Bringing cost transparency to the life annuity market

Transcription

1 Bringing cos ransparency o he life annuiy marke Caherine Donnelly Monserra Guillén Jens Perch Nielsen January 24, 2014 Absrac The financial indusry has recenly seen a push away from srucured producs and owards ransparency. The rend is o decompose producs, such ha cusomers undersand each componen as well as is price. Ye he enormous annuiy marke combining invesmen and longeviy has been almos unouched by his developmen. We sugges a simple decomposed annuiy srucure ha enables cos ransparency and could be linked o any invesmen fund. I has several aracive feaures: i i works for any heerogeneous group; ii paricipans can leave before deah wihou financial penaly; and iii paricipans have complee freedom over heir own invesmen sraegy. Keywords: Invesmen; Pensions; Pooled annuiy fund; Lifeime savings; Muual risksharing. Subjec Caegory and Insurance Branch Caegory: IM51, IB81. Corresponding auhor. Deparmen of Acuarial Mahemaics and Saisics, and he Maxwell Insiue for Mahemaical Sciences, Herio-Wa Universiy, Edinburgh EH14 4AS, Unied Kingdom. C.Donnelly@hw.ac.uk. Tel: Fax: Deparmen of Economerics, Riskcener-IREA, Universiy of Barcelona, Avinguda Diagonal 690, Barcelona, Spain. mguillen@ub.edu Cass Business School, Ciy Universiy, 106 Bunhill Row, London EC1Y 8TZ, Unied Kingdom. Jens.Nielsen.1@ciy.ac.uk

2 The annuiy overlay fund 2 1 Inroducion Over 40% of all privae indusry workers in he U.S. are saving for heir reiremen hrough a defined conribuion plan Bureau of Labor Saisics, 2012, Table 2. While he overall value of asses held in hese plans is immense, being approximaely $10 rillion in he U.S. in 2012 Towers Wason, 2013, individuals asse values can be small. For example, he median asse value held by hose age 55 or older in funds run by Vanguard, a large muual fund company, was around $ in 2011 Vanguard, A similar magniude of savings is repored in Poerba e al., 2011, Table 2 for people age 65 o 69 in he year Wih he lifeime guaraneed income offered by Social Securiy and defined benefi pension plans declining relaive o prereiremen income Webb, 2011, millions of individuals mus maximize heir reiremen income arising from heir defined conribuion plan savings. They canno afford o pay unnecessary charges and fees. Ye in he life annuiy conrac, which economic heory recommends as a significan componen of he opimal reiremen invesmen sraegy Yaari 1965, Davidoff e al. 2005, coss are hidden from he cusomer Blake 1999, Sewar We argue ha cos ransparency in life annuiies is very imporan, due o he generally irreversible and very long-erm naure of hese conracs, which poenially involves all of he life savings of individuals. Consumers have no idea if annuiy prices are fair, or if insurance companies are eiher making excessive profis or are grossly inefficien Carlin 2009, Del Guercio and Reuer 2013, Glode e al We presen a soluion o hese difficulies. We propose a decomposed annuiy srucure ha could be linked o any invesmen and ha enables all coss o be disclosed. Our aim is o improve he ransparency of he financial and insurance producs ha are offered o reirees. Greaer ransparency may also improve he financial regulaion of hese producs Kalemli-Ozcan e al., In he classical life annuiy conrac, called a fixed-payou life annuiy 1, he annuian is charged a single i.e. lump sum premium and in exchange receives a fixed income sream for life. The anicipaed ongoing coss are no disclosed explicily o he poenial annuian. Some 1 More specifically, i is a single premium immediae level annuiy wrien on a single life.

3 The annuiy overlay fund 3 insurance companies may charge explicily for sales commission and he iniial adminisraion coss of seing up he annuiy conrac. All he poenial annuian knows is he amoun of lifeime income ha her lump-sum reiremen savings will buy. To evaluae he worh of he annuiy compared o oher invesmens, he cusomer mus make a number of sophisicaed assumpions and complicaed calculaions. Generally, i is an irreversible conrac, so he cusomer mus rus ha he insurance company will coninue o pay he income sream over her fuure lifeime, which may be for decades. I is noable ha, worldwide, relaively few people volunarily annuiize heir reiremen wealh 2 Brown 2007, Michell and Piggo The main reason for he opaciy of life annuiy conracs is ha invesmen risk is combined wih moraliy risk and coss are no disclosed by he insurance company. As a consequence, life annuiies are no comparable on eiher an individual risk componen basis or on a cos basis. This inransparency has generaed a body of lieraure ha quesions if annuiies offer value-for-money o he annuian for example, Michell e al. 1999, Cannon and Tonks Typically, he auhors calculae he expeced value of a fixed-payou life annuiy, using wha hey believe o be a reasonable calculaion basis. Their esimaed prices are hen compared o hose quoed in he marke by insurance companies. The difference in he values gives an indicaion of he amoun of coss and profi expeced by he insurance companies during he conrac period. Unsurprisingly, given he sensiiviy of annuiy prices o he moraliy and invesmen reurn assumpions, here is a wide variaion in he resuls. For example, in Michell e al. 1999, Table 3 he annuiy prices quoed by insurance companies in he U.S. in 1995 are beween 74% and 94% of he auhors calculaed expeced values. A similar range is observed in he U.K. by Cannon and Tonks Wihou more informaion from insurance companies concerning heir annuiy calculaion basis, we can only hypohesize abou he reasons for he range of resuls. I may be due o he insurance companies assuming a differen calculaion basis han in he sudies. For example, he insurance companies may inves in riskier asses han hose assumed in he sudies, or hey may assume ha annuians live longer. I may be due 2 A phenomenon referred o as he annuiy puzzle. Recen reviews of he lieraure on he annuiy puzzle can be found in Brown 2009 and Lown and Robb 2011.

