Parameter Risks of Surplus Management Under a Stochastic Process *

Transcription

1 mee Risks of uplus Mgeme Ude ochsic ocess * Jeife L. Wg Depme of Risk Mgeme d Isuce Niol Chegchi Uivesiy, Ti Emil: jeg@ccu.edu. Ly Y. Tzeg Depme of Fice Niol Ti Uivesiy Emil: zeg@ms.cc.u.edu. Asc To hedge he iees-e isk gis fim s suplus, isuce compies commoly se he fim s sse duio equl o he de io imes he fim s liiliy duio. Hoeve, his segy focuses oly o he flucuio of iees es; i does o ddess y of he uceiy i he udelied fcos, hich guide he chges i iees es. This ppe fis ideifies pmee isks gis fim s suplus. We fuhe popose o use gol pogmmig o iege he diiol immuizio segy gis iees-e isk d he segies gis pmee isks. ice he gol pogmmig suggesed i ou ppe is ieged model of immuizio segies gis iees-e isk d pmee isks, he immuizio segy suggesed hee icludes clssicl immuizio segy s specil cse. Moeove, he esuls of ou simulio sho h, comped o clssicl immuizio, he gol pogmmig poposed i his ppe c educe sigificly he ovell isks gis isuce compy s suplus. Keyods: sse d liiliy mgeme, immuizio segy, pmee isks * The uhos gefully ckoledge he helpful commes of edio d o oymous efeees, d semi picips ul cofeece of Ameic Risk d Isuce Associio i 1. The ficil suppo fom Niol ciece Coucil of Ti is lso deeply ppecied.

2 . Ioducio My ppes Bieg, 1987; Gove, 1974; d Reio, 199 hve ecommeded usig clssicl immuizio seig he duio of sses equl o he sse/liiliy io imes he duio of liiliies fo immuizig iees-e isk gis isuce compy s suplus. To ecogize he sochsic ehvio of iees es s foud i he lieue, 1 Biys d Vee 1997 d Tzeg, Wg d oo hve exeded he diiol esech of suplus mgeme o he cse hee iees es follo sochsic pocess. The eseches hve foud h, ude sochsic pocess of iees es, he diiol mesueme of duio my misclcule he fim s isk d my equie fuhe modificio. Alhough his lie of esech hs povided my isighful segies fo sse-liiliy mgeme of isuce compies, mos ppes focus o chges i iees es i he cse of give pmees. Hoeve, isuce compy my usully eed o cope ih he eviome i hich oh iees es d ohe fcos guidig iees es could e ucei simuleously. Fo exmple, he cue iees es my flucue ecuse of me-eveig s ecogized y he lieue e.g., Vsicek, 1977; Cox, Igesoll, d Ross, O he ohe hd, log-em iees es could lso shif due o chges i my mcoecoomic policies. Fom isuce compy s poi of vie, chge i he ed of iees es could cuse eve moe sigific impcs o fim s suplus h he sochsic chges of he cue iees es. I his cse, isuce compies my hve o much ifomio o chceize hei pmee isks. 1 E.g., Vsicek, 1977; Doh, 1978; Cox, Igesoll, d Ross, 1979; Doh d Feldm, 1986, Ho d Lee, 1986; Ch e l., 199; d Heh, Jo, d Moo,

3 Aohe exmple is esimio eo i pmee esimes, hich isuce compies my hve moe ifomio o evlue hei pmee isks. The mges i isuce compies ypiclly use uised poi esimos fo pmees i he pocess. Hoeve, he mges lso ecogize h hee exiss esimio eo i pmee esimes. Thus, he pciioes my like o fuhe cool he isk cused y esimes sdd eos, eve hey hve ledy employed he uised esimos. I he ove o cses, he isuce compy my hve fe o some ifomio o mesue hei isk exposue o pmee isks. Bu, ihou y dou, he mges i he isuce compy should hve eed o fuhe cool uexpeced shock fom pmee isks. Thus, his ppe ieds o ivesige he pmee isks of suplus mgeme he iees es follo sochsic pocess. We employ he model poposed y Tzeg, Wg, d oo ecuse hei model is sho o e geel model of my ohe diiol models. Hoeve, ulike Tzeg, Wg, d oo, ho exmie he effecs of sochsic chge o cue iees es, e focus o he chges i he udelied pmee fcos h guide he pocess of he iees es. We fis ideify pmee isks gis fim s suplus d povide he mehods fo immuizig hose isks. Fuhemoe, e popose gol-pogmmig lgoihm o iege diiol immuizio segy gis iees-e isk d he segies gis pmee isks. ice he gol pogmmig suggesed i ou ppe is ieged model of immuizio segies gis oh iees-e isk d pmee isks, his immuizio segy icludes clssicl immuizio segy s specil cse. Moeove, he esuls of ou simulio sho h, comped o clssicl immuizio, he gol pogmmig poposed i his ppe c educe sigificly he doside pmee isks gis isuce compy s suplus.

