Faculy of Acuarial Science and Insurance Acuarial Research Paper No. 86 Opimal Funding and Invesmen Sraegies in Defined Conribuion Pension Plans under Epsein-Zin Uiliy David Blake Douglas Wrigh Yumeng Zhang Ocober 2008 ISBN 978--905752-5-7 Cass Business School 06 Bunhill Row London ECY 8TZ Tel +44 (0)20 7040 8470 www.cass.ciy.ac.uk
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Opimal Funding and Invesmen Sraegies in Defined Conribuion Pension Plans under Epsein-Zin Uiliy David Blake +, Douglas Wrigh and Yumeng Zhang # Ocober 2008 Absrac A defined conribuion pension plan allows consumpion o be redisribued from he plan member s working life o reiremen in a manner ha is consisen wih he member s personal preferences. The plan s opimal funding and invesmen sraegies herefore depend on he desired paern of consumpion over he lifeime of he member. We invesigae hese sraegies under he assumpion ha he member has an Epsein-Zin uiliy funcion, which allows a separaion beween risk aversion and he elasiciy of ineremporal subsiuion, and we also ake ino accoun he member s human capial. We show ha a sochasic lifesyling approach, wih an iniial high weigh in equiy-ype invesmens and a gradual swich ino bond-ype invesmens as he reiremen dae approaches is an opimal invesmen sraegy. In addiion, he opimal conribuion rae each year is no consan over he life of he plan bu reflecs rade-offs beween he desire for curren consumpion, beques and reiremen savings moives a differen sages in he life cycle, changes in human capial over he life cycle, and aiude o risk. Key words: defined conribuion pension plan, funding sraegy, invesmen sraegy, Epsein-Zin uiliy + Professor David Blake, Direcor of he Pensions Insiue, Cass Business School, Ciy Universiy; Dr. Douglas Wrigh, Senior Lecurer, Faculy of Acuarial Science and Insurance, Cass Business School, Ciy Universiy; # Yumeng Zhang, Faculy of Acuarial Science and Insurance, Cass Business School, Ciy Universiy, y.zhang- 5@ciy.ac.uk. The auhors are graeful o he Insiue of Acuaries for financial suppor.
Opimal Funding and Invesmen Sraegies in Defined Conribuion Pension Plans under Epsein-Zin Uiliy Inroducion. The role of he pension plan in allocaing consumpion across he life cycle A ypical individual s life cycle consiss of a period of work followed by a period of reiremen. Individuals herefore need o reallocae consumpion from heir working life when he lifeime s income is earned o reiremen when here migh be no oher resources available, excep possibly a subsisence level of suppor from he sae. A defined conribuion (DC) pension plan can achieve his reallocaion in a way ha is consisen wih he preferences of he individual plan member. There are hree key preferences o ake ino accoun. The firs relaes o he desire o smooh consumpion across differen saes of he naure in any given ime period. The second relaes o he desire o smooh consumpion across differen ime periods. Saving for reiremen involves he sacrifice of cerain consumpion oday in exchange for, generally, uncerain consumpion in he fuure. This uncerainy arises because boh fuure labour income and he reurns on he asses in which reiremen savings are invesed are uncerain. The plan member herefore needs o form a view on boh he rade-off beween consumpion in differen saes of naure in he same ime period and he rade-off beween consumpion in differen ime periods. Aiudes o hese rade-offs will influence he opimal funding and invesmen sraegies of he pension plan. In a DC pension plan, he member allocaes par of he labour income earned each year o he pension plan in he form of a conribuion and, hus, builds up a pension fund prior o reiremen. Then, on reiremen, he member uses a proporion of he accumulaed pension fund o purchase a life annuiy. The decisions regarding he conribuion rae 2
each year before reiremen (i.e., he funding sraegy) and he annuiisaion raio (i.e., he proporion of he fund a reiremen ha is used o purchase a life annuiy) are boh driven by he member s preference beween curren and fuure consumpion, as well as he desire o leave a beques, he hird key preference ha we need o ake ino accoun. Should he member die before reiremen, he enire accumulaed pension fund will be available o beques; afer reiremen, only ha par of he residual pension fund ha has no been eiher annuiised or spen can be bequesed. The invesmen sraegy (i.e., he decision abou how o inves he accumulaed fund across he major asse caegories, such as equiies and bonds) will influence he volailiy of he pension fund and, hence, consumpion in differen ime periods, and so will be influenced by he member s aiude o ha volailiy. In his paper, we invesigae he opimal funding and invesmen sraegies in a DC pension plan. To do his, we use a model ha differs radically from exising sudies in his field in hree key respecs. The firs key feaure of he model is he assumpion of Epsein-Zin recursive preferences by he plan member. This enables us o separae relaive risk aversion (RRA) and he elasiciy of ineremporal subsiuion (EIS). Risk aversion is relaed o he desire o sabilise consumpion across differen saes of naure in a given ime period (e.g., an individual wih a high degree of risk aversion wishes o avoid consumpion uncerainy in ha period, and, in paricular, a reducion in consumpion in an unfavourable sae of naure) and he EIS measures he desire o smooh consumpion over ime (e.g., an individual wih a low EIS wishes o avoid consumpion volailiy over ime, and, in paricular, a reducion in consumpion relaive o he previous ime period). 2 Thus, risk This research focuses on he invesmen and funding sraegies for a DC plan during he accumulaion sage and he only form of saving we allow is pension saving. Non-pension saving, housing-relaed invesmens and pos-reiremen invesmen sraegies are beyond he scope of his sudy. 2 The EIS is defined as 3