4 The annuiy overlay fund 4 o insurance companies coss, profi and risk capial requiremens, or i may be compeiive reasons. Wihou more informaion i is difficul o draw srong conclusions concerning he value-for-money of annuiies. The lack of informaion also means ha is no clear if annuiy prices quoed by insurance companies are compeiive, as hey can vary significanly across companies Michell e al. 1999, Cannon and Tonks Furhermore, even if he annuiy marke is compeiive, i does no follow ha consumers have low coss Orszag and Sigliz, For example, in he relaed muual fund marke, fees can be oo high e.g., see Crespo 2009 for he Spanish muual fund marke, and Gil-Bazo and Ruiz-Verdú 2009 for he U.S. marke and brokers can offer no angible benefis in exchange for high disribuion fees Bergsresser e al., Moreover, he annuiy markeplace is no as sraighforward as migh be imagined. Consider he annuiy rae, which is he raio of he annual income guaraneed for life by he insurance company o he single premium. Typically, he headline annuiy raes quoed in he popular press are for a single premium of $ An annuiy rae of 5% means ha he annuian receives $5 000 per annum in exchange for he upfron paymen of $ However, an insurance company ha offers he highes headline annuiy rae may no offer he highes annuiy rae for oher amouns of single premium. I may be a acical decision by he insurance company Harrison, 2012, or due o fixed coss incurred by selling each annuiy conrac, or simply a reflecion of he fac ha annuiy raes are no necessarily consan across same sex individuals of he same age; a wealhy man may have a higher expeced lifeime han a poor one, resuling in a lower annuiy rae for he former. The need for a ransparen annuiy marke is criical so ha individuals can make informed decisions on how o manage heir asses. They are required o make very complex decisions on how heir reiremen will be financed. For example, hey have o ake accoun of relaively concree facors such as Social Securiy benefis, housing, income from oher pension plans, as well as aking a view on unknowns like fuure inflaion, life expecancy and fuure healhcare coss. There are oher consideraions regarding he individual s qualiy of life, as well as he desire o bequeah money o ohers; see Smih and Keeney 2005 on making decisions abou invesmens in qualiy of life.

5 The annuiy overlay fund 5 Wih academic sudies able o give only a broad indicaion if he prices of life annuiies are fair, he abiliy of ordinary consumers o judge heir value is likely o be much lower. Many individuals are unaware of basic economics and finance Lusardi and Michell, 2011 and lack confidence in heir financial lieracy Graham e al., Furhermore, he simple life annuiy is in compeiion for reirees savings wih much more complicaed srucured producs. The laer include various financial and insurance opions and guaranees, which makes i difficul o ascerain if hey offer value-for-money Carlin e al., Indeed, aemps o value some of he moraliy opions in variable annuiies are he subjec of highly echnical academic papers e.g. see Milevsky and Promislow 2001 and Milevsky and Posner 2001, he laer finding ha marke prices for insurance risk charges are subsanially above heir heoreical values. If we can make he basic life annuiy conrac more ransparen, hen perhaps we can also improve he ransparency of hese more complicaed producs. We presen an annuiy overlay fund ha enables cos ransparency while giving one of he main benefis of he life annuiy, namely he pooling of moraliy risk across a group of people. I overcomes several disadvanages of he life annuiy. Cos ransparency. Wihin he proposed annuiy overlay fund, coss can be charged o each individual as hey occur. As invesmen risk is separaed from moraliy risk, coss can be aribued o each source independenly. For example, adminisraion coss, invesmen managemen fees and sales commission can be charged separaely o he consumer. If an individual believes ha he invesmen managemen fees are oo high, hen hey can swich o anoher fund manager Blake e al. 2013, Chrisoffersen e al Conrol over invesmens. Wih an annuiy overlay fund, each individual reains absolue conrol over heir own invesmens. They can decide how much o inves and how o allocae hose invesmens among any asse class. Conras his wih a life annuiy conrac, in which he individual no longer has any invesmens since he underlying asses are held by he insurer. Op in or op ou. An individual can decide o remove he annuiy overlay fund from all or some of heir asses a any ime. For he adminisraor of he annuiy overlay fund, his may

6 The annuiy overlay fund 6 be an incenive o keep he adminisraive coss low Bharah e al., Similarly, he paricipans can decide o add he overlay o more of heir asses a any ime. This flexibiliy does no occur wih a life annuiy conrac which is usually binding unil deah or, a bes, an exremely cosly conrac o exi. Tangible financial gains from pooling moraliy risk. Paricipans in he overlay receive financial paymens from he pooling of moraliy risk. The paymens are in addiion o any financial gains and are always nonnegaive while he paricipan is alive. Invesmen framing. The annuiy overlay allows he sharing of moraliy risk o be evaluaed in erms of yield like any oher invesmen decision. I may be a more aracive framing of he financial benefis o be gained from pooling moraliy risk han he naural consumpion frame of he life annuiy Agnew e al and Brown e al The annuiy overlay fund enables moraliy cross-subsidies, invesmen reurns and coss o be idenified individually and communicaed o he consumers. Furhermore, he overlay could be managed a a very low cos: as here are no guaranees, here are no reserving requiremens. The annuiy overlay fund is fundamenally differen o a life annuiy: he laer ransfers moraliy risk o an insurer, whereas he former pools moraliy risk among he paricipans in he srucure. Insead, i is a means of sharing he random flucuaions risk of moraliy. I does no guaranee an income unil deah and i does no proec agains longeviy risk, ha is he risk of under-esimaing how long you may be expeced o live. This means ha he annuiy overlay fund is no an insurance produc. Even hough he annuiy overlay fund allows moraliy risk o be separaed from invesmen risk, he moivaion is no o enable people o rade in he financial marke hemselves. Trading by individuals in he financial markes is fraugh wih problems for example, see Barber and Odean 2000a, Barber and Odean 2000b and Barber and Odean 2000c. Raher, he moivaion is o arrive a a ransparen marke in which people undersand wha hey are paying for and can deermine if he coss charged are reasonable, a marke in which consumers can more easily compare producs beween sellers and buy only wha hey need. The main purpose of he presen paper is o

