Received August 16, 2013; revised September 27, 2013; accepted October 26, 2013

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1 Journal of Mahemaical Finance 7-8 Published Online November (hp://wwwscirporg/journal/jmf) hp://dxdoiorg//jmf Opimal Variaional Porfolios wih Inflaion Proecion raegy and Efficien Fronier of Expeced Value of Wealh for a Defined Conribuory Pension cheme Joshua O Okoro Charles I Nkeki Deparmen of Physical cience Faculy of cience and Engineering Landmark Universiy Omu-Aran Nigeria Deparmen of Mahemaics Faculy of Physical ciences Universiy of Benin Benin Ciy Nigeria Received Augus ; revised epember 7 ; acceped Ocober Copyrigh Joshua O Okoro Charles I Nkeki his is an open access aricle disribued under he Creaive Commons Aribuion License which permis unresriced use disribuion and reproducion in any medium provided he original work is properly cied ABRAC his paper examines opimal variaional Meron porfolios (OVMP) wih inflaion proecion sraegy for a defined conribuion (DC) Pension scheme he mean and variance of he expeced value of wealh for a pension plan member (PPM) are also considered in his paper he financial marke is composed of a cash accoun inflaion-linked bond and sock he effecive salary of he plan member is assumed o be sochasic I was assumed ha he growh rae of PPM s salary depends on some macroeconomic facors over ime he presen value of PPM s fuure conribuion was obained he sensiiviy analysis of he presen value of he conribuion was esablished he OVMP processes wih iner-emporal hedging erms and inflaion proecion ha offse any shocks o he sochasic salary of a PPM were esablished he expeced values of PPM s erminal wealh variance and efficien fronier of he hree classes of asses are obained he efficien fronier was found o be nonlinear and parabolic in shape In his paper we allow he sock price o be correlaed o inflaion risk index and he effecive salary of he PPM is correlaed o inflaion and sock risks his will enable PPMs o deermine exens of he sock marke reurns and risks which can influence heir conribuions o he scheme Keywords: Opimal Variaional Meron Porfolios; Mean-Variance; Expeced Wealh Defined Conribuion; Pension cheme; Pension Plan Member; Iner-emporal Hedging erms; ochasic alary Inroducion his paper considers he OVMP sraegy under inflaion proecion expeced wealh and is variance for a DC pension scheme I was assumed ha he salary and risky asse are driven by a sandard geomeric Brownian moion he growh rae of salary of PPM is assumed o be a linear funcion of ime In his paper we focus on sudying he OVMP sraegy under inflaion proecion for a DC pension scheme In relaed lieraure [] and [] sudied opimal porfolio sraegy based on he non-gaussian models hey consruced opimal porfolios of variance swaps based on a variance gamma correlaed model he porfolios of he variance swaps are opimized based on he maximizaion of he disored expecaion given in he index of accepabiliy [] examined he raionale naure and financial consequences of wo alernaive ap- proaches o porfolio regulaions for he long erm insiuional invesor secors of life insurance and pension funds [] sudied he deerminisic life syling and gradual swich from equiies o bonds in accordance o prese rules his approach was a popular asse allocaion sraegy during he accumulaion phase of a defined conribuion pension plan and was designed o proec he pension funds from a caasrophic fall in he sock marke jus prior o reiremen hey showed ha his sraegy alhough easy o undersand and implemen can be highly subopimal [] developed a model for analyzing he ex ane liquidiy premium demanded by he holder of an illiquid annuiy he annuiy is an insurance produc ha is similar o a pension savings accoun wih boh an accumulaion and decumulaion phase hey compued he yield needed o compensae for he uiliy welfare

