Risk aversion and debt maturity structure

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1 Rsk avrson an bt matrty strctr Jorg Frnánz Rz* y Blanca Ccla García Mna* Fcha rcpcón: 6 mayo 2002; fcha acptacón: 4 fbrro 2003 Abstract: W sty th rlatonshp of rsk avrson an bt matrty strctr. n a mol n whch avrs slcton n fnancal markts crats a rol for th s of short-trm bt, w allow th possblty of borrowrs bng rsk-avrs. Ths crats a tra-off btwn rc xpct fnancng costs an hghr rsk an allows for th sty of th ffct of rsk avrson on optmal matrty strctr. W prov that, as rsk avrson ncrass, so os th prcntag of bt that s long-trm. Kywors: rsk avrson, bt matrty strctr, avrs slcton. Rsmn: En st artíclo stamos la rlacón ntr la avrsón al rsgo y la strctra vncmnto la a. En n molo n la slccón avrsa n los mrcaos fnancros ocasona la a corto plazo sa útl, prmtmos los ors san avrsos al rsgo. Esto cra na tnsón ntr la rccón los costos fnancros spraos y l amnto n l rsgo, y prmt s analc l fcto la avrsón al rsgo n la strctra óptma vncmnto la a. Probamos l porcntaj óptmo a largo plazo amnta conform s ncrmnta la avrsón al rsgo. Palabras clav: avrsón al rsgo, strctra vncmnto a, slccón avrsa. *El Colgo Méxco, Camno al Ajsco 20, Méxco, D. F. conomía mxcana. NUEVA ÉPOCA, vol. X, núm. 2, sgno smstr

2 Jorg Frnánz Rz y Blanca Ccla García Mna ntrocton Th mportanc of nrstanng bt matrty strctr has bn rcntly ganng rcognton n th fnancal conomcs ltratr an mportant progrss has bn ma n ts analyss. For nstanc, Barclay an mth 995 mphasz th mportanc of ths ss an prov an mprcal xamnaton of th trmnants of bt matrty at th frm lvl. At th contry lvl, rcnt xprncs n Mxco an othr vlopng contrs hav cas th analyss of ntrnatonal bt problms to mov byon th pr ss of bt sz to ncl mor tal aspcts of th fnancng procss nclng th bt matrty profl. n ths papr, w a to th ltratr on bt matrty strctr an focs on ts rlatonshp wth on borrowr charactrstc that has not hthrto rcv mch attnton n th ltratr, namly, th borrowr s rsk avrson. n ths papr, w rly on Flannry s 986 an Damon s 99, 993 vw on why thr s short-trm bt. Accorng to ths vw, short-trm bt can hlp allvat avrs slcton problms n fnancal markts. To s why, consr a staton n whch a frm has bttr nformaton than ts lnrs abot th projct t wants to nrtak. Usally, lnrs wll b abl to obsrv th prformanc of th frm an larn abot t rng th lf of th projct. Ths wll rc th ntal asymmtry of nformaton. Th rol of short-trm bt arss prcsly from ths fact. n, consr th choc of bt matrty. Th natral choc s for bt to matr at th sam tm at whch th projct wll rnr ts frts: to match th matrty of th bt wth th projct t s gong to fnanc. Apparntly, a msmatch btwn th matrty of th projct an th bt ss to fnanc t wol only crat rsks. Yt, som avantags ar obtan by fnancng th projct wth bt that matrs bfor th projct s gvn tm to proc nogh cash flows to rpay t. Th crcal pont s smply that lnrs larn mportant nformaton abot th frm bfor nogh tm has pass to allow for bt rpaymnt. Ths mpls that goo borrowrs may b wllng to borrow bt that matrs bfor t can b rpa ot of th projct s cash flows bcas thy know that at ths for nstanc Col an Kho 996 for an analyss that mphaszs th mportanc of bt matrty strctr n th Mxcan crss. Crmño, Hrnánz-Trllo an Vllagómz (200 an achs, Tornll an Vlasco 996 also al wth th rol of pblc bt strctr n th crss. 276

