risks Aricle The Role of Inflaion-Indexed Bond in Opimal Managemen of Defined Conribuion Pension Plan During he Decumulaion Phase Xiaoyi Zhang and Junyi Guo *, ID School of Mahemaical Sciences, Nankai Universiy, Tianjin 30007, China; zhangxiaoyi990@63.com * Correspondence: jyguo@nankai.edu.cn; Tel.: +86--3506430 These auhors conribued equally o his work. Received: 3 January 08; Acceped: 9 March 08; Published: March 08 Absrac: This paper invesigaes he opimal invesmen sraegy for a defined conribuion (DC) pension plan during he decumulaion phase which is risk-averse and pays close aenion o inflaion risk. The plan aims o maximize he expeced consan relaive risk aversion (CRRA) uiliy from he erminal real wealh by invesing he fund in a financial marke consising of an inflaion-indexed bond, an ordinary zero coupon bond and a risk-free asse. We derive he opimal invesmen sraegy in closed-form using he dynamic programming approach by solving he relaed Hamilon-Jacobi-Bellman (HJB) equaion. The resuls reveal ha, wih any level of he parameers, an inflaion-indexed bond has significan advanage o hedge inflaion risk. Keywords: inflaion-indexed bond; DC pension plan; sochasic opimal conrol; dynamic programming approach; HJB equaion. Inroducion and Moivaion An asse allocaion problem incorporaing inflaion risk for individual invesors has been sudied by many researchers. A closed form of opimal invesmen sraegy is given by Brennan and Xia (00), hen Munk e al. (004) obain he opimal sraegy in a model wih inflaion uncerainy by he dynamic programming mehod. Since differences in spending paerns and in price increases lead o unequal inflaion experiences, he work of Li e al. (07) considers an opimal invesmen and consumpion problem of households under inflaion inequaliy. Inflaion-indexed bond is defined as an financial insrumen ha delivers a defined payoff indexed by inflaion a mauriy ime, which can be uilized o hedge agains inflaion risk. For an invesmen company, Nkeki and Nwozo (03) find ha inflaion risk associaed wih invesmen could be hedged by invesing in inflaion-linked bond, wih some assumpions of sochasic cash inflows and ouflows of he company. Liang and Zhao (06) invesigae he efficien fronier and opimal sraegies of a family under mean-variance efficiency, and he work of Pan and Xiao (07) deals wih an opimal asse-liabiliy managemen problem. Boh works menioned above ake ino accoun he inflaion risk and consider he problem in he real marke insead of he nominal price. Such kind of bonds are applied in life insurance. Kwak and Lim (04) invesigae a coninuous ime opimal consumpion, invesmen and life insurance decision problem of a family under inflaion risk, which explici soluions are derived by using maringale mehod. Then Han and Hung (07) solve a similar invesmen problem of a wage earner before reiremen wih he mehod of dynamic programming approach. In order o hedge agains inflaion risk, an inflaion-indexed bond is inroduced in boh wo problems above. Risks 08, 6, 4; doi:0.3390/risks60004 www.mdpi.com/journal/risks
Risks 08, 6, 4 of 6 In order o guaranee consumpions for oneself afer reiremen, or for his or her beneficiaries (spouse, children or parens), a pension plan is inroduced as an organized and sysemaic financial insrumen o provide regular incomes. In he dimension of benefis, here are wo principal ypes of pension plans: defined conribuion (DC) and defined benefi (DB). In he DC case, benefis are generaed by he accumulaion of conribuions, and conribuions paid by he pension member are defined explicily. Opimal invesmen decision problems are sudied by many works for DC pension plan. For insance, he work of Sun e al. (07) deals wih he porfolio opimizaion problem by invesing he pension fund in a marke consising of various governmen and corporae bonds, allowing for he possibiliy of bond defauls. The work of Guan and Liang (06) sudies he sochasic Nash equilibrium porfolio game beween wo DC pension schemes under inflaion