Default options: Are the life cycle funds the solution? Mabrouk Chetouane PhD student, University of Paris-Dauphine LedA - SDFi Banque de France

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1 Defaul opions: Are he life cycle funds he soluion? Mabrouk Cheouane PhD suden, Universiy of Paris-Dauphine LedA - SDFi Banque de France December 2010 Absrac To ensure he susainabiliy of pension sysems, many governmens are encouraging heir ciizens o save for reiremen hrough funded DC schemes - boh occupaional and individual. The pension fund developmen across he developed and he emerging counries poses he quesion of he defaul opions. In his paper we aim o address wo key quesions. Firs, do lifecycle funds can be seled as defaul opion? How should individuals inves in asse classes wih differen risk/reurn properies over he lifecycle? We conclude ha lifecycle plans offer aracive feaures considering a mean variance framework as compared o oher defaul sraegies. Looking a he curren marke offer, hese can be srenghened by greaer focus in inflaion linked bonds.

2 Table of Conens 1 Inroducion and moivaion The lieraure The widespread defaul opion soluions The heoreical foundaions of he lifecycle asse allocaion Tesing defaul opions Defaul opions behavior: a simulaion exercise The framework Simulaion framework The risk indicaors The resuls Mone Carlo simulaion exercises Backesing he sraegies The background The resuls of he backess The riskless asse: a key issue for pension invesors A poenial candidae: he inflaion-linked bonds Using an ILB index as he riskless asse Enhancing he discussion: he porfolio insurance sraegies as a buffer Conclusion References Appendix A Daa calibraion Appendix B Sochasic dominance chars Appendix C Addiional resuls Lis of Figures Lis of Tables

3 1 Inroducion and moivaion Aging populaions and low economic performances have eroded he sae-run pay-as-you-go (PAYG) pension sysems. To ensure heir susainabiliy, he governmens are now adjusing he PAYG parameers (reiremen age, level of benefis and level of conribuions) reducing he generosiy of sae run schemes. Forced o reduce he generosiy of PAYG 1 or parially funded defined benefi pension models, many governmens are oday encouraging heir ciizens o save for reiremen hrough funded pension schemes - boh occupaional and individual - o compensae for he lesser fuure pension paymens from he sae sysem. For insance, he Japanese governmen inroduces in 2004 defined conribuion pension srucures wihin a pension sysem exclusively dominaed by defined benefi pensions. Comparable changes have been observed in Europe in paricular for counries where PAYG sysems are well developed as in Germany or France 2. For insance, he Briish governmen should inroduce he Naional Pension Saving Schemes in 2012 in which employees are forced o save for reiremen funding purposes unless hey are already enrolled ino an occupaional pension scheme (Cheouane, 2008). The funded pension plans have become a cenral building block in he redesign of pension sysems. According o he OECD pension saisics, oal pension asses in he OECD area sood a jus under 70% of he OECD GDP in Given a rapidly aging populaion and growing recourse o funding (pension reserves, ec) - no jus in he OECD bu a he global level oo - his marke is se o grow srongly in he coming years. Basically, wo ypes of funded pension schemes are proposed o employees in which o inves heir pension saving. On he one hand, he defined benefi pension plans (DB plans hereafer) gaher pension plans sponsored by a company and in which he pension benefi is promised o he fuure reiree, and depends on a se of parameers. On he oher hand, he defined conribuion pension plans - occupaional or individual - (DC plans hereafer) are pension schemes in which he pension benefi paid is a funcion of he amoun conribued and financial performances. The key difference beween boh pension schemes concerns he risk managemen. Regarding DC pension schemes, all he risks are borne by he individual invesors while hey were suppored by he sponsoring companies wihin a DB framework. The so-called "pension sorm" of increased longeviy, disappoining capial marke reurns, low long-erm ineres raes and a igher regulaory environmen has damaged he global pension landscape. In his conex, companies have seadily desered DB plans in favor of DC srucures. This shif owards funded DC plans raises imporan issues in erms of pension saving invesmens for individuals. 1 PAYG sysems provide a defined benefi sream and can hence be described as an unfunded (or noional) defined benefi plan (NDB). While a discussion hereof lies ouside he scope of his paper, i is ineresing o noe ha in addiion o he rend o shif o funded DC plans; noional defined conribuion (NDC) plans are also winning ground as a building block for pension reform. The resuling shif in risks shares a number of common poins wih he discussion above. 2 There are no auhenic DC pension plans in France bu invesmen vehicles which ac as DC plans. 3

4 Four risks underlying he pension paymen can be idenified: he longeviy risk, he human capial risk, he inflaion risk and he invesmen risk. Taking ino accoun marke informaion asymmeries and misaligned ineres raes, he risk is ha he DC model could resul in boh expensive and subopimal choices. A op of he lis of risks ransferred is he longeviy risk. Longeviy risk in reiremen is a socalled "pure" risk o he exen ha i involves a probabiliy of loss (he cos of living longer) bu wih no chance for a gain (assuming ha he individual is no longer earning any labor income). While he "law of large numbers" allows a cerain level of predicabiliy of longeviy for a oal populaion, a he level of he individual i is almos impossible o predic wihin any meaningful range. Hence, replacing a DB sysem by a DC sysem hus removes he "insurance" philosophy of he former. Turning o he second risk, we define human capial as he discouned value of all fuure labor income. In a funded DC plan, conribuions are defined as a share of labor income. As such, if here is a loss of labor income, for example due o a period of unemploymen, conribuions o he plan and hus fuure benefis will, all else being equal, be reduced. Noe ha o he exen ha he benefis of occupaional DB plans are based on labor income, he shif from occupaional DB o occupaional DC plans does no involve any ransfer of human capial risk, which was already carried by he plan beneficiary. The key variable for any pension plan beneficiary is no he nominal amoun of he pension paymen, bu he purchasing power hereof, his is he inflaion risk. In he firs insance, he reference measure for proecing fuure pension paymens is indexaion o consumer prices. In he second insance, he reference measure is wage inflaion. DB plans ofen offer some degree of proecion o boh risks o he exen ha pension benefis are ofen calculaed on he basis of a erminal salary, and hen indexed o consumer prices during reiremen 3. For DC plans, here is no explici inflaion proecion and as such his risk, ha can also be considered a "pure" risk, is fully ransferred o he individual. Invesmen risk disinguishes iself from longeviy, human capial and inflaion risks in ha i is a "speculaive" risk as opposed o a "pure" risk, and can hus yield eiher a gain or a loss. For funded DB plans - eiher in he sae or privae secor regime - he invesmen risk is carried fully by he plan sponsor, bu is ransferred o he individual under DC schemes. The quesion of how o manage he invesmen risk lies a he core of he discussion in he following secions. As seen from he discussion above, he move o funded DC schemes enails an imporan shif of risks o he individual. Individuals enrolled in a DC plan have o choose, according o heir risk aversion and heir ime invesmen horizon, an invesmen sraegy wihin a menu offered by he plan. Focusing on he invesmen choice process, behavioral finance sudies bring ineresing insighs o undersand he invesor psychology when he faces a large panel of choices. On he one hand, Shea and Madrian (2001) underline he role played by ineria wihin he reiremen plan invesmen decisions. This resul echoes hose found by Michell and Ukus (2004) who show ha only 10% of he Vanguard Group paricipans adjus heir conribuions allocaion every year. Besides, Lavigne and Legros (2005) underline ha individuals facing a se of pension invesmen soluion usually choose he defaul opion. Beshears e al. (2007) poin ou ha a significan proporion of pension invesors choose he defaul opion among he oher fund soluions 3 Pracices of cos-of-living adjusmen for pension benefis vary grealy from counry o counry and in some insance from plan o plan. 4

