u i re d . I n a d d i t io n ,t h e c o n v e x i t y o f t h e c u r v e s u g g e s t s t h a t th e p r e p a y m e n t p e r i o d i s m o s t s e n s i -t iv e f o r le s s e e s w i t h h i g h c r e d i t r is k . T h i s w o u l d s u g g e s t th a t v a r i a t io n s a c r o s sp r e p a y m e n t p e ri o d s s h o u ld b e m o r e p r o n o u n c e d w i th le s se e s a p p r o a c h i n g ' j u n k 'q u a l i t y c r e d i t ra t i n g s . I n a d d i t i o n , a s t h e v o l a t i l i t y o f t h e l e s s e e ( a x ) i n c r e a s e s ,t h e r e q u i r e d p r e p a y m e n t p e r i o d i n c r e a s e s .

6 . T h e v a l u a t i o n o f l e a s e c r e d i t i n s u r a n c e a n d g u a r a n t e e s

M a n y t y p e s o f i n s u r a n c e a r e a v a i l a b l e f o r l e a se t ra n s a c t io n s . O n e p a r t i cu l a rf o r m o f i n s u r a n c e t h a t p r o v i d e s a d ir e c t m e a n s o f e l i m i n a t i n g o r le s s e n i n g d e f a u l tr i s k i s c r e d i t i n s u r a n c e . L e a s e c r e d i t i n s u r a n c e p r o t e c t s t h e l e s so r, p a r t i a l l y o r


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S . R . G r e n a d i e r ~ J o u r n a l o1 F i n a n c ia l E c o n o m i c s 4 2 ( 1 9 9 6 ) 3 3 3 3 6 4 3 5 1

who l ly, aga ins t the loss o f pay m ents due to defau lt . Al though such pro tec t ionis ava i lab le in a var ie ty of fo rms , I focus on two spec i f ic examples o f l easecred i t insurance : rea l es ta te l ease guaran tees and c red i t enhancement p rov is ionsof l ease-backed secur i t i es .

W hen a land lord neg ot ia tes a l ease of comm erc ia l space wi th a smal l o rnewly es tab l i shed tenan t , a l ease guaran tee can be inc luded as an addendumto the lease . Al though the under ly ing lease i s an agreement be tween the land-lord and tenant , the guarantee wil l be between the landlord and a thi rd par ty.The th i rd par ty ( the guaran tor ) wi l l persona l ly guaran tee the ren t payments fo rthe ful l term o f the lease (W olfson , 1992) . A typical pro vis ion wil l s ta te : 'Th eguaran tor uncondi t iona l ly and wi thout rese rva t ion guaran tees tha t the tenan t wi l lfa i thful ly and punctual ly perform and fulf i l l a l l i t s obl igat ions , covenants , andagreem ents o f the lease . In the ev ent the tenan t defau l ts , then the guaran torguaran tees to pay ren t and a l l o ther moneta ry sums due to the land lord . '

Lease c red i t insurance i s imp or tan t in a recen t, bu t g row ing a rea o f f inanc ia lmarke ts : l ease-backe d securi t ies (LB S) . LB S are s im i la r to s tandard asse t -back edsecuri t ies contracts . A pool o f leases is bun dled and then sold to investors . Th ef i r s t such secur i t i es were i s sued by Sper ry Corpora t ion in 1985 . The vo lume ofsuch i s suances has been g row ing rap id ly. In 1992, Duff & Phe lps Cred i t Ra t ingCo m pan y a lone ra ted $690 mi l l ion of LBS. In 1993 , the num ber increased tojus t u nder $900 mi l l ion . W hi le few o f the i ssuances have been p ubl ic , a s teadyf low of transac t ions has been i s sued by sm al l and m id-s ized leas ing comp anies inthe pr iva te -p lacement marke t . S imi la r to o ther asse t -backed secur i t i es , LBS caninvolve insurance aga ins t c red i t loss in the form o f c red i t -enhanc em ent p ro v i -s ions . This insurance aga ins t defau l t m ay be fu ll o r part ia l . The metho d of c red itenhan cem ent m ay take a var ie ty of fo rms: overco l la te ra l iza t ion , cash reserves ,or le tters of credi t. Th us, w hi le the un derlying leases in the pool are subject todefau l t r i sk , the secur i t i es themse lves may be defau l t - f ree .

