Opimal Invesmen for a Defined-Conribuion Pension Scheme under a Regime Swiching Model An Chen, Lukasz Delong Universiy of Ulm, Insiue of Insurance Science Helmholzsrasse 20, 89069 Ulm, Germany an.chen@uni-ulm.de Warsaw School of Economics, Insiue of Economerics, Deparmen of Probabilisic Mehods Niepodleglosci 162, Warsaw 02-554, Poland lukasz.delong@sgh.waw.pl 1
Absrac We sudy an asse allocaion problem for a defined-conribuion (DC) pension scheme in is accumulaion phase. We assume ha he amoun conribued o he pension fund by a pension plan member is coupled wih he salary income which flucuaes randomly over ime and conains boh a hedgeable and non-hedgeable risk componen. We consider an economy in which macroeconomic risks are exisen. We assume ha he economy can be in one of I saes (regimes) and swiches randomly beween hose saes. The sae of he economy affecs he dynamics of he radeable risky asse and he conribuion process (he salary income of a pension plan member). To model he swiching behavior of he economy we use a couning process wih sochasic inensiies. We find he invesmen sraegy which maximizes he expeced exponenial uiliy of he discouned excess wealh over a arge paymen, e.g. a arge lifeime annuiy. Keywords: Exponenial uiliy maximizaion, macroeconomic risks, cerainy equivalen, Backward Sochasic Differenial Equaions. 2
1 Inroducion The ongoing shif from defined benefi o defined conribuion (DC) plans in many developed counries has pushed he opimal asse allocaion problem for a DC plan o he fron of risk managemen of occupaional reiremen plans. I is imporan o recognize ha a large par of he risk borne by a pension plan is sysemaic and depends on economic cycles. I is empirically observed ha he defici of pension benefis grows during an economic downurn, while i is more likely o have a surplus in an economic boom. In oher words, he financial posiion of a pension plan is srongly subjec o macroeconomic risks. 1 I is clear ha he variables like he mean and he volailiy of asse reurns (and consequenly he mean and he volailiy of pension funds) vary subsanially beween diverse economic saes. There are mixed empirical observaions abou how he volailiy changes wih economic siuaions, ye many researchers have shown he counercyclical behavior of he sock volailiy, see e.g. Schwer (1989) and Engle e. al. (2008). Moreover, macroeconomic variables like employmen, inflaion and ineres rae, which are srongly relaed wih he economic siuaion, have subsanial effec on he mean of sock reurns, see e.g. Campbell (1987) and Asprem (1989). Acuaries have developed ways o manage he risks relaed o economic and demographic indices in he long erm, bu o he bes of our knowledge, macroeconomic risks have no been modelled explicily in he acuarial lieraure on opimal asse allocaion for DC plans. In his paper we incorporae macroeconomic risks ino an asse allocaion problem for a DC pension scheme by considering a muli-sae regime-swiching financial model. We invesigae an economy which can be in one of I saes (regimes) and swiches randomly beween hose saes. The sae of he economy affecs he dynamics of he radeable risky asse and he conribuion process (and he salary income of a pension plan member). To model he swiching behavior for he saes of he economy, we use a 1 Unlike pension plans, he heory of bonus in life insurance refers o an enirely differen paradigm: policyholders ge only wha he realized ineres and moraliy over he rac period can susain (see e.g. Norberg (2001)). Hence, here is no environmen risk on he par of he company. 3
couning process wih sochasic inensiies. The ransiion inensiies depend no only on he curren sae of he economy bu also on he curren price of he risky asse. Hence, we model an effec in which no only he sock price is affeced by he ransiions beween he saes of he economy bu also he sock price deermines he ransiion inensiies, see Ellio e. al. (2011) for a financial moivaion of a so-called feedback effec. The asse allocaion problem for a DC plan differs from he sandard asse allocaion problem for an invesor (see e.g. Meron (1969), Meron (1971)) since we have o consider an addiional random sream of conribuions which flows o he pension fund. The amoun conribued o he pension fund by a pension plan member is modelled by a process which conains a radeable risk componen and non-radable risk componens. The radeable risk componen can be hedged wih he radeable risky asse, and he non-radeable risk componens represen unhedgeable coninuous flucuaions in he conribuion rae and he unhedgeable swiching behavior of he economy. The pension fund s objecive is o maximize he expeced exponenial uiliy of he discouned excess wealh over a arge paymen a he reiremen age. 2 The arge paymen can be chosen as a lifeime annuiy wih he benefi based on he he final salary of he pension beneficiary. Le us remark ha maximizing he exponenial uiliy of he excess wealh is relaed o minimizing he probabiliy ha he erminal wealh falls below he arge level. 3 In his paper we use Backward Sochasic Differenial Equaions (BSDEs) o solve he opimizaion problem. To solve our exponenial uiliy maximizaion problem we follow Hu e. al. (2005) and Becherer (2006) and adap heir echniques o our seing. We characerize he opimal invesmen sraegy for a DC pension scheme and he opimal value funcion for he opimizaion problem wih a soluion o a non-linear backward sochasic differenial equaion wih a Lipschiz generaor. From he mahemaical poin 2 In his paper, we assume here is no agency problem, i.e. he pension fund (manager)and he pension beneficiary share he same opimizaion objecive. 3 Aspoinedoue.g. bybrowne(1995),maximizingexponenialuiliyofhewealhaagivenerminal ime is inrinsically relaed o maximizing he survival probabiliy, which is equivalen o minimizing he probabiliy of ruin. If we maximize he excess wealh (erminal wealh minus a arge paymen), his is hen relaed o minimizing he probabiliy ha he erminal wealh falls below he arge level. 4
