Optimal Dynamic Asset Allocation with Capital Gains Taxes and Stochastic Volatility

Transcription

1 Opimal Dynamic A Allocaion wih Capial Gain Tax and ochaic olailiy Yuan-Hung Huku * Dpamn of Financial Opaion Naional Kaohiung Fi Uniiy of cinc and Tchnoy ABTRACT Thi pap appli h opion in h ax law o iniga h ffc of axaion of capial gain on h opimal dynamic conumpion and pofolio choic whn h i pdicabl aiaion in un olailiy. Fo a conai ino, und h lag ffc wih capial gain ax, w poid a ngai af-ax lag ffc on h inmpoal hdging dmand coming fom pu chang of ochaic olailiy wih h aumpion of h ngai alu of h inananou colaion bwn h unxpcd un on h ock and i ochaic olailiy. Moo, in a bad mak accompanid by high olailiy und h lag ffc, a conai ino will ha a ngai ga ffc of h ax opion on h inmpoal hdging dmand coming fom pu chang of ochaic olailiy. Kywod: Capial gain ax; Pofolio choic; ochaic olailiy; nmpoal modl, nmpoal hdging dmand EL claificaion: H4, G, C6 * Copondnc: Yuan-Hung Huku, Dpamn of Financial Opaion, Naional Kaohiung Fi Uniiy of cinc and Tchnoy,, uoyu Rd., Nanz Diic, Kaohiung 8, Taiwan, R.O.C.. Tl: x. 33; Fax: ; huku@ccm.nkfu.du.w.

2 . noducion n h cn ya, h ha bn om ach xploing h opimal dynamic a allocaion agi wih aiou ik, uch a olailiy ik, in ik o inflaion ik. Mon 97 wa h fi o conid h ffc of a ochaic inmn oppouniy in h analyi of opimal a allocaion agi fo long-hoizon ino. How, a a mpiical liau in h 99 ha dmonad ha om dg of a un i pdicabl. Bolll, Chou and Kon 99, Campbll, Lo and MacKinlay 997, and Campbll, Lau, Malkil and Xu ha hown ha ock mak un olailiy i no conan o im. inc hn, acadmic conomi ha mgd udying h ffc of un pdicabiliy on a allocaion agi. Bnnan, chwaz and Lagnado 997 and om of h cn ach fo hi aa xplo modl which xamin h opimal dynamic a allocaion agy whn h a aiabl follow ochaic poc. How, h i a y limid liau on h capial gain ax which apply o h opimal dynamic a allocaion agi wih im-aying olailiy ik. hil h a ubanial diffnc aco couni in boh h ll and ucu of capial incom ax, ino in many couni a gnally ubjc o a non-iial amoun of ax. hn hy ll cuii hy hold a a pofi, on may hink ha h ax on capial gain ha an appciabl impac on an indiidual conumpion and inmn dciion. Thfo, ax play an impoan ol in h dciion-making poc of indiidual concning hi conumpion and inmn plan. Th axaion of un on financial a al h bnfi of aing fo fuu conumpion and hu affc h ad-off bwn cun conumpion and inmn Dammon, pa and Zhang,. Th pupo of hi pap i applying h al opion in h ax law o iniga

3 h ffc of axaion of capial gain on h opimal dynamic conumpion and pofolio choic wih ochaic olailiy. Compuing h opimal conumpion and pofolio policy of an ino ubjc o capial gain ax i a challnging ak. Ou ach conibu o h liau on opimal a allocaion by xploing pcily how capial gain ax affc a allocaion wih ochaic olailiy. f a un o olailiy a im-aying, hi impli ha inmn oppounii a im-aying, oo. Mon 97, 973 how ha whn inmn oppounii a im-aying, dynamic hdging i ncay fo fowad-looking ino. Muli-piod o long-hoizon ino a concnd no only wih xpcd un and ik oday, bu wih way in which xpcd un and ik may chang o im. Dynamic a allocaion agi fo muli-piod o long-hoizon ino diff fom ho of ingl-piod ino bcau h fom dmand iky a no only fo hi ik pmia, bu alo fo hi hdging abiliy again ad chang in fuu inmn oppounii. Mon 969, 97, 973 how ha if inmn oppounii a aying oim, hn long-hoizon ino gnally ca abou hock o inmn oppounii and no ju abou walh ilf. Thy may k o hdg hi xpou o walh hock, and hi ca inmpoal hdging dmand fo financial a Campbll,. Rcnly, h ha bn om limid liau xploing and analyzing opimal dynamic pofolio choic wih olailiy ik Liu,, and Chacko and icia, 5. Thy ol fo h opimal conumpion and pofolio choic of long-hoizon ino whn h i pdicabl aiaion in ock mak un olailiy. How, h i no oh ach xploing boh h ffc of capial gain ax and ochaic olailiy on opimal pofolio choic. n addiion, whil aiou couni ha hi own ax law, h ax law in many couni uually ca a