The equi l ib r ium lease model in th i s paper can be appl ied to an ana lys i s o fcredi t insurance. Con sider a T -ye ar r isky lease wri t ten a t the equi l ibr ium rentP ( T ) , as def ined by Eq. (11) . The lessor securi t izes the lease and purchases in-surance to improve the secur i ty ' s c red i t ra t ing . The insurance cont rac t p romisesthat , should the lessee defaul t a t t imet*, the inves tor i s guaran teed a payoffof a t l eas t a f rac tion , ' o f the prom ised va lue of the rem ain ing lease pay me nts ,P(T ) (1 - e , .q r ~x / )/ ,r. S ince the amou nt recov ered on the under ly ing lease is(1 (o )Y[S( /* ) , T- t * ] , the inves tor i s assured of rece iv ing the grea te r o f(1 - ~ o ) Y [ S ( t ~ ) , T 1 ] and 7 . P ( T ) ( 1 e- c r- '* l ) r, , For th i s example , assumethat the insurer ' s obl igat ion is wi thout r isk .

Le t 7*(&X,t ; K 7) denote the value o f this insurance co ntract . I t mu st solvethe fol lowing par t ia l d i fferent ia l equat ion:

0 2 2 I ~ ~c,~ (S, t) S 7 ss + f~(S, x , t)o~(S,t )ox (x , t )SXq sx + ~or; (x , t ) x -~ v x

+z~,,(S,t)SCPs + ~,( X ,t)X q~ x + ~P, riP , (20 )


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3 5 2 S, R. Grenadier / Jour nal o f Finan cial Eco nom ics 42 199 6) 333~.364

Va l u e o f I n s u r an c e C o n t r a c t

40

30 ~. ...~ ,/= 1

2 0 = 8 5

10

I I I1.125 1.25 1.375 1.5

X K

F i g . 4. T h e e q u i l i b r i u m v a l u e o f c r e d i t r i s k i n su r a n c e .

T h i s f i g u r e s h o w s t h e e f f e ct o f c h a n g e s i n t h e l e v e l o f i n s u r a n c e ( 7 ) a n d t h e c u r r e n t d i s t a n c e t : omd e f a u l t X / K ) o n t h e v a l u e o f a c r e d i t r i s k i n s u r a n c e p o l i c y. A c r e d i t r i s k in s u r a n c e p o l i c y p r o m i s e st h e h o l d e r o f t h e l e a s e a p a y o u t o f a t l e a s t a f r a c ti o n ?, o f t h e v a l u e o f t h e r e m a i n i n g c o n t r a c t u a l

l e a s e p a y m e n t s . T h r e e c u r v e s p l o t t h e e f f e ct o f i n c r e a s i n g d i s t a n c e f r o m d e f a u l t o n t h e e q u i l i b r i u mi n s u r a n c e p r e m i u m : t h e l o p c u r v e i s f or a f u l l y i n s u r ed c o n t r a c t , t h e m i d d l e c u r v e i s f o r a c o n t r a c t

promis ing8 5 c o v e r a g e , a n d t h ebot tom c u r v e is f o r a c o n t r a c t p r o m i s i n g 7 0 c o v e r a g e . F o r e a c hl e v e l o f i n s u r a n c e 7 ,, i n c r e a s i n g t h e d i s t a n c e f r o m d e f a u l t l e a d s t o a d e c r e a s e i n th e i n s u r a n c e p r e m i u m .

H o w e v e r , t h e d i f f e r e n c e i n p r i c i n g i s o n l y s i g n i f i c a n t f o r t h e m o s t r i s k y l e a s e s . T h e d e f a u l t p a r a m e t e rva lu es are ~ ,. 0 , c~ = 0 .0 2, ~r~. = 0 .15 , aa 0 .15, p = 0 .5 , r = 0 .05 , co = 0 .2 , T = 5 , an d

S = I O .

sub jec t to the boundary cond i t ions

, v s , x , T ; T, ? ,)=o,

~ S , K , t ; , T, ? , )m a x [ ) P t ( T ) ( e -~ r- t ) ) l1 - - 1 - ~ ) v s , r - t ) , o .