7 The annuiy overlay fund 7 explain he srucure and operaion of he annuiy overlay fund, show ha i can be opimal o join he annuiy overlay fund, and invesigae he rade-off beween reurn and volailiy, from boh a heoreical and a numerical perspecive. We derive rules-of-humb o explain he rade-off, and find ha he spread of he age-wealh profile of he paricipans is very imporan. 2 The annuiy overlay fund: oy example We begin by illusraing he annuiy overlay fund wih a oy example ha communicaes he basic idea. Noe ha he oy example is unrealisic, as i assumes ha no financial reurn accumulaes on wealh, and i only approximaes he risk-sharing mechanism of he proposed fund; he correc, insananeous approach is deailed in Secion 3. Neverheless, he oy example demonsraes how he proposed fund allows people wih very differen characerisics o pool heir moraliy in an acuarially fair way. In he example, paricipans in he annuiy overlay fund agree o pool heir moraliy experience ogeher for one monh. Each paricipan has a fixed iniial wealh. The wealh of he paricipans who die during he monh is pu in a noional moraliy accoun. A he end of he monh, he money in he noional moraliy accoun is shared among all he paricipans, including hose who jus died during he monh. The paymen ha each paricipan receives from he noional moraliy accoun is proporional o heir individual moraliy rae and wealh. The annuiy overlay fund has a disincive feaure no shared by eiher he pooled annuiy funds which have been proposed and analyzed before in he lieraure e.g., see Donnelly e al. 2013, Piggo e al. 2005, Qiao and Sherris 2013, Richer and Weber 2011, Samos 2008, Valdez e al I allows individuals o exi he fund before deah, and o do so wihou any financial penaly. This is a key feaure ha disinguishes our pooled fund from all ohers. The reason why individuals can exi he annuiy overlay fund wihou paying a financial penaly is ha i is acuarially-fair a every insan in ime. Acuarial fairness is criical, paricularly when here is a finie number of heerogeneous

8 The annuiy overlay fund 8 Table 1: Characerisics of Alice and Bob a he sar of he monh. Name Wealh Probabiliy of dying in he nex monh Alice $ % Bob $ % members in he group. I means ha no single subgroup is subsidizing he remaining members. For example, as shown in Donnelly 2013, he group self-annuiizaion scheme proposed by Piggo e al resuls in he richer members of he group subsidizing he poorer members. Our proposed fund differs in anoher imporan way from oher pooled funds: in he annuiy overlay fund, paricipans have rue individual invesmen freedom. They can decide a any ime o change heir invesmen sraegy, again wihou paying any financial penaly. In he oher proposed pooled annuiy funds, he paricipans are forced implicily o follow he same invesmen sraegy as he people wih whom hey are pooling heir moraliy risk. The invesmen freedom becomes apparen when we move o he insananeous approach in Secion 3. Consider wo people, Alice and Bob, wih he characerisics shown in Table 1. Alice and Bob agree o ener he annuiy overlay fund for one monh. There are no oher paricipans. If Alice dies during he monh hen her wealh is pu in a noional moraliy accoun. The same rule applies o Bob if he dies. We assume hroughou he paper ha deahs occur independenly of each oher and ha here is no uncerainy abou he probabiliy of deah. A he end of he monh, he money in he noional moraliy accoun is shared among Alice and Bob in proporion o heir probabiliy of deah and wealh. Suppose Bob is he only one o die during he monh. When he dies, his wealh of $ is pu in he noional moraliy accoun. A he end of he monh, he money in he accoun is shared ou as follows. Alice ges $ $ % 40 = $ = $ $ % + $ % 41 This is Alice s acuarial gain from paricipaing in he fund for one monh. I is based on Alice s expeced wealh a risk due o her deah over he monh, relaive o Bob s expeced

9 The annuiy overlay fund 9 wealh a risk. I is a reurn due o sharing moraliy risk. Her wealh a he end of he monh is calculaed by adding her acuarial gain o her wealh of $ , giving her a oal wealh of $ a he end of he monh. Meanwhile, Bob ges he $1 220 ha is lef in he noional moraliy accoun. This can also be deermined by he allocaion mehod: $ $ % $ % + $ % = $ = $ Noe ha o calculae Bob s wealh a he end of he monh in he same way as we did for Alice, we deermine firs his acuarial gain as $ $1 220 = $ His acuarial gain is he sum of he amoun of his wealh ransferred o he noional moraliy accoun, due o his deah, and his share of he noional moraliy accoun a he end of he monh. Thus Bob loses $ as a resul of dying. His oal wealh a he end of he monh is he sum of his acuarial gain and his wealh of $50 000, giving a oal wealh of $1 220, as before. As Bob is dead, he money is paid o his esae. Alhough he sum paid o Bob s esae is non-rivial in he oy example, in pracice we do no expec ha he annuiy overlay fund operaes for only wo people. I is inended o enable a large group of people o pool heir moraliy risk. From he perspecive of he dying members, he annuiy overlay fund operaes similarly o cerain oher pooled annuiy funds and fixed-payou life annuiies. The sum paid o Bob s esae can be hough of as a balancing iem o make he annuiy overlay fund work for any group of heerogenous paricipans. Noice ha Alice s acuarial gain of $ exacly cancels wih Bob s acuarial gain of $ This is due o he fac ha no money is creaed by pooling moraliy risk; he wealh of he dead is simply re-disribued among all he paricipans. Repeaing he above calculaions across all possible scenarios, we obain Table 2 he amoun of money in he noional moraliy accoun a he end of he monh, Table 3 he acuarial

10 The annuiy overlay fund 10 Table 2: Noional moraliy accoun a he end of monh, which depends on who dies during he monh. Alice Bob alive dead alive $0 $ dead $ $ Table 3: Alice s and Bob s acuarial gains a he end of monh, which depend on who dies during he monh. Alice s acuarial gains are in normal ex and Bob s acuarial gains are in ialics. Alice alive dead Bob alive dead $0 + $ $0 $ $ $ $ $ gains of Alice and Bob and Table 4 he wealh of Alice and Bob a he end of he monh. We see from Table 3 ha, as long as Alice survives o he end of he monh, her acuarial gains are posiive. The same observaion holds for Bob and, indeed, holds more generally for any group. I is an imporan feaure of he fund since i is an incenive o join he fund. A he end of he monh, he surviving paricipans choose wheher or no o pool heir moraliy for anoher monh, and how much wealh hey wan o pool. This is a highly aracive feaure of our fund. I means ha individuals can wihdraw money according o heir needs. For example, hey may have long-erm care or large medical bills o pay. In comparison, convenional annuiies and oher pooled annuiy funds eiher do no permi exis for reasons oher han deah, or hey apply a severe financial penaly o any wihdrawn funds. Allowing he paricipans o leave he fund wihou financial penaly is a consequence of he expeced acuarial gains of Alice over all scenarios being zero, and similarly for Bob. In oher Table 4: Alice s and Bob s wealh a he end of monh, which depend on who dies during he monh. Alice s wealh is in normal ex and Bob s wealh is in ialics. Alice alive dead Bob alive dead $ $ $ $1 220 $ $ $ $25 610

