Efficient Annuitization: Optimal Strategies for Hedging Mortality Risk

Transcription

1 Efficien Annuiizaion: Opimal Sraegies for Hedging Moraliy Risk Jason S. Sco, John G. Wason, and Wei-Yin Hu* PRC WP Pension Research Council Working Paper Pension Research Council The Wharon School, Universiy of Pennsylvania 3620 Locus Walk, 3000 SH-DH Philadelphia, PA Tel: Fax: hp:// * The auhors hank William Sharpe, Geer Bekaer, Seve Grenadier, and Jim Shearer for many excellen commens and suggesions. Any remaining errors or omissions are he auhors' responsibiliy. The views expressed herein are hose of he auhors and no necessarily hose of Financial Engines. JEL Classificaions: D11, D91, E21, H55, J14, J26. Keywords: Annuiies, annuiizaion, Social Securiy, pensions, longeviy risk, insurance. Opinions and errors are solely hose of he auhors and no of he insiuions wih whom he auhors are affiliaed Pension Research Council. All righs reserved.

2 Efficien Annuiizaion: Opimal Sraegies for Hedging Moraliy Risk Absrac Two common explanaions for he dearh of volunary annuiizaion are beques moives and liquidiy demand, boh of which creae implici coss for each annuiized dollar. Whenever coss preven full annuiizaion, we demonsrae ha efficien annuiy allocaions concenrae annuiy-funded consumpion lae in life. This implies radiional immediae payou annuiies are inefficien relaive o recenly inroduced delayed payou annuiies which have survival-coningen paymens beginning years afer purchase. For ypical examples, a six percen delayed payou allocaion has uiliy comparable o a hiry-nine percen immediae annuiy allocaion. Since reirees appear averse o large annuiy purchases, delayed payou annuiies could significanly improve reiree welfare. Jason S. Sco (Corresponding Auhor) Financial Engines, Inc Embarcadero Rd. Palo Alo, CA T: F: jsco@financialengines.com John G. Wason Financial Engines, Inc Embarcadero Rd. Palo Alo, CA T: F: jwason@financialengines.com Wei-Yin Hu Financial Engines, Inc Embarcadero Rd. Palo Alo, CA T: F: whu@financialengines.com 2

3 Efficien Annuiizaion: Opimal Sraegies for Hedging Moraliy Risk 1 Inroducion Yaari [1965] heorized ha individuals wih access o acuarially fair annuiies and wihou a beques moive would opimally annuiize all of heir asses. Since his publicaion, i has been an open puzzle as o why individuals rarely volunarily choose o annuiize much, if any, of heir reiremen asses. Recenly, Davidoff, Brown and Diamond [2005] exended he puzzle by poining ou ha full annuiizaion did no require all he axioms of expeced uiliy maximizaion. Raher, since annuiies faciliaed consumpion a lower prices, all ha was essenial o he resul was a lack of a beques moive and a preference for more over less. Since he poenial benefi for fully annuiizing is subsanial 1, a significan amoun of research over he ensuing decades has aemped o explain he lack of annuiy demand. In broad erms, he research fell ino wo caegories. Firs, some researchers focused on why he benefis o annuiizaion were smaller han originally envisioned. Oher researchers hypohesized ha unaccouned-for coss associaed wih annuiizaion dampened demand. (See Brown and Warshawsky [2004] for an excellen overview of his lieraure.) Two explanaions have been proposed ha relae o reduced benefis from annuiizaion. Firs, some research has noed ha many people already have a subsanial fracion of heir wealh already annuiized in he form of promised pension benefis such as Social Securiy or employer-based defined benefi plans. (See, for example, Brown 1 Baseline annuiy benefis have been esimaed o increase effecive available wealh by 50% or more. (See, for example, Michell, Poerba, Warshawsky and Brown [1999].) 3

4 and Poerba [2000], and Michell e al. [1999].) The benefis of addiional annuiizaion are decreased if subsanial exising wealh is already annuiized. A second facor reducing he benefis o annuiizaion is he abiliy of families o pool heir moraliy risk. For example, Kolikoff and Spivak [1981] demonsrae ha a couple can capure close o half he poenial benefis associaed wih annuiizaion wihou acually purchasing any annuiies. While boh of hese facors could subsanially reduce he expeced benefis o annuiizaion, hey do no eliminae he benefi. Unless some coss associaed wih annuiizaion are inroduced, hese explanaions merely reduce he available gains wihou alering he opimaliy of full annuiizaion. Oher auhors ackle he annuiy puzzle from he cos perspecive. In Yaari s model, here are no downsides o annuiizaion. Wih only benefis and no coss, he opimal answer is full annuiizaion. Firs, some auhors have noed he original Yaari work assumes acuarially fair annuiy prices. If prices reflec insurance coss and he poenial for adverse selecion, hen depending on he pricing srucure, full annuiizaion may no longer be opimal (e.g., Friedman and Warshawsky [1988] and Michell, Poerba, Warshawksy and Brown [1999]). Anoher criical assumpion made in Yaari s analysis was he absence of beques moives. However, every dollar spen on annuiies implies one less dollar available for a beques. If individuals have a beques moive, hen full annuiizaion may again no longer be opimal. For example, Walliser [1999] calculaes ha, wih reasonable assumpions for risk aversion and beques moives, i is opimal o annuiize only 60% of wealh a age 65. Demand for liquidiy creaes anoher implici cos consideraion. If fuure expenses are uncerain, hen individuals would prefer o have a pool of liquid asses o 4

