Journal of Mahemaical Finance 3 3 3-37 hp://dxdoiorg/436/jmf33 Published Online February 3 (hp://wwwscirporg/journal/jmf) Opimal Porfolio raegy wih iscouned ochasic Cash nflows Charles Nkeki eparmen of Mahemaics Faculy of Physical ciences Universiy of Benin Benin Ciy Nigeria Email: nkekicharles3@yahoocom Received March 9 ; revised June ; acceped June 5 ABTRACT This paper examines opimal porfolios wih discouned sochasic cash inflows (C) The cash inflows are invesed ino a marke ha is characerized by inflaion-linked bond a sock and a cash accoun was assumed ha inflaionlinked bond sock and he cash inflows are sochasic and follow a sandard geomeric Brownian moion The variaional form of Meron porfolio sraegy was obained by assuming ha he invesor chooses consan relaive risk averse (CRRA) uiliy funcion The iner-emporal hedging erms ha offse any shock o he C were obained A closed form soluion o our resuling non-linear parial differenial equaion was obained Keywords: Opimal Porfolio; ochasic Cash nflows; nflaion-linked Bond; Variaional Form; neremporal Hedging Terms nroducion This paper consider he opimal porfolio sraegies wih valuing expeced discouned cash inflows was assumed ha he underlying asses and cash inflow process follow a sandard geomeric Brownian moion The invesmen of he invesor s C ino a cash accoun an inflaionlinked bond and a sock was considered The C and sock price were correlaed wih inflaion and sock marke risk n a relaed lieraure [] sudied an opimal invesmen problem in a coninuous-ime framework where he ineres rae follows he Cox-ngersoll-Ross dynamics They obained a closed form soluion for he opimal invesmen sraegy under a complee marke framework They assumed ha he invesor chooses CRRA uiliy funcion [] considered he process of finding he opimal porfolio opimal consumpion and he efficien fronier for a small agen in an economy They also considered financial marke ha composed of wo sources of uncerainies: an m-dimensional Brownian moion and a coninuous ime Markov chain They found heoreically ha he regimes of an economy have a significan impacs on he porfolio and consumpion decision as well as on he efficien fronier of a small invesor [3] sudied he presen value and overcome he difficuly of independence by reversing he order of he cash flow He found ha similar recursive formulas for he presen value is also applicable o he fuure value of he ex- peced reurns [4] considered he problem of consrucing a porfolio of finiely many asses whose reurns are described by a discree join disribuion They proposed a new porfolio opimizaion model involving sochasic dominance consrains on he porfolio reurn [5] sudied how porfolio composiion changes wih individual wealh [6] considered he presen value of expeced fuure conribuion ino he pension fund [7] invesigaed he sochasic dynamics of deposiory financial insiuion asses liabiliies and capial under he influence of macroeconomic facors They adoped dynamic programming echniques in heir opimizaion process They provided an analysis of he economic aspecs of he deposiory financial insiuion modeling [8] sudied a porfolio problem of a fund manager who wans o maximize he expeced uiliy of his erminal wealh in a complee financial marke He found ha he opimal porfolio is formed by hree componens: a speculaive an hedging and a preference-free hedging componen He obained a close form soluion o he asse allocaion problem [9] considered he opimal porfolio selecion problem wih porfolio consrains They derived he general uiliy funcion using he maringale approach They found ha using CRRA uiliy funcion opimal policies can be obained explicily when here are minimum capial requiremens ([]) sudied he opimal porfolio managemen in he accumulaion phase of a defined conribuion pension scheme They obained porfolio values wih hedging sraegies for a pension plan member n Copyrigh 3 cires