aversion and EIS are concepually disinc and, ideally, should be parameerised separaely. In his paper, we consider four differen ypes of member according o differen RRA and EIS combinaions, as shown in Table.. Table. Pension plan member ypes High RRA (risk averse) Low RRA (risk oleran) Low EIS (likes consumpion smoohing) risk-averse member who dislikes consumpion volailiy over ime low equiy allocaion, paricularly as reiremen approaches e.g., low income member wih dependans risk-oleran member who dislikes consumpion volailiy over ime high equiy allocaion a all ages e.g., low income member wihou dependans High EIS (acceps consumpion volailiy) risk-averse member who does no mind consumpion volailiy over ime low equiy allocaion, paricularly as reiremen approaches e.g., high income member wih dependans risk-oleran member who does no mind consumpion volailiy over ime high equiy allocaion a all ages e.g., high income member wihou dependans ϕ ( / c ) d c 2 / c ( ) ( ) U ( c ) / U ( c 2) c is consumpion in period and ( ) c ( 2) ( ) ( ) d ln c / c 2 = = d U c / U c 2 d ln ( / 2 ) U c U c where U c is he marginal uiliy of c, ec. The sign and size of he EIS reflecs he relaionship beween he subsiuion effec and income effec of a shock o a sae variable, such as an increase in he risk-free ineres rae. The subsiuion effec is always negaive, since curren consumpion decreases when he risk-free rae increases because fuure consumpion becomes relaively cheap and his encourages an increase in savings. The income effec will be posiive if an increase in he risk-free rae (which induces an increase in wealh) leads o an increase in curren consumpion; i will be negaive oherwise. If he income effec dominaes, he EIS will be negaive and an increase in he risk-free rae leads o an increase in curren consumpion. If he subsiuion effec dominaes (which is he usual assumpion), he EIS will be posiive and an increase in he risk-free rae leads o a decrease in curren consumpion. If he income and subsiuion effecs are of equal and opposie sign, he EIS will be zero and curren consumpion will no change in response o an increase in he risk-free rae: in oher words, consumpion will be smooh over ime in response o ineres rae volailiy., 4
Wihin he commonly used power uiliy framework, he coefficien of relaive risk aversion (RRA) is he reciprocal of EIS (see, for example, Campbell and Viceira (2002)). This resricion has been criicised because i does no reflec empirical observaions. For example, based on he consumpion capial asse pricing model, 3 Schwarz and Torous (999) disenangle hese wo conceps using he erm srucure of asse reurns. Using US daa, heir bes esimae for RRA is 5.65 (wih a sandard error of 0.22) and heir bes esimae of he IES is 0.226 (wih a sandard error of 0.008). Thus, a high RRA is associaed wih a low level of EIS, bu he esimaed parameer values do no have he reciprocal relaionship assumed by power uiliy. Blackburn (2006) also rejecs he reciprocal relaionship on he basis of a ime series of RRA and EIS parameers esimaed from observed S&P 500 opion prices for a range of differen expiry daes beween 996 and 2003. 4 The second key feaure of he model is he recogniion ha he opimal invesmen sraegy will depend no jus on he properies of he available financial asses, bu also on he plan member s human capial. A commonly used invesmen sraegy in DC pension plans is deerminisic lifesyling. Wih his sraegy, he pension fund is invesed enirely in high risk asses, such as equiies, when he member is young. Then, a some arbirary dae prior o reiremen (e.g., 0 years), he asses are swiched gradually (and usually linearly) ino lower risk asses such as bonds and cash. However, here has been no srong empirical evidence o dae demonsraing ha his is an opimal sraegy. If equiy reurns are assumed o be mean revering over ime, hen he lifesyle sraegy of holding he enire fund in equiies for an exended period prior o reiremen may be jusified, as he volailiy of equiy reurns can be expeced o decay over ime (as a resul of he effec of ime diversificaion ). However, here is mixed empirical evidence abou wheher equiy reurns are genuinely mean revering: Blake (996), Lo and Mackinley (988) and Poerba and Summers (988) find supporing evidence for he UK and US, 3 Breeden s 979 exension of he radiional CAPM which esimaes fuure asse prices based on aggregae consumpion raher han he reurn on he marke porfolio. 4 In paricular, Blackburn (2006) found ha, over he period 996 o 2003, he level of risk aversion changed dramaically whils he level of elasiciy of ineremporal subsiuion sayed reasonably consan. 5
while Howie and Davies (2002) and Kim e al (99) find lile suppor for he proposiion in he same counries. We would herefore no wish an opimal invesmen sraegy o rely on he assumpion of mean reversion holding rue in pracice. A more appropriae jusificaion for a lifesyle invesmen sraegy comes from recognising he imporance of human capial in individual financial planning. Human capial (i.e., he ne presen value of an individual s fuure labour income) can be inerpreed as a bond-like asse in which fuure labour income is he dividend on he individual s implici holding of human capial. Young pension plan members herefore implicily have a significan holding of bond-like asses and, hus, should weigh he financial elemen of heir overall porfolio owards equiy-ype asses. 5 Bu o dae, here has been no quaniaive research exploring he human capial dimension in a DC pension framework. This paper presens an ineremporal model o solve he life-cycle asse allocaion problem for a DC pension plan member. The model assumes wo asses (a risky equiy fund and a risk-free cash fund), a consan invesmen opporuniy se (i.e., a consan reurn on he risk-free asse, and a consan expeced reurn and volailiy on he risky asse) and sochasic labour income. We consider wo aspecs of labour income risk: he volailiy of labour income and he correlaion beween labour income and equiy reurns which deermines he exen o which labour income affecs porfolio choice (e.g., a posiive correlaion reduces he opimal asse allocaion o equiies). The hird key feaure of he model concerns he annuiisaion decision a reiremen. A member wih a srong beques savings moive will no wish o annuiise all he accumulaed pension wealh. In our model, he member chooses o annuiise a proporion of he accumulaed pension fund a reiremen by buying a life annuiy which will generae a reurn linked o bonds. We denoe his proporion he annuiisaion raio. This raio is chosen o maximise he expeced uiliy level a reiremen when annuiy 5 Noe his argumen migh no be appropriae for more enrepreneurial individuals whose paern of fuure labour income growh corresponds more o equiy han o bonds. 6
