Noional Defined Conribuion Pension Sysems in a Sochasic Conex: Design and Sabiliy Alan J. Auerbach Universiy of California, Berkeley and NBER and Ronald Lee Universiy of California, Berkeley and NBER December 2006 This research was suppored by he U.S. Social Securiy Adminisraion hrough gran #0-P- 98363--02 o he Naional Bureau of Economic Research as par of he SSA Reiremen Research Consorium. The findings and conclusions expressed are solely hose of he auhors and do no represen he views of SSA, any agency of he Federal Governmen, or he NBER. The auhors graefully acknowledge he excellen research assisance of Erin Mecalf and Anne Moore, he conribuions of Carl Boe in he developmen of he sochasic forecasing model, commens from Ed Palmer, Jason Seligman, Ole Seergren (none of whom is responsible for any remaining errors), paricipans in he NBER summer insiue, he 8 h annual RRC conference, and he Ocober, 2006 NBER Conference on Reiremen Research and suppor from Berkeley s NIA-funded Cener for he Economics and Demography of Aging. The research funded here builds on basic research funded by NIA gran R37-AG76.
Absrac Around he world, Pay-As-You-Go (PAYGO) public pension programs face serious long-erm fiscal problems due primarily o acual and proeced populaion aging, and mos appear unsusainable as currenly srucured. Some have proposed he replacemen of such plans wih sysems of fully funded privae or personal Defined Conribuion (DC) accouns, bu he difficulies of ransiion o funded sysems have limied heir implemenaion. Recenly, a new variey of public pension program known as Noional Defined Conribuion or Non-financial Defined Conribuion (NDC) has been creaed, wih he obecives of addressing he fiscal insabiliy of radiional plans and mimicking he characerisics of funded DC plans while reaining PAYGO finance. Using differen versions of he sysem recenly adoped in Sweden, calibraed o US demographic and economic parameers, we evaluae he success of he NDC approach in achieving fiscal sabiliy in a sochasic conex. (In a companion paper, we will consider oher aspecs of he performance of NDC plans in comparison o radiional PAYGO pensions.) We find ha he basic NDC scheme is effecive a prevening excessive deb accumulaion, bu does lile o preven significan asse accumulaion along many raecories and on average. Wih adusmen, however, he NDC approach can be made more sable. Keywords: Social Securiy, Reiremen, Deficis JEL nos. H55, J Alan J. Auerbach Ronald Lee Deparmen of Economics Deparmen of Demography Universiy of California Universiy of California 549 Evans Hall 2232 Piedmon Ave Berkeley, CA 94720-3880 Berkeley, CA 94720-220 auerbach@econ.berkeley.edu rlee@demog.berkeley.edu
Inroducion Around he world, Pay-As-You-Go (PAYGO) public pension programs are facing serious long-erm fiscal problems due primarily o acual and proeced populaion aging, and mos appear unsusainable as currenly srucured. All sric PAYGO programs (i.e., hose ha do no incorporae sizable rus fund accumulaions) can feasibly pay an implici rae of reurn equal o he growh rae of GDP (labor force growh plus produciviy growh) once hey are maure and in seady sae. This rae of reurn is ypically lower han he rae of reurn ha can be earned in he marke, eiher hrough low-risk bonds or hrough invesmen in equiies. The programs long-erm fiscal problems relae o a misalignmen beween hese low bu feasible raes of reurn and promised raes of reurn ha may once have been feasible bu no longer are so. The radiional plans are mosly defined benefi, and have been criicized for creaing srong incenives for early reiremen. More generally here is a concern ha he axes ha finance hese programs disor labor supply incenives hroughou life. Many also believe ha hese plans undermine moivaions o save, and, because hey are hemselves unfunded, hereby reduce overall capial accumulaion and consequenly lead o lower labor produciviy and slower growh. Recenly, a new variey of public pension program known as Noional Defined Conribuion or Non-financial Defined Conribuion (NDC) has been creaed and implemened by Sweden, wih firs paymens in 200. A number of oher counries have inroduced or are planning o inroduce NDC plans, including Ialy, Poland, Lavia, Mongolia and he Kyrgyz Republic, and proposed new plans for France and Germany have NDC aspecs (Legros, 2003; Holzmann and Palmer, 2005).
NDC programs differ in deail, bu he basic principle is ha hey mimic Defined Conribuion plans wihou acually seing aside asses as such plans do. Under an NDC program, a noional capial accoun is mainained for each paricipan. Balances in his accoun earn a rae of reurn ha is declared by he pension plan each year; and noional paymens ino his accoun are made over he enire life hisory o mirror acual axes or conribuions. Togeher wih he declared rae of reurn hese noional conribuions deermine he value of he accoun a any poin in ime. Afer a designaed age such as 62, a paricipan can choose o begin o draw benefis, which is done by using he accoun o purchase an annuiy from he pension plan. The erms of he annuiy will depend on moraliy a he ime he generaion urns 65 (for example) and on a rae of reurn sipulaed by he pension plan, which migh be he same rae of reurn used in he pre-reiremen accumulaion phase. NDC plans are seen as having many poenial advanages over radiional PAYGO sysems, bu our focus in his paper is on us one of hese poenial advanages, sabiliy. A plan of his sor appears srucured o achieve a considerable degree of fiscal sabiliy because he promised raes of reurn reflec he program s underlying PAYGO naure, raher han being marke-based, and he annuiy srucure should buffer he sysem from he coss of rising longeviy. Furher, in he even ha i begins o go off he racks, a braking mechanism can be incorporaed which auomaically modifies he rae of reurn, o help resore he plan o financial healh. Given he poliical difficulies of making frequen changes in PAYGO pension programs, he araciveness of an inherenly sable sysem is clear. In his paper, we use a sochasic macro model for forecasing and simulaing Social Securiy finances o examine he behavior of NDC-ype public pension programs in he conex of he US demography and economy. Given he srucure and sraegy of he sochasic model, 2
we can sudy he probabiliy disribuion of oucomes (benefi flows and raes of reurn) for generaions (birh cohors) of plan paricipans for he NDC program, as well as he overall financial sabiliy of he NDC sysem. The nex secion of he paper describes our sochasic forecasing model. In he following secion, we describe in some deail he Swedish NDC program and our adapaion of i o US economic and demographic condiions. We hen provide simulaions of his basic US NDC plan, as well as varians incorporaing modificaions of wo key aribues of he NDC plan, he mehod of deermining raes of reurn, and he brake mechanism applied when he sysem appears headed for financial problems. The Sochasic Forecasing/Simulaion Model The sochasic populaion model is based on a Lee-Carer (992) moraliy model and a somewha similar feriliy model (Lee, 993; Lee and Tulapurkar, 994). Lee-Carer models he ime series of a moraliy index as a random walk wih drif, esimaed over US daa from 950 o 2003. This index hen drives he evoluion of age specific moraliy raes and hereby survival and life expecancy. This kind of model has been exensively esed (Lee and Miller, 2003) and is widely acceped (Booh, 2006), and alhough we shall see ha he probabiliy inervals i produces for disan fuure life expecancy appear quie narrow, hese inervals have performed well in wihin-sample rerospecive esing. In a similar way, a feriliy index drives age specific feriliy, bu in his case i is necessary o prespecify a long erm mean based on exernal informaion. We se he long run mean of he Toal Feriliy Rae equal o he.95 birhs per woman, as assumed by he Social Securiy Acuaries (Trusees Repor, 2004, henceforh TR04). The esimaed model hen supplies he probabiliy disribuion for simulaed oucomes. Because i is fied on US daa, he feriliy model reflecs he possibiliy of subsanial baby boom and bus ype swings. 3