5 insure agains fuure needs. Since annuiy purchases are largely irreversible, every dollar spen on an annuiy is one less dollar available for unknown fuure expenses. As an example, some auhors have poined o expenses relaed o healh uncerainy as a poenial derimen o annuiy demand (e.g., Brown and Warshawsky [2004] or Brugiavini [1993]). Finally, a growing lieraure has focused on a marke imperfecion associaed wih many annuiy conracs in he Unied Saes. These annuiy conracs are no indexed o equiy markes and hus preclude access o any available equiy risk premium. Given his resricion, individuals may wish o reduce heir annuiy holdings or perhaps eliminae hem alogeher depending on heir risk olerance and he equiy premium relaive o annuiies (e.g., Kapur and Orszag [1999]; Sabile [2003]; Milevsky and Young [2003]; Dushi and Webb [2004]; Dus, Maurer and Michell [2005]; Kingson and Thorp [2005]; Horneff, Maurer, Michell, and Dus [2006]; and Koijen, Nijman and Werker [2006]). Similar o bequess and liquidiy, every incremenal dollar annuiized carries an addiional cos componen. In his case, he cos sems from having one less dollar available o inves in he equiy markes. Once coss are inroduced, he fundamenal problem of opimal annuiizaion changes dramaically. Firs, since every dollar annuiized incurs addiional coss, he opimal level of annuiizaion is generally a fracion of oal wealh. Second, coss imply individuals are concerned wih annuiy efficiency. Annuiy conracs ha can deliver larger benefis per dollar invesed (i.e., are more efficien) are always more desirable since hey can provide equal benefis a lower coss compared o less efficien annuiy conracs. Surprisingly, while he lieraure has explored coss associaed wih annuiy 5

6 purchases, no research has been done on annuiy efficiency. The almos universal assumpion has been ha buying an annuiy refers o purchasing an immediae annuiy whose paymens begin immediaely and are guaraneed for as long as he individual lives. This paper demonsraes ha annuiy conracs differ grealy in heir efficiency. We firs analyze opimal annuiy purchases assuming access o flexible sae-coningen annuiy conracs. Given his framework, we demonsrae ha he opimal allocaion of annuiy resources implies annuiized asses fund consumpion lae in reiremen and nonannuiized asses fund consumpion early in reiremen. Since he opimal allocaion of wealh involves a separaion beween non-annuiy and annuiy-funded years, immediae annuiies are generally only opimal when coss are low enough o allow full annuiizaion. The size of he inefficiency of immediae annuiies is surprising. For a ypical example, we find ha six percen of wealh efficienly allocaed o saeconingen annuiies can provide up o half of he available benefis from moraliy risk sharing. To receive a similar level of welfare, immediae annuiies would require an allocaion of hiry-nine percen of wealh. Unforunaely, sae-coningen annuiy conracs are a heoreical consruc no currenly available in he insurance marke. However, we demonsrae ha recenly inroduced delayed payou annuiy conracs can provide all, or subsanially all, of he benefis associaed wih full access o sae-coningen annuiy conracs. Because delayed payou annuiies are much more efficien han immediae annuiies, hey should be highly desirable o any individual facing annuiizaion coss from bequess, liquidiy concerns or asse allocaion consrains. I will be ineresing o see if hese new annuiy producs are successful in he marke. Theoreically, hey should be srongly preferred o 6

7 radiional immediae annuiies. Since hey have he poenial o provide a majoriy of he annuiy benefis wih a modes five o en percen annuiy allocaion, he usual cos-based explanaions for low annuiy demand (beques moives, liquidiy concerns, or marke imperfecions) are unlikely o apply o delayed payou annuiies. A lack of marke success for delayed payou annuiies would hus creae a new annuiy puzzle. The res of his paper is organized as follows. Secion 2 ses up he canonical problem of an individual enering reiremen wih a pool of wealh and an objecive o maximize lifecycle uiliy hrough he purchase of eiher bonds or annuiies. Secion 3 explores opimal consumpion assuming individuals are unconsrained in heir allowed allocaion of wealh o annuiies and have access o flexible sae-coningen annuiies. Secion 4 inroduces he concep of annuiy coss and considers he problem of opimal parial annuiizaion. Secion 5 analyzes he efficiency of differen annuiy producs a delivering he benefis of moraliy risk sharing. In addiion o a baseline analysis, he efficiency analysis considers wo exensions. Firs, he siuaion where he annuiy prices are acuarially unfair is analyzed. Second, he case where Social Securiy is a large fracion of oal wealh is considered. Secion 6 relaes he curren analysis o he exising lieraure, and Secion 7 concludes by summarizing he key findings including normaive suggesions for reirees sruggling wih longeviy consideraions. 2 Lifecycle Uiliy Problem Definiion Consider a reired individual ha has an accumulaed amoun of wealh available o suppor reiremen consumpion. This individual has wo flavors of invesmen opions available. The firs is zero coupon bonds. The prices for hese zero coupon bonds are given by B where: 7

8 B = price oday for $1 payou in year Annuiy producs are also available. For his analysis, we consider a hypoheical se of simple coningen claim annuiies ha pay ou $1 -periods ino he fuure, provided he individual is alive a ha ime. The price for each of hese annuiies is given by A where: A = price oday for $1 payou in year, condiional on survival o period These ypes of annuiies are someimes referred o as zero coupon annuiies due o heir similariy o zero coupon bonds or as Arrow annuiies due o heir sae-dependen payou srucure. We will adop he Arrow annuiy erminology o refer o hese coningen claim securiies. For simpliciy, assume our individual has no beques moive and wishes o allocae his wealh so as o maximize expeced uiliy. We also assume a his poin ha here is no inflaion, so dollar payous a any poin in ime are equivalen o consumpion unis. Finally, assume ha expeced uiliy is addiively separable in ime, and is given by he expression below: (1a) = max Π U ( c ) Ω =0 In Eq.(1a), Π is he probabiliy ha he individual is alive a period (as perceived by he individual planning a ime 0), is he individual s discoun facor for uiliy a period, U is he undiscouned uiliy of consumpion, assumed o have he same funcional form for all periods, and c is he consumpion a period. We assume ha he discoun facors 8

9 and he probabiliies Π are less han or equal o one, wih 0 = Π 0 = 1, and are sricly decreasing in. Le x be he amoun of consumpion from bonds, and y be he amoun of consumpion from annuiies. The oal consumpion a ime, c, is given by: ( 1b ) c = x + y. Furher, le V B equal he presen value of all bonds purchased, and V A equal he presen value of all annuiies purchased. We hen have: (1c ) (1d ) V V B A = = = 0 = 0 B x, A y. The presen value of all purchases mus equal he individual s curren wealh, W 0. This is he budge consrain: ( 1e) VB + VA = W0, Equaions (1) are a mahemaical program, bu he soluions are unbounded. The economics of he siuaion are apparen from he prices of he bonds and annuiies. Whenever he cos of an annuiy payou dollar in period, A, does no equal he cos of a bond payou dollar in period, B, hen here is an opporuniy for an individual o achieve infinie uiliy. For example, if B > A, hen an individual could shor sell B and purchase A in infinie quaniies o maximize uiliy. To eliminae his possibiliy, we rule ou shor sales from he analysis: i.e., ( 1f ) x, y 0. 9