risks. However, for DB ype pension scheme, benefis received afer reiremen are defined explicily by he rules of he scheme. There are also los of works abou opimal managemen on DB pension. For example, in order o minimize deviaions of he unfunded acuarial liabiliy, Josa-Fombellida and Rincón-Zapaero (00) consider opimal porfolio decision problem for an aggregaed DB pension plan in he framework of sochasic ineres rae. In he dimension of funding, here are also wo ypes of pension fund: pay as you go (PAYG) and funding. In a PAYG mechanism, conribuions paid by he acive affiliaes are direcly used as reired people s benefis. Opimal managemen in PAYG mechanism are inroduced by many researchers. For insance, by applying opimal conrol echniques, Haberman and Zimbidis (00) develop a deerminisic-coninuous model and a sochasic-discree model using a coningency fund. To guaranee he required level of liquidiy, Godínez-Olivares e al. (06) design opimal sraegies using nonlinear dynamic programming hrough changes in he key variables of pension sysem (conribuion rae, reiremen age and indexaion). Oher works in PAYG mechanism can be found in he works of Alonso-García and Devolder (06) and Alonso-García e al. (07). However, in a funding mechanism, conribuions paid by a group of people are invesed in he financial marke and will be used as heir own benefis. There are also many works in opimal managemen problems. The work of Li e al. (06) aims o derive opimal ime-consisen invesmen sraegy under he mean-variance crierion, by invesing pension wealh in a financial marke consising of a bank accoun and a risky asse (which price process saisfies he consan elasiciy of variance model). Oher works in funding mechanism are sudied by Sun e al. (06,07). As he invesmen of a DC pension plan involves quie a long period of ime, i seems implausible o ignore inflaion risk in he long run. Moreover, in a DC pension plan, benefis depend solely on he reurns of he fund s porfolio, so i is meaningful o proec inflaion risk in such kind of pension scheme. Yao e al. (03) solve a mean-variance problem by considering he real wealh process including he influence of inflaion. Okoro and Nkeki (03) examine he opimal variaional Meron porfolios wih inflaion proecion sraegy. Boh expeced values of pension plan member s erminal wealh and efficien fronier are obained in heir work. Generally, a pension scheme conains an accumulaion (conribuion) phase, which is he period before reiremen, and a decumulaion (disribuion) phase, which is he period afer reiremen. There are some applicaions of inflaion bond in pension plans which concenrae on he opimal managemen during he accumulaion phase. Zhang e al. (007) and Zhang and Ewald (00) invesigae an opimal invesmen problem by invesing an indexed bond, and presen a way o deal wih he opimizaion problem using he maringale mehod. In he work of Han and Hung (0), sochasic dynamic programming approach is used o invesigae he opimal asse allocaion for a DC pension plan wih downside proecion under sochasic inflaion, and he inflaion-indexed bond is again included in he asse menu o cope wih he inflaion risk. According o Chen e al. (07), an opimal invesmen sraegy for a DC plan member who pays close aenion o inflaion risks and requires a minimum performance a reiremen is solved by maringale approach. Anoher applicaions of inflaion-indexed bond can be found in Nkeki (08) and Tang (08).
