Annuity Decisions with Systematic Longevity Risk

Transcription

1 Annuiy Decisions wih Sysemaic Longeviy Risk Ralph Sevens This draf: November, 2009 ABSTRACT In his paper we invesigae he effec of sysemaic longeviy risk, i.e., he risk arising from uncerain fuure survival probabiliies, on he araciveness of differen ypes of annuiies. We consider a life-cycle framework wih expeced uiliy where an individual faces boh invesmen and longeviy risk. In conras o exising lieraure we allow no only for idiosyncraic, bu also for sysemaic longeviy risk. When comparing he expeced lifeime uiliy, condiional on he ype of annuiy which is purchased, we find for a 65-year old male ha (i) sysemaic longeviy risk reduces he araciveness of annuiies, (ii) when an immediae annuiy is purchased, he expeced lifeime uiliy is decreasing in he posponemen period, (iii) when in he fuure purchasing an immediae annuiies, he effec of he evoluion of he survival probabiliies on he opimal fracion of annuiized wealh is large, and (iv) he opimal annuiy o purchase a reiremen is a deferred annuiy which sars o pay afer only a shor deferral period. However, when he purchase of an annuiy wih he opimal deferral period is compared o he purchase of an immediae annuiy a reiremen dae, he uiliy gain is negligibly small. Keywords: Opimal life-cycle porfolio choice, Life annuiies, Asse allocaion, Longeviy risk, Marke price of longeviy risk, Opimal annuiizaion age. JEL Classificaions: C61, D14, D91, G11, G22, G23, J11. I hank Anja De Waegenaere, Joos Driessen, Berrand Melenberg, Theo Nijman, Kim Peijnenburg, Lisanne Sanders, Bas Werker, and seminar paricipans a Nespar and GSS for heir helpful commens and suggesions. Deparmen of Economerics and OR, Tilburg Universiy, CenER for Economic Research and Nespar, PO Box 90153, 5000LE Tilburg, The Neherlands. Phone: Fax: R.S.P.Sevens@TilburgUniversiy.nl.

2 1 Inroducion Our goal in his paper is o invesigae he opimal annuiy decision in a life-cycle model when here is sysemaic longeviy risk. Life expecancy has increased subsanially over he pas decades, and is expeced o increase furher in he fuure. However, here is considerable uncerainy regarding he exac developmen in fuure life expecancy. This uncerainy is called sysemaic longeviy risk. 1 Sysemaic longeviy risk can be modeled by allowing fuure survival probabiliies o be sochasic. In sudies invesigaing opimal annuiy decisions sysemaic longeviy risk is ypically ignored. However, is presence affecs he opimal life-cycle decisions in a number of ways. Firs, sysemaic longeviy risk is a non-diversifiable risk and herefore i will have a nonzero price of risk, complicaing he pricing of annuiies. Second, sochasic fuure survival probabiliies imply sochasic fuure annuiy prices, furher complicaing he opimal life-cycle decisions of he individual. We allow for a nonzero price of risk in he annuiy prices using risk-neural survival probabiliies, following Cairns, Blake, and Dowd (2006). Third, we allow for sochasic fuure survival probabiliies and opimal decisions which depend on he evoluion of hese survival probabiliies. To solve he opimizaion problem wih sochasic fuure survival probabiliies we follow Brand, Goyal, Sana-Clara, and Sroud (2005) and Carroll (2005), and using exensions proposed by Koijen, Nijman, and Werker (2009). In addiion o allowing for sysemaic longeviy risk, 2 we exend he curren lieraure on opimal annuiy decisions by no only invesigaing he possibiliy of invesing in an immediae annuiy bu also in a deferred annuiy. The exising lieraure on opimal annuiizaion in he conex of immediae annuiies bu wihou sysemaic longeviy risk is exensive. The lieraure was iniiaed by he seminal paper by Yaari (1965). Yaari (1965) and ohers (see, for example, Meron, 1983; and Davidoff, Brown, and Diamond, 2005) show ha an individual s opimal invesmen choice is o inves all his wealh in annuiies. This is shown in a sandard Modigliani life-cycle model of savings and consumpion wihou a beques moive, wih 1 Naurally, we also allow for idiosyncraic longeviy risk, which is due o a random individual remaining lifeime, condiional on given survival probabiliies. This is also referred o as non-sysemaic longeviy risk. 2 Cocco and Gomes (2009) also allow for sysemaic longeviy risk in a life-cycle model, bu in heir paper he individual maximizes he expeced lifeime uiliy in a seing wih only a risk-free asse and a longeviy bond, wihou annuiies or equiies as invesmen opporuniies. 2

3 as only invesmen opporuniy a risk-free asse and acuarially fair annuiies. The raionale behind his resul is ha he reurns from annuiies dominae he risk-free reurn, since he capial invesed in annuiies is allocaed only o he survivors. Alhough hese resuls sugges ha reirees will volunarily purchase annuiies, in mos counries very few acually do so (see, among ohers, Friedman and Warshawsky, 1990; Poerba and Wise, 1998; Moore and Michell, 2000; Büler and Teppa, 2007; and Dushi and Webb, 2004b). This annuiy puzzle has generaed a lo of lieraure aimed a solving his puzzle. 3 We show ha sysemaic longeviy risk reduces he araciveness of an immediae annuiy, hereby reducing he opimal level of annuiized wealh when purchasing an annuiy a reiremen dae. Thus, sysemaic longeviy risk seems o be an imporan ingredien in undersanding he annuiy puzzle. This paper also conribues o he lieraure on he opimal iming of he purchase of annuiies. Much research has been devoed o finding he opimal fracion of wealh invesed in annuiies and he bes iming for purchasing annuiies. Due o acuarial unfairness of annuiies posponing he purchase of an annuiy purchase may be raional, because he moraliy credi is oo low jus afer reiremen age. 4 Milevsky (1998) proposed posponing he annuiy purchase unil he moraliy credi is larger han or equal o he equiy risk premium. However, his annuiizaion sraegy would only be opimal for risk-neural individuals. Blake, Cairns, and Dowd (2003) found an opimal annuiizaion age in he range of 65 o 80, depending on individual characerisics such as risk aversion and beques moive. Milevsky and Young (2002) esimaed ha he real opion o annuiize remains valuable unil he age range 75-85, also depending on individual characerisics. Differen assumpions of he uiliy funcion have been made o find he opimal ime o purchase annuiies. These include he HARA uiliy (see Kingson and Thorpe, 2005) and he power uiliy (see Sabile, 2006). Ohers 3 For example, he evidence for he size of an individual s beques moive and he corresponding effec in a life-cycle model is mixed (see, for example, Yaari, 1965; Friedman and Warshawsky, 1990; Bernheim, 1991; Brown and Poerba, 2000; Hurd and Smih, 2001; and Vidal-Meliá and Lejárraga- García, 2006). Oher auhors have examined he sraegic beques moive (see, for example, Bernheim, Shleifer, and Summers, 1985). Anoher possible explanaion for he annuiy puzzle is he defaul risk of he annuiy issuer (see Babbel and Merrill, 2007) or he illiquidiy or irrevocabiliy of annuiies (see Sinclair and Smeers, 2004; and Peijnenburg, Nijman, and Werker, 2009). In addiion, behavioural effecs (see, for example, Hu and Sco, 2007; Brown, 2007; Brown, Kling, Mullainahan, and Wrobel, 2008; and Gazzale and Walker, 2009) may influence he decision o forgo volunary annuiizaion. 4 The moraliy credi is defined as he (yearly) excess reurn of an annuiy relaive o he reurn on he risk-free invesmen. The moraliy credi is formally defined in Secion 4. 3

