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1 Ciy Research Online Ciy, Universiy of London Insiuional Reposiory Ciaion: Zhang, Y. (2009). Opimal Plan Design and Dynamic Asse Allocaion of Defined Conribuion Pension Plans: Lessons from Behavioural Finance and Non-expeced Uiliy Theories. (Unpublished Docoral hesis, Ciy Universiy London) This is he acceped version of he paper. This version of he publicaion may differ from he final published version. Permanen reposiory link: hp://openaccess.ciy.ac.uk/203/ Link o published version: Copyrigh and reuse: Ciy Research Online aims o make research oupus of Ciy, Universiy of London available o a wider audience. Copyrigh and Moral Righs remain wih he auhor(s) and/or copyrigh holders. URLs from Ciy Research Online may be freely disribued and linked o. Ciy Research Online: hp://openaccess.ciy.ac.uk/ publicaions@ciy.ac.uk

2 Opimal Plan Design and Dynamic Asse Allocaion of Defined Conribuion Pension Plans: Lessons from Behavioural Finance and Non-expeced Uiliy Theories Yumeng Zhang A hesis submied for he award of he degree of Docor of Philosophy in Acuarial Science Deparmen of Acuarial Science and Insurance Cass Business School Ciy Universiy London, England June 2009 I

3 Conens Lis of Tables...VI Lis of Figures... VII Acknowledgemens...IX Declaraion...X Absrac...XI Seing he Scene.... Saemen of he Problem..... Why defined conribuion pension plans? The invesmen problem wih DC plans Dynamic asse allocaion Shor-erm invesmen sraegy (wo-period model) Long-erm invesmen sraegy (muli-period model) Financial and human wealh The consumpion problem Opimisaion mehods Behavioural issues The exising lieraure on DC invesmen Differeniaors Overview of he hesis Behavioural Feaures in DC Plan Arrangemen Inroducion Background Lieraure review Behavioural finance Repeaed gamble II

4 2.2.2 Prospec heory and loss aversion Menal accouning Myopic loss aversion (MLA) Framing effec Anchoring effec Overconfidence and confirmaion bias Defaul opion, Ineria and Procrasinaion Hyperbolic discouning Oher Applicaion o DC plan arrangemen Communicaion Decision o save Defaul and oher opions Fund selecion Fund performance daa Annuiisaion decision Conclusions Reexamining Behavioural Characerisics (UK Evidence) Inroducion Survey Subjecs Survey Design Mehod Resuls Loss aversion parameers Behavioural feaures Oher Discussion III

5 4 Loss Aversion Model Inroducion Asse Allocaion Problem for a DC Pension Plan Financial asses Labour income process Pension fund accumulaion and arge levels Formulaion of he arge driven process Opimisaion and numerical mehod Empirical analysis and parameer calibraion Loss aversion parameers Salary scale Parameer calibraion Simulaion and resuls The use of loss aversion uiliy in a arge driven model Baseline case Sensiiviy analysis Conclusions Recursive Uiliy Model Inroducion Allocaion of consumpion across he life cycle Epsein-Zin uiliy The model The model srucure and opimisaion problem Parameer calibraion Resuls Baseline case Sensiiviy analysis Conclusion IV

6 6 Conclusions Summary Fuure Developmens Bibliography... 5 Appendices A 0 Risk aiude survey A 02 Numerical mehod of loss aversion model A 03 Numerical soluion for he dynamic programming and inegraion process A 04 Opimal asse allocaion a differen ages A 05 Survival probabiliies able V

7 Lis of Tables Table Survey sample (966, all aduls aged 8+) Table 2 Loss aversion quesions Table 3 Esimaed loss aversion parameers Table 4 Esimaed subgroup loss aversion parameers Table 5 Financial knowledge: base rae Table 6 Financial knowledge: diversificaion Table 7 Framing effec Table 8 Behavioural quesions Table 9 Levels of replacemen raios under loss aversion and quadraic uiliy model (over 0k simulaions), wihou salary risk Table 0 Levels of replacemen raios under dynamic loss aversion model and 5-year lifesyle (over 0k simulaions) Table Reliabiliy of differen asse sraegies Table 2 Loss aversion curvaures Table 3 Final arge weigh (v=0.53, v2 =0.77 and lamda=3.4) Table 4 Only final arge case Table 5 Loss-aversion model Vs. Power uiliy Table 6 Loss-aversion model Vs. Prospec heory model Table 7 Pension plan member ypes Table 8 Baseline parameer values Table 9 RRA and EIS values for he differen ypes of plan member... 3 Table 20 The effec of he preference parameers on he mean opimal annuiisaion raio... 4 VI

8 Lis of Figures Figure : 5-year deerminisic lifesyle sraegy... 4 Figure 2: Markowiz mean-variance heory... 5 Figure 3: Financial and human wealh over he lifecycle up o reiremen... 9 Figure 4: Loss aversion value funcion... 7 Figure 5: Loss aversion value funcion Figure 6: Loss aversion value funcion Figure 7: Prospec heory probabiliy weighing funcion... 4 Figure 8: Survey sample (oal respondens: 966) Figure 9: Planning Reiremen Age (oal respondens: 269) Figure 0 Labour income process Figure Expeced pah of arge a age Figure 2 Salary scale (scaled o 00 a age 65) Figure 3 Loss aversion Vs. quadraic uiliy Figure 4 Hisogram (LA Vs. Quadraic uiliy model)... 8 Figure 5 Mean opimal equiy asse allocaion Figure 6 Hisogram of replacemen raio: baseline case Vs. 5-year lifesyle Figure 7 Sensiiviy: loss aversion coefficien (Lamda) Figure 8 Case :V< and V2< (S-shaped) Figure 9 Mean opimal equiy asse allocaion for case Figure 20 Case 2: v> and v2> (reverse S-shaped) Figure 2 Mean opimal equiy asse allocaion for case Figure 22 Hisogram of replacemen raios: baseline case Vs. Reverse S-shaped uiliy. 89 Figure 23 Case 3: v< and v2> (kinked power uiliy) Figure 24 Mean opimal equiy asse allocaion for case Figure 25 Hisogram of replacemen raios: baseline case Vs. kinked power uiliy... 9 Figure 26 Sensiiviy: final arge weigh Figure 27 Power uiliy case Figure 28 Loss aversion uiliy Vs. power uiliy VII