Risks 08, 6, 4 3 of 6 As he decumulaion phase of a DC pension scheme is also confroned wih inflaion risk, his paper applies he inflaion bond in his period and considers an opimal conrol problem, which coninuously decides weighs of invesmen in differen asses, including a zero coupon bond, an inflaion-indexed bond and a riskless asse, in order o maximize he erminal wealh wih he consideraion of he influence of inflaion. Anoher moivaion of our work is o invesigae wheher he invesmen efficiency is improved by he inflaion-index bond. The quesion is wheher he opimal uiliy funcion is increased wih he invesmen of he index bond. In order o do he comparaive sudy, we follow he definiion of he indexed bond price, see, for insance, Nkeki and Nwozo (03) or Han and Hung (07), bu find anoher SDE o describe is price. In our work, he price of an ordinary bond is jus a special case of he price of he indexed bond. The res of he paper is srucured as follows. Secion describes he financial marke wih sochasic ineres rae, sochasic price level and hree radable asses which are of ineres for our problem. The demographic paern is given by a drifed Brownian moion. Secion 3 solves an opimal invesmen problem wih invesmen in a complee marke including an inflaion-indexed bond, an ordinary zero coupon bond and a bank accoun. The closed form soluions of his sochasic conrol problem are given by solving he relaed HJB equaion. The counerpar, Secion 4 solves a similar problem wih he indexed bond excluded. Secion 5 gives he sensiiviy analysis. A las, Secion 6 compares he resuls given by Secions 3 and 4 and gives he conclusion. The comparaive sudy shows ha invesmen in he indexed bond has significan advanage o hedge inflaion risk.. Model Assumpions and Noaions.. The Financial Marke The insananeously nominal ineres rae R() is assumed o be sochasic. Here we use he Ornsein-Uhlenbeck process inroduced by Vasicek (997), i.e., R() saisfies he following differenial equaion: dr() = b ( a R() ) d + σ R dz () () where Z () is a sandard Brownian moion under probabiliy measure P, R(0) = R 0 and all parameers are assumed o be posiive consans. Assume here is a financial marke consising of hree radable underlying insrumens which are raded coninuously over ime and are perfecly divisible. In addiion, we assume ha here are no ransacion coss or axes in he conex. Borrowing and shor-selling are permied. a. A money marke accoun M() saisfies he following equaion: dm() M() = R()d () wih iniial value M(0) = M 0. b. A zero coupon nominal bond which pays one moneary uni a expiraion ime T 0, and is value B(, T 0 ) a ime can be wrien as he condiional expecaion under he so-called equivalen maringale measure corresponding o he arbirage free marke, under which e 0 R(u)du B(, T 0 ) is a local maringale: { B(, T 0 ) = E Q e T 0 R(u)du F (3)
Risks 08, 6, 4 4 of 6 Considering he maringale propery of he discouning process of B(, T 0 ) and following he work of Menoncin (008), we ge he dynamics of he bond price: db(, T 0 ) = B(, T 0 ) R() d + B(, T 0) R() B(T 0, T 0 ) = σ R dz Q () (4) where Z Q () is a sandard Brownian moion under measure Q. Le λ R be he marke price for he ineres rae risk, hen by Girsanov s heorem, dz Q () = λ Rd + dz { () is a Wiener process. The Radon-Nikodym derivaive is defined by Λ(T 0 ) = dq dp (Z [0, T 0 ]) = exp λ R Z (T 0 ) λ R T 0. Consequenly, one can ge he sochasic differenial equaion of B(, T 0 ) under he original measure P: db(, T 0 ) = ( R() + B B(, T 0 ) R ()σ ) Rλ R d + B R ()σ R dz () B(T 0, T 0 ) = (5) where B R () = B(,T 0) is he semi-elasiciy of he bond price wih respec o he ineres rae, R() B(,T 0 ) which is a negaive funcion of ime, because he bond negaively reacs o he shocks on ineres rae. Remark. Since he bond has a posiive premium compared wih he riskless asse, λ R is assumed o be negaive. c. Wih regards o he sochasic price level, we define he inflaion index (or he level of consumer price) process as: dp() P() = i()d + σ P dz () (6) wih P(0) = P 0 > 0. i() is a deerminisic funcion of ime represening he expeced rae of inflaion and σ P is he consan volailiy. Z () is anoher Brownian moion under he physical measure P, which generaes uncerainy in he price level and is independen of Z (). If we se λ p as he marke price of risk wih respec o dz (), hen dz Q () = λ Pd + dz () is a Wiener process, where Z Q () is a Brownian moion under he risk neural measure Q. P() has he explici form as he following: { P() = P 0 exp i(s)ds (σ P λ p + σ P 0 ) + σ PZ Q () As in he work of Han