3 Rsk avrson an bt matrty strctr tm som nw nformaton wll b known, showng that thy ar goo. Ths nw nformaton wll, n trn, allow thm to rfnanc thr bt at bttr trms. Th avantag of borrowng short-trm has to b wgh, howvr, aganst th rsks t ntals. Althogh goo borrowrs xpct that nw nformaton wll most lkly rval that thy hav goo projcts, ths fact can by no mans b takn for grant: to th xtnt that nw nformaton wll not compltly lmnat th asymmtry of nformaton, thr s a rsk that ftr nws wll amag goo borrowrs. Ths, short-trm bt mpls a hghr rsk than long-trm bt. n ths papr, n contrast to Flannry an Damon, w allow for frms to b rsk avrs. Ths mpls that th matrty choc has to balanc th bnfts of rfnancng at trms that allow for a bttr assssmnt of th alty of th projct aganst th costs of hghr rsk. W fn th matrty strctr that optmally solvs ths tra-off. Nxt, w ar n a poston to analyz th comparatv statcs of th optmal bt contract. W prov a rslt that s nttv: mor rsk avrs frms prfr to ss lss short-trm bt an mor long-trm bt than lss rsk avrs frms. W thn xtn or analyss to show that or man rslt xtns to mor complx sttngs. n partclar, w consr a scnaro whr thr s a rsk of a hk n markt ntrst rats rng th lf of a projct, as n Frnánz Rz (2002. Ths sttng captrs a staton whr ffrnt typs of nws arrv bfor th projct matrs, som of whch rfr to th frm s projct tslf whl othrs o not. From a formal pont of vw, or mol s closst to Flannry 986 an Damon 99, 993. Yt, thr ar mportant ffrncs. Frst, n th abov paprs, borrowrs ar rsk ntral whl hr th frm s rsk avrs. con, an closly rlat to th prvos ffrnc, ths paprs o not arss th ss of th ffct of rsk avrson on bt matrty strctr, whch s th focs of or papr. Thr, whl n ths paprs thr s only on kn of ntrmat nws, n ors, thr ar nws both abot th frm s projct an a varabl not rctly rlat to ts projct-markt ntrst rats. Or papr s also rlat to Frnánz-Rz, who sts a mol n whch a rsk avrs contry fnancs ts vlopmnt projct nr asymmtrc nformaton, an two typs of nws on of whch rcs th asymmtry of nformaton bcom known bfor th projct matrs. Bt Frnánz-Rz focss on th ablty of bt contracts to accomplsh 277

4 Jorg Frnánz Rz y Blanca Ccla García Mna th tasks prform by complt contracts. n contrast, n ths papr w focs on th rol of rsk avrson an ts ffct on bt matrty strctr.. Th Mol n ths scton w prsnt a mol n th sprt of Flannry 986 an Damon 99, 993. n ths mol a frm has to ras fns to nrtak a projct, an has prvat nformaton abot th alty of sch a projct. An mportant fatr of ths mol s that rng th lf of th projct, bfor t matrs, lnrs rcv nws abot som projct s charactrstcs whch hlp thm rc th ntal asymmtry of nformaton. A frm ns to ras an amont to nrtak a projct. Ths projct wll b vlop ovr thr pros, t = 0,, 2. At t = 0 th frm can borrow th amont n a compttv crt markt f t can prov nonngatv xpct rtrns to lnrs. At t =, nformaton abot th frm s projct arrvs. At t = 2 th projct s complt an procs cash flow. Thr ar two typs of projcts, an th frm has prvat nformaton abot th typ of projct t has. A frm wth a goo projct obtans an ncom of X >, whl a frm wth a ba projct obtans X wth probablty π an 0 othrws, wth π X <. Ths, nr symmtrc nformaton, lnrs wol not fnanc frms wth ba projcts. Yt, lnrs o not know f a frm has a ba projct. At at 0, thy assgn th frm a probablty f of havng a goo projct. Thrfor, thy assgn th frm a probablty = [ f f π] of obtanng at-two ncom al to X. By obsrvng th prformanc of th frm, crtors larn somthng abot th frm s projct, whch s rflct n th mol n th followng way: at at t = thr ar nws abot th projct that rc th asymmtry of nformaton btwn th frm an ts crtors. Ths nws can b goo, s = (an pgra of th frm s ratng taks plac or ba, s = (a owngra taks plac. Goo frms rcv ba nws wth probablty, an ba frms wth probablty r, wth < r. Crtors pat thr blfs abot th frm s typ pon obsrvng th ralzaton of s. Thy o ths by applyng Bays rl. Lt f ( f b th pat probablty accorng to Bays rl that th frm s goo gvn ba (goo nws. W hav: 278

