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THE IMPACT OF DC PENSION SYSTEMS ONPOPULATION DYNAMICS

Bonnie-Jeanne MacDonald* and Andrew J. G. Cairns†

ABSTRACT

This study investigates the risk inherent in defined contribution (DC) pension plans on an in-dividual and aggregate basis, based on U.S. data. Our aim is to gain insight into the consequencesof a DC pension scheme becoming the predominant pillar of retirement income for an entiresociety. Using the stochastic simulated output of a DC flexible age-of-retirement model, we firstdetermine the optimal investment strategies. We then examine the demographic retirement dy-namics of an entire population of DC pension plan participants.

We observe that even for the most risk-averse plan members there is a high level of uncertaintyin an individual’s age at retirement. At the aggregate population level, we find that this uncertaintydoes not get dampened to any great extent by a diversification effect. Instead, the central roleplayed by the market in determining retirement dates results in significant variation in the de-pendency ratio (the ratio of retirees to workers) over time. In addition, an attempt to amelioratethe outcome by introducing additional realistic features in the DC population modeling did littleto dampen this volatility, which suggests that countries dominated by DC schemes of this typemay, over time, be exposed to significant risk in the size of its labor force.

1. INTRODUCTION

The shift from defined benefit (DB) to definedcontribution (DC) pension plans is a prevailingphenomenon throughout the pensions world.Generally the income support system for retireesis composed of three parts: personal savings,occupational pension plans, and government-provided social security. We define a DC pensionplan as one that provides for each employee thedeposit of a certain percentage of pay into an in-dividual account that ultimately determines theirretirement benefit. At retirement, individualsmay wish to annuitize their savings or choose an-other medium of retirement funding. The DCpension design is, then, fundamentally an individ-

* Bonnie-Jeanne MacDonald, FSA, is a PhD candidate at the Depart-ment of Actuarial Mathematics and Statistics at the Maxwell Institutefor Mathematical Sciences, and the School of Mathematical andComputer Sciences, Heriot-Watt University, Edinburgh, Scotland,EH14 4AS, United Kingdom, [email protected].† Andrew J. G. Cairns, PhD, FFA, is Head of Actuarial Mathematicsand Statistics at the Maxwell Institute for Mathematical Sciences, andthe School of Mathematical and Computer Sciences, Heriot-WattUniversity, Edinburgh, Scotland, EH14 4AS, United Kingdom,[email protected].

ual savings account, except that the contribu-tions are made by more than the individual andemployer contributions increase the likelihoodthat an annuity must be purchased after retire-ment. Therefore, the first pillar of retirement sav-ings is inherently a DC plan. Second, amongemployer-sponsored pension plans in the UnitedStates, a strong trend toward DC pension planshas been in effect for over two decades (Ostasz-ewski 2001). The U.S. private DC plan market in-cludes the popular 401(k) plan, which has grownat such a rate that over the next 30 years it couldpotentially become the largest source of retire-ment wealth across the nation (Poterba, Venti,and Wise 2000). The common hypotheses thathave been put forth to explain the shift to DCplans include the following:

• The simplicity of DC plan designs• The reduction in risk to employers when under-

taking such a change in plan design• The opportunity for plan sponsors to reduce

their annual contributions• The rising costs associated with the govern-

ment’s increased regulation of DB plans and

18 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 11, NUMBER 1

• The superior portability properties of DC plans,which fit today’s more mobile and independentworkforce.

In addition to these hypotheses, Ostaszewski(2001) postulated that the weakening of realwage growth relative to real investment returnsin the United States has created a macroeco-nomic incentive for individuals and plan sponsorsto switch from DB to DC plans. Brown and Liu(2001) expanded on Ostaszewski’s study by ap-plying Canadian data to argue that the shift oc-curs when tax legislation and pension regulationincreasingly favor DC plan designs over DB plandesigns.

The third source of retirement income supportin the United States is Social Security. It, too, isthreatening to move toward a DC state pensionsystem design (Cogan and Mitchell 2003).1 InNorth America, financial security for the elderlyis increasingly becoming a major concern as theBaby Boomers approach retirement, thus chang-ing the demographic profile. For example, by2025, Canada’s aged dependency rate (the ratioof persons aged 65 and over to the populationaged 15–64) is expected to reach 38%, and theexpectation in the United States is not far behindat 34% (Turner and Watanabe 1995). As the pop-ulation ages, stress on the economy would ariseas unfunded pensions begin to be paid out, af-fecting Social Security and employer pensionplans. To control costs, many plan sponsors haveopted to switch their DB pension plans to DCpension plan designs; moreover, the UnitedStates has begun to debate seriously whether itshould replace its current Social Security pro-gram to include DC-style personal accounts. TheUnited States is not alone in considering this typeof structural pension reformation; Mitchell(1998) concluded that the DC pension plan de-sign caught the world’s imagination during the1990s. Approximately half of Latin America, aswell as eight countries in Eastern Europe, haveundertaken system restructuring to involve DCfeatures (Gill, Packard, and Yermo 2004). A com-mon purpose behind replacing a state system’sunfunded DB with a funded DC plan, where the

1 The U.S. president’s commission to strengthen Social Securityby including personal retirement accounts can be found atwww.csss.gov.

individual accounts are privately invested, is thecountry’s desire to reduce the government’s rolein economic life and to increase reliance on mar-ket institutions (Turner and Watanabe 1995).Turner and Watanabe also argued that privatizinga country’s pension system may serve to supportthe domestic financial market by increasing na-tional savings (and hence drive up real capital in-vestments), although the size of the effect is notagreed on by researchers. The problem in theUnited States, however, according to Gill and Ta-tucu (2005) in their study to draw lessons for theUnited States from the social security reforms inLatin America, is primarily the aging population.They wrote that the Latin American reforms re-flected a loss of faith in governments to act re-sponsibly in ensuring the promised pension ben-efits. This is not the case in the United States,where Social Security is well managed, and, al-though the current surplus is adequate to paynearly 80% of the promised benefits over the next75 years, the primary objective is to ensure thefuture financial solvency of the system (Gill andTatucu).

Consequently, the DC pension plan has grownfrom being a single and relatively unimportantsource of pension income to becoming two sig-nificant sources, with the potential of also becom-ing the third and final source of income for thenonworking elderly.

The practical application of a DC pension sys-tem has numerous drawbacks. DC pension planschemes are notorious for the uncertainty in thelevel of pension that they can provide. Researchand experience has shown that, with a fixed ageof retirement, it is difficult to predict accuratelythe pension income under a DC pension plan de-sign. Relative to a DB pension income bench-mark, the accumulated wealth in a DC plan canbe extremely risky (Blake, Cairns, and Dowd2001). Chile, which completely reformed its statepension plan to a DC system two decades ago,affords its citizens some protection from the un-certainty in their retirement income by ensuringa minimum pension income. The provision of aminimum pension income, however, could becriticized as an expensive welfare system and, toa large extent, could expose the governmentalpension sponsor to abuse and antiselection(Brown 1999). Furthermore, such a pension sys-tem could not be self-sustaining since a DC pen-

THE IMPACT OF DC PENSION SYSTEMS ON POPULATION DYNAMICS 19

sion system can duplicate the benefits offered bya DB state pension systems only through furtheradditional contributions of the state or the indi-viduals concerned. Other ancillary benefits suit-able for a DC system may include disability anddeath benefits, as well as a top-up for females,who would suffer an inherently lower pension in-come at the time of annuitization owing to theirlongevity. DC pension plan designs are frequentlyapplauded for their portability properties, butthis feature does not avail a state pension plan asthere is usually no need for a pension to be port-able, perhaps with the exception of emigrants.Under a DC pension plan, there is no redistribu-tion of wealth. Finally, DC pension plan partici-pants are responsible to pay onerous costs andfees, such as fund manager fees; in fact, admin-istrative expenses are higher than in a socializedsystem (Brown 1999). A recent World Bank re-port (Gill, Packard, and Yermo 2004) investi-gated the outcome of the Chilean government’shaving implemented a DC pension system, andthe Turner Report (Pensions Commission 2005)described the current situation in the UnitedKingdom, where the state pension plan also in-corporates private accounts. Both commentariesconfirm several of the predicted shortcomings ofimplementing a DC state pension system notedabove, such as the rising costs of Chile’s mini-mum pension income in consequence of low-income workers’ preferring not to save ratherthan have their minimum pension reduced. Be-sides Latin America and Britain, additional inter-national reform comparisons have been studiedto provide perspective to the current U.S. SocialSecurity debate. Simonovits (2005), while outlin-ing the relevant lessons for the United Statesfrom the Hungarian reforms, concluded that par-tial privatization of Social Security is not helpfuland does not solve the problems of SocialSecurity.

The debate around privatization of Social Se-curity has given rise to many arguments for andagainst the introduction of individual savings ac-counts. The arguments outlined above are gen-erally drawn from analogies to employer pensionplans (Blake, Cairns, and Dowd 2001; Brown andLiu 2001; Ostaszewski 2001), meaningful lessonsdrawn from the experiences of other countriesthat have undergone similar pension reforms(Gill and Tatucu 2005; Simonovitz 2005), and

general observations that stem from the currentpolitical, economical, and regulatory arena (Co-gan and Mitchell 2003; Mitchell 1998; Brown1999). (Please note that some of these studiesoverlap in two or even three categories.) Al-though extremely beneficial and relevant, thesepapers do not provide a clear picture of the im-pact of a ‘‘pure’’ nationwide DC plan, that is, theaggregate effect of a DC plan without the con-straints of additional regulations or the contem-porary complexities that may exist in the countrytoday. This study uses a bottom-up approach toemphasize and focus on the general effects of apure DC pension plan monopolizing the incomesupport system for the retired members of apopulation.