and Hung (0), we define an inflaion-indexed zero coupon bond, whose price a ime is denoed by I(, T 0 ), from which he invesor can ge P T0 unis of money a he mauriy ime T 0. By he fundamenal heorem of asse pricing, i is well known ha in an arbirage free marke he price of any asse coincides wih he expeced presen value of is fuure cash flows under he equivalen maringale measure Q. Thus, we have { I(, T 0 ) = E Q P T0 e T 0 R(u)du F T0 = P 0 e 0 i(s)ds (σ P λ P +σ P /)T 0 e σ PZ Q () E Q {e σ P ) (Z Q (T 0) Z Q () e T 0 R(u)du (7) (8) F Lemma. In Equaion (8), E Q {e σ P = E Q {e σ P (Z Q (T 0) Z Q () ) e T 0 R(u)du ) { (Z Q (T 0) Z Q () E Q e T 0 = e σ P (T 0 )/ B(, T 0 ) F R(u)du F (9)
Risks 08, 6, 4 5 of 6 Proof. Denoe F e T 0 = σ {Z Q (s), s, F R(u)du is independen of e σ P E Q {e σ P { = E Q e T 0 ) (Z Q (T 0) Z Q () (Z Q (T 0) Z Q () ) R(u)du = σ {Z Q (s), s { Since R() is independen of F, we have E Q e T 0, and se F = F F. Since and F, and F is independen of F, we have: e T 0 R(u)du F { ( ) (0) F E Q e σ P Z Q (T 0) Z Q () F. R(u)du { F = E Q e T 0 R(u)du F. F disappears in he second par because of he propery of independen incremen of a Brownian moion, and he las equaliy of (9) holds by he propery of exponenial maringale. A las, we ge a formulaion of I(, T 0 ) relaed wih B(, T 0 ): T0 I(, T 0 ) = P 0 e 0 i(s)ds σ P λ P T 0 e σ P / e σ PZ Q () B(, T 0 ) () By he chain rule of Iô s formula, he evoluion of I(, T 0 ) is described by he following sochasic differenial equaion: T0 [ ] di(, T 0 ) = P 0 e 0 i(s)ds σ P λ P T 0 e σ P / e σ PZ Q () db(, T 0 ) + e σ P / B(, T 0 )σ P e σ PZ Q () dz Q () Finally, we have: () di(, T 0 ) I(, T 0 ) = db(, T 0) B(, T 0 ) + σ PdZ Q () = ( R() + B R σ Rλ R + σ P λ P ) d + B R σ R dz () + σ P dz () (3) Thus, we ge a correlaion beween he price of he inflaion-indexed bond and ha of an ordinary zero coupon bond... The Demographic Paern As o he demographic paern, a meaningful and concise srucure is presened in he work of Zimbidis and Panelous (008). I is assumed ha he dynamic of number of deah follows a sochasic differenial equaion: dl() = θ()d + ν()dz 3 () (4) where θ() and ν() describe he insananeous drif and volailiy, respecively. Z 3 () is anoher sandard Brownian moion under measure P, which is independen of Z () and Z (). Remark. As for he demographic paern, more advanced srucure is considered in he work of Panelous and Zimbidis (009). The dynamic of he number of deah follows dl() = θ()d + ν()db H (), wih B H () a fracional Brownian moion, which is more significan since i can model observed long-range dependence of large family of sochasic processes in a simple way by he propery of long memory. Now se F = F F F 3, where F 3 resul in Lemma sill holds. = σ {Z Q 3 (s), s. I is no hard o show ha he 3. DC Pension Fund Managemen wih Invesmen of Inflaion-Indexed Bond In a DC pension scheme, see, for example, Zimbidis and Panelous (008), a plan member pays conribuions during his or her employmen period and beneficiaries of each pensioner (who dies a ime ) receive an accumulaed amoun, as a whole life assurance wih a deah benefi.
Risks 08, 6, 4 6 of 6 Here we consider a sochasic conrol problem wih finie ime horizon, in order o maximize he erminal expeced uiliy wealh and find opimal invesmen policies for asses in he decumulaion phase of he pension plan, from modeling ime = 0 o a suiable erminal ime = T(T T 0 ). Wih he invesmen of he ordinary bond and he inflaion-indexed bond, he financial marke is complee. The marke can be represened as he following marix form: db B di I R + B = R σ R λ R B R σ R 0 dz d + R + B R σ Rλ R + σ P λ P B R σ R σ P dz {{ Σ where he marix Σ is inverible. I is easily o ge he following equaion which describes he evoluion of he fund: (5) dx() = ( u BI () ubi ()) X() dm() M() + ubi () X() db(, T 0) B(, T 0 ) +u BI() X() di(, T 0) I(, T 0 ) X()c()( dl() ) [ = X() X(0) = X 0 +X() ( u BI ] ()( B R ()σ Rλ R + σ P λ P ) c()θ() d () + ubi ()) B R ()σ RdZ () + X()u BI()σ PdZ () X()c()ν()dZ 3 () R() + u BI () B R ()σ Rλ R + u BI (6) where c() is he weigh of benefi received immediaely by he beneficiaries, i.e., he benefi associaed wih he pension fund is assumed o be a deerminisic proporion c() of he pension wealh. Wih his assumpion, he pension sponsor pays more benefis wih a higher invesmen income, and pays less benefis wih a lower invesmen income, which makes he pension plan realisic and aracive o he individual. () and ubi () are weighs invesed ino he ordinary zero coupon bond and he inflaion-index bond a ime, respecively. A negaive value of u BI or u BI means ha he sponsor akes a shor posiion in he ordinary bond or he indexed bond, respecively, while a negaive value of u BI ubi means ha in order o buy ordinary bond or indexed bond, he sponsor borrows from he bank a he u BI rae R. Denoe u BI () = ( u BI (), ubi ()). I is called admissible if i saisfies he following condiions, and we denoe he se of all admissible conrols by Π.. u BI { () is progressively measurable wih respec o {F 0 ; T [(. E X() ( ) u BI () + ubi ()) B R σ ( R + X()u BI ()σ ) ( ) ] P + X()c()ν() d < ; 0 3. Equaion (6) has a unique srong soluion for he iniial value ( 0, R 0, X 0 ) [0, T] (0, ). As menioned in Secion, he conrol period for he pension fund is very long, hence he effec of inflaion becomes noiceable for he pension manager. Denoe he real wealh process including he impac of inflaion by X(), i.e., X() = X()/P(). By he chain rule, we have ha X() follows: dx() [ = R() + u BI X() () B R ()σ Rλ R + u BI()( B R ()σ Rλ R + σ P λ P σp ) c()θ() + σ P ]d i() + ( u BI () + ubi ()) B R ()σ RdZ () + ( u BI()σ ) (7) P σ P dz () c()ν()dz 3 () X(0) = X 0 /P 0 The pension sponsor would like o maximize he expeced uiliy of erminal real fund X(). Our opimal problem can be wrien as: [ ] max {u BI Π E U(X(T)) (8)
Risks 08, 6, 4 7 of 6 where U(x) is he uiliy funcion which describes he preference over wealh. Here we consider he ypical CRRA uiliy funcion, for which we can derive he explici form of he soluion, as follows: where γ is he relaive risk aversion. U(x) = x γ, γ > 0 and γ =, (9) γ Remark 3. For simpliciy, we do no consider any risk-based regulaory consrains such as hose in he Solvency II Direcive in his paper, hough i is more realisic in financial and insurance marke. To consider financial regulaion problems, for insance, Duare e al. (07) propose an asse and liabiliy managemen model wih a horough represenaion of a risk-based regulaion by applying he mulisage sochasic programming model. The problem can be solved by he dynamic programming mehod. Denoe V(, R, x) = E { U(X(T)) X() = x, R() = R as he value funcion. In sochasic opimal conrol heory, he HJB equaion accomplishes he connecion beween he value funcion and he opimal conrol, see Fleming and Soner (993) or Yong and Zhou (999). Denoe u BI () = ubi, ubi () = ubi, B R () = B R and so on, we have he associaed HJB equaion for he above problem as follows: where max {u BI Π Ψ(uBI, ubi ) = 0 (0) Ψ(u BI, ubi ) = V + V x x [ R + u BI + V xx x [( u BI + ubi B R σ Rλ R + u BI( B R σ Rλ R + σ P λ P σp ) cθ+σ P ) i] B R σr + (u BIσ P σ P ) + c ν ] +V R b(a R) + V RR σ R + V Rx x (u BI + ubi ) B R σ R = 0 () wih erminal condiion V(T, R, x) = x γ γ. V, V x, V R, V xx, V RR and V Rx denoe he firs and second order parial derivaives of V wih respec o, x and R, respecively. The maximizaion of Ψ(u BI, ubi saisfy he following necessary condiions: ) can be obained by he opimal funcional ubi Ψ(u BI, u BI ) = 0 dψ ) = 0 du BI dψ du BI (u BI and u BI, which () (u BI ) = 0 The firs order condiions expressed as feedback formulas in erm of derivaives of he value funcion are: u BI V x λ = R V xx x B R σ V Rx R V xx x B V x σ P λ P V R xx x σ P (3) u BI V x σ = P λ P + V xx x σ P Subsiuing Equaion (3) ino he HJB Equaion (0), we can finally ge he explici forms of he value funcion and he opimal invesmen sraegies. They are given in he following heorem.