5 Rsk avrson an bt matrty strctr f f = f f ( f r, ( f ( ( f( r. = (2 f W can calclat n a smlar mannr th contonal probablty that at-two ncom wll b X gvn ba (goo nws. Lt s not ths probablts by (. Th frm maxmzs ts xpct tlty E [(Y], whr Y s at-two ncom, nt of rpaymnts to crtors, an ( > 0, ( < 0. Captal markts ar compttv, an ar wllng to nvst as long as th xpct scont sm of nt rpaymnts from th frm als (or s hghr than 0.. Dbt matrty strctr Th frm can ras th fns to nrtak th projct by sng any combnaton of two ffrnt typs of bt, short-trm bt an long-trm bt, as n Flannry 986 an Damon 99, 993. Th frm can ss short-trm bt wth nomnal val that matrs at t =, aftr ntrm nws ar rlas, an long-trm bt wth nomnal val D, that coms aftr th projct s complt at t = 2. To rpay, th frm ns to com back to th crt markt, snc at th tm matrs th projct that has not yt proc any ncom. Ths, w assm that at t = th frm has accss agan to a compttv crt markt n whch t can ras fns promsng p to (X D, that s, th part of th rsorcs that wll not b n to rpay long-trm bt. Ths, at t =, th frm sss short-trm bt wth fac val 2 that coms at t = 2. W also assm that an D ar sch that th frm s always abl to ras th ncssary fns to rpay at t =. W assm for th momnt that th rsk-fr ntrst rat s constant throghot th lf of th projct, an for smplcty w tak ths ntrst rat to al 0. n an xtnson of th mol n th followng scton w wll consr th cas n whch ths ntrst rat may vary. 279

6 Jorg Frnánz Rz y Blanca Ccla García Mna Fgr llstrats th tmng of th ssanc an rpaymnt of th ffrnt typs of bt. Fgr Frm borrows matrs 2 matrs, D 2 ss 2 ss D 2 matrs t = 0 t = t = 2 W now n to stablsh svral rlatonshps btwn th ffrnt typs of bt. Frst, lt s fn how mch bt th frm mst ss at at to borrow th ncssary fns to rpay. Ths pns on th arrval of goo (s = or ba (s = nws abot th alty of th projct. To borrow at t =, th frm mst ss bt wth fac val 2 satsfyng: 2 = / f s =, (3 2 = / f s =. (4 To s why ths s so, consr for nstanc th cas s =, (th xplanaton for th othr cas s smlar. nc thr ar goo nws abot th projct (s =, fnancal markts xpct that t wll yl X wth probablty. Ths s th probablty wth whch th frm wll rpay th bt 2. Ths, th xpct rpaymnt from a bt wth fac val 2 s 2, whch mst al th amont borrow. Th frm maxmzs ts xpct tlty E[(Y], whr Y s at-two ncom nt of rpaymnts to crtors, an captal markts ar compttv. As n Flannry 986 an Damon 99, 993 w look for th poolng lbrm prfrr by goo alty frms. A goo alty frm chooss th amont of short-trm bt an long-trm bt D so as to solv th followng program (w omt th sbscrpt n : 280

7 Rsk avrson an bt matrty strctr Max, D θ X D ( X D θ sbjct to D [ f ( f π]. Th constrant of ths program nsrs that lnrs wll b wllng to ln th amont. ts lft-han s s th rpaymnt thy xpct to rcv from th frm: long-trm bt wth fac val D wll b rpa at t = 2 wth probablty = [ f - f π ], bcas ths s th probablty wth whch at-two cash flow wll b X. hort-trm bt wll b rpa for sr at t =. Ths rpaymnt has to b at last al to for lnrs to b wllng to ln. Lt s now look at th objctv fncton. t s th frm s xpct tlty from ts at-two ncom X nt of rpaymnts to crtors. Ths rpaymnts consst of long-trm bt wth fac val D, pls short-trm bt wth fac val 2. Consr th frst trm. Wth probablty thr wll b ba nws abot th frm s projct (s =. n ths cas, short-bt rpaymnts amont to 2 = /, as trmn by (4. Th xplanaton for th othr trm s smlar. W wll consr th famly of tlty fnctons wth a constant (Arrow-Pratt masr of absolt rsk avrson, r ( x = = θ. Ths tlty fnctons ar of th form ( x = a bxp( θ x, θ whr a hghr θ s assocat wth mor rsk avrson. Notc that ( > 0, ( < 0 mply b <0 an θ >0. Lt s assm, wthot any loss n gnralty, that a = 0 an b =, so that th frm chooss th bt strctr (, D that solvs th followng program. 28

