Optimal Discounted Portfolio, Expected Wealth and Strategic Consumption for a Defined Contribution Pension Scheme

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1 Global Journal of Scinc Fronir Rarch Mahmaic and Dciion Scinc olum 3 u 7 rion. Yar 3 yp : Doubl Blind Pr Riwd nrnaional Rarch Journal Publihr: Global Journal nc. USA Onlin SSN: & Prin SSN: Opimal Dicound Porfolio, pcd Walh and Sragic onumpion for a Dfind onribuion Pnion Schm By.. Nkki Uniriy of Bnin, Nigria Abrac - hi papr dal wih opimal dicound porfolio, pcd walh and ragic conumpion proc for a dfind conribuion D pnion chm. h aim of hi papr ar o find: h opimal dicound porfolio and opimal dicound conumpion choic o b adopd by h pnion plan mmbr PPM up o rirmn priod; h pcd dicound walh and arianc of h pcd walh for h plan mmbr. h financial mark i compod by a rikl and a riky a, and h ffci alary of h plan mmbr i aum o b ochaic. h pcd dicound walh and i arianc ar obaind. h dicound porfolio and conumpion proc of h plan mmbr ar obaind. i find ha par of h dicound porfolio alu i proporional o h raio of h prn alu of h dicound pcd fuur conribuion o h opimal dicound walh alu orim. i alo find ha hr i h nd for gradual ranfr of par of h porfolio alu in riky a o h rikl on again unforn hock. Kyword : Sragic dicound conumpion, opimal dicound porfolio, pnion plan mmbr, n hock, ochaic alary, dfind conribuion, pnion chm. unfor- GJSFR-F laificaion : MS : 9G Opimal Dicound Porfolio, pcd Walh and Sragic onumpion for a Dfind onribuion Pnion Schm Sricly a pr h complianc and rgulaion of : 3... Nkki. hi i a rarch/riw papr, diribud undr h rm of h rai ommon Aribuion- Noncommrcial 3. Unpord Licn hp://craicommon.org/licn/by-nc/3./, prmiing all non commrcial u, diribuion, and rproducion in any mdium, proidd h original work i proprly cid.

2 Rf. Blanchard, O. J. and Fichr, S. 989, Lcur on Macroconomic. h M pr. Opimal Dicound Porfolio, pcd Walh and Sragic onumpion for a Dfind onribuion Pnion Schm.. Nkki Abrac - hi papr dal wih opimal dicound porfolio, pcd walh and ragic conumpion proc for a dfind conribuion D pnion chm. h aim of hi papr ar o find: h opimal dicound porfolio and opimal dicound conumpion choic o b adopd by h pnion plan mmbr PPM up o rirmn priod; h pcd dicound walh and arianc of h pcd walh for h plan mmbr. h financial mark i compod by a rikl and a riky a, and h ffci alary of h plan mmbr i aum o b ochaic. h pcd dicound walh and i arianc ar obaind. h dicound porfolio and conumpion proc of h plan mmbr ar obaind. i find ha par of h dicound porfolio alu i proporional o h raio of h prn alu of h dicound pcd fuur conribuion o h opimal dicound walh alu orim. i alo find ha hr i h nd for gradual ranfr of par of h porfolio alu in riky a o h rikl on again unforn hock. Kyword : ragic dicound conumpion, opimal dicound porfolio, pnion plan mmbr, unforn hock, ochaic alary, dfind conribuion, pnion chm. W conidr opimal dicound porfolio and dicound conumpion problm in a dfind conribuory pnion chm. hi papr follow h work of [4] and [6]. n hir papr, hy conidrd h opimal porfolio and ragic conumpion in h lifcycl of a PPM in pnion chm. Bu, h ara of dicound porfolio, dicound conumpion and h arianc of pcd dicound walh of a PPM ar y o b conidrd o h b of my knowldg in h liraur of financial mahmaic and acuarial cinc. n hi papr, w aum ha h PPM conum coninuouly hroughou hi or hr lif im and conumpion rmina whn h indiidual di. n h liraur, [9] conidrd a numrical oluion a wll a analyical rul o h inrmporal conumpion problm for porfolio managmn. [7] amind a racabl modl of prcauionary aing in coninuou im and aumd ha h uncrainy wa abou h iming of h incom lo in addiion o h aumpion of nonochaic a rurn. [3] conidrd labour upply flibiliy and porfolio choic of indiidual lif cycl. hy conidrd h objci of maimizing h pcd dicound lifim uiliy by aumd ha h uiliy funcion ha wo argumn conumpion and labour/liur. [] adopd h quadraic uiliy funcion ha characrizd a linar marginal uiliy funcion. hy ard ha h uiliy funcion wa no araci in dcribing h bhaiour of indiidual oward rik a i impli incraing abolu rik 3 Global Journal of Scinc Fronir Rarch F olum X u rion Yar 43 Auhor : Dparmn of Mahmaic, Faculy of Phyical Scinc, Uniriy of Bnin, P. M. B. 54, Bnin iy, do Sa, Nigria. -mail : nkkicharl3@yahoo.com 3 Global Journal nc. US