C NKEK 3 his paper we consider he opimal porfolio and invesmen sraegies involving cash inflow valuaion over ime We obain explicily analyical soluion o our resuling HJB equaion The risk-free rae is assume o be deerminisic The remaining pars of he paper is srucured as follows: n ecion we described he srucure of he financial marke; n ecion 3 we consider he dynamics of he discouned presen value of expeced C process; n ecion 4 we consider he wealh process of he invesor; n ecion 5 we consider he valuaion of he discouned presen value of expeced C process of he invesor as well as he sensiiviy of he presen value of he C; n ecion 6 we consider he opimizaion program and opimal porfolio and opimal soluion of he invesor s wealh using CRRA uiliy funcion; ecion 7 concludes he paper The Model Le F P be a probabiliy space Le F F F : T where F s s: s The Brownian moions W W W is a -dimensional process defined on a given filered probabiliy space F F ( F) P T where P is he real world probabiliy measure he ime period T he erminal ime period W is he Brownian moion wih respec o source of uncerainy arising from inflaion and W is he Brownian moion wih respec o source of uncerainy arising from he sock marke is assumed ha he marke is arbirage-free complee and coninuously open beween ime period and T Financial Model Q The dynamics for he cash accoun wih he price a ime is given by dq rq d () Q ; where r is he shor erm ineres rae The sock price a ime is given by he dynamics: d d dw () s; where is he expeced growh rae of sock price and is he vola- iliy of sock The price of he inflaion-linked bond B is given by he dynamics: d B B r db d W (3) B b; where B is he volailiy of inflaion-linked bond is he marke price of inflaion risk is he inflaion index a ime and has ha dynamics: d q d dw where q is he expeced rae of inflaion which is he difference beween nominal ineres rae r real ineres rae r and is he volailiy of inflaion index ince he marke is complee we have ha B : (4) : (5) r Therefore he marke price of marke risk is given by : r where is he marke price of sock marke risk The exponenial process Z : exp W T is assumed o be a maringale We now define he sae-price densiy funcion by Z : Q Z exp T r 3 ynamics of ochasic Cash nflows The dynamics of he sochasic cash inflows wih price process is given by ; d kd dw where is he volailiy of he cash inflows and k is he expeced growh rae of he cash inflows is he volailiy arising from inflaion and is he volailiy arising from he sock marke Figure presens he simulaed diffusion pahs of (7) Figure was obained by seing k 99 5 36 d n and n Applying o ˆ lemma o (7) we obain exp k W (6) (7) (8) Copyrigh 3 cires
3 C NKEK Value of he C (in Naira)ae 5 5 5 3 4 5 6 7 8 9 Time Figure imulaed diffusion pahs of he sochasic cash inflows 4 The Wealh Process Le X be he wealh process and be he porfolio value a ime where is he porfolio value in inflaion-linked bond and is he porfolio value in sock a ime Then is he porfolio value in cash accoun a ime Therefore he dynamics of he wealh process is given by dx X X x X d() () db B dq X Q ubsiuing () () and (3) ino (9) we obain dx X r d X x X dw 5 The iscouned Value of C d (9) () n his secion we deermine he value of expeced discouned C efiniion : The discouned value of he expeced fuure C is defined as T u d () E u where E E F is he condiional expecaion wih respec o he Brownian filraion F and Zexp r is he sochasic discoun facor which adjuss for nominal ineres rae and marke price of risks for sock and inflaion-linked bond Proposiion : uppose is he discouned value of he expeced fuure C hen exp k r T kr Proof By definiion we have ha d () T u u E u (3) Applying change of variable on (3) we have T = E d (4) Applying parallelogram law and maringale principles on (4) we have Therefore E exp k r T exp k r d (5) negraing we have exp k r T kr (6) Figure was obained by seing k = 99 r = 4 5 36 9 4 3 6 and 8 A we obain he presen discouned value of fuure C o be f exp kr T kr r k and we allow T ie exp kr T lim lim T T kr r k r k r k (7) For a deerminisic case we have he presen discouned value fuure deerminisic cash inflows o be Copyrigh 3 cires
