Bond Immunization and Exchange Rate Risk: Some Further Considerations * Ivan Ivanov

Transcription

1 Bond Immunzaon and Exchang Ra Rsk: Som Fuh Consdaons * Ivan Ivanov Ramapo Collg of Nw Jsy 55 Ramapo Vally Road ahwah, Nw Jsy mal: vanov@amapo.du Jason ch, Ph.D. Assoca Pofsso of Fnanc School of Busnss Ramapo Collg of Nw Jsy ahwah, Nw Jsy 743 -mal: jhch@amapo.du Apl 5, 6 * Fnancal suppo fo hs sach pojc was povdd by a fnanc un gan fom h Ramapo Collg Foundaon.

2 Bond Immunzaon and Exchang Ra Rsk: Som Fuh Consdaons Absac Ths sach pojc sks o addss wo ccal poblms n h hoy of nnaonal bond pcng: how can xchang a sk b fomally ncopoad no sandad bond valuaon modls?, and how mus sags o mmunz bonds agans ns a and nflaon sk b modfd o also ncopoa xchang a sk? os of all, hs sudy analyzs h mahmacal pops of nnaonal bonds.g., Euobonds. A spcal consdaon s gvn o h wo mos mpoan chaacscs of db scus duaon and convxy and hough hm o h vaous ways o mmunz bonds and bond pofolos fom al ns, nflaon, and xchang a sks. Fogl 984 fomally addssd h ffcs of changs n nflaon and ns as on bond pcs. Unfounaly, xchang a sk dos no appa o hav bn fomally ncopoad no hs pvous modls. oov, w coc a mahmacal o n Fogl s analyss.

3 . Inoducon Ths sach pojc wll addss wo ccal qusons n h hoy of nnaonal bond pcng: how can xchang a sk b fomally ncopoad no sandad bond valuaon modls?, and can nnaonal bonds b mmunzd agans ns a, nflaon, and xchang a sks? Snc mos bonds povd fxd uns o nvsos n h fom of coupon paymns and pncpal, h pmay sk o a hold of a domsc bond s ha ns as and nflaon may ncas. Th consqunc s a ducon n h bond s mak pc du o a dcln n h puchasng pow of fuu coupon paymns and pncpal. Invsos n nnaonal bonds, howv, fac h addd sk ha should domsc ns as ncas, h valu of h cuncy wll appca agans h cuncy of h fogn bond. Thus, h valu of h fogndnomnad coupon paymns and pncpal wll also dcln n valu. ofo, sachs hav bn abl o fomally addss h ffcs of changs n nflaon and al ns as on bond pcs Fogl 984. Th lau has also mphaszd sk managmn chnqus and mmunzaon sags o mnmz boh of hs pacula sks. Unfounaly, xchang a sk dos no appa o hav bn fomally ncopoad no hs pvous modls.. Rdngon condons fo bond mmunzaon Ths sudy sks o analyz h mahmacal pops of h majo nnaonal bonds ssus such as fogn bonds and Euobonds. A spcal consdaon wll b gvn o h wo mos mpoan chaacscs of db scus duaon and convxy, and 3

4 hough hm o h vaous ways o mmunz bonds and bond pofolos fom h pvously mnond sks. Fank chll Rdngon 9 dnfd h wo condons fo mmunzng a bond pofolo also calld h Rdngon condons whch hav bn wdly usd and appld o managng bond pofolos n h nsuanc and bankng nduss. any savng and loans banks and oh fnancal nsuons bcam fnancally sssd dung h la-98s bcaus hy fald o adh o hs smpl condons. Fo xampl, a bank lvags uns by ssung sho-m labls dposs o fund long-m asss mogags. Whl hs sagy of mauy msmachng s faly boundd, s no an uncommon condon fo many fnancal nsuons mpl and Smonson 999. Rddngon fomally dfnd wo ncssay condons fo bond mmunzaon as follows: Th fs dvav of h asss wh spc o h ns a should b qual o h fs dvav of h labls wh spc o. Tha s changs n h asss a offs by changs n h labls: W A L, wh, W s h walh o h n psn valu of h cash flows, A s h psn valu of h asss, and L s h psn valu of labls a h sam pon n m. Rdngon s nal assumpon s ha A L D la Gandvll. Ths s also calld an xac mach of asss and labls Fogl. Dffnang h n valu of h cash flows n quaon wh spc o h ns a ylds h fs Rdngon condon: da dl da dl o d d d d 4