We assume that the individual will lengthen orshorten their working career depending on theiraccumulated pension savings in relation to theirexpected life span; in this way, participants candelay their retirement until a sufficient pensionfund has been accumulated. Rather than focus onthe accumulated wealth at a specified retirementage, we investigate the likely retirement age ofDC participants if they hope to maintain a fixedstandard of living at retirement, which would sus-tain them till death. The motivation behind thisassumption is that it is necessary for the primarysource of retirement income to provide a pensionsufficient to offer financial security to the elderlyand, therefore, facilitate the transition from em-ployment to retirement. Owing to uncertainty inits accumulated wealth, such a requirement couldnot be fulfilled by a pure DC pension plan if thepension delivery date is fixed. If there were rigidrestrictions on the worker’s age of retirement,dictated by either statute or company policy, itwould be difficult for a DC pension plan design towork on a large scale since inadequate pensionswould be commonplace, rather than the excep-tion, as is the case in a well-designed DB pensionscheme. Second, in a pure DC pension plan, aparticipant’s retirement would be entirely fi-nanced by their accumulated fund, and, asidefrom personal circumstances, there would be noembedded incentives to retire at any particularage. Lachance (2003) similarly justified a flexibleretirement date by noting that a fixed retirementage in the context of a DC plan, although com-mon and convenient, is inconsistent with the ab-sence of structural incentives in the retirement


decision as well as the mounting evidence thatthe performance of an individual’s investments af-fects their decision to retire.

Given that our objective is to observe the fullimpact of a DC state pension system, the realisticexistence of retirement age flexibility in a DC de-sign is central to our analysis of the demographicimplications of a nationwide DC plan. Further-more, allowing a flexible retirement date providesthe DC plan the unaided opportunity to fund fullyan adequate level of pension income, without theuse of top-ups or a minimum pension income. Weregard results bearing too high a probability ofretirement at very old ages (for example, aboveage 80) as a drawback of the DC pension designsince, in reality, illness and other factors couldinfluence an individual’s decision to retire if suf-ficient funds have not built up before reaching anelderly age.

Realistic modeling of the retirement savingsbehavior for a population is an ongoing process,since there is always room to add elements thatare more realistic. Accordingly, our currentmodel includes simplifying assumptions as well asa variety of realistic features. One important as-sumption is that we have taken as given the dy-namics of the stock market, which means that wedo not attempt to model the macroeconomic ef-fects of the mass actions of the DC plan memberssuch as mass demand for equities or liquidationof a particular asset. Clearly, we expect thatchanging retirement patterns would impact theprosperity of a country by impacting tax revenue,labor force growth, social programs, nationalsavings, and company profits. These outcomes,which we comment on in Section 3.2, in turnwould affect the prices of financial assets alongwith wages. The effect could contribute to bothpositive and negative feedback; thus, without itsinclusion, we are limiting ourselves to a slightlyartificial model, and the results of this reportcould be described as some worst-case scenarios.Having said that, we hope this paper will providean initial impression of what could occur, leadingto a wider discussion of nationwide DC pensionplan design.

The advantage of choosing a DC pension planfor the plan sponsor is that it shifts the risk of aninadequate pension from their hands to those ofthe individual. When a DC conversion occurs onthe national level, we propose that there are in-

trinsic risks not only to the economic well-beingof the plan participants, but as well to the entirenation. We hypothesize that introducing a na-tional DC pension plan as the primary source ofretirement income would result in the financialmarket’s condition strongly affecting the retire-ment pattern of the citizens, which could becontrary to the interests of the society at large.Consequently, the proportion of pensioners andworkers in the population could well be unpre-dictable and uncontrollable by the state, as wellas possibly detrimental to the economy.

Section 2 outlines our assumptions in model-ing the DC population, including a short descrip-tion of the steps undertaken to build and executethe population retirement model. Section 3.1 ex-plains how we initially use the stochastically sim-ulated results to select efficient portfolios for themembers of the population. We then discuss theretirement pattern behavior of the aggregategroup of DC participants in Section 3.2. Since theresults are contingent on the validity of themodel, Section 4 investigates their sensitivity tothe model’s simplified assumptions by assessingthe impact of adding more realistic features tothe population retirement savings model. In Sec-tion 5 we conclude by examining a DC plan de-sign feature that could potentially alleviate thepossible risks that a DC pension system poses onthe size of the labor force from one year to thenext.

2. MODEL AND ASSUMPTIONS

In this study we aim to model the DC pensionsystem in its ‘‘pure’’ form. The three sources ofretirement income—personal savings, social se-curity, and employer benefits—are treated as asingle traditional DC pension plan. This is donefor simplicity, but also to fulfill our aim to furtherthe understanding of the impact of a traditionalDC plan’s sustaining the retirement income of anentire population. We regard any additional fea-tures that differentiate a hybrid DC plan from itspure form as clouding our understanding, and,consequently, they are removed. There is neithera requirement to purchase an annuity contractafter retirement, a minimum pension guarantee,nor a mandatory age of retirement, either by stat-ute or by company policy. Every member withinthe population adopts a retirement strategy that


bases their retirement date on the size of the pen-sion income that can be supported by their re-tirement savings. In our pure DC plan populationmodel, annual contributions are made to eachparticipant’s individual retirement account, andthe member can direct his or her investments tofive different assets.

Despite the use of U.S. data, our DC populationmodeling assumptions are intended neither re-flect the current state of pension plans in theUnited States, nor project the effect of the pro-posed U.S. Social Security reforms. As Section 1discusses, the adoption of a state DC pension sys-tem is a growing reality in numerous countries,including the United States, and the shift withinoccupational pension plan scheme designs fromDB to DC is also a prevailing phenomenon. Werealize, nevertheless, that a collective shift in theUnited States among all DB providers to an iden-tical DC plan design is unlikely unless there ex-isted strict government regulation. We recognizethat variety among the employer pension plan de-signs would likely remain even if all occupationalpension schemes became the DC type, and that aDC state pension system would not have a puredesign, since it generally contains ancillary fea-tures and limitations on its participants. Al-though not realistic in practice, the pure DC de-sign approach in this study is intended to providea clearer picture of the overall impact of indi-vidual accounts fueling the retirement of themasses.

A DC accumulated pension income depends onsuch factors as the pension portfolio’s rate of re-turn, salary growth, annuity discount rate, and re-lationship between each rate specified. We as-sume that the pension fund is invested across fiveassets: equities, fixed-income bonds, index-linkedbonds, risk-free one-year bonds (cash), and index-linked cash. This study uses the Vasicek interestrate model (Vasicek 1977) to underpin the dy-namics of all the asset returns. Appendix A de-scribes the stochastic asset-return model. Theparticipant’s salary model incorporates a meritscale, the prevailing inflation rate, and real wagegrowth. The last two components are an integralpart of the stochastic asset model. The salarymodeling is elaborated on in the Appendix. Withrespect to the asset accumulation model, we haveapproached a macroeconomic problem using mi-croeconomic tools given that we do not model

the possible link among the retirement behaviorof the working population, asset demand, wages,and the financial market returns, which is ex-plained further in Section 1. Also in the asset ac-cumulation model, there are neither taxes, ex-penses, nor allowances for profit in the financialassets’ pricing and the management of the DCplan.

The asset allocation strategy of the DC partic-ipants in our study is static, which means thatparticipants maintain constant asset proportionsin their portfolios throughout the accumulationphase. This results in constant portfolio rebalanc-ing at the end of each year. Although a simplify-ing assumption, Blake, Cairns, and Dowd (2001)showed that a well-chosen static asset allocationstrategy performs substantially better than vari-ous common dynamic strategies, such as the pop-ular ‘‘lifestyle’’ strategy. The contributions madeto the DC pension plan are regarded as beingtruly invested, a method of funding known as afunded stated pension system.

Furthermore, the model includes demographicassumptions such as retirement decision-makingbehavior, mortality, contribution rate, meritgains on salary (noted above), age of plan enroll-ment, and the population’s age and gender dis-tribution. Sections 2.1, 2.2, 2.3, and 2.4 are ded-icated to four of these assumptions, while the restare detailed below.

We carry out a series of analyses. The first DCpopulation retirement model, in Section 3.1, isthe most basic. The participants enter the DCpension plan on their 25th birthday, have no de-pendents, have average annual merit gains of 2%,and make annual contributions of 10% of salaryat the beginning of each working year (see Sec-tion 2.3). Thus, the initial analysis assumes thatall members enter the plan at the same age, paythe same contribution rate, follow the same ca-reer path, and adopt the same investment and re-tirement strategies. Such a level of uniformity isunlikely to exist among actual population mem-bers. The purpose of choosing a simple model isthat it aids in identifying the efficient portfolios;nevertheless, its value in realistically portrayinga population’s retirement savings behavior isless obvious. Our intention in first investigatingthis extreme case is not to put this forward asa realistic representation of the risks we facein the future. Instead, our objective is mainly


exploratory, and we present it here as a potentialworst-case scenario. We will see that this extremesystem results in a considerable degree of varia-bility in the size of the working population.

In subsequent sections, we work away from thisworst case by introducing heterogeneity into thesystem and examining its effect. We improve themodel’s level of sophistication by developing het-erogeneity among the DC participants in their

• Investment strategy, in Section 3.2• Entry age to plan, in Section 4.1• Career flight path (high flyers/low flyers), in

Section 4.2 and• Contribution rate, in Section 4.3.

These realistic refinements are tested to assesstheir ability to dampen the severity of the simu-lated demographic outcomes that arise from thesimulation of the DC population.

2.1 Retirement Decision ModelThe reasons why people decide to retire on a par-ticular date are many. Factors influencing theretirement date of a DC plan member includeaccumulated wealth, health, age, preference forleisure time over work, direct pressure from em-ployer, and general peer pressure owing to socialcustoms (Brothers 1998).

In our study we base retirement on the level ofpension income that can be provided by the DCparticipant’s accumulated wealth. Pension in-come is determined by dividing the pension fundon the retirement day by the annuity factor(Wealth(t)/ ae�t(t)). The pension income dividedby the individual’s salary at retirement (Salary(t))produces the replacement ratio, RR(t):

Wealth(t)/ a (t)e�tRR(t) � ,Salary (t)

where t is time since entry into the plan and e isthe individual’s age of plan enrollment. The nu-merator and denominator are gross incomes andare not adjusted for taxes, making the replace-ment ratio a pre-tax measure. The annuity factor,ae�t (t), is the present value at the time of retire-ment, t, of one unit of an annuity for the remain-ing life of an annuitant aged e � t. Since we arenot assuming mandatory annuitization, moneymay not actually be withdrawn to purchase an an-

nuity at retirement. The philosophy, however,that an adequate amount of accumulated fundsis necessary for retirement is the same no matterwhat medium of retirement funding is used, andthe replacement ratio based on a fixed-income an-nuity is the selected measurement for adequacy.We also considered index-linked annuities as thefunding medium benchmark since they offer thebenefit of maintaining the purchasing power ofthe pension income. We chose fixed-income an-nuities since, to achieve the same initial pension,the higher cost of the index-linked feature wouldsimply drive up the retirement ages and the gen-eral conclusions would be unaltered. In addition,index-linked annuities (or voluntary annuitizationin general) are scarce among retirees in practicesince people typically prefer higher initial pen-sions. Brown and Warshawsky (2001) further ex-plored the explanation behind the reluctancy ofindividuals in the United States to annuitize theirDC fund, despite the potential benefits argued inDavidoff, Brown, and Diamond (2005). Brown andWarshawsky (2001) explained that there is a lackof inflation protection in the few annuity con-tracts purchased, suggesting to us that a fixed-income annuity assumption is preferable.

