Annuities in NDC by Juha Alho 1, Jorge Bravo 2, and Edward Palmer 3

Transcription

1 Draf (Unfinished) November 30, 2009 Annuiies in NDC by Juha Alho 1, Jorge Bravo 2, and Edward Palmer 3 1 Inroducion In a defined conribuion sysem he pension akes he form of a life annuiy, regardless of wheher money is acually accessible, as in a financial defined conribuion (FDC) pension scheme, or is only in he form of a pension righ noed on a personal accoun, as in a nonfinancial defined conribuion (NDC) scheme. The calculaion of he annuiy requires an esimae of life expecancy, appropriaely discouned. This creaes a longeviy risk for he insurer. In his paper we focus on he esimaion of he longeviy risk, and on ways of managing i wihin he conex of NDC pension schemes. Empirical sudies show ha pas forecass of moraliy have sysemaically underesimaed fuure declines. Biases can be reduced using appropriae saisical mehods, bu unpredicable random variaion remains. Alhough he period needed o calculae an annuiy a reiremen is relaively shor, hree o four decades, individuals making savings decisions while working have a longer planning horizon. In consequence, he biases and random variaion can have considerable consequences for he finances of a pension scheme. Assume, for example ha we can observe ex pos ha he esimae of a cohor s life expecancy used in calculaing he life annuiy from age 60 or 65 was one year shor of he oucome, where he esimae was 20 years and he oucome 21 years. This means he cohor lived en percen longer and ha oal paymens ended up cosing 10 percen more han 1 Joensuu Universiy, Finland 2 Universiy of Évora, Porugal 3 Uppsala Universiy, Sweden and Swedish Social Insurance Agency 1

2 projeced. This is equivalen o an annualized exra rae of reurn of abou a half a percen per year (Table 1). Table 1. Life expecancy error for an expeced life of 20 years Error expressed in years Toal error expressed as an annualized rae of growh Source. Palmer & Melkvis The sysem risks characerized by Table 1 are highly relevan, in view of he uncerainy of acual moraliy developmen. Figure 1, shows he developmen of he remaining period life expecancy a age 65 using Swedish daa. During i flucuaed more or less randomly beween 13 and 14 years. In he early 1940s somehing occurred ha led o an upward rend ha has characerized he pas 65 years. Moreover, here appears o be wo subperiods, he firs from 1940 o he 1970s, and he second from 1980 o he presen, he laer wih a sligh acceleraion in remaining life expecancy. 20 Figure 1. Life expecancy a age 65. Sweden e65 2

3 The figure provides an illusraion of he problem confroning a Swedish pension fund manager over a period of a cenury. Assume we begin wih a fla series resembling ha in he Figure 1 for he period Any saisical model based on a ime series wih his appearance canno perform beer han he underlying daa, which in his case signal no change. Neverheless, change occurs around 1942 and coninues hereafer. I may ake several years for a model of he underlying developmen of moraliy o capure he upward rend. The duraion of sysemaic underesimaion will depend on model characerisics and he esimaion period. A rigid model combined wih a long esimaion period will adap more slowly han a similar model based on a shorer esimaion period. A similar change occurs around 1980, when he rae of improvemen in life expecancy increases, which he saisical model will once again ake some ime o adjus o. If he divergence of acual from projeced life expecancy were a random process, financial equilibrium could be mainained when money lef by shor-lived cohors would finance cohors who live longer han expeced - given ha redisribuion of cohor surpluses is no prohibied by conrac or law. This no likely o happen, however, because forecas errors for consecuive cohors are auocorrelaed. Declining moraliy among persons 60 and older is a normal phenomenon, and acceleraing change has ypified many counries in he pas half decade. Japan s rapid improvemen in moraliy is he mos obvious example. We are used o hinking ha saisical models of hisorical ime series daa should be based on a leas 50 years of observaions (Box and Jenkins 1976). This is based on consideraions of model choice and accuracy of empirical daa characerisics. As suggesed by Figure 1, he use of shorer daa periods may seem aracive. Bu, shor daa periods lead o increased esimaion errors and erraic updaes, especially for long forecas horizons. The considerable concern among demographers abou he appropriae saring poin for a projecion, aess o his dilemma. This paper discusses crieria for choosing an appropriae projecion procedure for moraliy in counries, such as Sweden, where reliable long-erm daa exis. In many oher counries, lack of daa or poor qualiy of daa can be a major problem. 3

4 The cos of incorrec life expecancy projecions mus be borne by someone. This has implicaions on he economic welfare across generaions. A major argumen in favour of NDC pension schemes, as opposed o non-financial defined benefi (NDB) schemes, is ha NDC incorporaes life expecancy explicily in he calculaion of annuiies. Bu, if he assessmen of life expecancy is no sufficienly accurae, welfare losses will occur. If he life expecancy used o compue annuiies is sysemaically biased - eiher down or up - he disribuion of consumpion beween cohors and generaions will be affeced. Given ha forecas errors are ineviable, i is clearly desirable ha he procedure used o compue life expecancy is accurae on average so ha one can jusify residual redisribuions on he basis ha hey are random. The counerfacual is a procedure wih a endency owards inergeneraional redisribuion. Bu, neiher a procedure ha sysemaically benefis older cohors by underesimaing life expecancy nor a procedure ha sysemaically benefis younger cohors by being overly cauious is desirable. The ouline of he paper is as follows... 2 Annuiy models and sysem risks Longeviy risk exiss a an idiosyncraic and aggregae level. For he individual, i is he risk of ouliving his or her accumulaed wealh. In aggregae erms, longeviy risk is he risk ha he average member of a birh cohor will sysemaically live longer han prediced. Sandard annuiies are he classical ool in he managemen of individual longeviy risk. They are presen in DB and NDC social securiy sysems provided by governmens, DB benefi plans provided by corporaions hrough pension funds and life annuiies provided by insurance companies. In he absence of aggregae longeviy risk, he pooling mechanism implici in The Law of Large Numbers would be sufficien o make longeviy risk manageable for annuiy providers. In he presence of sysemaic longeviy risk, he provision of longeviy insurance for individuals becomes a difficul ask since exising insrumens do no allow annuiy providers o effecively hedge aggregae longeviy risk. In his secion we review he sandard fixed level annuiy model and discuss alernaive longeviy risk-sharing arrangemens considering variable annuiies. 4

5 2.1. Sandard annuiy model: fixed level annuiy As in financial defined conribuion schemes (FDC), life annuiies are a key componen in he disribuion phase of defined conribuion PAYGO (NDC) pension schemes. In a generic NDC sysem, he benefi is in he form of a fixed level annuiy, based on an individual capial balance a reiremen, payable for each individual s remaining life. In NDC i is also possible o index he annuiy wih he sysem s inernal rae of reurn. In NDC i is no possible o claim a lump sum a reiremen because he financing for he sream of paymens is dependen upon he fuure flow of conribuions. This conrass wih FDC schemes where individual asses are accumulaed in DC plans or individual saving accouns, which can offer hree main opions for he payou phase: lump-sum paymens, programmed wihdrawals and annuiies. Combined soluions involving any possible combinaion beween hese hree alernaives are of course possible and acually found in he markeplace. DC benefi paymens are given by B K a = (1) && x where a&& x is he sandard acuarial noaion for he presen value of a uni whole life annuiydue, deermined using he moraliy informaion and projecions available a ime = 0 and assuming a consan discree ineres rae of i per period. Using he equivalence premium principle (see, e.g., Gerber, 1997), a&& x is given by K( x) ω x 1 a && x = E υ = ( υ px) (2) = 0 = 0 where K( x) = [ T( x)] is he number of compleed fuure years lived by ( x ), also denominaed he curae fuure lifeime of x, p x denoes he probabiliy ha a person aged x will be alive 1 a age x +, υ = (1 + i) denoes he sandard discoun facor and ω is he limiing age of he life able. 5