of view he novely of he paper is ha we derive he soluion o he opimizaion problem in a new model and we invesigae properies of he soluion o he BSDE which arises in our calculaions. We would like o poin ou ha our model is differen from Hu e. al. (2005) and Becherer (2006). We consider a regime-swiching economy, i.e. a risky asse price dynamics driven wih a Brownian moion wih coefficiens depending on a couning process, and we allow for a sochasic flow of conribuions. Hu e. al. (2005) consider a risky asse price dynamics based on a Brownian moion, and Becherer (2006) considers a dynamics based on an absrac random measure, and he auhors do no consider a conribuion process. Our financial model is se up wih he goal ha he resuls derived in i can be beneficial for a manager of a DC pension scheme. We also would like o poin ou ha he soluion from Secion 11.1 from Delong (2013) canno be applied in our seing due o differences in he models, e.g. in Delong (2013) he coefficiens of he risky asse price dynamics do no depend on he jump process which is used o model an insurance risk. Alhough we are no able o model he economy very realisically in a long ime perspecive, he proposed framework and he soluion developed can sill give pension plan managers some insighs and guidance a a general, qualiaive level. We aim o model a plausible opimizaion problem of a DC pension scheme beneficiary in a fairly general seing which capures prevalen beliefs abou he workings of he marke (in macro). We do no recommend pension funds o adop he suggesed opimal invesmen sraegy quaniaively. There already exiss a sream of lieraure on opimal asse allocaion for pension funds. For insance, Gao (2008) sudies an asse allocaion problem under a sochasic ineres rae. Boulier e. al. (2001) incorporae a consrain ino an invesmen problem under which a guaraneed benefi is provided o a pension beneficiary. Blake e. al. (2012) invesigae an asse allocaion problem under a loss-averse preference. Cairns e. al. (2006) consider a sochasic salary income of a pension beneficiary and find he invesmen 5
sraegy which maximizes he expeced power uiliy of he raio of he erminal fund and he erminal salary. The closes o our research is he paper by Korn e. al. (2011), who invesigae a uiliy opimizaion problem for a DC pension plan wih a sochasic salary income and a sochasic conribuion process in a regime-swiching economy. Their main ineres lies in solving a filering problem since hey assume ha he saes of he economy are modelled by a hidden Markov chain. We would like o poin ou ha Korn e. al. (2011) assume consan volailiies of he asse reurns and he conribuion process and consan inensiies of he Markov chain. They provide an explici invesmen sraegy for a logarihmic uiliy. In his paper we consider more general dynamics of he radeable asse, he conribuion process and he Markov chain wih volailiies and inensiies depending on he (observable) saes of he economy, and we derive he opimal invesmen sraegy for an exponenial uiliy. Le us recall ha from he macroeconomic poin of view i is very imporan o assume counercyclical behavior of he volailiy of he sock and he dependence of he ransiion inensiy on he sock price, see Schwer (1989), Engle e. al. (2008), Ellio e. al. (2011). We would like o poin ou ha since we use BSDEs o solve our asse allocaion problem he resuls of his paper can be easily exended for a model wih more general - even non-markovian - dynamics. The paper is organized as follows. Secion 2 describes he financial marke under he macroeconomic risks and formulaes he asse allocaion problem. In Secion 3, we solve he invesmen problem and we find he opimal invesmen sraegy which maximizes he expeced exponenial uiliy of he discouned excess wealh over a arge paymen. In Secion 4, we presen some numerical resuls. Finally, Secion 5 provides some concluding remarks. 2 The model We deal wih a probabiliy space (Ω,F,P) wih a filraion F = (F ) 0 T and a finie ime horizon T <. We assume ha F saisfies he usual hypoheses of compleeness 6
(F 0 conains all ses of P-measure zero) and righ coninuiy (F = F + ). On he probabiliy space (Ω, F, P) we define an F-adaped, wo-dimensional sandard Brownian moion (W 1,W 2 ) = (W 1 (),W 2 (),0 T) and an F-adaped, mulivariae couning process N = (N 1 (),...,N I (),0 T). The one-dimensional Brownian moions W 1 and W 2 are independen. The one-dimensional couning processes (N 1,...,N I ) are no independen, and he dependence srucure is described in he sequel. We consider an economy which can be in one of I saes (regimes) and swiches randomly beween hose saes. For i = 1,...,I, he couning process N i couns he number of ransiions of he economy ino sae i. Furhermore, le J = (J(),0 T) denoe an F-adaped process which indicaes he curren sae of he economy. If he economy is in regime k {1,...,I} a he iniial poin of ime, hen he dynamics of he process J is given by he sochasic differenial equaion dj() = I (i J( ))dn i (), J(0) = k {1,...,I}. i=1 A pension plan manager manages a pension fund and rades in a financial marke. The financial marke consiss of a risk-free bank accoun and a risky asse. In he sequel we only consider discouned quaniies. Hence, he discouned value of he bank accoun is consan. We assume ha he dynamics of he discouned value of he risky asse S = (S(),0 T) is given by he sochasic differenial equaion ds() = µ(j( ))S()d+σ(J( ))S()dW 1 (), S(0) = s, (2.1) where (A1) (µ(i)) i=1,...,i is a sequence of real numbers and (σ(i)) i=1,...,i is a sequence of sricly posiive numbers, which describe he value of he drif µ and he volailiy σ of he discouned risky asse (2.1)if heeconomy isin regimei. The drif andhevolailiy of herisky asse depend on 7
