Continuous-time evolutionary stock and bond markets with time-dependent strategies

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1 Afcan Jounal of Busness Managemen Vol. 64 pp Febuay Avalable onlne a hp:// DOI:.5897/AJBM.5 ISSN Acaemc Jounals Full Lengh Reseach Pape Connuous-me evoluonay soc an bon maes wh me-epenen saeges Zhaojun Yang * an Feng Sh School of Fnance an Sascs Hunan Unvesy Changsha 479 P. R. Chna. Busness school Bejng Insue of Peochemcal echnology 6 P. R. Chna. Accepe Sepembe hs pape evelops a geneal connuous-me evoluonay fnance moel wh me-epenen saeges base on evoluonay game heoy. We show ha he connuous moel whch s a lm of a geneal scee moel s well-efne an f hee exss one compleely vesfe saegy n he mae hen hee s no suen banupcy. We suy n eal a eemnsc evoluonay bon mae an cefy ha a bon mae s evoluonay sable f an only f he oal euns acoss all asses ae he same. By hs way we eve an explc expesson fo he bon valuaon an pove an appoach o ecove he benchma nees ae fom an effecve bon mae. ey wos: Evoluonay fnance evoluonay bon mae connuous-me moel me-epenen saeges. INRODUCION he Dawnan pncple "Suvval of he Fes" s well nown o evoluonay bologss bu ahe less nown fo s applcably n fnancal mae heoy. Howeve hs majo pncple as apple o fnancal maes means nohng else ha hose aes wh he mos successful ang saeges wll omnae he mae a las afe an evoluonay pocess has aen place. hs evoluonay pocess can be unesoo as a pocess of aapaon an maon ahe han a pocess of nheance n evoluonay bology. Fom an evoluonay pon he mae s compleely eemne by he coesponng evoluonay sable ang saegy. Evoluonay fnancal mae moels have been consee n Blume an Easley 99 Evsgneev e al. 6 an Fame an Lo 999. he man pon n seng up an evoluonay fnancal mae moel s he specfcaon of an evoluonay ynamc whch eemnes he mae shaes of he elevan ang saeges n me. In geneal such a ynamc wll epen on he sochasc payoffs vens o pces of he unelyng asses as well as he ang saeges whch ae assume o be aape o he unelyng nfomaon. he moels evelope by Evsgneev e al. 6 an *Coesponng auho. E-mal: zjyang@hnu.eu.cn. el: Hens an Schen-Hoppé 5 assume ha sochasc vens o payoffs of he unelyng asses ae exogenously gven bu ha n conas o ohe moels he asse pces ae eemne by he ang saeges an a mae cleang conon. hs pape employs a smla appoach as n Evsgneev e al. 6 an Hens an Schen-Hoppé 5 bu ses up a moel n connuous me ahe han n scee me. he choce of connuous me bngs wh he usual echncal poblems whch le n he analycal fomulaon of he moel n pacula n a pobablsc famewo bu has he majo benef. In pacula he mehos fom classcal analyss such as PDE an sochasc calculus become applcable an pove poweful ools fo he soluon of poblems. I s heefoe necessay o exen scee me evoluonay Evsgneev e al. s 6 fnance moels o connuous me famewo. As a fs sep no hs econ Yang an Ewal 8 conse a moel n whch he ang saeges ae assume o be fx-mx ha s he elave buge facons ae consan n me. In hs moel Yang an Ewal 8 se up a connuous me sochasc ynamc whch escbes he evoluon of mae shaes n a populaon of fnely many ang saeges. Yang an Ewal 8 enfy evoluonay sable nvesmen saeges ha s hose saeges ha peven enans o he fnancal mae fom ganng wealh n he long

2 464 Af. J. Bus. Manage. un. Une he assumpon ha he elave vens ae fs-oe saonay an egoc Yang an Ewal 8 eve an evoluonay sable nvesmen ule. Fx saeges ae smple an also suable n some economc envonmens bu ae of couse oo escve. Along hs eseach lne Buchmann an Webe 7 eve a connuous me appoxmaon of he evoluonay mae selecon moel of Blume an Easley 99. Conons on he payoff sucue of he asses ae enfe ha guaanee convegence. I s shown ha he connuous me appoxmaon equals he soluon of an negal equaon n a anom envonmen. Fo consan asse euns he long-un asympoc behavo s scusse n eal. Howeve he pape by Buchmann an Webe 7 s subsanally smple han wha sue n hs pape. Fs Buchmann an Webe 7 conse a mae wh only sho-lve asses an hus capal gans ae ome whch s much less neesng. Seconly o guaanee convegence Buchmann an Webe 7 mpose an unnaual conon on he payoff sucue of he