6 In a generic NDC K represens he accumulaed noional capial a reiremen and he annuiy facor a&& x is calculaed using a cohor-specific prospecive life able reflecing life expecancy a reiremen. The annuiy can also incorporae a consan real rae of reurn deermined by he sysem s inernal rae of reurn, pension benefis in a NDC can generically be deermined according o (1)-(2). However, if he annuiy is fronloaded wih a rae of reurn here mus be a possibiliy o adjus for he ineviable divergence of his from he acual oucome. For example Sweden compues he annuiy based on real per capia wage growh of 1.6 percen and hen adjuss annuiies annually for deviaions of he oucome from his norm. Compared wih an FDC scheme, where in some cases gender specific life ables are considered, 4 in counries where NDC schemes have been inroduced he annuiy facor is calculaed using unisex life ables inroducing, hus, an explici redisribuion mechanism from hose who live a shorer life (he male populaion) o hose whose longeviy prospecs are longer (he female populaion). The rae of reurn plays a crucial role in calculaing sandard annuiies in boh FDC and in NDC sysems. Compared wih a FDC sysem, where he rae of reurn (echnical ineres rae) is fixed ex ane on a prudenial basis, i.e., in order o ensure ha he policy ne premium reserves are posiive, or a leas non-negaive, in an NDC scheme he rae of reurn for he payou phase is deermined in order o financially balance he sysem, i.e., in order o mach he presen values of overall sysem asses and liabiliies. Moreover, in order o mainain a fixed conribuion rae over ime, oal NDC sysem ne asses mus be non-negaive (Palmer 2006). Life annuiies consiue he unique financial insrumen ha accomplishes wo of he mos imporan objecives of a pension scheme: (i) proec he annuian from ouliving his or her resources; (ii) provide reiremen income for he remaining life of he pensioner and his or her dependans, offering hus longeviy insurance. In an NDC scheme, he annuiy provider can a firs accomplish his by pooling a large group of pensioners (a birh cohor) and by making careful assumpions abou he rae of reurn earned by he sysem asses (he norm ) or by compuing he annuiy based on no reurn and disribuing reurns as hey arise. The laer akes he form of normal indexaion. The pooling of pensioners means ha he capial of hose 4 In privae insurance, annuiy providers may even adop muli-decremen life ables, incorporaing addiional risk facors (smoker saus, benefi amoun, socio-economic caegory, ec.). 6

7 individuals who do no reach heir projeced remaining life expecancy, as calculaed from acuarial life ables, is redisribued among he survivors in he scheme. Those who live a shorer life han expeced will subsidize hose who live longer. Upon he deah of a pensioner, he balance of he original noional capial, ogeher wih invesmen reurns (collecively called moraliy credis or moraliy profis ), is redisribued among surviving pensioners. Compared wih oher forms of saving, hese moraliy credis have an opporuniy cos since he annuiy purchase is irreversible and he opion o leave beques is los. In an NDC scheme hese credis can become a componen of he capial balance ransformed ino an annuiy, and are deal wih his way in, for example, Sweden. There is an opion for he policy maker, however, o no include moraliy credis in he annuiy, which leaves an undisribued NDC surplus. To sum up, each annuiy paymen comprises hree componens: a proporion consising of he original acquisiion price NDC conribuions, a proporion consising of he capial reurn and a proporion consising of he moraliy profi released by he premaure (below average) deah of some pensioners. To illusrae he significance of longeviy insurance embedded opions wihin annuiies and DB pension schemes, consider he simple following example. Suppose ha, according o populaion life ables, here is a 20% chance ha a 85-year-old male will no survive unil his nex birhday. If 100 such males agree o ener ino a one year erm annuiy (a.k.a. onine) agreemen by conribuing wih 100 EUR each o a common fund, which will redisribue he capial and invesmen reurn (say 5% yield pa) amongs survivors, he funds will grow o 10,500 EUR by he end of he year. Of he iniial 100 males, 80 are expeced o survive, wih a raher small variance around he expeced value, leaving an average of 10,500/80= EUR per survivor. This leads o a oal reurn of 31.25%, far exceeding he 5% invesmen reurn used o accumulae he capial, because annuians have ceded conrol of heir asses in he even of deah and, as such, he inheriance capial is redisribued among survivors increasing he oal reurn on capial. The simple bu powerful algebra of he so-called moraliy credis behind fixed onines can be saed symbolically as follows: le 1 p x denoe he one-year probabiliy of survival for some currenly aged x and r he effecive ineres rae per year. Then, he expeced reurn for he 7

8 survivors will be given by Er 1+ r = > r, he expecaion being more likely he larger 1 px * ( ) 1 he number of members paricipaing in his risk-sharing scheme. The moraliy credis are defined by * ( Er ( ) r) and give an idea of he magniude of he exra reurn ha a selfannuiizer should a leas earn o keep up wih he annuian. Alhough in a real world he use of differen life ables, differen (including variable) ineres raes and he exension of he agreemen over many years as in he case of a life annuiy will lead o differen numerical resuls, he risk-sharing mechanism of onine insurance is he concep underlying all annuiies and DB pension plans. Through plain vanilla level annuiies, annuiy providers offer individuals proecion agains wo broad classes of risk: biomeric risks, such as longeviy and moraliy risks, and macroeconomic and financial marke risks, such as ineres rae, equiy, credi or liquidiy risks. There are however, risks wih sandard level life annuiies, boh in FDC and in NDC pension schemes. When an individual reires, he bears he ineres-rae risk in he sense ha a ha ime or in he period preceding he dae of he annuiy claim - marke or inernal raes of reurn may be low, causing reiremen annuiy benefis o be permanenly low. Afer he reires, he bears inflaion risk in he sense ha he real value of his pension benefis may drop due o subsequen unanicipaed inflaion. This is a real risk in many financial conexs bu in NDC he implici rae of reurn involved in poenial inflaion indexaion is exernalized in he form of price indexaion. Addiionally, he annuian is somehow exposed o sysemaic longeviy risk if longeviy improvemens are significanly higher han prediced, in which case he capial (solvency) reserves are consumed and he sysem becomes insolven. Annuiy providers (insurance companies and governmens) on heir side are faced wih wo major risk sources: ineres-rae risk and longeviy risk. Ineres-rae risk in he sense ha he observed rae of reurn and he corresponding invesmen earnings paern may be differen from he assumed consan rae i, causing a mismach beween asse cash flows and expeced liabiliy ouflows. In he financial lieraure, classic immunisaion heory (see, e.g., Redingon, 1952) suggess ha in order o proec an invesmen from small changes in he erm srucure of ineres raes, annuiy providers should back heir annuiy porfolios wih asses whose respecive duraions equal hose of annuiy liabiliies, and whose respecive convexiies are larger han hose of annuiy liabiliies. This is difficul in pracice, since in FDC schemes 8

9 long-erm bonds are no available in mos bond markes. Moreover, if real annuiies are o be provided, real long-erm bonds will have o be issued as well. This means ha FDC annuiy markes would definiely benefi from he issuance of long-erm governmen bonds. Evens in 2008 in some counries have shown, however, ha he qualiy of asses considered is also imporan, since he possibiliy of defaul is real and credi risk should be aken ino accoun. In NDC sysems he fulfilmen of firs-order (duraion) and second-order (convexiy) condiions for sysem immunisaion is even more difficul o aain. In fac, duraion and convexiy are ineres rae risk measures ha depend on discoun raes (yields), on he mauriy srucure of boh asses and liabiliies and on he rae of reurn and hey are no as easily manageable in NDC sysems as hey are in financial markes. The Swedish balancing mechanism essenially recognizes his difficuly by adjusing sysem liabiliies (accouns of workers and pensioners) o correc for adverse oucomes for insurer risks. Annuiy providers are faced wih aggregae longeviy risk, i.e., wih he risk ha enire cohors sysemaically live longer han anicipaed by projeced moraliy ables. In NDC sysems, he longeviy risk challenges he sysem financial equilibrium. For a counry wih an auomaic balancing mechanism his can rigger balancing (see, e.g., Sweden), i.e. a revision of accoun values, o achieve equilibrium.. In FDC schemes subsanial changes in moraliy paerns could seriously challenge he profiabiliy of insurance companies. Longeviy risk is a sysemaic risk and, hus, canno be eliminaed by diversificaion from he economy. This means ha annuiy providers assume he remaining sysemaic moraliy risk and guaranee ha he level benefis are paid regularly in deerminisic manner o hose alive. In exchange, in FDC sysems hey charge a risk premium in order o provide appropriae solvency capial and reduce he risk of defaul. An NDC scheme wih a balancing mechanism is designed o accommodae he effecs of unsysemaic demographic deviaions bu also bear sysemaic longeviy risks. NDC wihou explici balancing may neverheless mainain equilibrium saus by no disribuing he moraliy gains and/or no graning a full inernal rae of reurn o annuians. 5 So in boh NDC and FDC schemes he quesions will ulimae be: The quesion is, who should bear he longeviy risk? And, a wha price? 5 In principle, Lavia and Poland have chosen his approach. 9