hesaeofheeconomyasiisobserved inempiricaldaaandjusifiedbymacroeconomic heory, see he Inroducion. We commen on he dynamics (2.1). The dynamics for he discouned value of he risky asse can be moivaed in he following way. Le as assume ha he value of he bank accoun B = (B(),0 T) is modelled by he equaion db() = r(j( ))B()d, where he ineres rae r is sochasic and depends on he sae of he economy. Le he value of he risky asse S undisc = (S undisc (),0 T) saisfy he sochasic differenial equaion ds undisc () = µ(j( ))S undisc ()d+σ(j( ))S undisc ()dw(), where µ is he real-world drif of he asse. Then, he discouned value of he risky asse S() = e 0 r(j(s))ds S undisc () follows he dynamics ds() = ( µ(j( )) r(j( )))S()d + σ(j( ))S()dW() = µ(j( ))S()d + σ(j( ))S()dW(), which agrees wih (2.1). We now characerize he inensiies of he couning process. We assume ha (A2) for i = 1,...,I, he couning process N i has inensiy λ i (J( ),S()) where λ i : {1,...,i 1,i+1,...,I} [0, ) R is a bounded mapping. Consequenly, he compensaed couning processes Ñ i () = N i () 0 λ i (J(s ),S(s))ds, 0 T, i = 1,...,I, 8
are F-maringales. We remark ha λ i (j,s) denoes an inensiy of he ransiion of he economy ino sae i if he economy is in sae j and he discouned value of he risky asse is s. The dependence of he ransiion inensiy on he curren sae of he economy is obvious. The dependence of he ransiion inensiy on he risky asse is more sophisicaed. I models a so-called feedback effec in he marke under which no only he risky asse (he marke index) is affeced by he ransiions beween he saes of he economy bu also he risky asse (he marke index) deermines he ransiion inensiies, see he Inroducion. In he sequel we use he shor noaion: µ() := µ(j( )), σ() := σ(j( )), λ i () := λ i (J( ),S()), 0 T. Le T denoe he ime o reiremen for a pension plan member. Over he working lifeime, he nex T years, he pension beneficiary receives a salary income and par of his income is conribued ino he pension fund. We assume ha he salary income flucuaes randomly in ime. Le G := (G(),0 T) denoe he discouned value of he salary income. One possibiliy is o assume ha he dynamics of he discouned value of he salary income G is given by he sochasic differenial equaion dg() = µ G (J( ))G()d+σ G (J( ))G()(ρdW 1 ()+ 1 ρ 2 dw 2 ()), G(0) = g,(2.2) where (µ G (i)) i=1,...,i is a sequence of real numbers and (σ G (i)) i=1,...,i is a sequence of sricly posiive numbers which describe he value of he drif µ G and he volailiy σ G of he discouned salary (2.2) if he economy is in regime i, and ρ [ 1,1] is a correlaion coefficien which inroduces a correlaion beween he reurns of he salary income G and he radeable risky asse S. We should use he second Brownian moion W 2 o model he dynamics of he salary income o ake ino accoun he fac ha in realworld he reurns on salary incomes and radeable asses are no perfecly correlaed 9
and salary incomes canno be perfecly replicaed wih radeable asses (even in a onesae economy). We do no specify he dynamics of he salary income since we are more ineresed in a conribuion process. Le c := (c(),0 T) denoe a conribuion process, i.e. he discouned amoun which is conribued by he pension plan member ino he pension fund and is invesed for his/her reiremen. In a defined conribuion pension plan, he conribuion process c is linked o he discouned value of he salary income G. Usually he discouned conribuion paymen c() ino he fund is a consan proporion of he discouned salary income G(), i.e. c() = γg(), 0 T. (2.3) However, we do no have o specify he dynamics of he conribuion process. We only assume ha (A3) c := (c(),0 T) is an F-adaped, posiive, bounded process. I is worh noicing ha if c undisc denoes an undiscouned conribuion process and he ineres rae r depends on he sae of he economy, hen we can loose he original Markovian srucure if we deal wih he discouned conribuion c() = e 0 r(j(s))ds c undisc (). 4 The assumpion ha c is F-adaped means ha he random amoun conribued by he pension beneficiary o he pension fund conains a risk componen W 1, which can be hedged wih he risky asse S, and risk componens (W 2,N), which model unhedgeable coninuous flucuaions in he conribuion rae and he unhedgeable swiching behavior of he economy. We poin ou ha he assumpion ha c be bounded can be relaxed. We inroduce his assumpion since i simplifies he verificaion of he opimaliy of he soluion and, a he same ime, i is no resricive from a pracical poin of view. Le π = (π(),0 T) denoe he discouned amoun of money which is invesed 4 For a momen le us assume ha he boundedness assumpion for c is no in force. If c undisc is modelled by a CIR-like process wih coefficiens depending on he sae of he economy, hen (c,j,s) is no longer a Markov process even hough (c undisc,j,s) is a Markov process, he quadruple (c,. 0 r(j(s))ds,j,s) is now a Markov process. 10