2 J O OKORO C I NKEKI 77 loss which is induced by he inabiliy o re-balance and mainain an opimal porfolio when holding an annuiy [-8] considered he opimal design of he minimum guaranee in a defined conribuion pension fund scheme hey sudied he invesmen in he financial marke by assuring ha he pension fund opimizes is reribuion which is a par of he surplus which is he difference beween he pension fund value and he guaranee [9] sudied opimal invesmen sraegy for a defined conribuory pension plan hey adoped dynamic opimizaion echnique [] considered opimal porfolio sraegies wih minimum guaranee and inflaion proecion for a DC pension scheme [] sudied he opimal porfolio and sraegic lifecycle consumpion process in a defined conribuory pension plan his work is an exension of [] [] sudied he OVMP process of a PPM In his paper we inroduce addiional underlying asse which is he inflaion-linked bond ha will proec he invesmen agains inflaion risks in he invesmen profile [] considered opimal invesmen sraegy wih discouned cash flows In heir work he invesmen and he porfolios processes were proeced from inflaion risks by rading on he index bond asse ha is linked o inflaion risk over ime he remainder of his paper is organized as follows In ecion we presen he financial marke models and wealh process of a PPM ecion presens he expeced value of PPM s fuure conribuion he presen value of he expeced flow of fuure discouned conribuion sensiiviy analysis of he presen value of expeced fuure discouned conribuion and value of wealh process of a PPM In ecion we presen he opimizaion process and porfolio value of wealh of a PPM in pension scheme Also in his secion some numerical illusraions were made ecion presens he expeced wealh and variance of expeced wealh for a PPM up o erminal period Finally ecion concludes he paper Problem Formulaion Le F P be a probabiliy space Le F F : where and he Brownian moion F W s : s W W W is a -dimensional process defined on a complee probabiliy space F F P where P is he real world probabiliy measure and he erminal ime he vecors and z are he volailiy vecor for sock and inflaion index wih respec o W and W respecively changes in is he appreciaion rae for sock Moreover and z are he volailiies for he sock and inflaion index respecively referred o as he coefficiens of he marke and are progressively measurable wih respec o he filraion F We assume ha he invesor faces a marke ha is characerized by a risk-free asse (cash accoun) inflaion-linked bond and sock all of whom are radeable In his paper we allow he sock price o be correlaed o inflaion risk index and he effecive salary of he PPM o be correlaed o inflaion and sock his will enables us o check how sock marke risks can influence conribuion ino he scheme he Financial Model he dynamics of he underlying asses are given by () and () dc rcd C () d d dw () s d Z Z r z d () z d W Z z where r is he nominal ineres rae C is he price of he riskless asse a ime Z is he price of he inflaion-linked bond a ime is he inflaion index a ime and saisfies he dynamics d E q d dw E q is he expeced rae of inflaion which he difference beween he nominal ineres rae and he real ineres rae and is he volailiy of inflaion index is he sock price process a ime z is he inflaion price of risk is he correlaion coefficien and he volailiy marice now become he marke is complee since z herefore here exiss a unique marke price of risk vcor saisfying z r z z

3 78 J O OKORO C I NKEKI he effecive salary of he PPM is assume o follow he dynamics y d d dw () where is he volai- liy caused by he source of inflaion W and is he volailiy caused by he source of uncer- ainy arises from he sock marke W We assume ha he growh rae of he salary of PPM is linear and saisfies a a a is seen as expeced yearly growh of PPM s salary and economic growh effec or welfare of he PPM () define he buying power of PPM s salary a ime uppose ha () becomes d d y () define he deerminisic salary process of he PPM a ime In his paper we assume ha he salary process of he PPM is sochasic In he case of deerminisic salary we shall sae i herefore he flow of conribuions of he PPM is given by c where c is a proporion of PPM s salary ha he/she is conribuing ino he scheme We assume in his paper ha r Z are consans in ime We now define he following exponenial process which we assume o be Maringale in P : Z exp W which will be useful in he nex secion he Wealh Process of a PPM Le X be he wealh process and he porfolio process represens he proporion of wealh invesed in inflaion-linked bond and he proporion of wealh invesed in sock a ime hen is he proporion wealh invesed in he riskless asse a ime We now have he following definiion: Definiion : he porfolio process is said o be self-financing if he corresponding wealh process X saisfies d X X r c d X d W X x () where zz r he Expeced Value of PPM s Fuure Conribuion In his secion we consider he value of PPM s expeced fuure conribuion over he planning horizon We now make he following definiions Definiion : he discouned value of expeced fuure conribuion process is defined as E csds (7) s Zexp r where E E F is he condiional expecaion wih respec o he Brownian filraion F and is he s ochasic discoun facor which adjuss for nominal ineres rae and marke price of risk s Definiion : A porfolio process is said o b e admis- sible if he corresponding wealh process X saisfies s P X E c s ds heorem : Le () be he discouned value of expeced fuure conribuio n (PVFC) process of a PPM hen c exp (8) c exp c π exp Erfi Erfi cy π exp Erfi Erfi (9) where ar z Proof: ee Nkeki and Nwozo () Remark : he presen value of expeced fuure con - ribuion of a PPM is given by In his paper we consider he case of For he