10 These appear o be reasonable assumpions. Firs, here is no marke where an individual can sell an arbirary annuiy, hus implying a non-negaive y. Second, we would no expec a lender o allow borrowing a he bond rae using an annuiy as collaeral. 2 3 Unconsrained Opimal Consumpion Solving he mahemaical programming problem described by Eqs(1) yields he opimal consumpion pah c. We can reduce he opimizaion o an algebraic problem by wriing down he Karush-Kuhn-Tucker condiions for opimaliy (Bazaraa and Shey [1979]). Our resuls assume ha he invesor s uiliy funcion U(c) is monoonically increasing and concave wih marginal uiliy approaching infiniy as c approaches zero; i.e., our invesor is risk-averse. This assumpion leads o pahs for which here is posiive consumpion in any period ha he individual hinks hey have a non-zero probabiliy of being alive. We also assume ha U(c) is wice differeniable, and denoe derivaives wih respec o consumpion by a prime. Since U(c) is increasing and concave, we have U (c) > 0 and U (c) < 0 for all c. I hen follows ha he funcion U (c) has a unique inverse; we will pu his inverse funcion o good use o calculae opimal consumpion pahs. Furher, we will assume ha annuiy prices are such ha hey can be represened as a monoonically decreasing funcion, P, muliplied by he conemporaneous bond price. P is he pricing discoun received for agreeing o he annuiy conrac and can be viewed as he survivor funcion ha prices he Arrow annuiies. If his funcion equals he individual survivor funcion, Π, hen he annuiies are said o be acuarially fair. Laer 2 Yaari [1965] and Davidoff, Brown, and Diamond [2005] make less resricive assumpions abou he abiliy o borrow. Loosening his consrain needlessly complicaes our analysis wihou changing he fundamenal implicaions when annuiy purchases are consrained. 10

11 we will argue ha monoonically decreasing P is an exremely weak assumpion on annuiy prices. To summarize, he iniial assumpions for opimizaion are as follows: U(c) U (c) > 0 Twice differeniable Monoonically increasing U (c) as c 0 Infinie marginal uiliy a zero consumpion U (c) < 0 A = P B P < 1 P > P +1 Concave Annuiy pricing srucure Annuiy price always a discoun relaive o bonds Annuiy discoun funcion monoonically decreasing in ime Opimal consumpion a period is given by he formula: ( B, A ) U ( c ). ( 2a) Π U ( c ) = min 0 The inuiion for his equaion is sraighforward. The righ hand side of he equaion corresponds o he uiliy cos of purchasing an incremenal amoun of consumpion in period. The lef hand side of he equaion represens he marginal increase in uiliy from exra consumpion in period. Opimaliy requires a consumpion plan ha equaes hese wo quaniies. 11

12 Given he assumed pricing srucure for he Arrow annuiies, A < B for all 0. 3 We recover he sandard resul ha all consumpion is funded by annuiy purchases (x = 0 and c = y ). Eq. (2a) can be simplified o he following: (2b) U ( c ) P B = U ( c ) Π 0. Since U (c) has a well-defined inverse, we can always solve Eq.(2b) for he consumpion c, given he iniial consumpion c 0 and values for. The value of c 0 is hen deermined by imposing he budge consrain. Eq. (2b) also illusraes he well-known resul ha when annuiy pricing is acuarially fair (P = Π ) and he bond ineres rae equals he fuure uiliy discoun rae (B = ), hen consumpion is consan across ime. 4 Opimal Bond and Annuiy Allocaion wih Limied Annuiizaion The model described in he preceding secion predics full annuiizaion, because annuiies are a sricly cheaper way o fund consumpion in every ime period. However, here are several opporuniy coss associaed wih increasing levels of annuiizaion. These include fewer dollars available for bequess, fewer dollars available o mee unexpeced liquidiy demands, and more dollars in a resriced asse allocaion required by he annuiy conrac. These facors can each lead o parial annuiizaion, as a reiree would rade off he marginal benefi of an exra annuiy dollar agains he foregone opporuniies ha a liquid (bu more expensive) bond invesmen could saisfy. 3 Assuming period zero corresponds o immediae consumpion, hen A 0 = B 0 = 1. However, his creaes an indifference beween bond and annuiy based consumpion a period 0. We adop he convenion ha A 0 = 1-ε o break ies. This assumpion is immaerial o he overall resuls, bu has he advanage of reaining opimaliy for 100% annuiizaion (as would be he case in coninuous ime models). In effec, period zero consumpion occurs in he very near fuure wih an infiniesimal chance of deah over ha ime period. 12

13 Raher han pose a paricular funcional form for one of hese sources of annuiy opporuniy coss, we analyze he problem of opimal parial annuiizaion by inroducing he following consrain 4 : ( 1g) VA αw0. This consrained model s firs-order condiions involve a new radeoff: he marginal uiliy of an exra dollar of annuiy-funded consumpion is balanced agains he marginal cos imposed by he aggregae wealh consrain (V A + V B = W 0 ) plus he marginal cos of using one more dollar of he annuiy budge consrain ( V A αw0 ). The annuiy budge consrain's shadow price (is Lagrange muliplier) can be inerpreed as represening he muliple sources of opporuniy coss of annuiizaion discussed above. Whaever he exac source of hese opporuniy coss, i is necessary ha any opimal soluion for models ha lead o parial annuiizaion use efficien allocaions of annuiies, where efficien in his conex means ha he limied annuiy dollars provide he mos expeced uiliy. We now urn o he problem of efficienly allocaing annuiy wealh o maximize expeced uiliy given an available annuiy budge. Consraining he amoun of annuiies ha can be purchased forces he individual o spli heir wealh beween bonds and annuiies. The cenral quesion becomes wha annuiies are purchased when annuiy dollars are scarce? To begin o answer ha quesion, we firs inroduce he Bond - Annuiy Separaion Theorem. 4.1 Bond - Annuiy Separaion Theorem: 4 Davidoff, Brown, and Diamond [2005] use a model wih an equivalen consrain o describe he welfare gain from allowing annuiies, moving from α=0 o α=1. 13