Immigraion is aken as given and deerminisic, following he assumed level in TR04. Following Lee and Tulapurkar (994), hese sochasic processes can be used o generae sochasic populaion forecass in which probabiliy disribuions can be derived for all quaniies of ineres. These sochasic populaion forecass can be used as he core of sochasic forecass of he finances of he Social Securiy sysem (Lee and Tulapurkar, 998a and b, and Lee, Tulapurkar Anderson, 2003). Cross secional age profiles of payroll ax paymens and benefi receips are esimaed from adminisraive daa. The ax profile is hen shifed over ime by a produciviy growh facor which is iself modeled as a sochasic ime series. The benefi age profile is shifed over ime in more complicaed ways based on he level of produciviy a he ime of reiremen of each generaion. The real rae of reurn on special issue Treasury Bonds is also modeled as a sochasic ime series, and used o calculae he ineres rae on he Trus Fund Balance. The long run mean values of he sochasic processes for produciviy growh and raes of reurn are consrained o equal he cenral assumpions of TR04, bu he acual sochasically generaed oucomes will no exacly equal hese cenral assumpions, of course, even when averaged over a 00 year horizon. The probabiliy disribuions for he sochasic forecass are consruced by using he frequency disribuions for any variable of ineres, or funcions of variables of ineres, from a large number of sochasically generaed sample pahs, say 000, ypically annually over a 00 year horizon. Essenially, his is a Mone Carlo procedure. The sochasic sample pahs can equally well be viewed as sochasic simulaions, and he se of sample pahs can be viewed as describing he sochasic conex wihin which any paricular pension policy mus operae. The sochasic simulaion model is no embedded in a macro-model, and herefore does no incorporae economic feedbacks, for example o saving raes and capial formaion, and 4
hence o wage raes and ineres raes. For some purposes, his would be an imporan limiaion. However, he model has given useful resuls for he uncerainy of Social Securiy finances, and i should also give useful resuls in he presen conex. Once he sochasic properies of differen policy regimes have been sudied in his manner, i may be appropriae o exend he analysis o incorporae more general economic feedbacks in fuure work. A Sochasic Laboraory: Simulaing Saisical Equilibrium To dae, he sochasic Social Securiy mehod us described has been used solely for proecions or forecass, based on he acual demography and Social Securiy finances of he Unied Saes. However, i can also be used as a sochasic laboraory o sudy how differen pension sysems would perform in a sochasic conex divorced from he pariculariies of he acual US hisorical conex wih is baby boom, baby bus and oher feaures. This is he main sraegy we pursue in his paper, since we are hoping o find quie general properies of he NDC sysems. We build on he imporan earlier work by Alho e al. (2005). This approach also enables us o avoid dealing iniially wih he problems of he ransiion from our curren sysem o he new sysem. Insead we will analyze he performance of a maure and esablished sysem in sochasic seady sae. In laer work we hope o consider he ransiion and o accoun for he acual hisorical iniial condiions such as he curren age disribuion as shaped by he baby boom. The key feaure of a sochasic equilibrium is ha he mean or expeced values of feriliy, moraliy, immigraion, produciviy growh, and ineres raes have no rend, and he populaion age disribuion is sochasically sable raher han reflecing peculiariies of he iniial condiions. The basic idea is simple enough, bu here are a number of poins ha require discussion, as follows. 5
. Produciviy growh and ineres raes are already modeled as saionary sochasic processes wih prese mean values, so hese pose no paricular problem. 2. Ne immigraion is se a a consan number per period, following he Social Securiy assumpions (TR04). We rea immigraion as deerminisic and consan. 3. Feriliy is also modeled as a saionary sochasic process wih a long-erm mean value of.95 birhs per woman, consisen wih TR04. This is below replacemen level, so absen posiive ne immigraion, he simulaed populaion would decline oward zero and go exinc, wih he only possible equilibrium populaion being zero. Bu wih immigraion, here is some populaion size a which he naural decrease given a TFR of.95 will be exacly offse by he ne immigran inflow, and his will be he equilibrium populaion. The same principle applies in a sochasic conex. 4. According o he fied Lee-Carer moraliy model, he moraliy level evolves as a random walk wih drif. Firs, we noe ha unless he drif erm is se o 0, moraliy will have a rend. So in consrucing our sochasic equilibrium populaion, we will proec moraliy forward, wih drif, unil 200 and hen se he drif o zero hereafer. This ses equilibrium life expecancy a birh o be abou 87 years. Second, we noe ha a random walk, even wih zero drif, is no a saionary process. I has no endency o reurn o an equilibrium level, bu raher drifs around. Our sraegy is simply o se he drif erm o zero. This means ha we canno view he simulaed process as ruly achieving a saisical equilibrium, bu his is unlikely o cause any pracical problems. An alernaive would be o aler he model o make i ruly saionary by providing some weak equilibraing endency, e.g. replacing he coefficien of uniy on he previous level of moraliy in he process by 0.99. 6
5. We also need o generae an appropriae iniial sae for our sysem. We begin by consrucing a deerminisic sable populaion corresponding o he mean values of feriliy and moraliy for he given inflow of immigrans. Populaion size aduss unil he ne inflow of immigrans is equal o he shorfall in birhs due o below-replacemen feriliy. We hen sar our sochasic simulaion from his iniial populaion, bu we hrow ou he firs hundred years. We keep he nex five hundred years of sochasic simulaions as our experimenal se. Figure plos 5 sochasic sample pahs for he old age dependency raio defined as populaion 67+ divided by he populaion 2 o 66. Evidenly he simulaions ofen show very pronounced long erm variaions resuling from somehing like he baby boom and baby bus in he Unied Saes. 6. For our policy experimens, we have creaed a single se of 000 sample pahs or sochasic raecories. We will examine he performance of differen policies wihin he conex of his single se of sochasic raecories, so ha each mus deal wih he same se of random shocks, which makes heir performances more comparable. NDC Sysem Design As is well-known, he feasible inernal rae of reurn for a PAYGO sysem wih sable populaion srucure equals he rae of growh of he populaion (which equals he rae of growh of he labor force, in seady sae) plus he rae of growh of oupu per worker. Alernaively, his implici rae of reurn simply equals he growh rae of GDP, provided ha covered wages are a consan share of GDP. NDC sysems aim o mimic he srucure of funded DC sysems while mainaining fiscal sabiliy by using such an inernally consisen rae of reurn (wih due allowance for he non-seady sae conex) raher han a marke-based rae of reurn. 7
