1 Dynamic Porfolio Choice wih Deferred Annuiies Wolfram Horneff * Raimond Maurer ** Ralph Rogalla *** 200_final_Horneff, e al Track E Financial Risk (AFIR) Absrac We derive he opimal porfolio choice and consumpion paern over he life-cycle for households facing uncerain labor income, risky capial marke, and moraliy risk. In addiion o socks and bonds, he household have access o deferred annuiies. Deferred payou annuiies are financial conracs providing similar o social securiy or defined benefi pension plans lifelong income o he annuian afer a specified period of ime condiional on survival. We find ha deferred annuiies play a significan role in household porfolios and generae significan welfare gains. Our base case invesor wih high benefis from sae pensions and moderae risk aversion and moderae labor income risk sars purchasing deferred annuiies a age 40 and gradually increases heir porfolio share unil reiremen a age 65. Then, deferred annuiies already accoun for 78 percen of oal financial wealh, wih he res being invesed in socks. Facing low replacemen raes from sae pensions and a high exposure o labor income risk, he household will purchase more annuiies and earlier. Inroducing uncerainy wih respec o fuure moraliy raes has he same effec, i.e. invesors acively use deferred annuiies as a hedge agains longeviy shocks. JEL Codes: Keywords: D91 G11 J26 Porfolio Choice, Deferred Annuiies, Sochasic Moraliy * : Goehe Universiy Frankfur, Dep. of Finance, House of Finance (H23), Grueneburgplaz 1, 60323 Frankfur, Germany, Tel: +49 (0)69 798 33647, E-mail: horneff@finance.uni-frankfur.de ** : Goehe Universiy Frankfur, Dep. of Finance, House of Finance (H23), Grueneburgplaz 1, 60323 Frankfur, Germany, Tel: +49 (0)69 798 33647, E-mail: maurer@finance.uni-frankfur.de *** : Goehe Universiy Frankfur, Dep. of Finance, House of Finance (H23), Grueneburgplaz 1, 60323 Frankfur, Germany, Tel: +49 (0)69 798 33661, E-mail: rogalla@finance.uni-frankfur.de 2009 Horneff, Maurer, and Rogalla. All righs reserved.
2 1 Inroducion and Moivaion Previous research on dynamic porfolio choice over he life cycle argues ha purchasing annuiies wih immediae life coningen payous is imporan o finance consumpion of riskaverse households wih uncerain lifeime and no beques moive (see originally Yaari 1965 and more recenly Davidoff e al. 2005). Life annuiies are financial producs ha allow invesors o proec agains ouliving reiremen asses while enhancing expeced reurns hrough he moraliy credi. Despie his heoreical dominance, invesors around he world are relucan o volunary annuiize heir wealh. This discrepancy beween heoreically prediced and empirically observed behaviour is known as he annuiy puzzle. Argumens explaining his divergence include incomplee annuiy markes, beques moives, high coss charged by annuiy providers, and behavioural facors (Hu and Sco 2007, Yagi and Nishigaki 1993). Agains his background, he number of advocaes of embedding annuiizaion as a defaul opion in axshelered pension plans is increasing. In his conex, deferred annuiies more and more arac he aenion of policymakers, regulaors and financial inermediaries. Similar o an immediae annuiy, he provider of a deferred annuiy promises lifelong periodic payous o he annuian in exchange for a non-refundable premium. While paymens from immediae annuiies sar a he dae purchase, hose from deferred life annuiies only commence afer a cerain number of years (he deferring period) has passed, subjec o he individual s survival. Due o he discouning effec as well as he possibiliy ha he annuian perishes before payous sar, a deferred annuiy is much cheaper han an immediae annuiy wih idenical payous. According o Milevsky (2005a, p. 110), he comparably low price of deferred annuiies may help o overcome psychological barriers o volunary annuiizaion, as he poins ou ha engaging in irreversible financial ransacions - ha is annuiizaion - involving large lump sums will never be appealing o individuals regardless of (wheher hey grasp) he imporance of longeviy insurance. Cash flow paerns similar o hose of deferred life annuiies are provided by sae-organized social securiy sysems and occupaional pension plans of he defined benefi variey. In boh pension schemes, workers (repeaedly) conribue a cerain fracion of heir labor income during heir working life. In reiremen, hey receive pension benefis deermined by heir conribuions and working years for as long as hey live. Consequenly, in a world wih declining relevance of boh sae-organized social securiy sysems and occupaional defined benefi pensions, he relevance of deferred life annuiies offered in he privae marke migh increase in fuure. While immediae life annuiies have recenly been sudied o a large exen i, he role of deferred annuiies for privae households has no been considered in he lieraure on dynamic porfolio choice over he life cycle. Prior work on he role of life annuiies in realisically calibraed dynamic porfolio choice models wihin an incomplee marke seing includes Milevsky and Young (2007) and Horneff e al. (2008 a, b), who analyze he possibiliy of gradual annuiizaion for he case of fixed immediae payou annuiies. We exend his work in wo direcions: Firs, we incorporae life annuiies wih deferred payous in line wih hose proposed and analyzed in Milevsky (2005a) ii and derive he opimal consumpion, saving and porfolio choice paern for a CRRA uiliy maximizing household facing uncerain life ime, unspanned labor income, and risky capial marke reurns. Second, we exend his model by inegraing ime-varying moraliy raes ha can exhibi boh expeced improvemens in fuure life expecancy as well as sochasic variaions around his rend. This resuls in uncerainy wih 2