4 . Model of mee Risks Le CI d CO deoe he csh iflos d csh ouflos of isuce compy peiod. Le us ssume h he eu of he iees e follos he sochsic pocess suggesed y Cox, Igesoll, d Ross 1985 d c e expessed s d d dz, 1 hee is he spo e peiod d,, d e pmees 3. I he ove sochsic pocess, dz follos sdd Boi moio. is he dif e of he iees e d chceizes me-eveig pocess, hee d epese he momeum of he dif e d he me of he log-em iees e, especively. The sdd deviio of he iees e is popoiol o d is deoed y. Le d deoe he cue iees e d he cue vlue of oe-doll zeo-coupo od peiod. Fom Cox, Igesoll, d Ross 1985, α exp β, hee α [ e e 1 ], d β e e 1 1. My ohe sochsic models, such s Vsicek s 1979, c lso e used. Alhough ech model my hve is o segh, i ould e esie o pply he model ih closed fom soluio. 3 I pcices, ho sse-liiliy mges c do i ode o esime hese pmees coecly is impo issue. Ch e l. 199 suggesed geel mome mehod o esime he pmees i he 3

5 Usig Cox, Igesoll, d Ross mehod 1979, 4 e mesue he pese vlue of fuue csh flos of peiods y he mou of csh flos imes he cue pice of oe-doll zeo-coupo od,. Thus, he sses d liiliies of isuce compy, A d L, c e expessed s: A L CI, d CO. 3 The suplus of isuce compy is he equl o A L. 4 Like my diiol ppes, Tzeg, Wg, d oo hve poposed immuizio segy y seig Iees Re Immuizio. Alhough he iees e my chge sochsiclly, he immuizio segy of c poec he suplus of he fim, les loclly. Hoeve, he sochsic chge i he iees e is o he oly souce of isks gis fim s suplus. Le us ecll Equio 1: d d dz. 1 I he ove sochsic pocess, d epese he momeum of he dif e d he me of he log-em iees e, especively. The level of log-em iees es c e diffee fom he oe isuce compy uses ecuse of chge i goveme s ficil iees e pocess. 4 Li d Fees 1995 deived simil mehod of vluio o clcule he eseves fo isuce cocs. This mehod of vluio s lso used y Tzeg, Wg, d oo, ho ssumed h hee is sped eee he discou es of sses d liiliies. To focus o he iegio of pmee isks d iees e isk, fo he ske of simpliciy e ssume h he discou es of sses d liiliies e he sme. Hoeve, he mi esuls of he ppe sill hold fe elxig his ssumpio. 4

6 policies o simply ecuse of esimio eo. Hoeve, he segy of implicily ssumes h he pmees i he iees model do o chge. pecificlly, he isuce compy is ssumed o fce siuio, hee mges of he compy coside he ssumpio of he iees e pocess is cceple, u sill oy ou h he pmees i he iees e pocess my chge uexpecedly. I ohe ods, he pmees i he iees e pocess should e eed s viles, u ufouely he mges of he isuce compies migh hve o ide o oly lile koledge of he disiuio, he pe, o he pocess of hose pmees. To cope ih he pmee isks i suplus mgeme, sse-liiliy mges c mimic diiol immuizio segy d ge he sses d liiliies of he fim s follos: Momeum Immuizio, 5 Me Immuizio, d/o 6 Deviio Immuizio. 7 Me immuizio, momeum immuizio, d deviio immuizio, especively, e used o hedge he isks of chges i he log-em iees e level, he mgiude of he dif e, d he vice i he iees e. Oe dvge of usig Cox, Igesoll, d Ross 1985 model is h he pmees i hei model epese meigful chceisics of he iees e. Aohe dvge is h hei model povides explici soluios, such s Equio, fo he pice of oe-doll zeo-coupo od. Thus, pmee isks c e mesued esily y kig deivives ih espec o hose pmees. Hoeve, i is vey impo o ecogize h he exisece of pmee isks i suplus mgeme does o deped o he employme of y specific sochsic model fo iees 5