by he pension plan manager in he risky asse S. We call π an invesmen sraegy of he pension fund. We know ha we abuse he concep of he sraegy from he invesmen poin of view since in our seing π does no denoe he number of unis, neiher he fracion of he wealh, held in he sock. However, he number of unis or he fracion of he wealh which should be held in he sock can be calculaed from he discouned amoun of money π (if all oher financial quaniies are known). We inroduce he se of admissible invesmen sraegies. Definiion 2.1. A sraegy π := (π(),0 T) is called admissible, wrien π A, if i saisfies he condiions: 1. π : [0,T] Ω R is an F-predicable process, 2. K 1 () π() K 2 (), 0 T, where (K 1 (),K 2 (),0 T) are bounded, F-predicable processes. Pension plan managers usually face consrains imposed on invesmen sraegies. Received wisdom says ha he amoun invesed in he risky asse should decrease over ime as he pension beneficiary approaches he reiremen age. Shor-selling of asses is usually prohibied under he law. I is also reasonable o assume ha he limis se by he manager should depend on he curren sae of he economy, e.g. he pension plan manager is willing o inves more in he risky asse if he economy is booming and swiches o risk-free asses if he economy is busing. The se A defines invesmen consrains which he pension plan manager has o follow in he accumulaion period of he pension plan. Since we consider he discouned amoun π, he bounds in A are defined as general F-predicable processes, see he commen and he foonoe afer assumpion (A3). Noice ha if he pension plan manager ses limis for he amoun of money invesed in he risky asse, hen he discouned amoun π is bounded. Hence, assumpion 2 from Definiion 2.1 is reasonable. If needed, he upper bound can be chosen o be arbirary large and may no play a significan role in he invesmen decision. 11
We can now define he dynamics of he pension fund in he accumulaion period. The discouned value of he wealh of he pension plan member X π := (X π (),0 T) under an admissible invesmen sraegy π A is described wih he sochasic differenial equaion dx π () = π() ( µ()d+σ()dw 1 ())+c()d, X π (0) = x, (2.4) where x denoes an iniial capial invesed in he pension fund. We would like o commen on he dynamics (2.4). Le (π undisc,s undisc,c undisc,x undisc,πundisc ) denoe he amoun of money invesed in he risky asse, he value of he risky asse, he amoun conribued by he plan member and he value of he wealh process. Then, i is clear ha we should invesigae he dynamics dx undisc,πundisc () = π undisc () dsundisc () S undisc () +( X undisc,πundisc () π undisc () ) r()d +c undisc ()d, X undisc,πundisc (0) = x. By sandard calculus we can now derive he dynamics (2.4) for he discouned value of he wealh process X π () = e 0 r(s)ds X undisc,πundisc (), which is conrolled wih a process π describing he discouned amoun of money which should be invesed in he risky asse. We assume here is no agency problem, i.e. he pension plan manager and he pension beneficiary have he same opimizaion objecive. The pension beneficiary is ineresed in maximizing he expeced uiliy of he excess wealh over a pre-specified level a he reiremen age. We neglec moraliy risk, i.e. our problem is condiional on survival of he pension beneficiary hroughou he conribuion period. 5 By considering he excess 5 If we assume ha he ime of deah of he beneficiary is independen of he financial marke risk and he macroeconomic risk, hen he moraliy risk does no affec he opimal invesmen sraegy. More specifically, le he ime of deah be modelled by a random variable τ which is independen of he financial marke risk and he macroeconomic risk. The objecive for a DC pension plan akes he form sup E [ e α(xπ (T) F) 1{τ T} ], π A 12
wealh, we incorporae he fac ha he pension beneficiary is ineresed in achieving a arge paymen which ensures his lifeime reiremen. As he erminal pension fund migh fall below he arge level, he uiliy funcions like log uiliy and power uiliy are inapplicable here since hese funcions are exclusively defined for he posiive real line. In his paper we assume ha he pension beneficiary is ineresed in maximizing he expeced exponenial uiliy of he discouned excess wealh: supe[ e α(xπ (T) F) ], (2.5) π A where α > 0 is he risk aversion coefficien, and F denoes a arge discouned paymen for he pension beneficiary. Le us remark ha he exponenial uiliy is widely used and well moivaed in economics, finance, insurance and risk managemen, see e.g. Carmona (2008). As far as he arge is concerned we only assume ha (A4) F is F T -measurable, posiive and bounded. One possible example of he arge paymen F is F = κg(t)a(j(t)), (2.6) where κ is a fracion parameer, G(T) denoes he discouned salary of he pension beneficiary a he ime of reiremen, a denoes an annuiy facor for he lifeime annuiy which depends on he fuure sae of he economy J(T), e.g. on he erm srucure of ineres raes a he ime of reiremen. If he arge (2.6) is chosen, hen he pension beneficiary is ineresed in keeping his/her las salary income (or a fracion κ of he las salary income) as he lifeime annuiy benefi. Noe ha maximizing he exponenial uiliy of he excess wealh is in fac also relaed o minimizing he probabiliy ha he which by he independence assumpion is equal o sup E [ e α(xπ (T) F) ] P(τ T). π A 13
erminal wealh X(T) falls below he arge level F. Le us remark ha in our model he family {e αxπ (τ),f sopping ime τ [0,T]} is uniformly inegrable for any π A. I is easy o noice ha e αxπ (τ) = e αx α τ 0 π(s)µ(s)ds α τ 0 c(s)ds α τ 0 π(s)σdw 1(s) Ke τ 0 απ(s)σ(s)dw 1(s) 1 τ 2 0 απ(s)σ(s) 2ds, 0 τ T. (2.7) The family {e τ 0 απ(s)σ(s)dw 1(s) 1 τ 2 0 απ(s)σ(s) 2ds,F sopping ime τ [0,T]} is uniformly inegrable by he Novikov crierion, see Theorem III.45 in Proer (2004). Hence, from (2.7) we conclude ha {e αxπ (τ),f sopping ime τ [0,T]} is uniformly inegrable for any π A. 3 The soluion o he invesmen problem To solve our opimizaion problem (2.5), we rely on Backward Sochasic Differenial Equaions (BSDEs). We follow he approach from Hu e. al. (2005), Becherer (2006) and Chaper 11 in Delong (2013), o which he reader is referred o for furher deails. In order o use he heory of BSDEs we assume ha (A5) every (P, F) local maringale M has he represenaion M() = M(0) + Z 1 (s)dw 1 (s)+ Z 2 (s)dw 2 (s)+ 0 0 0 I U i (s)dñi(s), 0 T, i=1 wih F-predicable processes (Z 1,Z 2,U 1,...,U I ) which areinegrable inhe Iô sense. Le us remark ha his assumpion is saisfied if we define he probabiliy space and he driving processes in an appropriae way, see Crépey (2011) for deails. 14