income replaces labour income. The member invess he residual wealh ha is no annuiised in higher reurning asses in line wih he RRA. The member can draw an income from he residual wealh o enhance consumpion in reiremen, bu, unlike he life annuiy, he residual wealh can be bequesed when he individual dies. Before considering he model in more deail, we will review Epsein-Zin uiliy..2 Epsein-Zin uiliy The classical dynamic asse allocaion opimisaion model was inroduced by Meron (969, 97), and shows how o consruc and analyse opimal dynamic models under uncerainy. Ignoring labour income, in a single risky asse and consan invesmen opporuniy seing, he opimal porfolio weigh in he risky asse for an invesor wih a power uiliy funcion U ( W ) ( W) γ = ( ) coefficien of relaive risk aversion) is given by: µ α = [] γσ 2 γ (where W is wealh and γ is he where µ and 2 σ are he excess reurn on he risky asse and he variance of he reurn on he risky asse, respecively. The invesmen opporuniy se is assumed o be consan. Equaion [] is appropriae for a single-period myopic invesor, raher han a long-erm invesor such as a pension plan member. Insead of focusing on he level of wealh iself, long-erm invesors focus on he consumpion sream ha can be financed by a given level of wealh. As described by Campbell and Viceira (2002, p37), hey consume ou of wealh and derive uiliy from consumpion raher han wealh. Consequenly, curren saving and invesmen decisions are driven by preferences beween curren and fuure consumpion. 7
To accoun for his, Epsein and Zin (989) proposed he following discree-ime recursive uiliy funcion, 6 which has become a sandard ool in ineremporal invesmen models, bu has no hihero been applied o pension plans: ϕ ϕ γ = + ϕ γ ( β) β ( + ) V C E V [2] where V is he uiliy level a ime, β is he individual s personal discoun facor for each year, C is he consumpion level a ime, γ is he coefficien of relaive risk aversion (RRA), and ϕ is he elasiciy of ineremporal subsiuion (EIS). The recursive preference srucure in [2] is helpful in wo ways: firs, i allows a muliperiod decision problem o be reduced o a series of one-period problems (from ime o ime + ); and second, as menioned previously, i enables us o separae RRA and EIS. Ignoring labour income, for an invesor wih Epsein-Zin uiliy, here is an analyical soluion 7 for he opimal porfolio weigh in he risky asse given by: ( R V ) µ cov, α = + γσ γ σ + + 2 2 [3] This shows ha he demand for he risky asse is based on he weighed average of wo componens. The firs componen is he shor-erm demand for he risky asse (or myopic demand, in he sense ha he invesor is focused on wealh in he nex period). The second componen is he ineremporal hedging demand, which is deermined by he 6 Recursive uiliy preferences focus on he rade-off beween curren-period uiliy and he uiliy o be derived from all fuure periods. Kreps and Poreus (978) firs developed a generalised iso-elasic uiliy funcion which disinguishes aiudes o risk from behaviour oward ineremporal subsiuion. Following he KP uiliy funcion, Epsein and Zin (989, 99) proposed a discree-ime recursive uiliy funcion ha allows he separaion of he risk aversion parameer from he EIS parameer. Duffie and Epsein (992) hen exended he Epsein-Zin discree recursive uiliy in a coninuous-ime form called a sochasic differenial uiliy (SDU) funcion. 7 For more deails, see Meron (973) and Campbell and Viceira (2002). 8
covariance of he risky asse reurn wih he invesor s uiliy per uni of wealh over ime. Thus, ignoring labour income, he opimal porfolio weighs are consan over ime, provided ha he invesmen opporuniy se remains consan over ime (i.e., µ = µ and 2 2 σ= σ in [3]). In a realisic life-cycle saving and invesmen model, however, labour income canno be ignored. I is risky and canno be capialised and raded. Bu, allowing for labour income volailiy in he opimisaion process means ha an analyical soluion for he opimal asse allocaion canno be obained. To address his, he recen lieraure has employed a number of numerical mehods 8 o approximae he soluion of he dynamic porfolio opimisaion problem. In he presence of income risk, he opimal porfolio weigh in he risky asse is no consan, bu insead follows a lifesyle sraegy, as shown by Coco e al. (2005). This can be explained as follows: human capial or wealh can be hough of as he expeced ne presen value (NPV) of fuure labour income. Thus, an individual s labour income can be seen as he dividend on he individual s implici holding of human capial. The raio of human o financial wealh 9 is a crucial deerminan of he life-cycle porfolio composiion. In early life, as shown in Figure., his raio is large since he individual has had lile ime o accumulae financial wealh and expecs o receive labour income for many years. Given ha long-erm average labour income growh is of a similar order of magniude as average long-run ineres raes in he UK over he las cenury, as explained in Cairns e al. (2006), labour income can be hough of as an implici subsiue o invesing in he 8 By far, he mos popular approach is value funcion ieraion. Specifically, his involves he discreisaion of he sae variables by seing up a sandard equally-spaced grid and solves he opimisaion for each grid poin a he nex-o-las ime period. The expecaion erm in he resuling Bellman equaion is approximaed by using quadraure inegraion and hen he dynamic opimisaion problem can be solved by backward recursion. I is possible ha he accumulaed sae variable values from he previous ime period are no represened by a grid poin, in which case, an inerpolaion mehod (e.g., bilinear, cubic spline, ec.) mus be employed o approximae he value funcion. However, his approach requires knowledge of he disribuion of each of he shocks o he process, so ha appropriae quadraure inegraion (e.g., Gauss quadraure) can be used. Furhermore, his approach canno handle a large number of sae variables. To overcome hese limiaions, Brand e al. (2005) proposed a simulaion mehod based on he recursive use of approximaed opimal porfolio weighs. The idea is o esimae asse reurn momens using a large number of simulaed sample pahs, and hen o approximae he value funcion using a Taylor series expansion. If he reurn is pah-dependen, i is necessary o regress he reurn variable on he simulaed sae variables from previous ime period, before using he Taylor expansion wih condiional reurn momens. 9 In our model, he only source of financial wealh is pension wealh, and we use hese erms inerchangeably. 9