This study considers the retirement age as arandom stopping time, when the pension pur-chasable exceeds two-thirds the outgoing sal-ary. In other words, at the beginning of eachworking year, the replacement ratio is measuredso that

2–Retirement Age � min {e � t � RR(t) � }.3

This decision rule, thus, incorporates the reason-able view that retirement will be deferred untilsuch time as the member can afford retirement.

We model the retirement decision based on theaccumulated pension wealth of the participant aswell as their expected longevity. The two-thirdsrule does not explicitly allow for age dependencyin the decision, but age is taken into account in-directly through the annuity factor:

�

a (t) � P(x (t),t,t � s) P ,�e�t 1 s e�ts�0

where P(x1(t),t,t � s) is the price at time t of arisk-free zero-coupon bond that matures at timet � s and x1(t) is the instantaneous risk-free rateof interest at time t. The price formula for


P(x1(t),t,t � s), given in the Appendix, is basedon the Vasicek model. The annuity factor de-creases as the individual ages owing the conse-quential fewer number of payments expected tobe made due to higher expected mortality. Thelower the annuity factor, the higher the pensionincome and the likelihood that it exceeds 66.67%of the outgoing salary.

The two-thirds target is selected to representan adequate level of income for retirement, a tar-get value that is not largely different from theprevailing replacement ratio for singles in theUnited States and throughout the OECD coun-tries. Disney, d’Ercole, and Scherer (1998), in anOECD study on aging, measured the replacementratios across all OECD countries. Having onlygross U.S. income data, they determined that theaverage pre-tax salary replacement ratio of singlesin the United States is 62% when incorporatingall sources of disposable income in both the nu-merator and the denominator. Our pre-tax re-placement ratio benchmark is close to this actualaverage.

Moreover, the OECD study found that the re-placement ratios across all the OECD countriesexhibit a high degree of uniformity, with a typicalaverage of 70%, when the calculations are carriedout using all sources of retirement income net ofdirect taxes paid. According to the OECD study,a replacement ratio that is based on pre-tax in-come underestimates one that is net of taxes.It follows that, in the presence of a progressivetax schedule, the replacement ratio benchmarkwould exceed two-thirds once the taxes are de-ducted and draw nearer to the typical 70%.

In addition to our target’s consistency withcurrent data, the two-thirds replacement ratioalso falls within the range of an adequate pen-sion, given as a rough guide in The Handbook ofCanadian Pension and Benefit Plans (Greenan2002). Here an after-tax replacement ratio shouldfall between 60% and 70% to maintain a pen-sioner’s preretirement standard of living. Fur-ther, the Canadian Institute of Actuaries (1996)reported that an individual can preserve theirpre-retirement standard of living if their in-come replacement ratio falls between 60% and74%, where the exact level depends on theirunique earnings level since a higher rate is re-quired for low-income workers to satisfy minimalneeds.

2.2 Mortality ModelThis study uses the United States Life Tables2002 for females and males (Arias 2004), pub-lished by the National Center of Health Statistics.The data used to prepare these tables are, withinthe United States, final numbers of deaths foryear 2002, postcensal population estimates forthe year 2002, and data from the Medicare pro-gram of the Centers for Medicare and MedicaidServices.

The annual mortality rate for a participant agedx (qx) is a fixed blend of 50% of the male mor-tality rate and 50% of the female mortalitym(q )x

rate both determined from the U.S. life ta-w(q ),x

bles. To focus on total population dynamics, uni-sex rates are chosen to simplify the calculationprocedure even though this does not replicate thepopulation since females, having a longer life ex-pectancy than males, would have a higher weight-ing as the two genders age. Dealing separatelywith the males and the females by applying sex-distinct mortality functions would have alteredour results only very slightly and would not affectour general conclusions.

2.3 Contribution RateLike the contribution rate in our model, a 10%mandatory payroll tax pays for the individual ac-count plans in Chile. In the United States, Po-terba, Venti, and Wise (2005) summarized recentstudies that measured the mean 401(k) contri-bution rate, including both employee and em-ployer contributions. The reported values fromtheir references are between 8.7% and 12.6%. Fi-nally, Blake, Cairns, and Dowd (2001) assumed a10% rate while simulating DC individual accountssince it is a typical contribution rate of DC plansin the United Kingdom.

2.4 Simulation of an Entire PopulationWe use a stationary and stable population modelto simulate the demographics of a population.That is, there is no growth in the population size(a stationary population) and the population agedistribution is identical from one period to thenext (a stable population), (Bowers, Cairus, andDowd 1997). The model has 81 cohorts in thepopulation. At every point in time, the cohortsrange in age from 20 to 100. The relative size ofeach cohort aged x is labeled lx:


x�� dtt20l � e � p .x x�20 20

For example, the size of the aged 20 cohort attime t is 1, without loss of generality, and is sizesp20 by age 20 � s at time t � s.

From the output, each year of simulation cal-culates the dependency ratio (ratio of the numberof retirees to the number of workers) based onthe constant proportional sizes of each agegroup.

2.5 Dependency RatioWe measure the retirement dynamics of the so-ciety of DC pension plan participants with the de-pendency ratio, defined as the ratio of the num-ber of retirees to the number of nonretirees at orabove the age of 20. This second group of indi-viduals is referred to as workers, although theymay or may not have entered the pension plan:

# retired populationDependency Ratio � .

# working population

(Note: this is not the only definition of a depend-ency ratio. In Section 1 the aged dependency rateis defined as the ratio of persons aged 65 and overto the population aged 15–64. The reason behindthis deviation is that our purpose is to measurethe demographic labor force dynamics ratherthan the age structure of the population.)

2.6 Stochastic SimulationIn this paper we use stochastic simulation to in-vestigate the range of outcomes for a variety ofquantities of interest. The steps followed in ourmodel construction and simulation are as follows:

1. We begin with the asset model specificationand calibration. Asset return modeling in-cludes choosing an asset return model that isparameterized according to the realized U.S.returns and volatilities. Appendix A explainsthe derivation of the accumulation model, pa-rameter estimates, and sources of data.

2. We then carry out the demographic modelspecification and calibration. In this step weselect the mortality table, age of plan enroll-ment for the participants, design of the DCpension plan, and characteristics of the pop-ulation model, including their retirement sav-ings behavior. This aspect of the study is pre-sented in Sections 2–2.4.

3. The simulation calculates the pension wealthfor each DC participant so that, between timet � 1 and t, an individual’s pension account isaccumulated forward in the following manner:

Wealth(t) � (Wealth(t � 1) � 0.1

� Salary(t � 1))(1 � i(t)),

where i(t), the investment return betweentimes t � 1 and t and depends on the specifiedportfolio strategy. The wealth and salary arecarried forward for each of the simulated yearsuntil the member retires, when RR(t) �66.67%. Section 2.1 gives the details of the re-tirement model. For computational efficiencyand to aid comparisons among different strat-egies, we simulate the asset returns once anduse this same set of sample paths for each ofthe investment strategies.

4. Given the various assumptions, the programgenerates an empirical distribution of possibleretirement ages for each individual and de-pendency ratios for the entire population cor-responding to each particular investmentstrategy. To accompany each simulated out-come, relevant information regarding the pop-ulation and the financial market is alsogenerated.

3. RESULTS FOR AN ENTIREPOPULATION

Our first step in analyzing the results is to cal-culate the efficient portfolios in Section 3.1, and,from among them, we select three asset mixes torepresent the investment strategies of the mem-bers of the population in Section 3.2. In the sub-sequent simulations, we focus on relevant aspectsof the population’s retirement dynamics using cu-mulative distribution functions, time series plots,and scatterplots. These plots provide insight intothe consequences of DC pension plan schemes.Our analysis suggests that a DC pension systemcould have a strong impact on the labor force sta-bility of the population.

When measuring the value of the DC pensionpension plan on an aggregate level, our indicatorof success or failure is the dependency ratio, asSection 2.5 explains. The appropriate interpreta-tion of the dependency ratio in our results may


Table 1Dependency Ratio’s Simulated Average Value and Standard Deviation, along with Equivalent

Average Age of Youngest Group Retired for Each Asset

Asset

100%Index-Linked

Cash100%Cash

100%Index-Linked

Bond100%Bond

100%Equity

Dependency ratioAverageStandard deviation

0.25100.0557

0.28420.0564

0.31390.0507

0.34050.0987

0.76770.3051

Average equivalent age of youngest group retired 69.48 67.94 66.68 65.62 53.80

not be readily obvious. Our simulation assumesthat the entire working population uniformlyenrolls in a DC pension plan design and thata worker’s retirement is triggered by an ade-quate accumulated pension income; thus, a lowdependency ratio raises concern, as this indicatesthat elderly workers are financially unable to re-tire owing to the insufficiency of their DC pensionfund account. On the other hand, a high depend-ency ratio signifies that the DC pension plan isallowing workers to retire at young ages. Nor-mally a high dependency ratio is undesirablesince it is a symptom of an aging population. Agrowing proportion of the elderly and nonworkingmembers of the population would put a strain oneconomic programs such as Social Security andhealth care. Yet, considering that the distributionof ages is unchanging within our model, anincreasing dependency ratio is a positive out-come that measures the financial ability of indi-viduals to retire earlier. Since the plan is funded,a high dependency ratio would not incur costs forthe working population to provide for the finan-cial needs of the retirees. ‘‘Dependency ratio’’could be a misnomer in our study since only thoseworkers with a financially secured retirementbecome pensioners. Consequently, retired mem-bers do not require the financial welfare supportof the working population; nevertheless, theydo continue to rely on the workers to producethe necessary goods and services for theirconsumption.