We consider he BSDE Y() = F + T f(s)ds T Z 1 (s)dw 1 (s) T Z 2 (s)dw 2 (s) T I U i (s)dñi(s), 0 T, (3.1) i=1 where f is he generaor of he equaion which will be deermined in he sequel. The soluion o he BSDE (3.1) consiss of square inegrable processes (Y,Z 1,Z 2,U 1,...,U I ) such ha Y is F-adaped and (Z 1,Z 2,U 1,...,U I ) are F-predicable. We inroduce he process A π := (A π (),0 T) defined by A π () = e α(xπ () Y()), 0 T, π A. The process A π plays he key role in solving our opimizaion problem. I is sraighforward o noice ha E [ e α(xπ (T) F) ] = E [ e α(xπ (T) Y(T)) ] = E[A π (T)]. If A π is a supermaringale for every π A, hen we obain he inequaliy E[A π (T)] = E [ e α(xπ (T) Y (T)) ] A π (0), π A, (3.2) and if A π is a maringale for some π A, hen we derive he equaliy E[A π (T)] = E [ e α(xπ (T) Y(T)) ] = A π (0). (3.3) Combining (3.2) wih (3.3), we ge E [ e α(xπ (T) Y (T)) ] E [ e α(xπ (T) Y (T)) ], π A, 15
and we conclude ha he sraegy π is opimal and A π (0) is he opimal value funcion of he opimizaion problem (2.5). Therefore, we aim o find he generaor f of he BSDE (3.1), independen of π, such ha he process A π is a super-maringale for any π A and A π is a maringale for some π A. We show how o find (f,π ). From (2.4) and (3.1) we ge α(x π () Y()) = α(x Y(0)) ( ( ) ( α π(s)µ(s)+c(s)+f(s) ds+ π(s)σ(s) Z1 (s) ) dw 1 (s) 0 Z 2 (s)dw 2 (s) 0 0 i=1 We inroduce wo processes: 0 I ) U i (s)dñi(s), 0 T. D π () = απ()µ() αc() αf()+ 1 2 α2 (π()σ() Z 1 ()) 2 M π () =exp + 1 2 α2 (Z 2 ()) 2 { + + 0 0 0 I (αu i () e αui() +1)λ i (), 0 T, i=1 α(π(s)σ(s) Z 1 (s))dw 1 (s) αz 2 (s)dw 2 (s) I αu i (s)dñi(s)+ i=1 0 and we can wrie he candidae value funcion in he form: 0 1 2 α2 (π(s)σ(s) Z 1 (s)) 2 ds 1 2 α2 (Z 2 (s)) 2 ds } I (αu i (s) e αui(s) +1)λ i (s)ds, 0 T, 0 i=1 A π () = e α(x Y (0)) e 0 Dπ (s)ds M π (), 0 T. (3.4) Since M π is he sochasic exponenial of a local maringale, he process M π is a local maringale for any π A. Le us recall ha he value funcion derived from he opimal invesmen sraegy should be a maringale, whereas wih all he oher sraegies, he 16
value funcion should be a super-maringale. We choose π () = argmin { απµ()+ 1 } π A 2 α2 (πσ() Z 1 ()) 2, 0 T, αf () = min { απµ()+ 1 } π A 2 α2 (πσ() Z 1 ()) 2 αc()+ 1 2 α2 (Z 2 ()) 2 I (αu i () e αui() +1)λ i (), 0 T. (3.5) i=1 Wih his choice we have D π () = 0, 0 T, and D π () 0, π A, 0 T. Moreover, f is independen of π. We end up wih he BSDE Y() = F ( T + min { πµ(s)+ 1 } π A 2 α(πσ(s) Z 1(s)) 2 c(s)+ 1 2 α(z 2(s)) 2 ) 1 I (αu i (s) e αui(s) +1)λ i (s) ds α i=1 T Z 1 (s)dw 1 (s) T Z 2 (s)dw 2 (s) T I U i (s)dñi(s), 0 T. (3.6) In he las sep, we ransform he BSDE (3.6) wih a non-lipschiz generaor o a BSDE wih a Lipschiz generaor. We inroduce new variables: i=1 V() = e αy(), P 1 () = αv( )Z 1 (), P 2 () = αv( )Z 2 (), Q i () = V( )e αu i() V( ), i = 1,...,I, 0 T. (3.7) 17
Applying Iô s formula, we obain he BSDE V() = e αf ( ) T { + min απµ(s)v(s )+ 1 } π A 2 α2 π 2 σ 2 (s)v(s ) απσ(s)p 1 (s) αc(s)v(s ) ds T P 1 (s)dw 1 (s) T P 2 (s)dw 2 (s) T I Q i (s)dñi(s), 0 T, which we use o characerize he opimal value funcion and he opimal invesmen sraegy. We presen he main resul of his paper. i=1 Theorem 3.1. Assume ha (A1)-(A5) hold. Consider he BSDE V() = e αf ( ) T { + min απµ(s)v(s )+ 1 } π A 2 α2 π 2 σ 2 (s)v(s ) απσ(s)p 1 (s) αc(s)v(s ) ds T P 1 (s)dw 1 (s) T P 2 (s)dw 2 (s) T I Q i (s)dñi(s), 0 T. (3.8) a) There exiss a unique soluion (V,P 1,P 2,Q 1,...,Q I ) o he BSDE (3.8) such ha V is F-adaped and (P 1,P 2,Q 1,...,Q I ) are F-predicable. Moreover, V is sricly posiive, bounded away from zero and from above, (Q 1,...,Q I ) are bounded P da.e and (P 1,P 2 ) are square inegrable. i=1 b) The opimal value funcion of he uiliy maximizaion problem (2.5) is equal o e αx V(0) and he opimal admissible invesmen sraegy is given by { { π () = max K 1 (),min K 2 (), µ() ασ 2 () + P 1 () }}, 0 T. ασ()v( ) 18
Proof. a) Consider he funcion f π (s,v,p) = απµ(s)v + 1 2 α2 π 2 σ 2 (s)v απσ(s)p αc(s)v, (s,v,p) [0,T] R R,π A. Underourassumpionsiissraighforwardoconcludehaf π (s,v,p)islipschizconinuous in (v,p) uniformly in (s,π). Hence, f (s,v,p) = min π A f π (s,v,p) is also Lipschiz coninuous in (v, p) uniformly in s. Consequenly, here exiss a unique soluion o he BSDE (3.8), see Proposiion 3.2 in Becherer (2006) or Theorem 3.1.1 in Delong (2013). Since he generaor f of he BSDE (3.8) is Lipschiz coninuous, we can wrie dv() = ( L()V( )+H()P 1 () ) d +P 1 ()dw 1 ()+P 2 ()dw 2 ()+ I Q i ()dñi(), (3.9) i=1 where L() = f (,V( ),P 1 ()) f (,0,P 1 ()) 1{V( ) 0}, 0 T, V( ) H() = f (,0,P 1 ()) 1{P 1 () 0}, 0 T, P 1 () and L and H are bounded. Le us define an equivalen probabiliy measure wih he Radon-Nikodym derivaive dq dp F T = e T 0 H(s)dW 1(s) 1 T 2 0 (H(s))2ds. Changing he measure and aking he expecaion, see Proposiion 2.2 in El Karoui e al. (1997) or Proposiions 3.3.1, 3.4.1 in Delong (2013), we can deduce from (3.9) he 19