3 respec o fuure prices of deferred annuiies and individual survival probabiliies, which a raional invesor has o anicipae when making decisions on consumpion, saving and porfolio allocaion. We find ha deferred annuiies play a significan role in household porfolios and generae significan welfare gains. Our base case invesor wih high benefis from sae pensions and moderae risk aversion and moderae labor income risk sars purchasing deferred annuiies a age 40 and gradually increases heir porfolio share unil reiremen a age 65. Then, deferred annuiies already accoun for 78 percen of oal financial wealh, wih he res being invesed in socks. Facing low replacemen raes from sae pensions and a high exposure o labor income risk, he household will purchase more annuiies and earlier. Inroducing uncerainy wih respec o fuure moraliy raes has he same effec, i.e. invesors acively use deferred annuiies as a hedge agains longeviy shocks. In wha follows, we inroduce he life-cycle model applied o find he opimal consumpion and porfolio allocaion ino socks, bonds, and deferred annuiies. Here, we discuss preferences, risky labor income paerns, annuiy as well as capial marke specificaions, wealh accumulaion, and sochasic moraliy dynamics. Ou findings are presened in secion hree. We discuss he opimal life-cycle asse allocaion for our base case and in a scenario analysis we show expeced life-cycle profiles for various calibraions of key parameers. We coninue by analyzing he impac of ime-varying moraliy and he role of annuiies wih very high deferring age, before we sudy he welfare implicaions of adding deferred annuiies o he invesmen menu. A final chaper concludes. 2 The Model 2.1 Preferences In our sudy, we employ a ime discree model wih 0,..., T 1, where consiues he invesor s adul age (calculaed as he acual age minus 19) and T is he maximal age. We denoe p as he invesor s subjecive probabiliy o survive from unil + 1. Furhermore, s we assume ha he invesor s preferences are given by a ime-separable CRRA uiliy funcion defined over a single non-durable consumpion good. Le C be he consumpion level a ime he recursive definiion of he value funcion is given by 1 C s 1 E V 1 V p Here ρ is he level of relaive risk aversion and β is he personal ime preference discoun facor. We assume ha he household does no derive uiliy from bequeahing poenial heirs. iii Today s uiliy is given as he uiliy from consumpion and omorrow s discouned s uiliy from fuure consumpion. We have p 0 for he final period, i.e. in T equaion (1) T 1 p CT boils down o: VT, which gives us he erminal condiion for V T. From he final 1 p value, we can work backwards o find he opimal sraegies how o consume, how o inves in bonds, socks, and how o purchase deferred annuiies over he life cycle. (1) 3
4 2.2 Deferred Annuiy and Capial Markes The household has access o capial markes by invesing in riskless bonds and risky socks. The real bond gross reurn is consan over ime and given by R f, while he real gross risky sock reurn a ime is denoed by R. The risky sock reurns are assumed o be serially independen and idenically log-normally disribued wih an expeced reurn µ and volailiy σ s. In addiion o bonds and socks, he household can also purchase a each ime < K for an iniial non-refundable premium A deferred life annuiies wih consan lifelong real payous L K saring a he end of he deferring period a cerain age K. In our base case he end of deferring period is equal wih he reiremen age K = 65, i.e. when he individual receive sure social securiy benefis insead of risky labor income. Laer we consider also a siuaion whereby he annuiy would begin o sar payous a he advanced age of 85. The life annuiy does neiher conain any survival or esae benefis nor a cash value a any age. Using he acuarial principle of equivalence he annuiy provider calculaes he gross single premium as he presen value of expeced benefis paid o he annuian (including expense loadings), or formally A L h K (2) K where h is he annuiy facor for an individual wih adul age which is given by K 2 T K K 1 s a K 1 a s h 1 pu R f pu R f < K (3) u s 1 u K 1 Here δ is he expense or loading facor charged by he annuiy provider o organize he riskpool of annuians and he discoun facor R f is equal o he riskless bond reurn. The yearo-year survival probabiliies p used o price he annuiy are specified by a moraliy able a u and may be higher han he individual s subjecive survival probabiliies ( 4 a p u > p s u ). This allows o model asymmeric moraliy beliefs whereby he addiional price incremen is hough of as a compensaion for boh he adverse selecion problem in he privae annuiy markes (Brugiavini, 1993, Finkelsein and Poerba, 2004) and he macro longeviy risk (Cairns, Blake, and Dowd, 2006a). Adverse selecion in annuiy markes arises because individuals who believe hemselves o be healhier han average buy annuiies more likely. Macro longeviy risk refers o he risk of unexpeced changes of survival probabiliies. We consider a highly incomplee annuiy marke. The only annuiies available are deferred annuiies wih life-long consan payous, which can only be purchased during he deferral period, i.e. before paymens commence. Thus, he invesor neiher has access o deferred annuiies afer reaching he deferring age nor o immediae payou annuiies. We do no accoun for annuiies which payou a only one specific age and sae (as in he complee markes case in Davidoff, Brown, and Diamond, 2005) or on annuiies wih payous specified according o he performance of an underlying porfolio of risky asses (see Horneff, Maurer, Michell, and Samos 2010). The illiquidiy relaed o deferred annuiies adversely affecs he invesor s abiliy o reac o eiher adverse developmens of labor income or sudden declines in he sock marke. In reurn for he illiquidiy and he forgiven beques opporuniy, he household gains condiional on survival a spread over he payou from an invesmen in riskless bonds, also
5 referred in he lieraure as he moraliy credi. The spread comes abou because he funds of hose who die in he annuiy pool are disribued among he living members of a cohor. Therefore, a deferred annuiy simply consiues a separae asse class wih disincive risk and reurn characerisics. We rea he purchase of deferred annuiies as a porfolio choice problem by puing hem on an equal fooing wih equiy and bond invesmens. In he remainder, we model he annuiizaion decision essenially in a dynamic porfolio choice framework akin o Horneff, Maurer, and Samos (2008). To illusrae he reurn-enhancing effecs of he moraliy credi in deferred annuiies, we follow Milevsky (2005a, b) and presen he Implied Longeviy Yield (ILY) a various ages for annuiies wih deferring age 65 or 85. The ILY is he excess reurn over he discoun rae ha has o be generaed on he deferred annuiy premium in each year unil deferring age o be able o hen purchase an immediae annuiy ha pays he same benefi. Only in case excess reurns over he discoun rae generaed in he capial markes are equal or in excess of he ILY an individual would benefi from posponing annuiizaion unil he deferring age and hen purchasing an immediae annuiy. If he excess capial marke reurn falls shor of he ILY, an invesor ineresed in annuiy paymens from deferring age on is beer off by immediaely purchasing a deferred annuiy. Table 1 shows he ILY in basis poins (BP) based on our 2000 US Annuian Basic moraliy able (q 100 := 1) and a real annual discoun rae of 2 %. When purchasing a deferred annuiy due from age 65 (85), a woman aged 20 already locks in an ILY of 12 (84) BP. The moraliy credi increases wih age and so does he ILY. Buying he same annuiies a age 60 will provide her wih ILYs of 48 (207) BP. Paricularly when deferring unil age 85, i will be increasingly difficul o generae his excess reurn, making i profiable o purchase hese annuiies well before deferring age. As men on average are subjec o higher moraliy raes, heir ILY significanly exceed hose of women: for a deferring age of 65 (85) by abou 100 (50) percen. Table 1 here The long deferral periods combined wih he increases in reurn due o he aforemenioned moraliy credi have a significan posiive impac on he individual s budge and liquidiy. A ages 20, 40, and 60 prices for a deferred annuiy paying 1 from age 65 on are 6.94, 10.33, and 15.78. Purchasing an immediae annuiy offering he same benefi a age 65 would cos 17.84. Naurally, his effec is even more pronounced when annuiy paymens are deferred unil age 85. Then, a ages 20, 40, 60, and 80 prices for a life-long income of 1 are 1.20, 1.78, 2.72, and 5.31, while an immediae annuiy purchased a age 85 would cos 7.40. This emphasizes he insurance characer of deferred annuiies agains real longeviy risk. 2.3 Labor Income Process In order o undersand how he illiquidiy of deferred annuiies affecs he overall asse allocaion, we model ransiory and permanen income shocks. Previous lieraure on sraegic asse allocaion such as Bodie, Meron, and Samuelson (1992), Cocco, Gomes and Maenhou (2005), Heaon and Lucas (1997), and Viceira (2001) highlighed he relevance of considering unspanned labor income when analyzing he porfolio choice decisions of 5
6 households over he life cycle. The labor income Y is given by: Y exp( f ( )) PU, (4) P P N, (5) 1 where f() is used o recover he hump shape of he empirically ypically observed income profile over ime. Here, P is a permanen componen wih innovaion N. U is a ransiory shock. The logarihms of boh N and U are i.i.d normally disribued wih means zero and wih volailiies σ N, σ U, respecively. The shocks are assumed o be uncorrelaed. In reiremen ( ), we assume for he sake of simpliciy ha he individual receives consan pension paymensy exp f P, where ζ is he consan replacemen raio. The variable n u describes he correlaion beween he risky sock reurns and he permanen (ransiory) labor income shocks. 2.4 Wealh Accumulaion A he beginning of every period, he uiliy maximizing household under consideraion can decide on how o spread wealh on hand W across bonds B, socks S, new annuiies purchases A (as long as < K ), and consumpion C. Therefore, he budge consrain is B S A C K W, (6) B S C K where we refer o B + S as he value of financial wealh. The individual s disposable wealh on hand in + 1 is given by W 1 B R B R f f S R S R 1 1 Y L 1 1 Y 1 K K (7) where B R f + S R +1 denoes he nex period s value of financial wealh, L +1 is he sum of annuiy income which he invesor ges from all previously purchased annuiies and Y +1 is he labor income or, afer reiremen, he exogenously provided pension income. A any ime (< K ), he sae variable L K represens he previously accumulaed claims on annuiy payous due a adul age K, whereas from age K on, he sae variable L denoes he payous from previously purchased deferred annuiies. These sae variables follow he processes: L L 1 K 1 L L K A h K K (8) where A /h is he addiional annuiy paymen purchased in. We preven households from borrowing agains human capial and from selling annuiies. Boh resricions are binding because oherwise households would engage in highly leveraged sock posiions financed by shor posiions in bonds and/or annuiies in order o compensae he overinvesmen in human capial a young ages. Thus, in every year he opimal policy has o 6
7 saisfy: B, S, A 0. (9) 2.5 Moraliy Dynamics There is growing consen among academics and praciioners, ha moraliy raes over ime are neiher fixed nor following deerminisic rends bu raher follow sochasic processes (see Milevsky and Promislow (2001), Milevsky, Promislow, and Young (2009), Cairns, Blake, and Dowd (2006, 2008)). In our conex uncerain moraliy could be an imporan facor for he household s demand-paern of deferred annuiies. By buying deferred annuiies early in life he invesor is able o insure agains he risk of unexpeced declining moraliy raes occurring lae in life. Declining moraliy raes could have wo effecs: Firs, subjecive survival probabiliies used by he household o value fuure consumpion and cash flows are increasing. This makes cash flows sreams from annuiies already purchased earlier in life more aracive. Second, o reflec he new informaion abou unexpeced decreases in moraliy raes, he insurance company has o adjus he acuarial assumpions underlying he pricing of he annuiies. This makes he purchase of addiional annuiies lae in life more expensive for he household. To examine he implicaions of sochasic changes of moraliy raes on opimal life cycle consumpion, porfolio choice and he demand for deferred annuiies we use a sochasic moraliy model. One of he sandard approaches o modeling sochasic moraliy, widely spread among demographers and acuaries, is he framework developed by Lee and Carer (1992). In heir model, he naural logarihm of age-specific moraliy raes is described by a linear funcion of age-depending parameers and a laen, unobservable and periodspecific index variable: ln m x, ax bx k x,, (10) where m x, is he cenral deah rae for hose aged x in year, a x and b x are age-specific consans, k is he laen, ime-varying index and x, is an age-specific error erm wih zero mean and a variance e 2. The index variable k iself follows a random walk wih drif. Saing (10) differenly by aking he exponenial, he model can be inerpreed as ax b x k x, qx x m x, exp exp,, (11) where q x is he moraliy rae in he base year and x, is he cumulaive change in his moraliy unil year. While his is already a parsimonious approach o modeling sochasic moraliy, he curse of dimensionaliy renders is applicaion impracical in our dynamic sochasic programming framework, as he level of k represens an addiional coninuous sae variable. To circumven his problem, we assume ha insead of following a geomeric random walk wih drif, x, is described by a symmeric and bounded rinomial ree wih an odd number of n saes ( i ( n 1) 2,,0,,( n 1) 2 ), where he age-specific moraliy reducion facor in moraliy sae i is x,, i i x. Here, x is a ime-invarian, age-specific moraliy reducion facor adequaely calibraed o b x k from he Lee/Carer model, refraining from addiional agespecific shocks. The jump probabiliies are defined by 7