4 iuaion wh h axpay payoff fom a cou of acion mbl h payoff fom wiing a call opion o h gonmn. A a ul of h call-lik nau of h ino ax pay-off funcion, ino ha an incni o duc hi xpcd ax budn. Thi incni will ul in h adjumn of opimal dynamic a allocaion agi and h conumpion ul. n addiion, oh hing qual, h capial gain ax ym gnally impo a high budn on mo olail inmn han on l olail inmn wih h am xpcd un. n oh wod, h ax ym alo impo a high budn whn h mak i mo olail. no can duc hi xpcd ax budn by ducing h olailiy of hi capial gain. On way o duc capial gain olailiy i alo hough inmpoal hdging on h financial a, pcially whn facing an nionmn wih im-aying olailiy. find ha muli-piod ino alu a no only fo hi ho-m ik-un chaaciic, bu alo fo hi abiliy o hdg conumpion again ad hif in fuu af-ax inmn oppounii. Thu h ino ha an xa dmand fo iky a ha flc af-ax inmpoal hdging. n hi gnalizd inmpoal modl und h ochaic nionmn, Mon appoach 97, 973 could no b ud o di a clod-fom oluion by oling a nonlina diffnial quaion on h inmpoal hdging pofolio. Rcnly how, om of h liau ha bgun o wok on i, uch a h appoxima analyical oluion dlopd by Campbll and icia, Kogan and Uppal, and Chacko and icia 5. Th oluion a bad on pubaion of known xac oluion. Thy off analyical inigh ino ino bhaio in modl ha fall ouid h ill limid cla ha can b old xacly Campbll,. n hi pap, w u pubaion mhod o g lina appoxima oluion. mainly did h xplici oluion on a -lina xpanion of h 3

5 conumpion-walh aio aound i uncondiional man poidd by Campbll 993, Campbll and icia 999, and and Chacko and icia 5. Thi pap i oganizd a follow. cion dcib h modl ud and nionmn aumd in hi pap. cion dlop h modl of opimal conumpion policy and dynamic a allocaion agi wih im-aying olailiy and capial gain ax. cion poid analy of h modl ul and how capial gain ax affc a allocaion wih ochaic olailiy. Finally, concluion a gin in cion.. Th Modl. nmn Oppouniy n hi pap, w aum ha h ino in walh in adabl a only. Th a wo adabl a aailabl fo ading in h conomy. On of h a i a ikl mony mak fund, dnod by B wih a conan in a of. inananou un i db d. B Th ho a i aumd o b conan and ax-f in od o focu on h ochaic olailiy of h iky a. Th cond adabl a i a axabl iky ock. dno h pic of h iky financial a a im ; i inananou oal un dynamic a gin by d d dz, wh i h inananou xpcd a of un on h iky ock; and i h im-aying inananou andad diaion of h un on h iky a. dno ochaic aiabl wih a ubcip ; and l h condiional aianc of h 4