Risks 08, 6, 4 8 of 6 Theorem. The opimal uiliy and opimal invesmen sraegies are given by V(, R, x) = x γ γ eabi u BI λ () = R γ B R σ + R γ u BI σ () = P λ P + γ σ P ()+ABI ()R B R A BI() + σ P λ P γ σ P (4) where () = γ [ ] e b(t ) b A BI T A BI() α BI (s)ds (6) α BI () = γ ( λ γ R + (σ P λ P ) ) + ( γ)(σ P λ P cθ i) γ( γ)c ν +A BI()ba ABI () σr + γ γ ABI ()λ Rσ R + γ γ ABI () σr (7) (5) Proof. See Appendix A. Figure shows he ime-varying opimal weighs of u BI and ubi, from which we can see ha ubi increases exponenially wih ime, while u BI is independen of ime (wih B R = 0.5, λ R = 0.5, λ P = 0.5, γ = 0.5 and b = 0.5). In order o do some comparison wih Secion 4, we denoe V(, R, x) V BI (, R, x). Opimal Decision 3 4 5 6 7 u_ u_ 0 5 0 5 0 5 30 Figure. u BI () and u BI (). 4. DC Pension Fund Managemen wihou he Invesmen of Inflaion-indexed Bond In order o invesigae he role of he inflaion-linked bond in pension managemen, we consider anoher opimal porfolio selecion problem wih he indexed bond excluded in his secion. Here we abuse he noaion and se he wealh process again denoed by X(), bu he porfolio only consiss of a bank accoun and an ordinary T-bond. In his case he financial marke is incomplee: here is no
Risks 08, 6, 4 9 of 6 enough asses for hedging agains he inflaion risk, i.e., he sochasiciy Z. The financial asses in he marke can be summarized in he following marix form: db B = [ ] [ R() + B R σ R λ R d + ] dz B R σ R 0 {{ dz Σ (8) in which case he marix Σ is no inverible. Acually, here does no exis any linear combinaion of asses o replicae he inflaion risk. Similarly, he corresponding wealh process can be defined as: dx() = ( u B()) X() dm() M() + ub () X() db(, T 0) B(, T 0 ) X()c()( dl() ) [ ] = X() R() + u B() B R ()σ Rλ R c()θ() d + X()u B() B R ()σ RdZ () X()c()ν()dZ 3 () X(0) = X 0 (9) where u B () describes he weigh allocaed o he zero coupon bond. Again denoe he real wealh process including he impac of inflaion by X(), and by he chain rule, is dynamic is dx() X() = X(0) = X 0 /P 0 [R() + u B () B R ()σ Rλ R c()θ() + σ P i() ] d + u B () B R ()σ RdZ () σ P dz () c()ν()dz 3 () (30) We have exacly he same objecive funcion and he same opimizaion problem in Secions 3 and 4, excep ha he invesmen of an inflaion-indexed bond has been removed. Analogously, denoe u BI () = ubi, ubi () = ubi, B R () = B R and so on, he relaed HJB equaion is max {u B Π V + V x x ( R + u B B R σ R λ R cθ + σp i) + [ V xxx u B B R σ R + σ P + c ν ] +V R b(a R) + V RR σ R + V Rx x u B B R σ R = 0 where V(, R, x) is again he value funcion corresponding he opimal problem wih erminal condiion V(T, R, x) = x γ γ. By differeniaing wih respec o u B, he opimal weigh can be expressed in he form of he value funcion: u B V x λ = R V xx x B R σ V Rx (3) R V xx x Similarly, we can finally ge he explici forms of he value funcion and he opimal invesmen sraegy in he same way. They are given in he following heorem. Theorem. The corresponding value funcion and he opimal weigh for he T-bond saisfy: B R (3) V(, R, x) V B (, R, x) = x γ γ eab ()+AB ()R u B λ () = R γ B R σ + (33) A B R γ () B R
Risks 08, 6, 4 0 of 6 where A B () = ABI () and T A B() = α B (s)ds (34) α B () = γ γ λ R + ( γ)( cθ i + σ P ) γ( γ)(σ P + c ν ) +A B()ba + AB () σr + γ γ AB ()λ Rσ R + γ γ AB () σr Now we denoe ha A B () = ABI () A (). Proof. The proof is similar o ha of Theorem so we omi i here. Figure shows he ime-varying opimal weigh of u B, from which we can see ha ub increases exponenially wih ime (wih B R = 0.5, λ R = 0.5, λ P = 0.5, γ = 0.5 and b = 0.5). (35) Opimal Decision 0 3 4 u_ 0 5 0 5 0 5 30 Figure. u B (). 