5 offered. In he Unied Saes, 80% of individuals enrolled in a DC pension plans inves heir pension savings in his way. Similar proporion has been found in Ausralia (Basu & Drew, 2006). In he same vein Garnier and Thesmar (2009) repor he Swedish experience where governmen inroduced a funding pension sysem consraining he employees o save a lile fracion of heir compensaion. The pension plan members can choose beween a governmen reiremen funds and privae pension funds. A large majoriy of new paricipans op for he defaul opion (83% in 2001 and 96% in 2006). In addiion, hey reveal ha he defaul opion fund performances were much higher han hose proposed by asse managers. In his conex, he iniial invesmen decision plays a subsanial role as i deermines, in mos cases, he remaining asse allocaion over he enire saving horizon. In ligh of hese resuls, i appears ha fuure reirees are unable o make a reasoned invesmen choice. In addiion, he OCDE/IORP insiuions poin ou ha he lack of financial lieracy of individual invesors leads hem o make inadequae invesmen choices. This reinforces he necessiy for fund managers, or o a greaer exen for he regulaor, o supervise he pension plan paricipans in heir porfolio choice. In his conex he defaul opion choice is crucial. The invesmen menu proposed by DC plans always comprises a defaul opion which in mos cases corresponds o lifecycle funds. The lifecycle fund allocaes pension savings according o he age of he invesor. In oher words, he share of socks decreases as he invesor becomes older. The pension wealh is hen auo-piloed unil he reiremen dae. Taking ino accoun he success of he lifecycle soluion and he amoun of asses invesed for reiremen pensions, we ask how robus his pension soluion is, as compared o oher long erm invesmen sraegies? In oher words, does a lifecycle asse allocaion sraegy offer a superior risk-reurn oucome compared o oher defaul opions? Besides, he quesion of he riskless asse is a burning issue in building long erm invesmen soluions. Can he radiional long erm governmen bond yield fulfill his role or should we swich o inflaion linked bonds? In addiion, can oher invesmen mehods, such as porfolio insurance, be considered in his framework? Based on a Mone Carlo simulaion and backess and calibraed on he American financial markes, our empirical analysis pus forward he role of consan mix and lifecycle funds. Considering he risk olerance of he invesor and in ligh of several risk indicaors, aggressive lifecycle and consan mix sraegies can be seen as poenial candidaes for he defaul opion. While pure equiy funds always ouperform all oher defaul opions, he risk associaed wih his invesmen soluion is oo high o be considered as a defaul opion. The resuls also confirm he risk inheren o pure fixed income soluions in erms of replacemen rae. We find also ha auopilo sraegies provide beer oucome as compared o naive sraegies (equally weighed porfolio). These resuls are enhanced by he inroducion of inflaion-linked bonds. The inclusion of inflaion bond characerisics in he aggressive lifecycle and consan mix sraegies lead o replacemen raes comparable o hose of he pure equiy sraegies. Bu he risk inheren o hese sraegies is much lower han he pure equiy one. Finally, he firs aemps made o include a porfolio 5

6 insurance approach do no lead o conclusive resuls. Furher researches have o be performed in his field. The srucure of his paper is as follow. In secion 2, we review he heoreical and empirical sudies dealing wih he relevance of a defaul opion. In secion 3, we presen he background required for he Mone Carlo and he backess exercises. We also presen he performance indicaor which will be used o compare he invesmen sraegies. In secion 4, we repor and discuss he resuls. In secion 5, we will appreciae how relevan is he use of inflaion-linked bonds as he riskless asse. Likewise, he use of a porfolio insurance framework will also be considered. The las secion concludes. 2 The lieraure The recen figures of he Survey of Consumer and Finance (released in 2007), show ha reiremen is he firs saving moivaion of American households. Meanwhile, Thaler and Sunsein (2008) noice ha a large fracion of American households consider ha heir saving effor is insufficien o face reiremen issues. In his conex, he pension marke is expeced o gain ground over he coming years. This would encourage or reinforce he developmen of differen ypes of defaul pension soluions whose aim is o answer o he differen needs of he fuure reirees. 2.1 The widespread defaul opion soluions Benarzi and Thaler (2001), observe ha DC plan members porfolio choices do no follow he message delivered by he financial heory. They noice ha pension plan paricipans affec heir pension wealh in a naive way. Windcliff and Boyle (2005) pu forward a heurisic diversificaion rule called he 1/n pension invesmen puzzle. This simple rule describes a naive asse allocaion where he invesor breaks down is porfolio in as many equal proporions as here are available invesmen asse classes geing hus an equally weighed porfolio. Under a se of several condiions, he auhors show ha he equally weighed porfolio can proec he invesor agains very bad oucomes. When DC plan members are no able o choose an invesmen vehicle, he pension funds or he asse managemen indusry are obliged o propose defaul opions in heir panel of invesmen soluions. One of he mos widespread invesmen vehicle provided by he financial or he pension indusry is he consan mix soluion. The consan mix porfolio soluion is a paricular case of buy and hold sraegies in which he proporions invesed ino risky and safer asses remain consan over he invesmen horizon. Basically, consan mix sraegies involve socks and bonds, assuming ha bonds is he safer asse and equiies he risky asse. Le α be he fracion of risky asses in he porfolio, S he sock price index, B he riskless asse price index and W he financial wealh of he pension fund member. Hence he dynamics of W given by: é ê êë æ S B ö ù ç ( ) ú úû W = W- 1ê 1+ ç α + 1-α. S- 1 B çè ú -1ø 6