21)

The f ir st bou ndary cond i t ion ensures tha t the insurance pay m ent i s ze ro i f the re i sno defau l t . The second boundary cond i t ion i s the insurance payment upon defau l t :t h e p o t e n t i a l s h o r t f a l l b e t w e e n t h e a m o u n t r e c o v e r e d a n d t h e p r o m i s e d m i n i m u mpayout .

Th e cos t o f th is insurance po l icy i s pa id a t t ime ze ro and m us t equa l~ S ,X ,0;T, ; , ) , the solut ion to par t ia l d i fferent ia l equat ion (20) evaluated a t t = O. For thecase in which S and X evo lve as cor re la ted geom et r ic Bro wn ian m ot ions , a


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S.R. Grenadier /.hm rmd O/F#lancMI Economics 42 (1996) 333 364 353

so lu t ion i s p resen ted n t h e A p p e n d i x . T h i s s o l u t i o n c a n b e w r i t t e n a s

V J (& X , 0 ;7 , 7) B ( & X ; K 0 , 7 , 1 ~,)), ( 2 2 )

w h e r e t h e f i m c t i o nB ( S , X ; T, y l , Y 2 ,Ys) i s de f ined in the Append ix .F ig . 4 p lo t s the e ff ec t o f ch ang es in the l eve l o f i n su ran ce (7 ) and the cu r ren t

d i s t ance f ro rn de fau l t( X / K ) o n t h e v a l u e o f a c r e d i t ri sk i n s u r a n c e p o l i c y. T h r e ec u r v e s p l o t t h e e f fe c t o f i n c r e a s in g d i s t a n c e f r o m d e f a u l t o n t h e e q u i l i b r iu m i n -su ran ce p rem ium : the top cu rv e is fo r a 100% fu l ly in su red con t rac t (5, 1 ), t hem i d d l e c u r v e i s f o r a c o n t r a c t p r o m i s i n g 8 5 % c o v e r a g e , a n d t h e b o t t o m c u r v e isf o r a c o n t r a c t p r o m i s i n g 7 0 % c o v c r a g c . F o r e a c h l e v e l o f i n s u r a n c e 7 , a g r e a te rd i s ta n c e f r o m d e f a u l t le a d s to a d e c r e a s e in t h e i n s u r a n c e p r e m i u m . T h e p r e m i u m

is c o n v e x i n X / K , m e a n i n g t h a t t h e e ff e ct o f i n c r e a s i n g t h e d e g r e e o f c r e d it r is ko n t h e v a l u c o f th e i n s u r a n c e is m o s t p r o n o u n c e d f o r v e r y r i s k y l e a s es , i n a d d i -t io n , th e i n s u r a n c e p r e m i u m i n c r e a s e s w i t h th e l e v e l o f in s u r a n c e . H o w e v e r, t h ed i ff e rence in p r i c ing i s on ly s ign i l i can t l b r t he mos t r i sky l eases . AsA;/K r e a c h e saro un d 1 .25 , the effec t o f ; ' i s neg l ig ib le .

7 C r e d i t r i sk a n d t h e o p t i o n t o p u r c h a s e

A common c lause in l eases i s f i ) r t he l e s see to have the op t ion to pu rchasct h c a s s et a t th e e n d o f th e t e r m o f t h e l e a se f o r a p r e d e t e r m i n e d p r ic e . M c -C o n n e l l a n d S c h a l l h e i m ( 1 9 8 3 ) p r o v i d e a v a l u a t i o n o f t h is o p t io n . B e c a u s e t h eg r a n t i n g o f s u c h o p t i o n s is v a lu a b l e , t h e r en t a l p a y m e n t s m u s t b e a d ju s t e d u p -w a r d s .

In the con tex t o f c red i t r isk , these p u rch ase op t io ns a re e spe c ia l ly r e l evan t .B e c a u s e t h e s e o p t i o n s c a n b e e x e r c i s e d o n l y a t t h e e n d o f th e t e rm , t h e o p -t ions p rov ide an incen t ive lk )r t hc l e s see no t to de fau l t on the l ease . Th e in -t e n s i ty o f t h is i n c e n t i v e d e p e n d s o n t h e a t tr a c t iv e n e s s o f t h e p r e d e t e r m i n e d e x -e rc i se p r i ce , typ ica l ly se t a t t he exp ec ted va lue o f the un der ly ing a s se t a t theend o f the l ease te rm. F rom the v i ew po in t o f t he l e s so r, two l e s sees wi th equa lex an te c red i t r i sk wi l l l i ke ly have d i ff e ren t ex pos t l ease de fau l t r ea l i za t ionsd e p e n d i n g o n w h e t h e r t h e y s i g n a l e a s e w i t h o r w i t h o u t p u r c h a s eop t ions .