C NKEK 33 Presen Value of C (in Naira) ae 8 6 4 3 4 5 6 7 8 9 ime Figure The flow of he discouned value of C r k exp kr T k r Hence expk rt lim lim r k T T k r r k This shows ha as T converges o r k provided r k (8) For he deerminisic case i shows ha as T converges o r k provided r k Figure 3 was obained by seing k = 99 r = 4 5 36 9 4 3 6 and 8 Figure represens he flow of he discouned value of sochasic cash inflow in he invesmen a ime and Figure 3 represens he presen value of discouned fuure C a ime We now consider he sensiiviy analysis of Pro- posiion esablishes his fac Proposiion : Le kr hen T exp T Proof The resuls follow by aking he parial derivaives of wih respec o T r and k respecively T ; Presen Value of C (in Naira) ae 9 8 7 6 5 4 3 3 4 5 6 7 8 9 ime Figure 3 The presen discouned value of fuure C r T ; T ; r T k T r Table shows he sensiiviy analysis of he discouned value of he C Proposiion 3: uppose ha Proposiion holds hen d d r d dw (9) Proof Finding he differenial of boh sides of (6) and hen subsiue (7) we have exp T d kd dw expt d expt kd dw exp exp T d d T r d dw V be a value process a ime d r d dw d Le We defined Copyrigh 3 cires
34 C NKEK T Table imulaion of he sensiiviy analysis T 664 5 4359 785 596 7486 785 87 4857 7875 8738 857 4649 66858 857 3 397 379 5736 336 6349 487 58855 6349 4 435 93 3587 755 46 359 588 46 5 5548 9499 49 988 874 9685 4746 874 6 679 7697 96 6444 96 454 34638 96 7 777 5888 784 398 736 84 6495 736 8 848 47 4897 3543 588 7 837 588 9 9785 45 7 3996 86 76 3 86 7 4 495 4446 55 87 853 55 r k V as : V X () where X saisfy () and saisfy (9) Proposiion 4: Le V saisfy () X saisfy () and saisfy (9) hen dv r X X d X dw () Proof Finding he differenial of boh sides of () and hen subsiue in () and (9) he resul follows 6 Opimizaion of he Value of Wealh Process We define he general value funcion J v E u V X X x uv is he pah of where V efine o be he se of all admissible porfolio sraegy ha are F-progressively measurable ha saisfy he inegrabiliy con- T diions E u udu and le U V be a concave funcion in V such ha UV saisfies he HJB equaion sup U v E u V T X x supj v Therefore by applying ô lemma on () we obain he following HJB equaion: U v maxhv () subjec o: where v UT v HV ru rxu U x U x x x U x U U Le U v x (3) be he soluion of he HJB equaion () ince he uiliy funcion is concave and he value func- ion is smooh ie U VC R T hen () is well-defined Hence we have he following: HV Ux x U U from (4) we have U x xu U xu X x (4) (5) ubsiuing (5) ino () we obain he following HJB equaion: U rxux r U U U x U x U U UU U x x This is he resuling HJB equaion of our problem (6) Copyrigh 3 cires
C NKEK 35 Proposiion 5: The soluion of he HJB equaion (6) is of he form wih va U v A exp r T (7) AT Proof Finding he parial derivaives of va U v wih respec o x x and and hen subsiue ino (6) we obain he following: v A A rv A From (8) we obain Using (9) we obain v A (8) A exp r T (9) A v U v expr T v UT v (3) Figure 4 was obained by seing r 4 k 99 T 5 36 6 9 4 3 and 8 Figure 4 shows he expeced value of uiliy of wealh a ime for differen values of Here measures he level of risk he invesor is willing o ake Observe ha he smaller he value of he higher he expeced value of wealh and vice versa Therefore when 7 he expeced value of uiliy of wealh U vu v 789 when 8 U v 75 when 9 U v 94 and when U v 6487 Hence he lower he rae a which invesors are risk averse he more he wealh ha will accrue o hem and vice versa which is an inuiiv e resul Proposiion 6: uppose ha U V is he soluion of he HJB Equaion (6) hen he opimal porfolio in inflaionlinked bond sock and cash accoun are given by V (3) X X Expeced U iliy of Wea lh (in Naira) 3 5 5 5 Expeced Wealh for Gamma=7 Expeced Wealh for Gamma=8 Expeced Wealh for Gamma=9 Expeced Wealh for Gamma= 3 4 5 6 7 8 9 Time (in year) Figure 4 The expeced value of uiliy of wealh for differen values of γ r V X V X X X r Proof From (5) we have ha Bu V X X r (3) Copyrigh 3 cires