risk-free asse. Thus, younger individuals have a significan holding in his non-radable risk-free asse and, herefore, should allocae mos of heir financial wealh o he risky asse o keep he overall porfolio composiion consan, as suggesed by Equaion [3] above. As hey grow older, individuals accumulae more financial wealh and draw down human capial. 0 They should herefore rebalance heir financial porfolio owards riskfree asses as age increases. Figure. Decomposiion of oal wealh over he life cycle More recenly, life-cycle asse allocaion models wih a sochasic labour income process have been exended o include he use of a recursive uiliy funcion o allow a separaion of RRA and EIS by, for insance, Weil (989) and Campbell and Viceira (2002). By including a fixed firs-ime risky-asse enry cos and adoping Epsein-Zin uiliy, Gomes and Michaelides (2005) presen a life-cycle asse allocaion model o explain he 0 In our model, he value of human capial will be zero a he end of age 65. The salary is assumed o be paid a he sar of he year, so he human capial a 65 in Figure. is equal o he individual s final salary. 0
empirical observaions of low sock marke paricipaion and moderae equiy holdings for paricipans. Turning o DC pension plans, mos of he exising lieraure invesigaes heir opimal dynamic asse allocaion sraegy by assuming a fixed conribuion rae (e.g., 0% of salary per annum prior o reiremen) and maximising he uiliy of he replacemen raio (i.e., pension as a proporion of final salary) a reiremen (for example, Cairns e al. (2006)) or by minimising he expeced presen value of oal disuiliy 2 prior o reiremen (for example, Haberman and Vigna (2002)). The EIS is implicily assumed o be zero and here is no faciliy for adjusing he conribuion rae in response o changes in salary level or in asse performance. However, in pracice, mos DC plans allow members o make addiional volunary conribuions, and ofen se upper and lower limis on he conribuion rae per annum. Our aim in his sudy is o invesigae he opimal asse allocaion sraegy for a DC plan member wih Epsein-Zin uiliy, so ha an individual member s invesmen sraegy depends on he paern of preferred consumpion levels over he member s enire lifeime. We also derive he opimal profile of conribuion raes over he accumulaion sage of a DC plan. The res of he paper is srucured as follows. Secion 2 oulines he discreeime model wih Epsein-Zin uiliy, including he parameer calibraion process and opimisaion mehod used. In Secion 3, we generae simulaions of he wo key sae variables (i.e., wealh and labour income) and derive he opimal funding and invesmen sraegies for he DC pension plan; we also conduc a sensiiviy analysis of he resuls. Secion 4 concludes. Gomes and Michaelides (2005, page 87) argue ha he less risk-averse invesors have a weaker incenive o pay he fixed enry cos of equiy invesmen, and herefore sock marke paricipans in aggregae end o be more risk averse. 2 The disuiliy is normally defined using he deviaion of acual fund level from inerim and final arge fund levels.
2 The model 2. The model srucure and opimisaion problem We propose a wo-asse discree-ime model wih a consan invesmen opporuniy se. To ensure conformiy wih a DC pension plan, a number of consrains need o be specified: (i) pension wealh can never be negaive, (ii) in any year prior o reiremen, consumpion mus be lower han labour income, (iii) shor selling of asses is no allowed, (iv) members are no allowed o borrow from fuure conribuions. 3 Members are assumed o join he pension plan a age 20 (denoed ime = 0 below) wihou bringing in any ransfer value from a previous plan and he reiremen age is fixed a 65. We work in ime unis of one year and members are assumed o live o a maximum age ofω= 20. 2.. Preferences We assume he plan member possesses he discree-ime recursive uiliy funcion proposed by Epsein and Zin (989): ϕ ϕ γ γ W 20+ + ϕ γ 20+ ( β 20+ ) 20+ β 20+ 20+ 20+ + ( b V = + + 20+ ) p C E p V p b [4] γ 3 Some sudies have assumed ha he member can borrow from fuure conribuions (i.e., o incorporae a loan in he porfolio equal o he presen value of fuure conribuions). In his way, Boulier e al. (200) and Cairns e al. (2006) invesigae he opimal asse allocaion of DC pension plan wih guaraneed benefi proecion. However, here are argumens agains he use of his assumpion. In mos cases, his would no be allowed in pracice. Also, he loan amoun depends on assumpions abou he level of fuure conribuions and, in pracice, here can be a lo of uncerainy abou fuure conribuions. 2
where V 20+ is he uiliy level a ime (or age 20+ ), W 20+ is he wealh level a ime, C 20+ is he consumpion level a ime, γ is he coefficien of relaive risk aversion (RRA), ϕ is he elasiciy of ineremporal subsiuion (EIS), β is he discoun facor for each year, and p 20+ is he one-year survival probabiliy a ime (i.e., he probabiliy ha a member who is alive age 20+ survives o age 20+ + ). The parameer b is he beques inensiy and deermines he srengh of he beques moive. If a member dies during he year of age 20+ o 20+ +, he deceased member will give he remaining wealh a he end of he year, W 20 ++, a uiliy measure of γ ( ) ( ) b W b γ. Thus, a higher value of b implies ha he member has a 20++ sronger desire o beques wealh on deah. In he final year of age ( ω, ω), where we have p 0, equaion [4] reduces o: ϕ ϕ γ γ ω = Wω ϕ ω ω β b V = C + E ω b [5] γ which provides he erminal condiion for he uiliy funcion. 2..2 Financial asses We assume ha here are wo underlying asses in which he pension plan can inves: (i) a risk-free asse (i.e., a cash fund), and 3
(ii) a risky asse (i.e., an equiy fund). The risk-free asse yields a consan rae of ineres r, and he reurn on he risky asse in year is given by: R = r + µ + ε [6] 20+ 20+ where µ is he (consan) risk premium on he risky asse, and ε 20+ = σ Z,20+, where σ is he (consan) volailiy of he risky asse and Z,20+ is an independen and idenically disribued (iid) sandard Normal random variable Whils no necessarily corresponding wih he real world, he simplified assumpion of iid reurns on he risky asse considerably faciliaes he numerical mehod used. 2..3 Labour and pension income Before reiremen, he member receives an annual salary a he sar of each year and conribues a proporion π of his ino he pension plan a ime. We adop he sochasic labour income process used in Cairns e al. (2006) which is illusraed in Figure 2.. The growh rae in labour income prior o reiremen is given by: S S I = r + + σ Z + σ Z 20+ + 20+ 20+ I,20+ 2 2,20+ S20+ where r I is he long-erm average annual real rae of salary growh (reflecing produciviy growh in he economy as a whole), S 20+ is he career salary profile (CSP), or salary scale, a ime, so ha he erm ( ) S S S reflecs he promoional salary increase beween ime 20++ 20+ 20+ and ime +, σ represens he volailiy of a shock ha is correlaed wih equiy reurns, σ 2 represens he volailiy of he annual rae of salary growh, and [7] 4