3.1 Choosing Optimal InvestmentPortfolios

The choice of investment strategies plays an im-portant role in the retirement outcome for an in-dividual. In this section we evaluate the effects of

the investment strategies on the individuals byobserving their retirement age patterns and onthe population dynamics by measuring how thedependency ratio varies over time. The resultssuggest that, among the efficient portfolios, eq-uities perform impeccably well in the interest ofindividual members but pose a threat to the sta-bility of the population’s dependency ratio.

Using the initial DC population model whosehomogeneous assumptions are explained in Sec-tion 2, we are able to investigate a range of dif-ferent asset allocation strategies by assuming ev-eryone in the population adopts the samestrategy. In the simulations, we consider 581 dif-ferent investment strategies, each containing adifferent combination of bonds, cash, index-linked bonds, index-linked cash, and equities. Theportfolio’s exposure to bonds, index-linked bonds,and equities is tried in increments of 10% of thetotal portfolio, but only 20% increments for thecash and index-linked cash. The less precise in-crements of cash and index-linked cash are incon-sequential since they are absent from the result-ing efficient portfolios.

The homogeneity of the population with regardto their investment strategy results in there beingno ‘‘gaps’’ in the ages of the retirees. More spe-cifically, for each year of simulation, there is asingle age in which everyone at or above is retiredand everyone below is working. Every dependencyratio level has an equivalent age therefore, thatindicates the age of the prevailing youngest groupretired. For example, a dependency ratio of35.65% indicates that everyone at or above theage of 65 is retired. Table 1 presents the averagedependency ratios and their equivalent ages as-suming a 100% investment in each of the fiveassets.


Figure 1Simulated Empirical Cumulative Distribution Function of Retirement Age for Each Asset

0.0

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1.0

Pro

b A

ge o

f Ret

irem

ent <

x

100 % Index–Linked Cash 100 % Cash 100 % Index–Linked Bond 100 % Bond 100 % Equity

45 50 55 60 65 70 75

Age of Retirement, x

The results of the simulation suggest that theideal investment strategy in terms of offering cit-izens the earliest retirement age would be to al-locate the majority of the funds in equities, withthe remainder in bonds and index-linked bonds,while avoiding both cash and index-linked cash.According to Table 1, the performance of a pureequity portfolio exceeds all other asset allocationstrategies with a mean dependency ratio of76.77%. The inverse relation between the depend-ency ratio and the mean age of retirement neces-sitates that the equity portfolio carries the young-est mean retirement age from among theinvestment portfolios, which is just under age 54.In comparison, a pure index-linked cash portfolioproduces a mean dependency ratio of 25.1%,which corresponds to a mean retirement ageabove 69. Thus, from the individual investor’s per-spective, choosing equities would accelerate theirpotential retirement date.

An analysis of a DC member’s simulated retire-ment age provides further insight into the benefitof equities for the individual. In Figure 1, the em-pirical cumulative distribution function (CDF) ofa participant’s retirement age is graphed foreach asset. The CDFs illustrate the advantage of

holding an equity portfolio for an investor whotolerates some risk. (Each point on the CDFcurve shows the probability that a participant’sretirement age will fall below a particular level. Ifone curve lies more to the right than the othercurve, then that particular investment strategy islikely to produce higher ages of retirement thanthe other investment strategy.) The CDF gener-ated from an equity portfolio is distinctivelyshifted to the left. The index-linked bond portfo-lio is a distant second in terms of producing alower retirement age. On the right tail of the eq-uity portfolio CDF, after crossing the other CDFs,is where it is more likely that the equity portfoliowill deliver a later retirement. Furthermore, if aCDF curve rises more steeply between 0 and 1than another, such as the index-linked bond’sCDF relative to the equity’s, then that strategy isless volatile. The high crossover point (age 70)between the two CDFs suggests that, although in-vesting the majority of funds in equities createsless certainty in the age of retirement, it is stillthe best investment strategy since there is moreopportunity for early retirement and, except forthe worst-case scenarios, the individual wouldmost likely retire before or at the age that an


Figure 2Simulated Time Series Plot of Dependency Ratio for a Population of Equity Investors

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1.5

Dep

ende

ncy

Rat

io

100% Equity Portfolio Dependency Ratio

0 50 100 150 200 250 300

Time

index-linked bond investment strategy would havepermitted.

The benefit to a worker of increasing their riskyasset allocation as a result of having a flexibleretirement date has been ascertained in pre-vious studies. Lachance (2003) examined how aworker’s optimal portfolio choice is influenced bytheir capacity to adjust their retirement date asa function of market fluctuations. In her studyshe derived a closed-form solution for the opti-mal consumption and portfolio choice when aworker’s retirement is flexible. Utilizing this so-lution, she showed that more investment risk canbe assumed if a worker’s retirement date is flex-ible instead of being fixed.

Interestingly, these results are also consistentwith current asset allocation trends of DC pen-sion plan participants in the United States, whereDC pension plan investors are increasingly mov-ing to equity investments. Between 1983 and1996, U.S. members have increased the propor-tion of their DC pension plan assets in equitiesfrom 27% to 60% (Mitchell 1998).

Overall, an efficient investor should direct alarge proportion of their funds to equities, de-spite the less certainty in the age of retirement.

The fallacy of composition (Brown 1997) ar-gues that what could be good for the individual

could possibly not be good in aggregate: that is,although an investment portfolio could be opti-mal for an individual DC participant over theirlifetime, this same investment strategy could po-tentially not be the optimal solution for an entirepopulation over many lifetimes. One concern ona public policy level is the instability of the de-pendency ratio, as it would cause an unstableeconomy. Therefore, although riskier investmentsare beneficial to the individual, the additional vol-atility that is incurred in the dependency ratiocould be harmful to society and the economy asa whole. This is depicted by the erratic behaviorof the dependency ratio in Figure 2, which tracksthe dependency ratio for a population of equityinvestors over a 300-year simulation. Figure 3 dis-plays the corresponding age of the youngest re-tired member in the population. It, too, is tre-mendously irregular, and its range spans 38 years.If we are to do what is best from an aggregateperspective, we require a risk measure that re-flects the need for a stable dependency ratio. Thestandard deviation is an appropriate risk measurefor our purposes since it describes the stability ofthe dependency ratio.

Figure 4 plots the mean of the simulated de-pendency ratio against its standard deviation, cor-responding to each of the 581 asset strategies


Figure 3Simulated Time Series Plot of Youngest Group Retired in a Population of Equity Investors

50

60

70

80

Age

of Y

oung

est G

roup

Ret

ired

100% Equity Portfolio Youngest Retired

0 50 100 150 200 250 300

Time

detailed at the beginning of this section. The re-turn measure is the mean dependency ratioacross 4500 years of simulation, while the riskmeasure is the standard deviation of the simu-lated dependency ratios. A long simulation run ispreferred over multiple short runs since the latterwould likely be biased by initial conditions. Theplot gives an impression of the opportunity setbased on this measure of risk, traced out by the581 investment strategies. From the opportunityset, we can infer the efficient portfolios, which arethe portfolios that carry the lowest risk for eachgiven level of return.

It is useful to note several general aspects ofthe opportunity set plot in Figure 4:

• The indicators of success are high mean de-pendency ratios (members, from their perspec-tives, are happy because they are retiringearlier, on average) with low volatility (stablelabor force); thus, higher values on the y-axisand lower values on the x-axis are the preferableportfolios. Specifically, the points in the oppor-tunity set nearer to the top left are good.

• Since the population is discretized and allmembers follow the same strategy, the depend-ency ratio is restricted to a discrete set of val-

ues determined by the youngest retiree at anygiven time.

• Each cluster of points has a constant propor-tion invested in equities.

• The ‘‘X’’s mark the portfolios in each fixed-equity cluster on the efficient frontier. Thecompositions of the efficient portfolios, P0 toP100, are given in Table 2.

Figure 4 ranks the pure equity investmentstrategy as the most dispersed of the portfolios.It remains, nevertheless, as an efficient portfolio.Cash and index-linked cash appear to be a poorpension investment choice, as shown by their ex-clusion from every efficient portfolio listed in Fig-ure 4. Figure 4 also suggests that, among theefficient portfolios, investing in equities wouldelevate the mean dependency ratio while bothtypes of bond assets provide stability. Figure 5 de-scribes the breakdown among the assets by ex-tracting and enlarging the asset mixes with anequity exposure of 60% from the opportunity setin Figure 4 (box B). Looking from top to bottom,Figure 5a explains that increasing the portfolio’sproportion of index-linked cash would typically di-minish the mean dependency ratio without thebenefit of lowering the standard deviation. In con-


Figure 4Simulated Opportunity Set for a Population of DC Members Who Homogeneously Allocate Their

Funds in Specified Investment Portfolio

0.05 0.10 0.15 0.20 0.25 0.30

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Standard Deviation

Mea

n D

epen

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atio

P0

P10

P20

P30

P40

P50

P60

P70

P80

P90

P100

B(Figure 5)

Notes: This plot is generated for 581 different asset allocation strategies (small dots). The dependency ratio’s standard deviation is plottedagainst its mean. The efficient portfolios are marked by an ‘‘X’’ and the compositions of these portfolios, P0 to P100, are given in Table 2. Adashed line indicates the efficient frontier. Figure 5 contains an enlarged box B.

Table 2Portfolio Mix for Efficient Portfolios P0 to

P100 in Figure 4

MinimumRisk

Portfolio

%Index-LinkedCash

%Cash

%Index-LinkedBond

%Bond

%Equity

P0 0 0 70 30 0P10 0 0 60 30 10P20 0 0 50 30 20P30 0 0 40 30 30P40 0 0 30 30 40P60 0 0 20 30 50P60 0 0 10 30 60P70 0 0 0 30 70P80 0 0 0 20 80P90 0 0 0 10 90P100 0 0 0 0 100

trast, Figure 5b shows that (by looking from leftto right) a heightened exposure to cash typicallycauses the dependency ratio’s standard deviationto escalate without significant improvement in itsmean. Increasing the allocation to bonds (Fig. 5cand d) shows an opposite, but less clear, effect.Raising the proportion of the index-linked bonds(c) vaguely lowers the standard deviation, and in-creasing the weight in fixed-interest bonds (d)typically raises the mean dependency ratio. One,perhaps obvious, conclusion that we can takeaway from this is that while cash (fixed or index-linked) could be a good low-risk short-term in-vestment, our results suggest that it is not goodas a long-term component of a DC pension planfund.

