Internet Appendix to Simultaneous implication of credit risk and embedded options in lease contracts

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1 nn Appndix o Simlano implicaion of cdi ik and mbddd opion in la conac n i nn Appndix w poid () a mo compni lia iw of la conac alaion () mamaical poof of om fomla in main x () mpiical p fo implmnin o la a modl in in qaion (9) of main x and (4) compaai aic of o modl by nmical xampl. A. ia Riw of a onac Valaion any pap a iniad alaion of la conac wi ill and Upon (976) a minal wok o conc an qilibim modl in alin la conac. oninin a wok al pap (.. Scallim and connll (985) connll and Scallim (98)) popo a no-abia famwok o al a la conac y afomniond di dic-im modl o al la conac. n cona Gnadi (995) dlop a conino-im modl o al aio yp of la conac bad on a al-opion appoac. mi of i modl i a la m c i a fncion of xpcaion of f o-m la a and pply of lad a. Gnadi (995) a al of pmim fo opion mbddd in la conac a alway non-nai. Uin a al opion appoac Knyon and ompadi () imilaly inia ffc of idl im on alaion of la conac wi opion. y a al opion conidd in la conac poid flxibiliy o affc f ca flow fom lad a. ow Giaccoo al. (7) ow a pnali fo aly minaion of a conm aomoi la a ofn cd in c a way a mbddd cancllaion opion a lil al.

2 n alin mbddd opion in al a la conac Bow and Alb (998) bild a modl o al c conac wi a nmb of mbddd opion (.. nwal opion and pca opion) a a common in commcial la. Gnadi (5) f poid a am-oic aian of al opion famwok o al al a commcial la. i modl can analyz mo common lain aanmn c a mbddd opion o pca popy a nd of a la. Gnadi (5) popo a opion o pca lad a may b incldd in conac o alin incni of l and lo. opion poid an incni o l o mainain ndlyin a and o mak qid nal paymn. ow modl do no poid an inad famwok o al a nmb of mbddd opion imlanoly. Apa fom mbddd opion in la conac cdi ik of l may alo inificanly affc al of a la conac. Nl mo xiin modl alain la conac foc mly on dfal-f la. Gnadi (995) a cal cdi-ik modl wi an xonoly in dfal bai o al la conac and a a onc l dfal will lo all i o ponially alabl opion mbddd in a la conac. Gnadi (996) n dlop a nal famwok fo alin qilibim cdi pad on la bjc o dfal ik and xamin a aiy of la c c a mbddd la opion nd condiion of dfal ik and la cdi inanc conac. Gnadi (996) a dfal ik play an impoan ol in alaion of la conac and poin o a yild on la conac xcd yild on appoximaly qialn db-financin aanmn implyin a on imilaiy bwn la conac and jnk bond. n ano am of ac Ambo and Yildiim (8) xnd famwok of lapam and Gnnlin () by incopoain l cdi ik ino

3 alaion of la conac and apply Jaow and Yildiim () cd-fom modl o di m c of la a. nmical l a la m c a inificanly affcd by l cdi ik. n mpiical lia Scmi (4) inia cdi ik in ail la pofolio. foc on wo majo ik componn ndd fo imaion of lo diibion: pobabiliy of dfal and xpcd lo in dfal. mpiical idnc al a pyical collaal play a majo ol in cin cdi ik aociad wi la pofolio. B. amaical mma and Poof B. Poof of Popoiion Bfo poin Popoiion mma i inocd. mma Followin pcificaion of azad a and pobabiliy fncion of n dcibd in qaion (8) qaion () in mma can b win a: d R d R R R R B B d R d d d

4 4 R d R R d R B B d Sppo a R i iniial n of adjabl la wi a pca opion a cancllaion opion and dfal ik and a R i ncolad wi dnii of n and B. W can wi xpion in mma a: d d R R d R d R d R d B B d d R

5 R R R d R d d d B B d o po Popoiion ak lf i n o di iaion of Sinc d i.... i and n. nc d iaion of 5

6 6 Sinc d and B n w n d iaion of and 5

7 7 Sinc R d and n w and K K K K

8 8 9 8 K K 9 8 w 9 8. Similaly w w obain a R d R d. Fo 5 R d w

9 w a 5 R w fo all i 5 i i. d iaion of and 4 Sinc dfin d w bca and n

10 w and K K K K 9 8 K K w Similaly

11 w d Similaly d 4 d.