asses an so scly speang s connuous-me moel s no a lm of he moel of Blume an Easley 99. Lasly he assumpon of consan asse euns whch s much smple han he one suppose by hs pape s exemely escve an hus s saeges ae fx o me-nvaan as well. ang a sep fuhe no he connuous evoluonay fnance moel hs pape conses a moel n whch saeges ae me-epenen ahe han me-nvaan. A connuous me appoxmaon of he evoluonay mae selecon of a geneal scee-me moel s fs eve whch s a genealzaon of Evsgneev e al. 6. I s shown ha he connuous moel s well-efne an only f one of he saeges s compleely vesfe - hs conon almos mposes no escveness hee s no suen banupcy n he mae. Seconly a bon mae s sue n whch he ven pocess pa by each asse s eemnsc an pces an wealh vay ue o mae neacon. I s cefe ha a bon mae s evoluonay sable f an only f all he oal euns of bons efne n hs pape ae equal o each ohe alhough he same oal eun may change along he me. Any ohe mae can be nvae jus by a pofolo ha nvess all wealh n an asse whch pays off he lages oal eun. When nouce on he mae wh abaly small nal wealh hs pofolo nceases s mae shae a he ncumben s expense. By hs way he necessay an suffcen conons ae eve fo he evoluonay sable pofolo ule. I s shown ha a bon mae s evoluonay sable f an only f each bon s evaluae In a me-epenen saegy he facon of he emanng wealh afe consumpon whch he saegy assgns o he puchase of an asse n one peo may change wh me. On he conay n a fx me-nvaan o me-nepenen saegy he facon mus eep unchange. Of couse he me-epenen saegy s much moe easonable han he lae. by an mpope negal n whch he negan s a scoune value of he ven payoff wh he scoun ae beng mae consumpon paamee. A fomula s pove o evaluae he scoun ae o he mae consumpon ae base on he ang pces n a elave effecve bon mae. By hs way an appoach o compue benchma nees ae s pesene. he sucue of he acle s as follows. hs suy ses up a geneal scee-me mae selecon ynamc whee he lengh of he ang me neval s abaly posve. hen a connuous-me evoluonay soc mae moel s eve an scusse. Fuhemoe nvesgaes an evoluonay bon mae n whch he ven pocess s eemnsc afe whch wo fomulas o pce bons wee pove an he benchma nees ae was ecovee especvely. Fnally he man conclusons of hs pape ae summaze. CONINUOUS-IME MARE SELECION PROCESS WIH IME-DEPENDEN SRAEGIES Followng Lucas 978 hs pape nouces an nfne hozon asse mae moel. Fs a scee mae moel s esablshe wh an abay ang neval an hen a geneal connuous-me moel s eve by ang a lm of ha he ang neval conveges o zeo. Le Ω F P be a pobably space enowe wh F s an nfomaon he flaon F whee flaon fo mae pacpans up o me. All anom vaables an sochasc pocesses n hs pape wll be efne on hs base. Conse an asse mae wh long-lve asses an a sngle peshable consumpon goo. he asses ae nexe by Λ { L }. me s scee an enoe by Π { l l = L} whee s he lengh of he me neval. In hs mae hee ae fnely many playes nvesos an each playe plays one saegy. All playes o saeges nexe by = L I I compee wh each ohe fo he mae capal an he amoun of he wealh manage by saegy s F aape an enoe by w = L I. Each asse pays off a ven n cash o jus a sngle peshable consumpon goo as mple by Lucas 978. As seen fuhe whehe he ven s cash o goo oes no mae only f s suppose o be oally consume. he amoun of he ven pa by asse a peo s F aape an enoe by D. In hs pape he followng assumpon s mpose: Assumpon : All saeges have a common consumpon ae a each peo enoe by c

3 Yang an Sh 465 whch s F aape sasfyng < c < Π. he assumpon on common consumpon ae s obvously necessay n oe o compae pefomances among saeges snce one of he ams of he pape s o suy wha saegy wll omnae he mae a las. If on he conay he consumpon ae s ffeen hen he saegy omnaes he mae pobably also because consumes less. Moeove he assumpon ha < c < says evey saegy o playe mus consume a a scly posve consumpon ae a each peo bu s no peme o consume all hs wealh. Denoe by he facon of he emanng wealh afe consumpon w c whch saegy assgns o he puchase of he asse n peo. Fomally a saegy s a sochasc pocess whch s F aape. Fuhe he nex assumpon s mae: Assumpon : Fo evey saegy he facon nvese n each