8 p p 0 u u, pd, pr 1 0 i ( n 1) 2 pd i ( n 1) 2 i ( n 1) 2 i ( n 1) 2 where p u (p d ) [p r ] specifies he probabiliy o jump from sae i o i+1 (i o i-1) [i o i] as ime passes from o +1. This approach can be inerpreed as a generalizaion of he binomial ree model used in Milevsky and Promislow (2001). Assuming ha he probabiliies for upward and downward jumps are equal, i.e. p u = p d, his model generaes purely sochasic moraliy raes ha remain consan in expecaion. To recover he empirically observable increase in life expecancy, we superimpose a deerminisic rend over he rinomial ree. Hence, he age-specific moraliy reducion facor in moraliy sae i a 1 ime is ( 1. This se-up enables us o individually sudy he effecs of x,, i MT ) x,0, i sochasiciy and rends in moraliy raes on he individual s life-cycle behavior. p u p d (12) 2.6 Numerical Mehod and Model Calibraion Opimizaion problems of his ype canno be solved analyically due o he unradeable labor income, he irreversibiliy of annuiy purchases, and he shorselling resricions. Therefore we adop he sandard approach of dynamic sochasic programming o solve he household s opimizaion problem. The household maximizes (1) under budge and shor-selling resricions (6),(7),(8), and (9), whereby he choice variables in each year he household is alive are he demand for socks S, bonds M, new life annuiies A, and consumpion C. The opimal policy depends on five sae variables: he permanen income level P, wealh on hand W, annuiy payous from previously purchased annuiies L, age, and moraliy sae i. Firs of all, he curse of dimensionaliy can be parly miigaed by reducing he sae space by one sae variable as we exploi he scale independence of he opimal policy if we normalize he coninuous sae variables wih he permanen labor earnings componen P (see for example Cocco, Gomes, and Maenhou, 2005). We solve he problem in a four-dimensional sae space by backward inducion. The coninuous sae variables normalized wealh w and normalized annuiy payous l have o be discreized and he only discree sae variables are age and moraliy sae i. The size of he grid is 40(w)x40(l)x81()x9(i) (in our base case wih fixed moraliy, he sae space is reduced o hree dimensions). The grid we use is equally spaced for he logarihms of w and l since he policy funcions and value funcion are especially sensiive in he area wih low w or l. For each grid poin we calculae he opimal policy and he size of he value funcion. To provide numerical insigh ino our seup, we calibrae he sylized base case as follows: The saring age is se o 20, he maximum age o 100 (T = 81), and he reiremen age 65 (K = 46) is equal he end of he deferring period for annuiies purchase during working life. In addiion, we also sudy he case when annuiy payous sar only a age 85. The preference parameers are se o sandard values found in he life-cycle lieraure (e.g. Gomes and Michaelides, 2005): coefficien of relaive risk aversion ρ = 5 and he personal discoun facor β = 0.96. In our fixed moraliy scenarios, we use he 2000 US Annuian Basic moraliy able for pricing he annuiies and he 2000 US Populaion Basic moraliy able for evaluaing he uiliy from consumpion, boh for females. In our sochasic moraliy scenarios, we employ hese moraliy raes as he q x from (12). The probabiliy upward or downward jumps in moraliy is exogenously se o p u = p d = 10 percen, recovering he empirical observaion ha in many counries annuian moraliy ables are ypically updaed every decade. Based on he calibraions in Lee and Carer (1992), he age- 8
9 specific moraliy improvemen facor x is se o 5.77 percen o recover he average 1- period volailiy of moraliy raes and he moraliy rend facor is se o MT = 1.42 percen o recover he average 1-period moraliy improvemen. Furhermore, we assume ha changes in moraliy idenically affec boh, annuian as well as populaion ables. The expense rae for annuiies is se o δ = 0 in he base case, and 2,38% and 7,20% in he sensiiviy analysis. The deerminisic age-dependen labor income funcion f() for individuals wih high school educaion excluding college educaion and he replacemen rae ζ = 0.68 are aken from Cocco, Gomes, and Maenhou (2005). Our volailiy parameers σ u = 0.15 and σ n = 0.1 are in line wih he esimaes found by Gourinchas and Parker (2002). Furhermore, we selec a real ineres rae R f -1 of 2 percen, an equiy premium µ R f of 4 percen, and a sock volailiy σ of 18 percen. We choose correlaions 0 0. beween he sock reurns and he ransiory (permanen) income shocks of To gain furher insighs, we hen vary seleced parameers and conduc comparaive saic analyses. The addiional scenarios include low (ρ = 2) and high (ρ = 8) risk aversion, medium (ζ = 0.60) and low (ζ = 0.50) replacemen raes, no (σ u = σ n = 0) and high (σ u = 0.30, σ n = 0.20) labor income risk, as well as low (δ = 0.0238) and high (δ = 0.072) loading facors (Maurer, Michell, and Rogalla 2009). In wha follows, we presen he resuls of our opimizaion model assuming consan as well as sochasic moraliy dynamics in he case annuiy payous saring a he end of he working life. Nex, we urn o a siuaion where he individual can purchase only annuiy wih payous saring lae in life a age 85. Finally we presen he welfare gains for households having access o privae annuiy markes wih deferred payous. n u 3 Resuls 3.1 Deferred Annuiies under Consan Moraliy 3.1.1 Opimal Asse Allocaion In his secion, we urn o he opimal asse allocaions we obained for our fixed-moraliy base case from solving he Bellman equaion under he shor-selling resricions. For each age and level of cash on hand, i.e. remaining wealh afer consumpion, Figure 1 displays how he individual will opimally spread is financial means over he hree asse classes available, risky socks (Panel B), risk-free bonds (Panel B), and deferred life annuiies (Panel C), assuming ha no deferred annuiies were purchased before. Figure 1 here A young ages and wih low cash on hand, he individual will be fully invesed in socks seeking he opporuniy o cash in on he equiy premium. Wih lile wealh a sake and a long invesmen horizon remaining, here is lile need for diversifying ino less-risky asses. Wih rising cash on hand, however, he invesor will hold an increasing fracion of wealh in risk-free asses, even a lower ages. Here, in an aemp o insure agains risky labor income, young individuals will predominanly choose liquid bonds raher han deferred annuiies. I is noeworhy, however, invesors wih very high levels of cash on hand will devoe a measurable 9