7 Buy and hold or consan mix sraegies are based on asse class relaive performances observed over a long holding period. Gollier (2007) sudies he relaionship beween risky asse reurns and he holding horizon. He demonsraes ha long erm invesors should heavily inves on socks due o he negaive relaionship beween risk and sockholding. Likewise, Bec and Gollier (2007) consider his issue sudying he French case. Their invesigaions indicae ha he risk associaed wih equiies is acually significanly higher han he one of oher asses for shor invesmen horizons, bu i becomes saisically undisinguishable from he bonds rolled risk for horizons longer han 5 quarers and from he bonds held o mauriy afer around six years. This resul gives addiional suppor o he Samuelson criicism. Based on a fallacious lecure of he Law of Large Numbers, Samuelson indicaes ha he repeiion of an invesmen paern over many periods does no cause risk o wash ou in he long run (Gollier & Zeckhauser, 1997). In his conex, invesors can mainain a high exposure o equiies whaever heir posiion in he lifecycle. This resul receives addiional suppor considering he performance of riskless asses compared o equiies. Indeed, using a large sample (from 1802 o 2006), Siegel (2007) shows ha American sock reurns are on he one hand higher han he reurns of reasury bonds and cash asses and on he oher hand very sable, whaever he sub-period considered. Furhermore, Siegel (2007) signals ha real fixed income reurns were negaive on average beween 1946 and Davis (1995) also noes ha, beween 1967 and 1990, he sandard deviaion of fixed income invesmen reurns was close o he one of equiy reurns. Using he value a risk indicaor, Blake e al. (2004) show ha conservaive approaches increase he probabiliy of losses. Despie he heoreical and empirical evidences, his issue is sill subjec o conroversy. Bodie (2001) ackles his quesion by considering he coss of hedging agains he risk of lower earnings compared o risk free ineres rae. Bodie shows ha he riskiness of equiies increases as he invesmen horizon increases, wheher he sock reurns are mean revering or no. Enhancing he analysis of Bodie (2001), Lankassen and Propper (2007) find similar resuls. In addiion, he auhors poin ou he fac ha hese sudies neglec wo fundamenal aspecs regarding long erm invesors: he role played by he human capial and he link beween invesor age and risk aversion (addiional references are given in he nex secion regarding his wo issues). The inroducion of hese parameers in he long erm invesmen soluion design has opened he way for he developmen of lifecycle or arge dae soluions. A his sage, Malkiel (1996) proposes a simple rule which reconnecs he invesor s age o he porfolio choice over he lifecycle. Broadly speaking, an individual s equiy exposure should be equal o 100 minus heir age. For example, a 35 year old person holds 65% of he "risky" equiy porfolio and 35% of he "safe" fixed income porfolio. These producs have grown in populariy in recen years. In he US, 48.5% of 401(k) plans offered a lifesyling fund in up from a mere 12.1% in De faco, he lifecycle funds offer, proposed o DC plan members, has been diversified wih more or less pruden invesmen sraegies. The following chars give an illusraion of lifecycle funds soluion offered o 401(k) plan members in he Unied Saes. 7

8 Fracion of asses wihin he lifecycle sraegy 100 Fracion of socks: smoohed lifecycle Fracion of asses wihin he lifecycle sraegy Fracion of risk less asse:smoohed lifecycle 75 Fracion of socks: Pruen lifecycle Fracion of risk less asse: Pruen lifecycle Invesmen duraion 0 Invesmen duraion Figure 1: Smoohed and pruden lifecycle invesmen sraegy 2.2 The heoreical foundaions of he lifecycle asse allocaion Meron (1969) and Samuelson (1969) 4 showed ha under several resricive assumpions, he opimal porfolio srucure is independen of age. However, he inroducion of background risks, of less resricive assumpions on sock reurn characerisics, of addiional consrains such as liquidiy consrains, are differen ways used by he modern porfolio model o connec invesor age and is asse allocaion decision. Considering an ineremporal porfolio choice model, he inroducion of he purchasing power risk leads he opimal invesor o reduce is equiy exposure in favor of indexed invesmen producs. Likewise, he inroducion of mean revering sock reurns implies ha invesors may benefi from mean revering sock reurns. In his conex, younger invesors have o be aggressively invesed on socks and should reduce progressively heir equiy exposure as hey are aging. In addiion, Bodie, Meron and Samuelson (1992) inroduce human capial as a sae variable wihin a sochasic ineremporal framework. They find ha he fracion of an individual's wealh opimally invesed in equiy should decline wih age for wo reasons: young invesors can adjus heir labor supply wih greaer flexibiliy and heir human capial is greaer han ha of older persons and is generally less risky han many financial asses. While earnings and sock reurns are orhogonal in he shor run, Colin-Dufresne and Goldsein (2007) show however ha boh variables are coinegraed. In he long run, addiional income can be generaed by subsiuing risky asses o human capial asse. For he same reasons, Bakshi and Chen (1994) suppose he exisence of a lifecycle risk aversion ranslaing he fac ha he invesor s risk aversion changes hroughou his lifeime. In paricular, hey sugges ha risk aversion increases as he invesor is aging. Numerous sudies have addressed he relaionship beween risk aversion and invesor age. The invesigaions performed by Morin and Suarez (1983), Halek and Eisemhauer (2001) and Bellane and Green (2003) pu forward such a relaionship. Thereby, he propensiy o be exposed o risky invesmens is lower for older invesors han for younger ones, giving hus suppor o lifecycle invesmen soluions. A his sage, one may quesion he older invesor invesmen choices. Here again, various auhors addressed his issue using differen daasee and 4 Meron (1969) explored he LPS issue using a coninuous ime model whereas Samuelson (1969) did i in a discree ime framework. 8