asse s F aape an non-negave ha s fo almos evey sample pah an all. hee s a leas one compleely vesfe saegy ha s gven ha Π o fo he j L I wh connuous me moel hee s a { } j w > such ha < < fo almos evey sample j pah an all. Assumpon means ha sho sellng s pohbe hough whch s que possbly unnecessay o ge he man conclusons of hs pape. Meanwhle Assumpon also maes sue ha he pces of all asses ae scly posve hans o equaon whch s acually jus he eason why s assume. In hs famewo all he asses ae jus long-lve asses whch pay off vens n cash o pouce he same peshable consumpon goo say ml. he cash o peshable consumpon goo ae consume compleely an especally can no be use o enves. Fo hs eason he oal amoun of asses ae n he mae eeps consan. On accoun of a soc spl he followng assumpon s mpose whou loss of genealy: Assumpon 3: he supply of each asse s nomalze o. Accong o hs assumpon he mae-cleang pce enoe by ρ s gven by: I = ρ = w c. Denoe he aggegae mae wealh by W ha s I W = w hen he mae-cleang pce of = consumpon goo o cash enoe by eemne by: ρ ρ Π s W c = D whee D s he aggegae ven ha s D = D. In conas o Evsgneev e al. = 6 ae a nomalze pce fo cash ha s Followng hs s wen as ρ. W c = D whch s howeve a b moe afcal alhough he ffeence s no mpoan.. Fo a gven saegy pofle an he Le Λ { L } wealh w = L I he pecenage of all shaes ssue of asse ha saegy nvess ung peo s: π w = Π Λ. I j j j w = I s even ha: I π = = 3 = =. 4 Snce he change of a wealh esuls only fom vens an capal gans he wealh of saegy a he begnnng of peo + ha s w + s eemne by: w c w = ρ D + ρ ρ π = o accong o an 3 4 eemne by: w + ρ D ρ π = = +. 6 Whle he oal amoun of asses ae n he mae eeps consan he aggegae mae wealh W oes change wh me because of ang anomly. Howeve socal welfae has evenly no elaon o he level of he 5

4 466 Af. J. Bus. Manage. mae wealh W an a bg W only means hghe pces of he asses an consumpon goo o cash. Acually hee exs nfnely many soluons o he wealh ynamcs 6 snce s by homogeneous fo all wealh vaables ha s w Π = L I. In pacula any soluon mulple by an abay non-zeo numbe s sll a soluon. Fo hs eason an an economc conseaon he aggegae mae wealh W s aen as a numéae n he followng. Base on hs numéae he nomalze pces fo asse an consumpon goo ae p W an ρ p W especvely fo each peo. Fuhe he nomalze wealh o he mae shae of saegy s w W. Accongly follows ecly fom an 3 ha: c p = c = an =. I I π = p = Obvously he mae shae pocess s he mos mpoan nex ha ells how saegy pefoms. Fo example he mae shae pocess eemnes whehe he saegy s omnae o omnang. Consequenly he ynamcs of mae shae pocess s sue fuhe n eal. Smla o Yang an Ewal 8 by an elemenay manpulaon fo all follows fom an 6 ha: I j j + π = j= = c + c whee epesens he elave ven paymen of asse ha s. Equaon 8 s calle D D mae selecon equaon. Obvously hans o 8 f =. In oe o avo hs unneesng case s suppose houghou he pape ha L I. Fuhe > fo each { } s no ffcul o pove ha 8 s well-efne. Fomally he followng heoem hols. heoem Gven a saegy pofle an all he elave ven paymen pocesses { } Π ρ 7 8 Λ an pove ha Assumpon an hol hee exss a pah-wse unque mae shae pocess { } saegy = L I. Poof Π sasfyng 8 fo each A sech of a poof s gven hee Evsgneev e al. 6. In fac hs heoem can be pove by solvng 8 sep by sep fowa. A each sep say peo + 8 s a sysem of I lnea equaons n I vaables ha s { L I} an he coeffcen max can be + pove o be nveble by vefyng ha has a column omnan agonal hans o Assumpon an an 4. hus he heoem s pove. A connuous-me evoluonay fnance moel wh me-epenen saeges s esablshe n he followng by leng n 8. o acheve hs goal moe assumpons ae neee an shown hus: Assumpon 4: Fo all Π Λ { L I} an evey sample pah he followng lms exs an he equales hol: + lm + = lm + = an fuhe lm & 9 houghou hs pape "o" noaon s use fo evaves. Assumpon 4 seems oo escve bu acually no. he fs equaly of he assumpon says he elave ven pocess s connuous as well as sochasc whch escbes he economc phenomenon ha he elave ven changes gaually nsea of suenly. In aon s well-nown ha a connuous-me pofolo n mahemacal fnance s mosly lef connuous wh gh han