C o n s i d e r t h e v a l u e o f a n o p t i o n t o p u r c h a s e t h e u n d e r l y i n g a s s e t a t t h e e n do f the l ease [ 'o r t he f ixed exe rc i se p r i ce o f E . Th i s d i f fe r s f iom a s t anda rd ca llo p t i o n i n t h a t t h e o p t i o n m a y o n l y b e e x e r c i s e d s h o u l d t h e l e s s e e n o t d e t h u l t .T h e v a l u e o f t hi s o p t i o n ,H ( S , X . I ; T, E ) , w i l l o n c e a g a i n s a t i s f y t h e f o l l o w i n gequ i l ib r ium pa r t i a l d i f f e ren t i a l equa t ion :

I ~ ,a7(5,1 )5 2tt~-~ + p(S,A\ t )(r~(S, t ) a , (X , t )SXHs.+ ~ a ~ ( X t ). .V ~ H v v

- ~ , . ( S , t ) S t t s - ~ , ( X , t ) X H v+ H t - r H , ( 2 3 )


22/32

354 S. R. Grenadier~Journal o f Financial Econom ics 42 1996) 333 -364

Option Value

40

E = 180. . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..:iil....].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E = 220

0 5: I1 25 1 5 1 75 2

X/K

Fig. 5. The effect of credit risk on a purchase option's value.

This graph shows the value of an option to purchase the underlying asset, as both the lessec's initialcredit position and the exercise price varies The option allows the lessee to purchase the underlyingasset at the end of a five-year lease for a fixed exercisc price $E. This option may be exercised bythe lessee only if there is no default during the term of the lease. The oppoortunity to exercise apotentially valuable option provides an incentive for the lessee to avoid default. The top curve plotsthe value of the purchasc option with an cxercise pricc of $180 as the distance from the defaultthreshold increases. The middle curve is interpreted similarly, but with an exercise price of $200 (theexpected terminal asset value), and the bottom curve has an exercise price of $220 The value of

this option increases as thc exercise price falls and the probability of default declincs. As the ratioof the lessee's asset valuc to the default trigger, X/K, rises, the lessee's likelihood of default falls: ifX / K - I, the lessee will default immediately; ifATK --+ oo, the lease becomes riskless. For lessees onthe brink of default, the option is virtually worthless For lessees l:ar from the default boundary, theoption value is significant. The default parameter values are :q = 0, ~ = 0.02, a~ = 0.15, o-~ = 0.15,

-- 0.5, r = 0.05, ~,) - 0.2, 7' - 5, and S = 10.

subject to the boundary condi t ions

H S , X , T ; T , E ) =max [V (S ) - E, 0] ,

H S ,K , g ; T , E )= 0, (24)

where V S ) is the value of the underly ing asset , as det erm ine d in Eq. (4). Thef i rs t boundary condi t ion is the call opt ion pa yoff a t matur i ty. The second b oundary

condi tion reflects the fact that the value o f the opt ion falls to zero if the les seedefaults.

For the case in which S and X fol low corre la ted geometr ic Brow nian mot ions ,

a closed-form solution to partial differential equation (23) subject to boundaryconditions (24) and evaluated at t ime t = 0 is derived in the Appendix.

Fig. 5 plots the effect of the ratio of X/K on the value of the option to purc hasefor di fferent values of the exercise pr ice E. Since the expected value o f the asset a tthe end of the lease term is 200, the mi ddle curve represent s the typical protocolof se t t ing the opt ion exercise pr ice equal to the asset ' s expected terminal value.