4 have invesigaed he opimal gradual annuiy purchase paern during reiremen. For example, Kapur and Orszag (1999) and Horneff, Maurer, and Samos (2008) found ha gradual annuiizaion is opimal unil he moraliy credi is larger han he equiy reurn. In our seing we find ha for a 65-year-old individual posponing he annuiy purchase is uiliy-decreasing due o sysemaic longeviy risk. This difference wih he exising lieraure illusraes he imporance of sysemaic longeviy risk in he life-cycle opimizaion problem. This paper also conribues o he lieraure on he araciveness of deferred annuiies. The lieraure on he opimal deferral period of a deferred annuiy which is purchased a reiremen dae is no very exensive. Milevsky (2005) provides a descripion of (inflaion-linked) deferred annuiies, referred o as Advanced-Life Deferred Annuiies (ALDAs). This paper saes ha deferred annuiies, purchased a he reiremen dae, saring o pay afer a deferral period of around 15 o 25 years, are opimal. A deferred annuiies is opimal because such an annuiy provides longeviy insurance a a low price. Hu and Sco (2007) menion ha deferred annuiies may be more desirable for individuals han immediae annuiies, because he former overweigh small probabiliies. Dus, Maurer, and Michell (2005) show ha deferred annuiies can enhance he expeced payou and cu he expeced shorfall risk. In a seing wih only a risk-free asse and given rules of humb for he consumpion level, Gong and Webb (2009) show ha deferred annuiies provide longeviy insurance a a low cos. Horneff and Maurer (2008) find ha a deferred annuiy which sars income paymens a he age of 65 migh become more appealing han an immediae one when he loading facor is high enough. Bayrakar and Young (2009) find ha i is always opimal o purchase an immediae annuiy insead of a deferred one, when an individual s objecive is o minimize he probabiliy of financial ruin. We exend his lieraure in wo ways. Firs, we compare annuiies wih differen deferral periods. Second, we ake ino accoun ha he acuarial unfairness in annuiy prices may be due o a risk premium for sysemaic longeviy risk. We use a risk-neural pricing approach o model he risk premium in annuiies due o sysemaic longeviy risk. This resuls in annuiies wih a risk premium for sysemaic longeviy risk ha is dependen on he deferral period. We find ha he opimal deferral period is shor and ha he uiliy gain from purchasing an annuiy wih he opimal deferral period insead of an immediae annuiy is very small. Moreover, we find ha 4

5 when an individual purchases an annuiy wih a moderae deferral period (around 10 years) he can hold a subsanial amoun of liquid wealh wih a low reducion in he expeced lifeime uiliy. The paper is organized as follows. In Secion 2 we presen he preferences of he represenaive individual and describe he sochasic forecas models we use o forecas he probabiliy disribuion of fuure survival probabiliies. Secion 3 presens he parameer calibraion of he disribuions of he equiy reurns and he disribuion of he fuure survival probabiliies. The araciveness of an annuiy is affeced by he price and he paymen sream. Therefore, in Secion 4 we firs illusrae he effec of sysemaic longeviy risk on boh he price of a deferred annuiy and an immediae annuiy. In addiion, we illusrae he effec of sysemaic longeviy risk on he moraliy credi. In Secion 5 we deermine he opimal choices of an individual in he expeced lifeime uiliy model. We show he effec of differen annuiy choices and he effec of sysemaic longeviy risk on he opimal decisions of he individual. Robusness checks are subsequenly performed in Secion 6. Secion 7 presens he conclusions. 2 Preferences, survival probabiliies, and annuiies This paper invesigaes an individual s opimal fracion of wealh invesed in eiher a deferred annuiy or an immediae annuiy, and he effec of sysemaic longeviy risk on his decision. The opimal annuiy decision is deermined in a seing wih hree sources of risk: i) invesmen risk, caused by a random reurn in he equiy marke; ii) idiosyncraic longeviy risk, due o a random individual remaining lifeime (condiional on given survival probabiliies); iii) sysemaic longeviy risk, due o random fuure survival probabiliies. Secion 2.1 defines an individual s expeced lifeime uiliy funcion and describes he consrains he individual faces. The opimal choices of an individual depend on he probabiliy disribuion of fuure survival probabiliies which is described in Secion 2.2, and on he pricing of annuiies, which is described in Secion