10 We will ry o address his quesions wihin NDC pension schemes discussing some of he alernaive sraegies ha have recenly been proposed in he lieraure o manage longeviy risk wihin FDC sysems (for a deailed reamen of his subjec see, for insance, Blake e al. (2006a,b) and Bravo (2007)). There are roughly wo approaches in managing longeviy risk. The firs involves hedging he risk while reaining i, mosly hrough porfolio diversificaion. Insurance companies can reinforce porfolio diversificaion effecs wihin an annuiy porfolio and explore naural hedging opporuniies by combining porfolios wih complimenary cash flows (e.g., life insurance conracs and annuiies), reducing (bu no eliminaing compleely due o basis risk) he impac of negaive moraliy scenarios. Alernaive naural hedging soluions involve inernaional diversificaion, exploiing he evidence ha suggess ha moraliy shifs observed in differen counries are no perfecly correlaed, socioeconomic diversificaion, making use of he idea ha moraliy shifs affec differen socioeconomic groups differenly. Finally, diversificaion may be boosed hrough swap agreemens beween paries wih complimenary longeviy risk exposures. The second approach involves eliminaing (or a leas reducing) he risk by ransferring i o anoher counerpary. This can be achieved by conrac redesign, by resoring o liabiliy buyou s, by securiizaion mechanisms and, finally, by resoring o capial marke soluions, namely by issuing longeviy/survivor bonds and all sor of longeviy-coningen securiies (derivaives). In FDC sysems, some advocae ha governmens (or privae companies) should issue cohor survivor or longeviy bonds (LBs), which are essenially financial insrumens wih a sochasic mauriy in which fuure (principal and/or coupon) cash flows depend on he realizaion of a survivorship index of a seleced birh cohor. 6 Denoed I S x, he survivorship, index gives he proporion of he iniial reference populaion ha is sill alive a ime. The underlying idea beyond he use of his kind of bonds as hedging insrumens o he holders (annuiy providers and pension funds) is very simple: if he reference populaion lives longer han expeced, coupon paymens will be bigger han prediced and his will provide addiional cash flows for annuiy providers facing longeviy risk wihin annuiy porfolios. 6 See, e.g., Blake and Burrows (2001) and Blake e al. (2006a). 10

11 Palmer (2009) presens a model where he NDC asse is denominaed in NDC bond shares, which in he absence of a mechanism o disribue he risk o he annuians, means ha he governmen (he axpayers) bear his risk. In principle, his would creae longeviy bonds for NDC. This inroduces a redisribuive effec from annuians o axpayers in general. The alernaives are o ry o marke he NDC bonds or o disribue he risks wihin he insurance collecive, for example, on a birh cohor basis. Alhough in FDC schemes LBs are good candidaes for hedging aggregae longeviy risk, here are a number of argumens pro and agains he governmen issuance of LBs. Auhors like Blake and Burrows (2001), Blake (2003) and Brown and Orzag (2006) favour he governmen issuance of LBs invoking some of he classic argumens for governmen inervenion (public goods, exernaliies, marke failures, adverse selecion, inergeneraional issues, ec.). Specifically, he auhors argue ha his will suppress a marke failure moivaed by marke incompleeness (governmens acing as lenders of las resor), ha by spreading he risk over a huge amoun of axpayers his will eliminae he marke price of longeviy risk, ha Saes conribue o he increase in longeviy and ha his will allow a more efficien inergeneraional risk sharing. Regarding he inergeneraional issues, if financial markes have he capaciy o allocae risk across risk bearers and across ime only governmens have he unique power o allocae risk across boh curren (born) and fuure (unborn) generaions. In addressing his subjec, we should no neglec ha younger and older generaions are no in he same posiion o face he consequences of unexpeced increases in longeviy. In fac, as noed by Turner (2006) he human capial of he younger generaions provides hem wih a comparaive advanage in absorbing and oherwise adjusing o he new prospecs abou longeviy hroughou heir lifecycle unlike he senior generaions ha have exhaused heir human capial. Younger generaion can use is enire life-span o adjus labour supply, consumpion and saving paerns up o an increase in longeviy, somehing he older generaions canno. In addiion, he human capial of he younger generaions is likely o yield a higher rae of reurn han ha of preceding generaions. In pracice, voer preferences may deermine he resul. As populaions ge older he imporance of senior voer s enlarges, hus increasing he poliical pressure o a bias response in favour of he desires and aspiraions of curren generaions. The goal of NDC, however, is o as much as possible insulae he sysem from he poliical risk ha voer preferences may represen. We noe also ha an argumen agains LB s is ha he sae has already a significan long exposure o longeviy risk hrough 11

12 DB pension sysems and healh care sysems, and ha by issuing LBs saes would be diversifying only wihin he axpayers group (e.g. Dowd 2003; Turner 2006). In an NDC sysem, he governmen issuance of LBs as a longeviy risk hedging ool induces some redisribuive effecs. The reason is simple. In his case, he bigges annuiy provider and poenial LB buyer (governmen) is, simulaneously, he leading candidae o be he issuer. The marke for LBs is a marke where we should expec a clear srucural imbalance, wih naural excess of demand. Poenial issuers of LBs include, beyond governmens, public or privae eniies ha have a long posiion on longeviy risk, i.e., ha benefi from increasing longeviy (e.g., pharmaceuical companies, residenial car homes, housing associaions, bioech companies). In order o cope wih longeviy risk in NDC sysems we can, however, engineer alernaive capial marke soluions, for example a soluion in which governmens issue Inverse Longeviy Bonds (ILBs). ILBs are financial insrumens in which he (increasing) coupons are an inverse funcion of a survival index of a given cohor, I S x. These securiies would be comparable o convenional inverse floaers whose coupon paymens depend inversely on marke ineres raes. In his case, he bond would be designed o be a hedge o he issuer ha would benefi from a siuaion in which he survival index is higher han anicipaed (conversely, he buyer gains if he index is lower han anicipaed). The bond should be a long-erm bond wih sochasic mauriy equal o he ime of deah of he las survivor from he reference cohor, which means ha i would make paymens for as long as any member of he reference cohor is sill alive. The bond would hus be designed o proec he issuer agains any unanicipaed improvemens in moraliy up o he mauriy dae of he bond (cohor exincion). Alhough he araciveness of his bond o invesors is sill o be proven, one migh argue ha invesmen banks and hedge funds may be recepive o buy his bond and have a shor posiion on longeviy risk since is has low correlaion wih radiional marke risk facors. Thus, he combinaion of a low bea parameer, a poenially posiive alfa parameer and an accepable rae of reurn, including some compensaion for he marke price of longeviy risk, migh conver longeviy-linked securiies ino an aracive invesmen wihin a diversified 12

13 porfolio. Once he marke is developed, i would also arac oher marke sakeholders, namely speculaors being on fuure longeviy prospecs and arbirageurs. The issuance of LBs or ILBs in boh NDC and FDC pension schemes raises some problems ha have o be addressed. Firs he underlying asse of hese coningen claim securiies is a survivor index. As we know, indexes suffer from a number of problems: hey are consruced from daa published infrequenly and subjec o incurred-bu-no-repored errors, hisorical daa ypically needs o smoohed adding some model risk, indexes are subjec o inegriy and conaminaion risk, moral hazard problems may arise (daa providers have much earlier access o daa han invesors) and, finally, survivor indexes involve projecions of fuure moraliy ha are subjec o boh model and parameer risk. Addiionally, longeviy-linked securiies involve significan valuaion problems, eiher because we canno use sandard non arbirage argumens o price since he marke is incomplee, eiher because here are modelling issues o be deal, namely he adopion of a paricular (sochasic) represenaion of moraliy uncerainy, or because of esimaion issues, in paricular, of he marke price of longeviy risk Adjusable Annuiies: Alernaives Sandard fixed level annuiies provide a unique form of longeviy insurance. In spie of heir appeal in providing proecion agains ouliving one's own resources, empirical evidence shows ha in pension sysems in which annuiizaion is no mandaory he significance of annuiy markes is limied. This conradics economic heory ha predics ha, given some assumpions, raional individuals should purchase annuiies in he disribuion phase of heir life-cycle (Yaari (1965), Davidoff e al., (2005)). The possible reasons for he for small annuiy markes have been exensively discussed in he lieraure and are normally classified ino demand and supply consrains (see, e.g., MacKenzie (2002), Impavido e al. (2004), Sewar (2007) and Rusconi (2008)). On he demand side, reasons for no volunarily purchasing annuiies include he exisence and he imporance of subsiues, namely, governmen PAYGO pension benefis, he significance of he beques moive, he desire o save in more liquid asses in order o face unexpeced large expendiures (e.g., medical care, nursing), he high cos of annuiies due o 13