9 Figure 29 Hisogram of replacemen raio: baseline case Vs. power uiliy Figure 30 baseline Vs. prospec heory seing Figure 3: Decomposiion of oal wealh over he life cycle Figure 32: Labour income process... 3 Figure 33: Opimal equiy asse allocaion for he final ime period Figure 34: Mean of simulaed wealh, consumpion and labour income Figure 35: Mean of consumpion Figure 36: Human capial Figure 37: Opimal equiy asse allocaion prior o reiremen Figure 38: Opimal conribuion rae Figure 39: Mean opimal conribuion rae (wih lower limi of 5% per annum, doed)30 Figure 40: Mean opimal equiy asse allocaion (wih lower limi on conribuion rae of 5% per annum, doed) Figure 4: Mean opimal conribuion rae (for differen RRA/EIS combinaions) Figure 42: Mean opimal asse allocaion (for differen RRA/EIS combinaions) Figure 43: Mean opimal conribuion rae (for differen levels of beques moive) Figure 44: Mean opimal equiy asse allocaion (for differen levels of beques moive) Figure 45: Mean opimal conribuion rae (for differen levels of personal discoun facor) Figure 46: Mean opimal equiy asse allocaion (for differen levels of personal discoun facor) Figure 47: Mean opimal conribuion rae (for differen correlaion beween labour income and equiy reurns) Figure 48: Mean opimal equiy asse allocaion (for differen correlaion beween labour income and equiy reurns) Figure 49: Opimal equiy asse allocaion a age Figure 50: Opimal equiy asse allocaion a age VIII

10 Acknowledgemens My PhD research was funded by way of a bursary from he Deparmen of Acuarial Science and Insurance a Cass Business School, Ciy Universiy. I wish o sincerely hank my supervisor, Dr. Douglas Wrigh, for his consan encouragemen and guidance. I was very lucky o have Dr. Wrigh as my supervisor. In addiion o he research problem, I go much valuable professional advice from him on how o srucure and presen a paper wih more consideraions on poenial audience. The more pleasurable aspec was ha our discussions did no say resriced o my PhD. We discussed ravelling, books, fooballs, spors gambling, ec. This has become an imporan par of my PhD. I am also graeful o he following for heir various conribuions: Dr Russell Gerrard for his advice on he numerical mehods used in his hesis; Dr. Zaki Khorasanee and Dr. Iqbal Owadally for heir helpful commens on my ransfer repor in he firs year of my PhD. Las bu no leas I would like o hank my family and friends, wherever hey are, paricularly my Mum and Dad; and, mos imporanly, my wife Xue Han. The wriing of a hesis can be a lonely and isolaing experience. The simple hough ha she is always here is really a powerful source of inspiraion and energy o me. IX

11 Declaraion Powers of discreion are hereby graned o he Universiy Librarian o allow his hesis o be copied in whole or in par wihou furher reference o he auhor. This permission covers only simple copies made for sudy purposes, subjec o he normal condiions of acknowledgemen. X

12 Absrac The quesion of opimal asse allocaion sraegy for defined conribuion (DC) pension plans is addressed. A primary moivaion for his sudy is provided by he recen lieraure on behavioural finance and ineremporal life-cycle invesmen heory. In his hesis wo alernaive uiliy forms are considered: loss aversion and Epsein-Zin recursive uiliy. We develop a dynamic-programming-based numerical model wih uninsurable sochasic labour income and borrowing consrains. In he loss aversion case, members are assumed o be loss averse wih a arge replacemen raio a reiremen and a series of suiably defined inerim arge prior o reiremen. We also exend he ineremporal life-cycle saving and invesmen heory o he dynamic asse allocaion problem of DC pension schemes. A new approach o model conribuion and invesmen decisions wih focus on he member s desired paern of consumpion over he lifeime (based on Epsein-Zin uiliy preference) is proposed. The hesis draws on empirical evidence of salary scales and loss aversion parameers from UK households, wih labour income progress esimaed from he New Earnings Survey and loss aversion parameers esimaed on he basis of face-o-face inerviews wih 966 randomly seleced UK residens. XI

13 Chaper Seing he Scene. Saemen of he Problem Pension planning is probably he longes and mos imporan financial decision ha an individual has o make over heir lifeime. The invesmen and asse allocaion sraegy in relaion o pension funds has a profound effec on he capial marke. To employers, he process of choosing a pension provider is one of he mos imporan decisions an employer can make and one which can reflec direcly on business and saff reenion. To employees, even a small change in saving and invesmen decisions can have significan influence on heir reiremen sandard of living... Why defined conribuion pension plans? Basically, here are wo ypes of pension plans: Defined Benefi (DB) and Defined Conribuion (DC). Generally, in a Defined Benefi (DB) pension plan, he amoun of reiremen income of employees depends on he number of years hey worked for he employers and he level of heir salary when hey reire, i.e. heir final salary. A Defined Conribuion (DC) pension plan is also called a money purchase plan. The level of conribuions (made by employer and employee) is prese, bu he amoun of reiremen income is no. The invesmen performance and longeviy risks of he pension are borne by he employee, no he employer.