lms. Consequenly he pofolo s geneally no connuous le alone beng ffeenable. On he conay asse pces n hs seup ae enogenous nsea of exogenous as n mahemacal fnance an hey o change along wh he pofolo. Accongly once a pofolo s pefome he wealh manage by any playes o saeges wll be change n he same me hough a complcae ebalancng pocess fo mae-cleang. hs ebalancng pocess wll mae he amoun of wealh manage by any playes change afe ang whch s oally ffeen fom he self-fnancng ang assume by mahemacal fnance. As a esul evey aleaon of a pofolo wll ncu a song essance fom he mae an he spee of aleaon mpobably ges oo quc - ha s o say he pofolo s alee slowly an hus he ffeenal assumpons n 9 ae also accepable. Followng he seemly songly escve Assumpon 4 he nex assumpon s nuvely easonable:

5 Yang an Sh 467 Assumpon 5: he common consumpon ae fo each playe o saegy s a funcon of he me neval an he followng lms exs fo almos evey sample pah an all Π : c lm c an fuhe lm c =. hs assumpon says ha he longe he me neval he moe he consumpon ae an hee exss an nsananeous spee of he change of he consumpon ae whch s c fo all Π. Now s eay o eve he connuous-me evoluonay fnance moel wh me-epenen saeges. In oe o ge a eep nsgh he followng ex uns o he case I = whch s paculaly mpoan fom evoluonay game heoy. Une hs case only wo saeges compee wh each ohe fo he mae capal an he mae shaes sasfy ha + = fo all Π. Fo hs eason he ynamcs of only one mae shae say Π wll be sue n eal subsequenly he followng heoem s he man concluson of connuous-me mae selecon pocess wh me-epenen saeges n whch s calle connuous-me mae selecon equaon wh me-epenen saeges. heoem Suppose a saegy pofle { } Π = Λ s gven an Assumpon 5 hol: Fo all Π an = < < f an only f < < ; Fo almos evey sample pah: lm = lm = ; Fuhe fo almos evey sample pah he mae shae = s ffeenable an sasfes he followng anom ffeenal equaon: & + & + c = + = + c + = & Poof I suffces o vefy he conclusons abou he mae shae of saegy.fs I follows fom 8 an he case I = ha: c + π + c π = = + + c + c + π + + π = =. 3 As a esul pa of hs heoem s mmeaely pove hans o he assumpons of he heoem an: + = Π. 4 Fom 4 an 7 s conclue ha: π π π = = = + = + =. Consequenly one ges ha: π + π + = = = c + c + π + + π = c c =. 6 Nex le he me neval shn o zeo ha s hen by 9 an 4 he numeao conveges o zeo an he enomnao of he gh-han se of 6 conveges o: + = >. π π = = = + 7 Accongly he secon equaly of s eve. Fuhemoe one nfes fom 9 an 6 ha: c π π + + & = = = lm = + & = =. heefoe pa of he heoem s shown afe a smple subsuon fom 7. Mae selecon equaon ffes fom Iô Sochasc ffeenal equaon snce s negal can be wh nal value whle & = &. We only gve a sho poof fom nuon hee. Recenly Palczews an Schen-Hoppé pove a sc poof fo almos he same concluson.

6 468 Af. J. Bus. Manage. efne pah by pah as a Remann negal. emonsaes explcly ha he evoluon of each mae shae s eemne by all saeges as well as he elave ven an s well efne n he sense ha fo an abay nal value s soluon exss an s pah-wse unque. Wh a vew o povng hs concluson a geneal exsence an unqueness ae shown n he followng. Le: = & & + c + = x x h ω x x + x + x+ c an g ω x xh ω x hen can be equvalenly wen as: x& = g ω x o x& = xh ω x 8 n whch nal value = x = an x x. Geneally speang hee s a pah-wse unque soluon o 8 fo an abay nal value x [ nsea of only = x = [ ]. In oe o ge hs esul he nex assumpon nsea of Assumpon s suffcen: Assumpon 6: he consumpon ae an he evaves of he facon nvese n each asse ae almos suely pah-wse connuous fne an F -aape. ha s o say < c < & < an c & ae connuous funcons of fo all Λ { } an almos evey sample pah. hs assumpon nealy pus no exa escon fom an economc vewpon on accoun of he commens followng Assumpon 4. Now he exsence an unqueness of 8 ae fomally pesene hus. heoem 3 If Assumpon 6 hols fo a gven saegy pofle hen fo x [ an abay nal value wh x + x fo all Λ hee exss a pah-wse unque local soluon of anom ffeenal equaon 8. Poof Accong o he assumpon of he heoem hee s a c null se N F such ha fo evey ω N one can fn an open subse U sasfyng x U [ on whch funcon g ω x s