14 Claim: If he annuiy discoun funcion, P, decreases in ime, hen i is never opimal for annuiized payous o precede non-annuiized porfolio payous. Proof: Firs, assume an opimal soluion exiss ha consiss of purchasing bond payous for ime +n (x +n > 0), n periods afer an annuiy payou (y > 0). Nex, consruc an alernaive soluion derived by: a) Reducing he consumpion of period +n bonds by a small amoun (ε) b) Purchasing a similar amoun of consumpion in period +n via annuiies c) Reducing he consumpion of period annuiies by an amoun ha offses he purchase of annuiies in (b) and keeps he aggregae allocaion o annuiies consan. d) Purchase enough consumpion in period via bonds o offse he reducion in consumpion from (c) Denoing he alernae soluion wih an aserisk (*), he proposed alernae soluion deviaes from he presumed opimum by he following rades: ( 3a) x x * + n = + n ε ( 3b) y y * + n = + n + ε ( 3c) y * A = y ε A + n ( 3d) x * A = x + ε A + n 14

15 To assess wheher he alernae sraegy described by (3a) (3d) is feasible, we mus verify ha none of he consrains are violaed. Firs, he alernae complies wih he no shor sales consrains since he only reducions occur for y and x +n, boh of which are posiive by assumpion. Since ε can be arbirarily small, here exiss an ε small enough so he alernae sraegy complies wih he no shor sales consrain. The nex consrain o verify is he cap on aggregae wealh allocaed o annuiies. The aggregae allocaion of wealh o annuiies is unchanged from he assumed opimal soluion and he proposed alernae soluion (an amoun of annuiy wealh equal o ε A +n has been ransferred from period o period +n). Finally, we need o examine he budge consrain o deermine if he alernae soluion is feasible. Adding up he ne coss of deviaing from he presumed opimum yields he following impac on he budge consrain: Ne Coss = ε A +n / A (B A ) - ε (B +n A +n ) I suffices o show he ne coss are negaive o demonsrae ha he alernae soluion has higher expeced uiliy han he assumed opimal soluion. The alernae soluion would hen be one ha has equal consumpion as he assumed opimal, bu wih posiive wealh available for addiional consumpion. The ne coss are negaive when: (4a) ε A +n / A (B A ) - ε (B +n A +n ) < 0 (4b) (4c) A +n / B +n < A / B P +n < P Condiion (4c) is exacly he condiion posied in he heorem, hus he claim is proven. 15

16 The separaion heorem prohibis annuiy-funded consumpion from preceding bond-funded consumpion in opimal soluions. This resul is exremely useful in evaluaing siuaions wih consrains on he annuiy allocaion. All opimal plans have consumpion firs funded by he purchase of bonds and hen laer consumpion funded via he purchase of annuiies. 5 The separaion heorem relied upon he declining raio of annuiy o bond prices. Bu how reasonable is his assumpion? Firs, consider his assumpion in he case of acuarially fair annuiy pricing. In his case, he raio of annuiy o bond prices is given by: (5) P = A / B = Π B / B = Π Since survival probabiliies are sricly decreasing over ime, he separaion heorem holds given his pricing srucure. A very weak condiion for annuiy prices would be he assumpion ha A < B for all. If his condiion did no hold for a given, hen he bond in ha ime period would in essence dominae he annuiy. For equal or less money he bond would provide he same consumpion wihou requiring survival o collec. Rearranging he erms in Eq (4c) indicaes ha he condiion for separaion is similar. For arbirary annuiy pricing, he separaion heorem requires: (6) A +n < A ( B +n / B ) for all n Eq (6) illusraes wo differen ways of purchasing consumpion in period +n. The lef-hand side of Eq (6) purchases consumpion in period +n using he +n Arrow annuiy direcly. The righ-hand side of Eq (6) purchases +n consumpion by firs 5 For he discree model we analyze, here is he poenial for bonds and annuiies o fund consumpion in a unique ransiion year. The separaion heorem sill holds since prior o he ransiion year, bonds fund consumpion and aferward annuiies fund consumpion. 16

17 purchasing a period annuiy and hen invesing he proceeds in he bond marke beween and +n. The separaion heorem requires ha purchasing he annuiy direcly is he cheaper alernaive. Similar o he argumen ha A < B for all, if Eq (6) does no hold hen he period +n annuiy is effecively dominaed by a sraegy ha provides he same or more consumpion wih a weaker requiremen on individual survival. Thus, i srikes he auhors as very likely for Eq (6) o hold in any funcioning annuiy marke. We should noe ha while he prior discussion assumes ha riskless bonds underlie he annuiy conrac, he separaion heorem exends o annuiy conracs wih payous coningen on sochasic marke reurns. As demonsraed in Appendix A, as long as he annuiy discoun rae relaive o he price of he underlying securiy is declining, hen i is never opimal o have annuiy-based consumpion precede he consumpion from he underlying securiy alone. I is he pricing and no he sochasic naure of he reurns ha generaes he resul. 4.2 Consrained Annuiizaion and Opimal Consumpion We now reurn o he consrained opimizaion problem. The equaion ha governs opimal consumpion in he consrained case requires he inroducion of an addiional Lagrange muliplier o accoun for he annuiy consrain. In paricular he equaion is: A ( 2a') Π U ( c ) min B, = U ( c0 ). λ The new Lagrange muliplier is λ. I akes a value of one for he unconsrained case. However, for he consrained case, he value of λ is bounded beween zero and one. For a given value of λ, one can see how he separaion heorem is mainained. Given P is 17