7 es. Almos evey model of sochsic iees e is equied o esime cei pmees, hich my hve hei o meigs i eliy. Alhough e use Cox, Igesoll, d Ross 1985 model o demose ou mehodology, he ide of his ppe c e djused o fi io my ohe models. I should e lso ecogized h diffee ppoch should e used, if he isuce compy kos moe ifomio of pmees h e ssume i he ppe. If he disiuios d/o he pocesses of he pmees e ko, he his ifomio should e ieged o e-deive Equio 5. Of couse, he mehodology o simuleously cope ih oh iees e isks d pmee isks should lso e chged if Equio is o loge vlid. Recllig Equios, 3, d 4, he suplus of isuce compy c e eie s uplus,,,, 8 hee. deoes he suplus fucio. Le deoe he diffeece. By Tylo s expsio seies, he chge i he suplus cused y he chges i iees e d pmees i Equio 8 c e ppoximely expessed s: 5 We ppecie h efeees poi ou his ciicl emk. 6

8 Equio 9 is diecly deived fom Equio 8. Hoeve, i should e oed h chge of iees e could e fucio of chges i pmees i he oigil iees e pocess, Equio 1. Thus, Equio 9 should e oly cosideed s ppoximio of chge i suplus he e do o hve pecise koledge o he iecio eee iees e d pmees. Equio 9 povides some ioles fo he segies suggesed y Equios 5, 6, d 7. Fom Equio 9, e ko h E E E E E d sd sd sd sd sd, if he chges i iees e d pmees e ll idepede. Fuhe ssume h he fim keeps o void iees e isk, he E E E E d sd sd sd sd. If he isuce compy hve o ide of he chge i pmees, he he isuce compy my se,, d 6 Equio 9 oly povides he fis ode ppoximio of. The isuce compies c hve ee pefomce o immuizio if highe deivives e lso ke io cosideio. Dougls 199 d Chisese d oese 1994 suggesed, if mges expec he voliliy of iees es o e gee h h ppes i he em-sucue, he he fim s opiml ojecive ould e o mximize is covexiy of he suplus sujec o he zeo suplus duio d is udge cosis. Hoeve, Ggo d Johso 1994 d Be d Coppe 1997 hve demosed h mchig he covexiies of sse d liiliy does o lys impove he immuizio esuls. I his ppe, sice e ssume h he isuce compy kos eihe he disiuio o he pocess of hose pmees, e do o model i oh he iecio eee chge i iees e d pmees d highe deivives of chges i iees e d pmees.

9 o mke E d sd o elimie hei pmee isks. If he souce of pmee isk come fom esimio eos. Give h he poi esimos i he pocess e uised, e could coside h E E E. Thus, if he fim does o ke y isk o he chges i iees e, he he fim c keep d mke E. Assume h he chges i iees e d pmees e idepede, he he sdd deviio of he chge of he suplus could e ppoximed y sd sd sd. If he fim ould like o fuhe cool y isk o he chges i pmees, he he es segy is o se,, d. Thus, if he fim does o like o ke y isk o he chges i iees e d pmees, he he es segy is o keep s ell s,, d. epely, i my o e difficul fo mges o cope ih ech isk, such s o. Hoeve, immuizio segies my coflic ih ech ohe d/o my o eve e compleely compile. To iege he immuizio segies gis iees-e isk d pmee isks, e popose usig he gol-pogmmig lgoihm s follos: mi d 1 CI s.. d, 8