Lpschz connuous n he h agumen unfomly wh espec o he fs fo moe eals efe o he poof of heoem 4 below. Hence he heoem s pove hans o he well-nown Pca-Lnelöf heoem. Base on heoem 3 an Assumpon he exsence an unqueness of mae selecon equaon ae scusse nex an a moe nsghful esul s summe up n he followng. heoem 4 Fo a gven saegy pofle Assumpon an 6 hol: < < = hee Fo an abay nal value exss a pah-wse unque soluon of mae selecon equaon whch sasfes < < = fo all > ; Fo almos evey sample pah f = hen = fo = an ; f = hen = fo = an. Poof Accong o heoem 3 pa of he heoem s obvous an hus whou loss of genealy only pa fo = s shown n he followng. can be wen as: & = g ω o & = h ω 9 wh nal value < <. hans o heoem 3 an pa of he heoem he phase space of 9 s nclue n he neval [ ] fo an abay nal value < <. Fs hee exss a posve > such ha hee s a pah-wse unque soluon of he nal value poblem 9 fo a gven sample pah up o owng o heoem 3 an n pacula < < can be [ guaanee fo. Seconly he neval s exene as we as possble n he followng way. Fo a gven saegy pofle an an abay posve nege n efne soppng me: n ω nf Λ s.. < + < n o max{ c } } & & > n

7 Yang an Sh 469 whee he nfmum of he empy se s unesoo o be. Noe ha fo all x [ ] an < ω n + an mn{ } max{ } n x + x mn{ } x x max{ } an hus follows fom a lenghy bu elemenay calculaon ha fo all x [ ] < n ω an ω Ω 4 3 f x < 6n an h x > n. x ω Fo hs eason hee exss a pah-wse unque soluon of 9 up o by means of exensbly n ω of soluons. Moeove accong o he secon equaly of 9 he soluon sasfes: ω ω u = exp h u u > exp n u > 3 fo ha n ω. On he ohe han s even n < snce + = an one can also show > n he same way. Lasly can be vefe ha lm n n ω = a. s.. In fac f hee s a measuable se A F wh P A > such ha lm n n ω = τ ω < fo all ω A hen hee exss Λ an { } such ha + = o = o τ ω τ ω τ ω τ ω & = fo all ω A τ ω c τ ω. he las wo equales conac Assumpon 6 an he fs equaly leas o τ ω = ha s τ ω =.e. w τ ω = an w τ ω = an = o τ ω τ ω = whch conacs Assumpon. Hence he poof s fnshe. I follows ecly fom heoem 4 ha f one of he saeges s compleely vesfe hen hee s no suen eah o banupcy n he economy even some saeges ae no compleely vesfe. Fo example suppose n a mae hee ae only wo asses: One pays an anohe pays nohng all he me ha s = an = especvely. he c c >. consumpon paamee s consan.e. Saegy s compleely vesfe wh µ = an = µ < µ < all he me bu saegy s no wh = an =. Saegy s clealy vey ba snce nvess all he wealh n asse whch pays nohng all he me. Howeve he ba saegy wll no go banup foeve only f s nal wealh s scly posve ha s < <. In fac a smple calculaon leas o: µ exp c = > fo all + exp µ µ c alhough lm = an hus lm = ha s saegy omnaes he mae a las. EVOLUIONARY SABLE BOND MARE Mae selecon equaon s a ahe geneal connuous-me evoluonay fnance moel n whch s ffcul o answe wha s he "opmal" saegy snce he saeges ae me-epenen n a sochasc envonmen. In oe o avo a oo complex poblem suenly whle suyng hs geneal moel Yang an Ewal 8 eal wh he case of consan popoons saeges hough n a sochasc wol whch s subsanally smple han he geneal moel esablshe n subsequenly. Fo he same eason neveheless fom anohe pon of vew he followng ex focuses on a eemnsc bon mae bu eeps he saeges me-epenen. By a eemnsc bon mae we mean nohng ohe han he mae scusse above n whch he elave ven pocess s a eemnsc funcon of me. Whle funamenals ae fxe pces an wealh vay ue o mae neacon. hs mae appeas oo escve bu n fac nclues a vaey of maeable easuy secues ssue by he Une Saes Depamen of he easuy hough he Bueau of he Publc Deb: easuy blls easuy noes easuy bons an easuy Inflaon Poece Secues IPS. hs mae also nclues Accoun easuy bons ssue by he Chna Depamen of Fnance whch wee fs ssue n 997 bu he amoun nceases wh an exemely quc spee. In a eemnsc wol he ven s ece n avance an hus fo a gven saegy pofle s an onay ffeenal equaon. Le: & + & + c c + = + f ;. = +