9 mehodologies over various counries. Using numerous waves of he Survey of Consumer and Finance, Yoo (1994), Poerba and Samwick (1997), Heaon and Lucas (2000), Berau and McCuller (2000) find ha he fracion of riskless asses held by households increases wih he age of he household head. In oher words, hey highligh he exisence of a hump shaped paern beween he fracion of risky asses and he invesor s age profile. Comparable resuls have been found using he TIAA-CREFF survey (Bodie & Crane, 1997) and he daase exraced from he individual reiremen accouns asse allocaion (401k) daase (Agnew, Bladuzzi & Sunden, 2003; Holden & Vanderhei, 2009). 2.3 Tesing defaul opions Considering a mean-variance framework, Buler and Domian (1991) use simulaed daa and show ha equiy invesmens provide higher oucomes han lifecycle sraegies. To discriminae beween a se of financial vehicles, Kim and Wong (1997)use he sochasic dominance measuremen. They conclude ha he opimal sraegy mus be aggressively invesed in socks. In he same vein, Hickman e al. (2001) compare he performance of he 100 minus age rule o basic equiy funds. The simple rule only ouperforms he equiy index in 15% of cases. Likewise, Poerba e al. (2006) analyze he oucome disribuion of lifecycle and consan mix invesmen vehicles. They do no observe significan differences beween boh disribuions. In ligh of hese resuls, one could conclude ha lifecycle sraegies are no relevan as he defaul opion for long erm invesors. However, Hiber and Mowbray (2002) demonsrae ha lifecycle invesmen sraegies lead o a reducion of he oucome sandard deviaion. In addiion, Bodie and Trussard (2007) recommend he implemenaion of a safe arge dae fund which maches he pruden invesor. The Bodie and Trussard (2007) resuls confirm ha he one size fis all principle canno be applied wihou generaing subopimal siuaions 5. Anolin e al. (2009) signal ha DC pension schemes have experienced huge financial losses due o he financial downurn. In paricular, hey observe ha he financial crisis his aggressive invesmen sraegies. From an hisorical daabase, he auhor es differen defaul opions and reveal ha lifecycle sraegies, wih relaively high equiy exposure, offer higher replacemen raes han consan mix sraegies. Beyond his firs resul, hey insis on he fac ha lifecycle sraegies show higher resisance during financial downurn episodes. Accouning for he size of he DC pension schemes, and he increasing number of paricipans, Anolin e al. (2009) recommend redesigning he pension fund invesmen regulaion o avoid older workers and reirees being exposed o such risks. 3 Defaul opions behavior: a simulaion exercise 3.1 The framework We assume ha he fuure reiree begins o save for reiremen a 20 years old and ha he legal pension reiremen age is se a 65. Thus, he invesmen horizon is 45 years and we assume no early reiremen sysem. We consider ha he individual devoes a consan proporion of his earnings o face is 5 Vigna and Harberman (2002) conclude in he same way using a sochasic conrol approach. 9

10 reiremen. While he conribuion rae remains consan over he working period he wage growh rae is expeced o vary hroughou. Acually, he earnings growh rae varies wih he age of he individual. Basically, we observe a rapid increase in earnings a early ages (beween 20 and 34 years) and a progressive slowdown unil he reiremen age. The following char illusraes income earning changes in funcion of age in he Unied Saes average wage profil according he age o o o o o o 64 Figure 2: Unied Saes - Rebased average earning changes per age (sources: BLS) The employee conribues each year o he pension scheme a a fixed and predeermined rae. We assume ha he conribuion rae is se a 10% of he invesor s earnings. In line wih he DC pension plan behavior, he accumulaed conribuions are invesed in he financial markes hroughou financial vehicles seleced by he pensioners. We assume ha he reirees do no move from an invesmen vehicle o anoher during he invesmen period despie he absence of fees or ransacion coss. This assumpion is no limiaive as ineria affecs he household asse allocaion decision. Thus, he seleced invesmen vehicle a he beginning of he process remains exacly he same during 45 years Cumulaed conribuions Annual conribuions Figure 3: Saving and cumulaed conribuions o he DC plans (Sources: BLS and auhor s calculaions)

11 In his conex, he pension plan member financial wealh dynamics can be wrien as follows: i ( ) ù Wi,+ 1 = é êwi, 1+ r ú+ γa, ë û T T-1 i,t = é ù 0 ëê + + iúû + å j( + j ) + T i= 1 j= 1 W W 1 r γa 1 r γa, where W is he financial wealh, r i, he invesmen reurn, he conribuion rae and A he invesor s earning. Having esablished he conribuion profile of a DC plan member and hence his pension wealh, we briefly presen he soluion caegories proposed by DC schemes. We analyze he pure invesmen soluions, he consan mix soluions, he auopilo invesmen sraegy (or he 100 minus age soluion), he lifecycle funds and finally an invesmen soluion based on a naive approach. The following able gives furher deails regarding he defaul opions esed in his sudy: Iniial equiy exposure Average equiy exposure Average long erm bond exposure Average cash (or shor erm asse) exposure Naive divesificaion Equidiversified porfolio (1/n) Aggessive consan mix wih nominal bonds 70% 70% 30% - Pruden consan mix wih nominal bonds 40% 40% 60% - Vulgae "100 minus age" w/nominal bonds 75% 52% 48% - Aggressive life syling w/nominal bonds 100% 73% 27% - Pruden life syling w/nominal bonds 100% 22% 78% - Smoohed life syling w/nominal bonds* 100% 56% 44% - Pure equiies 100% 100% - - Pure nominal bonds 0% - 100% - Pure Money marke 0% % Table 1: Invesmen sraegies sample To perform our sudy, we need o simulae a sock price index, a cash index and finally a long erm bond index. The following secion is dedicaed o he presenaion of he insrumens used o simulae he pension defaul opions. 3.2 Simulaion framework We assume ha asse reurns are randomly disribued which implies no serial correlaion. Moreover, as we consider low frequency daa, namely, annual daa, we can hus assume ha sock reurns are no characerized by mean reversion (Basu, 2009). We asser in addiion ha financial asse real prices and yields can be represened by sochasic processes. In his conex, we choose o use a Mone Carlo ool o simulae a large number of marke daa. The shor erm real ineres rae is modeled as an Ornsein- Uhlenbeck process which corresponds o a mean revering process. This represenaion provides a saisfacory framework o describe real shor erm ineres raes. In our framework, we consider he real 11