14 Imporan changes are occurring in he overall occupaional pensions landscape. Tradiionally, DB plans have been he dominan form of occupaional pension provision in he UK, bu a combinaion of increasing regulaion and exernal marke facors have led a growing number of employers o close heir plans o new or even exising members and se up DC plans in heir place. Some of he facors include new accouning rules forcing schemes o value heir liabiliies a curren marke raes, he prolonged equiy bear marke of , a secular fall in ineres rae yields which are used o discoun pension liabiliies, and above all, longer life expecancy. The governmen is aiming o shif he curren 60:40 sae/privae pension secor raio o 40:60. An increasing number of workers now have o reply on defined conribuion schemes o provide heir fuure reiremen income, eiher hrough a scheme se up by heir employer or a personal pension as a group or individual arrangemen, which is primarily he reason of objecives of his research...2 The invesmen problem wih DC plans DC pension plans are designed o provide pensions on reiremen for members. In a DC pension scheme, he member conribues par of his or her labour income each year and builds up a pension fund before reiremen. A reiremen, he member annuiises par of he pension fund by buying a life annuiy. Unlike defined benefi (DB) pension plans, here is no guaranee offered by he employer ha a pension fund will pay ou a se amoun on reiremen. The invesmen performance risk of he pension is borne by he employee, no he employer. Members in a DC pension plan face he following risks: Inflaion risk. The risk ha he value of asses do no keep pace wih inflaion hereby eroding he purchasing power of he final pension; Annuiy risk. The risk ha as he member approaches reiremen he cos of purchasing an annuiy flucuaes significanly; 2

15 Invesmen risk. The risk ha he value of he member s asses drops significanly when very close o reiremen and here is lile scope for recovery Mos DC plans allow members a degree of choice abou how o inves heir conribuions. They also normally offer a defaul opion in which he member s conribuion is auomaically placed if he member does no acively choose a fund. However, many members show lile ineres in financial maers and have lile knowledge abou invesmen. They ofen readily accep he defaul opions. Choi e al. (2002) sudied he endency o accep scheme defaul opions in he US and found ha very few employees op ou of defaul arrangemens even when hey are free o do so. A similar endency o accep he defaul is found in he UK. Hewi Bacon and Woodrow, he pension consuling firm, found ha around 80% of members in UK DC schemes accep he defaul fund choice (Bridgeland 2002). As Blake e al. (2005, p4) claimed in heir sudies of UK sakeholder pensions, The vas majoriy of pension scheme members appear o passively accep whaever defaul fund he pension provider has chosen, bu here is lile consensus amongs providers as o wha he appropriae characerisics for a defaul fund are, despie he imporance of he choice in deermining pension oucomes. In his sense, sakeholder pension schemes can be characerised as a loery for he members. In pracice, a radiional deerminisic lifesyle invesmen sraegy (e.g. 5 year lifesyle, as shown in Figure below) is widely used by many DC pension plans as he defaul opion. In such a sraegy, he pension wealh is invesed enirely in high risk asses (e.g. equiies) when he member is young. Then, as he individual approaches reiremen, he asses are swiched gradually ino lower risk asses such as bonds and cash. This provides a pre-deermined swiching sraegy for members who do no wish o ake an acive role and represens a compromise beween risk and reward. 3

16 Figure : 5-year deerminisic lifesyle sraegy 00% Asse allocaion 80% 60% 40% 20% 0% T-5 T-4 T-3 T-2 T- T Equiies Bonds Cash Bu, recen research suggess ha a radiional deerminisic lifesyle invesmen sraegy sill leaves members wih considerable exposure o volailiy in he years preceding ha gradual swich. The range of poenial resuls (in erms of boh final fund levels and reiremen income) from a radiional lifesyle sraegy is huge. This makes i very difficul for a DC member o have any idea of wha level of reiremen income hey can expec. For DC members who seek greaer cerainy in heir reiremen planning, he asse sraegy adoped needs o be far more focused on achieving heir reiremen arges..2 Dynamic asse allocaion.2. Shor-erm invesmen sraegy (wo-period model) Modern finance heory sared wih he mean-variance analysis of Markowiz (952), which shows how invesors should selec asses if hey only care abou mean and variance of heir end-of-period wealh. His analysis is shown in Figure 2. 4

17 A riskless asse (cash) corresponds o a poin on he y-axis. The angen poin where he sraigh line ouches he curved line is he marke porfolio, which is he bes mix of risky asses. Invesors can pick a poin somewhere on his upward-sloping line depending on heir risk aiudes. This is also called muual fund separaion heorem. Figure 2: Markowiz mean-variance heory Expeced reurn Aggressive invesor Cash Moderae invesor Conservaive invesor Bonds Socks Bes mix of socks and bonds Risk If we assume ha invesors have uiliy defined over final wealh (i.e. making invesmen decisions o maximise he expeced uiliy of final wealh), in order o ge a racable soluion, several assumpions abou he uiliy funcion and disribuion of asse reurns need o be made. Some popular assumpions are he following: 2. Quadraic Uiliy, in which case, U ( W ) aw bw + = + +, absolue and relaive risk aversion increases wih wealh. No disribuional assumpions on asse reurns are needed. 2. Exponenial Uiliy, in which case, ( W ) ( W ) U θ, absolue risk + = exp + aversion is a consan θ, relaive risk aversion increases wih wealh. Asses reurns are assumed o be normally disribued. γ 3. Power Uiliy, in which case, U ( W ) = ( W ) /( γ ) + +, absolue risk aversion decline wih wealh and relaive risk aversion is a consan γ. Asse reurns are lognormally disribued. 5