5 Th scond dvav of h asss wh spc o h ns a should b ga han h scond dvav of h labls wh spc o, so ha W mans posv whn any nval of chang d: d A d L d A d L > o > d d d d 3 A bond s valu s hus shldd o mmunzd fom ns a changs f boh Rddngon condons a m. 3. Ins a and nflaon sk Accodng o h sandad Fsh quaon, h un on a sk-f nvsmn ncluds h al a of ns and h xpcd sho-m a of nflaon : 4 Fogl 984 xamns h ffc of boh of hs sks on h nvso s walh valuad a any hozon pon po o mauy and dnod as. A s h psn valu of h asss, L s h psn valu of h labls and W s h n psn valu of h cash flows, all of hm valuad a m. A and L can b gadd as asss and labls ha wll b acqud/ncud a m : W A L P, Y, L, C[ ] F[ ] C[ ] L, 5,wh P,, Y,, and L, s h psn valu of h bond, h nvsmn ncom and h valu of h labls, valuad a h m hozon. W can xnd Fogl s suls o connuous compoundng, wh w wll us D la Gandvll s dvaons of bond pcs and duaon: 5

6 T B C xp{ } 6 wh C s h bond s cash flow, and s h connuously compoundd ns a. Fo h sak of smplcy, w assum a fla yld cuv, wh h ns a s h sam gadlss of mauy. If w us h abov connuous compoundng modl ou walh quaon 5 bcoms: W A L Cxp{ } Fxp{ } Cxp{ } L, Bcaus duaon and convxy masus as ognally dvlopd by acuas, w dsgnd o addss mmunzaon of sk f db scus, only wsn Euopan and Japans govnmn bonds wh h hghs db angs a b consdd n hs pap. Bady bonds and oh dvlopng couns oblgaons a no o b akn no consdaon - snc hy hav a consdabl poon of dfaul sk - alhough hy psn an nsng fld fo a sach. On h oh hand, Gman Pfandbf Pap - whch a hghly lqud, vy low-sk Gman mogag bonds a lvan o hs analyss. Fabozz has obsvd ha h hav bn no cass of dfaul n Pfandbf Pap snc h fs ssu a cnuy ago Bond mmunzaon and xchang a sk socally h hav xsd dffn xchang a gms wh h managd-floa gm cunly favod by h Gman, Japans, and Amcan cnal banks. Accodngly, cuncs a gnally allowd o ad agans on 6

7 anoh whn som boadly dfnd ang bfo cnal banks nvn o y o sablsh xchang a pay. In cn ms, som govnmns nvn only n xm ccumsancs whl ohs follow a mo acv polcy. Unsabl conoms and hd wold couns usually pg h cuncy agans a majo cuncy.g., h dolla o a bask of cuncs whch a colad wh h conomc ccumsancs. In mos cass hs couns a pssud o follow monay polcs ha a smla o hos of h pggd cuncy. Ths can ca sous poblms fo h sall couny bu n aly s a b choc han hypnflaon and sous fnancal css Gandolfo. To ha xn, w wll now xamn how xchang a changs nflunc nnaonal bonds pcs. W dfn h valu of an nnaonal bond.g., a Gman govnmn bond a h hozon m, accodng o h followng quaon: V B 8 wh V s h mak valu of h bond n US dollas, s h uo-dolla xchang a and B s h mak valu of h bond dnomnad n uos a h hozon momn: B C[ ] F[ ] 9 Usng h sandad Fsh quaon 9 w can dfn a a o dscoun bond cash flows such ha: V C[ ] F[ ] By h abov dscusson w alady know ha ou n woh s qual o h dffnc bwn h oal psn valu of h asss and labls a h hozon, wh h asss a a combnaon of h psn valu of h bond and h nvsmn coupon ncom 7

8 a h m hozon. If w dfn and o b h xchang as bwn h domsc lgal nd and wo dffn fogn cuncs hn: W A L [ C[ ] F[ ] C[ ] ] L, Usng connuous compoundng w can ansfom.a no h followng quaon:.a [ Cxp{ } F xp{ } Cxp{ }] L,.B Equaons.A and.b show how boh asss and labls dpnd on h xchang a. Incopoang h fs Rdngon condon fom quaon ylds h followng suls: A L.A A L.B A L.C Bounday condon: A L. Th noaon h s: -al a of ns, -nflaon and -xchang a. No ha h bounday condon A L s quvaln o h Rdngon s nal assumpon, as dscbd abov. If w hn dffna paally wh spc o h al a and nflaon w g suls smla o Fogl s, howv now h xchang a has bn ncopoad no h modl. Asss and labls bcom funcons of h vaabls: h al a of ns, nflaon and xchang as. h addonal ffc of h 8