3 3 Global Journal of Scinc Fronir Rarch F olum X u rion Yar 44 arion. [3] concludd ha labour incom inducd h indiidual o in an addiional amoun of walh o h riky a. hy ablihd ha labour incom and inmn choic ar rlad, bu hy faild o analyzd h opimal conumpion proc of h inor. [4] udid h gnrally h opimal managmn of a dfind conribuion pnion plan whr h guaran dpnd on h ll of inr ra a a fid rirmn da. [5] conidrd opimal dynamic a allocaion ragy for a D pnion plan by aking ino accoun h ochaic faur of h plan mmbr lifim alary progrion and h ochaic propri of h a hld in accumulaing pnion fund. hy mphaid ha alary rik h flucuaion in h plan mmbr arning in rpon o conomic hock i no fully hdgabl uing iing financial a. hy furhr mphaizd ha wag-indd bond could b ud o hdg produciiy and inflaion hock. hy furhr ard ha uch bond ar no widly radd. hy rfrrd h opimal dynamic a allocaion ragy ochaic lifyling. hy compard i again ariou aic and drminiic lifyl ragi in ordr o calcula h co of adoping ubopimal ragi. hir oluion chniqu mad u of h prn alu of fuur conribuion prmium ino h plan. hi chniqu can b found in [], [4], [8]. Drminiic lifyling which i h gradual wich from quii o bond according o prn rul i a popular a allocion ragy during h accumulaion pha of D pnion plan and i dignd o proc h pnion fund from a caarophic fall in h ock mark ju prior o rirmn [], [5], [6]. [5] and [9] analyd nily h occupaional D pnion fund, whr h conribuion ra i a fid prcnag of alary. h claical dynamic lifim porfolio lcion in a coninuou im modl wa dlopd by [], []. [9] ud a dynamic programming approach o drid a formula for opimal inmn allocaion in a D chm and compard hr rik maur o analyzd h rminal n rplacmn raio achid by mmbr. hy uggd ha whn h choic of inmn ragy i drmind, rik profil of indiidual and diffrn rik maur ar boh imporan facor which hould b akn ino conidraion. n hi papr, w aim a finding h opimal dicound porfolio and opimal dicound conumpion choic o b adopd by h pnion plan mmbr PPM up o rirmn priod and h nir lifim; and h pcd dicound walh and arianc of h pcd dicound walh for h plan mmbr. h rucur of h rmaindr par of h papr i a follow. Scion prn h formulaion of h problm which includ h financial modl, walh proc and ochaic alary of a PPM. n cion 3, w prn h dicound walh and dicound conumpion proc of a PPM. Scion 4 prn h prn alu of dicound fuur conribuion of a PPM. n cion 5, w prn h aluaion of h dicound walh proc of a PPM. Scion 6 prn h opimal dicound porfolio and conumpion proc of a PPM. n cion 7, w prn h opimal pcd dicound walh aluaion and dicound conumpion proc of a PPM. Scion 8 prn h accumulad pcd dicound conumpion proc of a PPM. Finally, cion 9 conclud h papr. W conidr a coninuou-im financial mark whr hr ar wo inmn inrumn: a rikl and a riky a. h pric dynamic of h wo a ar gin, rpcily, by Rf 3. Bodi, Z., Mron, R.., and Samulon, W. F., 99 Labour upply flibiliy and porfolio choic in a lif cycl modl. Journal of conomic Dynamic and onrol, 6, Global Journal nc. US