18 .2.2 Long-erm invesmen sraegy (muli-period model) Inuiively, long-erm invesors are differen from shor-erm invesors, because long-erm invesors have a longer invesmen horizon and hey could change he porfolio decision when any new informaion arrives or simply jus as ime passes. However, according o he analysis of Markowiz, i seems ha all invesors who only care abou mean and variance of final porfolio wealh should hold he same porfolio of asses, no maer wheher hey have a shor-erm or a long-erm invesmen horizon. Is his really correc? The answer is yes, as long as he following condiions are me: ) he risky asse reurns are independen idenically disribued (IID); 2) invesmen opporuniy se is consan (i.e. consan risk-free rae, consan expeced reurn and consan volailiy); 3) invesors relaive risk aversion coefficien does no depend sysemaically on heir wealh. These poins have been undersood for many years. In a discree ime framework, Samuelson (963, 969) proposed a dynamic programming model o explain his. We will discuss his mehod laer bu he main idea is o use a series of wo-period problems o solve he muli-period opimisaion problem recursively. In he las ime period, he long-erm invesor acually faces a wo-period opimisaion problem and will become a shor-erm invesor. Since he asse reurns are IID and invesors risk aiudes do no change over ime, he resuls for all he wo-period maximisaion problems should be he same. In oher words, long-erm invesors should behave like shor-erm invesors in his case. Meron (969, 97) was he firs o apply his approach o a coninuous-ime opimal invesmen problem. The coninuous-ime models can be seen as he limi of he muliperiod discree-ime models when he ime period becomes very small. Meron s model suggess ha, in he single-risky-asse and consan-invesmen opporuniy seing, he 6

19 opimal porfolio weigh in he risky asse for an invesor wih power uiliy should equal 2 he risk premium ( µ ) divided by variance of he risky asse reurns ( σ ) imes he coefficien of relaive risk aversion (γ ): µ α = 2 γσ [] Anoher ineresing poin o noe abou long-erm invesors is ha hey are more concerned wih he sandard of living ha can be financed by wealh, raher han he final wealh level iself. Or, as explained by Campbell and Viceira (2002), hey consume ou of wealh and derive uiliy from consumpion raher han wealh (p37). For example, le us assume long-erm invesors have power uiliy on consumpion and only have wo asses o inves (one risky and one riskless). They face a wealh accumulaion process as follows: ( ) W + = ( W C ) ( α ) + rf + α ( + rh ) [2] where rf is he risk-free ineres rae; rh is rae of invesmen reurn of risky asse; W and C are he wealh and consumpion level a ime respecively. In his case, he opimal dynamic asse allocaion on risky asses ( α ) should maximise he expeced presen value (EPV) of oal fuure consumpion uiliy, i.e., max E i= 0 i β U ( C + i ) [3] where β is he ime discoun facor, represening he relaive weigh invesors pu on fuure consumpion. 7

20 An approximae closed form soluion is available for his problem. Campbell and Viceira (2002) showed ha, for above long-erm invesors wih power uiliy over consumpion, he opimal porfolio weigh should sill be he same as suggesed in[], if he consumpion-wealh raio is consan over ime..2.3 Financial and human wealh So far we have briefly reviewed he research hisory of long-erm porfolio choice wih financial asses only. In a realisic life-cycle saving and invesmen model, however, labour income is also imporan for long-erm invesors. Wih income risk, he opimal porfolio weigh is no consan and will follow a lifesyle sraegy. This can be explained. Human wealh can be undersood as he expeced ne presen value (NPV) of fuure labour income. An individual s labour income can be seen as a dividend on he individual s implici holding of human wealh. The raio of human o financial wealh is he crucial deerminan of life-cycle porfolio composiion. In early life, as shown in Figure 3 below, he raio is large because people have lile ime o accumulae financial wealh and expec o receive labour earnings for many years. Given ha he growh rae of labour income is close o he risk-free rae, labour income can be seen as an implici subsiue of riskless asse. Young individuals hold oo much in his non-radable riskless asse and herefore need o allocae mos heir financial wealh o risky asses o keep he overall porfolio composiion consan. When hey grow older, hey accumulae more financial wealh and have less human wealh lef (i.e. a smaller holding in his implici non-radable riskless asse). Thus, hey need o rebalance he porfolio and increase he weigh in he riskless asse. 8

21 Figure 3: Financial and human wealh over he lifecycle up o reiremen Wealh level Age Financial wealh Human wealh.2.4 The consumpion problem Mos economic decisions are ineremporal, as curren decisions made will affec fuure available choices. Decisions regarding he funding and invesmen sraegies adoped for reiremen savings are a classic example of his. The reiremen saving decisions made oday affecs no only an individual s curren level of consumpion bu also fuure consumpion possibiliies. In oher words, individuals face an ineremporal rade-off: if hey save more oday, hey mus consume less and hence heir curren uiliy declines, bu hey can hen consume more in he fuure (hereby increasing fuure uiliy). In fac, he mos basic objecive of a DC pension scheme (or all ypes of pension arrangemens) is o arrange consumpions over life cycle. The member s decision of conribuion rae, porfolio asse allocaion and he proporion of he accumulaed fund used o purchase an annuiy, are all driven by his or her preference beween curren and fuure consumpions. Thus, as a resul, he opimal asse sraegy of DC pension plans 9

22 should depend on he paern of consumpion levels over he enire lifeime, raher han jus focusing on he erminal pension wealh level a reiremen..2.5 Opimisaion mehods The inclusion of labour income will make he life-cycle invesmen model more realisic. As a radeoff, he opimisaion problem becomes very difficul (if no impossible) o solve analyically. Therefore, he recen lieraure uses a variey of numerical mehods o approximae he soluion of he dynamic porfolio opimisaion problem. Dynamic programming Dynamic programming was originally used in he 940s by Richard Bellman o solve discree-ime opimisaion problems. Since hen, i has become one of he mos fundamenal building blocks of numerical mehods in muli-period porfolio choice problems. The basic idea is o urn he muli-period opimisaion problem ino a series of wo-period opimisaion problems. A he hear of dynamic programming is he value funcion, which represens he maximum presen discouned value of he objecive funcion onward as a funcion of curren sae variables. For each period, going backward from he nex-o-las period o he beginning, he soluion is found by maximising he one-sep ahead expecaion of he approximaed value funcion. To explain he idea ino more deail, le us look a one simple example. A age, an invesor faces a long-erm muli-period porfolio choice problem o maximise he expeced presen value (EPV) of oal uiliy of consumpion ( U 65 ) a reiremen age 65. We simplify he problem and make he following assumpions: here are only wo financial asses o choose: one risk-free asse ( R ) and one risky asse ( R ); h f Dynamic programming mehod is also used in our models laer on o solve he opimisaion problem numerically. 0