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F o r e a c h v a l u e o f E , t h e o p t i o n p r i c e i n c r e a s e s w i t h th e r a ti o o f X / K . W h e nthe ra t io i s un i ty, the l e s see i s ce r t a in to de fau l t , and the op t ion i s wor th less .As the ra t io r ises , the p rob ab i l i ty o f de fau l t f a ll s to ze ro , an d the va lue o f thep u r c h a s e o p t i o n a p p r o a c h e s t h e v a l u e o f a s t a n d a r d ' n o - d e f a u l t ' c a l l o p t i o n o nt h e u n d e r l y i n g a s s e t . W h e n t h e l e s s e e ' s a s s e t v a l u e ( o r c a s h f l o w ) i s r e a s o n a b l yf a r a b o v e t h e d e f a u l t t r i g g e r v a l u e , t h e s t a n d a r d ' n o - d e f a u l t ' c a l l o p t i o n v a l u ew i ll b e a g o o d a p p r o x i m a t i o n o f t h e le a s e p u r c h a s e o p t i o n . H o w e v e r , w h e n t h el e s s e e ' s c r e d i t c o n d i t i o n i s v u l n e r a b l e , u s i n g a s t a n d a r d c a l l o p t i o n f o r m u l a c a nb e a v e r y p o o r a p p r o x i m a t i o n . A s t h e e x e r c i s e p r i c e i n c r e a s e s , t h e v a l u e o f t h eo p t i o n t o p u r c h a s e f a l l s . W h e n t h e l e s s e e ' s c r e d i t c o n d i t i o n i s h i g h l y v u l n e r a b l e ,h o w e v e r, t h e e f fe c t o f c h a n g e s i n t h e e x e r c i se p r i c e a r e q u it e m o d e r a t e . T h i s i sb e c a u s e t h e l i k e l i h o o d o f e x e r c is e , n o m a t t e r h o w a t tr a c t iv e t h e o p t i o n , f a ll s toz e r o d u e t o i m p e n d i n g d e f a u l t .

In o rde r to de r ive the equ i l ib r ium ren t ,P ( T ) , o n a T- y e a r r i s k y l e a s e w i t ha purcha se op t ion , w e wi l l use a va r i an t o f the s t and ard p u t ca ll pa r i ty r e la -t io n . I t w i ll p r o v e u s e f u l t o c h a r a c t e r i z e t h e v a l u e o f t w o a d d i t io n a l c o n t r a c t s.F i rs t, co ns id e r an o p t ion to se l l the l ease d asse t a t the end o f the l ease t e rm,c o n d i t i o n a l o n n o d e f a u l t h a v i n g o c c u r r e d . T h i s i s s i m p l y a p u t o p t i o n o n t h easse t wi th exerc i se p r i ce E and exp i ra t ion da te T, cond i t iona l on no de fau l t .D e n o t e t h e v a l u e o f s u c h a p u t o p t io n a sJ ( S , X , t ; K E ) . T h e v a l u e o f t h is p u top t ion mus t sa t i s fy an equ i l ib r ium par t i a l d i ffe ren t i a l equa t ion , which i s p ro -v i d e d i n t h e A p p e n d i x . S e c o n d , l e tN ( X , t ; T ) d e n o t e t h e v a lu e o f a z e r o - c o u p o nb o n d w h i c h p a y s 1 a t t im e T o n l y i f t h e r e i s n o d e f a u lt . T h e v a l u e o f t h isbond wi l l sa t i s fy an equ i l ib r ium par t i a l d i ffe ren t i a l equa t ion , a l so p rov ided in theA p p e n d i x .

N o w, w e c a n d e t e r m i n e t h e e q u i l i b r i u m r e n t ,P~'(T), o n a T- y e a r r i s k y l e a s ew i t h a p u r c h a s e o p t i o n . A s u s u a l , I w i l l p r e s e n t t w o e q u i v a l e n t m e t h o d s o f s e l li n gt h e u s e o f t h e a s se t. T h e f ir s t m e t h o d o f s e l li n g t h e u s e o f th e a s s e t i s to s i m p l yl e a s e th e a s s e t f o r T y e a r s u n d e r a l e a s e w i t h a p u r c h a s e o p t i o n . I f t h e l e s s e edefau l t s , then the l e s so r re - l eases the asse t fo r the rem aind er o f the t e rm (a f t e r de -duc t ing a c red i t loss ) to a r i sk less l e s see . Under th i s l eas ing po l i cy, the l e s sor se l l st h e u s e o f t h e a ss e t f o r T y e a r s a n d r e c e i v e s E - V [ S ( T ) ] i n y e a r T i f t h e o p t io ni s exe rc i scd . T he va lue o f th is por t fo l io i sF [ S , K , t ; P ( T ) , T ] H ( S , X , t ;K E ) .