8 47 Af. J. Bus. Manage. he mae selecon equaon n a eemnsc wol s ewen as: & = f ; an & = &. hee ae obvously only wo fxe pons n ha s an especvely accong o pa of heoem 4. Whou loss of genealy he sably of fxe pon s scusse below. Namely we nvesgae a bon mae whee hee ae wo playes: One s an ncumben o he omnan mae saegy saegy wh an nal mae shae an anohe s a muan saegy wh an nal mae shae. I s ou am o answe whehe he ncumben s evoluonay sable. A fs sgh he assumpon ha hee ae only such wo saeges n a mae seems unealsc bu n fac no. Fo example one can nepe saegy as he mae pofolo an saegy as an abay saegy say one playe by an nvual o even a fnancal nsuon. Fo hs eason whehe he ncumben s evoluonay sable equals o whehe he mae s effecve. An f he ncumben s evoluonay sable hen a "fa pce" of each asse can be eve by he followng 4. Accongly a suy on a mae even wh only such wo saeges can sll lea o a lo of neesng conclusons. I s clea ha he evoluon of s eemne by he sgn of funcon f ;. In pacula f he sgn s posve he mae shae of saegy wll ncease bu ecease on he conay. heefoe he followng efnon s pesene. Defnon : Saegy s calle evoluonay sable f hee exss a posve ε > such ha fo evey saegy say saegy wh an nal mae shae < < ε s mpossble o fn a me neval say a b such ha a b whle f ; > fo f ; fo a b c an ; On he conay saegy s calle evoluonay unsable f hee exss a me neval say a b an a saegy enoe by saegy wh an abay nal mae shae ha < < such f ; > fo a b whle f ; fo a b c an. I follows fom hs efnon ha an ncumben s evoluonay sable f an only f hee s no abage oppouny n he mae. Noe ha a he begnnng of he eny of a muan he mae shaes sasfy an hus he sgn of eemne by s lnea appoxmaon: f ; s f ; c + c +. & = Defnon : Fo a gven ncumben he oal eun of asse s efne by: c + & Γ Λ. Λ; { } 3 Γ s calle he oal eun because compleely eemne he nvesmen value of he asse as seen n Γ s a sum of he he followng heoem 5. In fac capal gan an he ven eun. o mae hs clea ecall 7 an an noe ha. hen one fns: p = + Λ 4 whch means he elave pce of an asse appoxmaely equals o he facon nvese n hs asse by he ncumben. Especally f he mae pofolo s aen as he unque saegy n a mae ha s = all he me hen he elave pce s jus he facon nvese by he mae pofolo accong o 4. I follows fom 4 ha & / p& / p an c / p. Clealy c p p& s a capal gan whle c p epesens a ven eun. 3 says ha he bgge he changng spee c of consumpon ae he moe epenen he oal eun on he elave ven bu he less epenen on he capal gan. Hence explans ha n a socey wh excess consumpon he ven pa by an asse shoul be much moe appecae. heoem 5 Fo a gven ncumben pove ha he oal eun of a mae wh he ncumben s no encal fo all asses a some me hs ncumben s evoluonay unsable. Poof. In fac f a muan nvess all wealh n an asse wh he lages oal eun a fs an hen gaually evse he pofolo o equal o he ncumben hen he muan wll gan he mae. Namely f ; s

9 Yang an Sh 47 scly posve n a small neghbohoo of an eep non-negave all he me. Hence hs heoem s shown fom Defnon. hs heoem means ha a mae s abage-fee only f he oal euns of an ncumben efne by 3 ae he same acoss all asses. o ae one sep fuhe nex we pove ha f he oal euns of an ncumben efne by 3 ae he same acoss all asses hen he ncumben s evoluonay sable.e. he mae s abage-fee: Assumpon 7: Fo a gven ncumben saegy he oal eun efne by 3 s encal fo all asses all he me. hs assumpon also appeas o be oo escve bu as pove n heoem 5 f he assumpon oes no hol hen hee exss an abage oppouny n he mae whch s no neesng o suy also n mahemacal fnance. By vue of Assumpon 7 he value Γ s nepenen of an s heefoe enoe by ν n he followng. Accongly one ges: c + & = ν Λ. 5 Noe ha: = = = = = & =. 6 Aggegang 5 ove asses one fns c c ν = c namely: + & =. 7 Consequenly une Assumpon 7 he lnea appoxmaon n s always zeo whch howeve oes no ecly mean saegy s evoluonay sable. Hence a moe eale analyss of he secon an even hghe oe appoxmaon s necessay. Fo hs pupose a secon-oe appoxmaon of funcon f ; s shown hus: & + c & = ; = + + c 3 f O = 8 whee a funcon h x s sa o be O g x f lm x h x / g x <. Equvalenly fo an abay emnal me one has: & + = + +. c c O = 9 heefoe f he age of a muan s o maxmze s mae shae a a gven emnal me ha s hen he opmal muan followng vaaon poblem: mn ; Λ { } c c = =.. = Λ s s a soluon of he & + 3 whee saegy s consane o be ffeenal accong o Assumpon 4. Regang hs poblem hee exss an ε opmal saegy shown fuhe. heoem 6 Le be a mnmum value of he se ε > { } Λ. hen fo any gven small an ε -opmal soluon o 3 s gven by: = δ ε = ; =. In aon δ ε 3 { Λ } whch passes δ ε an passes δ ε an whch δ ε an ae ffeenable a δ ε an whee δ ε s a suffcenly small posve numbe. he opmal objecve funcon equals o / / + whch can be scly negave snce ε s a suffcenly small posve numbe. Poof I follows fom 7 ha: c c = ε