12 w fo all i i 4 i Fo 6. d w w a w fo all i Similaly 6 i 4 i. d d w fo all i i 6 i Fo dfin. 7 B B d d d w

13 Sinc B n 8 9 B B 7 B B 4 4 B B B 7 B 7 B 7 B7 w B d Similaly B B d.

14 4 4 B B B B B B w fo all i i 7 i. d iaion of and 8 9 Rcall a nd ma Q d dw d and wio lo of naliy w nomaliz o. W n a W W. ln. fin Va 4 and n w 4

15 W nx a d 8 d 5 5. Fo R d 9 inc n w 5 5 and

16 6 K K K K K K w Similaly w

17 and o 9 R R 5 5 d d. R d w Fo w a R 5 5 w fo all i i 9 i. d. aild diaion of k k... i pa poid dfiniion of aiabl and diaion of k k.... 7

18 8.. an and Vaianc Fo in a poc call in qaion () a nd ma Q dw d a d. W coo a f a f a fo a. fin dw b f w a and if if a a a d b a fi wo momn of can n b pnd a: b f d Va. fin dw b f. Similaly fi wo momn of can b pnd a: b f d b Va a a a a a fo a. Fo ic-flow poc call in qaion (4) a nd ma Q dw d d w i mak pic of ik of ic-flow and d dw dw. Wio lo of naliy w am and n W W W W.

19 9 ln and dfin Va. Fo pic poc of lad a call in qaion (6) a nd ma Q dw d d w i mak pic of ik of ndlyin a and d dw dw. Similaly wio lo of naliy w nomaliz o and n a W W W W. ln and dfin Va. Fo poc of fnc indx call in qaion (8) a nd ma Q dw d d w i mak pic of ik of ndlyin a and d dw dw. Aain wio lo of naliy w am and n a W W W W. ln and dfin 4 Va 4. Sinc W W and W dfin W 5

20 W 6. W 7 Fo w obain a W W dw W dw W W dw dw W. Similaly fin 8 dw W dw. W dw W W 8 Va dw dw d. Similaly dfin 9 9 Va W W

21 Va... oaianc of Vaiabl n i cion w poid coaianc of aiabl in modl. Som of coaianc bwn aiabl will no b d in o diaion and will b omid. Fo daild diaion pla f o Ambo and Yildiim (8) and iao al. (8). dw fin d if i j and ij ji. d if i j ij dw i j i j o b o o 4 o 8 o 9 o o w d a d d d d d d fo a. a a a

22 o b o b 4 o b 5 o b 6 o b 7 o b 8 o b 9 o b o b d d d d d d d d d w b a a a d and a a a b d a a fo a. o o 4 o 5 o 6 o 7 o 8

23 o 9 o o 4 o 5 o 6 o 7 o 8 o 9 o o 45 o 46 o 47 o 48 o 49 o 4 o 58 o 59

24 o 5 o 68 o 6.9 o 6 o 78 o 7.9 o 7 o 89 o 8 o 9.. mpiical Sp fo mplmnin a Ra odl in qaion (9) Fo mpiical implmnaion of o modl w nd o obain ioical daa of in a inflaion a lad a pic and ic flow of lad a. Bo in a and inflaion a can b obaind fom conomic daa of U.S. Fdal R Boad. Fo xampl w can coo in a fo a -ya U.S. ay bill and aonally-adjd P ll in imaion. lad a pic can b akn fom all-anacion indic 4

25 obaind o U.S. Fdal oin Financ Ancy. daa coni of imad al pic and appaial. ow mak daa of ic-flow of lad a i difficl o obain. A poibl way o ima ic-flow i a on can obain daa of imila la conac wic do no conain any opion o poiion in mak and n ic-flow can b imad fom la n by qaion (9). W nx ima paam of inanano dif a ( and ) inanano olailii ( and ) and mak pic of ik ( and ) of ocaic poc in qaion () (5) and (7) by connional appoac c a maximal likliood imaion. Fo in a poc in qaion () paam of pd of adjmn ( a ) and olailiy ( ) can b imad in a imila way. inanano fowad a ( f ) can b obaind in mak daa of in a. Fmo w can ima colaion bwn ic-flow of lad a lad a pic in a and inflaion a ( and ) in ioical daa. Fo imain azad a paam mak daa of dfal a of la conac a of cancllin conac and a of pcain lad a alo nd o b acqid. A mniond abo cmlai xc n on ic-flow of lad a lad a pic and inflaion a p o im ( and ) can b obaind fom mak daa. paam of azad a fo likliood of dfal cancllaion Wbi: p:// 5