represenaion V() = E Q [ e αf e T L(s)ds F ], 0 T. (3.10) The asserion concerning he soluion V follows from boundedness of F and L. From he dynamics (3.8) we can also deduce ha I Q i ()(N i () N i ( )) = V() V( ), 0 T, i=1 and we can conclude ha each Q i is bounded P d-a.e. since only one couning process can jump a a ime and V is bounded. b) Since he BSDE (3.8) has a unique soluion (Y,P 1,P 2,Q 1,...,Q I ), he BSDE (3.6) has also a unique soluion defined by Y() = 1 α lnv(), Z 1() = P 1() αv( ), Z 2() = P 2() αv( ), U i () = 1 α ln ( Qi ()+V( ) V( ) ), i = 1,...,I, 0 T, and Y is F-adaped, bounded, (U 1,...,U I ) are F-predicable, bounded P d-a.e and (Z 1,Z 2 ) are F-predicable, square inegrable. I is obvious ha π A. We are lef wih provingheopimaliyprincipleforheprocessa π () = e α(xπ () Y()) = e α(x Y(0)) e 0 Dπ (s)ds M π () defined in (3.4). Since for any π A he process M π is a posiive local maringale (i is he sochasic exponenial of a local maringale) and D π () 0, 0 T, we can derive E[A π ( τ n ) F s ] = E [ e α(x Y(0))+ τn 0 D π (u)du M π ( τ n ) F s ] e α(x Y(0))+ s τn 0 D π (u)du E [ M π ( τ n ) F s ] = e α(x Y(0))+ s τn 0 D π (u)du M π (s τ n ) = A π (s τ n ), 0 s T, (3.11) where(τ n ) n 1 denoesalocalizingsequence forhelocalmaringalem π. Fromheuniform 20
inegrabiliy of he family {e αxπ (τ), F sopping imes τ [0,T]} for π A, see (2.7), and boundedness of Y we conclude ha he family {A π (τ), F sopping imes τ [0,T]} is uniformly inegrable for π A. Taking he limi n in (3.11), we obain he supermaringale propery of A π for any π A. For π we have D π () = 0, 0 T, and we obain he maringale propery for A π. For deails we refer o Hu e. al. (2005), Becherer (2006) and Chaper 11.1 in Delong (2013). We have characerized he opimal value funcion of our uiliy maximizaion problem (2.5) and he opimal invesmen sraegy wih he soluion o he backward sochasic differenial equaion (3.8). The BSDE (3.8) is a non-linear BSDE wih a Lipschiz generaor. In our general model he soluion o he BSDE canno be found in a closed form and we have o derive he soluion numerically. An efficien mehod o derive a soluion o a BSDE is o apply Leas Squares Mone Carlo which we discuss in Secion 4. In one case he soluion o he BSDE (3.8) has a nice closed-form represenaion. Proposiion 3.1. Le he assumpions of Theorem 3.1 hold. If c,f,k 1,K 2 and λ are independen of W 1, hen P 1 () = 0, 0 T, and { π () = = max V() = E [e αf e T K 1 (),min { K 2 (), µ() }} ασ 2 () ( απ (s)µ(s)+ 1 2 α2 (π (s)) 2 σ 2 (s) αc(s), 0 T, )ds ] F, 0 T. Proof. Since he coefficiens in fron of V and P 1 in he generaor of he BSDE (3.8) and he erminal condiion for he BSDE (3.8) do no depend on he Brownian moion W 1, we can choose P 1 () = 0. The form of π is obvious. The represenaion of V now follows from (3.10). In his case he BSDE (3.8) is a linear BSDE. Le us remark ha he lack of dependence of c,f,k 1,K 2,λ on W 1 means ha he conribuion process, he arge paymen, he invesmen limis and he ransiion inensiies are no relaed o he developmen of he risky asse S, i.e. hey do no have a radeable risk componen. 21
Le us now commen on he opimal value funcion and he opimal invesmen sraegy. We shall commen on he cerainy equivalen (CE) insead of he opimal value funcion. The use of he CE makes he quaniies easier o inerpre, because he CE expresses he expeced uiliy in moneary unis insead of uiliy unis. In our case we can define he cerainy equivalen as an amoun of a cerain capial which a pension plan member should receive a ime = 0 as an equivalen for an uncerain erminal wealh which arises from he opimally invesed conribuions in he financial marke. The equivalence of wealh is measured wih he exponenial uiliy of he discouned excess wealh over he arge paymen. Hence, he cerainy equivalen CE solves he equaion supe [ e ] α(xπ (T) F) = E [ e α(x+ce F)], π A and by Theorem 3.1 we ge CE = 1 α ln(v(0))+ 1 α lne[eαf ]. (3.12) Noice ha x+ce can be inerpreed as he cerainy equivalen of he discouned amoun which is annuiized by he pension plan member a he ime of reiremen T. We can also define he cerainy equivalen for he excess wealh CE excess as a soluion o he equaion supe [ e ] α(xπ (T) F) = e α(x+ceexcess), π A and we ge CE excess = 1 ln(v(0)). (3.13) α The opimal invesmen sraegy π consiss of wo pars. The firs par µ()/(ασ 2 ()) is he Meron invesmen sraegy, which is he opimal invesmen sraegy for an invesor who aims o maximize he expeced exponenial uiliy of he erminal wealh in a one- 22
sae economy wihou a conribuion process, a arge paymen and invesmen limis. The second par of he opimal invesmen sraegy P 1 ()/(ασ()v( )) is used by he pension fund manager, who rades he risky asse S, o hedge he radeable componen in he conribuion process, he arge paymen, he ransiion inensiies and he invesmen limis. Since(3.7)holds, wehavep 1 ()/(ασ()v( )) = Z 1 ()/σ()wherez 1 isheconrol processofhebsde(3.6). FromheheoryofBSDEs, seecorollary4.1inelkarouieal. (1997) or Theorem 4.1.4 in Delong (2013), we can deduce ha he process Z 1 defines he change in he value of he process Y resuling from changes in he risky asse S due o he movemen of he Brownian moion W 1. Since (3.7) and (3.13) hold, we can conclude ha he second par of he opimal invesmen sraegy is used by he pension fund manager o follow he opposie changes in he cerainy equivalen for he excess wealh resuling from changes in he conribuion process, he arge paymen, he ransiion inensiies and he invesmen limis due o he radeable risk componen W 1. 4 Numerical example This secion repors some numerical resuls based on he model developed in previous secions. We consider a wo-sae Markov-regime-swiching model. Sae 1 denoes economic boom and sae 2 denoes economic recession. The economy is in sae 1 a ime = 0. The salary income of a pension plan member is modelled by he sochasic differenial equaion dg() = µ G (J( ))G()d+σ G (J( ))G()(ρdW 1 ()+ 1 ρ 2 dw 2 ()), G(0) = g, and we choose a proporional conribuion rae, i.e. c() = γg(), 0 T. 23