12 shor erm ineres as a driver for bond, sock and inflaion linked bond indices. In his conex, he real shor erm ineres rae dynamics is hus wrien as follows: ( ) ( ) dr = a β -r d+ σ dw, r where parameers α, β and σ r are non-negaive and consan and r is he curren level of he real ineres rae. The parameer α represens he mean reversion degree wih which he ineres rae dynamics revers o an average value β. A high value of α ranslaes a rapid reurn of r o he long erm nominal ineres rae. The coefficien σ r is he diffusion of he sochasic process and W() is a Weiner process. The soluion of he previous dynamics is given by: ò ò (-a) -a( -s) -a( -s) = s 0 0 ( ) r r e βa e ds σ e dw β -r d, (-a) (-a) (-a) -a 0 ( ) s 0 r = r e + β 1- e + σ e e dw. ò The discreizaion of he previous relaionship leads o he following expression of he shor erm ineres rae dynamics which will be used o simulae our cash daa: (-2a) 1-e (-a) (-a) = ( ) ( ) r r e β 1 e σ ε. 2a Sock marke price pahs are usually modeled wih Geomerical Brownian Moion processes. Le S be he sock index price a ime. The GBM process and is exac discreizaion are wrien as follows: ds μ d σ dw, S S S = + ì ² æ σ ö ü S = S0expï r í - d+ σ dε ï ý. ç 2 ïè ø î ïþ where μ is a consan which represens he drif of he sochasic differenial equaion and σ is he diffusion associaed o he Brownian moion W(). In addiion, we choose o correlae he equiy marke incremens o he one of he real shor erm ineres rae, using he Cholevski decomposiion. This assumpion implies he following changes in he expression of he GBM: ds S S ( ) = μ d + ρdw + 1-ρ σ dw r 2 S r, ² ì æ σ ö ü r 2 S+ d = S exp ρ r í ï - dε + ( 1-ρ ) σ dε ï ý. ç 2 ï è ø î ïþ 12

13 In he same vein, we link he long erm bond incremens o he real shor erm ineres rae one. Hence, he dynamic of he bond index is given by: db B B ( ) = μ d + ρdw + 1-ρ σ dw r 2 B B, ì ² æ σ ö ü r 2 B B = B0expïρ r í - dε + ( 1-ρ ) σ dε ï ý. ç 2 ï è ø î ïþ The simulaions parameers are calibraed on he Unied Saes financial daa. The main difficuly lies in he choice of he mos relevan window for he calibraion of he processes. Keeping he purpose of his sudy in mind, we finally choose o calibrae our equaions using a rolling sample equal o he holding period, ha is, 45 years. We evaluae he average real rae of reurn and he average sandard deviaion over 45 years rolling periods going from 1850 o 2005 (or 110 overlapping periods), for a sock index, a cash index and a long erm bond index. The daa come from he Global Finance Daa and hey were provided by he Sociéé Générale Asse Managemen. The figures used o calibrae he sochasic process are given in he following able: Real Sock index yield¹ Real Cash index yield Real "nominal bond" index yield 3.3 The risk indicaors Mean 6,2% 1,6% 2,0% Sandard Deviaion 18,5% 4,9% 7,3% ¹ Real sock reurn accoun for real dividends ² 10 years inflaion expecaions are considered o build ILB index yield Table 2: Mone Carlo calibraion To compare he various invesmen vehicles, we use several measures of invesmen risk and reurn. The performance of an invesmen vehicle can be presened in differen ways. In our framework, we measure he invesmen yield by calculaing he number of reiremen years funded, aking he average earning over he working period as he amoun of pension benefi received. Le Γ i,t be he number of years paid by invesmen sraegy i. Then, we have E Γ i,t =. W This measure of he invesmen vehicle performance is debaable as pension fund members compare heir pension benefis o heir las earning. For insance, Poerba e al. (2006), Basu (2009), Anolin e al. (2010) use he reiremen wealh raio, which compares he erminal pension plan member earning o he erminal financial wealh, as a benchmark o evaluae invesmen sraegy performances. Considering his performance indicaor, we calculae he sandard deviaion: i,t 13

14 N 2 1 i = i i N å - i = 1 ( ) σ r μ. Moving on o he measures of risk, i mus be kep in mind ha he sandard deviaion gives he same weigh o negaive and posiive oucomes. To avoid his limiaion, we calculae he lower parial momen and focus in paricular on he semivariance indicaor, which was inroduced by Markoviz in (1959). The semivariance indicaor akes only ino accoun oucomes ha are below he average 6. Knowing he semivariance, we can evaluae he downside risk for each defaul opion. The lower parial momen, he semivariance and he downside risk expressions are given by he wo following relaionships: N N 1 γ 1 2 i = ( ( i i) ) i ( ( i i) ) Nå - = Nå - i= 1 i= 1 Lpm Max 0,r μ SVar Max 0,r μ, N 1 i = i i N å - i = 1 ( ( )) dws Max 0,r μ. To evaluae he probabiliy of losses, asse managers refer o he Value a Risk indicaor. For a predeermined level, he Value a Risk measures he poenial losses given an invesmen horizon. Assuming a Gaussian disribuion, he Value a Risk, for he risk hreshold α is given by: ( ) VaR h, p = ασ, i p h,i where h is he holding period. We admi however ha he use of he Value a Risk over such horizons is subjec o criicisms (see Blake e al., 2004, for addiional deails regarding his issue). In a second sage, we use relaive measures of risk and performance in order o build a ranking of he sudied invesmen vehicles. We sar by calculaing he Sharpe raio defined (Sharpe, 1966) by he following quaniy: ( κi, - Rf ) Sr i =, σ where R f is he risk free rae of reurn and, he invesmen reurn. In he same vein, we use he Roy raio (1952) in which he invesmen sraegy reurn is compared o a benchmark. When we consider he number of years of reiremen funded by he invesmen sraegies as our performance indicaor, he arge used as he minimum accepable reurn (MAR) is he life expecancy a 65 years old. The las figures of he Naional Vial Saisics Sysem (from he Ceners for Disease Conrol and Prevenion) show ha he life expecancy a 65, in 2005, reached 18.2 years. We use he performance of he pure long erm bond sraegies as well, as a benchmark o calculae he Roy raio. Broadly speaking, he Roy raio is given by: Γ i, ( Γ -Mar ) i, Roy i, =. σ Γ i, 2 6 Noe however ha he semivariance indicaor weighs idenically exreme losses and sligh losses. In oher words, his measure does no discriminae beween financial crashes and downwards financial markes for insance. 14