14 adverse selecion, he underesimaion of personal longeviy, myopia or lack of undersanding of he properies of annuiies, he lack of rus in financial insiuions (paricularly in Emerging Economies) or he insufficiency of ax incenives ha unbalance saving decisions in favour of buying annuiy proecion agains alernaive invesmens. On he supply side, he ype and scope of he limiaions o he developmen of annuiy markes is also significan. They include he lack of high-qualiy informaion on moraliy ables, which induces annuiy providers o price conservaively, he inexisence or insufficiency of asses wih which o back he long-erm promises and risks (ineres rae, longeviy, liquidiy risks) represened by annuiies, concerns regarding regulaory capial requiremens or wih he srengh of exising providers ha would make i difficul for new enrans o survive or, finally, he incomplee naure of annuiy markes in he sense ha hey do no offer proecion agains inflaion, hey lack equiy marke exposure, hey are illiquid and do no insure agains muliple shocks. The purchase of an annuiy is probably one of he mos imporan financial decisions in he life of an individual given is very long erm consequences. The perceived poor value of radiional annuiies moivaed he developmen of alernaive invesmen-linked reiremenincome soluions, he principal examples being income draw down programmes and variable (invesmen-linked) annuiies. For individuals wih low degrees of risk aversion, phased wihdrawals are considered an alernaive o he purchase of a whole life level annuiy in FDC sysems and could also be considered in a NDC scheme, alhough wihdrawals shorer han a whole life would have o be conciliaed wih he governmen s commimen o provide a social pension as he safey ne of las resor.. Phased wihdrawals conver a fund ino insalmens of income wihin prescribed limis. Numerous alernaive phased income draw down programmes could be envisaged, ranging from a fixed benefi level programme (unil he deah of he individual or he exhausion of funds) o fully variable programmes, in which he wihdrawals are linked, e.g., o life expecancy. Dus e al. (2003) analyse he implicaions of alernaive phased wihdrawal rules using German daa. The auhors consider a fixed benefi rule of wihdrawal ogeher wih hree alernaive rules ha generae variable benefis: a fixed percenage rule, a 1/ω rule, where ω 14

15 could denoe he highes aainable age in a life able or he life expecancy a he age of reiremen, and a 1/ Ee ( x( )) rule where Ee ( x( )) denoes he remaining life expecancy in a dynamic way. 7 Their simulaion resuls sugges ha a paricular mixed-rule seems o be more aracive. This rule involves underaking phased wihdrawals during he early period of reiremen, swiching laer o a sandard level annuiy. This rule offers relaively low expeced shorfall risk, enhanced payous early on in reiremen and longeviy insurance laer in life. The meris of delayed annuiizaion are sressed by Milevsky (1998, 2001) and Blake e al. (2003) on he grounds ha in his case reirees can earn a poenially higher rae of reurn from invesing a higher proporion of heir wealh in equiies raher han in bonds. In he absence of a beques moive, hey conclude ha i is opimal o swich fully o annuiies when he moraliy credis begin o exceed he equiy premium. Alhough he asse-allocaion argumen canno be used in a NDC sysem, phased wihdrawal programmes ha enhance early reiremen benefis could be jusified on he grounds ha hey are consisen wih ime preferences of individuals ha favour curren (early reiremen) consumpion versus fuure (laer in life) consumpion. On he oher hand, aking he overall governmen commimen for income and healh care suppor ino accoun, he drawback of his ha i encourages free riding in more advanced years. In fac, if individuals have reason o misrus he sinceriy of he governmen commimen in providing care services owards he end of a life longer han average his could provide a reason o defer paymens o laer years. When purchasing a radiional level annuiy here is a small probabiliy of losing all of he iniial invesmen (if deah occurs immediaely afer signing he conrac), while on he oher hand, and for he same reason, purchasing delayed deferred annuiies enails a high probabiliy of no receiving any of he promised payous. Hu and Sco (2007) realised his and ried o apply some resuls from behavioural finance and choice under uncerainy o explain he low demand for annuiies by reirees. People end o overweigh he imporance of exreme evens, paricularly hose whose oucomes are negaive. In his sense, immediae annuiies and delayed deferred annuiies have differen weighs aached o heir wors oucomes. A ypical risk adverse behavioural invesor will end o exacerbae he low probabiliy of premaure deah, which would resul in a near-loss of he iniial invesmen when compared o a higher probabiliy of deah in a (laer in life) deferred annuiy. The 7 This rule is used in he US during he disribuion phase of 401(k) plans, where he ax auhoriies ry o ensure ha reirees consume heir ax-qualified accouns insead of leaving hem as bequess. 15

16 auhors conclude ha annuiies which include some money-back guaranees (e.g., a guaranee period a lump sum a deah) could remove he single bigges consumer objecion o annuiies: If I die soon afer I reire, he annuiy provider will keep my fund. If we consider ha longeviy insurance can somehow be seen as equivalen o he purchase of a deferred annuiy, where he opimal deferred period is linked o he average life expecancy of he annuian, given ha only hose who live beyond he defermen period will be alive o collec he benefi paymens payable for heir lifeime, we can esimae ha reirees only need o spend a fracion of heir capial on longeviy insurance and use he remaining par in phased wihdrawals or oher similar programmes. I should be noed ha boh in FDC and NDC schemes he iming of annuiizaion impacs on annuiy paymens. To minimize he imporance of his problem, we may hink in soluions involving programmed annuiy purchases in he period leading up o reiremen, using some simple pracical rule (e.g., a consan rae of purchase) or adop more complex hedging sraegies employing derivaive securiies. Milevsky (2005) proposed a new deferred inflaion-adjused life annuiy conrac called advanced-life delayed annuiy (ALDA). The produc would be acquired a a given (young) age, would be paid over a long period by insalmens and would have no cash value or survival benefis. In his conrac, he defermen period can be seen as a deducible since he policyholder finances his consumpion unil some advance age (ranging from 80 o 90, say), afer which he insurer sars paying he annuiy provided he annuian is sill alive. Therefore, wha ALDA does is o ransform he consumer choice and consumer asseallocaion problem from a sochasic dae of deah o a deerminisic one in which he erminal horizon becomes he annuiy paymen saring dae. As he index is publicly available, he annuian is able o adjus his consumpion level during he deferred period. In an uncerain moraliy environmen, we can hink of alernaive indexing mechanisms for ALDA, for example by adjusing he benefi commencemen dae according o he evoluion of a given longeviy index. Consider, for example, a deferred life annuiy bough a he age of reiremen wih paymens saring a a given advance age (say 80), bu wih he opion o change (anicipae or delay) paymens according o observed longeviy improvemens. In his case, if observed longeviy improvemens are higher (lower) han expeced, he annuiy paymen commencemen dae could be delayed (anicipaed) for some period. Noe ha in he 16

17 case of deferred annuiies when produc locks in ino a level annuiy he remaining life expecancy will be much lower and so he problem of predicing moraliy far ino he fuure and he corresponding uncerainy will have been significanly reduced. Pure phased wihdrawal rules have some advanages in erms of flexibiliy, liquidiy and (possibly) bequeahing in he even of early deah, bu inroduce volailiy in benefis and, more imporanly, do no provide pooling of longeviy risk. In fac, in income draw down programmes here is no moraliy drag, i.e., here is no cross-subsidy from hose wih aboveaverage moraliy which means ha each one bears his own longeviy risk. The moraliy drag is equal o he proporion of he surviving cohor of annuians ha do no survive during he nex year. Phased wihdrawal programmes will be worhwhile only if he reurns on he annuiy funds exceed he annuiy yield by a margin big enough o cover boh he moraliy drag and exising charges on draw down programmes. Since he moraliy drag increases wih age and is cumulaive, i is increasingly unlikely ha alernaive invesmen will be able o do his so we can expec ha a some age i will be opimal o swich o an annuiy. In developed counries we have seen in recen years a significan shif in he way o pay benefis from DC schemes owards variable annuiies. Variable annuiies were inroduced in he US in he mid 1950s as saving accumulaion vehicles o compee wih muual funds and soon gained populariy because of he associaed ax benefis, various forms of insurance and he opion o conver accumulaed funds ino a life annuiy wih payous indexed o he evoluion of a diversified invesmen porfolio. Variable annuiies can ake many srucures bu he key denominaor is ha benefi paymens are no fixed over he conrac horizon bu change according o he evoluion of some inernal or exernal risk facors. The many differen ypes of annuiies can be classified ino some caegories according o heir 8 : Financing mehod: single or regular premiums, fixed or variable Timing of paymens: immediae, deferred, phased, annuiy-due Paymen frequency: monhly, quarerly, semi-annual, annual Coverage: single-life, join-life, join-survivor, reversionary, group 8 For a deailed classificaion see, e.g., Wadsworh e al. (2001). 17