6 iky ock ay ochaically o im. Fom h following ing, h inmn oppouniy i im-aying. aum ha h inananou aianc poc i d κ d dz, 3 wh h paam >, which dcib h long-m man of h aianc, κ, i h ion paam of h inananou aianc poc, i.. hi paam dcib h dg of man ion. dz and dz a wo in poc wih conan colaion. aum ha h ock un a colad wih chang in olailiy wih inananou colaion, which may b aumd o b ngai o capu h lag ffc o h aymmic ffc Glon al., 993. Th ngai colaion aumpion wih h man-ion on ock un olailiy can capu wo of h mo impoan fau dicud in h mpiical liau on h quiy mak. Each monay uni of ock old a om im i ubjc o h paymn of an amoun of ax compud accoding o h lai ax bai obd a h pio im. Thi pap aum ha h ax law ca a iuaion wh h ax pay payoff fom a cou of acion mbl h payoff fom holding a call opion. A a ul of h call-lik nau of h axpay ax funcion, and following ica and Papanicolaou 999 and Liu and Pan 3 and und h abo ing, w aum h alu of h al ax opion T, which i h funcion on h pic of h ock and on h olailiy of ock un a im, and will ha h following pic dynamic: dt [ λ T d dz dz, 4 wh λ dmin h ochaic olailiy ik pmium of h al ax opion, and < < and > a mau of h al ax opion pic niiiy o mall 5

7 chang in h undlying ock pic and olailiy, pcily. Thy mau h niiiy of h al ax call opion alu o infiniimal chang in h ock pic and olailiy, pcily. pcifically,, ;,,., Th al ax opion win on h ock wih h non-lina payoff ucu, K fo om ik pic K >, a δ δ δ < δ, and h ik pic K, in fac, i h ino ax bai fo h iky a obd a h pio im. Thu, h dynamic of h af-ax un on h iky ock would b d [ λ T d [ dz dz. 5. Pfnc aum ha h ino pfnc i cui and of h fom dcibd by Duffi and Epin 99. Rcui uiliy i a gnalizaion of h andad and im-paabl pow uiliy funcion ha paa h laiciy of inmpoal ubiuion of conumpion fom h lai ik aion Duffi and Epin, 99, Chacko and icia, 5. Thi man ha h pow uiliy i ju a pcial ca of h cui uiliy funcion whn h laiciy of h inmpoal ubiuion i ju h in of h lai ik aion cofficin. E [ f C, dδ, 6 δ δ wh f C δ, i a nomalizd agggao of ino cun conumpion C δ and uiliy ha h following fom: C f C,, 7 wh i h cofficin of lai ik aion, i h a of im pfnc δ 6

8 and i h laiciy of inmpoal ubiuion; hy a all lag han zo. Th ino objci i o maximiz h xpcd lifim uiliy by chooing conumpion and h popoion of h walh o in in h wo adabl a ubjc o h following inmpoal budg conain, d T n λ C d n dz n dz, 8 wh pn h ino oal walh, whil n a h facion of h ino financial walh allocad o h iky ock a im, and h ino inananou conumpion. C pn 3. Opimal Conumpion Policy and Dynamic A Allocaion agi wih Tim-aying olailiy and Capial Gain Tax Th main objcion of hi pap i o xplo h opimal dynamic a allocaion agi wih im-aying olailiy and capial gain ax. nad of a ingl piod ul, w alo wan o xplo h opimal inmpoal conumpion wih af-ax ochaic inmn oppouniy inducd by h ochaic olailiy. 3. A pcial Ca wih Uni Elaiciy of nmpoal ubiuion of Conumpion Th alu funcion of h poblm i o maximiz h ino xpcd lifim uiliy. Th pincipl of opimaliy lad o h following Bllman quaion fo h uiliy funcion. Und h abo ing, h Bllman quaion will aify 7

9 [,, up C n n n n C T n C f κ λ δ δ, n n 9 wh, dno h diai of. wih pc o walh,, and ochaic olailiy,, pcily. will u h imila noaion fo high diai a wll. alo no ha i h inananou colaion bwn h unxpcd un on h ock and i ochaic olailiy. Th fi-od condiion fo h quaion 9 a C, n [ [ λ. Th opimal dynamic a allocaion agy ha wo majo componn. Th fi m i h man-aianc pofolio wigh. Thi i fo an ino who only in in a ingl piod hoizon o und conan inmn oppouniy, h myopic dmand. Th cond m of h opimal dynamic pofolio allocaion i h inmpoal hdging dmand ha chaaciz dmand aiing fom h di o hdg again chang in h af-ax inmn oppouniy inducd by h ochaic olailiy. Thi m i dmind by h inananou a of chang in laion o h alu funcion. will dicu hi in mo dail la, bcau h fi-od condiion fo ou 8