9 xchang a can b nvsgad and can usful suls can b dvd as o whh mmunzaon s possbl. Noc ha n h cas wh asss and labls a boh funcons of h sam xchang a.., paally dffnang.a and.b wh spc o poducs a suaon wh h psn valu of h asss should xacly mach h psn valu of h labls, whch s a confmaon of ou bounday condon. If, howv, w a gvn dffn xchang as, hn w can paally dffna only wh spc o on of h gvn xchang as. As w wll show blow, hs suaon poss a sous challng o ou modl. 5. A cocon of a Fogl s sul Snc h nx scon of ou analyss dals xclusvly wh calculus basd applcaons, should b mnond h ha on should b vy caful whn cang ou paal dffnaon. Fogl mad h followng msak whn calculang h paal dvav of h nvsmn ncom wh spc o h nflaon a holdng all ls consan p. 53: Y C[ ] C[ ], whl h Fogl acual sul s C[ ] C[ ] [ ]. W gv h dvaon of h acual sul n h nx paagaph. Suppos w a gvn h nvsd ncom xpsson fom quaon 5: Y, C[ ] and ha w a askd o dffna paally wh spc o h nflaon a. ow do w pocd? On way o do hs s o spaa h lvan vaabl - so ha dffnaon s smplfd. Thus: 9

10 Y, C[ ] C[ ] [ ]. Now s as o s ha h only pnn pa of h las quaon s consan. nc: [ ] snc w a ang all ls Y C[ ] C[ ] [ ]. C[ ] [ ] 6. odl suls 6. Fs-Od Condons: Duaon Ths scon of h analyss wll focus on h mplcaons of noducng xchang a sk no h modl. In pacula, w xamn fou cass wh asss and labls a dnomnad n smla and dffng cuncs. W nnonally gno hdgng oppouns o undsco spcfc condons wh mmunzaon s hocally possbl and wh s pcludd. Cas I: Asss and labls a dnomnad n h sam fogn cuncy. Usng h xchang a walh quaon.a and paally dffnang wh spc o h nflaon a, usng.a: W A L { [ C[ ] F[ ] C[ ] ]} { L, } Af smplfyng h abov xpsson w g:

11 L, C[ ] [ ] F[ ] ] C[ ] [ ] Smlaly usng.a and.b w oban: 3.. W A L { [ C[ ] F[ ] C[ ] ]} { L, } Af smplfcaon: L, C[ ] [ ] F[ ] ] C[ ] [ ] 3.. Dffnang paally wh spc o h xchang a ylds us h followng sul usng.a and.c: W A L { [ C[ ] F[ ] C[ ] ]} { L, }, whch s quvaln o: C[ ] F[ ] C[ ] L, Th abov sul can also b wn as: A L 3..3 Ths condon shows how conomc n woh s mmunzd agans changs n any of h sk facos. In pacula, xchang a sk s pcludd whn boh asss and labls a dnomnad n h san cuncy. Cas II: Asss a dnomnad n a fogn cuncy and labls a dnomnad n h domsc cuncy.

12 In hs suaon, bcaus ou labls a dnomnad n h domsc cuncy and hus only asss a xposd o xchang a sks. Usng.A and.a w g h followng sul: L, [ C[ ] [ ] F[ ] ] C[ ] [ ] ] Smlaly usng.a and.b w oban: 3.. L, [ C[ ] [ ] F[ ] ] C[ ] [ ] ] 3.. Dffnang paally wh spc o h xchang a ylds us h followng sul usng.a and.c: W A L { [ C[ ] F[ ] C[ ] ]} { L, } C[ ] F[ ] C[ ] whch s quvaln o A 3..3 w nd up wh an mpossbl condon wh A. W can conclud ha mmunzaon s mpossbl bcaus, alhough h hd quaon of h sysm s vald n h mahmacal sns, mpls only a condon of xm ngav conomc valu. Cas III. Th asss and labls a dnomnad n dffn cuncs. Usng.A and.a w g h followng sul:

13 L, [ C[ ] [ ] F[ ] ] C[ ] [ ] ] Smlaly usng.a and.b w oban: 3.3. L, [ C[ ] [ ] F[ ] ] C[ ] [ ] ] 3.3. Dffnang paally fs wh spc o and hn ylds us h followng wo suls usng.a and.c: Rsul A: W A L { [ C[ ] F[ ] C[ ] ]} { L, } C[ ] F[ ] C[ ] ha s quvaln o A A Rsul B: W A L { [ C[ ] F[ ] C[ ] ]} { L, } ha s quvaln o L, B Agan, as n cas II. h s a conadcon such ha A o L. Thfo, w can conclud ha n hs cas mmunzaon s no possbl. 3

14 Cas IV: Th asss a dnomnad n h domsc cuncy and h labls dnomnad n fogn cuncy. Thus w hav and h labls vay wh. Usng.A and.a w g h followng sul: L, C[ ] [ ] F[ ] ] C[ ] [ ] Smlaly usng.a and.b w oban: 3.4. L, C[ ] [ ] F[ ] ] C[ ] [ ] 3.4. Dffnang paally wh spc o h xchang a ylds us h followng sul usng.a and.c: W A L {[ C[ ] F[ ] C[ ] ]} { L, }, whch s quvaln o: L, Agan, as n cas II. and cas III. w ach a conadcon L. W can also conclud ha mmunzaon s mpossbl h. Snc w hav a sysm of quaons, vy quaon n h sysm should b u f w wan o conclud ha h sysm s u. Thus, h abov suls lad us o h concluson ha h fs od condon mgh hold u only n cas I, wh boh h asss and h labls a dnomnad n h sam fogn cuncy. In h oh h cass mo complx fnancal nsumns han smpl bonds should b usd o sasfy h s Rddngon Condons of bond mmunzaon. Such nsumns can b oband by 4