4 db rb d, B, No ha ach conain all h P null of W dno by h of quar ingrabl adapd proc, and by h of quar ingrabl maurabl random ariabl, B pr, B ; 3 W aum ha h financial mark i arbirag-fr, compl and coninuouly opn bwn im and, i.., hr i only on proc aifying whr, i h mark pric of inflaion rik and and i h mark pric of ock rik. h ponnial proc µ d dw, S, ds S R. W allow r >, µ > r, o b conan. µ i h prdicabl -dimnional proc of c apprciaion ra in rlaion o h ock,, i a prdicabl proc of olailiy cor of ock, i olailiy of ock ariing from inflaion and i h olailiy ock ariing from h ock mark. W W i a, W ; R andard - dimnional Brownian moion on a filrd probabiliy pac Ω,, { wih W a null cor almo urly. W i h ourc of rik }, P [, ] of inflaion and W i h ourc of rik of h ock mark. W aum ha h filraion i gnrad by h Brownian moion and i righ coninuou, and L { } [, ] [, ] { } L L h proc adapd proc uch i aumd o b a maringal From and 4, w ha h dicoud facor o b { } ha [, ] [ ] d < i an F maurabl ar iabl uch ha.from, w ha, µ r, random [ ] < Z p W, L 4 3 Global Journal of Scinc Fronir Rarch F olum X u rion Yar 45 3 Global Journal nc. US

5 aking h diffrnial 5, w ha B h dynamic of h PPM ffci alary i gin by Z rd dw d Global Journal of Scinc Fronir Rarch F olum X u rion Yar 46 dy Y ω d dw, Y y>, whr Y i h alary of h PPM a im, ω i h pcd growh ra of alary of PPM and Y, i h olailiy cor of alary which i drin by h ourc of Y uncrainy of inflaion, W and h ock mark, W. h wo ourc of rik ar parial corrlad. W aum ha ω > i a conan and and i a conan cor. n hi cion, w conidr h dicound walh proc of h PPM in pnion chm. L porfolio proc ind in riky a a im and h conumpion proc a im, hn h pair i aid o b lf-financing if h corrponding walh proc X, [, ], aifi dx ds X X S db B cy d, whr rprn h proporion of h porfolio ind in cah accoun a im. Subiuing h a rurn in and ino 8, w obain h following dx r µ r cy d X dw, X hi i our ochaic diffrnial quaion which rprn h walh proc of h PPM a im. Uing 6 and 9, w ha h dicound walh proc o b d X c Y d X dw n hi cion, w prn h prn alu of pnion plan mmbr fuur conribuion in pnion chm. Dfiniion : h prn alu of h pcd dicound fuur conribuion proc i dfind a No cy d, 3 Global Journal nc. US

6 c> whr i h proporion of h alary conribud ino h pnion fund and i h condiional pcaion wih rpc o h Brownian filraion [ r],. Z p { } No i h ochaic dicoun facor which adju for nominal inr ra and mark pric of rik. Dfiniion : h pcd dicound fuur conumpion proc i dfind by W ak limi of ingraion in q. 3 from o bcau, w aum ha conumpion ar whn h PPM rird and conum hi/hr inmn bnfi ill h/h i dad. horm : L b h prn alu of pcd fuur conribuion ino h pnion fund, hn whr ϕ ω r Ψ u u du, [,. cy ϕ p[ ϕ ] Proof: Nwozo and Nkki. Find h diffrnial of bohid of q. 4, w obain d c Y d [ r d dw ] cy d Uing 6 and 5, w ha d dw Dfiniion 3: h alu of walh of h PPM i dfin a X. So ha h dicound alu of walh bcom X. hn, dynamic of h alu of h dicound walh of h PPM i obain a d d X d. Subiuing in and 6 ino 8, w obain h chang in walh of h PPM a follow: d d X dw Global Journal of Scinc Fronir Rarch F olum X u rion Yar 47 3 Global Journal nc. US