18 Paymen ype: level, sep-down, adjusable, escalaing, index-linked, limied price indexed, invesmen-linked, wih-profi, uni-linked, income draw down, markevalue-adjused, pooled annuiy fund Payou commimen: emporary, whole life, minimum guaranee, cerain, minimum guaranee and overlap, wih capial proecion, wih cash or insalmen-refund Embedded opions: GMAB (Guaraneed Minimum Accumulaion Benefis), GMIB (Guaraneed Minimum Income Benefis), GMWB (Guaraneed Minimum Wihdrawal Benefis), GMDB (Guaraneed Minimum Deah Benefis), GAO (Guaraneed annuiy opion) Oher feaures: heah-impaired, long-erm care, Gender-unisex In order o assess he adequacy of alernaive annuiy conracs o he goals of a pension scheme we mus be aware ha hey do no offer he same guaranees agains longeviy, invesmen, ineres rae, replacemen or annuiy rae risks (see Table 2). Table 2: Coverage agains reiree s income uncerainy by annuiy ype Annuiy Type of risk ype Longevi y Invesme n Inflaion Replacemen Annuiy rae Immediae erm cerain Medium High Low Low Low fixed Immediae life indexed High High High Medium High Pooled annuiy fund (GSA) High High Medium Low Low Phased wihdrawal Medium Low Medium High Medium Source: Michell e al. (2006). In an NDC pension scheme he pension akes he form of a level life annuiy. As a consequence, he annuiy provider assumes a shor posiion on longeviy risk. Since his ype of risk is non-diversifiable, we argued above ha one of he possible ways insurers can face aggregae longeviy risk involves ransferring i (parially or oally) o oher counerpary. In his secion we argue ha his can, in principle, be achieved by redesigning annuiy conracs, namely by adaping some of he feaures presen in some variable annuiies offered in he marke. The firs example was given above wih a varian of a conrac called ALDA in which 18

19 he benefi commencemen dae of a deferred annuiy conrac could be deermined by he evoluion of a given longeviy index. Anoher possibiliy would be given by an adjusable longeviy annuiy, a sandard level annuiy ha would rebase benefis periodically (say, every 5 years) in order o accommodae new informaion on he longeviy prospecs of he populaion. The conrac could be se o esablish a risk-sharing mechanism by which he profis/losses generaed by he new life ables would spli beween he annuian and he annuiy provider. Oher soluion would be o consider a sor of longeviy-index linked annuiy, an arrangemen by which he pension paymens would be linked o he evoluion of a survival index of a given populaion. In his case he paymens would decrease over ime which means ha in order o guaranee a cerain minimum hreshold reiremen income lower bounds mus be se. This ype of conrac would provide higher benefi paymen a incepion and a proecion agains individual longeviy risk. Wadsworh e al. (2001) develop an annuiy produc called an annuiy fund. The innovaive idea is ha he moraliy credis from hose who die in he pool are made explici and separaed from he invesmen elemen by means of he so-called survival credis. A he conrac incepion, he annuian acquires he righ o consume hroughou life a (susainable limied) rae of he fund as income, he rae being reviewed periodically, forfeiing his enancy on deah bu receiving survival credis while alive. A high ages, he fund rules lock he benefi ino a convenional annuiy. The periodic rae review reduces moraliy guaranees and inroduces greaer acuarial accuracy in pricing, while he survival credis mechanism could possibly conribue o change he public percepion abou he fairness of annuiies. The periodic review of survival credis in order o reflec moraliy experience limis guaranees o fixed periods, hus reducing he insurer exposure o longeviy risk. Richer and Weber (2009) develop a moraliy indexed annuiy in which annuiy providers reduce he risk of insufficienly funded conracs originaed by longeviy risk o a safe level (given an admissible shorfall probabiliy) by seing up a coningency fund. This fund is financed by he policyholder by means of an addiional surcharge which eiher increases he conrac premium or diminishes he benefis when compared wih an equivalen sandard level annuiy. The insurer deduces a cerain fixed amoun from each paymen o finance he ulimae coningency fund and so basically ses up a saving process in order o build a financial reserve. The auhors do no give any indicaion abou he way he expeced profis 19

20 from he pool will be passed back o he policyholders. However, we can in principle imagine a basic wih-profi srucure in which regular ( erminal ) bonuses are added as permanen (emporary) addiions o income. In (N)DC schemes, price indexaion of annuiies promises some absolue level of consumpion available o reirees, while wage indexaion of benefis mainains he relaive posiion of reirees wih respec o he working aged populaion, allowing reirees o benefi from produciviy gains of he working generaion. Pension sysems and variable annuiy conracs ofen include some proecion mechanisms in he form of minimum benefis and value/reurn guaranees. Guaranees suffer from moral hazard, since hey can encourage individuals o ake par in abnormal risk-aking behaviours. Reirees falling shor of savings o purchase sufficien, annuiized longeviy insurance, using financial asses accumulaed hrough he working years, bu wih a significan real wealh, can resor o reverse morgages as a vehicle o finance annuiy insurance. Reverse morgages (RMs) allow reirees o borrow agains he value of heir exising owner-occupied homes, receiving an income ha can be a lump sum, a line-of-credi, or a regular annuiy paymen. Conrary o an ordinary morgage, neiher ineres nor principal is repaid unil he house is sold or he borrower dies Pooled Annuiy Fund and Group Self Annuiizaion Risk pooling principle In sandard level annuiies annuiy providers bear boh sysemaic and idiosyncraic moraliy risks. Piggo e al. (2005) and Valdez e al. (2006) proposed an alernaive annuiy srucure in which pool paricipans are insured agains individual longeviy risk bu sill have o bear sysemaic longeviy risk. The idea is o consruc a pooled annuiy fund (PAF) considering groups or cohors of reirees and follow a Group Self Annuiizaion (GSA) sraegy. Pooled annuiy funds have many similariies wih sandard annuiies in ha he funds released by he paricipans who die premaurely (moraliy credis) are redisribued among he survivors, in ha paricipans renounce o beques or liquidiy moives, in ha he decision o purchase pooled annuiy fund unis is irreversible and in ha he advanages of invesmen 20

21 diversificaion are profied, bu presen a crucial difference: benefi paymens are linked o he moraliy experience of he group and, as such, leave annuians incomes and consumpion possibiliies exposed o he uncerainy associaed wih he moraliy risk of he group. Insiuions seing up a pooled annuiy fund ac only as managers of he accouns and, hus, do no bear eiher invesmen or moraliy/longeviy risk. This means ha pooled annuiy funds ransfer aggregae longeviy risk from annuiy providers o fund paricipans, hus offering an alernaive hedging soluion o he longeviy bonds originally proposed by Blake and Burrows (2001). Wih he necessary adjusmens, pooled annuiy funds could in principle become a par of boh a FDC and a NDC pension scheme. In NDC his would provide an alernaive o passing on he longeviy risk o he governmen (axpayers a large). In order o explain he way benefis would evolve over ime in an NDC scheme, we now briefly review he pricing mechanics of a pooled annuiy fund. 9 Wihou loss of generaliy, assume ha he pool sars (a ime = 0 ) wih an iniial size of l x homogeneous reirees in he sense of idenical age, gender and cohor, idenical moneary amouns and idenical risk exposures. We assume ha his annuiy arrangemen is offered o individuals in exchange for a single upfron premium K 0 (noional capial), based on heir accumulaed working-life conribuions. The annuiy provides an iniial level benefi B 0, paid once a year, calculaed using an annuiy facor accouning for expeced moraliy improvemens and a fla ineres rae. Given hese assumpions, and he bes esimae of fuure moraliy, he saring oal fund is F = l B a&& (3) 0 x 0 x In a pooled annuiy fund, he fuure value of annuiy benefis remains consan over he whole conrac unless deviaions from expeced moraliy raes are observed. If ha is no he case, i.e., if he number of hose surviving up o higher ages is differen from expeced, he remaining noional capial has o be redisribued among he remaining survivors. Le us assume firs ha he observed rae of reurn i of he NDC scheme behaves as expeced. A ime = 1 he oal fund comprises he iniial value less annuiy paymens accrued a he rae of reurn 9 For a deailed formal reamen see Piggo e al. (2005). 21