6 2.1 The individual s opimizaion problem In his secion we describe he opimizaion problem including he consrains faced by an individual who maximizes an expeced uiliy of lifeime consumpion. The invesmen choice consiss of he fracions of wealh invesed in a risky asse, a risk-free asse, and in an annuiy. We consider wo ypes of annuiies: i) An immediae annuiy which yields a nominal yearly paymen of 1, wih a final paymen in he year he insured dies; ii) a deferred annuiy which yields a nominal yearly paymen of 1, afer a deferral period of d years when he insured is sill alive, wih a final paymen in he year he insured dies. Le A (d) x, denoe an annuiy wih a deferral period of d years bough by an individual aged x a ime. Noe ha an immediae annuiy represens a special ype of a deferred annuiy, namely one wih a deferral period equal o one (i.e., d = 1). 5 We deermine an individual s lifeime expeced uiliy and opimal choices for wo cases, namely currenly purchasing a deferred annuiy wih a fixed deferral period d and posponing he purchase of an immediae annuiy unil a fixed ime s. 6 We assume ha he individual has an ineremporally separable, expeced lifeime consan relaive risk aversion (CRRA) uiliy funcion, wihou a beques moive. To avoid overloaded noaion, he ime a which we calculae he expeced lifeime uiliy, i.e., he base year, is se equal o zero, unless oherwise menioned. We assume ha he individual invess in an annuiy only once, a a fixed ime s 0, and invess in only one ype of annuiy, i.e., an annuiy wih deferral period d (wih d = 1 for an immediae 5 An annuiy can eiher be an ordinary annuiy or an annuiy-due. The difference beween he wo ypes of annuiies is ha an annuiy income paymens can eiher be a he beginning of a specified period (i.e., an annuiy-due) or a he end of he specified period (i.e., an ordinary annuiy). An annuiy wih d = 1 is an ordinary immediae annuiy and an annuiy wih d = 0 is an immediae annuiy-due. Noe ha using only a deferred annuiy wih a deferral period of zero years, one can obain he same paymen sream as wih a deferred annuiy wih a deferral period of one year. This occurs because he difference beween he wo annuiies is only a cerain immediae paymen. Under arbirage-free pricing he price of an annuiy wih an iniial paymen in he following year and an annuiy wih an immediae iniial paymen equals he level of he curren paymen. In his paper, when we refer o immediae annuiies, we mean an ordinary annuiy wih an iniial paymen in he following year, i.e., d = 1. 6 Noe ha one can also invesigae he effec of posponing he purchase of deferred annuiies. However, as we will argue in Secion 7, his will probably no be opimal due o he sysemaic longeviy risk. 6

7 annuiy). To invesigae he effec of he differen ypes of annuiies and he effec of posponing he annuiy purchase we calculae he opimal consumpion, invesmen, and annuiy choices condiional on he annuiy ype (i.e., condiional on d), and he ime when an annuiy is purchased (i.e., condiional on s). We obain he opimal annuiy choice by comparing he corresponding expeced lifeime uiliies. Le τ p x, be he probabiliy ha an x-year old a ime will survive τ years; le γ denoe he coefficien of relaive risk aversion; le β be he ime preference parameer (also referred o as he subjecive discoun facor); le W be he liquid wealh level in period ; le C be he consumpion level in period ; and le A be he annuiy income in year. An individual is characerized by his ime- age x, wealh level before annuiy income and consumpion, W, and he ime- sae variables corresponding o he annuiy income, A, and B. Now we consider a given ime s a which annuiies wih a deferral period of d years are bough, and deermine he opimal invesmen and consumpion choices. A ime he endogenous sae variables are W, A, and B and he exogenous sae variables a ime are denoed by he vecor X. 7 Le (C, w ) be he se of conrol variables in year, i.e., he ime- level of consumpion and he fracion of wealh afer annuiy income and consumpion invesed in equiy, respecively, and le a s (d) be an addiional conrol variable a ime s, i.e., he fracion of afer-consumpion wealh which in year s is invesed in an annuiy wih a deferral period of d years. The ime- expeced lifeime uiliy J of an individual is defined by: { J (x, W, A, B, X ) = max a s(d),{w τ,c τ } { τ = max {w τ,c τ } τ E [ τ 0 τp x, β τ (C +τ) 1 γ E [ τ 0 τp x, β τ (C +τ) 1 γ 1 γ ]}, if s ]}, if > s. A or before ime s, he individual maximizes his expeced lifeime uiliy wih he fracion of liquid wealh invesed in equiy, he consumpion, and he fracion of wealh invesed in annuiies a ime s as conrol variables. Afer ime s he individual does no purchase new annuiies, and hence he conrol variables are only he sequence of curren and fuure fracions of liquid wealh invesed in equiies, and he yearly consumpion 7 The exogenous sae variables depend on he evoluion of he fuure survival probabiliies up o ime. The evoluion of he survival probabiliies is described in Secion 2.2. The evoluion of an individual s informaion abou he disribuion of he fuure survival probabiliies is described in Appendix A. 1 γ (1) 7

8 levels. The wealh dynamics of he individual, for all τ 0, are given by: (W τ C τ ) (1 a s (d)) (1 + r rf + w τ (r τ r rf)), if τ = s, W τ+1 = (W τ + A τ C τ ) (1 + r rf + w τ (r τ r rf)), if τ s, (2) where r rf is he ime-independen risk-free reurn, and r τ is he (risky) reurn on equiy beween year τ and τ +1. The firs equaion corresponds o he wealh dynamics in he year in which he individual purchases an annuiy and he second corresponds o he wealh dynamics in he years in which he individual does no purchases an annuiy. The individual faces a sequence of shor-selling consrains and liquidiy consrains. These consrains imply ha an individual canno borrow agains fuure income. Hence, he objecive funcion for he individual, as represened in (1), is maximized subjec o he wealh dynamics in (2) and he following consrains: 0 a s (d) 1, (3) 0 w τ 1, for τ 0, (4) C τ W τ + A τ, for τ 0. (5) Equaions (3) (4) correspond o he no shor-selling consrains, and equaion (5) implies ha he individual canno borrow agains fuure income. Given ha he individual purchases an annuiy a ime s wih a deferral period of d years, he annuiy income level is, by definiion, given by: A, if s + d 1, A +1 = B, if = s + d 1, d > 1, a s(1) (W s C ) s), if = s, d = 1, V s ( A (1) x+s,s ( ) wih A 0 = 0, V s A (1) x+s,s he ime-s price of an immediae annuiy, and (6) B +1 = B, if s, a s(d) (W ( s C s) V s A (d) x+s,s ), if = s, 8