11 The annuiy overlay fund 11 words, here is a zero expeced gain from pooling moraliy over he monh. Thus, a he end of he monh, neiher Alice nor Bob have any furher acuarial obligaion o each oher and hus can ake heir money and go heir separae ways. The same approach can be used o pool moraliy risk among a large group of people. Indeed, we can hink of Alice as a proxy for an aggregae group of individuals. For example, she could represen a group of 100 individuals each wih wealh $ The oy example made he unrealisic assumpion ha he reurn on wealh is zero. We show in he sequel ha he fund can be made acuarially fair a all insans in ime, and no jus on a monhly basis, while allowing for invesmen reurns. 3 Theoreical operaion of he fund Here we show how he annuiy overlay fund operaes heoreically, which is on an insananeous basis. We prove ha he fund is acuarially fair, in he sense ha he expeced insananeous acuarial gains of each paricipan is zero a all imes. Consumpion is ignored because i does no affec he resuls. 3.1 Seup Suppose ha here are M N groups of individuals who paricipae in he annuiy overlay fund. We call he collecion of M groups he porfolio. Wihin he mh group here are L m 0 1 individuals age x m alive a ime 0 for example, we could have only one individual in each group so ha L m 0 = 1 for each m. Individuals wihin a group are homogeneous in he sense ha hey have he same moraliy characerisics, risk preferences and iniial wealh. We model he survival of he ih individual in group m by he Poisson process N m,i := {N m,i, 0}. We assume N m,i 0 = 0 for all m and i. If he ih individual in group m is alive a ime, hen N m,i = 0, and oherwise N m,i = 1. The rae parameer, called he force of moraliy or insananeous rae of moraliy, of he Poisson process N m,i is λ m a ime. Deahs are assumed o occur independenly of each oher, so ha he Poisson processes are independen processes.

12 The annuiy overlay fund 12 Denoing by N m he number of deahs which have occurred up o and including a ime in he mh group, we have he relaionship N m := L m 0 i=1 N m,i. 1 Define he number of people alive a ime in he mh group as L m = L m 0 N m. Then N m := {N m, 0} is a Poisson process wih rae λ m L m a ime. As deahs occur independenly, he processes N 1,..., N M are independen. The financial marke consiss of wo raded asses: a risky asse and a risk-free asse. The risk-free asse has price B a ime wih dynamics db = rb d, 2 wih consan risk-free rae of reurn r > 0. The price process S of he risky asse is driven by a 1-dimensional sandard Brownian moion Z, so ha a ime i has dynamics ds = S µd + σdz, S 0 > 0 consan, 3 wih µ > r consan and σ > 0 consan. The Brownian moion and Poisson processes are defined on he same complee probabiliy space Ω, F, P and are independen processes. Wih N P denoing he P-null ses in he probabiliy space, he informaion a ime 0 is represened by he filraion F = σ{n 1,1 s,..., N 1,L1 0 s,..., N M,1 s,..., N M,LM 0 s, Z s, s [0, ]} N P. 4 In oher words, a each ime, i is known which individuals have died in each group and he price of he risky asse a all imes up o and including a ime. We assume ha individuals have provided for any desired bequess in advance of commiing any asses o he annuiy overlay fund, for example by buying a life insurance policy or commiing less han 100% of heir asses o he fund.

13 The annuiy overlay fund Theoreical operaion on an insananeous basis The pool of M groups of individuals paricipae in he annuiy overlay fund. In addiion o joining he fund, paricipans inves in he financial marke. For simpliciy, we assume here ha paricipans only exi he fund due o heir own deah, alhough his assumpion can be relaxed wihou changing he resuls. Denoe he wealh a ime of each paricipan in he mh group who is alive a ime by W m, for any 0 and for each m = 1,..., M. If an individual in he mh group dies during he shor ime inerval, hen her wealh W m is pu in he noional moraliy accoun. Le U represen he amoun of money which has passed hrough he noional moraliy accoun up o ime. The amoun of money which is pu in he noional moraliy accoun during he shor ime inerval, is wrien mahemaically as du = M W dn m m. 5 m=1 The amoun du is hen shared ou a ime among all he paricipans who were alive a ime. The amoun allocaed o each paricipan is proporional o heir individual wealh and force of moraliy. Thus each paricipan in he kh group who was alive a ime receives a paymen a ime of amoun λ k W k M m=1 W m λm Lm du 6 from he noional moraliy accoun. The paymen, which we call a moraliy credi is made irrespecive of wheher or no he paricipan is alive a ime. Formally we calculae he acuarial gains of each individual due o heir paricipaion in he fund over he ime inerval,. This allows us o separae he gains due o invesmen in he financial marke from he acuarial gains due o sharing moraliy risk. We denoe by G k,i he oal acuarial gains up o ime of a fixed individual i in he kh group. Allowing for he wealh of hose dying being ransferred ino he noional moraliy accoun, he change in he