As under any pension sysem, an individual goes hrough wo phases under an NDC scheme, corresponding roughly o periods of work and reiremen. During he work phase, he individual s payroll axes (T) are credied o a virual accoun ypically referred o as he individual s noional pension wealh (NPW). Like he individual accoun under an acual DC plan, his accoun has a saed value ha grows annually wih conribuions and he rae of reurn on prior balances; for an individual, his evoluion is represened by: i () NPW + = NPW ( + r ) + T where i r is he rae of reurn credied o each individual s exising balances. Unlike he individual accoun balance under he DC plan, NPW is only a virual balance and he rae of reurn is based on he sysem s inernal growh rae. Once an individual reires, he or she receives an annuiy based on he value of noional pension wealh a he ime of reiremen. The Swedish sysem normally bases r i on he conemporaneous rae of growh of he wages per worker, which we call g, raher han he oal growh of wages, which would also accoun for he growh rae of he work force, which we label n. The noional accouns of individuals in Sweden also receive an annual adusmen for so-called inheriance gains, represening a redisribuion of he accoun balances of deceased cohor members. Tha is, he rae of reurn o he cohor as a whole, which we denoe r, equals g, where r=g<r i. 2 Upon reiremen in he Swedish sysem, he individual s NPW is convered ino an annuiy sream based on conemporaneous moraliy probabiliies and an assumed real rae of Our characerizaion of he Swedish sysem relies on several sources, including Palmer (2000) and Seergren (200a, 200b). 2 Accouns in Sweden are also reduced annually o accoun for adminisraive coss. In our simulaions, we ignore hese adusmens and he underlying coss. 8
reurn of.6 percen. Leing he superscrip denoe he generaion ha reaches reiremen age in year, he annuiy in year for an individual reiring ha year, implicily from he formula, x, may be solved for (2) T T ( s ) ( s ) = (.06) +, s = (.06) +, s s= s= NPW P x x P where NPW is he individual s noional pension wealh in he year of conversion, P, is he probabiliy of survival from year unil year s, assessed in year, and T is he maximum life span. In subsequen years, he individual s annuiy level is increased or decreased according o wheher he acual average growh of wages per worker, denoed r = g above, exceeds or falls shor of.6 percen. If wage growh coninues a.6 percen hen he annuiy level would remain consan in real erms hroughou he individual s life. However, if he realized growh of wages per worker in year were acually.3 percen, he annuiy would be 0.3 percen lower in real erms in year + han in year. If r were.3 percen in every year, he annuiy would fall in real erms a a rae of 0.3 percen per year. s The Brake Mechanism The sysem us described incorporaes an adusmen mechanism aimed a keeping benefis in a range ha can be suppored by growh in he payroll ax base. However, he adusmen is no perfec. Firs, alhough benefis are adused annually for changes wage growh, hey are no adused afer reiremen o reflec changes in moraliy proecions. More imporanly, he cohor rae of reurn r in he Swedish sysem is based on he growh rae of he average wage, g. While his approach migh be more comprehensible from an individual worker s perspecive, basing he cohor s rae of reurn insead on he growh rae of he covered 9
payroll, n+g, would auomaically ake ino accoun anoher deerminan of he sysem s capaciy o provide benefis, he growh rae of he work force Finally, as illusraed in he Appendix using a simplified version of he NDC plan, even an NDC plan wihou hese problems is no assured of annual balance if n+g varies over ime. This is in line wih he analysis of Valdes- Prieo (2000), who observed ha, under cerain condiions, an NDC plan migh be sable in a seady sae, bu will no be so in he shor run. Alhough i was anicipaed by is designers ha he Swedish sysem would neverheless be quie sable, hey added o he sysem a balance mechanism 3, which we call a brake, ha would slow he growh rae of noional pension wealh and reduce he level of pension benefis in he even of a hrea o he sysem s financial sabiliy, as measured by a balance raio b based on he sysem s condiions, (3) b = F + C NPW + P The numeraor of he balance raio is mean o accoun for he sysem s asses, and is he sum of wo erms. The firs erm in he numeraor (F) equals he financial asses of he sysem (negaive if he sysem has financial deb); he second erm in he numeraor (C) is a so-called conribuion asse equal o he produc of a hree-year moving median of ax revenues and a hree-year moving average of urnover duraion, which is he average expeced lengh of ime beween he paymen of conribuions and he paymen of benefis, based on curren paerns. If he economy were in a seady sae, he conribuion asse would provide a measure of he size of he pension liabiliy ha conribuions could susain. 3 The balance mechanism is described in Swedish Pension Sysem (2005), pp. 38-39. 0
The denominaor is he pension sysem s liabiliy, equal o he sum of wo componens. The firs componen of he denominaor (NPW) is aggregae noional pension wealh for generaions no ye reired; he second componen (P) is an approximaion of commimens o curren reirees, equal o he sum over reired cohors of curren annual paymens o each cohor muliplied by ha cohor s so-called economic annuiy divisor, roughly he presen value of heir annuiies calculaed using he assumed.6 percen real reurn. 4 This balance measure can be calculaed enirely from observed values and does no involve any proeced values. This has he advanage of reducing he risk of poliical manipulaion, bu a disadvanage in no aking advanage of all informaion available a he ime in compuing he conribuion asse and he liabiliy o curren reirees, basing hem simply on curren condiions. The balance raio is no a perfec measure of he sysem s financial healh. For example, he wo componens of he asse measure are based on inconsisen rae-of-reurn assumpions, he financial componen being assumed o yield a marke rae of reurn and he conribuion asse being valued using he sysem s implici rae of reurn. However, one would sill expec a higher value of he balance raio, in general, o be associaed wih a more viable sysem. We refer o he Swedish balance mechanism as a brake because is effec is o preven he excessive accumulaion of deb, bu no of asses; i applies only when he sysem is underfunded, as indicaed by he balance raio, bu no when i is overfunded. I was undersood when he sysem was designed ha his poenially could lead o he accumulaion of surpluses, bu no formal mechanism was pu in place o deal wih his. One could imagine a sysem wih a 4 For furher deail, see Swedish Pension Sysem (2005), page 70.