23 The risk-less asse is assumed o yield a consan ineres rae r f p.a. The reurn on he risky asse ( R ) is assumed o be normally disribued wih mean r + µ, and volailiy of σ ; f he invesor has a wealh level of h W a age ; he invesor has a salary level of Y a age ; he growh rae of salary is given by I = ri + Z 2 where ri is he annual growh rae of average salary 2 and Z ~ N(0, ) ; 2 σ 2 γ C he invesor has ime-separable power uiliy ( U ( C ) = ) on consumpion a γ age ; he invesor is allowed o rebalance her porfolio annually; he invesor will consume all available wealh in he las period. The opimisaion problem is Max E α, C i= 0 i β U ( C + i ), subjec o he consrain ha W = ( W + Y C ) [( α )( + rf ) + + α ( + r + f µ + Z)], where α is he asse allocaion 2 in risky asses a age, β is he ime discoun facor, and Z ~ N(0, ); σ We sar from he nex-o-las period, i.e. he period from age 64 o age 65. Specifically, a age 64, he value funcion is J ( W Y ) Max[ U + βe ( J )] 64 64, α64 =, where γ W65 J 65 =. γ As an imporan sep of sochasic dynamic programming, we need o discreise he sae variables (wealh level and income level, in his example) firs. For example, wealh and labour income can be discreised ino 00 and 0 even grids, respecively, in compuaion, so ha we can calculae he opimal conrol variables ( α, C ) and value funcion for each grid poin on he 00 by 0 marix. To approximae he expecaion erm in he value funcion, by far, he mos popular approach is quadraure inegraion. Gauss-Hermie quadraure is used o discreise shocks

24 (normally disribued variables for he risky asse reurn and salary growh rae, in his example) ino several (e.g. 7) nodes, and he procedure of discreising Z and Z 2 is o subsiue 2Z, m and 2Z2, n for hem respecively. So, E 7 7 ( J ) J ( W, Y ) f ( Z, Z ) dz dz π w w [ J ( 2Z, 2Z W Y )], where + = Z, 2, ;, m Z, n 2 + m n +, + m= n= w, Z,m w and Z,m, Z 2,n are he Gauss-Hermie quadraure weighs and nodes. Z 2,n As soon as we ge opimal asse allocaion and value funcion for each grid poin a age 64, we can hen use hem o solve he opimisaion problem wih he same 00 by 0 marix for previous ime period, i.e. J ( W Y ) Max[ U + βe ( J )] 63 63, α 63 =. I is very likely ha he accumulaed sae variable values from he previous ime period are no on a grid poin, in which case, some inerpolaion mehods (e.g. bilinear, cubic spline, ec. ) are employed o approximae he value funcion ( J 64, in his case). The ieraion process is hen repeaed backward unil he beginning. The above numerical mehod is called value funcion ieraion. However, his approach requires knowledge of he disribuion of all he shocks, so ha appropriae quadraure inegraion (e.g. Gauss quadraure) can be used. Furher, i canno handle a large number of sae variables. To overcome hese limiaions, Brand e al. (2005) propose a simulaion-based mehod based on recursive use of approximaed opimal porfolio weighs. The idea is o esimae asse reurn momens wih a large number of simulaed sample pahs hen approximae he value funcion using a Taylor series expansion. Also, if he reurns are pah-dependen, i would be necessary o regress he reurn variable on he simulaed sae variables from he previous ime period, before using he Taylor expansion wih condiional reurn momens. In a coninuous-ime framework, Bellman exends earlier work wih William R. Hamilon and Carl G. J. Jacobi and derives Hamilon-Jacobi-Bellman (HJB) equaion, which represens he fundamenal parial differenial equaion obeyed by he opimal value funcion. This hen becomes he so called sochasic opimal conrol heory. 2

25 Since mos of he models in my PhD research are based on a discree-ime seing, coninuous-ime models are ou of he scope of his hesis. Coninuous-ime models areas appear o be a fruiful area of fuure research. The bigges benefi of using coninuousime models is ha he mahemaical calculaions are easier in coninuous ime (and herefore i migh be easier o ge closed-form soluions). Bu if we need o use numerical mehods o solve he problem, discree-ime and coninuous-ime models are in fac very similar in erms of mehodology..3 Behavioural issues In a DC pension plan, he invesmen performance risk is borne by he employee, no he employer. The paricipans have more responsibiliies in erms of deciding how much o save and how o inves he funds. DC plans have radiionally been regarded as employee-direced wih employees seen as he acive agens and he employer hough o play a minimal decision-making role. Bu, in fac, pension plan design is no a neural vehicle wihin which paricipans make heir own raional choices based on raional expecaions. Insead, paricipans have a srong endency o choose defaul opions in DC schemes. They can be easily influenced by plan design, boh in he saving area and in he invesmen decision-making as well..3. The exising lieraure on DC invesmen Cairns (994) reviewed and divided he objecives of defined conribuion (DC) pension research ino wo caegories: (a) members are old he likely fuure amoun of pension and expec he arge o be aained, in which case, we need he acual pension o be as close o his level as possible; or (b) members are old he acual pension only a he dae of reiremen, in which case, we would need o maximise he expeced uiliy of he ne replacemen raio a reiremen. 3