8 (Program Jorg Frnánz Rz y Blanca Ccla García Mna Max xp θ X D xp θ X D sbjct to D [ f ( f π ]. As s shown n th appnx, th comparatv statcs of th solton to program la to th followng proposton: Proposton. Whn th rsk-fr ntrst rat s constant along th lf of th projct, an ncras n th gr of rsk avrson θ translats nto an optmal fnancal contract comprsng lss short-trm bt an mor long-trm bt D. W llstrat ths proposton by showng th optmal bt strctr for ffrnt grs of rsk avrson, whn f = ½, = ¼, r = ½, π = ½, =, an X = 2.. An xtnson: rsk of chang n ntrst rats n ths scton w consr an xtnson of th abov mol that consrs th arrval of nws not rlat to th projct tslf rng th lf of th projct, as n Frnánz Rz (2002. Mor prcsly, thr may b a hk n ntrst rats rng th lf of th projct. o, th mol s mof as follows: At t =, nformaton abot th frm s projct an abot markt ntrst rats arrv. Ths, thr ar two typs of nws. Frst, as n th prvos scton, thr ar nws abot th projct that rc th asymmtry of nformaton btwn th frm an ts crtors. Bt now thr ar also nws abot rsk-fr ntrst rats. Thy may rman constant or ncras: th on-pro rsk-fr ntrst rat (that s, th ntrst rat on a loan that wll b rpa for sr rmans at 0 wth probablty λ, an ncrass to > 0 wth probablty λ. Th possblty of a hk n ntrst rats mofs th rlatonshps btwn th ffrnt typs of bt. Mor prcsly, th amont of bt 282

9 Rsk avrson an bt matrty strctr Fgr 2. Fracton of short-trm bt as rsk avrson vars whn = 0.40E0.20E0.00E0 / ( D 8.00E E E E E Rsk avrson Fracton of Tabl. Optmal vals of short-trm an long-trm bt whn = 0 as rsk avrson vars Thta Fracton of D E E E E E E E th frm mst ss at at to borrow th ncssary fns to rpay pns now not only on th arrval of goo (s = or ba (s = nws abot th alty of th projct, bt also on whthr th ntrst rat rmans at 0 (n = c, or rss to > 0 (n = h. To borrow at t =, th frm mst ss bt wth fac val 2 satsfyng: 283

10 Jorg Frnánz Rz y Blanca Ccla García Mna 2 = / f s =, n = h, (5 2 = / f s =, n = c, (6 2 = / f s =, n = h, (7 2 = / f s =, n = c. (8 To s why ths s so, consr for nstanc th cas s =, n = h (th xplanaton for th othr cass s smlar. nc thr ar goo nws abot th projct (s =, fnancal markts xpct that t wll yl X wth probablty. Ths s th probablty wth whch th frm wll rpay th bt 2. Ths, th xpct rpaymnt from a bt wth fac val 2 s 2. Thrfor, short-trm bt 2 offrs an xpct (gross rat of rtrn of 2 /, whch mst al th (gross ntrst rat, from whr 2 = / follows. Gvn ths rlatonshps, w wll look agan for th contract prfrr by goo alty frms among th poolng lbrm contracts. Th amonts of short-trm bt an long-trm bt D ar ths chosn so as to solv th followng program: Max, D λ θ X D λθ X D λ θ X D λθ X sbjct to D [ f ( f π ] [ λ ( λ ( ] [ λ ( λ ( ] D Lt s ntrprt ths program. As bfor, th constrant nsrs that lnrs wll b wllng to ln th amont an ts lft-han s s th rpaymnt thy xpct to rcv from th frm: long-trm bt wth fac val D wll b rpa at t = 2 wth probablty = [ f f π], bcas wth ths probablty at-two cash flow wll b X. hort-trm. 284