more symmeric brake ha raises benefis and pension accumulaions when he sysem is overfunded, and we consider such a sysem below. For he Swedish sysem, he balance mechanism is acivaed only when he balance raio b falls below.0, and says in effec unil a es of fiscal balance is saisfied. While he mechanism is acive, wo hings happen each year. Firs, cohor pension wealh accumulaes no a a gross rae equal o (+g ), bu insead a a rae equal o (+g )b, where b is he balance raio. Second, he gross rae of growh used o adus he pension benefis of reirees is also se equal o (+g )b, meaning a greaer likelihood of a real decline in pension benefis for any given cohor, since real benefis grow a a gross rae of (+g )b /.06. The balance mechanism remains in effec as long as he produc of balance raios from each year of he episode remains below.0. Tha is, if he balance raio firs falls below.0 in year, hen he balance mechanism coninues s o apply in year s> if b <. 0. Once his produc exceeds.0, he balance mechanism is v= v aken off, wih a new episode beginning he nex ime he balance raio again falls below.0. This design has several implicaions. Firs, because he balance mechanism is removed when he produc of balance raios firs moves above.0, he balance raio mus exceed.0 in he year he balance raio is removed. Second, as he balance raio may say below.0 for some ime, here may be several years during he episode for which b >.0. Third, as menioned above, he balance mechanism is asymmeric, in ha i applies only when b firs falls below.0; pahs on which b sars above.0 and rises well above his level are subec o no exernal adusmen. Fourh, here may be recurren episodes during which he balance mechanism is in effec. Fifh, while he balance raio as defined in (3) can be negaive (if financial deb exceeds he conribuion asse), he balance mechanism is no meaningful for b < 0, for his would call for more han complee confiscaion of pension wealh and benefis. 2
As o he logic of he es imposed o deermine when he balance mechanism is removed, based on he produc of balance raios, his approach ensures ha he balance mechanism has no long-run impac on he level of benefis. Tha is, he cumulaive gross rae of reurn from he firs year in which he brake applies, say, hrough he year, say T, in which he produc of he balance raios reaches.0 and hence he brake mechanism is removed, is (+g )b *(+g + )b + * *(+g T )*b T = (+g )* *(+g T )* (b * * b T ) = (+g )* *(+g T ). Even hough here is no impac on he long-run level of benefis, he brake mechanism has he capaciy o reduce deb accumulaion by keeping benefis below heir regular long-run pah for some ime. Adaping he NDC Sysem o he US Conex We have already oulined he basic srucure of he Swedish NDC sysem, bu here are various deails o be specified in adaping he sysem o he US conex. Conribuions Wha proporion of payroll is o be conribued? For comparabiliy o our curren Social Securiy sysem, we assume he OASI ax rae of 0.6 percen, applied o he fracion of oal wages below he payroll ax earnings cap. Raes of Reurn Rae of reurn assumpions are required in wo places in he NDC sysem, for use in accreions of Noional Pension Wealh and in conversion of Noional Pension Wealh ino an annuiy sream upon reiremen. The Swedish plan ses he firs of hese raes equal o he growh rae of average wages, which should roughly equal he growh rae of produciviy. I ses he second rae equal o.6 percen, aken o be he expeced rae of produciviy growh, and hen aduss annuiies up or down in response o variaions in he acual growh of he 3
average wage. Sweden does no accoun for he growh rae of he labor force bu in principle i should be included since i is a componen of he rae of reurn o a PAYGO sysem. Noe ha even if he growh rae of he labor force is no included, demography will sill influence he oucomes for generaions hrough he back door, because if he sysem begins o go ou of fiscal balance hen he brake will be applied. For he US sysem, we ake he long-run mean rae of growh of he real covered wage o be. percen, following he Social Securiy assumpion (TR04), as described below. We will refer o his inerchangeably as he produciviy growh rae or he growh rae of wages, g, alhough hese are acually somewha differen conceps. 5 We have implemened NDC in wo ways for he Unied Saes, once wih rae of reurn based only on wage growh (g), and once wih he rae of reurn based on boh wage growh and labor force growh (n+g). These will presumably disribue risk in differen ways across he generaions. In sochasic equilibrium populaion growh is near zero in any case, on average, bu demographic change will cerainly occur along simulaed sample pahs. Annuiy Calculaions We assume ha annuiizaion of NPW occurs a age 67, he normal reiremen age o which he US sysem is currenly in ransiion. We use he same rae of reurn for accumulaions of NPW as in convering he accoun balance a reiremen ino an annuiy sream. Tha is, we use eiher he growh rae of wages (g) or he growh rae of wages plus labor force (n+g) in boh cases. As in he Swedish sysem, we se he paern of he annuiy sream o be consan in real 5 We noe ha he growh rae of produciviy (oupu per hour of labor) may oversae he growh rae of covered wages, as is explicily aken ino accoun by he US Social Securiy Adminisraion. The growh rae of covered wages will be affeced by changes in he supply of labor per member of he populaion of working age and by sex, boh labor force paricipaion and hours worked per paricipan, and by shifs in he populaion age disribuion. I will also be affeced by he proporion of compensaion ha is given in preax fringe benefis. 4