22 F = ( F l B )(1 + i) = l B ( a&& 1)(1 + i) (4) 1 0 x 0 x 0 x o Redisribuing his capial among he acual l x + 1 remaining survivors for heir expeced fuure lifeime, including he moraliy credis inheried from non-surviving members, he fuure value of annuiy benefis becomes, afer some algebra B F p = = B o 1 x 1 o 0 o lx 1a&& + x+ 1 px (5) where p x and p o x denoe, respecively, he expeced (bes esimae) and realized survival probabiliies for an individual aged x a ime = 0 in he ime inerval ( +, 1). Proceeding inducively, a any ime in he fuure he benefi paymen will be deermined by B B p B p B p ( + j) 1 o o x+ 1 x x+ j 0 = 1 = 0 = 0 o o px+ 1 p o x j= 0 px+ j( 0 + j) (6) The exension o he case where observed raes of reurn (i o ) are differen from he iniial fla rae is sraighforward. 10 In his case, he annuiy benefi a ime is given by B o p 1+ i o (7) 1+ i o o x+ 1 = B 1 px+ 1 From (7) i is clear ha he annuiy paymens a ime can be deermined from previous benefi payous muliplied by wo adjusmen facors: he firs is a moraliy correcion facor ha is based on he raio beween expeced and observed survivorship raes; he second is an ineres rae adjusmen ha is relaed o he relaion beween observed and expeced invesmen earnings rae. From (7) i is also clear ha a higher han expeced realized rae of reurn leads o higher annuiy benefis and ha in a scenario where he number of hose surviving o age x + is 10 Furher exensions of hese formulae o he case o varying conribuions and annuiy payous, muliple-cohors and evolving expecaions are sraighforward and can be found in Piggo e al. (2005). In addiion, an inflaion adjusmen facor can be inroduced (see, e.g., Creighon e al., 2005). 22

23 sysemaically higher han iniially expeced benefi paymens will ineviably drop in order o preven fund imbalance. This conrass wih radiional life annuiy conracs ha guaranee a level paymen for he remainder of recipien's lifeime independenly of fuure moraliy developmens. The explanaion is inuiive. Since he remaining funds will have o be spread across a larger survivorship group, he moraliy credis inheried by each surviving paricipan will be lower ha expeced hus reducing he benefi payou. In NDC a GSA would disribue he risk o he insured cohor raher han o fuure cohors or o axpayers a large. Pooled annuiy funds inroduce an auomaic adjusmen mechanism o financial and demographic risk facors in order o ensure ha he fund mainains is balance beween asses and liabiliies. Generic NDC sysems can also include an auomaic balancing mechanism (e.g., Sweden), bu in his case he correcion is made hrough he rae of reurn earned by workers and pensioners accoun values, i.e., he profis/losses induced by he adjusmen mechanism are shared beween acive and inacive generaions. In addiion, in NDC wih a balancing mechanism i may ake decades for sysemaic longeviy risks o be borne by he NDC. GSA offers hus a clear choice o he unclear passing on of he longeviy error o fuure workers. As opposed o he unclear pass hrough of he Swedish balancing mechanism, in a pooled annuiy fund he auomaic balancing mechanism for he longeviy risk is explici and impacs each paricipan s benefi in a proporional way. The poenial profis/losses resuling from changing demographic and financial risks affec only he group of pensioners. Of course, his can be quesioned in he conex of a mandaory public pension scheme since, has we ve said above, human lifeime is finie and senior generaions have already exhaused mos of heir human capial and have limied ime and abiliy o adjus o new condiions. A ypical risk adverse behavioural reiree will end o overweigh he uiliy of unexpeced losses observed a very old ages and will challenge he fairness of he sysem across cohors. Compared o sandard level annuiies, pooled annuiy funds include opion feaures ha, when negleced, could significanly under(over)esimae annuiy payous. To see his, assume wihou loss of generaliy ha ineres raes are consan over ime. From (6) we can see ha benefi paymens a ime can be expressed as: 23

24 o px min( B0, B ) if < 1 o px o px f( B ) = B0 if = 1 o px o px max( B0, B ) if > 1 o px (8) Equaion (8) shows, for insance, ha in a scenario of longeviy risk he benefi paymen is o given by he curren value of he reference fund disribued among he acual x 1 l + remaining survivors capped by is incepion value B 0. In financial erms, in his case he benefi can be expressed in erms of he final (mauriy) payoff of an European pu opion wih srike price equal o he annuiy benefi a incepion, i.e., o { } f ( B ) = B max B B ;0 o 0 0 (9) From (9) we see ha in his case he annuian holds acually a shor posiion in an European pu opion wih srike equal o B 0 mauring a ime, whose exercise is coningen on he realized survivorship experience. Moreover, his opion is wha we call in financial erms auo-puable in his sense ha he exercise decision is auomaic. If, for he conrary, he acual number of survivors is less han iniially esimaed, benefi paymens a ime are floored by he annuiy benefi a incepion and can be expressed in erms of he final payoff of an European call opion wih srike equal o B 0, i.e., o { } f ( B ) = B + max B B ;0 o 0 0 (10) In his case, he annuian holds a long posiion in an European call (auo-callable) opion wih srike equal o B 0 and mauriy a ime. Equaions (9) and (10) show ha GSA pooled annuiy conracs include opion feaures ha, up o our knowledge, have never been considered in he design and pricing of hese arrangemens. In fac, annuiy providers end o adop an over-simplified approach and compleely ignore embedded opions, resoring o consolidaed acuarial echniques for pricing (and hedging) he conrac. Afer all, in his case all acual losses/profis are beared by he remaining survivors. 24

25 Financial opions are asymmeric conracs in which he holder (buyer) has he righ (bu no he obligaion) o buy or sell he underlying asse a some ime in he fuure. This allows he holder of he opion o exercise he conrac only when i is profiable o do so, whereas he wrier (seller) of he conrac is a weaker posiion since is obligaions are coningen o he buyer s decision o exercise or no he opion. For his privilege, he opion buyer has o pay o he opion wrier an up fron, non-reurnable premium (opion price). Taking his argumen ino he pooled annuiy fund arena, we argue for example ha annuiies including an opion o reduce benefi paymens in he fuure if he acual survivorship raes are higher han expeced should pay an addiional benefi (equal o he opion premium) when compared o sandard level annuiies, since in his case he annuiy provider holds a series of European pu opions ha allow hem o reduce annuiy benefis if he underlying asse (raio of survivorship raes) closes ou below he srike level. On he oher hand, if acual longeviy developmens are worse han iniially expeced, benefi paymens in a pooled annuiy fund will increase due o a higher inheriance effec since he accumulaed funds will be spread across a smaller surviving group. Compared o sandard life annuiies, in his case life insurance companies will no be compensaed for he los reserves. In a symmerically designed pension sysem, i.e., one ha always ensures a zero ne presen value and a balance raio (asses/liabiliies) of uniy, we can hus argue ha individuals assessing he possibiliy of annuiizing heir wealh bu wih disbelief abou heir longeviy prospecs may feel araced o paricipae in a pooled annuiy fund if hey were given he chance o increase annuiy paymens if heir predicions are confirmed. In oher words, individuals willing o have his opion should pay an addiional premium (or accep a lower iniial benefi when compared o sandard annuiies) o hold a series of call opions on he underlying asse. The above consideraions assume ha individuals are risk-neural regarding heir pension level. If we assume, for he conrary, ha pensioners are risk averse in he sense ha hey overweigh he imporance of a decrease in heir pension benefis, we can envisage an asymmerically designed pooled annuiy fund which, in a siuaion where acual longeviy improvemens were acually worse han expeced, would canalize he addiional reserves originaed by exra moraliy credis o a buffer-fund (solvency capial) ha would serve o smooh benefi reducion adjusmens necessary o balance he sysem in a longeviy risk 25