Z,20+ 2 is an iid sandard Normal random variable. Equaions (6) and (7) are subjec o a common sochasic shock, Z,20+, implying ha he correlaion beween he growh rae in labour income and equiy reurns is given by ( 2 + 2 ) σ σ σ. 2 Figure 2. Labour income process Following he work of Blake e al. (2007), we use a quadraic funcion o model he CSP: 2 4 S20+ = + h + + h 2 + + 3* [8] 45 45 45 On reiremen a age 65, he member is assumed o annuiise a proporion k of he accumulaed pension fund by buying a life annuiy, where k, he annuiisaion raio, is chosen o maximise he member s uiliy level a reiremen. The amoun of annuiy income received depends on he accumulaed wealh level a reiremen, he annuiisaion raio and he price of a life annuiy. In his model, he price of a life annuiy is calculaed 5
using he risk-free reurn on he cash fund, so i is fixed over ime and no annuiy risk is considered. Afer reiremen, he member invess he residual wealh ha is no annuiised. Reiremen income herefore comes from wo sources: he annuiy and possible wihdrawals from he residual fund unil deah. 2..4 Wealh accumulaion Before reiremen, he growh in he member s pension wealh will depend on he invesmen sraegy adoped, he invesmen reurns on boh he risk-free asse and he risky asse, and he chosen conribuion rae. The conribuion rae a ime is given by = ( ) π Y C Y (for 0 44 ), 20+ 20+ 20+ 20+ where Y 20+ is he labour income level a ime. We require he conribuion rae o be non-negaive, so ha Y20+ C 20+ before reiremen. The conribuion rae is allowed o vary over ime, so ha consumpion in any period can adjus o changes in income level and invesmen performance. We also need o impose he resricion W 20 0 (for 0 00 ), o ensure ha he + wealh level is always non-negaive a each age over he life cycle. A proporion, α 20 +, of he member s pension accoun is assumed o be invesed in he risky asse a ime. Then, for 0 43 (i.e., up o and including he year prior o reiremen), we have he following recursive relaionship for he wealh process: ( π ) α ( µ ε ) W20+ + = W20+ + 20+ Y20+ + r + 20+ + 20+ [9] As menioned above, we assume ha ha shor selling of asses is no allowed and herefore impose he resricion ha 0 α 20 +. 6
A he sar of he year in which he member is aged beween 64 and 65, he member receives he final salary paymen and makes he final conribuion o he pension fund. So, we have: ( π ) α ( µ ε ) W 65 = W 64+ Y 64 64 + r + 64 + 64 [0] A he end of his year, he member reires and chooses he annuiisaion raio k, giving a residual wealh on reiremen a exac age 65 of ( ) W = k W. The annuiisaion 65 65 raio k is chosen o maximise he uiliy level a reiremen. This conrol variable does no appear in he uiliy funcion, bu raher in he wealh consrain in he reiremen year. Afer reiremen, he member invess a proporion α 20 + (for 45 ) of he residual wealh (i.e., ha which was no annuiised) in he risky asse and receives annuiy paymens raher han labour income a he sar of each year, provided ha he member is sill alive. This implies ha he recursive relaionship for he wealh accumulaion process a his sage of he member s life cycle is given by: k W 65 W20+ + = W20+ + C20+ + r+ α 20+ ( µ + ε 20+ ) aɺɺ 65 where aɺ ɺ 65 is he price of a life annuiy a age 65 (and, hence, k W 65 aɺɺ 65 represens he annual annuiy income afer reiremen). [] Finally, we mus consrain consumpion afer reiremen such ha ( ) C W + k W aɺɺ. 20+ 20+ 65 65 2..5 The opimisaion problem and soluion mehod The model has hree conrol variables: he asse allocaion a ime, α 20 +, he consumpion level a ime, C 20+, and he annuiisaion raio a reiremen, k. 7
The opimisaion problem is hen: subjec o he following consrains: (i) for 0 43, we have: max ( ) E V 20+ α 20 +, C20 +, k a) a wealh accumulaion process saisfying: ( π ) α ( µ ε ) W20+ + = W20+ + 20+ Y20+ + r + 20+ + 20+ 0, b) an allocaion o he risky asse saisfying 0 α 20, and c) a conribuion rae saisfying π 20 + 0; + (ii) for = 44, we have: a) a wealh accumulaion process saisfying: ( ) ( ) ( π ) α ( µ ε ) W 65 = k W 64+ Y 64 64 + r + 64 + 64 0, b) an allocaion o he risky asse saisfying 0 α64, c) a conribuion rae saisfyingπ 64 0, and d) an annuiisaion raio a age 65 saisfying 0 k ; (iii) and, for 45, we have: a) a wealh accumulaion process saisfying: k W 65 W20+ + = W20+ + C20+ + r+ α 20+ ( µ + ε 20+ ) 0, aɺɺ 65 b) an allocaion o he risky asse saisfying 0 α 20, and + k W 65 c) consumpion saisfying C20+ W20+ +. aɺɺ 65 8
The Bellman equaion a ime is: (, ) J W Y 20+ 20+ 20+ W ϕ γ max ( β 20 ) 20 β + + 20+ 20+ 20+ + ( b = p C + E p J + p20+ ) b α20+, C20+, k γ ϕ ϕ γ γ 20+ + [2] An analyical soluion o his problem does no exis, because here is no explici soluion for he expecaion erm in he above expression. Insead, we mus use a numerical soluion mehod o maximise he value funcion and derive he opimal conrol parameers. We use he erminal uiliy funcion a age 20 o compue he corresponding value funcion for he previous period and ierae his procedure backwards, following a sandard dynamic programming sraegy. To avoid choosing a local maximum, we discreise he conrol variables (i.e., asse allocaion, consumpion and annuiisaion raio) ino equally spaced grids and opimise hem using a sandard grid search. As an imporan sep in implemening he sochasic dynamic programming sraegy, we need o discreise boh he sae space and shocks in he sochasic processes (i.e., equiy reurn and labour income growh) firs. Wealh and labour income level are discreised ino 30 and 0 evenly-spaced grid poins, respecively, in he compuaion. 4 Also, he shocks in boh he equiy reurn and labour income growh processes are discreised ino 9 nodes. 5 4 Clearly, he choice concerning he number of nodes is subjecive, bu we fel ha his choice represens an appropriae rade-off beween accuracy and speed of compuaion. 5 Again, 9 nodes represens a balance beween accuracy and compuing ime, and is a sandard seing in he exising lieraure. 9