T S(0) G(0) γ K 1 K 2 ρ x α 1 100 10 0.1 0 60 0.5 200 0.1 Table 1: The values of he sae-independen parameers. The iniial capial in he pension fund is x = G(0)a(1), where a(1) denoes a lifeime annuiy facor calculaed based on macroeconomic assumpions for sae 1 (e.g. based on he erm srucure of ineres raes in economic boom). The choice of he iniial capial implies ha if he pension fund is annuiized hen i can be convered o a lifeime annuiy wih he benefi equal o he curren salary of he pension beneficiary. The arge paymen for he pension plan is of he form F = G(T)a(J(T)), hence he pension beneficiary is ineresed in a lifeime annuiy wih he benefi equal o his/her las salary. We fix he parameers for he numerical example as in Tables 1-2. We assume ha shor-selling of he risky asse is prohibied for he pension fund and he upper bound for he invesmen sraegy is 150% of he Meron opimal invesmen sraegy in sae 1 for he exponenial uiliy (K 2 = 1.5 µ(1)/(σ 2 (1)α)) = 60). We choose a moderae correlaion ρ beween he risky asse and he salary process. I is reasonable o assume ha he drif is higher in sae 1 han in sae 2, and he volailiy is higher in sae 2 han in sae 1. The lifeime annuiy facor is higher in sae 2 han in sae 1 (e.g. due o he fall of ineres raes in recession). In order o guaranee ha he conribuion process is bounded we inroduce an upper bound on he conribuion and we redefine c() = min{c(),20}. 24
sae i µ(i) σ(i) µ G (i) σ G (i) λ(i) a(i) sae 1 0.04 0.10 0.03 0.02 1 20 sae 2 0.01 0.20 0 0.06 2 22 Table 2: The values of he sae-dependen parameers. We would like o compare he cerainy equivalens (3.12), he excess wealh X π (T) F and he replacemen raios X π (T)/F under he opimal invesmen sraegy π in differen scenarios. We have o solve he BSDE (3.8). Le us commen how he soluion (V,P 1 ), which we need o define he opimal invesmen sraegy and he cerainy equivalen, can be derived numerically, see Chaper 5.1 in Delong (2013) for deails. Firs, we inroduce a pariion 0 = 0 < 1 <... < i <... < n = T of he ime inerval [0,T] wih a ime sep h. Nex, he soluion can be defined by he recursive relaion: V j (T) = e αg(t)a(j), j = 1,2, P 1,j ( i ) = 1 [V h E J(i+1 )( i+1 ) ( W 1 ( i+1 ) W 1 ( i ) ) G( i ) = g,j( i ) = j], j = 1,2, i = 0,...,n 1, [ V j ( i ) = E V J(i+1 )( i+1 ) ( { + min απµ(j)v J(i+1 )( i+1 )+ 1 } 0 π K 2 2 α2 π 2 σ 2 (j)v J(i+1 )( i+1 ) απσ(j)p 1,j ( i ) ) ] αγgv J(i+1 )( i+1 ) h G( i ) = g,j( i ) = j, j = 1,2, i = 0,...,n 1, (4.1) see Bouchard and Elie (2008). Finally, he expecaions in (4.1) are esimaed by he Leas Squares Mone Carlo mehod, i.e. are esimaed by fiing regression polynomials a each poin ( i ) i=0,...,n 1 wih a dependen variable G( i ) based on a generaed sample of (G( i ),J( i ) i=1,...,n, see Longsaff and Schwarz (2001). In Tables 3-4 we find he cerainy equivalens CE = 1 ln(v(0))+ 1 α α lne[eαf ], he expeced excess wealh E[X π (T) F] and he expeced replacemen raios E[X π (T)/F] compued for diverse parameer combinaions. We have he following observaions: The higher he invesmen limi K 2 or he higher he fracion γ of he salary conribued ino he pension fund, he higher he cerainy equivalen, he expeced 25
µ G (1) = 0.03 µ G (1) = 0.04 µ G (1) = 0.06 CE 3.722 3.706 3.688 σ G (2) = 0.04 σ G (2) = 0.06 σ G (2) = 0.1 CE 3.043 3.722 7.462 λ(2) = 1 λ(2) = 2 λ(2) = 3 CE 3.675 3.722 3.788 a(2) = 20 a(2) = 22 a(2) = 25 CE 3.312 3.722 4.386 K 2 = 30 K 2 = 60 K 2 = 100 CE 3.060 3.722 3.851 γ = 0.1 γ = 0.2 γ = 0.3 CE 3.722 4.849 5.875 ρ = 0.1 ρ = 0.5 ρ = 0.9 CE 2.149 3.722 6.081 α = 0.01 α = 0.1 α = 0.15 CE 3.257 3.722 7.396 Table 3: The cerainy equivalens under differen parameer combinaions. Oher parameers are specified in Tables 1-2. excess wealh and he expeced replacemen raio. This conclusion is obvious. If he pension plan manager has less consrains on he invesmen policy and more conribuions o inves, hen he can apply he sraegy which is closer o he opimal unconsrained sraegy, he can opimally inves more funds in he marke and he can achieve a higher erminal wealh (he arge paymen is no affeced in his scenario). The higher he ransiion inensiy λ(2) from economic recession o economic boom, he higher he cerainy equivalen, he expeced excess wealh and he expeced replacemen raio. We can noice ha a higher ransiion inensiy λ(2) implies ha he economy recovers faser from he recession and, consequenly, he pension plan manager receives higher conribuions and earns higher reurns on he pension fund over longer ime periods. Moreover, under a higher ransiion inensiy λ(2) here is a lower probabiliy ha he economy will be in recession a he ime of reiremen 26
µ G (1) = 0.03 µ G (1) = 0.04 µ G (1) = 0.06 E[X π (T) F] -8.140-9.766-13.043 E[X π (T)/F] 0.964 0.956 0.942 σ G (2) = 0.04 σ G (2) = 0.06 σ G (2) = 0.1 E[X π (T) F] -8.145-8.140-8.404 E[X π (T)/F] 0.963 0.964 0.964 λ(2) = 1 λ(2) = 2 λ(2) = 3 E[X π (T) F] -10.127-8.140-6.714 E[X π (T)/F] 0.955 0.964 0.970 a(2) = 20 a(2) = 22 a(2) = 25 E[X π (T) F] -1.635-8.140-17.919 E[X π (T)/F] 0.993 0.964 0.928 K 2 = 30 K 2 = 60 K 2 = 100 E[X π (T) F] -9.151-8.140-7.706 E[X π (T)/F] 0.959 0.964 0.966 γ = 0.1 γ = 0.2 γ = 0.3 E[X π (T) F] -8.140-7.079-6.067 E[X π (T)/F] 0.964 0.969 0.974 ρ = 0.1 ρ = 0.5 ρ = 0.9 E[X π (T) F] -8.392-8.140-8.091 E[X π (T)/F] 0.961 0.964 0.964 α = 0.01 α = 0.1 α = 0.15 E[X π (T) F] -7.874-8.140-8.576 E[X π (T)/F] 0.964 0.964 0.961 Table 4: The expeced excess wealh and he expeced replacemen raios under differen parameer combinaions. Oher parameers are specified in Tables 1-2. 27