10 amoun of heir means o purchasing deferred annuiies, acceping o wai for anoher 45 years before receiving any payoffs from his invesmen. Wih rising age and cash on hand, we see he ypical life-cycle paern of decreasing fracions of sock invesmens. As human capial depreciaes, he invesor is more willing o purchase bondlike financial asses, wih deferred annuiies becoming he predominan form of risk-free invesmen as he individual ages due o an increasing moraliy credi. By he age of 60 deferred annuiy purchases have compleely crowded ou bonds for any level of cash on hand. Excep for very low levels of cash on hand, purchases of deferred annuiies skyrocke a age 64, resuling in a sharp decline of sock holdings. This can be aribued o wo facs. Firs, wih deferred annuiies beginning o pay benefis a age 65, he deferral period has shrunk o is minimum. More imporan, however, is he fac ha age 64 provides he las chance for he individual o purchase any annuiies. As can be seen in Panel C, annuiy purchases from age 65 on are zero by definiion and cash on hand has o be disribued among socks and bonds only. Hence, i comes as no surprise, ha individuals, who haven purchased annuiies unil age 65 and hus won hold any in he fuure, will be significanly invesed in bonds, resuling in he sharp increase in opimal bond holdings a age 65. Bond holdings increase wih boh, cash on hand as well as age, reflecing he decreasing invesmen horizon. Neverheless, even individuals wih very high wealh levels will unil he end of he life-cycle inves abou 30 percen of heir cash on hand in socks. For hose wih very low remaining financial wealh i is even opimal o be fully invesed in equiies. 3.1.2 Expeced Life-cycle Profile We now analyze he simulaed disribuions of he relevan choice and sae variables by conducing an exensive Mone Carlo analysis based on he opimal feedback conrols derived above. Drawing 50,000 independen sochasic scenarios, we compue he expeced life-cycle profile for our sylized base case wih risky labor income. Resuls are summarized in Figure 2, where Panel A shows he developmen of labor income, consumpion, liquid savings, annuiy purchases and annuiy income and Panel B shows he expeced allocaion of wealh o equiies, bonds and annuiies over he individual s life-span. Figure 2 here We firs aend o Panel A of Figure 2. The expeced rajecory of labor income exhibis he empirically observed hump-shape unil reiremen a age 65 when i is replaced by a risk-less pension of 68 percen of final labor income. Early in he life-cycle consumpion falls shor of labor income, helping o build up a significan level of liquid savings. This nes egg peaks a 55 when savings are abou 6 imes he average labor income. From age 40 on, he individual sars purchasing deferred annuiies in expecaion and coninues o do so unil reiremen. Thus, he individual is iniially willing o wai for 25 years before receiving he firs pay-offs from he annuiy invesmen. Wih annuiy purchases gradually rising over he individual s fories and fifies and coninuously increasing consumpion exceeding labor income from age 49, liquid savings gradually decrease unil age 64, when he individual akes he final chance and shifs a significan fracion of liquid wealh ino deferred annuiies, which hen commence o provide a second sable income from reiremen age 65 on. Consumpion coninues o gradually increase during reiremen, peaking a age 83 and marginally 10
11 decreasing hereafer, ye no falling shor of consumpion in he mid-fifies unil he end of life. Wih no ineres in bequeahing any financial means, his high level of reiremen consumpion is finance by coninuously depleing liquid savings over he remaining lifecycle. Turning o Panel B of Figure 2, we obain he well-known resul ha he individual is fully invesed in equiies early in life, due o he low level of available savings, he mainly bond-like characerisics of human capial and expeced excess reurns on equiy invesmens. Wih depreciaing human capial, he individual s appeie for bond-like invesmens increases. Thus, from he mid-hiries, he allocaion o equiies decreases and a small bu increasing fracion of wealh is invesed in liquid bonds. Alhough deferred annuiies are calculaed a he same rae of reurn, liquid bonds are preferred over annuiies as he individual needs o insure agains oday s risky labor income, while he annuiies would only provide an addiional sable income from reiremen age onward. Only over ime, he excess reurn on annuiies resuling from an increasing moraliy credi is sufficienly high o overcompensae he annuiies illiquidiy and deferral. Wih annuiy purchases saring a age 40, he allocaion o boh annuiies and bonds increases over he fories while he equiy weigh is slowly reduced. A age 45, equiies sill make up 92 percen of he porfolio, he allocaion o bonds has increased o five percen and hree percen of he wealh is invesed in deferred annuiies. Wih abou 6 percen, he allocaion o bonds peaks a age 49. From ha age on, he growing moraliy credi, decreasing exposure o labor income risk, and he shrinking deferral period increase he appeal of annuiies, which compleely crowd ou bonds unil he lae fifies. Unil age 55, he annuiy share has already increased o 26 percen, while he allocaion o equiies (bonds) had dropped o 71 (3) percen. Massive purchases of deferred annuiies a he end of he working life urn his relaion upside down and enering reiremen a age 65, 78 percen of he individual s wealh is invesed in annuiies wih he remaining 22 percen remaining in equiies. Wih liquid savings remaining virually consan over he following decade, he fracion of overall wealh invesed in equiies slighly increases due o depreciaing annuiy wealh. Unil age 75, he equiy share rises o 26 percen, and a age 85 equiies sill accoun for 24 percen of overall asses. Only hen does he allocaion o equiies begin o coninuously decline unil he end of he life-cycle. 3.1.3 Impac of Cenral Risk Parameers Having discussed he resuls for our base case in he previous secion, we now direc our focus o analyzing he impac of varying cenral risk parameers on expeced life-cycle asse allocaion. In paricular, we will focus on alernaive parameerizaions of he individual s risk aversion, he exogenous pension replacemen rae, he level of labor income risk and he loadings charged by he annuiy provider, varying only he respecive parameer and oherwise mainaining he base case calibraion. Resuls are presened in Table 2, which shows he age of iniial purchases of deferred annuiies and he allocaion o equiies, bonds, and deferred annuiies as well as he level of liquid savings for various ages from 45 o 85. In addiion o ha, Figure 3 provides more deailed insighs ino he allocaion o deferred annuiies over ime, comparing for each parameer dimension our base case and wo alernaive calibraions. Table 2 here Figure 3 here 11