erms, based on he growh rae and moraliy proecions a he ime of he original annuiy compuaion. Unlike he Swedish sysem, we use he acual growh rae (eiher g or n+g) as of reiremen, raher han an assumed long-run value (in he Swedish case,.6 percen). The annuiy calculaion can eiher be se once a he ime of reiremen, or i migh be updaed during he benefi period o reflec changes in he implici rae of reurn, as is done in Sweden. We have programmed boh possibiliies, referring o one as updaing and he oher as no updaing. Wha moraliy schedule is used o compue he annuiized income sream? Once again, his can be based on condiions a he ime a generaion reires (as is done in Sweden), or i can be revised during he benefi period, an approach ha Valdes-Prieo (2000) refers o as a CREF-syle annuiy. We have done i boh ways, bundled ino he updaing and no updaing programs. Because we wish o deermine he exen o which he NDC plan can be made sable, we presen he resuls for he updaing version below. However, he difference beween he wo versions is minor in our simulaions, so he lack of updaing for moraliy experience in he acual Swedish sysem is unlikely o be a significan source of insabiliy. The Brake As explained earlier, he Swedish program has a brake bu a limied acceleraor ha applies only unil he impac of he brake on he level of benefis has been reversed. If surpluses begin o accumulae, here is no mechanism o raise benefis or reduce axes relaive o wha is called for by he basic sysem. In our NDC program, we have incorporaed his brake, bu we also consider how much beer i does han a simpler asymmeric brake ha applies only when b < (i.e., a brake mechanism wihou he Swedish sysem s provision for bringing benefis back o heir long-run levels). In anoher version we use a symmeric brake wih an acceleraor ha raises he rae of reurn and raises curren benefis when he fiscal raio exceeds uniy. 5
A second change we implemen is in he design of he brake mechanism iself. As discussed above, he brake in he Swedish sysem muliplies he gross reurn implied by wage growh, r, by he curren balance raio of sysem asses o sysem liabiliies. Tha is, when he brake is in effec, he adused ne rae of reurn, a r, is given by a (4) r ( + r ) b = A low values of b, his mechanism implies a near confiscaion of pension wealh, a no very desirable oucome if one is rying o spread fiscal burdens among generaions. We herefore consider a generalized version of he balance mechanism in which (4) is replaced by: a (5) r ( + r )[ + A( b )] = where r and b are defined as before and A [0,] is a scaling facor. Seing A = resuls in a brake like ha in (4); when A <, full confiscaion will resul only when b reaches /A < 0. Seing A = 0 eliminaes he brake mechanism, and a posiive value of A ha is oo small will sill fail o provide adequae financial sabiliy. In our simulaions below, we use a value of A = 0.5, meaning ha he mechanism is welldefined for values of b above -. This value of A was large enough o ensure ha virually none of he 000 raecories, each 500 years long, ever encounered he lower bound on b for NDC ype sysems wih r = g (only 2 of 000 raecories for he asymmeric brake case and 7 of 000 for he symmeric brake case). Even for a much lower value of A =.2, he lower bound is basically irrelevan for raecories wih r = n+g and binds along only relaively few raecories for NDC sysems wih r = g (5 for he asymmeric brake and 47 for he symmeric brake). 6
Iniial Condiions As discussed above, we sar our simulaions wih a populaion srucure based on a deerminisic version of our demographic model, and hen run he economy for a hundred-year pre-sample period o ge a realisic disribuion of demographic characerisics for he sochasic version of he model, which we hen simulae over a period of five hundred years. We also use his iniial hundred-year period o esablish he iniial condiions for he NDC sysem. As of he beginning of he acual simulaion period, and for each raecory, we calculae each working cohor s NPW based on is earnings during he pre-sample period and he relevan growh raes (g or n+g) used in compounding NPW accumulaions. For each reired cohor, we calculae annuiy values in he same manner. Finally we assume an iniial sock of financial asses equal o 50 imes he average primary defici in he firs year of he model based on g wih no brake. (This is roughly he level ha would be needed o service he primary defici while mainaining a consan asses-payroll raio.) Defining he Policy Scenarios We simulae eigh versions of he NDC sysem, differing as o wheher he rae of reurn is based on he produciviy growh rae, g, or he growh rae of wages, n+g, he ype of brake used in aemping o achieve fiscal sabiliy (none, Swedish, asymmeric, symmeric), and he srengh of he facor, A, used o modify he brake adusmen. In our proecions, he mean rae of growh of he real covered wage is assumed o be. percen per year, following he assumpions of he Social Securiy Trusees. The long run growh rae of he proeced populaion is close o 0 in he sochasic equilibrium we generae, so he growh rae of covered wages is also abou. percen per year. The inernal raes of reurn for individual cohors along any given raecory of our sochasic proecions should herefore end 7
o flucuae around his cenral value if he sysem mainains financial sabiliy. 6 In addiion o considering he inernal raes of reurn (IRRs) under each NDC varian, we are also ineresed in he financial sabiliy ha each sysem provides. We measure financial or fiscal balance using he raio of financial asses o payroll, where in he figures below a negaive value indicaes deb. Noe ha he numeraor of his expression includes only financial asses, no he conribuion asse ha is used in compuing he balance raio. Simulaion Resuls Consider firs he performance of he NDC sysem based on r = g, roughly he Swedish approach wihou a brake mechanism. As shown in he firs row of Table, his sysem provides a mean inernal rae of reurn of.07 percen, close o he rae of. percen for a sysem in which all variables are consan a heir mean values. Noe ha he IRR is a highly nonlinear funcion of he variables, so ha he mean IRR across sochasic raecories need no be close o he IRR of he means of he raecories. The median IRR also equals.07 percen for his scheme. However, he need for a brake is quie eviden from Figure 2, which shows he disribuion of asses-payroll raios for he sysem s firs 00 years of operaion. The median raecory has essenially no accumulaion of deb or asses. Bu, wih no brake, some raecories lead o accumulaion of deb levels nearly 40 imes payroll, clearly an unsusainable level. Indeed, a deb-payroll raio of nearly 0 is presen afer 00 years in one-sixh of all raecories, so his problem is no one limied o exreme draws from he disribuion of oucomes. In addiion, several raecories involve subsanial accumulaion of asses relaive o payroll. 6 The Social Securiy Trusees assumpion abou GDP growh is.5 percen (his is from TR04, where i resuls from.6 percen produciviy growh plus 0.2 percen growh in oal employmen plus he GDP deflaor of 2.5 percen minus he CPI deflaor of 2.8 percen). Bu his is no in sochasic equilibrium, he populaion growh rae is no near 0, and he raio of covered payroll o GDP is changing over ime. 8