and he erminal wealh is more likely o be compared wih he arge paymen coningen on he lower annuiy a(1). However, a higher ransiion inensiy λ(2) also implies ha he arge paymen which is coningen on he salary process is likely o be higher since he salary processes rises in he economic boom. We observe ha he increase in he conribuions and he reurns of he pension fund in he economic boom compensae he increase in he arge paymen. The higher he annuiy facor a(2), he lower he expeced excess wealh and he expeced replacemen raio. This agrees wih inuiion since he erminal wealh is compared wih a higher arge paymen. However, i is imporan o realize ha in our model he arge paymen affecs he opimal invesmen sraegy since π depends on he soluion (V,P 1 ) o he BSDE (3.8) wih he erminal condiion F. Wecanconcludehaahigherargepaymen F, whichcanbepariallyhedgedwih he risky asse, forces he pension plan manager o inves more in he risky asse in order o achieve a higher erminal wealh and compensae he pension beneficiary for he decrease in he uiliy resuling in a higher arge paymen. Wih such an inerpreaion we can now jusify he observaion ha he higher he annuiy facor a(2) is, he higher he cerainy equivalen is. In spie of a higher wealh achieved by he pension plan manager, he higher arge paymen decreases he expeced excess wealh and he expeced replacemen raio as already noiced. Thehigherhevolailiyσ G ofhesalaryprocess, hehigherhecerainyequivalen. A higher volailiy σ G increases he chance ha more conribuions will be invesed in he pension fund, he erminal wealh will be higher and he arge paymen will be higher. Using he inerpreaions we previously deduced, we can conclude ha he cerainy equivalen should indeed increase in σ G. On he oher side, noice ha a higher volailiy σ G also increases he chance ha less conribuions will be invesed in he pension fund, he erminal wealh will be lower and he arge paymen will be lower. The enire effec of he volailiy σ G of he salary process on 28
he expeced excess wealh and he expeced replacemen raio is negligible. The higher he correlaion coefficien ρ beween he radeable risky asse and nonradeable salary process, he higher he cerainy equivalen, he expeced excess wealh and he expeced replacemen raio. The sronger he risky asse is correlaed wih he salary process, he easier i is for he pension plan manager o replicae he arge paymen coningen on he salary process and o achieve a higher excess wealh, and mos imporanly, a higher erminal wealh. The impac of he correlaion coefficien on he expeced excess wealh and he expeced replacemen raio is small. However, i is also worh poining ou ha he higher he correlaion coefficien ρ is, he lower he variance of he excess wealh is. The sandard deviaions for he excess wealh are 12.049, 11.423, 9.740 for ρ = 0.1, 0.5, 0.9. This observaion indicaes an obvious conclusion ha he invesmen porfolio hedges he arge paymen more effecively if ρ is higher. The higher he risk aversion coefficien α, he lower he expeced excess wealh and he expeced replacemen raio. This paern agrees wih inuiion. The more risk averse he pension plan manager, he smaller amoun he invess in he risky asse, and consequenly, he is likely o achieve a lower erminal wealh (he arge paymen is no affeced in his scenario). The higher he risk aversion coefficien α, he higher he cerainy equivalen. This is surprising a he firs sigh in he view of he fac ha he expeced erminal wealh under he opimal invesmen sraegy decreases in α (since he amoun invesed in he risky asse is lower for a more risk averse manager). This is rue for a given arge paymen F. However, in order o achieve he opimal value funcion V(0), which is used o define he cerainy equivalen, he pension plan manager opimally adjuss he invesmen sraegy o follow he arge paymen F, whereas he cerainy equivalen is no invesed in he marke. Hence, i is more likely ha a shorfall in he excess wealh CE F arises, which is more severe in erms of he uiliy for higher risk aversion coefficiens. In order 29
o compensae a more severe shorfall in he excess wealh CE F for higher risk aversion coefficiens, a higher cerainy equivalen is required for higher α. The higher he drif µ G of he salary process, he lower he expeced excess wealh and he expeced replacemen raio. The higher drif µ G implies ha he amouns conribued o he pension fund are likely o be higher and he arge paymen, which is coningen on he salary process, is likely o be higher. I seems ha he increase in he conribuions does no compensae he increase in he arge paymen. Hence, he expeced excess wealh and he expeced replacemen raio decreases in µ G. We also observe hahehigher hedrifµ G ofhesalaryprocess, helower hecerainy equivalen. Thisisnoclearinheviewofhefachaheexpeced erminalwealh under heopimalinvesmen sraegyincreases inµ G (since moreconribuionsflow ino he pension fund). Ye, he cerainy equivalen does no solely depend on he expeced erminal wealh under he opimal invesmen sraegy, i also depends on he arge paymen. As previously noiced, he excess wealh decreases in µ G and, consequenly, heopimalvaluev(0)increases inµ G. Thepay-offE[e αf ]increases in µ G and he oal oucome is ha he cerainy equivalen 1 ln(v(0))+ 1 α α lne[eαf ] decreases in µ G. We remark ha he impac of he drif of he salary process on he cerainy equivalen is small. 5 Conclusion This paper looks ino he invesmen behavior of a defined conribuion pension plan in an economy wih macroeconomic risks. We have considered an economy which can be in one of I saes and swiches randomly beween hose saes. We have found he opimal invesmen sraegy which maximizes he expeced exponenial uiliy of he discouned excess wealh over a arge paymen, e.g. a arge lifeime annuiy. The opimal value funcion of our opimizaion problem (he cerainy equivalen) and he opimal inves- 30
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