9 wih B 0 = 0, and V s (A (d) x+s,s) he ime-s price of an annuiy wih a deferral period of d years. The sae variable B does no play a role when d = 1. To obain he opimal consumpion and invesmen choices we use a simulaionbased mehod which can deal wih many exogenous sae variables, proposed by Brand, Goyal, Sana-Clara, and Sroud (2005) and by Carroll (2006). In addiion, we include several exensions which were proposed by Koijen, Nijman, and Werker (2009). In Appendix A he mehod used o obain he opimal consumpion and invesmen choices is described. 2.2 Survival probabiliies As can be observed from equaion (1) he opimal life-cycle choices of an individual depend on fuure survival probabiliies of he individual. In his paper fuure survival probabiliies are sochasic. In his secion we describe he modeling of he probabiliy disribuion of fuure survival probabiliies. We use he model proposed in Cairns, Blake, and Dowd (2006). This CBD model is aracive because i uses only a few parameers o obain a good fi of he moraliy probabiliies and i has already been exended o include a marke price of sysemaic longeviy risk in an empirically jusified mehod. In he CBD model he moraliy curve is a special case of he Perks models (see, for example, Perks, 1932, and Benjamin and Pollard, 1993). Le q x, be he ime- one-year moraliy probabiliy for he cohor aged x a ime. In he CBD model he logi of he one-year moraliy probabiliy is modeled as: ( ) qx, log = k (1) + x k (2) + ǫ x,, (7) 1 q x, where k (i) = [ k (i), k (i) +1,... ] for i {1, 2} are sochasic processes wih being he firs year of moraliy daa, and ǫ x, an he age- and ime-specific idiosyncraic residual assumed o be independen and idenically disribued (i.i.d.) normally disribued wih zero mean and age-specific variance. We esimae he process for moraliy probabiliies using moraliy daa. Le be he las year of he moraliy daa. Then, in order o projec he fuure moraliy probabiliies he individual needs he fuure values of he sochasic processes k = 9

10 [ ], k (1) k (2) for >. Following Cairns, Blake, and Dowd (2006) we assume ha he individual forecass hese sochasic processes by a wo-dimensional random walk wih drif: k +1 = k + µ + C N, (8) where µ is a consan 2 1 vecor, C is a consan 2 2 upper riangular marix, and N is a wo-dimensional sandard Gaussian process. For longeviy risk i is common o include no only process risk, i.e., he risk arising from he random process N, bu also parameer risk (see, for example, Cairns, Blake, and Dowd, 2006). There is a consensus in he lieraure ha he exclusion of parameer uncerainy, given a specificaion like (7) (8), would lead o a significan underesimaion of longeviy uncerainy. 8 Le D = k k 1, and D = [D +1,...,D +n ] wih n =. To incorporae parameer risk in he parameers µ and V = C C we use he Jeffrey s prior as in he CBD model, a noninformaive prior disribuion, which is a common prior for he mulivariae Gaussian disribuion in which boh µ and V are unknown: p (µ, V ) V 3/2, where V is he deerminan of he covariance marix V. The poserior disribuion for (µ, V D) a ime τ saisfies: ( V 1 D Wishar τ 1, τ 1 V 1 τ ), (9) µ V, D MV N ( µ τ, τ 1 V ), (10) where µ τ and V τ are he maximum likelihood esimaes of he parameers of he sochasic process based on he revealed informaion up o ime τ. A any ime τ > hese 8 Likewise, he invesor migh allow for parameer uncerainy in he equiy process. Since our focus in his paper is on he effec of sysemaic longeviy risk, we assume ha he invesor only accouns for process risk in he equiy process. 10

11 parameers are esimaed by: µ τ = 1 τ V τ = 1 τ τ =+1 τ =+1 D, (11) (D µ τ ) (D µ τ ). (12) The one-year survival probabiliies are given by: p x, = 1 q x,, where p x, denoes he probabiliy ha an x-year-old a ime- will survive a leas anoher year. Le τ p x, p x, p x+1,+1 p x+τ 1,+τ 1 denoe he ime- probabiliy ha an x-year-old a dae- will survive a leas anoher τ years. 2.3 Annuiy prices The price of an annuiy depends on he probabiliy disribuion of an individual s remaining lifeime. The acuarially fair price (i.e., he expeced discouned cash flows) of an annuiy can be deermined using he probabiliy disribuion of he sochasic fuure survival probabiliies which is described in Secion 2.2. The acuarially fair value of an annuiy does no incorporae he effec of sysemaic longeviy risk on he price he annuiy. Idiosyncraic longeviy risk is diversifiable (i.e., he risk becomes negligible when he he porfolio size is sufficienly large) and hus will no be priced in an efficien marke wihou arbirage opporuniies. In conras, sysemaic longeviy risk does no decrease wih he porfolio size, and may hus lead o a risk premium. There is srong empirical evidence ha he marke price of annuiies exceeds he acuarially fair one (see Michell, Poerba, Warshawsky, and Brown (1999) for he US marke and Frinkelsein and Poerba (2002) for he UK marke). To accoun for his, he exising life-cycle model lieraure commonly uses a loading facor. Commonly, he loading facor is seen as a ransacion cos, (see, among ohers, Michell e al., 1999). However, he acuarial unfairness migh be due o he price of sysemaic longeviy risk raher han ransacion coss, which is also menioned in Milevsky and Young (2007). The premium for sysemaic longeviy risk in an annuiy migh be significan 11

12 which implies ha he acuarial fair value migh be a significan underesimaion of he marke price. 9 More imporanly, a loading facor independen of he deferral period of an annuiy does no necessarily properly reflec he risk premium for sysemaic longeviy risk. No liquid marke exiss ye for sysemaic longeviy risk (see Blake, Cairns, and Dowd, 2008). Therefore, i is difficul o calibrae he marke price of sysemaic longeviy risk. The exising lieraure proposes differen approaches o obain a fair value for annuiies when sysemaic longeviy risk exiss. These approaches include he uiliy maximizaion principle (see Malamund, Trubowiz, and Wührich, 2008); he Sharpe raio approach (see, for example, Milevsky, Promislow, and Young, 2006, 2008; Bayrakar, Milevsky, Promislow, and Young, 2009; and Bauer, Börger, and Ruß, 2009); he Wang ransform (see Lin and Cox, 2005; Cox and Lin, 2007; Denui, Devolder, and Goderniaux, 2007; and Lin and Cox, 2008), and risk-neural pricing (see Cairns, Blake, and Dowd, 2006). An excellen overview of differen pricing mehods is given in Bauer, Börger, and Ruß (2009). In his paper we use he risk-neural approach o calculae he risk premium for sysemaic longeviy risk. The risk-neural approach is based on long-esablished financial economic heory and saes ha, if he overall marke is arbirage-free, here exiss a risk-neural measure such ha he price of an annuiy equals he expeced discouned paymens under he risk-neural measure. Due o marke incompleeness many risk-neural risk measures migh exis. Therefore, we shall assume ha an individual is acing in an equilibrium seing, and ha his equilibrium selecs a marke consisen (unique) risk neural measure. Following he exising lieraure on life-cycle models we also use, as alernaive o he risk-neural approach, a loading facor for pricing annuiies. As loading facor we ake 7.3%, which is in line wih Michell e al. (1999), and commonly used in he life-cycle lieraure (see, for example, Horneff, Maurer, and Samos, 2008). Hence, his paper considers annuiy marke prices ha exceed he acuarially fair ones which are modeled using: 9 For example, he Solvency II projec (Group Consulaif Acuariel Europeen, 2008) requires he valuaion of annuiies using a marke-o-model approach. The approach proposed in he Solvency II guidelines leads o a valuaion of an annuiy which is approximaely 6% o 9% higher han he real-world expeced discouned cash flows of an annuiy for an annuian aged 65, due o sysemaic longeviy risk (see Olivieri and Piacco, 2008). 12