15 Invesmen sraegies can also be seleced using heir empirical cumulaed disribuion funcion. Using he sochasic dominance properies, we can deermine graphically he invesmen sraegy which sochasically dominaes he ohers. Le S i and S j be wo invesmen sraegies. X sochasically dominaes Y a he firs order if: ò b E éu( S ) u ( w) df ( w) Eéu( S ) ù ë ù= û ê ú= u ( w) df ( w ). ë û i Si 1 j Sj a a The firs order sochasic dominance may appear resricive in mos cases. Hence, we consider he second order sochasic dominance definiion. S i sochasically dominaes S j a he second order if: w w ò Si ò a a ò () () Î[ ] F s ds F s ds for w a,b. Sj In he coming secions, we will presen and discuss he resuls from he Mone Carlo simulaions using boh radiional measures and he indicaors menioned above. In a second sage, we presen he resuls derived from he backess using he same indicaors. 4 The resuls 4.1 Mone Carlo simulaion exercises As we consider rolling periods, we generae 300 pahs for each marke variables geing hus ime series, leading o observaions for each invesmen sraegies. The Mone Carlo simulaions show, wihou any surprise, ha he porfolio enirely invesed in equiies finances he highes number of reiremen years (Table 3). Secondly, i can be observed ha invesmen sraegies based on a heavy exposure o sock markes lead o higher oucomes and ouperform he oher defaul opions. For insance, an invesor who invess his conribuion in equiy invesmen vehicles over he enire working period generaes a erminal financial wealh which corresponds o 15 years of earnings. The simulaions indicae ha aggressive lifecycle invesmen vehicles provide beer oucomes compared o he aggressive consan mix sraegy. Besides, he performance of he auopilo or linear lifecycle sraegy is very close he aggressive consan mix one which is unexpeced as he proporion of bonds in much lower han in he previous one. b 15

16 Mean Min Max Quarile 1 Quarile 3 Naive divesificaion 8,18 1,89 24,55 5,99 10,00 Pure nominal bonds 6,93 0,92 42,41 3,58 8,51 Pure Money marke 6,97 1,28 23,75 4,96 8,49 Pure equiies 14,57 1,64 144,90 8,17 18,61 Aggressive consan mix wih nominal bonds 10,97 1,62 55,26 7,05 13,67 Pruden consan mix wih nominal bonds 8,74 1,29 36,04 5,51 10,95 Aggressive life syling w/nominal bonds 12,83 1,64 144,90 7,83 16,21 Smoohed life syling w/nominal bonds 9,86 1,64 55,57 6,59 12,12 Pruden life syling w/nominal bonds 7,83 1,62 36,32 4,86 9,56 Vulgae "100 minus age" w/nominal bonds 10,25 1,64 50,00 6,81 12,66 Table 3: Mone Carlo Simulaion resuls for he invesmen vehicles in erms of years of reiremen funded This resul suggess ha here is a benefi in ime diversificaion. Excluding equiies, pure invesmen sraegies provide he lowes performance in erms of funded years of reiremen. Naïve diversificaion of he porfolio, frequenly used as he las resor invesmen vehicle by pension plan members, presens reurns comparable o pruden approaches. Furhermore, he pure long erm bond soluion provides he wors oucome. Looking closer a he oucome disribuion, we can noice ha he dispersion of he firs quarile is small while i is much more pronounced for he hird quarile Roy raio Roy raio (MAR = life (MAR = Bond expecancy a 65) porfolio asse) Sharpe Raio Naive divesificaion 3,301 0,411 0,241 Aggessive consan mix wih nominal bonds 1,327 0,741 0,248 Pruden consan mix wih nominal bonds 2,151 0,407 0,175 Vulgae "100 minus age" w/nominal bonds 2,038 0,254 0,247 Aggressive life syling w/nominal bonds 1,484 0,822 0,236 Pruden life syling w/nominal bonds 4,543 0,383 0,206 Smoohed life syling w/nominal bonds 1,764 0,620 0,238 Pure equiies 0,391 0,823 0,207 Pure nominal bonds 2,270 Pure Money marke 3,975 0,013 0,109 Table 4: Oher performance measures Mone Carlo simulaions Table 4 gahers relaive performance indicaors, i.e. he Roy raio in which we consider alernaively he life expecancy a 65 and he long erm bond yield as he minimum accepable reurn, and he Sharpe raio 7. While he previous resuls poin ou he weakness of pure long erm bond invesmen sraegies, he resuls derived from he Roy raio calculaions pu his observaion ino perspecive. Acually, he conribuions invesed in pruden lifecycle producs or in a 100% cash opions give he weakes performances while aggressive equiy opions ensure he highes reurn. This normalized indicaor shows 7 The Sharpe raio is calculaed considering he invesmen reurn, in a lieral sense, and he riskless ineres rae a each dae. 16