14 The annuiy overlay fund 14 acuarial gains a ime of individual i in he kh group is given as dg k,i = λ k W k M m=1 W m λm Lm du W k, if individual i dies during,, λ k W k M m=1 W m λm Lm du, if individual i is alive a ime, 0, if individual i is dead a ime. 7 As he change in he acuarial gains dg k,i is due o paricipaion in he fund over he shor ime inerval,, we refer o he gains as he insananeous acuarial gains. Since individuals mus be alive a ime in order o paricipae in he fund over he ime inerval,, hey can no have any acuarial gain a ime if hey are dead a ime. A ime, any individual who is sill alive can coninue o paricipae in he fund for anoher insan in ime, if hey choose o do so. Proposiion 3.1. The expeced insananeous acuarial gains for a paricipan in he annuiy overlay fund are zero a all imes, i.e. for individual i in he kh group, E dg k,i F = 0, 8 for all 0 and for each i = 1,..., L k 0 and k = 1,..., M. Proof. See Appendix A. Proposiion 3.1 is a consequence of paricipans pooling heir moraliy risk only insananeously. This sands in conras o producs like life annuiies for which annuians pool heir moraliy risk over heir lifeime, and hus canno exi eiher before deah or wihou being charged an onerous financial penaly. In he annuiy overlay fund, a paricipan can exi wihou financial penaly, leaving wih he full value of heir wealh. However, even hough he expeced acuarial gains are zero, he incenive o join he annuiy overlay fund is ha he acuarial gains for a paricipan who survives are always nonnegaive. Proposiion 3.2. Condiional upon survival, he expeced insananeous acuarial gains for

15 The annuiy overlay fund 15 individual i in he kh group are E dg k,i F, N k,i = 0 = λ k W k 1 λ k W k M m=1 W m λm Lm d, 9 for all 0 and for each k = 1,..., M. Proof. See Appendix A. Corollary 3.3. Condiional upon survival, he expeced insananeous acuarial gains for a paricipan in he annuiy overlay fund are nonnegaive a all imes. Corollary 3.3 shows ha, as long as a paricipan survives, hey do no lose financially from paricipaing in he fund. This is an imporan poin and i is a key difference beween he annuiy overlay fund and a life annuiy. I means ha he naural frame for he annuiy overlay fund is an invesmen frame, which considers is risk and reurn feaures. In conras, he naural frame for evaluaing he life annuiy is a consumpion frame, which focuses on wha can be consumed over ime. However, many individuals may prefer o evaluae he life annuiy in an invesmen frame Brown e al., Having paid a known single premium a he sar of he conrac, he individual may ask if hey can live long enough o make back heir original invesmen Hu and Sco, For example, consider an individual who pays a single premium of $ o buy a life annuiy income of $5 000 per annum, paid a he end of each year unil he individual dies. If he individual dies in he sixh year afer purchase, hen hey have received 5 paymens of $ From he individual s perspecive, he annuiy s inernal rae of reurn is -33.5% per annum 3. The individual has o live a leas 20 more years in order for he annuiy o break even, and live more han 26 years o have a reurn of 2% per annum or higher. If living long enough o benefi financially is a crierion for buying an annuiy, hen i may no look like an aracive invesmen o people who under-esimae heir fuure lifeime. Tha may be rue for a large number of people. For example, in a survey of people age 45 years 3 Of course, a guaranee can be purchased in conjuncion wih he life annuiy so ha he annuiy income is guaraneed for, say, 10 years. However, as a guaranee can also be purchased in conjuncion wih he annuiy overlay fund, i is no useful o consider guaranees in he analysis here.

16 The annuiy overlay fund 16 o 80 years, Greenwald and EBRI 2012, Figure 15 repor ha 41% of he surveyed group guessed a personal fuure life expecancy ha was 5 years or more below heir expeced fuure life expecancy, based on a populaion moraliy able suiable for heir age and sex. This issue does no occur wih he annuiy overlay fund. Is srucure means ha he individual gains an explici financial paymen while alive due o he pooling of moraliy. They do no lose any of heir money from pooling moraliy risk unil hey die, unlike in he life annuiy where he loss occurs a he sar of he conrac. The annuiy overlay fund may be more aracive o individuals simply because of he invesmen framing of he moraliy gains. Addiionally, observe ha he annuiy overlay fund is closer in spiri o he acuarial noes 4 inroduced and analyzed in Yaari 1965, han he life annuiy. Alhough a group of people benefi from moraliy gains in a life annuiy conrac, he gains o each individual can only be appreciaed by using a lifeime approach, which involves assigning probabiliies o each fuure possible lifeime. I requires a sophisicaed and absrac calculaion. Wih he annuiy overlay fund, surviving individuals have an annual reurn ha is a leas as big as he reurn from invesmen in he financial marke. They are no required o use a lifeime probabiliy model o appreciae he financial benefis of pooling moraliy Pracical consideraions We have presened he annuiy overlay fund in is mos general form, allowing people o leave whenever hey choose. Indeed, i is highly unlikely ha he fund could realisically be operaed wihou some resricions. However, he main poin is ha, as he proposed fund is acuarially fair a every insan in ime, i is very flexible and can be adaped o any required resricions. For example, if he fund has a paricular purpose, such as o pool moraliy risk from any cause of deah, hen allowing individuals o exi a any ime, or wihou paying a financial penaly, would no be advisable; individuals have more informaion on heir own healh han he oher paricipans in he fund. These poins aside, one may wonder how o implemen he moraliy risk-sharing mechanism 4 An acuarial noe pays ou a a fixed ime upon survival o ha ime. A similar conrac is he Arrow annuiy defined in Davidoff e al However, o swich o a consumpion frame hey do need such a model.

17 The annuiy overlay fund 17 in pracice. For example, we migh know someone s dae of deah bu no heir exac ime of deah. This would imply ha he disribuion of money from he noional moraliy accoun should be done a mos daily. We can imagine ha broadly he implemenaion seps could be: An age- and ime-dependen force of moraliy funcion is assigned o each paricipan upon joining he annuiy overlay fund. This may incur an iniial charge o each paricipan. The wealh of paricipans a he sar of each day is recorded. Upon he noificaion of a deah among he paricipans, he wealh of he dead paricipan is liquidaed and disribued among he paricipans, using a discreized version of equaion 6. The calculaion is done as a he dae of deah, using he wealh and he force of moraliy appropriae o each paricipan a sar of he dae of deah. However, he amoun of money o be disribued from he noional moraliy accoun mus clearly be he curren liquidaed wealh of he dead paricipan. The moraliy credis are paid o he surviving paricipans, eiher as cash or invesed in line wih a paricipan s chosen invesmen sraegy. The moraliy credi due o he dead paricipan is paid o heir esae. Each year, paricipans receive an invesmen saemen deailing heir curren individual wealh, how much hey gained from heir invesmens over he year, he amoun of any moraliy credi paid o hem, and coss such as invesmen managemen fees, adminisraion coss, and so on. Addiionally, each paricipan could receive annual informaion on how much moraliy credi hey can reasonably anicipae from he annuiy overlay fund over he nex year, based on he composiion of he annuiy overlay fund and he paricipan s wealh and invesmen sraegy o dae. Thus we do no sugges ha paricipans are supplied wih deails of each oher s wealh and force of moraliy, bu ha hey are given an indicaion of he fuure moraliy credi ha hey may receive from he annuiy overlay fund.