Now consider he NDC sysem wih he Swedish brake in place. As one would expec, imposing such a brake reduces he mean IRR, as shown in he second row of Table. The median IRR, hough, acually rises slighly. 7 As Figure 3 shows, he lower ail of asses-payroll oucomes is raised, as also expeced. Indeed, he Swedish balance mechanism is exremely successful in prevening excessive deb accumulaion. Afer 00 years, he 2.5 h percenile of he asse-payroll disribuion is us -0.26, a deb-payroll raio of us over ¼. Bu, even wih he periods of acceleraion ha exis while i is in effec, he balance mechanism does lile o resrain he accumulaion of asses. Indeed, by cuing off he lower ail of he asse-payroll disribuion, he balance mechanism raises boh median and mean asses-payroll raios so ha boh are subsanially posiive afer 00 years. Indeed, even he 83.3 rd percenile is slighly higher, as heading off deb makes he subsequen accumulaion of asses levels more likely along a given raecory. This upward shif in he disribuion of asses over ime has an ineresing impac on he disribuion of inernal raes of reurn by cohor. Figure 4 shows his disribuion for he Swedish brake us discussed, for all cohors wih complee lifeimes during he 500-year simulaion period. For early cohors, he lower ail of he disribuion of inernal raes of reurn is quie low due o a chance of prolonged downward adusmen of NPW and benefis in he case of low raes of growh. Over ime, he likelihood of such an oucome falls; wih prior asse accumulaion, even a series of bad draws wih respec o he growh rae are less likely o drive he balance raio below.0. 7 This resul is possible because he brake mechanism has periods during which he growh raes of NPW accumulaion and benefis are acceleraed relaive o he basic sysem wihou a brake. Thus, even hough he brake mechanism is designed o reduce benefis overall while in effec, i may have disribuional effecs across cohors ha reduce he reurns of cohors wih IRRs already below he median, while raising he reurns of hose a he median. 9
The Swedish balance mechanism is asymmeric, aking effec only once he balance raio falls below.0. Once in effec, however, he Swedish mechanism does provide for cach-up periods of faser growh. A simpler mechanism would eliminae he cach-up phase and apply simply whenever b <. How differen would his even more asymmeric sysem be from he Swedish sysem? Tha is, how significan is he cach-up phase of he Swedish balance mechanism? Figure 5 shows he asse accumulaion paern under his simple asymmeric brake. As one would expec, he disribuion is shifed upward relaive o he Swedish sysem, bu only slighly. Consisen wih his upward shif, he mean and median inernal raes of reurn are a bi lower for his plan han for he Swedish sysem (compare he second and hird lines of Table ). Thus, he Swedish sysem performs very much like a purely asymmeric brake, doing very well a avoiding deb accumulaion bu acually increasing he possibiliy of significan asse accumulaion. To limi asse accumulaion in he presence of a brake, a symmeric brake mechanism is needed, o increase accumulaions and annuiy benefis when he sysem s fiscal healh is assured, no us o compensae for pas periods of cubacks. A symmeric brake would always be in effec, adusing benefis and NPW up or down according o expression (4) or (5) regardless of wheher b is below or above.0. Implemening a symmeric version of he brake leads o more generous benefis for some raecories, and hence higher mean and median IRRs, as he fourh row of Table shows. (I also eliminaes he rend in inernal raes of reurn seen in Figure 4.) The disribuion of asses-payroll raios is similar for he lower ail as under he Swedish and asymmeric brakes, bu he upper ail has been pulled down by he brake s symmery, wih he 97.5 h percenile asses-payroll raio us below.2 afer 00 years. Furher, boh he mean and median asses-payroll raios say close o zero. Thus, he NDC sysem can be 20
made o be quie sable financially, in boh direcions, by applying he brake no only when he balance raio is oo low bu also when i is oo high. This sabiliy holds over he longer erm as well, as shown in Figure 6, which exhibis he disribuion of asses-payroll raios over 500 years for he symmeric brake scheme. Anoher modificaion of he brake mechanism is in is srengh raher han is symmery. While rapid adusmen of benefis and NPW growh may provide sabiliy, i may also concenrae he burdens of fiscal adusmen on relaively few cohors. We plan o explore he quesions of risk-sharing and disribuion in subsequen research, bu we can consider here he impac of a more gradual adusmen process on he sabiliy of he brake mechanism. Figure 7 shows he disribuion of asse-payroll raios when he brake is symmeric bu he adusmen parameer, A, is se equal o 0.5 raher han. One can see from a comparison of Figures 6 and 7 ha more gradual adusmen does widen he disribuion by abou a hird, bu he impac is limied. Anoher poenial modificaion of he NDC sysem involves he compuaion of he rae of reurn for NPW accumulaions and annuiy compuaions. Even if he average populaion growh rae is zero, his growh rae can flucuae, and wih his flucuaion he abiliy of he NDC sysem o cover benefis. Thus, building populaion growh ino he rae of reurn should provide greaer sysem sabiliy, ceeris paribus. Figures 8, 9, and 0, and he las hree rows of Table, presen asses-payroll disribuions and IRRs for NDC sysems based on r = n+g for he no-brake, Swedish-brake and symmeric-brake (A = 0.5) varians. The impac of his change in he mehod is mos easily seen by comparing Figure 8, he raecory of deb under he NDC sysem wih no brake and wih r = n+g, and Figure 2, he deb raecory under he sysem wih no brake and r = g. While he asses-payroll disribuion sill 2