10 d, d, d. hee d is he isk posiio he fim kes d,,, d e he eighs of pmee isks d iees-e isk, especively. Give he isuce compy s liiliy schedule CO, i is oh oig h,,, d e ll lie fucios of sse llocio CI, hich is he decisio vile of Equio 1. Thus, mgeme c solve Equio 1 y lie pogmmig. The iole of Equio 1 is h mges mke he opiml llocio of fim s sses d liiliies o cope simuleously ih pmee isks d iees-e isk gis fim s suplus. If he opiml soluio of Equio 1 is d *, he he segies gis pmee isks d iees-e isk e compleely compile. If he opiml soluio of Equio 1 is gee h zeo, he mges c lso esily ko ho much isk hey ke ude vious isk fcos. By mes of hei expeiece d judgme, sse-liiliy mges c fuhe djus he eighs eee pmee isks d iees-e isk ccodigly. The smlle he vlue of he eigh give i isk, he sice he immuizio segy gis he 9

11 udelied isk he mges ied o ke. 7 Fo exmple, mges c use he segy seig d o impleme he clssicl immuizio gis iees-e isk. Fuhemoe, y seig log ih he ppopie eighs fo ohe pmee isks, mges o oly immuize fim s iees-e isk u lso cool he fim s pmee isks. Thus, he model suggesed y Equio 1 c e cosideed s geel model of diiol clssicl immuizio segy, sice i icludes clssicl immuizio segy s specil cse.. imulio To ivesige pmee isks i suplus mgeme, e cosuc hypoheicl isuce compy give expeced clims. The lce shee d clims schedule fo he hypoheicl isuce compy e sho s Exhiis 1 d. Exhii 1: Blce hee of Hypoheicl Isuce Compy $14,13,74 $13,63,74 $5, Fo he ske of simpliciy, e fuhe ssume he fim is u-off cse, d he liiliies e o e pid ou ove e yes, s sho i Exhii. Exhii : Clims chedule of Hypoheicl Isuce Compy eiods Csh Ouflos 1 $354, $675, 3 $989, 4 $1,417, 5 $1,73, 7 Oe y o deemie he eighs is o se hem popoiol o he sdd eos of esimos. 1

12 6 $,57, 7 $,48, 8 $,83, 9 $3,19, 1 $3,55, Le us ssume h he iees e follos Cox, Igesoll, d Ross pocess 1985, hee 5%,. 1,. 5, d. 3. ice isuce compies my eed o fulfill miimum solvecy mgis d my o e le o oo moey i el pcice, ou simulios coside o ddiiol solvecy cosis d e expessed s: j CI CO 1,, j 1,...,1, 8 d 11 j CI,,...,1. 1 Le us se d 1. Thus, he opiml llocio of csh flos c e geeed y Equio 13 d is sho i Exhii 3. mi CI d s.. d, d, d,, A L, 8 The isuce compy is ssumed o e le o eives is e csh flos ech peiod i he sme ivesme pofolio d he miimum solvecy mgi is 1,. 11

13 j CI CO 1,, j 1,...,1, d j CI,,...,

14 Exhii 3: Opiml Csh Iflos Allocio Usig he Gol-ogmmig Mehod eiods Csh Iflos 1 $1,348,89 $33,68 3 $1,4,38 4 $1,575,733 5 $1,155,584 6 $3,587,633 7 $1,963,884 8 $,65,836 9 $,446,475 1 $4,67,16 The clssicl immuizio segy c e implemeed y he soluios h sisfied Equio 4 d. Hoeve, o void he polems of muliple soluios, he opiml csh flos ude clssicl immuizio e geeed y mximizig he covexiy 9 of he fim sujec o Equios 4, 11, d 1 d. The soluio of he clssicl immuizio segy is sho i Exhii 4. Exhii 4: Opiml Csh Iflos Allocio Usig he Immuizio egy eiods Csh Iflos 1 $86,447 $6,44 3 $69,34 4 $3,163,553 5 $186,618 6 $3,947,679 7 $388,69 8 $,569,774 9 $3,65,38 1 $3,684,377 9 The mi esuls sill hold if he clssicl immuizio segy is chose y ohe ciei. 13