18 The annuiy overlay fund 18 The moraliy funcions are updaed periodically o allow for unanicipaed changes in moraliy. We have shown acuarial fairness holds insananeously in a heoreical model. In pracice, performing he calculaions daily, as suggesed above, should give a reasonable approximaion o coninuous ime and hence acuarial fairness. A criical quesion is when could acuarial fairness break down in a non-rivial way in he real world. Poenial pifalls include: Incorrec choice of moraliy model for he paricipans, for reasons ha may be due o moral hazard, adverse selecion or incorrec assessmen by he fund adminisraors. Large changes in he wealh of he paricipans over he course of a day. This could be allowed for by a suiable adjusmen o he calculaion of he moraliy credis, such as using average wealh value of he paricipans over he day, if he daa is available, or by having a fund in which all paricipans have he same invesmen sraegy. In general he choice of he forces of moraliy will depend on he condiions placed on enering and exiing he fund. We do no consider in his paper wha resricions should be placed on a fund o mee a paricular purpose. Neiher do we explore he addiional issue of adverse selecion, which is a problem also faced by annuiy providers. However, observe equaion 6, which shows he share of he noional moraliy accoun paid o each paricipan in he kh group. We see ha he relaive values of he forces of moraliy are more imporan han he absolue values. Thus we need a moraliy model which accuraely capures he relaive differences in moraliy among paricipans, raher han heir absolue differences, so ha he noional moraliy accoun is shared ou equiably. Furhermore he moraliy model can be updaed frequenly o reflec curren moraliy, since he money in he noional moraliy accoun is shared ou immediaely. Thus we can allow for longeviy improvemens and oher variaions in moraliy hrough ime, somehing which is no possible for many convenional life annuiies.

19 The annuiy overlay fund 19 4 The infinie annuiy overlay fund and is wider connecions We have already observed ha, compared o invesmen in he financial marke alone, i is raional for an invesor wih no beques moive and who prefers more money o less, o join he annuiy overlay fund; his is he essence of Corollary 3.3. Here we describe an idealized version of he annuiy overlay fund, called he infinie annuiy overlay fund, in which here are infiniely-many paricipans in each group. The infinie annuiy overlay fund is srongly conneced o boh he classical life annuiy conrac and a paricular ype of pooled annuiy fund, as we show in Secion 4.2. Wheher he infinie annuiy overlay fund can be used as a saisfacory approximaion o a specific finie annuiy overlay fund depends on he number of paricipans and heir wealhmoraliy profile. Our resuls in he sequel sugges ha, for a suiably diversified fund, he numbers of paricipans may be in he hundreds raher han he housands for his approximaion o be reasonable. However, we emphasize ha acuarial fairness coninues o hold in he annuiy overlay fund regardless of he number of paricipans and he heerogeneiy of he group. This is a very imporan poin which should no be disregarded as mere acuarial nipicking, paricularly for he relevance of he proposed fund o a real-world applicaion. 4.1 Descripion of he infinie annuiy overlay fund Here we deermine he acuarial gains in he infinie annuiy overlay fund. Consider an individual who has no beques moive. Suppose he individual i joins he annuiy overlay fund and is assigned o he kh group. Proposiion 4.1. Condiional upon survival, he variance of he insananeous acuarial gains for individual i in he kh group is Var dg k,i F, N k,i = 0 for all 0 and for each k = 1,..., M. 2 λ k W k M = M m=1 W mλm Lm m=1 W m 2 2 λ m L m W k λ k d. 10

20 The annuiy overlay fund 20 Proof. See Appendix A. Furher assume ha a ime > 0, each group in he annuiy overlay fund has exacly he same number of members, so ha L := L 1 = L 2 = = L M > 0. In ha case, he insananeous acuarial gains of he chosen individual, assuming hey are alive a ime, are from 7, dg k,i = λ k W k L M m=1 W m λm du 11 Now le he number of members in each group end o infiniy. From Proposiion 3.2 we ge E dg k,i F, N k,i = 0 λ k W k d as L. 12 From Proposiion 4.1, Var dg k,i F, N k,i = 0 0 as L. 13 Thus here is no volailiy in he insananeous acuarial gains as he number of paricipans in each group ends o infiniy. In an infinie annuiy overlay fund, deahs occur coninuously, which releases a coninuous flow of money ino he noaional moraliy accoun. As his is shared among infiniely-many paricipans, heir individual wealh increases a a coninuous rae equal o heir own force of moraliy, wih zero volailiy. In his perfec pool, he volailiy of reurn on wealh arises solely from invesmen in he financial marke. To see how he acuarial gains in he infinie annuiy overlay fund affec he wealh dynamics of he paricipans, assume he financial marke deailed in Secion 3.1. Consider an individual i who is a member of he kh group in he annuiy overlay fund. Denoe by π he amoun of he individual i s wealh invesed in he risky sock a ime. Then ignoring consumpion, heir wealh dynamics are dw k = rw k + µ rπ + λ k W k d + σπ dz. 14 The benefi of joining he infinie annuiy overlay fund is seen in he addiional erm λ k W k, an