13 i) risk-neural survival probabiliies; ii) a loading facor, which is independen of he deferral period of an annuiy. Especially for deferred annuiies he mehod used o price annuiies is imporan. Using our risk-neural survival probabiliies he acuarial unfairness (measured by he fracion of he price of an annuiy due o he risk margin for sysemaic longeviy risk) in deferred annuiies is an increasing funcion of he deferral period. In conras, using a loading facor he acuarial unfairness is independen of he deferral period. To illusrae ha he resuls are robus for hese differen ypes of pricing annuiies, we will use boh mehods (i.e., he risk-neural pricing mehod and he loading facor) o price annuiies. Le us now describe he risk-neural mehod o obain he price of an annuiy. Recall ha τ p x, is he probabiliy ha an x-year old a ime will survive a leas τ years, and r rf is he risk-free reurn. Le E Q [ ] denoe he ime- expecaion under he risk-neural measure. Then he ime- marke price of an annuiy V (A (d) x,) is given by: 10 where 1 (x,) τ V (A (d) x,) =E Q [ τ d ( ) ] τ 1 1 (x,) τ = ( ) τ 1 E Q 1 + r rf [ τ p x, ], (13) 1 + r rf τ d is an indicaor which equals one if he individual wih age x a ime is alive a ime + τ, and zero oherwise. As can be observed from equaion (13), o calculae he price of an annuiy using he risk-neural approach we need he risk-adjused expecaion of fuure survival probabiliies. The probabiliy disribuion of fuure survival probabiliies is described in Secion 2.2. To include he marke price of sysemaic longeviy risk we follow he mehod proposed in Cairns, Blake, and Dowd (2006). In his mehod he risk-adjused pricing measure Q(λ) is modeled using an adjusmen in he dynamics of he sochasic process k. Le λ = [λ 1 λ 2 ] be he vecor represening he marke price of sysemaic longeviy risk, which is assumed o be ime-independen. The dynamics of he process k under he real-world measure is described in equaion (8). The process k under he 10 In case of uncerainy in he insananeous forward rae curve i has been shown ha, under he assumpion of independence ( ) of he insananeous forward rae and survival probabiliies, equaion (13) τ 1 sill holds, replacing 1+r on he righ hand by he ime- price of a zero-coupon bond wih face rf value 1 mauring in year + τ (see, for example, Cairns, Blake, and Dowd, 2006). 13

14 risk-adjused measure Q(λ) is given by: k+1 = k + µ + C (N λ), = k + µ + C N, (14) where µ = µ C λ. Whereas he individual updaes he parameers µ and V coninuously, we assume ha he parameer λ is no updaed over ime. 3 Parameer calibraion 3.1 Financial marke In his secion we describe he financial marke reurn processes. We assume ha, besides differen ypes of annuiies, he financial marke consiss of a risk-free asse, and a risky asse. The yearly reurn on he risk-free asse is se a 4% and assumed o be ime-independen. Hence, we have ha r rf = Define S as he ime- sock price, assuming ha here are no dividends. 11 Then r S +1 S 1 is he yearly equiy reurn beween year and + 1. The sock price is modeled as a Brownian moion wih drif: ds = µ S S d + σ S S dz, where µ S r rf + λ S σ S and σ S are model parameers, wih λ S he parameer for he marke price of equiy risk, and Z is a sandard Brownian moion. Following he lifecycle lieraure we se λ S equal o and σ S equal o 0.158, resuling in an expeced yearly excess equiy reurn equal o 4% and a sandard deviaion of he yearly equiy reurn equal o 17% (see, for example, Gomes and Michaelides, 2005; Yao and Zang, 2005; and Cocco, Gomes, and Meanhou, 2005). The equiy risk premium of 4% is lower han he hisorical one, which is very common in his lieraure. The lower-hanhisorical reurn is an adjusmen in order o ake ransacion coss ino accoun, mos of which are in he form of muual fund fees. Due o he high dimensionaliy of modeling he ransacion coss explicily (as is, for example, done in Heaon and Lucas, 1996) i 11 Noe ha when here are dividends, he reurn dynamics are no affeced when he dividends are reinvesed in equiies. 14

15 is common o use his shorcu represenaion of a lower expeced reurn o ake ino accoun ransacion coss. In Secion 6 as a robusness check we se λ S equal o resuling in an expeced yearly equiy excess reurn equal o 7%. This leads o an equiy risk premium which is in line wih hisorical resuls (see, for example, Brennan and Xia, 2002). 3.2 Sysemaic longeviy risk To esimae he parameer values of he sochasic processes k (1) and k (2) we use age-, gender-, and ime-specific moraliy probabiliies for he Unied Saes, obained from he Human Moraliy Daabase. 12 Figure 1 displays he parameer values of he sochasic process for males in he US from = 1970 o = 2006, by fiing (7). The value of k (1) displays he ime- general level of moraliy. The general decrease over ime in he level of he sochasic process k (1) implies ha here is generally a decrease in he level of he moraliy probabiliies over ime. The general increase over ime in he value of k (2) implies ha he moraliy reducion is generally lower a higher ages. Figure 1: Esimaed parameer values of he sochasic processes k. k (1) (year) k (2) (year) This figure displays he esimaed parameer values of he sochasic processes k (1) and k (2). The lef panel displays he esimaed parameer values of he sochasic processes k (1), he righ panel displays he esimaed parameer values of he sochasic processes k (2). The sochasic processes are esimaed using US male moraliy daa from 1970 o Using US male moraliy probabiliies from 1970 o 2006 we obain he esimaes 12 Available from he Human Moraliy Daabase: 15