26 scenario. 11 We can even, a leas in heory, imagine a sysem in which here are upper and lower bounds o his buffer fund admiing, inclusively, he possibiliy of a sysem negaive buffer fund, suppored by exernal ax revenues. The exisence of a floor and a ceiling would delimiae he poenial volailiy of pension benefis. Alhough he pracical implemenaion of his arrangemen is a challenge for fuure research, in he below secion we advance some innovaive ideas. The pooled annuiy fund is an arrangemen by which paricipans are proeced agains individual moraliy longeviy risk bu leaves annuians paymens and consumpion possibiliies exposed o he uncerainy associaed wih he group s longeviy. In a world wih no uncerainy, sandard analysis of consumer behaviour saes ha he ime profile of consumpion depends on heir subjecive discoun raes (for consumer impaience) relaive o he real reurn on capial. When he discoun facor for consumer impaience is higher (lower) han he real reurn on capial, he consumpion should be expeced o fall (increase) over ime. Pooled annuiy funds inroduce uncerainy on he rae of reurn on savings individuals will obain by purchasing an annuiy conrac. Because of his addiional risk, he rae of reurn of a pooled annuiy fund should be a leas equal o he expeced reurn offered by a convenional annuiy. The inroducion of uncerainy in annuiy benefis will have implicaions in consumpion/saving behaviour according o he pensioner s degree of risk aversion. Economic heory predics, for insance, ha a raional consumer wih consan relaive risk aversion (CRRA) will, for any expeced reurn, end o save more han oherwise would be he case in he absence of uncerainy. The significance of precauionary saving is crucial in pooled annuiy funds since he rae of reurn on capial depends on he moraliy credis inheried by hose who survive which are, in his case, uncerain. If we assume ha pensioners do have a CRRA, he uncerainy regarding fuure longeviy improvemens would mean ha hey would have o adjus heir consumpion profile, namely by reducing heir consumpion in he early years of he annuiy so ha a precauionary reserve is available should he cohor s moraliy rae be effecively lower han prediced by he moraliy projecion model. This means ha hey could somehow accep an annuiy conrac ha increases paymens over ime and ha pays he amoun a pensioner would raionally choose o fund his consumpion needs. 11 This argumen is no new and is acually behind he Auomaic Balance Mechanism of he Swedish Pension Sysem. 26

27 Valdez e al. (2006) use a simple model of consumer behaviour o conclude ha in pooled annuiy funds he adverse selecion problem presen in radiional life annuiy markes is minimized. Samos (2008) analyses he consumpion and porfolio decisions for a finie number of paricipans in a pooled annuiy fund and concludes ha he opimal policy depends on he number of pool members and on age. The firs resul is rivial since he bigger he pool he more moraliy credis can poenially be inheried in he fuure. The increase in age is explained by he fac ha as reirees become old, heir survival probabiliy diminishes hus increasing heir ime preference owards curren consumpion. Annuiy Porfolio Losses A differen way o undersand he opion-like feaures of annuiy conracs is o analyse he relaion beween survival probabiliies and annuiy porfolio losses. Consider once again a classic life annuiy conrac wih level paymen B 0. The loss on he underlying annuiy porfolio a ime is defined as l x ( () 0 [ () 0] ) (11) L= I B EI B + i i i= 1 where Ii() = 1τ i > is an indicaor funcion ha jumps from 1 o 0 a he ime of deah τ i of he annuian. Noe ha EI [ ( )] = p. Losses on he porfolio are he amoun ha he annuiy i x paymens a ime exceed he expeced paymens. For a given populaion survival probabiliy p x, he disribuion of he number alive a ime is binomial ( ) lx+ ~ Binomial lx, px px (12) As recognized by Lin and Cox (2005), here are wo sources of uncerainy in he porfolio loss a ime. This firs is due o uncerain lifeimes given he acual moraliy raes. The second is aribued o he sochasic naure of survival probabiliies. Given his, he oal variabiliy in he porfolio is he uncondiional variance of he compound binomial disribuion 27

28 ( ) [( ) ] [( ) ] Var lx+ = E Var lx+ px + Var E lx+ px (13) For large porfolios of annuians, he main source of uncerainy will come from changes in moraliy raes impacing all lives in he porfolio raher han he variabiliy in he number of deahs a a paricular age given he moraliy rae. In oher words, he randomness in l x+ will mainly be due o he uncerainy in p x. Annuiy providers pool moraliy risks by he Law of Large Numbers, i.e., lim E Var[( lx+ ) px] 0. However, from (13) we can conclude ha he assumpion ha lc moraliy and longeviy risks can be diversified away by wriing a large number of policies in incorrec if we ake ino accoun he dynamics of he underlying moraliy raes, i.e., Var E[( lx+ ) px] 0 Moraliy dynamics is influenced by a complex seing of socioeconomic facors, biological variables, governmen policies, environmenal effecs, healh condiions and social behaviours. Since he fuure moraliy is acually unknown, here is always he likelihood ha fuure deah raes will urn ou o be differen from he projeced ones and hus moraliy shocks can desroy he insurance pooling mechanism. For example, for an annuiy porfolio he risk is ha he annuians will sysemaically live longer han expeced a he policies incepion. Sysemaic moraliy risk canno be eliminaed by diversificaion and hus should have a marke price. The porfolio loss in equaion (11) can be wrien as ( ) L = l B p o p + (14) x 0 x x Redisribuing among he remaining survivors we have o L lxb + 0 px p o x = ( px px) = B0 max ;0 o o o lx+ lx+ px p x = B0 max 1 ;0 o px (15) 28

29 Equaion (15) shows ha he loss inheried by each surviving policyholder includes an opion feaure ha depends on raio of survivorship raes. To be more specific, he loss has an embedded pu opion wih srike equal o uniy and underlying equal o he raio beween esimaed and acual survivorship raes. To value his opion we can resor o radiional discree-ime (Binomial) or coninuous-ime approaches (Black-Scholes), wih proper adjusmens for an incomplee markes siuaion. Alernaively, we can resor o Mone-Carlo simulaion echniques. 2.4 Pooled Annuiy Fund incorporaing longeviy risk-sharing mechanisms and longeviy insurance Pooled annuiy funds (PAF) presen an ineresing framework for he decumulaion sage of an NDC sysem bu, a leas in is pure forma, generae a number of poenial sources of disuiliy ha need o be ackled. Firs, in a scenario of longeviy risk, a pure PAF is expeced o pay decreasing annuiy benefis. The possibiliy ha annuians can see heir paymens dropping below a reasonable value may spread disconen hrough pensioners who weren' aware of he poenial impac of fuure moraliy improvemens on heir "apparenly guaraneed" income, which may feel hey didn' receive any compensaion for acually being shor in a pu opion conrac. The dispersion of fuure moraliy (and invesmen) raes is ignored in his arrangemen and no allowance is made for risk aversion and for differen subjecive discoun raes beween workers and pensioners. Second, as in oher variable annuiy conracs he annuian does no know in advance he rae of reurn of he pool, hence i carries some risk. Pooled annuiy funds wihou addiional guaranees are srucured so ha individuals share boh moraliy and invesmen risk in upside and downside imes. Third, in a limiing siuaion here will be no paymens for hose who survive beyond he highes aainable age assumed in he life able used o price he conrac, i.e., individuals migh end up wih no resources o fund consumpion. This violaes he wo basic posulaes of a proper pension scheme: o proec he individual from ouliving heir resources; o provide reiremen income for he remaining life of he pensioner offering longeviy insurance. 29

30 Fourh, annuiy providers in his case do no bear any kind of risk, eiher acuarial or financial, eiher sysemaic of idiosyncraic. In realiy, in his case here is no need o use a risk bearer since he funds are periodically reallocaed o he surviving annuians, based on he previous paymen adjused for any deviaions in moraliy and ineres rae from expecaions. The apparen advanage of a PAF over self-insuring is ha he risk exposure is no immediae in ha is borne by he pool and i's smoohed ou by he annuiy provider over a long ime horizon. In addiion, since moraliy and invesmen reurns are largely uncorrelaed, here is some chance ha a negaive reurn on invesmens may be parially offse by a posiive "inheriance effec" or vice-versa. In oher words, he effecs of he overall risk exposure migh be miigaed and posponed a he individual level when considered in a pooling srucure. In a PAF all annuiy porfolio losses are shared among he insured cohor (raher han among fuure cohors or axpayers a large), who have already exhaused heir human capial and have hus limied abiliy o permanenly adjus heir consumpion, saving and labour paerns o new prospecs abou longeviy. Finally, in is simples form, a PAF fund does no give pool member's access o he principal invesmen nor o any accumulaed fund. This means ha, similar o sandard annuiies, he arrangemen does no cover beques moives. To cope wih hese problems, a number of differen soluions could be in heory designed. For example, we can imagine a PAF iniiaed a he age of reiremen x combined wih a mandaory conversion ino a level life annuiy a some advanced age x max old). In his case, he annuiy paymens a ime will be given by: > x (say 80 years B0 for = 0 o λ o p x i B = B 1 for = 1,...,( xmax x 1) o px i o B 1 for = xmax,..., ω (16) Where λ = 0 sands for he case where only demographic deviaions are considered and λ = 1 for he case where boh risk facors are aken ino accoun. From (16) we can see ha in his case annuiy benefis would be allowed o flucuae according o moraliy and invesmen deviaions from projeced values up o some advanced age, wih he remaining longeviy risk (implici in he level annuiy acquired a age x max 30 ) being backed by governmen (i.e., by