10 47 Af. J. Bus. Manage. Consequenly applyng he ule fo negaon by pas 3 s equvalen o: mn ; Λ = c c c c + = =.. = Λ. s + 3 Obvously he opmal saegy s myopc. In aon he objecve funcon s quac an he consane conons ae convex. hus hee exss a unque globally opmal soluon o 3. Moe conceely hee opmzaon poblems ae solve below: he fs one s: mn ; Λ = = s.. = Λ n whch hee s a unque global soluon - ha s = Λ an he opmal value of he objecve funcon s /. he secon one s: max ; Λ = = s.. = Λ n whch hee s a unque global soluon - ha s = whle { } = Λ an he opmal value of he objecve funcon s s:. he las one mn ; Λ c c c c + = =.. = Λ s 33 n whch hee s a unque global soluon.e. = Λ an he opmal value of he objecve funcon s zeo. Accongly on accoun of ha he saeges mus be ffeenal he heoem s pove. hs heoem shows ha he opmal muan s almos he same as he ncumben excep ha whle appoachng o he emnal me he muan gaually bu qucly nvess all he wealh n only one asse of whch he facon of he wealh he ncumben assgns o he puchase s mnmum a he emnal me. I s also shown ha he opmal objecve funcon s scly negave. hs mples fom 9 ha he mae shae of he muan s scly nceasng a he emnal me alhough eeps unchange po o he en. Howeve ha oes no epesen he ncumben s evoluonay unsable. o mae hs pon clea one noes ha f he muan plays he ε -opmal saegy gven by 3 he oal eun on asse mus be he smalles among all asses a he emnal me n he new mae pofolo nclung he nvesmen of he muan afe he eny. As a esul Assumpon 7 oes no hol anymoe a me an he only asse he muan puchases has he leas value fo nvesmen snce no aonal nveso wll buy a he puchase pce of he muan accong o heoem 5. Meanwhle f he pofolo of he muan eeps nvaan he gh-han se of s scly negave a leas n a sho peo afe he emnal me an so he muan s mae shae wll scly ecease. Fo example suppose an nsuonal nveso buys an asse on a lage scale ung a sho peo. Unoubely he asse pce wll sho up an hs mae shae wll ncease. Howeve hs phenomenon wll no las long an he pce wll efnely fall. Afewas he wll pobably lose wha he gane. Accongly anohe conon s mpose on saegy playe by he muan. he am s o mae sue ha he new exene mae pofolo afe he eny sasfes Assumpon 7 sll a he emnal me - ha s o say he consan = Λ s ae n he followng. heefoe he vaaon poblem 3 s change no:

11 Yang an Sh 473 mn ; Λ c c = = = s.. Λ = Λ. & + 34 Base on heoem 6 he followng heoem s even an hence he poof s ome. heoem 7 If Assumpon 7 hols hen hee s a unque soluon of 34 whch s gven by: = Λ. 35 he opmal objecve funcon an f ; equal o zeo fo all [ ]. Fo an abay saegy whch s ffeen fom he saegy gven by 35 he objecve funcon an f ; a some me [ ] ae scly negave n a small neghbohoo of. Hence he ncumben sasfyng Assumpon 7 s evoluonay sable. Noce ha he emnal me s abaly selece an n pacula f le hen one conclues 35 hols anyme. heefoe he followng coollay s ec fom heoems 5 an 7: Coollay : An ncumben s evoluonay sable f an only f he pofolo of he ncumben sasfes Assumpon 7. EXPLICI EXPRESSION FOR BOND VALUAION AND BENCHMAR INERES RAE hs scuss poves an expesson fo bon valuaon f he mae s abage-fee. Owng o Defnon a mae s abage-fee f an only f he ncumben ha s he mae pofolo s evoluonay sable o equvalenly f an only f Assumpon 7 hols accong o Coollay. hus hee s suppose ha Assumpon 7 hols. Clealy Assumpon 7 leas o 7 whch yels by negaon: u c u c u c s = exp u exp s u. u o equvalenly: u c u u s c c = exp u + exp s u. u 36 Followng ha he man esul hee s pove subsequenly. heoem 8 A bon mae s evoluonay sable f an only f each bon elave pce s gven by: c s p u = c u exp s u Λ; u 37 On he conay f he bon pces ae collece aleay n an evoluonay