10 poblm a no xplici oluion unl w know h complicad indic uiliy funcion. ubiuing h fi-od oluion back ino h Bllman quaion, w g [ C C f λ κ [, λ. conjcu ha h xi a oluion of h funcional fom, whn, and ubiu i ino quaion, hn h odinay diffnial quaion will ha a oluion of h fom xp. Raanging ha quaion, w ha h quaion fo, and af collcing m in, and. poid h full dail in Appndix. a now abl o obain h indic uiliy funcion and h opimal conumpion ul and dynamic a allocaion agy wih im-aying olailiy and capial gain ax whn. Th indic uiliy funcion i xp,. 3 Th ino opimal conumpion-walh aio and h opimal dynamic a allocaion agy a C, 4 9

11 n [ [ λ. 5 How, fo h im bing, w df oling hi modl inc hi oluion i mly a pcial ca of ou modl ing whn. n h nx cion, w will u pubaion mhod o find h gnal oluion o ou modl. 3. Appoxima Clod-Fom oluion by Pubaion Mhod Th baic ida bhind h u of pubaion mhod i ha of fomulaing a gnal poblm, on h condiion ha w find a paicula ca ha ha a known oluion, and hn uing ha paicula ca and i oluion a a aing poin fo compuing appoxima oluion o naby poblm. n many financial conomic modl, dmining h unknown funcion play a ky ol in conomic analyi und h aumpion of a gin funcional fom. How, h mo gnalizd h modl i, h mo difficul i i o find a clod-fom oluion, pcially in h ca of an inmpoal conumpion and pofolio choic poblm wih ochaic nonlina paial diffnial quaion. n pi of hi, hi iuaion ha y cnly bgun o chang a a ul of al lad dlopmn. On of h dlopmn ha inold h u of pubaion mhod in om pcial ca wh oluion a did fo compuing appoxima oluion ha will hlp mak conomic analyi mo xplici. Th mhod off analyical inigh ino ino bhaio in modl ha fall ouid h ill-limid cla ha can b old xacly Campbll,. udd and Guu 997,, Kogan and Uppal, Campbll and icia 999, and, and Chacko and icia 5 c. ha ud hi appoach o ol dynamic conomic o financial modl. n h maind of hi pap, w will apply

12 pubaion mhod o ol ou modl. n h conx of ou poblm, h inigh w obain i ha h oluion fo h cui uiliy funcion whn poid a connin aing poin fo pfoming h xpanion. apply h in h piou cion a ou aing poin and compu ou modl aound hi oluion. ihou h icion of, h Bllman quaion can b xpd a h following quaion by ubiuing quaion ino quaion and conjcuing h xi a oluion of h funcional fom,, λ κ [ λ [. 6 To implify, w can mak h anfomaion Φ, and gi h following non-homognou odinay diffnial quaion, λ κ Φ Φ Φ Φ Φ Φ Φ Φ Φ [ [ Φ Φ λ [. 7

13 Unfounaly, h abo quaion canno b compud in clod fom. Ou appoach i o obain an aympoic appoximaion o quaion 7, wh h xpanion i by aking a -lina xpanion of h conumpion-walh aio aound i uncondiional man a hown in h pap of Campbll 993, Campbll and icia 999, and and Chacko and icia 5. Fom h anfomaion Φ, w can g h nlop condiion of h quaion, C C Φ xp{ } xp{ c w}. 8 Thn, uing a fi-od Taylo xpanion of xp{ c w } aound h xpcaion of c w, w can wi Φ xp{ E c xp{ E c w } xp{ E c w } { E c w } [ c w E c w w } xp{ E c w } c w φ φ c w. 9 ubiuing quaion 9 ino h quaion 7 and guing hi quaion ha a oluion of h fom Φ xp, and fom hi gud oluion, quaion 8 can find ha c w { [xp }. A uch, w can xp quaion 7 a

14 [ λ κ φ φ [ λ. Raanging h abo quaion, w ha h following h quaion fo, and, [ κ [ [, φ κ φ λ λ, 3 3