4 J O OKORO C I NKEKI 79 case of see [7] ensiiviy Analysis of Presen Value of PPM s Fuure Conribui on (PVPPMFC) In his subsecion we consider he sensiiviy analysis of PPM s p resen value fuure conribuion for and for sochasic salary We inend o find he elasiciy of PVPPMFC wih respec o change in is dependen pa- rameers In a sequel we se Firs we find he parial derivaive of wih respec o cye π e π e πe Erfi () π( ) Erfi ; he parial derivaive of wih respec o c : y c πe Erfi Erfi () ; he parial derivaive of wih respec o y : c y πe Erfi Erfi () ; he parial derivaive of wih respec o a : cye a π () e πe π Erfi ; he parial derivaive of he parial derivaive of wih respec o : cy e () wih respec o r are respecively given as; z r e cy e cy z e πerfi π () e e e πerfi π () cy ze π e πe πerfi (7) cy e (8) e e π Erfi π cy e π e πe π Erfi cy z e e πe Erfi π We se (o obain he values in able by varying each of he parameers) a 9 r 8 z 9 9

5 8 J O OKORO C I NKEKI y 8 c able gives he values of he parameers and he corresponding change in PVPPMFC I is observed ha he PVPPMFC is posiively sensiive o all he parameers excep c and y ha is consan overime (or we say ha PVPPMFC is inelasic in c and y ) Bu PVPPMFC is negaively sensiive in We observe ha as increases by he PVPPMFC decreases by abou 9 We also observed ha among he sensiive parameers some are more sensiive han he oher s For insance a small change in will lead o a very high proporionae change in PV PPMFC ie an increase of by uni will bring abou an average increase of PVPPMFC by unis I is furher observed ha an increase of by one uni will lead o a very small change in PVPPMFC Increasing a by will lead o abou an average of 9 increases in PCPPMFC imilarly an increase in r by will lead o a proporionae increase in PVPPMFC by imilarly an increase in z by will bring abou proporionae increases in PVPPMFC Again an increase in by will lead o abou 7 proporionae increases in PVPPMFC Finall y an increase in by will lead o abou proporionae increases in PVPPMFC When increase by he PVPPMFC increases alongside wih We herefore conclude ha PVPPMFC is elasic wih respec o change in he parameers he elasiciy of PVPPMFC wih respec o change in he above parameers will enable fund managers and invesors o value asses flow of conribuions and invesmen flow ha are o be receive some ime in fuure From () we obain he criical ime horizon for he presen value of PPM s fuure discouned conribui on as follows: log ecy (9) Lemma : he dynamics of he discouned value of fuure conribuion process is given by where H A d zdw d d c π Erfi Erfi a A H A exp exp Proof: ee Nkeki and Nwozo () Value of PPM s Wealh Process Le he value of PPM s wealh be given by V a ime () is V X () Finding he differenial of boh side s of () and subsiuing in () and () we obain dv X r d X W d he Opimizaion Process and Porfolio Value of a PPM () In his secion we consider he opimal porfolio process able imulaion of he sensiiviy analysis of PVPPMFC a a r r z z

6 J O OKORO C I NKEKI 8 We define he general objecive funcion x uv is he pah of V gi Defi V o b J V Eu V X where ven he porfolio sraegy ne e he se of all admissi- ble porfolio sraegy ha are FV - progressively meas- urable and le uv be a concave v alue funcio n in V such ha hen U V V sup J V V U V subjec o: where sup Eu V X x saisfies he HJB equaion V U sup H v () v U v x H v rx x U U x Uxx x Ux U Finding he parial derivaive of H v wih respec o he opimal conrol and seing i o zero we obain U U x x () ubsiuing () ino () we obain he HJB equaion () we have U rxux U xu xx M U x M U U U U U x x x xx U where M Prop osiion: Le U vr U v be he soluion o he HJB Equaion () hen xx xx () M R R rxv R v M v R ; R X X M M X () (7) Proof: We commence by obaining he following parial derivaives: U v R R xx Ux v R U v R U v R U v R x U v R ubsiuing he parial derivaives above ino () we obain M R rxv R R v R M v R Again subsiuing he parial derivaives ino () we obain M X M X X herefore he porfolio value in he riskless asse is obain as M M X (8) X X he porfolio value (7) is made up of wo pars he firs par is he OVMP value and he second par is he ineremporal hedging erm ha offse any shocks o he sochasic salary of he PPM a ime I is observe in (8) ha he flow of ineremporal erm is a gradual ransfer fund from he risky asses o he riskless one over ime he ineremporal hedging erm ineresingly is a funcion of financial marke behaviour and volailiy vecor of he effecive salary process of a PPM I is also a raio of he PPM discouned value of expeced fuure conribuions of a PPM o he opimal wealh process his sraegy is srongly recommended for PPMs of whom conribuions are made compulsory his will ensure ha he inflaion risks in he porfolio of members are hedged We herefore say ha (7) is a porfolio wih