31 axpayers and/or workers). The periodic benefi review reduces he moraliy guaranees, inroduces greaer acuarial accuracy in pricing and reduces he insurer exposure o longeviy risk. In erms of pracical implemenaion, we can imagine single-cohor annuiy funds which would require ha each year a new PAF is offered o an age and, poenially, gender or socioeconomic group, being closed hereafer for new subscripions. In alernaive, we can consider mulicohor funds ha would allow for new paricipans (belonging o differen generaions) o come ino he pool over ime so ha pooling effecs are appropriaed by muliple cohors. An alernaive arrangemen would be o consider demographic uncerainy only and se up a capped-paf (possibly combined wih a level life annuiy a some advanced age) bu in which pensioners bear only a fracion α, α [0,1], of he annuiy porfolio losses if he acual survivorship raes confirm o be higher han expeced. In his case, he annuiy paymens a ime will be given by: B px B0 if 1 o px = p x px B0 1 α max 1 ;0 if < 1 o p o x px (17) In his case pensioners give up some upside poenial in exchange for some downside proecion implici in he risk-sharing mechanism. For he cases in which he PAF is no combined wih a level annuiy a some advanced age we can conceive ha α is decreasing over ime so ha an increasing fracion of he unexpeced porfolio losses generaed by longeviy risk is borne by governmen. Anoher soluion would be o assume ha annuians should only bear he sysemaic par of he longeviy risk, whereas he annuiy provider should offer insurance agains he random flucuaions of moraliy around is projeced values. In his siuaion, we can hink in a symmeric PAF in which pension benefis are revised downwards/upwards only when observed survival probabiliies are beyond some upper/lower bound, e.g., he level corresponding o upper/lower bounds of he of he disribuion funcion of he survival probabiliy. Denui and Dhaene (2007) apply he concep of comonooniciy o derive 31

32 approximaions for he sochasic survival probabiliies in he Lee-Carer framework. The upper p and lower u x0 p bounds for he survival probabiliy of an individual aged x 0 a l x0 ime 0 are approximaed by p exp exp ( Z), Z ~ N(0,1) (18) 1 u x = δ 0 j μj + σ j j= 0 2 ( ( ) ) 1 p exp exp r ( ) Z 1 r ( ), Z ~ N(0,1) (19) 1 l 2 x = δ 0 j μj + j σ j + j σ j j= 0 2 where r( ), i = 0,..., 1, is given by i r i = 1 j= ( ) δ exp μ β β Cov[ κ, κ ] j j x+ j x+ i 0+ j 0+ i ( + ) σ δ δ exp μ μ β β Cov[ κ, κ ] i j k j k x+ j x+ k 0+ j 0+ k j= 0 k= 0 and where δ j = exp( α x + j), μ j = βx + j( κ + j + jθ) and ( + ) j x0 j j σ = β σ. In his case, he annuiy paymens of he PAF a ime would be given by: B0 if p p p l p x0 o l B = B0 1+ η max 1 ;0 if px 0 px o 0 p x p l o u x0 x0 x0 u x0 u o B0 1 α max 1 ;0 if px 0 px o 0 p x (20) where we include an addiional parameer η, η [0,1], o accoun for a possible differen disribuion of annuiy porfolio profis. This srucure limis he volailiy of pension benefis over ime since only deparures from he projeced confidence inervals of he survival probabiliy would be considered. In oher words, he coss/profis of incorrec life expecancy projecions is borne by annuians only o he exen ha he deviaions correspond o sysemaic moraliy deparures from he moraliy projecion model. 32

33 3 Daa analyic aspecs of moraliy change 3.1 Underesimaes in he Pas Forecasing moraliy is generally believed o be easier han forecasing feriliy, le alone migraion. Ye, a hisorical look shows ha demographers and acuaries have sysemaically underesimaed life expecancy for many decades (e.g., Oeppen and Vaupel 2002). To see how he underesimaion can come abou, we consider a 1926 forecas of Swedish moraliy, produced by a noed exper in mahemaical saisics, professor Sven D. Wicksell ( ). Comparing age-specific moraliy daa from o moraliy in , he noed a dramaic decline, and assumed ha a mos a similar improvemen could sill ake place. The decline had been 50% in ages 0-15 and The sagnaing developmen in ages was aribued o uberculosis and accidens 12 and was reaed as a passing phenomenon. In ages he decline was 45%, bu in older ages i decreased o 15% in age 95 (Wicksell 1926, pp. 92, 105, 123). Wicksell s assumpion was ha moraliy would iniially decline fas, and hen slow down so ha he lowes possible level would be reached asympoically. This would happen in he young ages around 1960, in he working ages in , and in he oldes ages by The implicaion was ha life expecancy from birh (boh sexes combined) would increase from he value of 58 o a maximum of 69.5 years. I is eviden ha Wicksell considered his forecas o be a bold one. He characerized he assumed improvemens as radical and noes ha a colleague (Prof. Brinsman) hough ha he maximum would be 65. I was known a he ime ha a cenury earlier life expecancy from birh had been 40, so Wicksell expeced 11.5 more years. I seems clear ha no one enerained seriously he possibiliy ha here could be an addiional = 18 years o life, 12 Thus, he acciden hump exised in Sweden a cenury ago already! 33

34 during he cenury o come. In rerospec we know ha he average of he Swedish male and female life expecancies in 2009 was 81 years. Thus, he increase was = 23 years. Examples from oher counries, and oher periods show ha Wicksell was bold, among his conemporaries (e.g., Modeen 1934, Whelpon e al. 1947). Even very recen examples indicae (cf., Keilman e al. 2008) widely held suppor for a law of diminishing reurns : ever-increasing resources are hough o be needed o achieve improvemens similar o hose in he pas. However, when one examines he causes of why Wicksell s forecas of moraliy urned ou o be oo high, wo facors sand ou: (1) relaive declines in he moraliy of young and working ages have coninued, (2) raes of decline in older ages have increased. Conrary o he forecas of Wicksell and many ohers, no recangularizaion of he survival curve has occurred. Simply allowing pas declines o coninue would have made moraliy forecass more accurae (e.g., Alho 1990). A paricularly elegan mehod of doing ha is o use he bilinear model of Lee and Carer (1992). However, as he sandard applicaion of he mehod does no allow for (2), here should be room for improvemen. 3.2 On Paerns of Moraliy Decline A noable feaure in Wicksell s forecas was he assumpion ha changes in he dynamics of moraliy would occur laer in older han in younger ages. This can be undersood as a recogniion ha (2) is possible. We will use daa from Sweden and seleced oher counries o invesigae his empirically. Using daa from he Human Moraliy Daabase, we have compued esimaes raes of decline for ages 45+ in 15 counries. In his exercise, our emphasis is on Sweden, and he oher counries are primarily used o provide a check as o he generaliy of he conclusions. 34

35 Le m(x,) be he age-specific moraliy rae 13 in age x during year. For all counries, daa for x = 45,,99 were obained for years = For many counries, only a shorer series was available. To smooh he paerns of change, he daa were firs aggregaed o 5-year age groups and 5-year periods, and averaged. Thus, we had age groups j = 1,,11 corresponding o ages 45-49,, 95-99, and periods i = 1,, 21, corresponding o years ,, , for he counries ha had he complee daa. The moraliy measure was hen defined as j i 1, m ij = 1 m ( x, ) 25 x = ( j 1) = (i 1) The rae of decline in moraliy in age j a ime i was hen defined as d ij = log(m ij ) log(m i+1, j ). Raes of decline, even when compued from such aggregaed measures, are sill fairly erraic for descripive purposes. Thus, we furher averaged hem in ages (j = 1, 2, 3, 4), (j = 5, 6, 7, 8), and (j = 9, 10, 11). Figure 1 displays he raes of decline for Sweden. 13 This is he raio of he number of deahs in age x during year o he person years lived in age x during year. The esimaes are based on combined male and female daa. 35

36 Figure 1. Average rae of decline of moraliy in Sweden, during 20 five-year periods ha span he years , for age-groups (doed), (dashed), and (solid). Several hings are obvious. Firs he raes of decline do no appear o have been consan during he pas cenury, in Sweden. I is logically correc, wihin he Lee-Carer framework, o inerpre he variaion around he mean, in each age group, as random and unpredicable, bu if his is done, i mus be accouned for he error esimaes. Alernaively, our inerpreaion may be ha, in line wih Wicksell s assumpions, here were sysemaic 36