26 and pca in qaion (9) o () ji i fo j B and (5 paam in oal) can b imad by a impl lin ion mod. Onc all paam a obaind w can bi m ino qaion (9) o calcla iniial n R of la conac. Wil ioical daa of lad a pic in a and inflaion a a ai o obain mak daa of ic-flow of lad a a wll a dfal a of la conac a of cancllin conac and a of pcain lad a a mc mo difficl o acqi. Fmo coy a of n in wic l dfal xci cancllaion opion o xci pca opion ( and ji ji B i fo j R and i ) a alo difficl o obain. nc in nx cion w poid nmical xampl o illa m c of la a.. ompaai Saic n i cion w conin nmical analyi pnd in main x and poid compaai aic of in-a olailiy mak pic of ik azad a coy a and colaion bwn in a and ic-flow on la-a m c. nmical analyi al al inin popi of diffn yp of la. Fi wn colaion bwn in a and la ic-flow i poii an inca in in a olailiy ca a dclin in conidd la-a m c wic i conin wi Ambo and Yildiim (8). Scond an inca in mak pic of ik l in a ap dclin in On may ioical coy a of n in wic l dfal xci cancllaion opion o pca an opion fo implmnaion. 6

27 conidd la-a m c. i impac inca wi maiy of la conac. id la-a m c if pwad a azad a inca. impac of azad a on la-a m c inca wi maiy of la conac. akn o l a lo will ai iniial n a a ik pmim wn likliood a l dfal o xci opion mbddd in la conac bcom i. dail dicion a in a follow:.. impac of n Ra Volailiy xibi ow impac of in a olailiy σ on m c of an adjabl-a la conac wi dfal ik an mbddd cancllaion opion and a pca opion. W am a colaion bwn in a and la a flow i poii. al of modl paam a am a o in abl xcp a diffn ll of σ a in: σ =.5..5 and.9. [n xibi ] n xibi w ob a an inca in in a olailiy ca a dclin in conidd la-a m c wic i conin wi Ambo and Yildiim (8). impac of in a olailiy on conidd la-a m c i no inifican nil maiy of conidd la conac i lon an fi ya. i impac inca wi maiy of conidd la conac wic i alo conin wi l of lapam and Gnnlin () and Ambo and Yildiim (8). Wil la n i aociad moly wi ic-flow fom lad a lapam and Gnnlin () alo find a in a olailiy a a dic ffc on la n and in of i ffc i oppoi o a of 7

28 colaion bwn in a and ic-flow. A in a olailiy inca i can m c of in a and by m c of la n. ap of conidd la-a m c in xibi a nl qi diffn fom o poidd by Gnadi (995 5) lapam and Gnnlin () and Ambo and Yildiim (8). Wn impac of in a olailiy i analyzd conidd la-a m c a a conx ap (fo σ =.5 and.) o i naly a ai lin (fo σ =.5 and.9). n cona Gnadi (995 5) lapam and Gnnlin () and Ambo and Yildiim (8) po conca ap of la-a m c. Fmo lapam and Gnnlin () a a a mp-ap la-a m c in Gnadi (995 5) can b l of in a ncainy. ow w canno poc conca o mp-apd la-a m c d by Gnadi (995 5) lapam and Gnnlin () and Ambo and Yildiim (8) nl w i an naonabl al of σ bca all of m analyz impac of in a olailiy on a ik-f la wil la conac conidd i no ik-f. Wn dfal ik mbddd cancllaion opion and pca opion a incopoad in adjabl-a la conac imlanoly in o modl azad a of xciin mbddd opion dpnd iidly on in a. nc impac of in a olailiy on conidd la-a m c i mo complicad an a on a ik-f la. A olind in Jaow and Yildiim () in pacic in a ffc on azad a of dfal and xci of cancllaion opion cold b poii. n cona in a ffc on azad a of xciin Fo xampl w can obain a conca-apd la m c if σ i amd o b. and a mp-apd la m c if σ i amd o b.. Ambo and Yildiim (8) am a =. and σ =.7 o obain a mp-apd la m c. 8