36 C NKEK Therefore V X r X V X X r V X X V X X r V X X implies ha V (33) X X () r V X V X X r V X X X (34) (35) From (33) he firs erm represens he classical porfolio sraegy while he second erm represens he ineremporal hedging sraegy ha offse any shock o he C a ime From (34) r V X represens he classical porfolio sraegy a ime and represens he ineremporal X hedging erm ha offse shock resuling from he C a ime From (35) observe ha hese hedging erms can be ransfer o cash accoun a ime or i can be reinves in sock and in inflaion-linked bond a ime Figure 5 was obained by seing r 4 k 99 T 5 36 6 9 4 3 8 and 5 The opimal porfolio value in inflaion-linked bond a ime is obained as 6 (or 6%) Figure 6 was obained by seing r 4 k 99 T 5 36 6 9 4 3 8 and 5 The opimal porfolio value in sock a ime is obained as 56 (or 56%) Figure 7 was obained by seing r 4 k 99 T 5 36 6 9 4 3 8 and 5 The opimal porfolio value in cash accoun a ime is obained as 656 (or 656%) n Figures 5-7 we se r 4 k 99 T 5 36 6 9 4 3 8 and Figure 5 shows he porfolio value in inflaion-linked bond We found ha he opimal porfolio value in inflaion-linked bond a ime is 6 (or 6%) Figure 6 shows he porfolio value in sock We found ha he opimal porfolio value in sock a ime is 56 (or 56%) Figure 7 shows he porfolio value in cash Copyrigh 3 cires
C NKEK 37 Figure 5 Porfolio value in inflaion-linked bond Figure 6 Porfolio value in sock Figure 7 Porfolio value in cash accoun accoun We found ha he opimal porfolio value in cash accoun a ime = is 656 (or 656%) 7 Conclusion The opimal porfolio sraegy X wih discouned sochasic cash inflows was considered was assumed ha he cash inflow sock and inflaion-linked bond are sochasic and follow a sandard geomeric Brownian moion The sensiiviy analysis of he presen value of he discouned cash inflows was carried ou in his paper and he resuls are presened in Table Analyical soluion o he resuling HJB equaion was obained was found ha he smaller he value of (which measure he level of risk he invesor is willing o ake) he higher he expeced value of wealh and vice versa The opimal porfolio values in sock inflaion-linked bond and cash accoun were obained The resuling opimal porfolio values in sock and inflaion-linked bond were found o involve ineremporal hedging erms ha offse any shock o he C REFERENCE [] G eelsra M Grasselli and P Koehl Opimal nvesmen raegies in a CR Framework Journal of Applied Probabiliy Vol 37 No 4 pp 936-946 doi:39/jap/484374 [] O Cajueiro and T Yoneyama Opimal Porfolio Opimal Consumpion and he Markowiz Mean-Variance Analysis in a wiching iffusion Marke 3 unbbr/face/eco/seminarios/sem83pdf [3] A Zaks Presen Value of Annuiies under Random Raes of neres 3 hp://academicresearchmicrosofcom/publicaion/6447 5/presen-value-of-annuiies-under-random-raes-of-ineres [4] encheva and A Ruszczynski Porfolio Opimizaion wih ochasic ominance Consrains AM Journal on Opimizaion Vol 4 No 3 pp 548-566 doi:37/5634458 [5] Blake Efficiency Risk Aversion and Porfolio nsurance : An Analysis of Financial Asse Porfolios Held by nvesors in he Unied Kindom Economic Journal Vol 6 No 438 996 pp 75-9 doi:37/3554 [6] A Zhang ochasic Opimizaion in Finance and Life nsurance: Applicaions of he Maringale Mehod Ph Thesis Universiy of Kaiserslauen Kaiserslauern 7 [7] J Mukuddem-Peerson M A Peerson and M choeman An Applicaion of ochasic Opimizaion Theory o nsiuional Finance Applied Mahemaics ciences Vol No 8 7 pp 359-385 [8] P Baocchio Opimal Porfolio raegies wih ochasic Wage ncome: The Case of a efined Conribuion Pension Plan Working Paper Universié Caholique de Louvain Louvain-la-Neuve [9] B H Lim and U J Choi Opimal Consumpion and Porfolio elecion wih Porfolio Consrains nernaional ciences Vol 4 9 pp 93-39 [] C Nkeki On Opimal Porfolio Managemen of he Accumulaion Phase of a efined Conribuory Pension cheme Ph Thesis Universiy of badan badan [] C Nkeki and C R Nwozo Variaional Form of Classical Porfolio raegy and Expeced Wealh for a efined Conribuory Pension cheme Journal of Mahemaical Finance Vol No pp 3-39 doi:436/jmf5 Copyrigh 3 cires