15 [ κ λ φ κ φ φ λ φ φ, 4 wh can b old o h quadaic quaion, can b old o h quaion 3 gin, and can b old o h quaion 4, gin and. A uch, w can now g h indic uiliy funcion and h opimal conumpion ul and h opimal dynamic a allocaion agy wih capial gain ax in h ochaic nionmn wihou conain whn. Th indic uiliy funcion i Φ, xp. 5 Th ino opimal inananou conumpion-walh aio i C xp. 6 Th opimal dynamic a allocaion agy wih capial gain ax i 4

16 n [ [ λ. 7 Now w ha xplicily old h poblm of h dynamic a allocaion agy fo long-hoizon ino wih im-aying olailiy and capial gain ax. n h nx cion, w will poid analy of ou ul. 4. Analy of h Modl Rul and How Capial Gain Tax Affc A Allocaion wih ochaic olailiy Th opimal dynamic a allocaion agy can b paad ino wo componn: h myopic componn, and h inmpoal hdging componn. Fi, h dpndnc of h myopic componn i impl. i an affin funcion of h cipocal of h im-aying olailiy and dca wih h cofficin of lai ik aion. inc olailiy i im aying, h myopic componn i im aying, oo. n oh wod, h myopic componn i imply linkd o h af-ax ik-and-un adoff aociad wih pic ik. Th high h capial gain ax a would lad o h high dla of h al ax opion, and hi will dca h af-ax un. And hnc dca h myopic componn in h opimal dynamic a allocaion fo h iky ock. n addiion, w know ha h capial gain ax ym impo a high budn on mo olail iky ock han on l iky ock wih h am xpcd un. Thi pap how hi phnomnon by h ga of h al ax opion. Th high h ga of h al ax opion, i.. h high h niiiy of h ax budn o infiniimal chang on h ock un olailiy accompanid by h, h high h inca of h ax budn wih pc o > 5

17 h inca in h ock un olailiy, and h low h af-ax un on h iky ock will b. And hnc h ino will dca h myopic componn of h a allocaion on h iky ock. Th inmpoal hdging componn of h opimal dynamic a allocaion i an affin funcion of h cipocal of h im-aying olailiy, wih cofficin and. hil i h oluion o h quadaic quaion, i h oluion o h quaion 3 gin, and i h oluion o h quaion 4, gin and. hn > fo h cofficin, h quaion ha wo al oo of oppoi ign accoding o h quadaic quaion hoy. And h alu funcion i maximizd only wih h oluion aociad wih h ngai oo of h diciminan of h quadaic quaion, i.. h poii oo of quaion. > can immdialy b hown ha. > inc, i man ha h ign of h cofficin of h inmpoal hdging dmand coming fom pu chang in im-aying olailiy i poii whn >. can fuh paa h inmpoal hdging dmand ino h ffc. Fi, if w don inoduc any capial gain ax conidaion, and inad h holding ock i ax-f, h inmpoal hdging componn fo h iky ock will coni of only h colaion ffc o lag ffc. Th inmpoal hdging componn of h opimal a allocaion fo iky ock wihou capial gain ax i affcd by h inananou colaion bwn h unxpcd un and chang in ochaic olailiy of h iky ock. f <, i man ha h unxpcd un on h iky a i low h mak iuaion i bad, and hn h > a of h mak uncainy will b high. inc whn >, h ngai 6

18 inananou colaion bwn unxpcd un on h iky ock and i ochaic olailiy impli h ino will ha ngai inmpoal hdging dmand du o chang olly in h olailiy of h iky a, which lack h hdging abiliy again an inca in olailiy. imila dicuion a found in Liu and Chacko and icia 5. How, in ou gnalizd modl, h conidaion of capial gain ax wih im-aying olailiy complica h inmpoal hdging componn on a allocaion fo long-hoizon ino. n h piou cion w aum a al ax opion who pic xpou i poii >, and olailiy xpou i poii >, wihou any lo of gnaliy. Fom ha, w how ha und h lag ffc fom h ngai colaion bwn olailiy of h iky ock and i pic hock <, w will ha wo capial gain ax ffc in h inmpoal hdging componn fo h iky ock, h ax-opion dla ffc >, and h ax-opion ga ffc <. Thi impli ha und h colaion ffc i.. whn h unxpcd un on h iky ock i low h mak iuaion i bad, and h mak uncainy i high, h low unxpcd un on h iky ock and h high uncainy of h mak a du o h high olailiy of h iky ock will mak capial gain ax play a impoan ol in h inmpoal hdging dmand du o h dla ffc and h ga ffc, and a conai ino will ha a poii componn on h inmpoal hdging dmand coming fom h ax-opion dla ffc, and a ngai componn on h inmpoal hdging dmand coming fom h ax-opion ga ffc. Fo a conai ino, if h don impo any ax and hold only h iky ock, h will dca h holding of h iky ock ia h inmpoal hdging componn du o h lag ffc und high olailiy accompanid by low unxpcd un on h iky ock. 7