11 Rsk avrson an bt matrty strctr bt wll b rpa for sr at t =, whch xplans th scon trm n th lft-han s. Th rght-han s s th opportnty cost of. Wth rspct to th objctv fncton, t s agan a goo frm s xpct tlty from ts at-two ncom X nt of rpaymnts to crtors. t has now for trms bcas thr can b for ffrnt combnatons of at-on nws. Lt s ntrprt th frst trm. Wth probablty λ thr wll b ba nws abot th frm s projct (s = an a hk n ntrst rats (n = h. n ths cas, short-bt rpaymnts amont to 2 = /, as trmn by (7. Th xplanaton for th othr thr trms s smlar. Whn sng a tlty fncton wth constant absolt rsk avrson, th bt strctr (, D solvs th followng program: (Program 2 Max λxp θ X D λ xp θ X D λxp θ X D λxp θ X D sbjct to [ f ( f π ] [ λ ( λ ( ] [ λ ( λ ( ] D Th comparatv statcs of th solton to program (2 la (s appnx to th followng proposton: Proposton 2. Whn th rsk-fr ntrst rat may rs at t =, an ncras n th gr of rsk avrson θ translats nto an optmal fnancal contract comprsng lss short-trm bt an mor long-trm bt D. W llstrat ths proposton sng th sam vals as n th xampl aftr proposton, pls λ = ½ an = ¼.. 285

12 Jorg Frnánz Rz y Blanca Ccla García Mna Tabl 2. Optmal vals of short-trm ( an long-trm bt (D whn = ¼ as rsk avrson vars thta /(D D E E 04.50E E E 02.42E E 02.36E 0.30E E 0 2.9E 0.7E E E E E 0 5.4E E E E E E E E E E E 0 Fgr 3 shows th proporton /(D. V. Conclsons W vot ths scton to smmng p or man fnngs an to commnt on two sss not alt wth n or mol that srv attnton. n ths papr w hav focs on th rol of rsk avrson on bt matrty strctr. W hav rawn on th argmnt ma by Flannry 986 an Damon 99, 993 among othrs, that avrs slcton problms n fnancal markts can xplan a vrgnc btwn bt matrty an projct matrty, to bl a mol amnabl to sty th rlatonshp btwn rsk avrson an bt matrty strctr. Th a lang to th s of short-trm bt n th basc mol s that goo borrowrs prfr short-trm bt bcas t allows thm to rfnanc at a tm whn thr prformanc has shown that thy ar n goo. Ths bnft of short-trm bt has to b balanc aganst th hghr rsk t crats. W hav prov n ths basc mol th nttv rslt that as rsk avrson ncrass so os th proporton of total bt that s long-trm. W hav xtn th basc mol to allow for ncrtanty n varabls not rctly rlat to th projct to b fnanc. Ths, w hav consr a staton n whch rng th lf of a projct thr ar two kn of nws. On of ths nws rcs th asymmtry of nformaton btwn th borrowr an th lnr. Th othr on os not rctly rlat to th projct. W hav takn ths scon varabl to b th on-pro rsk-fr ntrst rat. W hav fon that n ths 286

13 Fgr 3. hort-trm bt as a fracton of total bt as rsk avrson vars whn = ¼.40E00 Rsk avrson an bt matrty strctr.20e00.00e00 / ( D 8.00E0 6.00E0 4.00E0 2.00E0 0.00E Rsk avrson Fracton of xtn sttng or prvos rslt contns to hol: Mor rsk avrs borrowrs fn t optmal to contract a hghr proporton of long-trm bt. W now al wth two sss not arss n th mol that srv som commnt. Th frst on calls for an xtnson of th mol that s bst nrstoo by consrng a staton whr th ntrmat sgnal s so ba that both th frm an ts lnrs know that th projct s slss. To al wth ths ss, w col xtn th mol to allow for laton. n othr wors, to allow for th projct to b cancl bfor t matrs. Althogh cancllaton of th projct bfor ts matrty wol com at a cost a partal loss of th ntal nvstmnt t wol avo a worst-cas scnaro. Ths has mplcatons for th optmal bt matrty strctr, snc frm s lablts shol b sgn to mak ffcnt s of ftr nformaton an ths may mply sttng th frm s lablts so as to mak possbl canclng th projct f th nws ar sffcntly ba. Whn comparng th avantags an savantags of short-trm vrss long-trm bt, 287