29 pca opion cold b nai. Fom l ppci wn in a bcom i / mi a mo difficly o fnd la paymn and wold b n nwillin o pca of lad a. Wil in a olailiy poxi fo ik of conomic condiion la-a m c mo a in a olailiy inca bca fom lo ppci la n appa o b a laily abl incom fo lo. fo wn ncainy in conomic condiion i i lo may ca a low la n o dca pobabiliy of l dfal o of xciin mbddd opion and inad p laily abl ca flow... impac of ak Pic of Rik xibi ow impac of mak pic of ik on m c of an adjabl-a la conac wi dfal ik an mbddd cancllaion opion and a pca opion. lapam and Gnnlin () dmona a ap of la-a m c i y nii o ll of mak pic of ik. al of modl paam a am a o pnd in abl xcp a w fo diffn combinaion of λ λ and λ : () λ =.75 λ =.5 λ =.; () λ =.5 λ =. λ =.; () λ =. λ =. λ =.4; and (4) λ =.45 λ =. λ =.6. W coo fi combinaion of λ λ and λ o b a alf of al in balin ampion fo mak pic of ik and id and fo combinaion o b wo im and im of al in balin ampion pcily. cond combinaion i xacly al in balin ampion fo mak pic of ik. [n xibi ] n xibi w ob a an inca in mak pic of ik l in a dclin in conidd la-a m c and a impac inca wi 9

30 maiy of la conac. Fmo wn mak pic of ik bcom i lop of la-a m c will can fom pwad lopin o downwad lopin. i i conin wi l of lapam and Gnnlin () and Ambo and Yildiim (8)... impac of azad Ra xibi ow impac of azad a on m c of an adjabl-a la conac wi dfal ik an mbddd cancllaion opion and a pca opion. al of modl paam a am a o in abl in in main x xcp a w i diffn ca of azad a: () alf of balin azad a; () balin azad a; and () wo im balin azad a. [n xibi ] Wn likliood a l dfal o xci mbddd opion in la conac bcom i lo will iniily ai n a compnaion fo ik. A xpcd in xibi w ob a la-a m c if pwad wn azad a inca. oo impac of azad a on la-a m c inca wi maiy of la conac. Ambo and Yildiim (8) alo ow imila l wn xaminin impac of a nan cdi ik on la-a m c..4: impac of Rcoy Ra xibi 4 illa impac of coy a on m c of an adjabl-a la conac wi dfal ik an mbddd cancllaion opion and a pca opion. al of modl paam a am a o in abl xcp a w i fo diffn ca of coy a: () coy a = ;

31 () balin coy a; () im balin coy a; and (4) coy a = %. [n xibi 4 ] n xibi 4 w a iniial n bcom low a coy a inca indicain a lo will qi a mall ik pmim if coy a i i. i inin o ob a la-a m c nd fo diffn coy a a mally paalll. qaion (9) in main x ow a iniial n R B i nially a lina fncion of and B wic a amd o b a an in balin ca. Sifin balin coy a l in a copondin momn in la-a m c b kp lop ncand. No a R R R and R a amd o b in balin ca. ow iniial n R i no a lina fncion of R R R and R inc fo paam appa in dnominao of i-and id in qaion (9) of main x. o xamin alnai pcificaion of coy a w am a R =. = RO =. = B = R =. = R =.5 = and B =. W dno i pcificaion a alnai coy a. xibi 5 ow impac of alnai coy a on la-a m c. al of modl paam a am a o in abl xcp a fo diffn ca of coy a a d: () balin coy a; () im balin coy a; () alnai coy a; and (4) im alnai coy a. [n xibi 5 ] xibi 5 ow a la-a m c nd fo alnai pcificaion of coy a a no lon mally paalll. i coy a will ill na low la-a m c b diaion a qi