does no fully sabilize, is range is much smaller, especially in he lower ail. The (2.5, 97.5) range of oucomes is now (-2, +8) insead of (-35, +9). Sill, a brake is needed o preven evenual deb explosion along some pahs, and he Swedish brake accomplishes his, as in he previous case of r = g. Again, basing pension calculaions on n+g raher han g also subsanially reduces he variaion in asse-payroll raios, as one can see from a comparison of Figures 3 and 9. Even hough he upper range of he disribuion has been lowered, hough, a symmeric brake is sill needed o sabilize he up side. Thus, even for he case of r = n+g, a symmeric brake is a sine qua non for sysem sabiliy in he very long run. Figure 0 shows he raecory for he symmeric brake wih r=n+g and A=0.5. As under he plan wih r=g picured in Figure 7, he disribuion of oucomes is quie accepable over even 500 years. A comparison of he wo figures indicaes ha using n+g in calculaing he rae of reurn is paricularly effecive a prevening deb accumulaion, a resul ha was also eviden in he earlier comparisons. Sources of Insabiliy As we have seen, he basic NDC sysem, even wih Swedish-syle ne brake, is financially unsable. Even he NDC sysem based on seing he rae of reurn r = n+g requires he applicaion of a symmeric brake o head off subsanial asse or deb accumulaions on some raecories. Wha is causing such insabiliy? One can consider he impac of some sources of uncerainy by eliminaing ohers from he simulaions. In our basic model, uncerainy arises from demographic and economic changes, he laer consising of flucuaions in he ineres rae and he rae of produciviy growh. 8 These economic flucuaions, i urns ou, are an imporan source of he NDC sysem s insabiliy. 22
Figure 2.a and 8.a presen 00-year disribuions of deb-payroll raios for boh versions of he NDC sysem (g and n+g) wih no brake, corresponding o Figures 2 and 8 and differing from he sysems depiced in hose figures only in ha produciviy growh and he ineres rae are held consan a heir mean values. Wih only demographic flucuaions presen, hese new figures show, he disribuions of asse-payroll raios are subsanially narrowed. Under he NDC(g) sysem, he (2.5, 97.5) percenile range a 00 years shrinks from (-35, +9) o (-22, +8); under he NDC(n+g) sysem, he same range shrinks from (-2, +8) o (-0.5, +2.5). Thus, even wih he growh rae g incorporaed in he rae of reurn used in he NDC sysem s calculaions, his process does no come close o neuralizing flucuaions in ha growh rae. Alhough hese comparisons may seem o indicae ha demographic flucuaions are an unimporan source of uncerainy relaive o produciviy growh and ineres raes, oher comparisons show he dramaic improvemen in sabiliy ha resuls from including demographic change, n, in he sysem rae of reurn. For example, comparing Figure 8a o Figure 2a indicaes ha insabiliy as measured by he widh of he 95% range is narrowed by a facor of 0 in he n+g sysem relaive o he g sysem, and comparison of Figure 8 o Figure 2 indicaes ha his range narrows by a facor of 7 even when he economic variaion is included as well. Clearly boh he economic and demographic variaion are imporan sources of uncerainy in hese NDC sysems. Conclusions We have considered he financial sabiliy of differen varians of a sysem of Noional Defined Conribuion accouns, using demographic and economic characerisics of he Unied Saes. In subsequen work, we will consider oher aspecs of NDC sysems, noably heir abiliy 8 The ineres rae maers because he NDC approach is no a pure PAYGO sysem. Wih nonzero values of financial asses, he rae of reurn on hese asses maers. 23
o smooh economic and demographic risks among differen generaions. Among our findings here are:. A sysem similar o ha currenly in use in Sweden, which bases raes of reurn on he growh rae of average wages and uilizes a brake o adus he rae of reurn during periods of financial sress, ensures effecively agains excessive deb accumulaion bu, very much like a simpler asymmeric brake, leads on average o considerable asse accumulaion. 2. Only a symmeric brake, which raises raes of reurn during periods of financial srengh, can avoid considerable accumulaions of financial asses on some pahs. The brake can be more gradual han under he Swedish sysem and sill provide a sable disribuion of oucomes. 3. An NDC sysem in which raes of reurn are based on oal raher han per capia economic growh is inherenly more sable han he basic NDC sysem, wihou reference o he brake mechanism in use. 4. A considerable share of he volailiy in he financial performance of NDC sysems is aribuable o economic, raher han demographic, uncerainy. Evidenly sochasic simulaion of he sysem s finances can reveal aspecs of is performance ha are no oherwise obvious, and can assis in improving sysem design. This promises o be a valuable use for sochasic simulaion models of pension sysems. 24
References Alho, J.M., J. Lassila and T. Valkonen. 2005. Demographic Uncerainy and Evaluaion of Susainabiliy of Pension Sysems. In Non-Financial Defined Conribuion (NDC) Pension Schemes: Concep, Issues, Implemenaion, Prospecs, edied by R. Holzmann and E. Palmer, 95-5. Washingon D.C.: World Bank. Booh, Heaher. 2006. Demographic forecasing: 980 o 2005 in review. Inernaional Journal of Forecasing 22(3): 547-58. Holzmann, R. and E. Palmer, Ediors. 2005. Non-Financial Defined Conribuion (NDC) Pension Schemes: Concep, Issues, Implemenaion, Prospecs. Washingon D.C.: World Bank. Lee, Ronald. 993. Modeling and Forecasing he Time Series of US Feriliy: Age Paerns, Range, and Ulimae Level. Inernaional Journal of Forecasing 9: 87-202. Lee, Ronald and Lawrence Carer. 992. Modeling and Forecasing U.S. Moraliy. Journal of he American Saisical Associaion 87(49): 659-67, and Reoinder, same issue, 674-675. Lee, Ronald and Timohy Miller. 200. Evaluaing he Performance of he Lee-Carer Approach o Modeling and Forecasing Moraliy. Demography 38(4): 537-549. Lee, Ronald and Shripad Tulapurkar. 994. Sochasic Populaion Proecions for he Unied Saes: Beyond High, Medium and Low. Journal of he American Saisical Associaion 89(428): 75-89. Lee, Ronald and Shripad Tulapurkar. 998a. Sochasic Forecass for Social Securiy, In Froniers in he Economics of Aging, edied by David Wise, 393-420. Chicago: Universiy of Chicago Press. Lee, Ronald and Shripad Tulapurkar. 998b. Uncerain Demographic Fuures and Social Securiy Finances. American Economic Review 88(2): 237-24. Lee, Ronald D., Michael W. Anderson, and Shripad Tulapurkar. 2003. Sochasic Forecass of he Social Securiy Trus Fund Repor for he Social Securiy Adminisraion (January) posed on he web sie of he Office of he Acuary. Legros, Florence. 2003. Noional Defined Conribuion: A Comparison of he French and German Poin Sysems, paper presened a he World Bank and RFV Conference on NDC Pension Schemes, Sandhamn, Sweden (Sep. 28-30, 2003). Palmer, E. 2000. The Swedish Pension Reform Model Framework and Issues, World Bank s Pension Reform Primer Social Proecion Discussion Paper no. 002. 25