T h e s e c o n d m e t h o d i s f o r t h e l e s s o r t o f o r m t h e f o l l o w i n g p o r t f o l i o : s e l l t h cu s e o f th e a s s e t f o r T y e a r s b y l e a s i n g t o a r is k l e ss l e ss e e , p u r c h a s e E z e r o -c o u p o n b o n d s p a y i n g 1 a t t i m e T o n l y i f n o d e f a u l t o c c u r s , w r it e a n o p t i o nt o se ll t h e a s se t a t t im e T f o r E w h e r e t h e o p t io n m a y o n l y b e e x e r c i s e d i fn o d e f a u l t o c c u r s , a n d w r i te a n o p t i o n to p u r c h a s e t h e a s se t a t t im e T f o r 0w h e r e t h e o p t i o n m a y o n l y b e e x e r c i s e d i f n o d e f a u l t o c c u r s . T h e v a l u e o f t h ispor t fo l io i s Y(S,T ) + E ,~ (X ,t ; T ) - J ( & X , t ;T, E ) - J ( S , X , t ; K 0 ) . T h e p a y o f f o ft h is p o r t f o l i o is p r e c i s e l y t h a t o f th e f ir st m e t h o d : t h e l e s s o r r e c e i v e s p a y m e n t sf o r s e ll i n g th e u s e o f t h e a s s e t f o r T y e a r s p l u s a p a y m e n t o fE - V [ S ( T ) ] inye ar T i f the op t ion i s exe rc i sed .


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356 X R. G remtdier / . lourna/ o f F imm cia l Econom ics 42 (1996) 333 364

S i n c e b o t h o f t h e s e m e t h o d s o f l e a s in g p r o d u c e i d e n ti c a l p a y o ff s , t h e l e a s e r at ep 0 ( T ) m u s t b c s e t a t t i m e z e r o t o c q u a t e t h e i r v a l u e s . T h u s , i n e q u i l i b r i u m P ( T )m u s t b e s e t t o s a t i s f y t h e f o l l o w i n g c o n d i t i o n :

F i X , K , i; i f ( r ) , T ] - H ( S , X , t ; r , E )

- - Y ( S , T ) + E N ( X , t ; T ) - J ( S , X , t ; T , E ) J ( S , X , t ; T , O ) . 25)

A l t e r n a t i v e l y, t b r a g i v e n ( h i g h e r ) r e n tP ( T ) , i t i s s imple to so lve fo r the equ i -l i b r i u m e x e r c i s e p r i c e , E , o n t h e e m b c d d e d c a l l o p t i o n .

8 P e r c e n t a g e l e a s e s a n d c r e d i t r i sk

I n t h e U . S . , th e p e r c e n t a g e l e as e i s t h e d o m i n a n t m e t h o d o f l e a s in g r e ta il s p a c ein sho pp ing mal l s . Th i s i s c l ea r ly an im por tan t sec to r o f the l eas ing lna rke t , w i tha l m o s t 3 8 , 0 0 0 s h o p p i n g c e n t e r s i n t h e U . S . a c c o u n t i n g f o r o v e r $ 7 0 0 b i l li o nin re ta i l sa les in 1991 (Shopp ing Cen te r Di rec t ions , 1992) . The ren t s pa id byd i ffe ren t c l a sses o f ma l l t enan t s v a ry subs tan t i a l ly. In 1992 , the me dian ren t pe rs q u a r e l o o t p a i d b y d e p a r t m e n t s t o r e s i n s u p e r r e g i o n a l s h o p p i n g m a l l s w a s S 1 . 9 5( D o l l a r s a n d C e n t s o f S h o p p i n g C e n t e r s , 1 9 9 3 ) . F o r h o m e f u r n i s h i n g s t o re s t h cm e d i a n r e n t w a s $ 2 5 , a n d l b r j e w e l r y s t o re s t h e m e d i a n r e n t w a s $ 4 2 . I n a d d i -t io n , t h e a v e r a g e r e n t p e r s q u a r e f o o t f o r s t o r es t h a t w e r e m e m b e r s o f n a ti o n a lc h a i n s w a s o n l y a r o u n d h a l f a s h ig h a s th a t l b r i n d e p e n d e n t s to r e s. U n d c r ap e r c e n t a g e l e a s c , t h e s h o p p i n g c e n t e r o w n c r c o l l e c t s a f i x e d b a s e r e n t p l u s ap c r c e n t a g c o f t h e s t o r e ' s s a le s , p r o v i d e d t h e s t o r e ' s s a l e s e x c e e d a g i v e n t h r e s h -o ld . In a de fau l t- li ee con tex t , su ch c on t rac t s hav e been an a lyz ed b y B cn jan l in ,B o y l e , a n d S i n n a n s ( 1 9 9 0 ) a n d G r e n a d i e r ( 1 9 9 5 ) . B e c a u s e p e r c e n t a g e l e a se sa r e m o s t c o i n m o n o n t h e s m a l l e r m a ll s h o p s , t h c i m p a c t o f c r e d i t r is k m a y b ec o n s i d e r a b l e .