7 8 J O OKORO C I NKEKI inflaion proecion sraegy he soluion o () can be obained numerically Numerical Example Le a 9 r 8 z y 8 c we have he following figures: Figures - are obained for and for deerminisic salary process Figure shows he porfolio value of a PPM invesed in inflaion-linked bond for a period of years given ha he wealh is beween o million imilarly Figure shows he porfolio value of a PPM invesed in sock for he same period and he same wealh Figure shows he porfolio value of a PPM in cash accoun I is observed ha he porfolio values in sock and inflaion-linked bond are nonnegaive while in cash accoun is negaive over ime We herefore recommend (based on hese numerical examples) ha he porfolio value in cash accoun should be shoren by he negaive value o finance he risky asses (ie more money should be borrowed from cash accoun o finance he invesmen in sock and inflaion-linked bond) I is also observed ha he wealh generaed by sock is higher han ha of inflaion-linked bond his sugges ha more fund should be invesed in s ock ha inflaion-linked bond o ensure maximum reurns for members a reiremen Figure shows he porfolio value in sock when and for he ca se salary sochasic Figure shows he porfolio val ue in ock when for salary so- chasic and randomness is re laxe d Figure shows he porfolio value in cash accoun when and salary sochasic Figure 7 shows he porfolio value in cash accoun when salary sochasic and randomness is relaxed Figure 8 shows he porfolio value in inflaion-linked bond when and salary is sochasic Figure 9 shows porfolio value in inflaion-linked bond for salary sochasic and randomness is relaxed Figures 8 are esablished for and for sochasic salary process hese figures are made up of several shocks and pahs hese shock pahs made i difficul o make useful decisions hence he need for us o relax he randomness associaed wih he porfolios for effecive decision making We now have ha he following Figures 7 9 are special cases of Figures 8 respecively imilar behaviors exhibied by Figures - were also exhibied by Figures 7 9 he major difference is ha he porfolio value under sochasic salary yields higher reurns ha he deerminisic salary his makes sense since i is expeced ha he higher he risk aken he higher he expeced wealh Ineresingly from he numerical examples he amoun ha was gradually ransferred from he risky asses o cash accoun seems o have been re-invesed back o he risky asses overime Figures - show he porfolio values for deerminisic salary and for in inflaion-linked bond sock and cash accoun respecively We observed ha under his sraegy he enire fund should be invesed in sock for maximum wealh for he PPM a reiremen Figures - show he porfolio values for sochasic salary and in inflaion-linked bond sock and cash accoun overime I should be noed ha he shocks associaed wih he porfolios arise from he salary risks of he PPM overime Expeced Wealh and Variance of he Expeced Wealh In oher o deermine he expeced wealh of he PPM a ime we find he mahemaical expecaion of () as follows: Porfolio Value ime (in year) For Bea= for Deerminisic alary Wealh Figure he porfolio value in inflaion-linked bond for and salary deerminisic lue Porfolio Va ime (in year) For Bea= for Deerminisic alary Wealh Figure he porfolio value of a PPM in sock for and salary deerminisic

8 J O OKORO C I NKEKI 8 For Bea= for Deerminisic alary - Porfolio Value ime (in year) Wealh Figure he porfolio value of a PPM in cash accoun for and salary deerminisic Figure he porfolio value in cash accoun When and salary sochasic For Bea= for ochasic alary bu normalized - Porfolio Value Figure he porfolio value in sock When and salary sochasic For Bea= for ochasic alary bu normalized - ime (in year) Wealh Figure 7 he porfolio value in cash accoun When and salary sochasic randomness is relaxed Porfolio Value ime (in year) Wealh Figure he porfolio value in sock When and salary sochasic and randomness is relaxed Figure 8 he porfolio value in inflaion-linked bond when and salary sochasic

9 8 J O OKORO C I NKEKI For Bea= for ochasic alary bu normalized 9 8 Porfolio Value 7 ime (in year) Wealh Figure 9 Porfolio value in inflaion-linked bond When and salary sochasic and randomness is relaxed Figure he porfolio value in cash accoun for and deerminisic salary Figure he porfolio value in inflaion-linked bond for and deerminisic salary Figure he porfolio value in inflaion-linked bond for and for sochasic salary Figure he porfolio value in sock for erminisic salary and de- Figure he porfolio value in sock for and salary sochasic