7 3 Global Journal of Scinc Fronir Rarch F olum X u rion Yar 48 n hi cion, w drid h opimal dicound porfolio and dicound conumpion proc of a PPM undr dynamic programming principl. W aum ha h PPM choo powr uiliy funcion in maimizing h pcd uiliy of h rminal dicound walh and conumpion proc. h choic of h powr uiliy which i a linar combinaion of wo powr uiliy funcion. h fir rm i wih rpc o h walh proc whil h cond i wih rpc h conumpion proc i moiad by h fac ha pnion chm ar in gnral larg inmn conpani who ragic plan ar wih rpc o h iz of fund hy ar managing. W now gi h opimal dicound porfolio and dicound conumpion proc for pnion fund plannr a im. h PPM problm i o choo an admiibl ragy o a o maimiz h pcd uiliy of accumulai conumpion and rminal dicound walh,, U, up J,,, whr W aum ha U i conca and,, [ ] U,, R R, i h PPM prfrnc dicoun facor, Π, of admiibl porfolio ragy ha ar ingrabiliy condiion, U J,,, U u, u du U, F <, d d <, X, progrily maurabl, ha aify h dno h condiional pcaion a im gin h iniial ndowmn, and h uiliy funcion i akn a,, U,< <. urn ou ha U, aifi h following Hamilon Jacobi Bllman quaion U U U U p U. No 3 Global Journal nc. US

8 hi yild h HJB quaion for h alu funcion, Η, U, No Subjc o: whr U,< <. Η, U U U U p Hnc, w ha h following opimal dicound porfolio and opimal dicound conumpion, rpcily a follow: * * U p[ ] du d whr,. U, U Subiuing q., and 9 ino q., w ha U U U U U 9 3 Global Journal of Scinc Fronir Rarch F olum X u rion Yar 49 U U p U, U U p p 36 3 Global Journal nc. US

9 W aum ha in hi papr ha 3 and U, Q Finding h parial driai of 3 wih rpc o h following: Q U,.,,,, 3, w ha No Global Journal of Scinc Fronir Rarch F olum X u rion Yar 5 U U U U U U Q Q Q Q Q Q Q W obr ha h aumpion of concaiy of U urn ou o b ru, a U <. U U Subiuing h parial driai ino 36 w ha < Q Q Q 4 can b ol numrically. inc, p 4 Q Suppo ha U, 5 hn 5 bcom Q Q p Q 6 3 Global Journal nc. US

10 Soling 6, w ha whr, 3 Global Journal nc. US 5 Global Journal of Scinc Fronir Rarch olum X u rion Yar 3 F Q Q, 4 3 ψ ψ Q r r ψ and hrfor, h opimal dicound profolio for h PPM a im i obaind a 7. ψ, * * * * X X X, * * * X X X h opimal dicound conumpion proc * of h PPM a im, gin ha 5 hold, i obaind * r h g 8 whr 4 3 ψ ψ g No

11 3 Global Journal of Scinc Fronir Rarch F olum X u rion Yar 5 and A im, 8 bcom whr and g r h. * ψ r ψ g h A im, 8 bcom * g h, whr and h r g h ψ ψ ψ ψ 3 4. ψ 3 4 Now, for < <, w ha inuiily ha h growh ra GR of h opimal pcd dicound conumpion i obain a GR r., hi i rfrrd o a h ulr quaion for h inrmporal maimizaion of dicound conumpion undr uncrainy. h poii rm capur h uncr No 8 3 Global Journal nc. US

12 3 Global Journal nc. US 53 Global Journal of Scinc Fronir Rarch olum X u rion Yar 3 F ainy of h financial mar. Whn h financial mark i riky, i will induc inor o hif conumpion or im. Suppo hr i no iniial walh, hn 3 bcom From 7, h fir rm i h ariaional form of h claical porfolio ragy whil h la rm i an inrmporal hdging ragy ha off any hock o h ochaic alary of h PPM. A, w ha n hi cion, w prn h opimal pcd alu of walh of h PPM in pnion plan a im. From 9, w ha ha r GR, *. dw d d dw d d d d d d d Soling, 33 and 34, w ha d 35 No

13 54 Global Journal of Scinc Fronir Rarch olum X u rion Yar 3 F No 3 Global Journal nc. US 36 whr, d Κ Κ d d. ϕ ϕ ϕ ϕ ω ϕ ω ϕ ω c c d c Κ d d ar A, w ha d ar. Κ d d h accumulad pcd dicound conumpion proc of h PPM up o rirmn priod i gin by Ψ du u d h g GR d g d h GR 39