3.2 Variety in the Population’sInvestment Strategy

We now move towards a more realistic scenario.Having established the efficient investment


Figure 5Enlargement of Box B in Fig. 4

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% Index–Linked Cash Efficient Portfolio

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0% 20%

0% 20% 40%

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0% % Cash Efficient Portfolio

0.188 0.190 0.192 0.194 0.188 0.190 0.192 0.194

Standard Deviation Standard Deviation

(c) (d)

0%

0%

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10%

10%20%

20%30%

40%

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10%20%

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% Index–Linked Bond Efficient Portfolio 0%

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30% 40%

% Bond Efficient Portfolio

0.188 0.190 0.192 0.194 0.188 0.190 0.192 0.194


Notes: Each plot contains the same points; the numbers provided, however, tell us about the proportions invested in (a) index-linked cash,(b) cash, (c) index-linked bond, and (d) bond. The efficient portfolio (P60) is marked by an X.

strategies for the DC members of the population,we first determine three investment strategies ap-propriate for both the individual members andthe entire population. We then examine the dy-namics of the retirement behavior within a pop-ulation whose members follow these three in-vestment strategies. Our analysis shows severe

volatility in the dependency ratio, which is pri-marily driven by the market’s performance.

To add realism to the simulation, we expandthe population’s investment strategies from a sin-gle arbitrary homogeneous investment portfoliochoice to three well-performing portfolios in linewith current market trends. Specifically, in this


second DC population model, we assume thatequal proportions of each age group will allocatetheir wealth to a low-risk fund (Portfolio A), amedium-risk fund (Portfolio B), and a high-riskfund (Portfolio C); each fund is detailed below.Once Portfolios A, B, and C become the invest-ment strategies for their respective one-third ofnew entrants, those members will maintain thesame proportion of assets over their entire work-ing life, classified earlier as a static asset alloca-tion strategy. Thus, we assume that plan assetsare rebalanced annually to maintain predeter-mined proportions in each asset class:

Portfolio A: P20 (20% Equities, 50% Index-Linked Bonds, 30% Bonds)

Portfolio B: P60 (60% Equities, 10% Index-LinkedBonds, 30% Bonds)

Portfolio C: P100 (100% Equities).

A perceived need of this study is to incorporaterealistic features that would help minimize thevolatility of the dependency ratio. This objectivestems from the severity of the results presentedlater in this section. For this reason the selectionof the asset mixes is based on their efficient port-folio status as exhibited in Figure 4 as well as thefollowing:

• The portfolios are consistent with the currentmarket trends for DC schemes in the UnitedStates according to Mitchell (1998), who pro-nounced equities as being the most popular in-vestment choice with an average asset alloca-tion of 60%, but with some diversification withbonds and other assets (Portfolio B).

• If a participant could tolerate risk, we establishin Section 3.1 that it is in their best interest toallocate their funds solely into equities (Port-folio C) on account of the large decrease in theexpected retirement age.

• There could also exist investors who are ex-tremely risk averse and whose concerns lie inthe 99% percentile of the retirement age dis-tribution. As Section 3.1 explains, investorswho require elevated levels of assurance thatthey will retire prior to a particular age shouldincrease their exposure to bonds, since portfo-lios with a high bond weighting deliver betterresults in the worst-case scenarios. Portfolio Asatisfies the needs of such investors, as well asinserting additional diversity to the portfolio se-lection of the population.

It is useful to look at the whole of the depend-ency ratio distribution under the second popula-tion model, using histograms, CDFs, time seriesplots, and scatterplots. The standard deviationrisk measure associated with each investmentstrategy is helpful in Section 3.1 in deciding be-tween assets mixes; nevertheless, having decidedon efficient portfolios, it is the plots in the com-ing sections that help us to penetrate the DC pen-sion plan and to understand its impact on theworkforce dynamics.

To examine various aspects of the population’sretirement patterns, it is helpful to have a feel forwhat key factors influence the dynamics of thedependency ratio. We investigate its relationshipwith a simple geometrically weighted average ofthe past annual investment returns (specifically,we attach a weight of 5% to the most recent fundreturns and add this to the 95% weight of theprevious year’s weighted average). We find thiscoefficient of 5% to be approximately optimal interms of maximizing the correlation with the de-pendency ratio. Displaying the asset performanceusing this smoothing technique reflects the in-creasing importance of the most recent fund re-turns on the DC participants’ accumulatedwealth. The total pension fund of an entire pop-ulation with a heterogeneous investment strategy(composed of Portfolios A, B, and C) is labeled‘‘Portfolio ABC.’’ Similarly, ‘‘Portfolio B’’ signifiesthe aggregated fund of the portion of the popu-lation who invest their funds in Portfolio B. Thereturn on Portfolio ABC is equal, therefore, toone-third of the return on Portfolio A, one-thirdof the return on Portfolio B, and one-third of thereturn on Portfolio C.

Over the span of 300 simulated years, Figure6a tracks the volatile dependency ratio (solidline) for the heterogeneously invested populationand the equally volatile smoothed investment re-turn (dashed line) of Portfolio ABC. It is worthnoting that since the time span of the simulationis relatively short, the time series plots will ap-pear different from one simulation trial to thenext. We will, therefore, focus on observationsthat are consistent across all the trials executed.

The results demonstrate the important effectthe smoothed investment return has on the de-pendency ratio. Their harmonious movement isdisplayed in Figure 6a, where a double y-axis fa-cilitates their comparison. Unsurprisingly, during


Figure 6Simulated Time Series and Scatterplot of Dependency Ratio and Smoothed Investment Return

(a)

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1.0

0.04 0.06 0.08 0.10 0.12 0.14

Exponentially Smoothed Investment Return of Portfolio ABC

Notes: (a) Simulated time series plot of the dependency ratio (left-hand scale) for a population with a heterogeneous Portfolio ABC investmentstrategy and the exponentially smoothed investment return of Portfolio ABC (right-hand scale), and (b) a scatter plot of these two values foreach simulated year.


Figure 7Simulated Time Series Plots of Dependency Ratio and Smoothed Investment Return

(a)D

epen

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atio

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Notes: (a) Simulated time series plot of the dependency ratio for a population invested in Portfolio ABC and for a population invested inPortfolio B. (b) Smoothed investment return plots of Portfolios A, B, and C.

bull markets, members are able to retire earlier,thus causing the dependency ratio to rise. Like-wise, a bear market drives down the dependencyratio. The scatterplot in Figure 6b illustrates thehighly positive correlation between the depend-ency ratio and the smoothed interest rate; spe-cifically, the correlation coefficient is over 70%.

The next striking conclusion is that diversifyingthe asset allocation decisions among the partici-pants does little to reduce the significant fluctu-ation of the dependency ratio, which is controlledprimarily by the unpredictable performance ofthe market. From the same 300-year simula-tion, Figure 7a plots the dependency ratio for a


Table 3Long-Term Correlation among Smoothed

Investment Returns of Each Portfolio, Basedon 8000-Year Simulation

Portfolio A Portfolio B Portfolio C

Portfolio A 1 0.87 0.79Portfolio B 0.87 1 0.99Portfolio C 0.79 0.99 1

Table 4Long-Term Correlation among Annual Log

Returns of Efficient Investment Assets, Basedon 8000-Year Simulation

AssetIndex-Linked

Bond Bond Equity

Index-linked bond 1 0.40 �0.17Bond 0.40 1 0.09Equity �0.17 0.09 1

Figure 8Time Series Plot of Ages of New Retirees for Each Year of Simulation

0 50 100 150 200 250 300

5060

7080

Time

Ret

iring

Age

Gro

up

Portfolio APortfolio BPortfolio C

Notes: Each symbol indicates an age group retiring at that particular time. Plotted age groups are divided according to their investmentstrategy through the use of a different symbol for each portfolio choice. To facilitate viewing, a thin line connects the age groups with thesame investment strategy and who retire in the same year.

population with a heterogeneous investmentstrategy of Portfolio ABC and for the members ofthe population with a homogeneous investmentstrategy of Portfolio B. The minimal impact onthe dependency ratio is due to the similar move-ment of the smoothed investment return of eachportfolio, which Figure 7b illustrates. Simulatingover an extended time period reveals that thefluctuations of the smoothed rates are highly cor-related. This is in spite of the low correlation be-tween the annual log returns on index-linkedbonds and equities of �17%. Table 3 lists the cor-relation coefficient among the smoothed invest-ment returns of each portfolio, and Table 4 dis-

plays the correlation coefficients between theannual log returns of the relevant assets.

The inconsistency in the ages of retirementfrom one year to the next within every investmentstrategy is surprising when we consider the iden-tical nature of each member; to be more precise,each retiree has identical investment portfolios,levels of contribution, employment histories,mortality statistics, and retirement decision-making behavior. Figure 8 displays the ages of thenewly retired members within each year. In termsof offering a young age of retirement, the supe-rior performance of the equity investment is ap-


parent. The age plot of the portfolio with the leastamount of equities is indeed more stable (dia-monds), but it consistently delivers the highestages of retirement, causing the stability to be apoor tradeoff for the near certainty of a delayedretirement. In contrast, the pure equity portfolioproduces ages of retirement (triangles) muchmore variable, but consistently lower than theother investment strategies.

For each year of simulation in Figure 8, the ver-tical clusters of symbols, followed by gaps of noretirements, indicate that members collectivelydecide to retire or not retire at the same timebecause of their common dependency on the fi-nancial market’s performance. The pattern ofretirements revealed by Figure 8 suggests thatthere could be

• An unstable demand for different types of finan-cial assets if individuals choose to revise theirportfolios at retirement and adopt a less riskyinvestment strategy and

• An unstable demand for financial assets in gen-eral owing to variation in the relative numberof workers and retired members. A growing pro-portion of workers would create an increaseddemand for those assets that make up Portfo-lios A, B, and C. In contrast, a large populationof retired members would result in a greatersupply of assets as pensioners sell financial as-sets to support retirement consumption.

For example, there could be a greater demand forequities when there are fewer retirements. Figure8 demonstrates this scenario between years 47and 71. Here there is a gap of 11 years, which isalmost immediately followed by a gap of 12 years,during which time no members of the Portfolio Cgroup retire. A single triangle at year 58 repre-sents the retirement of the only retiring agegroup holding Portfolio C during nearly a quarterof a century. Once retirement becomes afforda-ble, there could be a massive shift from equitiesto bonds. In Figure 8 such an event occurs in ourscenario beginning in year 218, where the steepvertical of triangles indicates a multitude of re-tirements among Portfolio C holders. During thefollowing 10 years, the retirement age drops fromage 80 to age 58. In year 227 alone, seven Port-folio C cohorts (ages 58–64) retire.