26 If equiy reurns are mean-revering, o hold equiy for a long period before reiremen can be jusified because he volailiy of equiy reurns will be lower over a longer ime period. Bu Howie and Davies (2002) finds limied evidence on mean-reversion of UK equiy prices (or ime diversificaion, which is mahemaically he same hing). Also, as discussed earlier, o have a fixed asse allocaion is no appropriae for long-erm invesors. DC pension plan members normally have a 20 o 30 year invesmen horizon and expec o receive many years of pension income afer reiremen. Thus, i is also necessary o consider non-pension asses 2 (e.g. labour income) and have a more dynamic invesmen soluion. Wih regard o he dynamic asse allocaion problem for DC pension plans, mos of he exising lieraure invesigaes he opimal dynamic asse sraegy of DC pensions by assuming a fixed conribuion rae (e.g. 0%, 2%) and maximising he expeced uiliy of he erminal 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 he oal disuiliy 3 unil reiremen (for example, Haberman and Vigna (2002)). Haberman and Vigna (2002) derived a dynamic-programming-based formula for he opimal invesmen allocaion in DC schemes and considered hree risk measures in analyzing he final ne replacemen raio: he probabiliy of failing he arge, he mean shorfall and he value a risk (VaR). They suggesed ha he risk profile of he individual and he rade-off beween differen risk measures of he downside risk are imporan facors o be aken ino consideraion. According o heir research, only risk averse members should adop a lifesyle sraegy and he swiching poin depends on he degree of risk aversion of members and he ime o reiremen, i.e. he more risk averse, he sooner he swich; he longer he accumulaion period, he laer he swich. 2 In real life, oher non-pension asses such as housing also plays an imporan role in he member s financial planning. However, his is ou of he scope of our models in his hesis. 3 The disuiliy is normally defined over he deviaion of acual fund level from inerim and final arge fund levels. 4

27 Cairns e al. (2004) invesigaed a model incorporaing asse risk, salary risk and ineresrae risk and proposed a new form of erminal uiliy funcion by using he plan members final salary as a numeraire. They showed ha he use of a sochasic asse allocaion sraegy (which hey called sochasic lifesyle) could enhance he welfare relaive o deerminisic lifesyle and benchmark mixed funds sraegy..3.2 Differeniaors Applicaion of behavioural feaures in DC plan design As argued by Michell and Ukus (2004), defined conribuion (DC) plans can provide real reiremen securiy, bu only if paricipans uilise hem appropriaely and make opimal invesmen decisions. There is growing evidence suggesing ha here are key behaviour barriers prevening paricipans from doing so. One obvious soluion o dealing wih significan behavioural barriers o he effecive use of DC plans for reiremen provision is o offer some form of educaion o paricipans, bu inelligen plan design is also required when some paricipans show lile ineres in financial maers and readily accep defaul opions. Represening an alernaive way of looking a financial marke, behavioural finance research is quie helpful for pension plan design. Behavioural finance is a combinaion of psychology and economics ha invesigaes wha happens in markes in which some of he agens display behavioural limiaions and complicaions. Mos behavioural sudies have an empirical componen in common and show a high predicive value. The ferilisaion of economics wih psychology and empirical evidence makes i ineresing and promising in he pension plan design field. We believe behavioural sudies are useful ools o improve boh he design of pension schemes and he efficiency of communicaion. Loss-aversion dynamic asse allocaion The concep of risk and risk aversion are cornersones of economic modelling. Wihin he expeced uiliy framework, he only explanaion for risk aversion is ha he uiliy 5

28 funcion for wealh is concave. People are assumed o be risk-averse expeced uiliy maximisers and make raional choices based on raional expecaions. To dae, he invesmen problem of DC pension plans has been addressed principally for life cycle expeced uiliy opimizers. However, as will be discussed laer in Chaper 2, observed behaviour appears o invalidae expeced uiliy heory as a descripive model. Insead he use of loss aversion (LA) uiliy seems quie promising and helpful in modelling he opimal dynamic asse allocaion of DC pension funds. The idea of loss aversion was firs proposed by Kahneman and Tversky (979) wihin he framework of prospec heory 4. As one of is disinguishing feaures, he loss aversion value funcion is defined on gains and losses of wealh relaive o a reference poin, raher han absolue levels of oal wealh (as is he case wih he radiional ideas of uiliy heory). As shown in Figure 4 below, he wo key properies of he loss aversion value funcion are: (i) i is S-shaped (i.e. convex below he reference poin and concave above i), implying ha individuals are risk seeking in he domain of losses and risk averse in he domain of gains; and (ii) i is asymmeric (i.e. seeper below he reference poin han above, because of he effec of he loss aversion raio λ ), implying ha individuals are more sensiive o losses han o gains. 4 Kahneman and Tversky (979) developed his heory o remedy he descripive failures of subjecive expeced uiliy (SEU) heories of decision making. 6

29 Figure 4: Loss aversion value funcion Uiliy Value funcion Loss Gain Reference level Wealh We can incorporae his behaviour research finding ino he invesmen soluion of DC pension plans. In a DC pension plan, prior o reiremen, he members can be hough o have a final arge fund level a reiremen and a series of consisen inerim arges before reiremen. Members are assumed o be loss averse wih respec of hese arges (which define he reference poins in he loss aversion framework above) and make asse allocaion decisions o maximise he sum of expeced presen value (EPV) of loss aversion value funcion a each age unil reiremen. Consumpion problem Mos economic decisions are ineremporal, as he curren decisions made will affec he fuure available choices. Decisions regarding he funding and invesmen sraegies adoped for reiremen savings are a classic example of his. The reiremen saving decisions made oday affec no only an individual s curren level of consumpion bu also fuure consumpion possibiliies. In oher words, individuals face an ineremporal 7