17 ha hese wo sraegies do no succeed o provide a sufficien financial wealh o mee DC plan member needs over heir remaining lifeime. Considering he bond porfolio asse as a benchmark, he Roy raio (2 nd column) signals he low performance of he auopilo opion. The Sharpe raio, in which he difference beween he defaul opion reurn and he risk free rae (here he cash asse) is normalized by he sandard deviaion, challenges he previous picures. Acually, he Sharpe raio associaed o he pure equiy soluion is subsanially lower han he aggressive consan mix and lifecycle fund. Bu, more surprisingly, he auopilo and he smoohed lifecycle sraegies display a Sharpe raio comparable o he one of he aggressive soluions. In ligh of hese findings, hese invesmen sraegies appear mos relevan for long erm invesors. Sandard deviaion Semi variance Risk shorfall Naive divesificaion 3,036 0,079 0,280 Aggessive consan mix wih nominal bonds 5,447 0,130 0,360 Pruden consan mix wih nominal bonds 4,405 0,065 0,256 Vulgae "100 minus age" w/nominal bonds 4,916 0,079 0,280 Aggressive life syling w/nominal bonds 7,175 0,158 0,398 Pruden life syling w/nominal bonds 2,343 0,062 0,250 Smoohed life syling w/nominal bonds 4,726 0,083 0,287 Pure equiies 9,282 0,231 0,480 Pure nominal bonds 4,964 0,016 0,127 Pure Money marke 2,825 0,076 0,276 Table 5: Risk indicaor based on he Mone Carlo simulaions To refine our judgmen, hese resuls have o be considered in ligh of risk parameers. Moving on o risk indicaors (repored in Table 5), pure equiy and aggressive lifecycle invesmen sraegies presen much higher sandard deviaion han he oher invesmen sraegies. Mos surprisingly, holding pure nominal bond porfolio does no proec he invesor agains bad oucomes, as in Blake (2004). The lowes sandard deviaion is observed for he pruden lifesyling invesmen sraegy. The assessmen is quie differen when he risk indicaor only comprises negaive oucomes. In his conex, he pure long erm bond invesmen vehicle appears o be he safes defaul opion while he pure equiy porfolio is sill he riskies. Comparing now he lifesyling opions o he consan mix approaches in erms of risk, he aggressive consan mix opions show beer figures while risk adverse profiles should prefer naïve diversificaion or smoohed lifesyling porfolios indifferenly. 17

18 Value a risk 10% 5,0% 2,5% 1,0% Naive divesificaion 3,89 5,03 5,86 7,07 Aggessive consan mix wih nominal bonds 6,97 9,02 10,51 12,69 Pruden consan mix wih nominal bonds 5,64 7,29 8,50 10,26 Vulgae "100 minus age" w/nominal bonds 6,29 8,14 9,49 11,45 Aggressive life syling w/nominal bonds 9,18 11,88 13,85 16,72 Pruden life syling w/nominal bonds 5,63 7,29 8,49 10,25 Smoohed life syling w/nominal bonds 6,05 7,82 9,12 11,01 Pure equiies 11,88 15,37 17,91 21,63 Pure nominal bonds 6,35 8,22 9,58 11,57 Pure Money marke 3,62 4,68 5,45 6,58 Table 6: Risk indicaor - Value A risk We finally end he analysis by paying aenion o he Value a Risk measure. The Value a Risk indicaes he probabiliy o see he invesmen opion reurn below a cerain level, considering various confidence levels and rolling periods of 45 years. In oher words, here is a 5% chance of seeing he pure equiy reurns goes below he 15 years level. Likewise, here is a 5% chance of seeing he naïve diversificaion providing a replacemen level equivalen o 5 years of earning. Repored in Table 6, he evaluaions confirm he resuls given by he previous indicaors. A a 5% level of risk, we noe ha he lowes Value a Risk is reached for cash invesmen vehicles whereas he highes is reached for pure equiy. The Value a Risk poins ou he superioriy of lifecycle funds over he consan proporion ones. In ligh of his indicaor, he auopilo sraegy provides beer oucome han he naïve diversificaion and he smoohed lifecycle fund soluions. To conclude, he Mone Carlo simulaions bring ineresing and unexpeced insighs. The simulaion exercises pu in perspecive he superioriy of he pure equiy invesmen soluion. Correced from he risk indicaor, he aggressive lifecycle sraegies provide comparable oucomes o he mos aggressive soluions and higher replacemen levels as compared o he aggressive consan mix. In he same vein, he auopilo lifecycle fund ouperforms pure fixed income sraegies, which provide he lowes oucomes, and he equally weighed porfolio invesmen. Finally, risk oleran and risk adverse invesors can find in he lifecycle funds a relevan answer o he reiremen funding challenges. 4.2 Backesing he sraegies The background Considering he previous invesmen vehicles, we propose o es he relevance of he resuls wih a hisorical daabase which sars in We hus apply each successive 45 year periods of marke hisory o he individual, given he lifeime earning profiles and applying he same consan conribuion rae of 10%. As previously, he labor supply is assumed o be fixed wih no opions o aler pension savings in response o financial marke performances. Furhermore, we do no ake ino accoun he probabiliy o 18

19 be in employmen as i is assumed ha he conribuions are paid every year. The daa used in he model are hisorical daa for he US equiy marke, long-erm nominal governmen bonds and shor-erm money marke bills. The backess are performed on real raes of reurn, calculaed using he American consumer price index. The overall idea is hus o es how he described invesmen sraegies did over all he 45 year periods of marke daa from 1850 o Given ha he final 45 year period sars in 1960, he simulaion covers a oal of 110 overlapping periods. In line wih he previous simulaion exercise, we use a similar indicaor o compare and evaluae he differen invesmen opion oucomes The resuls of he backess Unsurprisingly, he fixed income sraegies offer he safes opion when long erm bonds are used (Table 7). Neverheless, he relaive securiy offered by his sraegy has a cos in erms of reurns as he nominal bond porfolio can only cover seven years of reiremen. However, he money marke opion appears decidedly less safe han inuiion may have suggesed. This is because money marke insrumens provide neiher ineres rae hedge due o heir shor mauriy nor any inflaion hedge. The more surprising oucome of hese simulaions is ha he aggressive life syling sraegy appears superior o he aggressive consan mix in a significan way measured in reurns. Conversely, he pruden lifecycle invesmen is less profiable han he consan mix one. In ligh of hese oucomes, lifecycle approaches could play he role of defaul opions for DC plans members. These backess confirm he superioriy of auopilo sraegies over naive sraegies in erms of replacemen rae as well. This resuls has however o be aken wih cauion as we do no inroduce he risk in our discussion. Mean Min Max Quarile 1 Quarile 3 Sandard deviaion Naive diversificaion 10,04 3,16 19,24 7,04 12,18 4,11 Pure nominal bonds 7,27 1,74 13,91 5,46 9,40 3,12 Pure money marke 7,73 2,21 15,92 4,59 11,18 4,21 Pure equiies 16,57 3,06 37,49 8,18 12,85 9,22 Aggressive consan mix wih nominal bonds 13,59 3,71 26,96 7,98 18,38 6,17 Pruden consan mix wih nominal bonds 10,63 3,03 19,67 7,48 13,29 4,25 Aggressive life syling w/nominal bonds 14,98 3,06 33,32 8,35 20,91 7,46 Smoohed life syling w/nominal bonds 11,82 3,06 24,39 7,78 14,36 4,91 Pruden life syling w/nominal bonds 8,86 3,06 18,01 6,44 11,43 3,62 Auopilo life syling w/nominal bonds 12,50 3,44 24,82 7,82 16,30 5,33 Table 7: Invesmen sraegy backess: Reurns in erms of years of reiremen funded. Looking a he sandard deviaion indicaor, he auopilo sraegy presens, as expeced, a higher risk profile compared o he naive sraegy and he correced indicaors of risk end o confirm his resul. In addiion, we noice ha he auopilo and smoohed lifesyling sraegies are as risky as he aggressive consan mix, even hough heir exposure is significanly lower. This observaion reduces he ineres of such opions for reiremen. Noe moreover ha he resuls highlighed by he Mone Carlo simulaion regarding he pure fixed income are confirmed by he backess exercises. 19