The lack of stability of the dependency ratio isworrisome and could have far-reaching effects, as

is the DC pension system’s inability to retire theparticipants at systematic and reasonable ages.The first concern is the late retirement risk forplan members. If such a case did occur in reality,factors other than finances could force retire-ment, such as illness or disability, thus causinginsufficient pensions and hardship for such retir-ees. In other words, the DC pension schemewould fail these elderly participants. Second, theswings in the labor force could affect the coun-try’s economy. A comprehensive understanding ofthis is outside the scope of this study, but a fewof the repercussions could include the following:

• According to the results, the participants maketheir retirement decisions in large numbers.Recall that the simulation model does not con-sider the interrelations among the sectors ofthe economy and their effect on asset prices. Asuccessful market would generate the retire-ment of the masses, leading toward a rise inasset liquidation, while a suffering marketwould encourage workers to delay their retire-ment and continue saving. Market dynamicscould be influenced by the irregular retirementpatterns, and there could possibly be marketequilibrium upset, as Section 1 brieflydiscusses.

• Second, the benefit of a high dependency ratiois that participants are able to retire early. Alarge number of retirements could, neverthe-less, cause a labor shortage. It could also re-duce tax revenue, because when somebody re-tires, his or her income would generallydecrease.

• On the flip side, unemployment could increaseduring times of a low dependency ratio. To ex-plain further, if the older citizens cannot affordto retire, then they would need to cling ontotheir jobs. If there was a fixed supply of jobs,this could cause unemployment to rise amongyounger members of society who cannot pene-trate the workforce. If the elderly were forcedto retire, they would not have a sufficient pen-sion. This could create additional elderlypoverty or reliance on state-funded welfareprograms.

• Finally, the yo-yo effect of the dependency ratioinsinuates that a DC pension design does an in-credibly poor job in terms of balancing theeconomy’s production with consumption. For


example, if the source of labor declines in size,then the society’s production would suffer whilethe consumption would remain level. The antic-ipated repercussion of such a scenario is priceinflation (Brown, Damm, and Sharara 2001).

Overall, it would be difficult for the governmentto manage a fluctuating dependency ratio. Theproduction and consumption of goods, the taxrevenue, and the necessary social programs wouldbe as unpredictable as the stock market.

4. DAMPENING THE DEPENDENCYRATIO VOLATILITY

We recognize that the instability in the depend-ency ratio could be due to the combination of asimple model and a strict retirement rule. So ourinitial observations could portray a form of worst-case scenario. If this is the case, then refining themodel and adding realistic features should hope-fully dampen the volatility. We see in Section 3.2that adding diversity in the participants’ invest-ment choice does not successfully dampen the de-pendency ratio owing to the long-term correla-tion among the assets. In this section weintroduce further heterogeneity into the modelto identify what, if any, aspects of a DC systemcontribute to greater stability in the dependencyratio.

We first discuss the theoretical basis for eachof the additional features. Following this, we pre-sent the dependency ratio outcome resultingfrom the inclusion of each individual modifica-tion. We also experiment with the aggregate im-pact of incorporating the combination of modifi-cations in Section 4.5. We find that none of thespecified model modifications ameliorate the de-pendency ratio volatility.

4.1 Multiple Ages of EntryA fixed age of plan enrollment for all membersof the population is unrealistic since, althoughthere could exist an age after which it is manda-tory for working citizens to contribute to a statepension plan, it is unlikely that all participantswould have entered the workforce by that age.Participants undoubtedly begin employment at avariety of ages that could exceed the mandatoryage of pension plan enrollment. We now observethe impact of incorporating multiple ages of en-

try into the DC plan. To do so, we assume thatone-third of the population enters the plan at age20, one-third at age 25, and the final third atage 30.

We make the assumption that individuals willmaintain consistency in their plans for retire-ment, irrespective of their ages of entry. All elsebeing equal, a participant would expect to retireat an older (younger) age if they began saving forretirement at a later (earlier) age. We assume,therefore, that an individual will increase theircontribution rate if they begin saving at a laterage than the norm or will reduce their contribu-tions if they enter the plan at a younger age. Theimplication of this assumption is that the averagedependency ratio and retirement age of each en-try age group are approximately equal.

To target the same average retirement ageamong the three groups, participants who enterthe plan at age 20 should reduce their savings to8.25% of salary per annum (a 1.75% decrease),those who begin at age 25 should continue tocontribute 10%, and 30-year-old entry aged par-ticipants should save at the higher rate of 12.5%(a 2.5% increase).

4.2 Multiple Levels of MeritWe could upgrade the authenticity of the wagesimulation by incorporating additional variety inthe career paths. This could be done by varyingthe average annual growth that accounts formerit increases, which is currently fixed at 2%.

The participants’ wage growth modeling isoutlined in Appendix A. In summary, the wagegrowth is made up of two components: generalwage inflation and merit increases. General wageinflation is simulated stochastically and affectseach member of the population in an identicalmanner. We model merit increases deterministi-cally and as a function of the individual em-ployee’s years of service, granting more signifi-cant merit increases during the early years ofemployment.

In our first DC population model, the meritcontribution to salary growth amounts to an av-erage annual rate of 2%. If we raise this value to3%, we could introduce employees with flourish-ing careers (high flyers). Similarly, we could in-clude less successful workers by decreasing theassumption to 1% (low flyers). We could test theeffect of multiple career paths by having one-third


Figure 9Merit Scale Function When Average Annual Merit Growth Is 1%, 2%, and 3%

0 10 20 304 0

0.0

0.5

1.0

1.5

2.0

Years of Service

Mer

it In

crea

se

Low Flyer (1%)Medium Flyer (2%)High Flyer (3%)

of the population fall under each category (lowflyers, medium flyers, and high flyers). Figure 9plots the merit model function at the three as-sessed levels.

4.3 Multiple Contribution RatesBringing in several contribution rates among theparticipants could also enhance the retirementsavings portion of the model. In reality, a vastlydifferent retirement savings pattern prevailsacross the U.S. population. Wise (2003) discussedthe great variation in the savings behavior of U.S.residents on account of public and employer pol-icies, as well as their social and economic envi-ronment. We strengthen the model by experi-menting with three contribution levels across thepopulation: 9%, 10%, and 11%.

4.4 The Effects on the DependencyRatio

The model refinements do not appear to improvethe stability of the dependency ratio. Figure 10contains the opportunity sets of each of the pop-ulation simulations involving the three model im-provements. Section 3.1 outlines the method ofsimulation, except we are now considering a re-duced number of investment portfolios. There are

20% increments for each available asset, totaling126 portfolios. Figure 10a is the benchmark plot;that is, it is a less detailed version of Figure 4since it is based on the original set of assump-tions from the first model except with a reducednumber of executed investment strategies. Theefficient frontier from Figure 10a acts as a pointof reference when assessing the effect of the mod-ifications; therefore, it is represented by a thinsolid line in Figures 10b, c, and d.

Adding variety to the pension plan entry agesin plot b, the contribution rates in plot c, and themerit scales in plot d produces dependency ratiosthat are virtually identical in shape and value tothat produced under the homogeneous scenario,as shown in plot a. This does not imply, however,that the dependency ratio’s standard deviation isunmoved by each individual modification. For ex-ample, under the bond and equity investmentstrategies, the dependency ratio is directly cor-related with the entry age. Table 5 lists the stan-dard deviations for each entry age generated bythe bond and equity portfolios, including thestandard deviation of the aggregate population’sdependency ratio. Comparing the three entry ageoutcomes under each portfolio, it appears thatincreasing the population’s entry age reduces thedependency ratio’s volatility. This is a reasonable


Figure 10Fig. 4 with Three Separate Model Improvements

(a) (b)

Mea

n D

epen

denc

y R

atio

Mea

n D

epen

denc

y R

atio

0.3

0.4

0.5

0.6

0.7

0.3

0.4

0.5

0.6

0.7

P20

P40

P60

P80

P100

Mea

n D

epen

denc

y R

atio

Mea

n D

epen

denc

y R

atio

0.3

0.4

0.5

0.6

0.7

0.3

0.4

0.5

0.6

0.7

P0

P20

P40

P60

P80

P100

P0

0.05 0.10 0.15 0.20 0.25 0.30 0.05 0.10 0.15 0.20 0.25 0.30


(c) (d)

P20

P40

P60

P80

P100

P0

P20

P40

P60

P80

P100

P0

0.05 0.10 0.15 0.20 0.25 0.30 0.05 0.10 0.15 0.20 0.25 0.30


Notes: Similar to Fig. 4, except considering a reduced number of investment portfolios in plot (a) and including the following model improve-ments: (b) multiple ages of entry, (c) multiple contribution rates, and (d) multiple merit scales. To facilitate comparison, the efficient frontierfrom plot (a) (thin solid line) is drawn in each of the plots (b), (c), and (d), along with their respective efficient portfolios (dashed line).

result since an older age of plan enrollment, ac-companied by a larger contribution rate, shouldshorten the participants’ exposure to the stockmarket fluctuations. For example, consider theextreme case in which all participants beginsaving at a very late age (for example, age 60), at

which time they make an enormous contribution.If the participants make a large enough contri-bution so that they are able to retire immediately,they would have no exposure to the stock marketfluctuations. In this exaggerated example, thereis no fluctuation in the dependency ratio since


Table 5Simulated Dependency Ratio Standard

Deviation of Aggregate Population and ofEach Individual Entry Age Group within the

Population

Asset

EntryAge

of 20

EntryAge

of 25

EntryAge

of 30Total

Population

100% bond 0.106 0.099 0.086 0.099100% equity 0.358 0.305 0.248 0.305

everyone above age 60 is consistently retiredwhile everyone below is not. The results also showthat it is a population of equity investors whosedependency ratio volatility is most sensitive tothe age of enrollment assumption.

Increasing the population’s entry age and con-tribution rate to achieve stability in the depend-ency ratio is contrary to the objective of thisstudy, since our aim is to understand the impactof a realistic DC pension plan over the careersand generations of an entire population. The rel-evance of this exercise is to examine whether in-cluding multiple ages of entry would reduce theoverall volatility of the aggregate population’s de-pendency ratio. The conclusion is that the threemodel enhancements do not diminish the severityof the dependency ratio’s volatility.