Seergren, O. 200a. The Auomaic Balance Mechanism of he Swedish Pension Sysem A Non-echnical Inroducion. Wirschafspoliishe Blaer 4/200: 399-349. Also.: hp://www.forsakringskassan.se/sprak/eng/publicaions/dokumen/au007.pdf Seergren, O. 200b. Two Thousand Five Hundred Words on he Swedish Pension Reform. For he Workshop on Pension Reform a he German Embassy, Washingon D.C. Swedish Pension Sysem. 2005. Annual Repor. Valdes-Prieo, Salvador. 2000. The Financial Sabiliy of Noional Accoun Pensions. Scandinavian Journal of Economics 02(3), 395-47. 26
Appendix: Benefis and Taxes under a Simple NDC Plan This appendix illusraes he relaionship beween benefis and axes a a given poin in ime under a simple version of he Noional Defined Conribuion scheme in which he inrinsic rae of reurn is based on he growh rae of covered wages. Consider he relaionship beween axes and benefis a any given dae under a simplified version of an NDC sysem under which he rae of reurn used o accumulae noional pension wealh and o calculae annuiies, r, is equal o he conemporaneous growh of covered wages. Taxes a ime are: + + 2 + L (A) T = τ ( W + W +... + W ) where W + is covered wages in year for he enire cohor ha will reire years hence and L is he number of years ha individuals work. For simpliciy, assume ha each reired cohor receives in benefis he annual real reurn on is noional pension wealh of r, so ha he cohor s NPW will say consan in real erms afer reiremen, and is annual payou is consan as well. 9 Then aggregae benefis a dae will equal: (A2) B = r ( NPW + NPW...) + 9 This assumpion implies ha a cohor s benefis per capia grow as he cohor s populaion declines and indeed approaches infiniy as he generaion dies off, which is obviously unrealisic. We impose i here only for purposes of exposiion 27
28 where NPW is he noional pension wealh in he year of reiremen for cohor reiring in year -. The noional pension wealh a reiremen for cohor - is: (A3) + + + + + = = 2 ) (... ) ( L l l L r W r W W NPW τ Combining expressions (A2) and (A3) and comparing he resuling expression wih expression (A), we can see ha a sufficien condiion for axes and benefis o be equal is ha, for all k beween and L, (A4) = = + + = 0 ) ( k l l k k r W W τr τ or, ha axes paid by workers k years away from reiremen equal benefis aribuable o earnings a he same age for all reirees. We have assumed ha r s equals he growh rae of covered wages beween daes s- and s. If we assume in addiion ha his growh rae is shared by he enire age-wage disribuion (i.e., ha he relaive age-disribuion of covered wages remains fixed), hen expression (A4) can be rewrien as: (A5) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 0 0 0 = + + + + + + = + + = = = = = = = = = + + = = = + k l l m m k q q k l l m m k p p k k k l l m m k k r r r r r r r rw r r rw W
The las line of expression (A5) is saisfied if r is consan over ime, which reflecs he underlying consisency of using he growh of covered wages as a rae of reurn for he NDC sysem. If r varies over ime, hough, expression (A5) will generally no hold. For example, suppose k =, corresponding o wages in he year prior o reiremen. Then he las line of (A5) reduces o: (A6) r ( + r ) ( + ( + r ) + ( + r ) +...) 2 = From his, we can see ha if he curren growh rae used o compue annuiies, r, is greaer (less) han he growh raes of covered wages during he accumulaion phase, hen he expression on he lef-side will be greaer (less) han and axes on earnings for hose in he year prior o reiremen will be inadequae (more han adequae) o cover benefis for reirees based on earnings in he year prior o reiremen. Alhough he resuls are more complicaed for values of k >, he poin is ha variaions in r over ime can cause he NDC sysem o run deficis or surpluses, he variaion being larger he larger is he variaion in he growh rae of covered wages. This variaion in deficis occurs even under he assumpion of a fixed covered earnings age profile; relaxing his assumpion adds ye anoher poenial source of variaion in he sysem s annual deficis. 29
Table. Average Inernal Raes of Reurn Simulaion Mean IRR Median IRR NDC (g) No Brake.007.007 NDC (g) Swedish Brake.000.009 NDC (g) Asymmeric Brake, A=.0093.005 NDC (g) Symmeric Brake, A=.022.030 NDC (g) Symmeric Brake, A=.5.006.030 NDC (n+g) No Brake.00.03 NDC (n+g) Swedish Brake.00.03 NDC (n+g) Symmeric Brake, A=.5.033.034 Source: Calculaed from sochasic simulaions described in ex.
Figure. Raio of Reirees o Workers, 5 Sample Pahs.2 Raio of old (>66) o young (2-66) 0.8 0.6 0.4 0.2 0 0 50 00 50 200 250 300 350 400 450 500 Year Financial Asses/Payroll 20 0 0-0 -20 0.025 0.67 0.5 0.833 0.975 mean Figure 2. Financial Asses/ Payroll (r=g, no brake) -30-40 0 0 20 30 40 50 60 70 80 90 00 Year
Figure 3. Financial Asses/Payroll (r=g, Swedish brake) 20 Financial Asses/Payroll 5 0 5 0.025 0.67 0.5 0.833 0.975 mean 0-5 0 0 20 30 40 50 60 70 80 90 00 Year Inernal Rae of Reurn 0.025 0.02 0.05 0.0 0.005 0 0.025 0.67 0.5 0.833 0.975 mean Figure 4. Inernal Raes of Reurn (r=g, Swedish brake) -0.005-0.0 0 50 00 50 200 250 300 350 Year
20 Figure 5. Financial Asses/Payroll (r=g, asymmeric brake, A= ) Financial Asses/Payroll 5 0 5 0.025 0.67 0.5 0.833 0.975 mean 0-5 0 0 20 30 40 50 60 70 80 90 00 Year Financial Asses/Payroll 2.5 2.5 0.5 0 0.025 0.67 0.5 0.833 0.975 mean Figure 6. Financial Asses/Payroll (r=g, symmeric brake, A= ) -0.5-0 50 00 50 200 250 300 350 400 450 500 Year
Financial Asses/Payroll 2.5 2.5 0.5 0 0.025 0.67 0.5 0.833 0.975 mean Figure 7. Financial Asses/Payroll (r=g, symmeric brake, A=.5) -0.5-0 50 00 50 200 250 300 350 400 450 500 Year Financial Asses/Payroll 0 8 6 4 2 0.025 0.67 0.5 0.833 0.975 mean Figure 8. Financial Asses/Payroll (r=n+g, no brake) 0-2 0 0 20 30 40 50 60 70 80 90 00 Year
Figure 9. Financial Asses/Payroll (r=n+g, Swedish brake) Financial Asses/Payroll 0 8 6 4 2 0.025 0.67 0.5 0.833 0.975 mean 0-2 0 0 20 30 40 50 60 70 80 90 00 Year Financial Asses/Payroll 2.5 2.5 0.5 0 0.025 0.67 0.5 0.833 0.975 mean Figure 0. Financial Asses/Payroll (r=n+g, symmeric brake, A=.5) -0.5-0 50 00 50 200 250 300 350 400 450 500 Year
Financial Asses/Payroll 20 0 0-0 -20 0.025 0.67 0.5 0.833 0.975 mean Figure 2a. Financial Asses/ Payroll (r=g, no brake, consan ineres, growh raes) -30-40 0 0 20 30 40 50 60 70 80 90 00 Year Financial Asses/Payroll 0 8 6 4 2 0.025 0.67 0.5 0.833 0.975 mean Figure 8a. Financial Asses/Payroll (r=n+g, no brake, consan ineres, growh raes) 0-2 0 0 20 30 40 50 60 70 80 90 00 Year