16 of he parameers in equaions (11) and (12): µ = V =Ĉ Ĉ = , where Ĉ can be recovered from a Choleski decomposiion of V. Recall ha he base year is se equal o 0. To forecas he disribuion of he fuure moraliy probabiliies we ake as he saring value of k 0 he esimae k corresponding o = 2006, i.e., k 0 = k = [ ]. The moraliy probabiliies are forecased by simulaing he parameers µ and V for each pah. Le MA be he maximum aainable age, which is se a 110 years. Then, given he simulaed parameers µ and V, we simulae he pahs of k for each fuure ime period, i.e., for = 1,...,MA x, given he assumed values of k 0. The disribuion of fuure moraliy probabiliies is obained by he simulaed values of he sochasic processes k and he simulaed residuals {ǫ x, x {x,...,ma}, {1,..., MA x}}. Each such pah gives us a pah of he fuure moraliy probabiliies. 3.3 Pricing sysemaic longeviy risk In Secion 2.3 we menioned ha he marke price of an annuiy exceeds he acuarially fair one, which migh be due o sysemaic longeviy risk. Since here is no liquid marke for sysemaic longeviy risk i is difficul o calibrae he risk-neural survival probabiliies using empirical daa. Alhough lile informaion is available, i is reasonable o assume ha sysemaic longeviy risk leads o a posiive risk premium. 13 This implies ha he physical expecaion of he discouned cash flows is lower han he risk-adjused expecaion. The risk adjused process for survival probabiliies, as defined in equaion (14), 13 There exiss also naural counerparies of sysemaic longeviy risk in annuiies, for example deah benefi providers. However, i is sill likely ha here is a posiive risk premium, because shor (for example annuiies) exposure is larger han long (for example deah benefis) exposure (source: American Council of Life Insurers, U.S. Life Insurance Moodys Saisical Handbook, Augus 2006; Pension Markes in Focus, OECD, Ocober 2006; and Moody s U.K. Life Insurance Indusry Oulook, January 2007). 16

17 depends on he parameer λ. Cairns, Blake, and Dowd (2006) calibraed as parameer value λ = [ ], using he EIB/BNP longeviy bond, which was announced in November We will use his calibraed value of λ for pricing annuiies. 15 Using he calibraed parameer λ = [ ] in he risk-neural pricing we obain a marke price for an immediae annuiy which is 7.35% higher han he acuarially fair one. Using he empirical prices of immediae annuiies Michell e al. (1999) found a loading facor of 7.3%. 16 This migh imply ha using only he risk-adjused process Q(λ) wih λ = [ ], one could explain he price observed in he real-world by only using he risk-neural pricing mehod for sysemaic longeviy risk. One migh argue ha he price of sysemaic longeviy risk using he calibraed parameer from Cairns, Blake, and Dowd (2006) (i.e., λ = [ ] ) migh be an overesimaion of he real one, since he longeviy bond was wihdrawn prior o issue. As an alernaive, we assume ha he individual posulaes a uniform disribuion on λ = [λ 1 λ 2 ], i.e., λ 1 = λ 2 U (0, 0.175). Noe ha his sochasic λ leads o a lower risk premium for sysemaic longeviy risk han he calibraed λ = [ ]. In summary, for he pricing of annuiies we disinguish hree cases: i) he calibraed parameer on he EIB/BNP longeviy bond, i.e., λ = [ ] ; ii) a sochasic λ, wih λ 1 = λ 2 U (0, 0.175); iii) a loading facor, which is se equal o 7.3%, irrespecive of he deferral period. The las case is included in order o compare our resuls wih he exising lieraure on life-cycle models. As we will show in he following secion, for deferred annuiies he risk-neural pricing mehod implies ha he risk premium as fracion of he acuarially 14 The EIB/BNP longeviy bond was wihdrawn prior o issue. One of he reasons why his issue was unsuccessful migh be he price of he longeviy bond indicaing ha his calibraed λ overesimaes he price of sysemaic longeviy risk. However, here are several design issues (see Blake, Cairns, and Dowd (2008) for an exensive invesigaion of he failure of his longeviy bond) which migh explain why he bond was wihdrawn prior issue. 15 The longeviy bond was based on publicly available Office for Naional Saisics (ONS) daa on English and Welsh moraliy for a cohor of males aged 65 in Tuljapurkar, Nan, and Boe (2000), among ohers, have shown ha he moraliy developmen in wesern counries has similar paerns. This indicaes ha ha he driving forces for he decline in moraliy may be he same in wesern counries, which implies ha he price of longeviy risk would be similar for wesern counries. Hence, i migh indicae ha he marke price of risk (λ) is approximaely he same in wesern counries. 16 This holds for immediae annuiies. Since we do no have informaion on he loading facor for deferred annuiies we do no know wheher his also holds for deferred annuiies. 17

18 fair price of an annuiy premium is increasing in he deferral period. By deermining he opimal choices in he life-cycle model in a seing where annuiies are priced using a consan loading facor we show ha our resuls also hold when he acuarial unfairness in annuiies is independen of he deferral period of a deferred annuiy. 4 The effec of longeviy risk on annuiy prices This paper exends he exising lieraure on life-cycle models in wo ways, namely by including sysemaic longeviy risk and by allowing an individual o purchase a deferred annuiy. Clearly, he opimal annuiy decision also depends on he curren and fuure prices of he differen annuiies. Therefore, before invesigaing he opimal annuiy decision in a uiliy framework seing in Secion 5, we invesigae in his secion he effec of sysemaic longeviy risk on he price of immediae and deferred annuiies. Sysemaic longeviy risk affecs he price of annuiies in wo ways. Firs, sysemaic longeviy risk leads o a risk premium in he price of annuiies. Second, i leads o uncerain fuure survival probabiliies and hus o sochasic fuure annuiy prices. We deermine he curren price of deferred annuiies and he probabiliy disribuion of he price of immediae annuiies purchased a ime s > 0, using he model o forecas he disribuion of fuure survival probabiliies. For illusraive purposes we consider a 65-year-old male a ime = 0 and se λ = [ ] (see Secion 3.3). Figure 2 displays he effec of sysemaic longeviy risk on he curren price of deferred annuiies, as a funcion of he deferral period. The lef panel displays he price as a funcion of he deferral period (solid curve, he expecaion under he Q-measure), he acuarially fair price (dashed curve, he expecaion under he P-measure), and he risk premium for he sysemaic longeviy risk of he annuiy (dashed-doed curve, he difference in he expecaion under he Q- and he P-measure). The righ panel displays he fracion of he price of he annuiy which is due o he risk premium, i.e., he risk premium as fracion of he price, as a funcion of he deferral period. 18