Figure 11 further illustrates the unchangingvolatility of the dependency ratio by revisiting theresults of the second model in Figure 6a and in-cluding the upgrades. We continue to assumethat the population invests across the three port-folios that Section 3.2 describes. Heterogeneityis incorporated within the entry ages in Figure11a, contribution rates in Figure 11b, and careerpaths in Figure 11c. The dependency ratios re-sulting from the adjusted assumptions are plottedwith dotted lines. The original path of the de-pendency ratio, which Figure 6a shows, continuesto be represented by a solid line as a benchmarkfor comparison. The dependency ratio’s behavioris nearly identical despite the improvements.

4.5 Collective Impact of All ModelImprovements

The final test is the simultaneous incorporationof all the model refinements. As with each indi-vidually applied modification, we evaluate their

aggregate impact by first assuming that one-thirdof the population invests in Portfolios A, B, andC. Thereafter, we incorporate the model improve-ments in the following manner:

• The members within each third of the popula-tion are subdivided into three groups, whereone group makes contributions of 9%, anotherof 10%, and the last of 11%.

• After this, nine groups exist in the population,each of which is further subdivided into threeentry age groups, whereThe first group enters the pension plan at age

20, while reducing their contribution rate by1.75%

The second group enters at age 25, while main-taining their contribution rate and

The remaining group enters at age 30, whileincreasing their contribution rate by 2.5%.

To summarize the three contribution rate levelsassociated with each entry age, the threegroups of investors are each divided into thefollowing nine categories:

ContributionLevel

EntryAge

of 20

EntryAge

of 25

EntryAge

of 30

Low 0.0725 0.0825 0.0925Medium 0.09 0.10 0.11High 0.115 0.125 0.135

• Finally, each of the 27 groups is further parti-tioned into three career path groups: high fly-ers, medium flyers, and low flyers.

The result is 81 groups, each possessing differentretirement savings characteristics. Figure 12 dis-plays the simulated outcome. Once more, the de-pendency ratio behavior is nearly unchanged.

5. RISK MANAGEMENT WITHIN THE DCPLAN DESIGN

In this section we briefly examine a feature thatcould be included in the DC system design to re-duce the potential dependency ratio fluctuations.A method of possibly adding stability to the de-pendency ratio is to put a restriction on equitiesin the personal accounts, but we find that therewould be several drawbacks of such a policy.


Figure 11Fig. 6(a) with Three Separate Model Improvements

0 50 100 150 200 250 300

0.2

0.4

0.6

0.8

1.0

(a)

Time

Dep

ende

ncy

Rat

io

Original AssumptionsMultiple Ages of Entry

0 50 100 150 200 250 300

0.2

0.4

0.6

0.8

1.0

(b)

Time

Dep

ende

ncy

Rat

io

Original AssumptionsMultiple Merit Scales

0 50 100 150 200 250 300

0.2

0.4

0.6

0.8

1.0

(c)

Time

Dep

ende

ncy

Rat

io

Original AssumptionsMultiple Contribution Rates

Notes: Simulated time series plot of the dependency ratio from Figure 6(a) (thin solid line), with the following separate model improvements(dashed line): (a) multiple ages of entry, (b) multiple contribution rates, and (c) multiple merit scales.


Figure 12Fig. 6(a) with Three Additive Model Improvements

0 50 100 150 200 250 300

0.2

0.4

0.6

0.8

1.0

Time

Dep

ende

ncy

Rat

io

Original AssumptionsMultiple Ages of Entry,Contribution Ratesand Merit Scales

Notes: Simulated time series plot of the dependency ratio from Figure 6(a) (thin solid line), with the collective addition of multiple ages ofentry, multiple contribution rates, and multiple merit scales (dotted line).

Figure 4 ranks equities as the largest source ofdependency ratio volatility. We also discover inFigure 7a that the instability of the dependencyratio is not diminished when using equities as atool for diversifying the asset allocation decisionsamong the participants (this is on account of thelong-term correlation of the assets). In view ofthese two findings, a straightforward method toreduce the dependency ratio volatility would beto eliminate equities from the permitted invest-ments in the individual DC accounts. Such a fea-ture could be implemented with government pol-icies that show preferential tax treatment towardfixed-income bonds and index-linked bonds only.

Over the same 300-year simulation in Figure6a, Figure 13 presents the dependency ratio fora population that does not allocate their pensionsavings to equities. Assuming that they continueto invest efficiently, the appropriate portfolio ac-cording to Figure 4 is P0, which we now refer toas Portfolio D. As expected, there is an impressiveimprovement in the behavior of the dependency

ratio relative to the heterogeneously investedpopulation (Fig. 13a) and to each of the otherhomogeneously invested populations (Fig. 13b).The stability in the dependency ratio created bysuch a policy would likely be accompanied by nu-merous negative repercussions, including thefollowing:

• A delay in the nationwide average retirementage: Eliminating equities should tighten the de-pendency ratio’s movement about its mean, butthe downside is that the mean exhibited in Fig-ure 4 is relatively low and corresponds to anaverage retirement age just under age 67.

• A rise in the overall costs: Increased contribu-tions is a means of reducing the average retire-ment age, but this would create a more expen-sive pension plan.

• Inadequate supply of bonds: There would alsoneed to be a large supply of high-quality bonds,which may not be the case since companieswould always need equity capital.


Figure 13Simulated Time Series Plots of Dependency Ratio for a Population of Portfolio A, B, C, D or ABC

Investors

(a)0.

2 0.

4 0.

6 0.

8 1.

0

Dep

ende

ncy

Rat

io

Portfolio ABC Portfolio D

0 50 100 150 200 250 300

Time

(b)

0.5

1.0

1.5

Dep

ende

ncy

Rat

io

Portfolio A Portfolio B Portfolio C Portfolio D

0 50 100 150 200 250 300

Time

Notes: Simulated time series plot of the dependency ratio for a population invested in Portfolio D, and for a population invested in (a) PortfolioABC, (b) Portfolio A, B, or C.


If a restriction on the permissible assets wasincluded in the DC plan design features,there could potentially be a reduction to thedependency ratio oscillations. Nevertheless, thiscould be tied in with several negative aspects thatoutweigh the benefits.

6. CONCLUSION

The drawbacks of implementing a DC pension sys-tem at the state level have been discussed anddocumented in numerous studies. In a DC plandesign, investment, inflation, and mortality riskare transferred completely to individual workersrather than being shared across the populationand over generations. Our study tackles the hugeunknown effect of a national retirement savingspool on the economy and the labor force. We be-gan with an extreme scenario that gave rise toconsiderable instability in the proportion of work-ers from one year to the next. We then consideredrealistic model improvements to determine whichaspects of a DC pension system add stability. Inreality, there exists great variation in the retire-ment savings behavior across a population; there-fore, we introduced additional heterogeneity inthe modeling of the population’s investmentstrategies, contribution rates, entry ages into thepension plan, and career paths. We found thatnone of these dampened the volatility. We ob-served that restricting the participants’ invest-ment options to low-risk portfolios would providesome stability; this option, nevertheless, has sig-nificant shortcomings, including increased costsand problems with the supply of relevant assets.

Our flexible age of retirement model resultssuggests that, if a DC pension system were in-troduced to an entire society to serve as theirprincipal salary replacement in retirement, thefinancial market would have an exceptional im-pact on the proportions of retirees and workersfrom one year to the next. Hence, we propose thatthe significant fluctuation in the market’s per-formance could produce corresponding swings inthe population’s workforce demographics. Fur-ther to the detriment of the society’s labor forcestructure, the unpredictability of the financialmarkets could produce ambiguous and unmana-geable retirement ages, which could lead to per-sonal hardship and anxiety for the individual DCmember. The volatile demand for financial assetscould potentially upset market equilibrium, while

the unstable aggregate retirement pattern couldbe disastrous not only at the individual level, butalso for the economic health of the entirepopulation.

APPENDIX: ACCUMULATION MODEL

In this section we describe the arbitrage-free sto-chastic model used in this study for modeling thedynamics of the wage growth and the asset ratesof return. The specific model chosen to simulatesome of the economic processes is the Vasicekmodel (Vasicek 1977). This is a one-factor modelfor the term structure of interest rates within acontinuous-time framework. We first describe theunderlying economic processes. Following this,we present the wage model and the stochastic dif-ferential equations for the available assets forinvestment.

A.1. UNDERLYING ECONOMICPROCESSES

Throughout this section we use the general no-tation x1(t) to ensure notational compactness ofthe stochastic differential equations (SDEs). Theeconomic processes are numbered as follows:

1. x1(t) Instantaneous risk-free nominal rate ofinterest at time t

2. x2(t) The log total return on equities fromtime 0 to time t

3. x3(t) Instantaneous risk-free real rate of inter-est at time t

4. x4(t) The consumer price index (CPI) loggrowth from time 0 to time t

5. x5(t) The log real return on wages from time0 to time t.

We will sketch the stochastic differential equa-tion (SDE) for each economic process under twoprobability measure models—the risk-neutral andthe real world—since the risk-neutral measure isrelevant for pricing bonds and the real world isneeded for simulation purposes.

Following the assumption that the instantane-ous risk-free rate of interest, x1(t), follows theVasicek model, its SDE under the risk-neutralmeasure Q is as follows:

dx (t) � � (� � x (t)) dt˜1 1 1 1

5

˜� � dW (t), (A.1)� 1j jj�1


where (t), . . . , (t) are independent standard˜ ˜W W1 5

Brownian motions under the risk-neutral proba-bility measure, Q. In the model:

• �1j: the local volatility associated with risk j(i.e., (t))Wj

• : the risk-neutral long-term mean rate�1

• �1: the rate at which the rate of return revertsback to its long-term mean

• �j: the market price of risk associated with thesource of risk j (see eq. [4])

• �1: the real-world long-term mean rate, where

�1 � � (see equation A.1).� �1j j5� �˜1 j�1 �1

Without loss of generality, we choose to param-eterize the model so that �ij � 0 for j � i. Thegeneral parameters in the following equationsmaintain, however, these parameters for nota-tional convenience.

Returning to equation (A.1), we continue bytransferring from Q to the real-world measure Pby replacing with dWj(u) � �jdt. There-˜dW (u)j

fore, the SDE for x1(t)under measure P is asfollows:

dx (t) � � (� � x (t)) dt˜1 1 1 1

5

� � (dW (t) � � dt)� 1j j jj�1

� � (� � x (t)) dt1 1 1

5

� � dW (t), (A.2)� 1j jj�1

where Wj(t) is a standard Brownian motion underthe real-world probability measure P and

� �1j j5� � � � . If � � 0�˜1 1 j�1 11�1

(and � � � � � � � � 0),12 15

then typically �1 is less than zero. This ensuresthat investments in fixed-interest bonds (whichare risky in the short term) attract a positivepremium.