The expeced uiliy level a ime is hen compued using hese nodes and he relevan weighs aached o each (i.e., Gauss quadraure weighs and inerpolaion nodes). 6 The advanage of using his se of nodes is ha he sae variables can be compued more quickly and precisely; however, because we have a fine grid on he conrol variables and a much coarser grid on he shocks, we may have some sae variable values ouside of he grid poins in he nex ime period. In his case, cubic spline inerpolaion is employed o approximae he value funcion. While his approach does no significanly reduce he accuracy of he resuls obained, use of a much finer grid for he shocks in he equiy reurn and labour income growh processes would significanly increase compuing ime, as menioned previously. For each age 20+ prior o he erminal age of 20, we compue he maximum value funcion and he opimal values for he conrol variables a each grid poin. Subsiuing hese values in he Bellman equaion, we obain he value funcion of his period, which is hen used o solve he maximisaion problem for he previous ime period. Deails of he dynamic programming and inegraion process are given in Appendix. The compuaions were performed in MATLAB. 7 2.2 Parameer calibraion We begin wih a sandard se of baseline parameer values (all expressed in real erms) presened in Table 2.. The consan ne real ineres rae, r, is se a 2% p.a., while, for he equiy reurn process, we consider a mean equiy premium, µ, of 4% p.a. and a sandard deviaion, σ, of 20% p.a.. Using an equiy risk premium of 4% p.a. (as opposed o he hisorical average of around 6%) is a common choice in recen lieraure (e.g., Fama and French (2002), Gomes and Michaelides (2005)). This more cauious assumpion reflecs he fac ha he hisorical equiy risk premium migh be higher han can reasonably be expeced in fuure, and hus will reduce he weigh given o equiies in he opimal porfolios obained. We 6 For more deails, see Judd (998, page 257-266). 7 hp://www.mahworks.com/producs/malab/. The code is available on reques from he auhors. 20
use he projeced PMA92 able 8 as he sandard male moraliy able, and hence, using a (real) ineres rae of 2% p.a., he price of a whole life annuiy from age 65 is 5.87. Asse reurns Table 2. Baseline parameer values Preference parameers risk-free rae, r 0.02 RRA, γ 5 equiy premium, µ 0.04 EIS, ϕ 0.2 volailiy, σ 0.2 beques inensiy, b discoun facor, β 0.96 Moraliy Labour income process moraliy able PMA92 saring salary, Y 20 average salary growh, r I 0.02 Annuiy volailiy of shock correlaed wih equiy reurns, σ 0.05 annuiy price, ɺɺa 5.87 volailiy of annual rae of salary 65 (based on growh, σ 2 0.02 PMA92 and a risk-free rae of 0.02) h -0.276 h 0.75835 2 We sar by presening resuls for wha migh be considered as a relaively sandard plan member, wih RRA of γ = 5, EIS of ϕ = 0.2 and discoun facor β = 0.96. 9 As menioned above, he beques inensiy, b, plays an imporan role in life-cycle saving and invesmen. We se b equal o uniy in he baseline case (which represens a moderae level of beques saving moive). We laer conduc a sensiiviy analysis on hese parameer values. The saring salary is normalised on uniy. All absolue wealh and income levels are measured in unis of he saring salary. In line wih pos-war UK experience, he annualised real growh rae of naional average earnings is assumed o be 2% p.a. wih a 8 PMA92 is a moraliy able for male pension annuians in he UK based on experience beween 99 and 994; here, we use he projeced raes for he calendar year 200, i.e., he able PMA92(C200), published by he Coninuous Moraliy Invesigaion (CMI) Bureau in February 2004. 9 This parameer consellaion is common in he lieraure (e.g., Gomes and Michaelides (2004)). The values of RRA and EIS are also consisen wih power uiliy for he baseline case. 2
sandard deviaion of 2% p.a. (i.e., r= I 0.02 and σ 2= 0.02 ). Following he work of Blake e al. (2007), we esimae he CSP parameers h and h 2 using average male salary daa (across all occupaions) repored in he 2005 Annual Survey of Hours and Earnings. The esimaed values are h = 0.276 and h 2 = 0.75835 (see [8]). 3 Resuls 3. Baseline case 3.. Opimal asse allocaion assuming no beques moive or labour income risk As suggesed by equaion [3] above, he opimal porfolio composiion should be consan when here is no beques moive and labour income risk is ignored. Figure 3. shows he opimal equiy weigh for he final ime period (i.e., year of age 9 o 20) wih no beques moive. As expeced, we can see ha when he accumulaed wealh level is large (and labour income is small in comparison), he opimal asse allocaion is close o he resul suggesed in equaion [3], so ha we have: µ cov( R20, V20 ) α = + 0.2 γσ 2 γ σ 2 This resul shows ha we can approximae an analyical soluion numerically reasonably accuraely, hereby jusifying he use of our grid search numerical mehod. 3..2 Simulaion oupu The oupu from he opimisaion exercise is a se of opimal conrol variables (i.e., asse allocaion, α 20 +, and consumpion level, C 20+ ) for each ime period and he opimal annuiisaion raio, k, a reiremen age 65. We generae a series of random variables for boh he equiy reurn and labour income shocks, and hen generae 0,000 independen simulaions of wealh and labour income levels. 22
Figure 3. Opimal equiy allocaion for he final ime period Opimal equiy allocaion 0.8 0.6 0.4 0.2 0 Wealh Annuiy Figure 3.2 shows he simulaion means of labour income and opimal wealh and consumpion levels for ages 20, 2,, 20, and Figure 3.3 shows he consumpion profile on a larger scale. We have beques and reiremen saving moives in his model. In he early years of he life cycle (i.e., up o age 45 or so), wealh accumulaion is driven by he beques inensiy (i.e., he exen of he desire o proec dependans if he member dies) and by he aiude o risk (i.e., he degree of aversion o cuing consumpion in unfavourable saes). 20 Consumpion increases smoohly during his period. Then, as he member ges older, he reiremen moive becomes more imporan as he member recognises he need o build up he pension fund in order o suppor consumpion afer reiremen. From age 45 o he reiremen age of 65, he reiremen savings moive dominaes and he pension fund grows significanly. As a resul, consumpion remains almos consan during his period. 20 This will become clearer in secions 3.2. and 3.2.2. 23