18 decreasing in ime and given λ beween zero and one, bonds are iniially purchased unil P = λ. Afer ha poin, annuiies fund consumpion. In Secion 5 we explore he efficiency wih which various annuiy bundles provide benefis compared o full access o Arrow annuiies. A consideraion in he efficiency analysis is how closely can he annuiy bundle mimic he desired consumpion pah opimally seleced wih Arrow annuiies? As we will see, increasing or consan annuiy-based consumpion pahs will generally resul in beer performance for a given annuiy produc space. When Arrow annuiies are available, he ime pah for consumpion is defined by Eq. ( 2a '). Rearranging erms yields: (2b') U ( c0 ) U ( c ) = λ P Π B This equaion is valid over he consumpion region funded by annuiy purchases. 6 Opimal allocaions will equae he marginal uiliy of consumpion in period o he produc of he hree erms in Eq. ( 2b '). If he produc of hese erms is increasing over ime, he consumpion pah will decrease due o he concaviy of U. The firs erm is simply a posiive scalar. I does no influence wheher annuiy-funded consumpion is rising, falling or consan over ime. The second erm is he raio of wo discoun facors. The firs discoun facor, P, measures he discoun received, relaive o bonds, for purchasing consumpion via annuiies. The second discoun facor, Π, represens he individual discoun facor for survival o period. Acuarially fair annuiy prices would equae hese wo discoun facors, eliminaing his erm from influencing he ime pah of consumpion. The hird erm is he raio of he bond discoun rae o he individual 6 For he bond-funded region, he P and λ erms drop from he above equaion. 18

19 discoun rae for fuure uiliy. To summarize, for ime periods where annuiies fund consumpion, (7a) Annuiy-funded consumpion c is a decreasing funcion of P B Π During he ime frame where bonds are used o fund consumpion (i.e. he early years), min(b, A / λ) = B. During hese years, (7b) Bond-funded consumpion c is a decreasing funcion of Π B We will refer back o hese relaionships when discussing wheher a paricular annuiy produc space is capable of replicaing he desired ime pah of consumpion. 5 Efficiency Loss: Wha if Arrow Annuiies are no Available? The previous analysis hypohesized he exisence of Arrow annuiies, which faciliaed an arbirary annuiy-based consumpion paern. In realiy, Arrow annuiies are no available for purchase. Insead, he only available annuiy conracs are hose ha involve bundles of Arrow annuiies. The following analysis assesses he efficiency loss associaed wih annuiy bundling. 5.1 Annuiy Produc Spaces We consider five differen annuiy produc spaces ha allow increasingly flexible annuiy-based consumpion paerns. Throughou he analysis, we assume ha annuiy bundles do no carry discouns or premiums relaive o he cos of purchasing he given paern of annuiy payous via Arrow annuiies. Annuiy Produc Space #0: No Annuiies / Bonds Only 19

20 This is he mos resricive produc space. No annuiies are available for purchase in any form. This opimizaion provides a helpful baseline o compare resuls from he oher siuaions. We follow he convenion in he lieraure of reporing an Annuiy Equivalen Wealh or AEW. (See, for example, Brown and Warshawsky [2004], Brown and Poerba [2000] or Brown, Michell and Poerba [1999] for addiional background on AEW.) Throughou his analysis, he AEW answers he quesion: How much aggregae wealh is required for an individual wih no access o annuiies o be indifferen o a siuaion wih $100 in wealh and he given (poenially consrained) access o annuiies? 7 To define AEW mahemaically, le Ω i (w,α) Max uiliy given wealh w, α percen annuiies and produc space i. Then AEW is always defined by he following equaion: (8) Ω 0 (AEW,0) = Ω i ($100,α) For example, suppose we are considering he case where a person has access o Arrow annuiies and hey are willing o allocae 10% of heir asses o annuiies. Suppose furher ha he repored AEW was $142. This would indicae ha opimizing assuming $100 in wealh, access o Arrow annuiies and a 10% consrain on annuiy purchases provides he same expeced uiliy as opimizing wih $142 and no access o annuiies. 8 Annuiy Produc Space #1: Immediae Annuiies 7 In general, he AEW is a funcion of wealh. However, in all siuaions where we repor AEW, we also assume CRRA uiliy. Given CRRA uiliy, he AEW raio is wealh independen. We adop a $100 wealh level for ease of exposiion. 8 The lieraure generally assumes he righ hand side of Eq. (8) corresponds o full annuiizaion. Here, he definiion of AEW is modified o be consisen wih our focus on parial annuiizaion. 20

21 This is also a fairly limied annuiy produc space. There is only a single annuiy available for purchase. This annuiy is a bundle of all he Arrow annuiies and i pays $1 in all periods where he annuian is alive. An immediae annuiy is assumed o have he following characerisics: Payou srucure = $1 payou in every year, condiional on survival o period Price = =0 A Given his annuiy produc space, he only annuiy-funded consumpion paern available is consan consumpion. To achieve any oher consumpion paern requires he use of bonds o fill in he differences beween desired and annuiy-funded consumpion. Annuiy Produc Space #2: Delayed Purchase Annuiies Several auhors have invesigaed he poenial of delaying he purchase of an immediae annuiy. (See, for example, Kapur and Orszag [1999], Dushi and Webb [2004], Milevsky and Young [2003], Dus, Maurer and Michell [2005] and Horneff, Maurer, Michell, and Dus [2006] who also provide a nice summary of his lieraure.) Our curren framework imagines all decisions as occurring a ime 0. To assess he poenial benefi from delaying he immediae annuiy purchase, we inroduce differen annuiy producs ha replicae he cos and consumpion paern achieved by delaying he purchase of an annuiy. Before describing he payou and pricing srucure for a delayed purchase annuiy, le us firs consider he assumed pricing for a delayed purchase Arrow annuiy. Suppose 21