The SDE for x2(t) under Q is51 2dx (t) � x (t) � � dt�� 2 1 2j2 j�1

5

˜� � dW (t),� 2j jj�1

where �2j is the local volatility associated withrisk j.

The value of equities as the price of a tradableasset, meaning that the asset pays no dividends,is represented by S(t), where S(t) � Thex (t)2S(0)e .SDE for S(t) under Q is

5

˜dS(t) � S(t) x (t) dt � � dW (t) .�� 1 2j jj�1

We transfer from Q to the real-world measure Pby replacing with dWj(u)� �j dt. Therefore,˜dW (u)j

the SDE for x2(t) under measure P is

5 2�2jdx (t) � x (t) � � � � dt�� 2 1 2j j 2j�1

5

� � dW (t), (A.4)� 2j jj�1

where �2j�j is the equity risk premium over5�j�1

the risk-free rate of interest.The SDE for S(t) under measure P is

5

dS(t) � S(t) x (t) � � � dt�� 1 2j jj�1

5

� � dW (t) .� �2j jj�1

Under the assumption that the instantaneousrisk-free real rate of interest, x3(t), follows theVasicek model, its SDE under the risk-neutralmeasure Q is as follows:

5

˜dx (t) � � (� � x (t)) dt � � dW (t).�˜3 3 3 3 3j jj�1

Similar to x1(t), the following are the parametersused in the SDE for x3(t):

• �3j: the local volatility associated with risk j• the risk-neutral long-term mean real rate� :˜3

• �3: the rate at which the rate of return revertsback to its long-term mean

• �3: the real-world long-term mean real rate,

where �3 � �� 3j j5� .�˜3 j�1 �3

Under measure P, the SDE for x3(t) is


dx (t) � � (� � x (t)) dt˜3 3 3 3

5

� � (dW (t) � � dt)� 3j j jj�1

5

� � (� � x (t)) dt � � dW (t),�3 3 3 3j jj�1

where �3 � �� 3j j5� .�˜3 j�1 �3

The SDE for the log growth of the consumerprice index, x4(t), under the risk-neutral measureQ is

51 2dx (t) � x (t) � x (t) � � dt�� 4 1 3 4j2 j�1

5

˜� � dW (t),� 4j jj�1

where �4j is the local volatility associated withrisk j.

Similarly, the SDE for x4(t) under P is

5 2�4jdx (t) � x (t) � x (t) � � � � dt�� 4 1 3 4j j 2j�1

5

� � dW (t).� 4j jj�1

The value of the consumer price index (CPI) attime t, represented by C(t), is C(t) � .x (t)4C(0)e

We model the logarithmic real return on wagesin a similar fashion as the other four assets. Thatis, under the risk-neutral measure Q:

5

˜dx (t) � � dt � � dW (t),�˜5 5 5j jj�1

where �5j is the local volatility associated with riskj and is the risk-neutral long-term mean real�5

salary growth.We arrive at the SDE for x5(t) under the real-

world measure P by setting �5 � � �5j�j5� �˜5 j�1

and replacing with dWj(u) � �j dt, which˜dW (u)j

gives

5

dx (t) � � dt � � dW (t).�5 5 5j jj�1

A.2. THE WAGE GROWTH

The growth in an individual’s wage is generallymade-up of general wage inflation and meritincreases. We assume that general wage inflationis decomposed into two parts: price inflationand real wage growth. The wage level at time tfor an individual who begins employment at time0 is:

m(t) x (t)�x (t)�x (0)�x (0)4 5 4 5Y(t) � Y(0)em(0)

C(t)m(t) x (t)�x (0)5 5� Y(0)e ,C(0)m(0)

where Y(t) is the individual’s wage level and m(t)represents the merit component of the wagegrowth and is a function of the worker’s lengthof employment.

Between times t � 1 and t, this model assumesthat the general wage inflation depends on thatyear’s annual inflation, x4(t) � x4(t � 1) (Wilkie,1995), a long-term mean real rate �5, and a ran-dom component that is independent of the firstfour sources of risks. In other words, we assumethat �5j � 0 for j � 1, 2, 3, 4.

We also add to the salary growth a determin-istic merit increase that depends on the individ-ual’s years of service. We assume that an aver-age employee will receive merit increases thatamount to an annual average of 2% over their ca-reer. If an average working career is 40 years andthe annual overall merit increase is 2%, an indi-vidual’s salary would be expected to grow by 121%from the merit increases alone. It is generally as-sumed that merit increases are more significantduring the earlier part of one’s career and leveloff as one becomes more senior. Following fromthis, we model the merit scale using an exponen-tial function with an exponential coefficient of�0.1 to create a relatively gradual rate of change.We need to translate the function on the y-axisto allow for a total lifetime merit increase of 1.81.The merit function is plotted in Figure 9 and isgiven by

�0.1tm(t) � 1.81 � e ,

where t is years of service.


A.3. AVAILABLE ASSETS FORINVESTMENT

In addition to equities, the available assets for in-vestment are the following:

• Cash: a one-year zero-coupon risk-free bond• Index-linked cash: a one-year index-linked zero-

coupon risk-free bond• Fixed-interest bond: an irredeemable bond,

which pays 1 at the end of each future year and• Index-linked bond: an irredeemable index-

linked bond, which pays C(T)at the end of eachfuture year T.

If one unit is invested in each of the assets at time0, then their values at time t are represented by

• Cash fund, G(t)• Index-linked cash fund, B(t)• Fixed-interest bond fund, F(t) and• Index-linked bond fund, Q(t),

with the reinvestment of any coupon income.According to the Vasicek model, the price at

time t of a risk-free zero-coupon bond that ma-tures at time T is given by

(A (t,T)�B (t,T)x (t))1 1 1P (x (t),t,T) � e ,1 1

where

�� (T�t)11 � eB (t, T) � ,1 �1

A (t, T) � (B (t, T) � (T � t))1 1

5 5 2 22 � B (t, T)� 1j 11j� � � .� �˜� �1 22� 4�j�1 j�11 1

Similar to P1(x1(t),t,T), the price at time t of arisk-free zero-coupon bond that matures at timeT and yields a real rate of return is given by

(A (t,T)�B (t,T)x (t))3 3 3P (x (t),t,T) � c ,3 3

where

�� (T�t)31 � eB (t, T) � ,3 �3

A (t, T) � (B (t, T) � (T � t))3 3

5 52 2 2� � B (t, T)3j 3j 3� � � .� �˜� �3 22� 4�j�1 j�13 3

At time t, the price of a zero-coupon index-linked bond that matures at time T is

C(t)P (x (t),t,T),3 3

where C(t) is the value of the consumer price in-dex at time t and P3(x3(t),t,T) signifies the ex-pected risk-neutral growth component of thebond that is attributed to the guaranteed realrate of return. Therefore, an investment of

$P3(x3(t),t,T) at t will return at T.C(T)

$C(t)

The rate of return on each of the funds betweentime t and t � 1 is as follows:

G(t)G(t � 1) � ,

P (x (t),t,t � 1)1 1

C(t � 1)B(t � 1) � B(t) ,

C(t)P (x (t),t,t � 1)3 3

F(t � 1) � F(t)�

1 � P (x (t � 1), t � 1, T)� 1 1T�t�2 ,�

P (x (t),t,T)� 1 1T�t�1

C(t � 1)Q(t � 1) � Q(t)

C(t)�1 � � P (x (t � 1), t � 1, T)T�t�2 3 3 .� �� P (x (t), t, T)T�t�1 3 3

Additionally, the annuity factor for retirementage e � t at time t, ignoring expenses, is given by

�

a (t) � P (x (t),t,t � s) p .�e�t 1 1 s e�ts�0

One disadvantage in the Vasicek model is that itallows for the instantaneous risk-free rate of in-terest to become negative, which is somewhat un-realistic, particularly when calculating the priceof an annuity. Therefore, the instantaneous risk-free rate of interest, x1(t), used in calculating thefair annuity value has a floor of 0%.

A.4. PARAMETER ESTIMATES ANDDATA

This model has been calibrated using U.S. dataprovided partly by Professor David Wilkie andpartly from the following public sources:


• U.S. Federal Government Treasury nominalsecurities, which can be found at the U.S.Government Federal Reserve Web site (www.federalreserve.gov/releases/h15/data.htm) and

• The Bureau of Economic Analysis, U.S. De-partment of Commerce (these tables can beaccessed from www.bea.doc.gov/bea/dn/nipaweb/SelectTable.asp).

The limited U.S. index-linked bond data havebeen supplemented by U.K. data, which are avail-able online at the Heriot-Watt/Faculty andInstitute of Actuaries Gilt Database (see www.ma.hw.ac.uk/�andrewc/gilts/).

The model parameter estimates are thefollowing:

Parameter �1 �1 �11 �1 �21 �22 �2

Estimate 0.051 0.15 0.0185 �0.152 �0.011 0.156 0.328

Parameter �3 �3 �31 �32 �33 �3

Estimate 0.027 0.56 0.0075 0 0.0086 �0.419

Parameter �41 �42 �43 �44 �4 �5 �55

Estimate 0 �0.013 0 0.0022 �0.066 0.01 0.0205

7. ACKNOWLEDGMENTS

The authors would like to thank David Wilkie forhis helpful discussions of the paper and for pro-viding some of the data used in the Appendix. Wethank those who contributed to the discussionson an earlier version of this paper, as presentedat the AFIR Colloquium held in Zurich, Septem-ber 2005. Finally, the authors wish to thank ananonymous referee and Pat Currie for their help-ful comments on an earlier version of the paper.

This paper was completed during the PhD workof Bonnie-Jeanne MacDonald at Heriot-Watt Uni-versity. She wishes to acknowledge the financialsupport of Heriot-Watt University, the BritishCouncil, the Spencer Education Foundation, theAmerican Society for Quality, and the NorthAmerican Society of Actuaries.

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Discussions on this paper can be submitted untilJuly 1, 2007. The authors reserve the right to replyto any discussion. Please see the Submission Guide-lines for Authors on the inside back cover for instruc-tions on the submission of discussions.

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