14 Jorg Frnánz Rz y Blanca Ccla García Mna t shol b notc that th frst on s bttr pp to al wth ths statons, snc t forcs th frm to com back to th captal markts bfor th projcts matr. f ths markts assssmnt of th alty of th projct s ngatv, th frm wll b nabl to rfnanc ts bt. Ths, n trn, wll forc th frm to lat ts projct an wll allow nvstors to rcovr som of thr ntal nvstmnt. On th othr han, by sng short-trm bt w wll fac th rsk of nffcntly latng th projct. mmng p, ang ths xtra mnson wol both mak th mol mor ffclt an nrch th analyss. Th scon commnt has to o wth lty sss. Th strctr of th mol os not consr thm an on col thnk of at last two ways n whch thy col b a. Frst, thr s no lty prmm prsnt n th atons that scrb th rlatonshp btwn long-trm an short-trm ntrst rats. con, th frm os not fac th possblty of a lty shock at any pont rng th lf of th projct. Thy ar smplfyng assmptons that mak th mol mor tractabl. Ths two sss srv attnton. Yt, by abstractng from thm w hav bn abl to solat on mportant channl throgh whch rsk avrson affcts th optmal bt matrty strctr. Rfrncs Barclay, M. J. an C. W. mth Jr. 995, Th Matrty trctr of Corporat Dbt, Jornal of Fnanc, nm. 50, pp Crmño, R., F. Hrnánz-Trllo y A. Vllagómz (200, Rgímns Cambants, Estrctra Da y Fragla Bancara n Méxco, Estos Económcos, núm. 6. Col, H. an T. J. Kho 996, A slf-flfllng mol of Mxco s bt crss, Jornal of ntrnatonal Economcs, nm. 4, pp Damon, D. W. 99, Dbt Matrty trctr an Lty Rsk, Qartrly Jornal of Economcs, nm. 06, pp , Bank Loan Matrty an Prorty whn Borrowrs can Rfnanc, n Mayr, C. an X. Vvs, 993, Captal Markts an Fnancal ntrmaton, Cambrg Unvrsty Prss. 288

15 Rsk avrson an bt matrty strctr Dornbsch, R. 989, Dbt Problms an th Worl Macroconomy n achs, J., Dvlopng Frm Dbt an Economc Prformanc, Th Unvrsty of Chcago Prss, Chcago. Frnánz-Rz, Jorg (2002, Optmal Fnancal Contractng an Dbt Matrty trctr nr Avrs lcton, Estos Económcos, núm. 7. Flannry, M. J. 986, Asymmtrc nformaton an Rsky Dbt Matrty Choc, Jornal of Fnanc, nm. 4, pp achs, J., A. Tornll an A. Vlasco 996, Th Mxcan pso crss: n ath or ath fortol?, Jornal of ntrnatonal Economcs, nm. 4, pp Appnx Proof of Proposton t s a partclar cas of proposton 2, whn = 0. Proof of Proposton 2 To prov proposton 2, w frst solv program (2: Max λxp θ X D λ xp θ X D λxp θ X D λxp θ X D sbjct to [ f ( f π ] [ λ ( λ ( ] [ λ ( λ ( ] D Notc frst that th constrant hols wth alty, othrws w col cras thr or D, ncras th objctv fncton an stll satsfy ths constrant. Ths, w hav that. 289

16 290 ( ( [ ] ( (, wth, ( ( π λ λ ϕ = = ϕ π λ λ = f f f f D an program (2 can b wrttn as. ( xp ( xp ( xp ( xp ( ϕ θ λ ϕ θ λ ϕ θ λ ϕ θ λ = X X X X G Max W hav that ( ( ( ( ( ( ( ( ϕ θ θ θ ϕ λ θ ϕ λ θ ϕ λ θ ϕ λ = X G xp xp xp xp xp Jorg Frnánz Rz y Blanca Ccla García Mna

17 29 an ( (. 0 xp xp ( xp ( xp < ϕ θ ϕ λ θ ϕ θ ϕ θ λ ϕ θ ϕ λθ ϕ θ ϕ θ λ = X X X X G nc ths scon rvatv s ngatv (ach of ts for trms s ngatv, th followng frst-orr conton charactrzs an ntror maxmm:. xp xp xp xp 0 θ ϕ λ θ ϕ λ θ ϕ λ θ ϕ λ = Applyng now th mplct fncton thorm w hav, aftr som smplfcatons, that θ = θ < 0, whch provs th proposton. Rsk avrson an bt matrty strctr

19 Nota

Fakultät III Wirtschaftswissenschaften Univ.-Prof. Dr. Jan Franke-Viebach

Fakultät III Wirtschaftswissenschaften Univ.-Prof. Dr. Jan Franke-Viebach Unvrstät Sgn Fakultät III Wrtschaftswssnschaftn Unv.-rof. Dr. Jan Frank-Vbach Exam Intrnatonal Fnancal Markts Summr Smstr 206 (2 nd Exam rod) Avalabl tm: 45 mnuts Soluton For your attnton:. las do not