5. Sensiiviy Analysis In his secion, we make a sensiiviy analysis o show he relaionship beween opimal invesmen sraegies and some parameers in Secion 3. Unless oherwise saed, values of parameers in Secions 5 and 6 are as follows: B R = 0.5, λ R = 0.5, λ P = 0.5, γ = 0.5, b = 0.5, = 0 and T = 30. Figure 3 gives he relaionship beween he volailiy and he opimal weigh of he ordinary zero coupon bond. Wih suiable parameers chosen, he sponsor holds a shor posiion of he ordinary bond. The opimal invesmen decision a iniial ime u BI (0) becomes lower, equivalenly, he absolue value of u BI (0) becomes higher wih a higher σr. I means ha he ordinary bond becomes more aracive (even in a shor posiion) when he value of σ R increases, since he premium of he ordinary bond B R σ Rλ R increases wih a rising σ R. Figures 4 and 5 show he relaionship beween he volailiy and he opimal weigh of he inflaion-indexed bond, wih posiive λ P (= 0.5) and negaive λ P (= 0.5), respecively. In he case λ P = 0.5, he opimal invesmen sraegy a iniial ime u BI (0) becomes higher wih an increasing volailiy σ P, since he premium of he indexed bond σ P λ P becomes higher, which means ha he indexed bond becomes more aracive o invesors. However, in he case λ P = 0.5, he premium σ P λ P decreases wih a rising σ P, which makes he indexed bond less aracive, so u BI (0) decreases wih σ P.
Risks 08, 6, 4 of 6 u_^{bi.5.0 0.5 0.0 0.40 0.45 0.50 0.55 0.60 sigma_r Figure 3. The impac of σ R o u BI (0). u_^{bi 0.0 0.5.0.5 0.3 0.4 0.5 0.6 0.7 sigma_p Figure 4. The impac of σ P o u BI (0) wih posiive λp. u_^{bi 4.5 5.0 5.5 6.0 0.3 0.4 0.5 0.6 0.7 sigma_p Figure 5. The impac of σ P o u BI (0) wih negaive λp.
Risks 08, 6, 4 of 6 6. Comparaive Saics and Conclusions In order o invesigae wheher he inflaion-indexed bond has an significan influence o he invesmen efficiency, a comparison beween Secions 3 and 4 is shown in he following heorem: Theorem 3. Se = 0. We have V BI (0, R, x) V B (0, R, x), which means ha he maximum expeced uiliy of he real erminal wealh by invesing in an inflaion-indexed bond is higher han ha of a porfolio only consising of an ordinary zero coupon bond and a money marke accoun. Proof. α BI α B = γ γ (λ P σ P ) + γ( γ)σ P + ( γ)σ P(λ P σ P ) = γ ( ) (36) ( γ)σp λ P 0 γ We have α BI α B, hus A BI A B, and he resul follows. Remark 4. From he firs equaion in (36), we have a higher V BI /V B wih a higher λ P σ P. A raional explanaion migh be as follows: Since λ P = i σ P σ P, λ P > σ P is equivalen o i > σ P + σ P, i.e., when he expeced inflaion rae is significanly low, here is no remarkable advanage o inves in he inflaion-indexed bond. Probably we can conclude ha he hedging is no significan. Now we invesigae he influence of he difference beween λ p and σ p o value funcions. I is clear ha in Figure 6, when λ p is greaer han σ p, he raio beween he value funcion corresponding he opimal problem wih inflaion-indexed bond and he value funcion corresponding he opimal problem wihou he indexed bond is almos increasing exponenially wih he increasing of λ p σ p. When we change he value of σ p o 0.48 and 0.46, i is easy o conclude ha he curve increasing faser wih σ p becomes smaller. Remark 5. The impac of erminal ime T: In Figure 7, he raio beween wo value funcions again increasing exponenially wih he increasing of he erminal ime. A raional explanaion of his phenomenon is ha he invesmen becomes riskier wih a longer ime inerval, and he hedging becomes more necessary by an indexed bond. We make a conclusion as follows: This paper considers, by means of dynamic programming approach, he opimal invesmen sraegy for he decumulaive phase in DC ype pension schemes under inflaion environmen. The objecive is o deermine he invesmen sraegy, maximizing he expeced CRRA uiliy of he erminal real wealh in a complee marke consising of an inflaion-indexed bond, a zero coupon bond and a riskless asse. The explici soluion of he opimal problem is derived from he corresponding HJB equaion. In order o invesigae he role of he indexed bond, we also solve anoher opimal invesmen problem in an incomplee marke wih he indexed bond excluded. Wih any level of parameers, we find ha he value funcion in he complee marke is never lower han he value funcion in he incomplee marke, i.e., he maximum expeced uiliy of he erminal wealh by invesing in an inflaion-indexed bond is never lower han ha of a porfolio wihou he index bond, hus we may conclude ha an indexed bond definiely has significan advanage o hedge inflaion risk.