19 Figure 2: Price of deferred annuiies. Price Risk margin (% of marke price) Age a firs paymen (65+d) Age a firs paymen (65+d) The lef panel of his figure displays componens of he price of a deferred annuiy as a funcion of he deferral period. The solid curve corresponds o he marke price of a deferred annuiy; he dashed curve o he acuarially fair price, i.e., he expeced discouned cash flows; and he dashed-doed curve o he risk premium for sysemaic longeviy risk. The righ panel of his figure displays he risk premium for sysemaic longeviy risk as a percenage of he price of a deferred annuiy. From Figure 2 we observe ha, as expeced, boh he price of an annuiy and he risk premium are decreasing funcions of he deferral period. This occurs because a longer deferral period reduces he (expeced) number of paymens o be made. The decline in he risk premium for sysemaic longeviy risk is small for shor deferral periods, and large for long deferral periods. 17 This occurs because sysemaic longeviy risk for paymens o be made for shor duraions is much smaller han i is for long duraions. Finally, we observe ha he risk premium as fracion of he price of an annuiy is an increasing funcion of he deferral period. The uncerainy in he survival 17 The risk premium for a deferred annuiy wih a long deferral period is high. This occurs because here is a large amoun of uncerainy in he probabiliy of surviving unil advanaged ages condiional on being alive a he age of 65. In addiion, he risk premium is high due o he skewness of he disribuion of he probabiliy of surviving. 19

20 probabiliies is greaer for survival probabiliies farher in he fuure, which leads o (relaively) higher risk premiums for paymens which have a longer mauriy. Insead of purchasing a deferred annuiy he individual can also pospone he purchase of an immediae annuiy. The income sream of a deferred annuiy wih deferral period d can be mimicked by he following sraegy: when he individual is alive a ime d 1 he purchases an immediae annuiy (wih d = 1), and when he individual is no alive a ime d 1 he does no purchase any annuiies. Alhough he income sream of a deferred annuiy can be mimicked using an immediae annuiy, when currenly purchasing a deferred annuiy he price is known, whereas he fuure price for an immediae annuiy is currenly sochasic. As new moraliy informaion becomes available, i can be consuled before seing he annuiy prices. Therefore, when he annuiy purchase is posponed here is generally less uncerainy abou he developmen of fuure survival probabiliies. This reduces he risk premium for longeviy risk for an annuiy, hus making i more aracive. However, a disadvanage of posponing he purchase of he annuiy is ha here is currenly uncerainy abou wha he fuure price of an annuiy will be. Figure 3 displays he median (dashed curve) and 95% confidence inervals (dasheddoed curves) of he discouned price of an immediae annuiy a dae s, i.e., V s (A (1) ( ) s,65+s) 1 s, 1+r as a funcion of he posponemen period s. For comparison, he curren price rf of a deferred annuiy wih a nominal yearly paymen of one, as a funcion of he deferral period (solid curve) is also displayed in Figure 3. 20

21 Figure 3: Presen value of annuiy prices. 15 Presen value of he marke price of an annuiy Age a firs paymen (65+s+d) This figure displays seleced quaniles of he presen value of he dae-s price of an immediae annuiy and he price of a deferred annuiy as a funcion of he age a which he iniial paymen is made. The solid curve corresponds o he curren marke price of a deferred annuiy; he dashed curve corresponds o he presen value of he median marke price of an immediae annuiy purchased a ime s; and he dashed-doed curves correspond o he presen value of he 95% confidence bounds of he marke price of an immediae annuiy purchased a ime s. From Figure 3 we observe ha i is generally cheaper o currenly purchase a deferred annuiy han o pospone he purchase of an immediae annuiy. Currenly purchasing a deferred annuiy insead of posponing he purchase of an immediae annuiy has he advanage ha some of he buyers will no survive unil he payoff phase. This can be observed from he following relaion beween he deferred annuiy price and immediae annuiy price: V ( A (d) x, ) = E Q [ ( ) d 1 1 ( ) (d 1p 1 + r rf x, V +d 1 A (1) x+d 1,+d 1 + (1 d 1 p x, ) 0) ]. The effec ha par of he buyers of a deferred annuiy are no alive a ime d 1 21

22 dominaes he effec of a risk premium for sysemaic longeviy risk which is generally lower when he momen of purchase is posponed. Noe ha, even for an individual wihou any beques moive, i migh sill improve uiliy o pospone he purchase of an annuiy insead of currenly purchasing a deferred annuiy, since he money invesed in deferred annuiies canno be invesed in equiies, which reduces he capial gains from he equiy risk premium. Le us finally discuss he effec of sysemaic longeviy risk on he araciveness of annuiies. The araciveness of annuiies as an invesmen opporuniy is due o pooling: i.e., he individuals who live longer han expeced are subsidized by hose who do no. This reallocaion of he conribuions of hose who die o hose who survive, is referred o as he moraliy credi advanage, see, for example, Milevsky (1998), Milevsky and Young (2002), and Horneff, Maurer, and Samos (2008). The moraliy credi (MC) is defined as he reurn from currenly purchasing an annuiy and selling i (a marke price) he following year in excess of he risk-free reurn. This is equivalen o he excess reurn of purchasing an annuiy insead of posponing is purchase o he succeeding year. In a seing wihou sysemaic longeviy risk he moraliy credi is defined as: ( V +1 MC (, d, x) = A (max{0,d 1}) +1,x+1 V (A (d),x) ) + 1 d=0 ( 1 + r rf) (15) = 1 + rrf 1p x, ( 1 + r rf), (16) where 1 d=0 is an indicaor funcion which equals one if d = 0, and zero oherwise, and 1p x, is he deerminisic one-year survival probabiliy. Because 1 p x, is beween zero and one, he moraliy credi is always posiive. This occurs due o he risk-sharing principle, i.e., in he following year he annuiized wealh is re-allocaed o he survivors. In a seing wih sysemaic longeviy risk he moraliy credi from (15) equals: MC (, d, x) = s max{d 1,0} EQ +1 [ sp x+1,+1 ] (1 + r rf) s + 1 d=0 s d EQ [ s p x, ] (1 + r rf ) s ( 1 + r rf). (17) Compared o he moraliy credi in a seing wih deerminisic moraliy probabiliies, he moraliy credi in a seing wih sochasic moraliy probabiliies differs in wo 22