14 Soling 39, w ha No whr Ψ f w allow o nd o infiniy, w ha ψ ψ 3 4 α α α lim αψ. limψ Ψ u du ψ 3 u du ψ 4. hi how ha hroughou h lif-cycl of h PPM, hi or hr accumulad pcd dicound conumpion will rmain nonmpy. f Ψ i nonngai, i impli ha h PPM i unabl o finihd hi or hr conumpion bfor h or h di. f Ψ i ngai, i impli ha h PPM i finihd hi or hr conumpion why ill ali. f Ψ i zro, i impli ha h priod h PPM finih hi or hr conumpion wa h priod h or h di. nringly, h naur of Ψ aboluly dpnd on h alu of,. hi impli ha h ll of rik h PPM ak will dpnd on how larg or how mall h alu of h accumulad dicound conumpion will b and conumpion ra. W obr ha a or, opimal pcd dicound conumpion will b unboundd. h opimal pcd dicound conumpion will b ngai if 3 Global Journal of Scinc Fronir Rarch F olum X u rion Yar 55 <, 3 Global Journal nc. US

15 and non ngai if >. 3 Global Journal of Scinc Fronir Rarch F olum X u rion Yar 56 hi papr dal wih opimal dicound porfolio, pcd walh and ragic lif-im conumpion proc for a dfind conribuory pnion chm. h pcd dicound walh and i arianc ar obaind. h dicound ariaional claical porfolio and conumpion proc of h plan mmbr ar ablihd. h accumulad dicound conumpion proc of h PPM hroughou hi or hr lif-cycl wa ablihd. wa found ha par of h dicound porfolio alu i proporional o h raio of h prn alu of h dicound pcd fuur conribuion o h opimal dicound porfolio alu orim. wa found ha hr i h nd for gradual ranfr of par of h porfolio alu in riky a o h rikl on again unforn hock.. Blak, D., Wrigh, D. and Zhang, Y., 8, Opimal funding and inmn ragi in dfind conribuion pnion plan undr pin-zin uiliy. Dicuion papr, h pnion niu, a Buin School, iy Uniriy, UK.. Blanchard, O. J. and Fichr, S. 989, Lcur on Macroconomic. h M pr. 3. Bodi, Z., Mron, R.., and Samulon, W. F., 99 Labour upply flibiliy and porfolio choic in a lif cycl modl. Journal of conomic Dynamic and onrol, 6, Boulir, J. F, Huang, S. J. and aillard, G.,, Opimal managmn undr ochaicinr ra: h ca of a procd dfind conribuion pnion fund. nuranc: Mahmaic and conomic, ol. 8, pp airn, A. J. G., Blak, D. and Dowd, K., 3, Sochaic lifyling: Opimal dynamic a allocaion for dfind conribuion pnion plan. RS Dicuion papr ri, h Uniriy of oingham. 6. ajuiro, D. O. and Yonyama,., 3, Opimal porfolio, Opimal conumpion and h Markowiz man-arianc analyi in a wiching diffuion mark. Working papr. 7. Flming, W. and Rihl, R., 975, Drminiic and ochaic opimal conrol. Springr-rlag, Nw York. 8. Korn, R. and Krkl, M.,, Opimal porfolio wih fid conumpion or incom ram. Working papr, Uniriy of Kairlaurn. 9. Maon, R. and Wrigh, S.,, h ffc of uncrainy on opimal conumpion, Journal of conomic Dynamic and onrol, 5, 85-. No 3 Global Journal nc. US

16 No. Mron, R.., Lifim porfolio lcion undr uncrainy: h coninuou-im ca. Riw of conomic and Saiic, 969, ol. 5, pp Mron, R.., 97, Opimum conumpion and porfolio rul in a coninuouim modl. Journal of conomic hory, ol. 3, pp Nkki,.., On opimal porfolio managmn of h accumulaion pha of a dfind conribuory pnion chm, Ph.D hi, Dparmn of Mahmaic, Uniriy of badan, badan, Nigria. 3. Nkki..and Nwozo. R,, ariaional form of claical porfolio ragy and pcd walh for a dfind conribuory pnion chm, Journal of Mahmaical Financ,, pp Nwozo. R. and Nkki,.., Opimal porfolio and ragic conumpion planning in a lif-cycl of a pnion plan mmbr in a dfind conribuory pnion chm, ANG nrnaional Journal of Applid Mahmaic, 44, pp och, P., A racabl modl of prcauionary aing in coninuou im. conomic lr, 5, 87, Global Journal of Scinc Fronir Rarch F olum X u rion Yar 57 3 Global Journal nc. US