15 Fuhemoe, diec compiso of he chges i fim s suplus vlue eee he gol pogmmig mehod d he immuizio segy ude diffee pmees,,, d ill help o compe he pefomce of hese o segies fo leive cicumsces, especively. I el pcice, isuce compy usully hs o ide o oly lile koledge of he disiuio, he pe, o he pocess of hose pmees. Thus, i ou simulio, e ssume he isuce compy kos he cue vlue of pmees.1,.5 d.3 u does o ko ho hey ill chge sisiclly. To demose he poi of he uexpeced chges, e y o simule leive cses such s chges fom.1 o.1 o. chges fom.1 o.5 o.7 d chges fom.3 o.1 o.5. To keep he iel cosisecy of he ppe, e do o simule ou esuls y ssumig, d follo cei disiuios. If he isuce compy kos he disiuio of, d, hey should ke i io cosideio o deive he picig fomul. ice e employ equio hough he ppe, e mke ou simulio cosise o ou model. The isuce compy c use ou poposed gol pogmmig o cool he isk cused y he uexpeced chges of he pmees. Tle es 1 d i he Appedix epo he pecege diffeece i chges i fim s suplus vlue eee he gol-pogmmig mehod d he immuizio segy. The esuls idice h he pefomce of he gol-pogmmig mehod does o uiveslly domie h of he clssicl immuizio segy 1. Hoeve, i is oh oig h he segies of oh clssicl immuizio d gol pogmmig e cosevive ivesme segies, hich ied o loe he doside isk ised of giig pofis. Thus, he esuls of Tle es 1 d could e cosideed s de-off eee 1 Iuiively he moe volile hese pmees e, he moe likely h he gol pogmmig ppoch ill domie he clssicl immuizio ppoch. We ppecie h efeee poi ou his ciicl emk. 14

16 coollig isk d kig isk fo mkig pofis. Ou simulio esuls suppo h he poposed gol pogmmig model hs he dvges ove he clssicl immuizio model s log s, d e dom vile h ill chge uexpecedly. Moeove, he esuls of Tle es 1 d sho h he pefomce of he gol-pogmmig mehod geelly domies h of he clssicl immuizio segy he d e smll s ell s he is lge. Thus, e c coclude h he gol-pogmmig mehod could help isuce compies o educe he impc of ufvole pmee shocks sigificly.. Discussios Model isks s oe ype of pmee isks We demose fuhe h model isks could e cosideed s oe ype of pmee isks. Accodig o Wg d Hug, ho hve ivesiged he so-clled model isk i suplus mgeme, model isk is evlued he he mge of fim implemes he immuizio segy ude he ssumpio h he iees e follos Vsicek s pocess 1977, he he iees e cully follos Cox, Igesoll, d Ross pocess I fc, oh pocesses c e ieged io moe geel sochsic pocess s follos: d p d dz 14 Thus, i is ovious h he model isk ivesiged y Wg d Hug is he pmee isk of p. Alhough Equio 14 icludes sevel ell-ko sochsic pocesses fo iees Re, 11 o evey model c povide closed-fom soluio fo, hich is esseil fo 11 E.g., Vsicek, 1977; Doh, 1978; Cox, Igesoll, d Ross, 1979; Doh d Feldm, 1986; Ho d Lee, 1986; Ch e l., 199; d Heh, Jo, d Moo,

17 16 implemeig he lgoihm i Equio 1. If does o hve closed-fom soluio, e co use deivives such s Equios 5, 6, d 7 o mesue he pmee isks. Thus, e fuhe popose mehod o ypss he o-closed-fom-soluio ie cused y Equio 14. If he iees e is ssumed o follo moe geel pocess s Equio 14, he suplus of he fim c e eie s CO CI ] [. 15 Thus, he gol-pogmmig lgoihm i Equio 1 c e eged s d CI mi 16 d CO CI d CO s.., d CO CI d CO d CO CI d CO, p p d p CO p CI d p CO, d CO CI d CO. Oviously, he gol pogmmig c e opeed if e ko,,, p, d. Alhough e my o hve closed-fom soluio of o deive hose coefficies diecly, hey c e esimed y simulio. Thus, y he foh cosi i Equio 16 e c iclude he model isk idiced y Wg d Hug s oe ype of pmee isks.