23 ways. Firs, from equaion (17) we observe ha he moraliy credi is sochasic insead ( ) of deerminisic, because V +1 A (max{0,d 1}) depends on he evoluion of he moraliy +1,x+1 probabiliies unil ime + 1. Second, he moraliy credi is dependen on he deferral period d in a seing wih sysemaic longeviy risk, whereas i is independen of he deferral period in a seing wihou sysemaic longeviy risk. This is due o he fac ha he price of an annuiy in he following year depends on he change in he disribuion of fuure survival probabiliies due o revealed moraliy informaion beween ime and ime + 1. This change may be differen for differen ages, resuling in various changes o he risk-adjused expeced discouned cash flows of he differen annuiy paymens. Figure 4 displays seleced quaniles of he disribuion of he moraliy credi as a funcion of he age of he individual, for immediae annuiies (i.e., d = 1). Figure 4: Moraliy credi for immediae annuiies MC (%) Age This figure displays seleced quaniles of he disribuion of he moraliy credi for immediae annuiies under he real-world measure, as a funcion of he age of he individual. The solid curve corresponds o he median moraliy credi for immediae annuiies under he real-world measure and he dashed curves correspond o he 95% confidence bounds of he moraliy credi for immediae annuiies under he real-world measure. From Figure 4 we observe ha in a seing wih sysemaic longeviy risk, in conras 23

24 o a seing wihou sysemaic longeviy risk, he moraliy credi can be negaive. A negaive moraliy credi implies ha i is cheaper for he individual o currenly inves in he risk-free asse and purchase an immediae annuiy in he following year han o currenly purchase an immediae annuiy. This can occur due o sysemaic longeviy risk, which migh lead o a change in he disribuion of fuure survival probabiliies when new moraliy informaion is revealed. Noe ha from age 72, he effec of he posiive moraliy probabiliy is generally larger han he effec of he new informaion on moraliy probabiliies, leading o a posiive moraliy credi wih a probabiliy of more han 97.5%. Moreover, recall ha he expeced excess reurn of equiy is se a 4%, which is lower han he median of he moraliy credi for he ages above 80. As discussed in Milevsky and Young (2002) in a seing wihou sysemaic longeviy risk, when he moraliy credi is higher han he equiy risk premium i is opimal for an individual o annuiize all his wealh, because annuiies yields a higher expeced reurn wih less uncerainy. 5 Opimal life-cycle choices In his secion we quanify he effec of he choice of he deferral period and he ime a which an annuiy is purchased on an individual s expeced lifeime uiliy. Boh he price and he payoff sream of an annuiy influence he expeced lifeime uiliy. Recall from Secion 2.1 ha he individual is a raional expeced lifeime uiliy maximizer wih a CRRA uiliy funcion. The individual s preference parameers of he CRRA uiliy are se equal o values used in he life-cycle lieraure (see, for example, Gomes and Michaelides, 2005): γ = 5 and β = The individual is a male currenly aged 65, who faces longeviy risk and, when he invess in equiies he faces invesmen risk. We assume ha he individual invess only once in an annuiy and only in one ype, i.e., eiher an immediae annuiy (d = 1, wih s 0), or a deferred annuiy wih a fixed deferral period d (wih s = 0). We quanify he araciveness of he differen ypes of annuiies by he cerainy equivalen consumpion. The cerainy equivalen consumpion is he yearly consumpion level CEC for which he uiliy of his consumpion sream equals he uiliy given he opimal choices condiional on he ype of annuiy, J 0 (x, W 0, 0, 0, X 0 ). Hence, he 24

25 cerainy equivalen consumpion is deermined by he following equaion: E 0 [ τ 0 ] τp x,0 β τ (CEC)1 γ = J 1 γ 0 (x, W 0, 0, 0, X 0 ), (18) wih J 0 (x, W 0, 0, 0) as defined in (1). In our resuls we compare he uiliy (quanified by he cerainy equivalen consumpion) obained by he opimal consumpion and invesmen choices o he uiliy of he consan consumpion level ha arises from currenly invesing all afer-consumpion wealh in annuiies. We refer o his invesmen sraegy as he fully annuiized (fa) sraegy. The corresponding cerainy equivalen ( )) consumpion in his sraegy equals CEC fa = W 0 / (1 + V 0, where he denominaor equals he price of a yearly consumpion of one, i.e., he curren consumpion plus he price of an immediae annuiy. In Secion 5.1 we invesigae he opimal fracion of wealh invesed in a deferred annuiy and he corresponding cerainy equivalen consumpion condiional on he immediae purchase of a deferred annuiy, i.e., s = 0, for differen deferral periods. In Secion 5.2 we invesigae he opimal fracion of wealh invesed in an immediae annuiy and he corresponding cerainy equivalen consumpion condiional on posponing he purchase of an immediae annuiy, i.e., d = 1. We assume ha he individual purchases an immediae annuiy only once. A (1) Purchasing a deferred annuiy a reiremen dae In his secion we invesigae he effec of he deferral period on he expeced lifeime uiliy of an individual, condiional on currenly (s = 0) purchasing a deferred annuiy. We maximize he individual s expeced lifeime uiliy as given in (1) given consrains (2) (5) for annuiies wih d = 1,..., 15 (i.e., differen deferral periods, including an immediae annuiy), respecively and s = 0 (i.e., immediaely purchasing an annuiy). Firs, le us invesigae he effec of he deferral period on an individual s expeced lifeime uiliy. Figure 5 displays he cerainy equivalen consumpion relaive o he cerainy equivalen consumpion in he fully annuiized sraegy, as a funcion of he deferral period. The figure also illusraes he effec of he pricing mehod of annuiies (i.e., using a risk-neural approach or using a consan loading facor of 7.3%) on he 25