22 our reiree decides o inves in a bond for periods and hen purchases an Arrow annuiy ha pays ou in +n periods. This sraegy would ne he reiree $1 in period +n a a price equal o B (A +n / A ). 9 The period delayed purchase annuiy would hen have he following characerisics: Payou srucure = $1 payou in every year +n, given survival o +n, n 0. B Price = A n= 0 A + n If he desired annuiy-funded consumpion paern is increasing in ime, hen hese annuiies provide a more flexible produc space o mee he demand. Relaive o immediae annuiies, delayed purchase annuiies could provide wo addiional benefis. Firs, delayed purchase annuiies can provide a more flexible annuiy-based consumpion paern. Second, hey poenially provide a more efficien way o allocae scarce annuiy dollars. This is in conras o he lieraure where he equiy risk premium generally plays he major role in driving he demand for delays. In ha lieraure, annuiizing asses precludes capuring he equiy risk premium, so generally invesors delay some annuiy purchases unil he annuiy premium provided by moraliy risk sharing equals he equiy risk premium. Our focus is no how he underlying asses are invesed (e.g., bonds or socks), bu raher how he differen annuiy conrac srucures are able o deliver efficien longeviy proecion o reirees unwilling o fully annuiize. 9 This assumes ha an Arrow annuiy purchased a period for payou a period +n carries a price of A +n / A. For example, acuarially fair annuiy pricing and a fla erm srucure would conform o his assumpion. More generally, any saic erm srucure beween period and +n is allowable. 22

23 Annuiy Produc Space #3: Delayed Payou Annuiies Poenially one of he mos imporan innovaions in he annuiy produc space is he delayed payou annuiy. In Sepember 2004, MeLife inroduced heir Reiremen Income Insurance SM produc ha was described in a press release as: an annuiy ha provides guaraneed lifeime income and is designed o generae income saring a a laer age, for example an individual s 85h birhday, when oher income sources may be running low. [MeLife, 2004] This produc is essenially a bundle of Arrow annuiies ha sar a a fuure dae insead of saring immediaely. 10 These producs generae he same payou sream as a delayed purchase annuiy wihou he uiliy loss associaed wih bond invesing prior o period. While Arrow annuiies allow he opion of arbirary allocaions o annuiies over ime, he separaion heorem implies opimal annuiy allocaions are zero unil a cerain poin in ime and hen suppor all consumpion afer ha poin in ime. Given his opimal pah of Arrow annuiy demand, i is feasible ha delayed payou annuiies could provide all of he benefis of Arrow annuiies even hough hey require bundled purchases. In fac, as long as he period-by-period growh in desired annuiy-funded consumpion exceeds he period-by-period growh in he paymens from he delayed payou annuiy, hen delayed payou annuiies provide all he benefis of Arrow annuiies. We demonsrae his resul more formally in Appendix B. 10 In March 2006, The Harford Financial Services Group inroduced a similar delayed payou annuiy produc described as a fixed paymen annuiy which guaranees lifeime income saring a a fuure poin and coninuing for he res of one's life, no maer how long ha migh be. 23

24 This annuiy produc space allows a delayed payou annuiy indexed o each ime period. The h delayed payou annuiy pays ou saring years in he fuure and coninuing unil deah. The period delayed payou annuiy would hen have he following characerisics: Payou srucure = $1 payou in every year +n, given survival o +n, n 0. Price = = n 0 A + n Noice how his produc provides he same payous as he delayed purchase annuiy, bu a a lower cos. Noice also, ha if opimal consumpion based on Arrow annuiies is non-decreasing, hen hese delayed payou annuiies can provide idenical consumpion a idenical coss. I is in his sense ha hey can subsiue for Arrow annuiy availabiliy. Annuiy Produc Space #4: Arrow Annuiies This is he mos flexible annuiy produc space considered. This annuiy produc space conains he annuiy building blocks and hus can replicae or improve on any of he soluions based on he previous annuiy produc spaces. The annuiy produc spaces are lised in order of increasing desirabiliy. Each subsequen annuiy produc space eiher allows more flexibiliy or provides equal flexibiliy a a lower cos compared o he previous annuiy produc space. If AEW i refers o he AEW achieved given annuiy produc space i is available, hen for he above annuiy produc spaces he following holds: AEW i AEW j, i > j 24

25 Appendix C deails he addiional opimizaion consrains required o solve he uiliy maximizaion given access o he various annuiy produc spaces. 5.2 Efficiency Analysis: Baseline, Acuarially Unfair Pricing and Social Securiy To faciliae an efficiency analysis of he various annuiy produc spaces, i is necessary o make assumpions regarding he specific uiliy funcion and discouning parameers relevan o our reiree. The firs assumpion concerns he uiliy funcion, and is common across all of he cases invesigaed. In paricular, we assume he uiliy funcion for our reiree is he following power uiliy funcion: (9) U ( c ) = c 1 γ 1 γ 1 Where gamma (γ) acs as he risk aversion parameer. The second assumpion we will make across all of he scenarios evaluaed is ha he discoun rae for fuure uiliy,, is jus offse by he bond ineres rae, B. In addiion, we assume ha he annuiy discoun and bond ineres rae is 3%. In oher words, we assume: 1 = B = ( 10), for all 1.03 We now urn our aenion o he evaluaion of efficiency losses associaed wih various annuiy produc spaces in specific siuaions. 25

26 Baseline Analysis: As a baseline analysis, we will assume annuiy prices are acuarially fair. Tha is, we will assume ha P = Π. 11 Given his assumpion, we recover he sandard resul ha opimal annuiy-based consumpion is consan over ime. This can be readily seen in Eq (2b') where he marginal uiliy of consumpion is consan over ime. For he bondfunded years, opimal consumpion is decreasing wih ime; decreasing survival probabiliy leads one o purchase less bond-based consumpion in laer periods, even hough he rae of ime preference equals he bond rae of reurn. Evaluaing (7b) wih he baseline assumpions implies ha he bond-funded consumpion ime pah is a decreasing funcion of 1/Π. Given a declining survival funcion, consumpion during he bond-funded years is also declining. For hese assumpions, opimal consumpion decreases while i is funded by bonds and hen says consan once consumpion is funded by annuiies. Only annuiy produc spaces ha can replicae his opimal pah of declining bond-funded consumpion followed by consan annuiy-funded consumpion can poenially provide equivalen benefis o Arrow annuiies. How does our reiree benefi from he inroducion of various annuiy produc spaces? Figure 1 illusraes he AEW generaed from allocaing increasing amouns of annuiy dollars uilizing he various annuiy produc spaces. For his example, he reiree is assumed o be male, age 65 wih a risk aversion parameer (γ) equal o four. Wih hese assumpions, he unconsrained AEW is $154. Since he opimal unconsrained 11 Generally, he GAM-94 moraliy able is assumed o price he annuiies. In conras o SSA ables, GAM-94 caps moraliy a 50% for he very aged. This cap creaes implausible rends in he survival raios for GAM vs. SSA populaions. To eliminae his peculiariy from he analysis, we universally adop he age 100+ moraliy assumpions from he SSA wih improvemens from he SSA Acuarial Sudy #120. Given he exremely low probabiliy of survival o hese ages, his assumpion is immaerial o he resuls. 26