20 Sandard Semi deviaion variance Naive diversificaion 4,11 2,90 1,70 Aggressive consan mix wih nominal bonds 6,17 3,59 1,90 Pruden consan mix wih nominal bonds 4,25 2,70 1,64 Auopilo life syling w/nominal bonds 5,33 3,51 1,87 Aggressive life syling w/nominal bonds 7,46 4,58 2,14 Pruden life syling w/nominal bonds 3,62 2,42 1,56 Smoohed life syling w/nominal bonds 4,91 3,52 1,88 Pure equiies 9,22 5,72 2,39 Pure nominal bonds 3,12 1,95 1,40 Pure money marke 4,21 3,16 1,78 Table 8 : Backess - Alernaive risk measures Downside risk Moving on o he relaive performance indicaors, repored in Table 9, we observe ha whaever he invesmen opion considered, he Roy raio (in he firs column) is negaive. This ranslaes he fac ha here is no invesmen opion which succeeds in generaing a sufficien amoun of financial wealh o ensure an income over he remaining lifeime of DC members. The wo oher indicaors reveal ha boh aggressive consan mix and lifecycle funds hardly ouperform he pure equiy sraegies. Roy raio Roy raio (MAR = life (MAR = Bond expecancy a 65) porfolio asse) Sharpe Raio Naive divesificaion -1,98 0,68 0,75 Aggressive consan mix wih nominal bonds -0,75 1,02 1,20 Pruden consan mix wih nominal bonds -1,78 0,79 0,85 Auopilo life syling w/nominal bonds -1,07 0,98 1,19 Aggressive life syling w/nominal bonds -0,43 1,03 1,21 Pruden consan mix wih nominal bonds -2,58 0,44 0,55 Smoohed life syling w/nominal bonds -1,30 0,93 1,14 Pure equiies -0,18 1,01 1,20 Pure nominal bonds -3,51-0,05 Pure Money marke -2,49 0,11 - Table 9: Backess - alernaive measures of performance The backes exercises confirm he main resuls derived from he pure simulaion exercise. While pure equiy sraegies remain he mos profiable for DC plan members, he backess pu in perspecive heir supremacy as aggressive opions provide comparable reurns, in erms of reiremen years, for a slighly lower level of risk. We noice ye ha hisorical daa give addiional suppor o he fac ha fixed income soluions are no he safes soluions and do no succeed in offering a sufficien income o cover he longeviy risks borne by DC plan paricipans. Keeping in mind he conclusion of Bodie and Trussard (2007) who insis on he one size does no fi all principle, he backward simulaions sugges ha he bes defaul opion soluion for risk adverse invesors is he pruden consan mix approach. On he oher hand, he lifecycle soluion could be an advisable defaul opion in paricular for pension paricipan whih a high risk olerance. 20

21 A his sage, we only consider he radiional asse classes in he consrucion of he defaul opion. In he nex secion, we exend his analysis by discussing he choice of he riskless asse. We subsiue he long erm bond asse by an inflaion-linked bond asse. We also discuss he relevance of porfolio insurance ools and provide preliminary resuls. 5 The riskless asse: a key issue for pension invesors The deerminaion of he riskless asse for he consrucion of invesmen opions for DC plan members remains a key issue 8. Long erm nominal governmen bonds are usually considered as he riskless asse. However, his asse class does no hedge agains he inflaion risk which is one of he main concerns for long erm invesors. Some economiss susain he idea of a new inflaion regime characerized by a higher and/or more volaile inflaion rae. Besides, he recen developmen of he financial markes gives an addiional suppor o his insigh. Indeed, some argues ha he curren moneary policy (Quaniaive Easing and low ineres raes) may lead o higher inflaion pressures. Oherwise, he relaive scarciy of naural resources and o a lesser exen of he labor force should drive price indices up. Accouning for he fac ha long erm invesors are in essence inflaion risk averse, his conex should lead hem o increase he demand for proecion agains he inflaion risk. One possible answer is o consider he inflaionlinked bonds as he riskless asse in he srucuring heir porfolio. 5.1 A poenial candidae: he inflaion-linked bonds Broadly speaking, an inflaion-linked bond is a fixed income produc which pays a fixed real ineres rae over he enire bond lifeime and in which he principal is compleely hedged agains he price increase. The inflaion-linked bond provides a perfec guaranee agains he risk of purchasing power losses as compared o radiional long erm fixed income producs. While he economy is facing deflaion, he invesor is ensured o ge back his iniial capial. These inflaion hedging producs appeared for he firs ime a he beginning of he eighies in he Unied Kingdom (1981). Since hen, all he developed counries have issued inflaion-linked bonds. Several reasons led governmens o issue inflaion-linked bonds (Beleski, 2006; La Bruslerie, 2002): By issuing inflaion-linked bonds, governmens can reduce he cos of financing heir deb and defici; in addiion, inflaion-linked bonds (ILB) issuances faciliae he maching beween governmen asses and liabiliies as boh ax incomes and governmen spending are highly correlaed o he inflaion dynamics; in a high inflaionary environmen, he issuance of ILB signals o he marke he governmen willingness o incurve and sabilize inflaion pressures. 8 The financial downurn, sared in 2007, has progressively affeced he sovereign deb secor. Consequenly, numerous European counries have been downgraded (Greece, Ireland, Porugal and Spain) from January Despie he deerioraion of he siuaion, in paricular, he probabiliy of see European counries being in a defaul siuaion remains exremely weak or nil. These roubles cause erraic changes of long erm ineres raes. This implies managemen of he ineres rae risks. 21