18 Alhough Equio 14 epeses fmily of iees-e models i he lieue, hose models e clssified s oe-fco models, sice hey model oly he sochsic ehvio of he sho es. To cpue possile chges i pmees i oe-fco models, my ppes hve poposed muli-fco iees e models. 1 Wih moe degees of feedoms muli-fco models c usully expli he chge i iees es moe pecisely. Hoeve, mos muli-fco models e ofe difficul o pply ecuse hey usully do o hve closed-fom soluios d esimio of pmees i hose models could e edious. Thus, he gol-pogmmig lgoihm i ou ppe could seve s compomise eee heoy d pcice. O oe hd, o mke he segy echiclly cle, he model employs oe-fco model o chceize he ehvio of iees es. O he ohe hd, gol pogmmig esues h ll pmee shifs e ude cool. Whe mke pices e diffee fom he pediced vlue ccodig o he heoeicl sochsic pocess To pply he lgoihm of his ppe, i is impo fo sse-liiliy mges o hve ifomio of he pocess o he iees e d he pices of zeo-coupo ods. Hoeve, i my o e ecessy o hve he ifomio i his ode. I he simulio, e ssume h isuce compies c lloce hei fuue csh flos y pices expeced s heoeicl pedicios. Ufouely, his my o e he cse i eliy. Le us ssume h fim uses hisoicl d o iees es o esime he udelied sochsic pocess of he iees e, s suggesed y Ch e l The fim fuhe employs he sochsic pocess o clcule he pices of zeo-coupo ods. Hoeve, he mke pices 1 E.g., Be d chz, 198; Fog d Vsicek, 1991; Logsff d chz, 199; Che d co, 199; d Adeso d Lud,

19 of he ods my o e s pediced i heoy d, heefoe, he fim my o e le o sfe csh flos fom oe peiod o ohe s heoeicl pices. Oe y o cope ih his issue is o use he mke pices of ods ised of he heoeicl pices o discou csh flos i he udge cosi. Rhe h usig he esimed iees e y he sochsic pocess o clcule he pices of zeo-coupo ods, e c use he mke pices of zeo-coupo ods o esime he sochsic pocess of he iees e. ice he discou fcos i Equio 15 come fom mke pices, i, ideed, mkes he llocio of he csh flos fesile. The fim fuhe uses od pices o esime he udelied sochsic pocess of he iees e kig he fom of p d d dz. The polem ih his ppoch is h he esimed iees-e pocess my o hve close fom fo evluig he compive sics fo leive isks. Fouely, usig he simulio suggesed i he ove secio, e c ovecome his possile ie. The sochsic pocess of discou fcos fo liiliies Fo he ske of simpliciy, e hve ssumed so f h hee is o sped eee he discou es of sses d liiliies. Thus, e employ he sme se of zeo-coupo od pices o discou fuue csh iflos d ouflos. Hoeve, i my o e ppopie fo isuce compies o evlue fuue csh ouflos s egive vlues of csh iflos, sice his ivolves ll kids of uceiies. Fuhemoe, csh ouflos geeed y diffee lies of usiess my o e ppopie fo usig he sme discou es ecuse diffee ypes of isuce usiesses my ivolve dmiclly diffee isks. Thus, i could impove he effecs of immuizio o esime he sochsic pocesses of es of eus fo sses d liiliies sepely, s suggesed y Tzeg, Wg, d oo. 18

20 Hoeve, hee my o exis complee secody mke fo he csh ouflos of isuce compies. Thus, h e cll od pices fo csh ouflos my o e osevle fo mges i isuce compies. Theefoe, e my eed o esime he sochsic pocess of es of eus of fim s liiliies fis. Fouely, he isuce idusy usully ecods loss ios d comied ios, hich c e employed s poxies fo he es of eus of fim s liiliies.. Coclusios I his ppe, e poposed o use gol pogmmig o iege he diiol immuizio segy gis iees-e isk d he segies gis pmee isks. ice he gol pogmmig suggesed i ou ppe is ieged model of immuizio segies of iees-e isk d pmee isks, his immuizio segy icludes clssicl immuizio segy s specil cse. We hve lso demosed h he lgoihm of gol pogmmig c e exeded o cope ih model isk i suplus mgeme. evel pcicl issues i implemeig he immuizio segy hve ee discussed d possile soluios hve ee poposed. Moeove, he esuls of ou simulio sho h, comped o clssicl immuizio, he gol pogmmig poposed i his ppe c educe sigificly doside pmee isks gis isuce compy s suplus. 19