P e r c e n t a g e l e a se s a r e a s p e ci a l c a s e o f a m o r e g e n e r a l f o r m o f l e a s in g s t ru c t u r ein w h i c h r e n t is t ie d t o s o m e m e a s u r e o f t h e i n t en s i ty o f a n a s s e t ' s u s e. E x -a m p l e s i n c h i d e c a r le a s c r at e s l in k e d t o m i l e a g e , c o p y m a c h i n e r e n t s l in k e d t ot h e n u m b c r o f c o p i e s , a n d c o m p u t e r l e a se s l i n k e d t o C P U c y c l e s . S m i t h a n dW a k e l n a n ( 1 9 8 5 ) p r o v i d e a d i s cu s s i o n o f t h e ra t io n a l e u n d e r l y i n g s u c h l e a s in ga r r a n g e m e n t s .

T h e m o d e l c a n b e a p p l i e d t o v a l u c t h e p e r c e n t a g e l e a s e u n d e r c o n d i t i o n s o fpo ten t i a l de fau l t . The con t rac t spec i f i e s a base ren t , R s~, wh ich m us t be p a idr e g a r d l e ss o f s al e s p e r f o r m a n c e . I n a d d i t io n , s h o u l d s a l e s r is e a b o v e a t h r e s h o l d

~ , a p e r c e n t a g e p o f th e l e v e l o f sa l e s a b o v e t h e th r e s h o l d is p a id t o t h e l a n d lo r d .A s s u m e t h a t t he s t o r e ' s s a l e s a r e p r o p o r t i o n a l t o th e a s s e t v a l u e o f th e s p a c e ,i< . V [ S ( I ) ]. T h u s , t h e l e a se p a y m e n t ( f l o w ) m a d e a t ti m e1, R l ( t ) , c o n t i n g e n t o n


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S.R . Grenadier Journa l o f Financial Economics 42 (1996 ) 333 364 357

no defau l t hav ing occur red , i s

R P ( t ) = R B + m a x 1 0 0

Re + 100 m ax V [S (t) ] - ~c

The ren ta l payment on a pe rcen tage l ease (con t ingen t on no defau l t hav ingoccur red ) i s equa l to the sum of a f ixed pay m ent , R e , p lus the_payoff on p~c /100purchase op t ions on the under ly ing asse t , wi th exerc i se p r iceS/tc and exp i ra t ionda te t. The va lue o f such an op t ion was de r ived in Sec t ion 6 : a t time ze ro the

va lue o f such a purchase op t ion i sH ( S , X , O ; t , S / t ) .For the case in which Sand X fo l low cor re la ted geo me t r ic Bro wn ian m ot ions , a c losed - fo rm so lu t ion fo rH (S,X, 0;t , E ) i s p resen ted in the Appendix , and i t s p roper t i e s were ana lyzed inSect ion 7 .

The v a lue o f the pe rcen tage l ease ,LP (S ,X , t ; T, p , RB ,S ) ,i s therefore equal tothe va lue o f a s t andard r i sky l ease wi th ren ta l R B , p lus the t ime- in tegra l ( sum )of purcha se op t ions . Us ing the va lue o f a s tandard l ease (F ) de r ived in Sec t ion2 and the va lue o f a purchase op t ion ( H ) de r ived in Sec t ion 7 , the va lue o f thepay m ent f low f rom a T -yea r pe rce n tage l ease , a s o f t ime t - 0 , i s

L P ( S ' X ' O ; T ' p ' R ~ ' S ) F ( S ' X ' O ; R S ' T ) + 1 0 0 H S ,X ,O ;t ,d r.