12 In our base case, we assumed our invesor o have CRRA uiliy wih a risk aversion coefficien of = 5. We now analyze invesmen decisions for an individual wih lower ( = 2) and for one wih higher ( = 8) risk aversion. As one would expec, he individual wih low risk aversion has lile appeie for eiher saving or risk-less invesmens, remaining fully invesed in equiies virually unil reiremen. Deferred annuiies are purchased only from age 63 onward and even hen hey only make up for an insignifican fracion of wealh; 0.8 percen a age 65, while our base case invesor would already hold 78.2 percen of oal wealh in annuiies. Wih ongoing depleion of liquid savings, he allocaion o annuiies seadily increases, bu even a age 85 a mere of 4.6 percen of oal wealh is held in annuiies. Only hen does his fracion begin o dramaically increase unil liquid savings are compleely exhaused in he early nineies, resuling in an annuiy fracion of 100 percen. By conras, our highly risk averse individual will save significanly more, sar purchasing deferred annuiies already from age 28 on and inves liquid asses much more conservaively. A age 45, deferred annuiies accoun for 39.7 percen of oal wealh, while 48.2 (12.1) percen are being invesed in socks (bonds). A his ime, he invesor holds 9.3 imes he iniial labor income as liquid savings, only marginally less han he 10.6 imes he (idenical) iniial labor income of our base case invesor, who in urn only holds 3 percen of overall wealh in deferred annuiies. A reiremen age 65, our highly risk averse individual holds 93.4 percen of oal wealh in annuiies and he remaining liquid savings (1.7 imes he iniial labor income) are fully invesed in equiies (6.6 percen of oal wealh). Even in reiremen, his individual will iniially coninue saving, resuling in again increasing liquid asses 2.8 (2.6) imes he iniial labor income a age 75 (85). Saving combined wih depreciaing annuiy wealh leads o again increasing allocaions o socks and bonds 21.1 percen and 0.2 percen a age 85 before he allocaion o annuiies increases again due o depleion of liquid asses. We now urn our aenion o variaions of he (exogenous) replacemen rae, i.e. he fracion of pension paymens from an exernal pension sysem wih respec o final year labor income. In our base case, his replacemen rae was = 68 percen. As i is widely believed ha his figure will decrease in fuure, we analyze wo alernaive seings wih lower replacemen raes of 60 and 50 percen. Reducing he replacemen rae (wihou compensaing he individual, e.g. hrough higher labor income) resuls in marginally increased iniial savings, earlier annuiy purchases and more conservaive invesmen behavior. While our base case invesor will sar purchasing deferred annuiies from age 40, individuals wih a replacemen rae of 60 (50) percen will already inves in annuiies a age 39 (38). A age 45, he allocaion o annuiies increases from 3 percen in he base case o 4.3 (6.4) percen wih he medium (low) replacemen rae. A he same ime, he equiy fracion drops from 92.1 percen o 90.5 (88.3) percen, while bonds make up for 5.1 (5.3) percen of oal wealh compared o 4.9 percen in he base case. These paerns are mainained over he whole life-cycle. Annuiy holdings increase from 78.2 percen o 80.6 (83.0) percen a reiremen, while he allocaion o equiies drops from 21.8 o 19.4 (17.0) percen. A age 85, 78.5 (80.9) percen of oal wealh is held in annuiies compared o 76.2 percen in he base case, while he equiy exposure is reduced from 23.6 o 21.3 (18.9) percen. Nex, we analyze he implicaions of differen levels of labor income risk. In our base case, he volailiy of he permanen labor income shock was σ n = 0.1 and he ransiory shocks had a volailiy of σ u = 0.15. Now, we look a an individual ha does no face any labor income risk, i.e. σ u = σ n = 0, and an invesor who faces very risky labor income wih double he volailiies of he base case, i.e. σ u = 0.3 and σ n = 0.2. In case labor income has no volailiy, he individual does no only have a predicable income during working life bu also a foreseeable pension, as his is based on final salary. Thus, i comes as no surprise ha he 12
13 need o insure sufficien pension income by early and more heavily invesing in deferred annuiies is less developed han in a siuaion wih labor income risk. As a resul, he individual will only purchase annuiies from age 53, when he rising moraliy credi of he annuiy invesmen is increasingly able o over-compensae is illiquidiy. As in our base case, he allocaion o annuiies coninuously increases hrough he fifies and early sixies and jus before reiremen, he invesor will shif a significan fracion of liquid savings ino annuiies, which reach a maximum share of 68.7 percen of oal wealh a reiremen age 65. Over he firs decade of reiremen, depreciaing annuiy wealh resuls in decreasing porfolio shares (63.9 percen a age 75), as savings remain almos consan. Wih liquid asses being depleed over he nex decades, he allocaion o annuiies again increases oward he end of he life-cycle. Having a secure labor income also influences he allocaion of liquid asses. While in our base case bonds play a cerain role from he midhiries o he mid-fifies, he individual will now be virually fully invesed in equiies, as here is no need o hedge risky labor income. The opposie is rue for our individual ha faces high labor income risk. Already early in he life-cycle, he invesor has a srong appeie for risk-free invesmens, boh liquid an illiquid, as human capial shows many characerisics of equiies raher han hose of bonds. Annuiies are being purchased from age 24 and reach a share of 22.7 percen of oal asses a age 45. A he same ime, he individual holds anoher 31.3 percen in bonds, while equiies only accoun for 46 percen of overall asses. As in all oher cases, annuiies crowd ou bonds unil reiremen, when he allocaion o annuiies reaches a emporary maximum of 82 percen, wih he remaining 18 percen being invesed in equiies. Wih reiremen, risky labor income is replaced by riskless pension income and invesmen paerns mimic hose in he cases wih less-risky labor income. Equiy shares firs rise wih depreciaing annuiy wealh (22.0 percen a age 75) and laer drop as liquid asses are coninuously depleed. Finally, we are ineresed in how he picure changes when we give up he (unrealisic) assumpion of cos-free deferred annuiies. Insead we inroduce expense loadings of 2.38% and 7.2 percen, he former being represenaive for comparably cheap deferred annuiies wihin company pension plans, he laer represening marke loadings of privae annuiy providers. As in our base case, he individual will sar purchasing deferred annuiies a age 40 for boh our alernaive expense raes, and also a age 45, here are only negligible differences in saving and invesmen paerns over he hree alernaive seings. While savings are virually unchanged, he allocaion o deferred annuiies marginally drops from 3.0 percen in our base case o 2.9 (2.7) percen for low (high) loadings. Consequenly, equiy holdings are slighly higher under high expense raes (92.3 percen) compared o he base case (92.1 percen). Bond holdings remain unchanged. The marginal impac of inroducing loadings a young ages resuls from he fac ha deferred annuiies, which only become due 25 years in he fuure, are hen comparably cheap. Adding a 7.2 percen levy on he annuiy price hus has a low absolue impac. As he individual ages, however, deferred annuiies wih loadings become increasingly more expensive in absolue erms compared o he case where no expenses are levied on hem. Consequenly, he individual is less inclined o purchase hese producs. Thus, a reiremen, individuals ha face low (high) expense raios only hold 74.6 (67.2) percen of heir oal wealh in annuiies compared o 78.2 percen in he base case. In urn, hese individuals keep more of heir liquid savings (5.2 (6.7) compared o 4.5 imes he iniial labor income) and inves more heavily in equiies (25.4 (32.8) compared o 21.8 percen of oal asses). This paern is mainained over he res of he life-cycle. 