27 consumpion is consan over ime, a reiree willing o inves all wealh in an annuiy can achieve he maximum AEW wih any of he annuiy produc spaces. Figure 1 illusraes mos of he key conceps in our analysis. Firs, since opimal annuiy-based consumpion is consan wih Arrow annuiies, delayed payou annuiies provide equivalen benefis as Arrow annuiies. The abiliy of delayed payou annuiies o provide all or a leas he vas majoriy of benefis achievable via Arrow annuiies, he heoreically ideal annuiy produc se, is a pervasive resul in our analysis. A second sriking feaure of Figure 1 is he inefficiency of immediae annuiies. Delayed payou and Arrow annuiies provide significan moraliy risk sharing benefis from he firs few dollars annuiized. Immediae annuiies, in paricular, and o a lesser exen delayed purchase annuiies require subsanially more wealh allocaed o annuiies o generae similar benefis. Figure 1a explores he efficiency issue in more deail. Figure 1a repors essenially he same daa as Figure 1; however, he y-axis is normalized o range from 0% ($100 AEW) o 100% ($154 AEW). The y-axis now corresponds o he percenage of poenial annuiy benefis delivered. Finally, Figure 1a only graphs he benefis associaed wih he firs 50% of wealh allocaed o annuiies. The efficiency differences beween he annuiy produc spaces are remarkable. Consider he amoun of wealh required o achieve 50% of he oal benefi available from annuiizaion, which in his case is abou $27 in addiional AEW. Arrow and delayed payou annuiies require 6% of wealh allocaed o annuiies o deliver half he benefis of annuiizaion. Wih delayed purchase annuiies, half he maximum benefi requires four imes he allocaion or 24% of wealh. Immediae annuiies are even less efficien. They require our reiree o ie up 27

28 39% of wealh o achieve a similar level of benefis. A reiree who has a beques moive or is simply uncerain abou his fuure liquidiy demands may find i very hard o commi almos 40% of heir wealh o annuiies. Tha same reiree may be willing o allocae 6% o a delayed payou annuiy since i is such a small fracion of exising wealh and i generaes a benefi equivalen o increasing consumpion by 27%. If implici coss such as beques moives, liquidiy concerns, or a limied annuiy invesmen universe are core facors limiing annuiy purchases, he adven of he delayed payou annuiy could significanly impac annuiy desirabiliy and improve reiree welfare. Table 1 repors informaion for scenarios ha differ wih respec o risk aversion and gender. In paricular, daa are repored for he following four scenarios: Scenario #1: Male, 65, γ = 4, Figure 1 and Figure 1a Scenario Scenario #2: Male, 65, γ = 2 Scenario #3: Male, 65, γ = 1 Scenario #4: Female, 65, γ = 4 For each scenario, resuls are repored assuming 5%, 10% and 20% of wealh are allocaed o annuiies. For each level of annuiy allocaion, a number of saisics are repored. Firs, he AEW for he various annuiy produc spaces is shown. Nex, he age a which Arrow payous begin is repored. For Arrow annuiies, all consumpion afer his age is funded by annuiies due o he separaion heorem. The final wo daa elemens repored correspond o he percenage of oal benefi achieved by he Arrow annuiy, and he level of immediae annuiizaion required o mach he Arrow benefi. 28

29 The general paerns repored in Figure 1 and 1a are apparen in he oher scenarios repored in Table 1. Arrow and delayed payou annuiies are much more efficien compared o immediae or delayed purchase annuiies. The maximum AEW benefi declines wih decreasing levels of risk aversion. The maximum AEW assuming log uiliy (γ equal o one) is $134 compared o $154 for γ equal o four. Scenario #4 considers a 65 year-old female wih a γ parameer of four. This scenario highlighs he impac of increasing life expecancy. Compared o a similarly risk-oleran 65 year-old male, he female generaes a lower AEW from access o annuiy markes since lower moraliy implies lower benefis from annuiizaion, ceeris paribus. 12 While he oal benefi is reduced for individuals wih lower moraliy raes, he efficiency provided by Arrow annuiies is increased. In his example, a five percen annuiy allocaion provides 47% of he oal annuiy benefi for he male, bu 50% of he benefi for a female. In addiion, he female would need o allocae 38% of wealh o immediae annuiies o receive he same benefi, bu he male would only need o allocae 36%. Privae Moraliy Analysis The analysis so far has assumed ha he implied moraliy able ha he annuiy provider uses o price he annuiies is he same as he moraliy able used by he individual o discoun fuure uiliy, or in oher words, annuiy prices are acuarially fair. However, i is impossible for an insurance company o offer an acuarially fair annuiy o everyone. Coss associaed wih offering annuiies, capial requiremens, and adverse selecion issues all resul in annuiy prices ha are less favorable compared o acuarially 12 This may seem couner-inuiive: one migh hink ha females have longer lives o insure, and hus should benefi more from he availabiliy of annuiies. One way o undersand how longeviy affecs annuiy demand is o consider an infiniely lived individual. Since survival is guaraneed, hey would be indifferen beween bonds and annuiies. They receive no benefis from annuiizaion. 29