( 2 7 )

In equ i l ib r ium, the va lue o f sel l ing the use o f the asse t fo r T y ears under apercen tage l ease m us t equa l the va lue o f se l ling the use o f the asse t under as tandard T- ye ar r iskless lease ,Y ( S , T ) . T h e r e f o r e , e q u i l i b r i u m c o m b i n a t i o n s o f( p , R ~ , S ) are solut ions to the fol lowing equal i ty :

( 2 8 )(S , T ) - - LP( X ,O; T, p , Rg , S ) .

9 C o n c l u s i o n

1 der ive a mo de l tha t p rov ides a un i f i ed ap proac h to the eq u i l ib r ium va lua t ion o fleases sub jec t to de fau l t r i sk . Us ing an op t ion-pr ic ing approach , the mode l a l lowsfor a s tochast ic service f low from the leased asset , as wel l as for the s tochast icoccur rence an d conseq uences o f de fau l t. The m ode l i s fl ex ib le enoug h to pe rm i tthe de te rmina t ion o f equ i l ib r ium ren ta l r a tes under a wide va r ie ty o f rea l is t icleas ing s truc tu res. Such s t ruc tu res inc lude secur i ty depos i ts , up- f ron t p rep aym ents ,l ease c red i t insurance con t rac t s , embedded lease purchase op t ions , and percen tageleases.


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3 5 8 S . R . G r e n a d i e r ~ J o u r n a l o ] F i n a n c i a l E c o n o m i c s 4 2 ( 1 9 9 6 ) 3 3 3 ~ 6 4

Severa l ex tens ions o f the mo de l w ould p rove in te rest ing . F i rs t, a ri cher mo de lwou ld inc lude a t r ea tment o f the asse t supp ly sec to r o f the indus t ry. Tha t i s, ra the rthan t ak ing the equ i l ib r ium se rv ice f low (and asse t va lue) as exogenous , a morerea li s ti c t r ea tment wo uld inc lude a cons idera t ion o f the equ i l ib r ium cons t ruc t ionresponse o f supp l ie r s . Second , the m ode l cou ld be em pi r ica l ly t e s ted on ac tua ll ease con t rac ts . T he ava i l ab i l ity o f l a rge and re l i ab le samp les o f ind iv idua l l easecontracts i s current ly qui te d i ff icul t to obta in . The model suggests that equi l ibr iumlease ra tes wi l l be sensi t ive to the s tochast ic processes under lying both the leasedasset and the less ee s f i rm value . In addi t ion, the term s o f the co ntract (e .g . ,m a t u r i t y, e m b e d d e d o p t i o n s , p r e p a y m e n t , a n d s e c u r i t y d e p o s i t p r o v i s i o n s ) m u s tbe ca re fu l ly t aken in to account in the empi r ica l spec i f i ca t ion .

A p p e n d i x

A . 1. E x p l i c i t s o l u t i o n s j o r l e a se c o n t r a c t s

In th i s sec t ion o f the A ppend ix , I p resen t exp l ic i t so lu t ions fo r va r ious l eas-ing con t rac t s p resen ted in the mod e l fo r the case in which S and X evo lve ascor re la ted geomet r ic Brownian mot ions , i . e . ,

d S - ~ , , , S d t a , S d z , , ,

d X - ~ . , X d t + a x X d z , - ,( 2 9 )

and where p i s the ins tan taneous cor re la t ion coeff i c ien t be tween the Wiener p ro-cesse s dz, a nd dz. ,. . U nd er this spec ification ,S ( t ) a n d X ( t ) wil l be log-normalrandom var iab les .

T h e r i s k h , s s h a s e

Eq. (2) def ines the value o f a T -ye ar r iskless lease by the funct ionY ( S , T ) .Using the p roper t i e s o f log-norm al va r iab les , the expec ted va lue o f the in tegra lcan be wri t ten as

S [1 e r x, .Y (

``leasing and risk``

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