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22 Chisese, ee Ove, d Bje G. oese., 1994, Duio, Covexiy, d Time Vlue. The Joul of ofolio Mgeme, Wie, Cox, Joh C., Joh E. Igesoll, d ephe A. Ross., 1979, Duio d he Mesueme of Bsis Risk. Joul of Busiess 5, , 1981, A Re-Exmiio of Tdiiol Hypoheses Aou he Tem ucue of Iees Res. The Joul of Fice 36, , 1985, A Theoy of he Tem ucue of Iees Res. Ecoomeic 53, Doh, L. Ui., 1978, O he Tem ucue of Iees Res. Joul of Ficil Ecoomics 6, Doh, Michel U., d Dvid Feldm., 1986, Equiliium Iees Res d Mulipeiod Bod i illy Osevle Ecoomy. The Joul of Fice 41, Dougls, L. G., 199, Bod Risk Alysis: A Guide o Duio d Covexiy Ne Yok Isiue of Fice. Fog, H. Giffod, d Oldich A. Vsicek., 1984, A Risk-Miimizig egy fo ofolio Immuizio. The Joul of Fice 39, Ggo, Louis, d Leis D. Johso., 1984, Dymic Immuizio Ude ochsic Iees Res. The Joul of ofolio Mgeme, pig, Gove, M.A., 1974, O Duio d Opiml Muiy ucue of he Blce hee. Bell Joul of Ecoomics d Mgeme cieces 5, Heh, D. C., Roe A. Jo, d Ade Moo., 199, Bod icig d he Tem ucue of Iees Res: A Ne Mehodology fo Coige Clims Vluio. Ecoomeic 6, Ho, Thoms., d g Bi Lee., 1986, Tem ucue Movemes d icig Iees Re Coige Clims. The Joul of Fice 41,

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24 Wg, Jeife L. d Rchel Hug,, Model Risks fo uplus Mgeme Ude ochsic ocess. Joul of Acuil cice 1, Zeios,. A., M. R. Holme, R. Mckedll d C. Vssidouzeiou., 1998, Dymic Models fo Fixed-Icome ofolio Mgeme Ude Uceiy. Joul of Ecoomic Dymics d Cool,

25 Tle e 1. Appedix The ecege Diffeece i Chges i uplus Usig he Clssicl Immuizio egy R % % % R % -1.47% % R % % %.3 R % % % R % % -.763% R % % % R % % % R % -.763% % R % % -.377% R % % % R % -.154% -.5% R %.6767%.631%.5 R % % % R % -.5%.4918% R %.631% 1.748% R % % -1.% R.7 -.5%.4918%.4514% R.7.631% 1.748% % R % % % R % 1.994% 1.49% R.3.714%.6847%.6336%.7 R % % % R % 1.49% % R %.6336% % R % % % R % % % R % % -.7% Tle e The ecege Diffeece i Chges i uplus Usig he Gol- ogmmig Mehod R.3.88%.59%.159% R.3.48%.4%.6% R.3.31%.91%.5%.3 R.5.59%.159%.7% R.5.4%.6% -.378% R.5.91%.5%.697% R.7.159%.7%.87% R.7.6% -.378% -.8% R.7.5%.697%.798% R %.183%.113% R % -.513% -.113% R %.1861%.4%.5 R.5.183%.113% -.38% R % -.113% -.686% R %.4%.3368% R.7.113% -.38%.413% R % -.686% -.367% R.7.4%.3368%.3483% R % -.5%.749% R.3 -.6% -.38%.44% R.3.591%.537%.551%.7 R.5 -.5%.749%.19% R %.44%.79% R.5.537%.551%.165% R.7.749%.19%.67% R.7.44%.79%.4% R.7.551%.165%.17% 4