30 rade-off: if hey save more oday, hey mus consume less and hence heir curren uiliy declines, bu hey can hen consume more in he fuure (hereby increasing fuure uiliy). In fac, he mos basic objecive of a DC pension scheme (or all ypes of pension arrangemens) is o plan consumpions over life cycle. The member s decision of conribuion rae, porfolio asse allocaion and he proporion of he accumulaed fund used o purchase an annuiy, are all driven by her preference beween curren and fuure consumpions. Thus, as a resul, he opimal asse sraegy of DC pension schemes should depend on he paern of consumpion levels over he enire lifeime, raher han jus focusing on he erminal pension wealh level a reiremen..4 Overview of he hesis The hesis consiss of six chapers and is srucured as follows: Chaper inroduces he background of he invesmen problem of DC pension scheme, including a brief review on exising lieraure in he area of opimal asse allocaion problem. Research moivaions are discussed in chaper as well. In Chaper 2, we review he behavioural feaures ha are relevan o he work of DC plan design and communicaion. This is necessary especially given ha some approach aken in curren DC plans may be counerproducive in encouraging reiremen saving and helping members make appropriae invesmen decisions. Chaper 3 is devoed o a survey we did wih sponsorship from Disribuion Technology Ld. The survey was conduced on a face-o-face inerview basis from 4 h April 2005 o 9 h April A oal of 966 responses were received. The resuls help us o invesigae how people s aiudes o risk vary during long-erm financial decision making. This 8

31 chaper provides he empirical evidence of behavioural uiliy parameers on UK households. This is followed, in Chaper 4, by incorporaing he survey resuls ino a loss-aversionbased model o invesigae he opimal invesmen sraegy for DC scheme members. Members are assumed o be loss averse wih a arge fund level a reiremen and a series of suiably defined inerim arges prior o reiremen, and are assumed o make asse allocaion decisions wih he aim of maximising he expeced presen value (EPV) of heir oal loss aversion value funcion over he period unil reiremen. Chaper 5 is based on a differen research idea, ineremporal saving and invesmen. I buil an ineremporal invesmen model for DC pension plan, in which case, he plan members conribuion and invesmen decisions all depend on heir preferences of consumpion levels over heir enire lifeime. Epsein-Zin preference is used o separae risk aversion and elasiciy of ineremporal subsiuion (EIS). The effecs of risk aversion and EIS on conribuion rae and asse sraegy over he life cycle are also considered. The hesis concludes wih Chaper 6, which highlighs he main conribuions of he research, explains some of he shorcomings of he models, and looks o he fuure of asse allocaion sraegy of DC pension plans. 9

32 Chaper 2 Behavioural Feaures in DC Plan Arrangemen 2. Inroducion 2.. Background Given ha in many counries, social securiy pensions are eiher non-exisen or provided a only very low levels, and given ha employers are moving away from providing defined benefi (DB) pension plans, i is increasingly becoming he responsibiliy of individuals o make adequae reiremen provision for hemselves. An increasing number of workers now have o rely on defined conribuion (DC) plans o provide heir fuure reiremen income, eiher hrough a plan se up by heir employer or a personal pension as a group or individual arrangemen. DC plans have radiionally been regarded as employee-direced wih employees seen as he acive agens and employer hough o play a minimal decision-making role. In fac, pension plan design is no a neural vehicle wihin which paricipans make heir own raional choices based on raional expecaions. As argued by Michell (2004), Being good a reiremen savings requires accurae esimaes of uncerain fuure processes, including lifeime earnings, asse reurns, ax raes, family and healh saus, and longeviy. In order o solve his problem, he human brains as a calculaing machine 20

33 would need o have he capaciy o solve many decades-long ime value of money problem, wih massive uncerainies as o sochasic cash flows and heir iming. Many workers do no have paricularly firm convicions abou heir desired saving behaviour. They can be easily influenced by plan design, boh in he saving area and in he invesmen decision-making as well. According o he annual DC/AVC Survey of Hewi Bacon & Woodrow s (2004), only 3% of employers hink ha DC members have a good undersanding of he funding levels required o build sufficien savings for reiremen. In US, according o survey by John Hancock insurance company (2003), 42% of he respondens said hey had lile or no invesmen knowledge, a furher 38% saed hey were somewha knowledgeable, only 20% of regarded hemselves as knowledgeable invesors. Alisair Byrne (2004b) did a similar survey in UK on he members of a mid-sized occupaional pension plan and found ha he resuls were broadly consisen wih he US findings in ha many employees show limied knowledge and ineres in heir pension arrangemens. According o he survey by Office of Fair Trading (997), half of he respondens agreed or srongly agreed ha I have found all he informaion I have seen, and he advice I have received, on pensions very confusing. All he above evidence shows ha he plan design and invesmen opion offering will have subsanial implicaions o he DC pension plan members. If DC pension plan members have behavioural biases and are no raional agens, hen we are in fac ransferring he long-erm invesmen risk o many individuals who canno make opimal decisions. This will has a poenially cosly long-erm consequences o he economy. Thus, i is imporan ha invesmen arrangemens in a DC pension plan are carried ou wih he knowledge of members poenial biases and errors in decision making. Represening an alernaive way of looking a financial marke, behavioural finance is a combinaion of psychology and economics ha invesigaes wha happens in markes in which some of he agens display human limiaions and complicaions. 2