21 The annuiy overlay fund 21 increase in he wealh due o he pooling of moraliy risk wih infiniely-many oher people. 4.2 Connecion of he infinie annuiy overlay fund o oher annuiies The acuarial gains in he heerogeneous infinie annuiy overlay fund are idenical o hose in he homogeneous infinie pooled annuiy fund, analyzed by Samos This can be seen by comparing equaion 14 wih Samos 2008, equaion In he laer fund, here are an infinie number of paricipans who are independen and idenical copies of each oher. The wealh of he deceased are shared equally among all he survivors. Boh Donnelly e al and Samos 2008 analyze his ype of pooled annuiy fund. Consequenly, he welfare analysis of Samos 2008 can be applied direcly o he infinie annuiy overlay fund. His analysis shows significan uiliy gains for individuals paricipaing in an infinie annuiy overlay fund compared o a pure wihdrawal plan. The welfare gains of he annuiy overlay fund compared o a fixed-payou annuiy depend on he individual s level of risk aversion: an individual wih low o moderae levels of risk aversion would derive greaer uiliy from joining he annuiy overlay fund compared o buying a fixed-payou annuiy, whereas he siuaion is he reverse for an individuals wih a high level of risk aversion. We refer he ineresed reader o Samos 2008 for he precise deails. We can also connec he infinie annuiy overlay fund wih a life annuiy. Suppose ha a ime 0, he individual i invess $w in he risk-free asse and joins he kh group of he infinie annuiy overlay fund. She consumes her wealh coninuously a he consan amoun c per annum. Then he dynamics of heir wealh process W as long as she is alive, are dw k = rw k + λ k W k c d, 15 subjec o W k 0 = w. Those familiar wih life insurance reserving may recognize 15 as he dynamics of he reserve held by he insurance company for a single life annuiy wih annual paymen $c paid coninuously, when moraliy risk is fully diversified. I is a version of he celebraed Thiele s differenial equaion Dickson e al., 2009, Secion Thus equaion 6 Noe ha Samos 2008 uses π o denoe he proporion of wealh invesed in he risky asse, whereas we use i here o denoe he amoun of wealh.

22 The annuiy overlay fund ells us ha he wealh of a surviving paricipan in he infinie annuiy overlay fund maches he reserve held by an insurance company agains each of is annuiy policies, when hey boh use he same assumpions and he annuiy income paid by he insurance company maches he paricipan s consumpion amoun c. 5 Analysis of he finie annuiy overlay fund Here we consider an annuiy overlay fund in which here are only a finie number of members in each group. I is imporan o consider how he heerogeneiy among he paricipans can affec heir acuarial gains. For a member of he annuiy overlay fund here are wo sources of wealh volailiy: he invesmen marke and he membership of he fund. We assume ha a member is indifferen o he source of volailiy. For example, hey do no care wheher heir wealh has increased due o a share dividend paymen or due o anoher member dying. We wan o analyze he impac of a heerogeneous fund in erms of he moraliy-wealh profile of he fund on he wealh volailiy of a paricipan in he fund, while allowing for he paricipans o inves heir wealh in a financial marke. I may be ha he volailiy due o deahs occurring in he fund is no significan compared o volailiy from he financial marke. We assume ha members moraliy disribuion is known. While he expeced reurn on wealh due o sharing moraliy risk in he annuiy overlay fund is consisen wih he disribuion, he acual reurn may differ due o volailiy in he deahs in he fund. We compare paricipaion in he annuiy overlay fund o membership of a benchmark fund called he moraliy-linked fund a more exensive discussion of he moraliy-linked fund is provided by Donnelly e al In he moraliy-linked fund, wealh volailiy arises from he invesmen marke only. The random moraliy credi of he annuiy overlay fund is replaced by a deerminisic moraliy-linked ineres rae paid by an insurer. In his conex, he insurer is analogous o an annuiy provider: hey are indirecly pooling he moraliy of he members of he benchmark fund. The deerminisic moraliy-linked ineres rae ha he insurer pays on a member s wealh

23 The annuiy overlay fund 23 is equal o he member s force of moraliy bu wih a reducion o allow for coss. Noe ha, in his secion, we use he word coss in a differen sense o earlier. The coss are wha he insurer of he moraliy-linked fund charges o he individual o remove he laer s moraliy risk. We emphasize ha he moraliy-linked ineres rae is a deerminisic ineres rae. Exacly as in he annuiy overlay fund, members of he moraliy-linked fund are free o inves heir wealh in he financial marke as hey choose. The coss are he ool ha we use o analyze he differences beween he annuiy overlay fund and he moraliy-linked fund. Definiion 5.1. The insananeous breakeven coss applying a ime are he coss such ha, for equal insananeous volailiies of reurn on he wealh, a surviving individual has he same insananeous expeced reurn on wealh from he annuiy overlay fund as from he moraliylinked fund a ime. The idea is ha we calculae firs he volailiy of reurn on wealh for a paricipan in he annuiy overlay fund, given ha some proporion of heir wealh is invesed in a risky financial asse. Nex we calculae he proporion of wealh ha an idenical member of he moraliylinked fund would have o inves in he risky asse in order o have he same volailiy of reurn on wealh. The proporion should be higher for he member of he moraliy-linked fund since hey have volailiy from he financial marke only. Finally, we calculae he coss such ha he expeced reurn for he wo individuals is he same, allowing for he differen proporions of wealh invesed in he risky asse. These are he insananeous breakeven coss. If he acual coss charged by he moraliy-linked fund are higher han he insananeous breakeven coss, hen an individual can obain a higher expeced reurn from he annuiy overlay fund for he same amoun of volailiy of reurn on wealh. In Secions 5.1 and 5.2 we wrie down he expeced reurns and volailiy of a chosen individual in he annuiy overlay fund and moraliy-linked fund. This allows us o wrie down a mahemaical expression for he insananeous breakeven coss in Secion 5.3.