sable mae hen he scoun ae consumpon paamee s ecovee by: c = p& p Λ. Poof 38 In hs bon mae he mae pofolo can be consee as he unque saegy say saegy wh. heefoe we conclue fom 4 ha p = fo Λ. Le n 36 hen 37 s eve snce: c uu lm exp =. Lasly 38 follows ecly fom 7 an he equaly p = fo all Λ. Snce p = as shown n he afoe poof he fs pa of he heoem accos wh economc nuon. I says ha n an evoluonay sable bon mae one shoul ve wealh acoss asses accong o he elave asse pces ha s he abage-fee pces whch equal o he scoune values of he elave vens wh he scoun ae jus beng he consumpon paamee c. Howeve he concluson of he secon pa appeas moe neesng. Fo example he benchma nees ae s exemely mpoan n fnance nusy howeve how o fx s open o my bes nowlege. In oe o appoach hs poblem he h pa of heoem 8

12 474 Af. J. Bus. Manage. suggess ha n a elavely effecve ha s evoluonay sable bon mae he elave bon pces ae collece fs an hen he consumpon paamee c ae ecovee by 38. hs paamee shoul be a goo canae fo he benchma nees ae. CONCLUSIONS hs pape evelops a connuous-me evoluonay fnance moel wh me-epenen saeges fom whch an evoluonay sable bon mae s sue. Fs a geneal scee-me evoluonay fnance s esablshe wh an abay ang me neval. A connuous-me moel s eve by leng he me neval convege o zeo. I s shown ha he connuous-me moel s well-efne an hee s no suen banupcy n he geneal mae only f all he asse pces eep posve. Secon as a specal case of he geneal moel a bon mae wh eemnsc ven payoffs s scusse n eal. hs bon mae s no so afcal as one may conse a fs sgh snce nclues a vaey of maeable easuy secues say easuy blls easuy noes easuy bons an easuy Inflaon Poece Secues IPS ssue by he Une Saes Depamen of he easuy. hs mae also nclues Accoun easuy bons ssue by he Chna Depamen of Fnance. I s cefe ha a bon mae s evoluonay sable whch mples he mae s abage-fee f an only f he oal eun of each asse n he mae s encal all he me. Fuhe he abage-fee pce of each bon s eve whch equals an mpope negal wh he negan beng a scoune value of he ven payoff. he scoun ae s encal fo all bons an equals he mae consumpon paamee. o my bes nowlege hee s no an accepe appoach o evaluae he scoun ae o he benchma nees ae alhough hs ae s val fo asse pcng. Equaon 38 a leas poves a new aemp o aac hs poblem. Fo example f one ges bon pces n a elavely effecve o evoluonay sable bon mae hen he scoun ae o he benchma nees ae can be ecovee fom 38. In pacula along hs eseach lne mus be vey neesng o have hs moel ese wh mae aa. Snce hs s no a smple wo an wll lea o anohe pape so we leave n fuue eseach. Fnally alhough a geneal connuous-me evoluonay soc mae wh me-epenen saeges s esablshe n hs pape we afewas focus on an evoluonay bon mae n whch he funamenals ae eemnsc an boh pces an wealh vay ue o mae neacon. Clealy a pofoun suy on connuous-me evoluonay soc mae n a sochasc wol s moe neesng an of couse moe challengng. hs wo s also lef n fuue eseach. ACNOWLEDGEMENS he eseach fo hs pape was suppoe by Socal Scence Funs of Hunan Povnce n Chna Pojec No. YBA58 by Naonal Naual Scence Founaon of Chna Pojec No an No an by Docoal Fun of Mnsy of Eucaon of Chna Pojec No. 6. I am gaeful o Pofesso laus Rene Schen-Hoppé an Pofesso Ja an Yan. hs pape was nae by Pofesso Schen-Hoppé whle I vse he Unvesy of Lees n 5 an he man wo was one whle I vse Chnese Acaemy of Scences n 7. A subsanal evson s mae n hs veson an all emanng eos ae mne. he auhos ae gaeful o he anonymous efeee fo he commens. REFERENCES Blume L Easley D 99. Evoluon an mae behavo. J. Econ. heoy 58: 9-4. Buchmann B Webe S 7. A connuous me appoxmaon of an evoluonay soc mae moel. In. J. heoecal Appl. Fnance : Evsgneev IV Hens Schen-Hoppé R 6. Evoluonay sable soc maes. Econ. heoy 7: Fame JD Lo W 999. Fones of fnance: evoluon an effcen maes. Poceengs of he Naonal Acaemy of Scences 96: Hens Schen-Hoppé R 5. Evoluonay sably of pofolo ules n ncomplee maes. J. Mah. Econ. 4: Lucas R 978. Asse pces n an exchange economy. Economeca 46: Yang ZJ Ewal CO 8. Connuous me evoluonay mae ynamcs: he case of fx-mx saeges. Inves. Manage. Fnan. Innov. 5: 34-4.