Afer reiremen, here is a large fall in consumpion compared wih he period immediaely prior o reiremen and hereafer consumpion remains sable for he remainder of he member s lifeime. Evidence presened in Banks e al (998) shows ha consumpion ends o fall afer reiremen. Par of his is explained by he fac ha workrelaed expenses such as ravel and clohes no longer need o be incurred. Par is explained by he fac ha precauionary balances need o be mainained o pay for lumpy expendiures such as car repairs or home mainenance, given ha working o pay for hese expenses is no longer an opion. Par is also explained by he fac ha reired people end o say a home more and so spend less on high-cos iems, such as resauran meals and holidays. We do no aemp o model hese complex issues in a formal way. Insead, we ry o capure hem in he pos-reiremen consumpion consrain ( ( )) C W + k W aɺɺ (i.e., for 45 20+ 20+ 65 65 ), given in secion 2..4 above. Consumpion afer reiremen is herefore subjec o a cash-in-hand consrain and canno exceed he sum of pension income and unannuiised wealh. The consrain is igher he higher he fracion, k, of pension wealh he member chooses o annuiize a age 65. Given ha k is a conrol variable, Figure 3.3 shows he opimal reducion in consumpion in reiremen when k is chosen opimally (i.e., o maximise E( V 64) ). The size of he fall in consumpion a reiremen when here is a cash-in-hand consrain will be influenced by he level of non-pension wealh, such as discreionary savings or housing. Since non-pension wealh does no need o be annuiized, i acs as a buffer ha can be used o smooh life-cycle consumpion. Non-pension wealh is ouside he scope of he presen paper. 24
Figure 3.2 Mean of simulaed wealh, consumpion and income Figure 3.3 Mean of consumpion 25
Figure 3.4 shows he expeced NPV of oal fuure labour income (i.e., human capial). We can see ha human capial increases unil abou age 35. This is because of he very high rae of salary growh in he early years (relaive o he discoun rae applied o fuure labour income). Figure 3.4 Human capial Figure 3.5 shows six possible opimal asse allocaion profiles for equiies a each age before reiremen, α 20 +. Each profile coincides wih a paricular quanile from he disribuion of oucomes from 0,000 simulaions.these profiles are consisen wih he invesmen sraegy called sochasic lifesyling, firs oulined in Cairns e al (2006), wih a high equiy weighing a younger ages and a gradual swich from equiies o he risk-free financial asse as he reiremen age approaches. Prior o around age 35, he member should inves all financial wealh in he risky asse, because he implici holding in he non-radable riskless asse (i.e., human capial) is increasing, as illusraed in Figure 3.4. However, afer age 35, human capial sars o decline. The member should hen begin o rebalance he financial wealh porfolio owards he risk-free financial asse 26
o compensae for he decline in human capial. This is because he risk-free financial asse and human capial are subsiues, wih he degree of subsiuabiliy inversely relaed o he correlaion beween labour income growh and equiy reurns, ( 2 + 2 ) σ σ σ. 2 The invesmen sraegy is known as sochasic lifesyling because he opimal equiy weighing over he life cycle depends on he realised oucomes for he sochasic processes driving he sae variables, namely labour income and he risky financial asse. The profiles have a similar shape which can be characerised as hree connecing (and approximaely) linear segmens. The firs is a horizonal segmen involving a 00% equiy weighing (approximaely) from age 20 o an age somewhere in he range of 40-47. The second is a seep downward segmen ha involves a reducion in equiies o somewhere beween 0-40% over a seven year period. The hird is a more genle downward sloping segmen ha reduces he equiy weighing o somewhere beween 0-0% by he reiremen age. I is imporan o noe ha he profiles in Figure 3.5 are no, however, consisen wih he more radiional deerminisic lifesyling sraegy, which involves an iniial high weighing in equiies wih a predeermined linear swich from equiies o cash in he period leading up o reiremen (ypically he preceding 5 or 0 years). 27
Figure 3.5 Opimal equiy allocaion prior o reiremen Figure 3.6 plos six quaniles from he disribuion of opimal conribuion raes, corresponding o he opimal asse allocaion sraegies shown in Figure 3.5. Considering he profile corresponding o he mean, he iniial annual conribuion rae a age 20 is jus under 8% p.a.. I hen decreases seadily o below % by age 35. This fall reflecs a radeoff beween he beques moive and risk aversion, on he one hand, and he increase in human capial, on he oher. Prior o age 35, when labour income is growing very rapidly and human capial is increasing, he member wishes o increase consumpion and does so by reducing he conribuion rae ino he pension plan, despie being boh risk averse and having a beques moive. 2 Afer age 35, however, labour income growh slows down and human capial begins o decline, and he reiremen savings moive sars o become imporan. The conribuion rae hen increases seadily o almos 5% p.a. by age 48. Labour income flaens ou afer age 48 (see Figure 2.) and he conribuion rae hen remains roughly consan unil reiremen. 2 Noe ha pension conribuions will fall by less han he conribuion rae since income is growing. 28
Figure 3.6 Opimal conribuion rae The mos ineresing finding from Figure 3.6 is ha he opimal conribuion rae in a DC pension plan is age-relaed, raher han consan. I exhibis a U-shaped paern beween age 20 and age 48 and i is only (approximaely) consan afer age 48. As a consequence, he conribuion rae is very variable, ranging from, in he case of he mean profile, below % (a age 35) o almos 5% (a ages 48 and above). The oher quaniles have a very similar shape: hey are relaively close o he mean during he U-shaped phase, bu have a wider range of pos-age-48 consan raes (ranging beween 0 and 20%). An age-relaed paern of conribuion raes is no common in real-world DC plans: for example, in he UK, here is ypically a fixed sandard (combined employer and employee) conribuion rae varying beween 8 and 0% (GAD (2006, Table 8.2)). Alhough age-relaed conribuion raes are no common, minimum conribuions are more so. Figure 3.7 illusraes he mean opimal conribuion rae over he life cycle when a lower limi of 5% p.a. is imposed on he conribuion rae (he original unconsrained mean conribuion rae profile is shown for comparison). In his case, he member accumulaes greaer pension wealh when young and, herefore, can afford a lower 29