34 2..2 Lieraure review There has been considerable amoun of research carried ou in he applicaions of behavioural sudies in capial marke, bu relaively few of hem deal wih pension design and he implicaions o he work of pension acuaries. By reviewing several behavioural feaures, Michell and Ukus (2004, p30) illusraed how behavioural research of he las few years had fundamenally challenged he ways in which plan sponsors, reiremen service providers and policy makers should hink abou reiremen plan design in he fuure. They proposed several insighs in pension plan design: behavioural research challenged he noion ha workers are raional, auonomous and can exercise unbiased judgmen in heir reiremen plans; sponsors and policymakers can affec members saving and invesmen decisions by choosing differen defaul srucures; some approach aken in curren DC plans may be counerproducive in encouraging reiremen saving; educaion in DC plans has is effecive limis. The plan members endency o choose defaul opions in DC plans ells us ha he plan design has a significan impac on scheme members. Blake e al. (2005, p4) claimed in heir sudies of UK sakeholder pension, The vas majoriy of pension scheme members appear o passively accep whaever defaul fund he pension provider has chose, bu here is lile consensus amongs providers as o wha he appropriae characerisics for a defaul fund are, despie he imporance of he choice in deermining pension oucomes. In his sense, sakeholder pension schemes can be characerised as a loery for he members. Some researchers ried o use he findings in behavioural sudy o improve he pension plan design. Taylor (2000) discussed he implicaions of behavioural finance on acuarial work including anchoring effec, prospec heory, framing, myopic loss aversion, overconfidence and menal accouning 5. Sykes (2004) discussed he ideas of behavioural finance and heir applicaion o he areas in which acuaries work, also highlighed several issues by means of examples. 5 These conceps will be explained and discussed laer on in his chaper. 22

35 Much of he oher exising lieraure explaining saving behaviour in DC pension plan focuses on how paricipaion and saving raes vary according o plan design. Choi e al. (2002) invesigae he role of ineria in defaul opions and found ha employees choices can be easily swayed by he defaul opions esablished by heir employer. Papke (2004) finds ha having he abiliy o direc he asse allocaion of conribuions o an employersponsored saving plan leads o a large 36 percen poin increase in he probabiliy of paricipaing. Bu his is no o say, having a greaer number of funds available should make members in DC schemes more willing o inves. Acually, Iyengar, Huberman and Jiang (2003) show a srong negaive relaionship beween he number of funds offered in DC schemes and average paricipaion raes. They find ha increasing he number of funds offered by 0 lead o a.5 o 2 percenage poin decline in he paricipaion rae. Also, behavioural sudies find evidence ha when an employee is offered a number of funds o choose, here is a bias owards dividing he money evenly among he funds offered. The asse allocaion an invesor chooses will depend on he arrays of funds offered in he reiremen plan. For example, employees may evenly allocae asse when hey are offered one equiy fund and one bond fund. Bu if anoher equiy fund were added, he allocaion o equiies would jump o wo hirds. As menioned by Van Der Sar (2004), mos behavioural sudies have an empirical componen in common and show a high predicive value. I is maybe rue ha mos of he behavioural research findings can no provides a sound basis for a normaive heory of asse allocaion. However, as will be discussed in his chaper, we believe behavioural sudies are definiely useful ools o design DC pension plans and improve he efficiency of communicaion. As claimed by Sykes (2004), financial expers should be concenraing heir effors on idenifying inefficiencies and designing he necessary producs o offse hem, raher han rying o decide which heory is a beer explanaion of curren behaviour. In his chaper, I choose some disinguishing feaures ha seem paricularly relevan o DC pension arrangemen and hen discuss heir possible applicaions in he differen processes of DC pension arrangemen. This chaper will proceed as follows: in Secion 23

36 2.2, all relevan behavioural feaures are reviewed; applicaions o DC pension arrangemen are elaboraed in Secions 2.3 and concluded in Secion Behavioural finance 2.2. Repeaed gamble The lieraure on repeaed gambles can be raced o 963. Samuelson asked his colleague wheher he would be willing o accep he following be: a 50 percen chance o win $200 and a 50 percen chance o lose $00. The colleague urned him down and offered his raionale as: I won be because I would feel he $00 loss more han he $200 gain. However, he colleague announced ha he was happy o accep 00 such bes. Afer ha, many academics ried o invesigae why individuals make decisions as Samuelson s colleague does. They argued individuals end o perceive and evaluae changes of wealh (gains and losses) raher han final wealh posiions as assumed in he expeced uiliy framework. More imporanly, hey are more sensiive o losses han o gains. To explain he effec, if we assume Samuelson s colleague has a value funcion as follows, x,if x 0 v( x) = { 3x,if x < 0 [4] where is x he poenial oucome. he would rejec he loery wih 50% chance of $200 gain and 50% chance of $00 loss. This can also helps o explain why Samuelson s colleague is willing o accep 00 such bes. For example, when here are wo consecuive such bes, he acually faces anoher 24

37 loery wih 25% chance o win $400, 50% chance o win $00 and 25% chance o lose $200. According o he above value funcion, he would feel indifferen o 2 gambles and be willing he ake a sequence of more han 2 gambles. Benarzi and Thaler (999) sudied he decision making of muliple plays of a gamble or invesmen and showed repeaed play of a posiive expeced value gamble is more aracive if hey are shown as he explici resuling disribuion of possible oucomes. They ried o apply he finding o reiremen invesing and found ha subjecs were willing o inves up o 90% of heir invesmen funds in socks when hey were shown disribuions of long-urn reurns raher han one year. However, he above value funcion [4] does no fully capure he empirically observed aiude owards risk. Individuals display diminishing sensiiviy in boh gain and loss. Kahneman and Tversky (979, 992) discussed he issue in deail and make i popular hrough heir prospec heory Prospec heory and loss aversion As one of is disinguish feaures, in he prospec heory, here is a value funcion defined on gains and losses relaive o a reference poin, raher han absolue levels of oal wealh. Tversky and Kahneman (979,992) suggesed a value funcion as follows: { v x if x 0 V ( x) =, v2 λ( x) if x < 0 [5] where λ, 0 v and 0 v2. This funcion reflecs loss aversion via parameer λ and diminishing sensiiviy via parameer v, v 2. As shown in 25