19 How, und h lag ffc wih capial gain ax, h ngai inmpoal hdging componn will b paially off by h poii dla ffc of h al ax opion fo <. Th n lag ffc, which w m h af-ax < lag ffc, on h inmpoal hdging dmand coming fom pu chang of ochaic olailiy i. Thi componn of h inmpoal hdging dmand i alo ngai fo h aumpion of h ngai alu of h inananou colaion bwn h unxpcd un on h ock and i ochaic olailiy. Th conidaion of h capial gain ax will dca h abolu alu of hi componn. Th poii dla ffc on ax opion i inuii bcau und h lag ffc, i.. h low unxpcd un on h iky ock wih h high un olailiy on h iky ock, h inca of h holding of h ock will no inca ax budn, und h bad mak. Thfo, h capial gain ax ffc will poid om off ffc on h lag ffc of h ngai inmpoal hdging dmand. How, du o h bad mak accompanid by high olailiy und h lag ffc, a conai ino will ha an ngai ga ffc of h ax opion on h inmpoal hdging dmand coming fom pu chang of ochaic olailiy fo >. Thi ul i ha whn h capial gain ax impo a high budn on mo olail inmn han on l olail inmn wih h am xpcd un, i will nd o cau ino o alloca hi capial o flow away fom iky ock and owad ikl bond. Thfo, w will ha an xa ngai inmpoal hdging dmand fom h ga ffc of ax opion. 5. Concluion Alhough aiou couni ha hi own ax law, h ax law in many couni 8

20 uually ca a iuaion wh h axpay payoff fom a cou of acion mbl h payoff fom wiing a call opion o h gonmn. A a ul of h call-lik nau of h ino ax pay-off funcion, ino ha an incni o duc hi xpcd ax budn. Thi incni will ul in h adjumn of opimal dynamic a allocaion agi and conumpion ul. Th pupo of hi pap i applying h al opion in h ax law o iniga h ffc of axaion of capial gain on h opimal dynamic conumpion and pofolio choic wih ochaic olailiy. Ou ach conibu o h liau on opimal a allocaion by xploing pcily how capial gain ax affc a allocaion wih ochaic olailiy. Th opimal dynamic a allocaion agy can b paad ino wo componn: h myopic componn, and h inmpoal hdging componn. Th myopic componn i imply linkd o h af-ax ik-and-un adoff aociad wih pic ik. can fuh paa h inmpoal hdging dmand xplicily. Fo a conai ino, if h don impo any ax and hold only h iky ock, h will dca h holding of h iky ock ia h inmpoal hdging componn du o h lag ffc und high olailiy accompanid by low unxpcd un on h iky ock. How, und h lag ffc wih capial gain ax, h ngai inmpoal hdging componn will b paially off by h poii dla ffc of h al ax opion. Th n lag ffc, which w call h af-ax lag ffc on h inmpoal hdging dmand coming fom pu chang of ochaic olailiy i alo ngai und h aumpion of h ngai alu of h inananou colaion bwn h unxpcd un on h ock and i ochaic olailiy. n hi pap, w how ha a bad mak accompanid by high olailiy und 9

21 h lag ffc, a conai ino will ha a ngai ga ffc of h ax opion on h inmpoal hdging dmand coming fom pu chang of ochaic olailiy. Thi ul i ha whn h capial gain ax impo a high budn on mo olail inmn han on l olail inmn wih h am xpcd un, hi will nd o cau ino o alloca hi capial away fom iky ock and owad ikl bond. Thfo, w will ha an xa ngai inmpoal hdging dmand fom h ga ffc of ax opion.