24 The annuiy overlay fund Finie annuiy overlay fund As before, we assume ha here are M N groups of individuals in he annuiy overlay fund. Each surviving paricipan in he kh group has wealh W k, force of moraliy λ k and invess a proporion p k of heir wealh in he risky asse a ime. The remaining proporion of wealh 1 p k is invesed in he risk-free asse. Thus he wealh W k of an individual i in he kh group in he finie annuiy overlay fund has he dynamics dw k = r + p k µ r W d k + σp k W dz k + dg k,i, 16 subjec o W0 k = wk 0 > 0. The firs wo erms on he righ-hand side are due o he invesmen in he financial marke. The hird erm, dg k,i, represens he insananeous acuarial gains from paricipaion in he fund. Condiional on individual i surviving o ime, her insananeous expeced reurn on wealh is calculaed from he dynamics given by equaion 16 and Proposiion 3.2 o be E dw k W k F, N k,i = 0 = r + p k µ r + λ k 1 W k λ k M m=1 W m λm Lm d, 17 in which we recall ha L m represens he number of individuals in he mh group who are alive a ime. Similarly, he insananeous variance of he reurn on wealh condiional on individual i surviving o ime is Var dw k W k F, N k,i = 0 = σp k 2 + λ k 2 M m=1 W m 2 λ m L m W k 2 λ k M 2 m=1 W mλm Lm d. The same decomposiion is seen for he insananeous expeced reurn on wealh and he insananeous variance of reurn on wealh: here is a componen due o individual i s invesmen in he financial marke, and a componen from her acuarial gains. 18

25 The annuiy overlay fund Moraliy-linked fund wih coss Suppose insead ha he individual i decides o join he moraliy-linked fund, which is sold by an insurer. As long as she survives, a moraliy-linked ineres rae is paid by he insurer on her wealh, less he coss which are specified below. In he moraliy-linked fund, le p k be he proporion of wealh invesed in he risky asse a ime by individual i. The remaining proporion of wealh 1 p k is invesed in he risk-free asse. The coss ha he insurer charges o individual i are represened by a k. As long as individual i survives, her wealh W k has he dynamics d W k = r + p k µ r W k d + σ p k W k dz + 1 a k λ k W k d, 19 subjec o W 0 k = wk 0 > 0. The erm 1 ak λ k W d k is he rae a which he moraliy credi is paid by he insurer o individual i. Noe ha for a k = 0, he wealh dynamics for he surviving members of he moraliy-linked fund mach hose of an infinie annuiy overlay fund, in which here are an infinie number of members in each group of he annuiy overlay fund 7. The insananeous expeced reurn on wealh condiional on individual i being alive a ime is E d W k W k F, N k,i = 0 = r + p k µ r + 1 a k λ k d. 20 The insananeous variance of he reurn of he wealh, condiional on individual i being alive a ime, is Var d W k W k F, N k,i = 0 = σ p k 2 d. 21 Unlike in he annuiy overlay fund, here is no uncerainy abou he moraliy credi for he paricipan 8 ; he insurer pays i o individual i as long as individual i is alive. The only source of volailiy for a survivor in he moraliy-linked fund is he financial marke. 7 I can be shown ha he infinie annuiy overlay fund coincides wih he pooled annuiy fund analyzed in Donnelly e al and Samos Insead, i is borne by he insurer. Addiionally, he insurer is exposed o model risk if he moraliy index is no represenaive of he paricipan s acual moraliy.

26 The annuiy overlay fund Insananeous breakeven coss Here we calculae he insananeous breakeven coss. For ease of noaion, we use bold noaion o denoe a vecor of lengh M. For example, p = p 1,..., p M, W = W 1,..., W M and so on, where we use o denoe X he ranspose of he vecor X. We also define he useful shor-hand noaion S k w, l, λ := M m=1 wm 2 λ m l m w k 2 λ k M m=1 wm λ m l m 2, 22 for all w, λ, l R M + R M + N M and for each k = 1,..., M. Lemma 5.2 Insananeous breakeven coss. Suppose an individual i, who is in he kh group of he annuiy overlay fund, invess he proporion p k of her wealh in he risky asse. To have he same insananeous volailiy of wealh in he moraliy-linked fund, she mus inves he proporion p k p, W, λ, L of her wealh in he risky asse, wih p k p, w, λ, l := p k 2 λ k 2 1/2 + S k w, l, λ, 23 σ for all p, w, λ, l R M + R M + R M + N M. Then he insananeous breakeven coss are a k = a k p, W, λ, L, wih a k p, w, λ, l := µ r λ k [ p k p, w, λ, l p k] w k λ k + M m=1 wm λ m l, 24 m for all p, w, λ, l R M + R M + R M + N M. Proof. To show 23, equae he insananeous volailiies, given by equaions 18 and 21, and rearrange. To show 24, equae he insananeous expeced reurns, given by equaion 17 and equaion 20, and rearrange o find a k. Thus he breakeven coss a which he expeced reurns from he funds are equal decomposes ino wo componens, one due o he financial marke and he oher due o he pooling of moraliy. The firs erm on he righ-hand side of equaion 24 represens he exra expeced

27 The annuiy overlay fund 27 reurn from higher invesmen in he risky asse in he moraliy-linked fund. The second is he fracion of he money in he noional moraliy accoun received by he paricipan a ime. However, i is difficul o undersand from equaion 24 he main facors affecing he breakeven coss since he proporion of wealh invesed in he risky asse in he moraliy-linked fund also depends on he wealh-moraliy profile of he annuiy overlay fund. To undersand hese, we apply a Taylor series expansion o 24 o ge he firs-order approximaion o he breakeven coss a k p, W, λ, L µ r 2σ 2 λ k p k S k W, λ, L + W λ k k M m=1 W. 25 mλm Lm The firs-order approximaion suggess ha he spread of he wealh weighed by he expeced number of deahs in each group, as approximaed by S k W, λ, L, is a criical facor in he deerminaion of he breakeven coss. The reason is ha a high value of S k W, λ, L indicaes a higher volailiy in he amoun and iming of money ha is credied o he noional moraliy accoun. We explore he impac of heerogeneiy in he numerical illusraions nex. 5.4 Numerical illusraions We explore he impac of heerogeneiy in he annuiy overlay fund by comparing i wih he moraliy-linked fund he benchmark fund, for hree differen heerogeneous porfolios. As he analysis is done over an insan in ime, we do no need o consider consumpion. The resuls sugges ha a here only has o be a few hundred paricipans in he porfolio for he breakeven coss o be very low, allowing for moderae heerogeneiy among he paricipans, bu b severe heerogeneiy in he porfolio may invalidae he above conclusion. Therefore, heerogeneiy needs o be sudied furher in he conex of annuiy overlay funds. The low breakeven coss are very ineresing. They sugges ha a group of a few hundred individuals, who are willing o accep volailiy in he reurn on wealh from deahs, can obain