Risks 08, 6, 4 3 of 6 V_BI/V_B 3 4 5 6 sigma_p=0.50 sigma_p=0.48 sigma_p=0.46 0.40 0.45 0.50 0.55 0.60 lambda_p Figure 6. The impac of λ P. V_BI/V_B 0 000 4000 6000 8000 0000 000 0 5 0 5 30 T Figure 7. The impac of T. Acknowledgmens: This work was suppored by he Naional Naural Science Foundaion of China (No.5789). Auhor Conribuions: The wo auhors conribue equally o his aricle. Conflics of Ineres: The auhors declare no conflic of ineres. Abbreviaions The following abbreviaions are used in his manuscrip: DC DB CRRA HJB PAYG Defined conribuion Defined benefi Consan relaive risk aversion Hamilon-Jacobi-Bellman Pay-as-you-go
Risks 08, 6, 4 4 of 6 Appendix A The proof of Theorem. Subsiuing Equaion (3) ino he HJB Equaion (0), we ge: V + V x x R V x ( λr + (σ P λ P ) ) + V x x(σ P λ P cθ i) + V xx V xx x c ν VRx σ V R = 0 xx +V R µ + V RR σ R V xv Rx V xx λ R σ R (A) We may ry ha he soluion of V(, R, x) has a form as follows: V(, R, x) = x γ γ eabi ()+ABI ()R (A) wih erminal condiion V(T, R, x) = x γ γ. Differeniaing i, we ge: V V R = ( A BI ) () + A BI ()R V Vx = γ V x V xx = γ( γ)x V = A BI () V V RR = A BI () V V Rx = γ x A BI () V (A3) Subsiuing Equaion (A3) ino Equaion (A), and arranging i by order of R, we have: [ R A BI +A BI ] () ba BI () + ( γ) γ ( () + λ γ R + (σ P λ P ) ) + ( γ)(σ P λ P cθ i) γ( γ)c ν +A BI () b a + ABI () σr + γ γ ABI ()λ Rσ R + γ γ ABI () σr = 0 (A4) wih erminal condiions A BI (T) = ABI (T) = 0. The above equaion saisfies for every R, so i is equivalen o he following wo equaion sysems: A BI () + + A BI () b a + ABI A BI (T) = 0 A BI () ba BI () + ( γ) = 0 A L(T) = 0 γ ( λ γ R + (σ P λ P ) ) + ( γ)(σ P λ P cθ i) γ( γ)c ν () σr + γ γ ABI ()λ Rσ R + γ γ ABI () σr = 0 The resuls hold by solving he above ordinary differenial equaions. By subsiuing he value funcion ino he firs order condiions in Equaion (3), he opimal invesmen weighs are hus obained. References Alonso-García, Jennifer, and Pierre Devolder. 06. Opimal mix beween pay-as-you-go and funding for DC pension schemes in an overlapping generaions model. Insurance: Mahemaics and Economics 70: 4 36. Alonso-García, Jennifer, María del Carmen Boado-Penas, and Pierre Devolder. 07. Adequacy, fairness and susainabiliy of pay-as-you-go pension sysems: Defined benefi versus defined conribuion. The European Journal of Finance 7. doi:0.080/35847x.07.39949. Brennan, Michael J., and Yihong Xia. 00. Dynamic asse allocaion under inflaion. The Journal of Finance 57: 0 38. (A5) (A6)
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