30 fair prices. This secion analyzes he siuaion when he individual has privae informaion abou heir moraliy. In paricular, he individual is assumed o have average life expecancy (represened by he Social Securiy populaion average moraliy ables), bu he annuiy prices are assumed o be based on above-average life expecancy (represened by he Group Annuiy Moraliy or GAM ables). These wo moraliy ables have fairly significan differences. For example, he median life expecancy using he populaion average ables for a 65 year old male is a lile above 17 years. Using he GAM 94 moraliy ables, he median life expecancy increases o jus over 20 years. Figures 1, 1a and Table 1 repor he AEW benefi for access o he various annuiy produc spaces assuming he same moraliy able, GAM 94, is relevan for pricing and uiliy discouning. Figures 2, 2a and Table 2 repor similar informaion assuming he individual discouns using populaion average moraliy insead of GAM 94 moraliy. Comparing Table 1 wih Table 2 illusraes he impac of changing from acuarially fair pricing o privae moraliy inferior o pricing moraliy. Before exploring he changes, firs consider wha has no changed. Annuiies are sill he cheapes mehod for aaining fuure consumpion, hus full annuiizaion is sill he opimal unconsrained soluion. In fac, annuiy prices have no changed a all, so he annuiy prices sill saisfy he requiremens for he separaion heorem. However, several hings have changed. Firs, Eq (2b) ha governs opimal annuiy-funded consumpion is now: (2b') U ( c ) P B = U ( c ) Π P = Π 0. 30

31 P represens he pricing survival probabiliy and Π represens he privae survival probabiliy used o discoun fuure consumpion. Even wih he assumpion ha he bond ineres rae equals he individual discoun rae, opimal annuiy-funded consumpion is no longer consan. The improvemen in annuiy prices is no longer sufficien o offse he discouning associaed wih survival. For he region funded by bonds, he consumpion ime pah is sill a decreasing funcion of 1/Π. For he region funded by annuiies, he ime consumpion ime pah is a decreasing funcion of P / Π. Given he moraliy assumpions, P / Π is always greaer han one and is a decreasing funcion of unil age 100. Afer 100, he moraliy assumpions converge and he survival raios sabilize. The concaviy of U hus implies opimal consumpion is declining prior o age 100 and fla afer age 100. Since opimal annuiy-based consumpion is decreasing given access o Arrow annuiies, none of he annuiy bundles are subsiues for full access o Arrow annuiies even in he unconsrained case. Turning o Figures 2, 2a and Table 2, he surprising resul is jus how lile has changed especially over he region where relaively small amouns of wealh are annuiized. A reducion in life expecancy decreases he demand for fuure consumpion. Since he benefi of annuiies is cheap access o fuure consumpion, he maximum annuiy benefi has been reduced. While he maximum benefi has been reduced, he efficiency of delayed payou and Arrow annuiies is relaively unchanged. The maximum benefi has been reduced from $154 o $149 for he baseline reiree. Figure 2 looks remarkably similar o Figure 1 in erms of he relaive efficiency of he various annuiy produc spaces. As illusraed in Figure 2a, o ge half he benefis available from moraliy risk sharing, a reiree would need o allocae 6% o Arrow or delayed payou 31

32 annuiies, 24% o delayed purchase annuiies, or 36% o immediae annuiies. Recall ha he analogous numbers for acuarially fair annuiies were 6%, 24% and 39%, respecively. Similar o he reiree facing acuarially fair pricing, a reiree wih privae moraliy raes markedly inferior o he pricing moraliy can sill garner a significan improvemen in AEW ($124.5) from a fairly modes amoun spen on annuiies (6% of wealh). Essenially, he benefis from he firs few dollars opimally spen on annuiies are so significan ha even relaively poor pricing does no subsanially impac he desirabiliy of annuiizaion. A second sriking feaure of he privae moraliy analysis is he abiliy of delayed payou annuiies o deliver benefis comparable o Arrow annuiies. Wih hese assumpions, opimal annuiy-funded consumpion declines unil age 100 and hen flaens. However, delayed payou annuiies are no able o deliver declining consumpion. Since delayed payou annuiies can no longer replicae he opimal Arrow soluion, he AEW for delayed payou annuiies will be lower han he Arrow annuiy AEW. While he AEW for delayed payou annuiies is indeed lower, he degree of AEW loss is immaerial as illusraed in Figures 2, 2a and Table 2. To beer undersand his resul, Figure 2b compares he opimal consumpion pahs assuming a 20% allocaion o eiher Arrow or delayed payou annuiies. As expeced, he opimal consumpion pah when Arrow annuiies are available is declining while consumpion is bond-funded, coninues o decline afer annuiies sar funding consumpion and finally levels off afer age 100. Over ages where Arrow annuiies fund consumpion, consumpion declines from $5.81 o $5.03 represening abou a 13.5% reducion in consumpion. The consumpion ime pah wih delayed payou annuiies is 32

33 differen. During he bond-funded region, he payou is virually idenical o he Arrow payou. 13 However, during he annuiy funded region, he soluions diverge. The Arrow annuiy soluion coninues o decline a a fairly similar rae o he bond-funded region. The delayed payou annuiy has a subsanial decline during he firs year fully funded by he annuiy, bu is hen required o be non-decreasing afer ha poin. Examining he relaive AEW for each case; however, clearly makes he poin ha he wo consumpion sreams are very similar in erms of desirabiliy. The AEW using Arrow annuiies is $ while he AEW using delayed payou annuiies is $ For hese assumpions, access o cheap fuure consumpion is much more imporan o delivering a high AEW han he abiliy o mach a given consumpion pah. Social Securiy Analysis Table 1 repors he benefis of annuiizaion assuming all wealh currenly available for invesmen is in eiher bonds or annuiies. However, many individuals hold a significan porion of heir reiremen wealh in he form of Social Securiy benefis. Table 3 invesigaes he impac on he analysis of assuming 50% of wealh is already invesed in an immediae annuiy. 14 Having 50% of wealh pre-allocaed o an immediae annuiy will clearly reduce he benefi of addiional annuiizaion. However, he basic resul ha Arrow and delayed payou annuiies provide a significanly more efficien pah o annuiy benefis sill holds. 13 The delayed payou consumpion is acually a fracion of a penny higher during he bond-funded years. 14 While he fracion of wealh held by pre-reirees in he form of Social Securiy varies by wealh level, he median level is approximaely 50% (Moore and Michell [2000]). Previous research focuses on a 50% allocaion as indicaive of he Social Securiy scenario (see, for example, Michell, Poerba, Warshawsky, and Brown [1999]). 33