conribuion rae as reiremen approaches. Furher, because of he higher accumulaed pension wealh, he member swiches o he risk-free asse earlier, as shown in Figure 3.8 (he original unconsrained mean opimal asse allocaion is shown for comparison).. Neverheless, here is a cos from imposing his consrain: expeced uiliy a age 20 drops by around % (from 2.349 o 2.329). Figure 3.7 Mean opimal conribuion rae (wih lower limi on conribuion rae of 5% per annum, shown as doed line) 30
Figure 3.8 Mean opimal equiy allocaion (wih lower limi on conribuion rae of 5% per annum, shown as doed line) 3.2 Sensiiviy analysis In his secion, we conduc a sensiiviy analysis on he key parameers in he model. 3.2. RRA and EIS We begin by examining simulaion resuls for plan members wih differen RRA and EIS. As shown in Table 3. (see also Table.), we have four ypes: low RRA and low EIS (Type ): o risk-oleran member who dislikes consumpion volailiy over ime o e.g., low income member wihou dependans low RRA and high EIS (Type 2): o risk-oleran member who does no mind consumpion volailiy over ime o e.g., high income member wihou dependans 3
high RRA and low EIS (Type 3): o risk-averse member who dislikes consumpion volailiy over ime o e.g., low income member wih dependans high RRA and high EIS (Type 4): o risk-averse member who does no mind consumpion volailiy over ime o e.g., high income member wih dependans The baseline case in Secion 3. deal wih Type 3 (highlighed in Table 3.), a member wih a high RRA and a low EIS (i.e., γ = 5 and ϕ = 0.2 ). Table 3. RRA and EIS values for he differen ypes of plan member RRA, γ EIS, ϕ Type 2 0.2 Type 2 2 0.5 Type 3 5 0.2 Type 4 5 0.5 Figure 3.9 shows he differen paerns of opimal conribuion raes corresponding o hese four ypes. For risk-oleran members wih low RRA (i.e., Types and 2), conribuions prior o age 50 are negligible. From age 50 or so, he reiremen savings moive kicks in and conribuions ino he pension plan begin. The conribuion rae is lower for Type (low EIS) han for Type 2 (high EIS) members (by approximaely 3-4% p.a.). This lower reiremen savings inensiy is he resul of a sronger aversion o cuing consumpion: due o he lower EIS level, cus in consumpion needed o fund he pension plan are more heavily penalised in he uiliy funcion of Type members han of Type 2 members. 32
Figure 3.9 Mean opimal conribuion rae (for differen RRA/EIS combinaions) As a resul of he lower mean conribuion raes (paricularly a younger ages), riskoleran members, ceeris paribus, accumulae a lower level of pension wealh. They herefore need (and are, of course, willing o accep) a much higher average equiy weighing (and he corresponding equiy premium) in he financial porfolio in an aemp o produce he desired level of reiremen savings. As shown in Figure 3.0, he mean equiy allocaion decreases only gradually and sill exceeds 70% a reiremen. The riskoleran member wih a low EIS level (i.e., Type ) chooses a higher equiy weighing a reiremen (by approximaely 0%) han he risk-oleran member wih a high EIS level (i.e., Type 2). This is because, as discussed above, he annual conribuion rae is lower and he reliance on he equiy premium correspondingly greaer. 33
Figure 3.0 Mean opimal equiy allocaion (for differen RRA/EIS combinaions) For risk-averse members wih high RRA (i.e., Types 3 and 4), Figure 3.9 shows ha he beques moive leads o much larger iniial conribuions (in excess of 7% p.a. a age 20 on average) han for members wih low RRA, and he reiremen savings moive leads o a significan rise in conribuions afer age 40, again compared wih low RRA members. For hose risk-averse members wih low EIS (i.e., Type 3), he variabiliy of he conribuion rae over he working life is much less pronounced han is he case for hose risk-averse members wih high EIS (i.e., Type 4). The conribuion rae is also lower paricularly a very young and very old ages (e.g., by around 5% p.a. a boh age 20 and 60 on average). The explanaion is he same as given above (i.e., he relucance of members wih a lower EIS level o olerae cus in curren consumpion o fund fuure consumpion in reiremen). The fall in he conribuion rae o an average of around % beween age 20 and age 35 for risk averse members is explained by he increase in human capial over his period which increases he desire for curren consumpion a he expense of saving for reiremen. 34
As a resul of higher conribuion raes, risk-averse members accumulae a much higher level of pension wealh and, as can be seen in Figure 3.0, swich away from equiies much earlier (from abou age 40) and hold less han 0% of he pension fund in equiies a reiremen on average. For a given level of risk aversion, Figure 3.0 shows ha a lower EIS leads o a higher equiy weighing during he laer sages of he accumulaion phase of he pension plan. A firs sigh, his finding seems counerinuiive: surely individuals wih a lower IES would prefer a lower equiy weighing in heir pension fund and hence more sable conribuions over ime? Consaninides (990), for example, appeals o habi formaion (i.e., he complemenariy beween consumpion in adjacen periods) for inducing a srong desire for sable consumpion and a correspondingly low demand for equiies. However, in our model, he reiremen saving moive is imporan, since he only source of pos-reiremen consumpion in our model comes from he pension plan. During he accumulaion phase, a member wih a low EIS is no willing o reduce curren consumpion in order o increase plan conribuions (o mee he reiremen saving moive), bu is willing o use a higher equiy weighing and he corresponding equiy premium in an aemp o generae he necessary reiremen savings. Figure 3.0 shows ha he reiremen savings effec dominaes he habi formaion effec in deermining he equiy weighing in he laer years of he accumulaion phase (Gomes and Michaelides (2005) derived he same resul). 3.2.2 Beques moive Invesors wih a desire o beques wealh o heir dependans on deah would be expeced o save more han hose who do no. Figure 3. shows he mean opimal paern of conribuion raes for he baseline case of γ = 5 and ϕ = 0.2 and differen beques inensiies. When he beques moive is absen (b = 0), members consume almos all of heir earnings in he early years of heir working lives, and have very negligible conribuions ino heir pension plans. By conras, members wih a high beques inensiy (b = 2.5) make very high conribuions in he early years (4% a age 20). They also 35