3.2 Deferred Annuiies under Time-Varying Moraliy 13
14 As discussed above, he assumpion ha moraliy raes remain consan over he individual s whole life-cycle is somewha unrealisic. No only does empirical evidence show clear rends in moraliy over ime, i is also apparen ha moraliy raes exhibi some exen of sochasiciy. These facors may well affec he individual s annuiizaion decisions, as boh annuiy prices over ime as well as individual life expecancy will vary over ime. Thus, in his secion, we revisi our base case from secion 3.1 and analyze he impac of ime-varying moraliy modeled as described in secion 2.5. Again, our represenaive invesor has CRRA uiliy wih a risk aversion coefficien of = 5, faces medium labor income risk, will reire a age 65 wih an exogenously provided pension of 68 percen of he final salary, and is unil reiremen offered deferred annuiies wihou loadings. Our findings are presened in Table 3 and Figure 4, boh of which, for ease of comparison, also resae our base case resuls from secion 3.1 where moraliy is consan over ime. Table 3 here Figure 4 here Firs, looking a he case wih sochasic bu no rending moraliy, we find ha he individual will begin purchasing deferred annuiies from age 37, hree years earlier ha in he consan moraliy scenario. While earlier in he life-cycle he individual prefers liquid asses as an insurance agains labor income flucuaions, i faces an increasing dispersion of possible moraliy levels as ime goes by. The invesor will herefore prepone annuiy purchases compared o he consan moraliy case in an effor o avoid possibly higher annuiy prices laer in life. A age 45, annuiies already make up for 6.3 percen of he invesor s porfolio, more han double he fracion our base case invesor holds in annuiies. A he same ime, under sochasic moraliy, liquid savings are higher by 0.9 imes he iniial labor income, indicaing ha he individual measurably reduces consumpion in he lae hiries and early fories o finance annuiy purchases. As in our base case wih consan moraliy, liquid asses are primarily invesed in equiies (88.7 percen) while bonds only play a small role (5 percen). The annuiy fracion seadily increases o abou 50 percen in he early sixies, and jus before reiremen, he individual will shif a subsanial fracion of liquid savings ino annuiies. Hence, a age 65, annuiies make up for 82 percen of he invesor s porfolio, 4 percen more han in he fixed moraliy scenario, while he invesor sill holds liquid savings of 4.2 imes he iniial labor income, abou as much as in our base case. As in our consan moraliy scenario, he remaining liquid asses are held in equiies. Buying more annuiies and earlier, resuls in expeced annuiy benefis ha exceed hose in he consan moraliy scenario by abou 17 percen. We nex urn o our world wih deerminisically rending moraliy. Here moraliy raes for all ages coninuously drop by 1.42 percen per year. While in our pure sochasic se-up here is a significan chance of moraliy raes remaining consan over some ime, he invesor in his scenario anicipaes ha moraliy raes will decline even from he beginning of he lifecycle, driving up boh prices for deferred annuiies as well as he invesor s life expecancy. As annuiies in any period are priced a he prevailing moraliy raes, he invesor has a srong incenive o annuiize early in life. A he same ime, he individual again faces risky labor income, which canno be hedged wih annuiies, as paymens only commence a reiremen. Thus, he invesor has o pospone annuiizaion in an effor o build up a buffer sock of liquid asses ha can assis in bolsering labor income shocks. Trading-off in expecaion increasing annuiy prices and over ime declining labor income risk, he invesor will annuiize earlier han in our consan moraliy case as well as our pure sochasiciy case. Annuiy purchases 14
15 commence a age 35 and by age 45, deferred annuiies make up for 10.4 percen of he individual s porfolio. The annual increase in he annuiy fracion, however, is lower han in our wo previous cases, despie he fac ha he value of previously purchased deferred annuiies will increase wih decreasing moraliy, indicaing ha he individual is less inclined o purchase deferred annuiies laer in he working life as heir price is increasing coninuously. This becomes mos apparen jus before reiremen, when las-order annuiy purchases significanly fall shor of hose in boh previous cases. A age 65, annuiies only accoun for 64.1 percen of he individual s porfolio, wih he res being held in equiies. Liquid savings in urn amoun o 8 imes he iniial labor income, almos wice he sum we found in our wo earlier scenarios. The individual is aware of he fac ha declining moraliy raes require a bigger reiremen nes-egg. Annuiies, however, appear o be oo expensive a ha ime. Hence, while he individual begins o purchase deferred annuiies early, benefis fall shor of hose in he consan moraliy case by abou 24 percen and by abou 35 percen of hose in he purely sochasic scenario. In our final scenario, we combine he wo previously analyzed paerns of ime-variabiliy of moraliy and assume ha moraliy raes are sochasic and addiionally follow he deerminisic, downward sloping rend. As we saw earlier in his secion, superimposing sochasiciy on deerminisic (previously, deerminisic mean consan) moraliy behavior led o he individual being ineresed in buying more annuiies and earlier. No surprisingly, we derive a similar resul now. When combining he deerminisic rend wih sochasic moraliy developmen, we find ha purchases of deferred annuiies commence a age 34, one year earlier han in he pure deerminisic rend scenario and asse allocaion hroughou he lifecycle is also iled more oward annuiies. A age 45, 13.4 percen of