32 mall. W no a if l dfal o cancl conac n lo can la a wi am poiion o mak b lo a facion of ial al a anacion co. ow n if coy a qal % al of mbddd opion and pmim fo dfal ik a ill incopoad in iniial n of nw la conac. pobabilii of nw l dfal and xciin mbddd opion inca wi im o maiy o do al of mbddd opion and pmim fo dfal ik. fo la a c main pwad lopin. ombinin xibi 4 and xibi 5 w concld a coy a a a laily mall ffc on conidd la-a m c..5. impac of olaion bwn n Ra and Sic flow xibi 6 ow impac of colaion bwn in a and ic-flow ρ on m c of an adjabl-a la conac wi dfal ik an mbddd cancllaion opion and a pca opion. al of modl paam a am a o in abl in main x xcp a diffn ll of ρ a d: ρ =.5 and -.5. [n xibi 6 ] n xibi 6 w ob a an inca in colaion bwn in a and ic-flow ca a dclin in la-a m c. oo impac on la-a m c inca wi maiy of la conac b i impac i ininifican nil maiy of conidd la conac i lon an fifn ya. 4 n cona o Ambo and Yildiim (8) w find a la-a m c pn a conx ap w 4 Followin Gnadi (996) w alo xamin impac of o colaion bwn aiabl dicd in i pap incldin colaion bwn ic flow and lad a pic. ow impac of o colaion i nliibl.

33 colaion bwn in a and ic-flow i poii nai o zo. Rfnc. Aawal S. B. W. Ambo. an and Y. Yildiim m Sc of a Ra wi ndono fal i and nan apial Sc: oy and idnc Jonal of Financial and Qaniai Analyi Ambo B. W. and Y. Yildiim di Rik and m Sc of a Ra: A Rcd Fom Appoac Jonal of Ral a Financ and conomic Bow G. W. and J.. Alb Picin of mbddd Opion in Ral a a onac Jonal of Ral a Rac lapam. and Å. Gnnlin Rnal xpcaion and m Sc of a Ra Ral a conomic Knyon. and S. ompaidi Ral Opion in ain: ffc of dl im Opaion Rac Giaccoo. G.. Goldb and S. P. d Val of mbddd Ral Opion: idnc fom onm Aomobil a onac Jonal of Financ Gnadi S. R Valin a onac: A Ral-Opion Appoac Jonal of Financial conomic Gnadi S. R ain and di Rik Jonal of Financial conomic Gnadi S. R An qilibim Analyi of Ral a a Jonal of Bin

34 . Jaow R. and Y. Yildiim Valin fal Swap nd ak and di Rik olaion Jonal of Fixd ncom connll J. J. and J. S. Scallim Valaion of A ain onac Jonal of Financial conomic ill. and. Upon ain Byin and o of apial Sic Jonal of Financ Scallim J. S. and J. J. connll A odl fo minaion of "Fai" Pmim on a ancllaion nanc Polici Jonal of Financ Scmi. di Rik in ain ny Jonal of Bankin and Financ

35 xibi : mpac of n Ra Volailiy al of modl paam a am a o in abl xcp a diffn ll of σ a in: σ =.5..5 and.9. xibi : mpac of ak Pic of Rik al of modl paam a am a o in abl xcp a fo diffn combinaion of λ λ and λ a in: () λ =.75 λ =.5 λ =.; () λ =.5 λ =. λ =.; () λ =. λ =. λ =.4; and (4) λ =.45 λ =. λ =.6. 5

36 xibi : mpac of azad Ra al of modl paam a am a o in abl xcp a w i diffn ca of azad a: () alf of balin azad a; () balin azad a; and () wo im balin azad a. xibi 4: mpac of Rcoy Ra (a) al of modl paam a am a o in abl xcp a fo diffn ca of coy a a in: () coy a = ; () balin coy a; () im balin coy a; and (4) coy a = %. 6

37 xibi 5: mpac of Rcoy Ra (b) al of modl paam a am a o in abl xcp a fo diffn ca of coy a a in: () balin coy a; () im balin coy a; () alnai coy a; and (4) im alnai coy a w w am R =. = RO =. = B = R =. = R =.5 = and B =.8 in alnai coy a. 7

38 xibi 6: mpac of olaion bwn n Ra and Sic flow al of modl paam a am a o in abl xcp a ll of ρ a in: ρ =.5 and