10 J O OKORO C I NKEKI 8 Figure he porfolio value in cash accoun for and salary sochasic dv rv M V r M d M V d W aking he mahemaical expecaion (9) we have dev M olving () we have where dv r E V r M E d e e s (9) () EV v f sds () f s r M s E s r () M M V f V d M M V dw aking he mahemaical expecaion of () we have de V M M EV f EV d Inegraing () we have where e e e s s e f s f e dd s s EV v v f s Efficien Fronier M M ds () In his subsecion we presens he efficien fronier of he hree classes of asses A we have where ve e EV EV v e s f s d s s s s e v e e f s ds f s f e d ds herefore he variance of he porfolio is obained as () Var V E V E V () v e E V () is he expeced erminal wealh of he PPM and () is variance of he expeced erminal wealh of he PPM he efficien fronier of he hree classes of asses is obain as e Var V v E V ve E V ve EV ve v EV v F EV ve e e

11 8 J O OKORO C I NKEKI where e F v herefore he efficien fronier (ha is nonlinear and have parabolic shape) of he porfolio process is w here i implies ha EV v F e i If i V V F EV v e his shows ha fund can be borrowed from he opimal wealh a ime for year Conclusion In his paper we sudied he opimal variaional Meron porfolios wih inflaion proecion sraegy for a defined conribuion Pension scheme he presen values of PPM s fuure conribuion and sensiiviy analysis of he presen value of he conribuion were esablished he opimal variaional Meron porfolio processes wih iner-emporal hedging erms and inflaion proecion ha offse any shocks o he sochasic salary of a PPM are obained Furhermore expeced values of PPM s erminal wealh and variance as well as he efficien fronier were obained REFERENCE [] L Cao and Z F Guo Opimal Variance waps Invesmens IAENG Inernaional Journal of Applied Mahemaics Vol No Pages [] L Cao and Z F Guo Dela hedging hrough Delas from a Geomeric Brownian Moion Process Proceedings of Inernaional Conference on Applied Financial Economics London June- July [] P E Davis Porfolio Regulaion of Life Insurance Companies and Pension Funds Oxford Universiy Press Oxford hp//wwwoecdorg/daaoecd//9/88pdf [] A J G Cairns D Blake and K Dowd ochasic Lifesyling: Opimal Dynamic Asse Allocaion for Defined Conribuion Pension Plans Journal of Economic Dy- namics & Conrol Vol No pp hp://dxdoiorg//jjedc9 [] Brawne A M Milevsky and alisbury Asse Allocaion and he Liquidiy Premium for Illiquid Annuiies he Journal of Risk and Insurance Vol 7 No pp 9- hp://dxdoiorg// [] G Deelsra M Grasselli and P Koehl Opimal design of he Guaranee for Defined Conribuion Funds hp://wwwamazoncom/opimal-design-guaranee-defin ed-conribuion/dp/br9i [7] G Deelsra M Grasselli and P Koehl Opimal Invesmen raegies in he Presence of a Minimum Guaranee Insurance: Mahemaics and Economics Vol No pp 89-7 hp://dxdoiorg//7-87()-7 [8] G Deelsra M Grasselli and P Koehl Opimal Design of he Guaranee for Defined Conribuion Funds Journal of Economics Dynamics and Conrol Vol 8 No pp 9- hp:/ /dxdoiorg//jjedc [9] C R Nwozo and C I Nkeki Opimal Invesmen raegy for a Defined Conribuory Pension Plan in Nigeria Using Dynamic Opimizaion echnique udies in Mahemaical ciences Vol No pp - [] C R Nwozo and C I Nkeki Opimal Invesmen and Porfolio raegies wih Minimum Guaranee and Inflaion Proecion for a Defined Conribuion Pension cheme udies in Mahemaical ciences Vol No pp [] C R Nwozo and C I Nkeki Opimal Porfolio and raegic Consumpion Planning in a Life-Cycle of a Pension Plan Member in A Defined Conribuory Pension cheme IAENG Inernaional Journal of Applied Mahemaics Vol No p 99 [] C I Nkeki On Opimal Porfolio Managemen of he Accumulaion Phase of a Defined Conribuory Pension cheme PhD hesis Universiy of Ibadan Ibadan [] C I Nkeki and C R Nwozo Variaional Form of Classical Porfolio raegy and Expeced Wealh for a Defined Conribuory Pension cheme Journal of Mahemaical Finance Vol No -9 hp://dxdoiorg/ /jmf [] C I Nkeki and C R Nwozo Opimal Invesmen under Inflaion Proecion and Opimal Porfolios wih ochasic Cash Flows raegy o appear in IAENG Journal of Applied Mahemaics Vol No p

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