22 Appndix Th diaion of h pcial ca fo opimal dynamic a allocaion agy wih capial gain ax and im-aying olailiy whn conjcu h xi a oluion of h funcional fom, whn, and ubiu i ino quaion, λ κ [ λ [. A Th abo odinay diffnial quaion ha a oluion of h fom xp, o A can b xpd a [ λ κ λ. A Raanging h abo quaion, w ha h following h quaion fo,

23 and, [ κ, A3 λ κ, λ. A4 [ κ λ κ λ. A5 Fom quaion A3, w ha:

24 a ac b b 4 ±, A6 wh a b κ [ c Fom hi ul, w can g h indic uiliy funcion and h opimal conumpion ul and opimal dynamic a allocaion agy whn. 3

25 Rfnc Babi, N. C.,, ning fo h long un whn un a pdicabl, ounal of Financ, 55, pp Bolll, T., R. Y. Chou and K. Kon, 99, ARCH modling in financ, ounal of Economic, 5, pp Bnnan, M.., E.. chwaz, and R. Lagnado, 997, agic a allocaion, ounal of Economic Dynamic and Conol,, pp Campbll,. Y., 993, nmpoal a picing wihou conumpion daa, Amican Economic Riw, 83, pp Campbll,. Y., and L. M. icia, 999, Conumpion and pofolio dciion whn xpcd un a im aying, ualy ounal of Economic, 4, pp Campbll,. Y., and L. M. icia,, ho hould buy long-m bond? Amican Economic Riw, 9, pp Campbll,. Y., and L. M. icia,, agic A Allocaion: Pofolio Choic fo Long-Tm no Oxfod Uniiy P, Oxfod, U.K. Campbll,. Y.,, A picing a h millnnium, ounal of Financ, 55, pp Campbll,. Y., M. Lau, B. G. Malkil and Y. Xu,, Ha indiidual ock bcom mo olail? an mpiical xploaion of idioyncaic ik, ounal of Financ, 56, pp. 44. Campbll,. Y., A.. Lo and A. C. MacKinlay, 997, Th Economic of Financial Mak Pincon Uniiy P, Pincon, N. Chacko, G., and L. icia, 5, Dynamic conumpion and pofolio choic wih ochaic olailiy in incompl mak. Riw of Financial udi, 8, 4

26 pp Dammon, R., C. pa and H. Zhang,, Opimal conumpion and inmn wih capial gain ax, Riw of Financial udi, 4, pp Dammon, R., C. pa and H. Zhang, 4, Opimal a locaion and allocaion wih axabl and ax-dfd ining, ounal of Financ, 59, pp Duffi, D., and L. G. Epin, 99, A picing wih ochaic diffnial uiliy, Riw of Financial udi, 5, pp Glon, L. R., R. agannahan and D. Runkl, 993, On h laion bwn h xpcd alu and h olailiy of h nominal xc un on ock, ounal of Financ, 48, pp udd, K. L., 998, Numical Mhod in Economic MT P, Cambidg, MA. udd, K. L., and. M. Guu, 997, Aympoic mhod fo aggga gowh modl, ounal of Economic Dynamic and Conol,, pp.5 4. udd, K. L., and. M. Guu,, Th conomic ffc of nw a: an aympoic appoach, oking Pap Hoo niuion, anfod Uniiy. Kogan, L., and R. Uppal,, Rik aion and opimal pofolio polici in paial and gnal quilibium conomi, NBER oking Pap, No. w869. Liu,., and. Pan, 3, Dynamic diai agi, ounal of Financial Economic, 69, pp Liu,.,, Dynamic pofolio choic and ik aion, oking Pap Uniiy of Califonia, Lo Angl. Liu,.,, Pofolio lcion in ochaic nionmn, oking Pap Uniiy of Califonia, Lo Angl. Mon, R. C., 969, Lifim pofolio lcion und uncainy: Th coninuou im ca, Riw of Economic and aiic, 5, pp

27 Mon, R. C., 97, Opimum conumpion and pofolio ul in a coninuou-im modl, ounal of Economic Thoy, 3, pp Mon, R. C., 973, An inmpoal capial a picing modl, Economica, 4, pp ica, K. R., and G. C. Papanicolaou, 999, ochaic olailiy, mil and aympoic, Applid Mahmaical Financ, 6, pp