15 cang a pofolo conssng of bonds and opons, bonds and fuus, o a combnaon of hs. Ths cass can also b gnalzd o suaons of connuous compoundng nsad whch a psnd n Appndx A. 6. Scond Od Condons: Convxy Th scond od condons wll b such ha h lascy of h asss s ga han ha of h labls: A L > 4.A A L > 4.B A L > 4.C If w ak h scond dvavs of h xchang a quaons.a and.b, w s ha h abov sad sysm of quaons 4.A, 4.B, 4.C can nv hold u, smply bcaus n all of h fou cass boh A L and hav valus of zo. Thus, yng o sasfy 4.C w ach h conadcon >. Th mplcaon s ha mmunzaon s no possbl f w hav an nnaonal pofolo wh h asss o h labls, o boh of hm vay wh xchang as. 7. Concluson Whl xchang a sk can b fomally ncopoad no h basc bond valuaon modl, h do no xs sasfacoy hocal condons fo smpl bond pofolo mmunzaon wh asss o labls, o boh a dnomnad n fogn cuncs. W hav showd ha an nvso dalng wh nnaonal bonds o such 5

16 pofolos canno fully mmunz hs poson agans advs changs n al ns, nflaon and xchang as. Only n cas I, wh boh of h asss and labls a dnomnad n h sam fogn cuncy, paal pocon of h pofolo mgh b achvd whou h hlp of mo complx fnancal nsumns. owv, fo h scond Rdngon condon o hold u n cas I, mo complx asss should b usd. A cla lmaon of hs analyss s ha w hav no noducd hdgng oppouns no hav w xplod mpcal ss of hs modls whch a all lf fo fuu nvsgaons. 6

17 Appndx A. Boh asss and labls vayng wh h sam xchang a.. Asss dnomnad n a fogn cuncy and labls dnomnad n h domsc mdum of xchang. 3. Asss and labls dnomnad n dffn fogn cuncs. 4. Asss dnomnad n h domsc cuncy and labls dnomnad n a fogn cuncy. W consd h sub cass A, B, C fo ach of h abov fou cass. Thy cospond o.b h xchang a walh quaon usng connuous compoundng whn dffnad usng h sysm of quaons.a,.b and.c.. A L C F C, } xp{ } xp{ } xp{ 5... B, } xp{ } xp{ } xp{ L C F C 5... C, } xp{ } xp{ } xp{ L C F C, whch cosponds o: A L

18 . A L C F C, ] } xp{ } xp{ } xp{ [ 5... B, ] } xp{ } xp{ } xp{ [ L C F C 5... C } xp{ } xp{ } xp{ C F C, whch s quvaln o: A 5..3 Ths, as w mnond abov, s no fasbl. 3. A L C F C, ] } xp{ } xp{ } xp{ [ B, ] } xp{ } xp{ } xp{ [ L C F C C } xp{ } xp{ } xp{ C F C A 8

19 O L, B, whch s quvaln o: A o L and as w mnond abov hs s conadcon. 4. A L C F C, } xp{ } xp{ } xp{ B, } xp{ } xp{ } xp{ L C F C C L, no possbl

20 Bblogaphy D La Gandvll, Olv,. Bond Pcng and Pofolo Analyss, nd don. Th IT Pss, London. Fabozz, Fank J., ooad Choudhy, dos, 3. Th andbook of Euopan Fxd Incom Scus, s don. John Wly & Sons, Inc, Nw Yok. Folg,. Russll, 984. Bond Pofolo Immunzaon, Inflaon, and h Fsh Equaon, h Jounal of Rsk and Insuanc 5, No. Jun., 984, Gandolfo, Gancalo,. Innaonal Fnanc and Opn-Economy acoconomcs, s don. Spng, Nw Yok. mpl, Gog., Donald G. Smonson, 998. Bank anagmn: Tx and Cass, 5 h don. John Wly & Sons, Inc, Nw Yok. shkn, Fdck, 3. Th Economcs of ony, Bankng and Fnancal aks, 7 h don. Addson Wsly Publshng Company. Rddngon, Fank. 95. Rvw of h Pncpl of Lf-Offc Valuaons, Jounal of h Insu of Acuas, 78, No. 3, 95, Smh, Roy, do, 3. Global Bankng, nd don. Oxfod Unvsy Pss, USA.