he individual s wealh is invesed in annuiies, hree percenage poins more han in he pure deerminisic rend scenario. Liquid savings a ha ime are up by one imes he iniial labor income, indicaing ha he individual reduced consumpion early in he life-cycle even more han in he previous case. A reiremen, he allocaion o annuiies exceeds ha in he previous case by 4 percenage poins, while liquid savings are virually he same. Purchasing more annuiies and earlier, resuls in annuiy benefis exceeding hose in he deerminisic moraliy rend case by 21 percen. Compared o he consan moraliy scenario, however, annuiy income sill falls shor by abou 9 percen. In conclusion, we find ha negaive moraliy rends resul in earlier bu in oal fewer annuiy purchases, as annuiy prices coninuously increase. A he same ime, inroducing sochasiciy wih respec o fuure moraliy raes will induce risk averse individuals o purchase more annuiies and earlier, hence, insuring agains he possibiliy of adverse annuiy price developmens as well as he risk of no having purchased enough annuiies o mainain a high and smooh consumpion sream should moraliy raes decrease in reiremen. 3.3 Deferred Annuiies as Pure Longeviy Insurance So far, we assumed ha deferred annuiies provided benefis from reiremen age 65 on. Alernaively, deferred annuiies migh be purchased o insure agains he risk of real longeviy, as for example pu forward by Milevsky (2005). I is argued, ha in a world were DB pension plans are increasingly being replaced by DC-ype schemes, he relucance o proec agains longeviy by annuiizing migh be overcome by offering comparably cheap producs ha only offer longeviy proecion a very high ages, i.e. when individuals need i he 15
16 mos. Along his line of hough, German Rieser plans, ax-qualified supplemenary privae pension plans available since 2002, require mandaory annuiizaion from age 85 onward. Agains his background, we in his secion analyze he opimal life-cycle invesmen paerns in a world ha provides deferred annuiies ha sar paying benefis from age 85. In paricular, we will again aend o he four scenarios from he previous secion, and analyze he consan moraliy scenario (i.e. he base case from secion 3.1) as well as he scenarios wih purely sochasic, deerminisically rending, and sochasically rending moraliy. Resuls are summarized in Table 4 and Figure 5. Table 4 here Figure 5 here The individual facing consan moraliy sars purchasing deferred annuiies from age 42, acceping o wai for 43 years before receiving any payoffs from his invesmen. The allocaion o deferred annuiies slowly rises from 0.8 percen a age 45 o 12.3 percen a reiremen age 65. During reiremen, he fracion of oal wealh invesed in deferred annuiies increases rapidly, reaching 30 percen a age 75, and by he age of 85, when he deferred annuiies begin o pay benefis, he individual is fully invesed in hem. The exponenial growh of he annuiy fracion resuls, on one side, from coninuously increasing annuiy purchases, on he oher side, wih a shrinking deferral period, pension claims increase in value due o decreasing discoun facors and increasing (condiional) survival probabiliies. As deferred annuiies will only provide paymens from age 85 on, he individual faces a significan reducion of periodic income a reiremen. To bolser his income shock and o bridge he gap unil annuiy paymens will commence, he individual has o rely on is liquid savings. A reiremen, liquid savings amoun o 20 imes he iniial labor income a age 65, coninue o rise o 20.5 imes a age 70, and are hen depleed unil age 85. The allocaion of liquid savings o equiies and bonds exhibis he well-known life-cycle paern. Equiy holdings coninuously drop from 92.9 percen a age 45 o 46.8 percen a age 75. While in secion 3.1 he role of bonds was diminishingly small, hey are of much more appeal in his se-up. The decreasing invesmen horizon, depreciaing human capial, and he need o compensae he income shock a reiremen wih oher sources of sable income drives he bond fracion up from 6.3 percen a age 45 o 26.4 percen a age 65. While he allocaion o bonds remains quie sable over he subsequen decade (23.4 percen a age 75), bond holdings are hen also depleed unil age 85. As we derived in he previous secion, in a world wih purely sochasic moraliy, he individual will again buy more deferred annuiies and earlier. Purchases commence a age 37 and when he deferred annuiies begin paying benefis a age 85, he individual is again virually fully invesed in annuiies. Prior o ha, allocaions o deferred annuiies and bonds as well as liquid savings are slighly higher han in he consan moraliy case. While he difference appears o be small in erms of overall porfolio weighs, purchasing more annuiies and earlier resuls in expeced benefis ha exceed hose in he consan moraliy case by abou 21 percen. In he deerminisic moraliy rend scenario, he individual will already begin purchasing deferred annuiies a age 33, anicipaing increasing fuure annuiy prices and higher individual benefis due o rending moraliy improvemens. As hese deferred annuiies are significanly cheaper han hose in secion 3.2, he individual can annuiize earlier a lower cos and sill build up sufficien liquid asses o insure agains labor income risk. By he age of 45, deferred 16
17 annuiies already accoun for 9.5 percen of oal wealh. As in secion 3.2, we find ha while he fracion of wealh invesed in deferred annuiies in he sochasic moraliy scenario is always above is consan moraliy scenario counerpar before paymens begin, he rae of increase in he annuiy fracion is lower. From age 45 o 65, he annuiy fracion only doubles o 20.3 percen, while i increased by facor 15 in he fixed moraliy case. Allocaion o annuiies coninues o rise unil age 85 when annuiies sar o pay benefis and no furher annuiies may be purchased. Wih liquid savings comparable o hose in he consan moraliy case, increased annuiy weighs resul from reducions in boh equiy and bond holdings. As in he previous secion, increasing annuiy prices due o decreasing moraliy raes have a significan impac on he annuiy benefis. While he allocaion o annuiies significanly exceeds ha in he consan moraliy case, annuiy income a age 85 falls shor by abou 20 percen. Finally, we again urn our aenion oward he scenario ha combines sochasic moraliy wih moraliy rends and derive familiar resuls. Annuiy purchases commence even earlier, a age 31. Liquid savings as well as allocaions o annuiies and bonds are up compared o he purely deerminisic case, while allocaions o equiies are lower before he deferring age. Again, differences in he allocaion o deferred annuiies appear o be of minor imporance. As we have seen before, however, his small deviaion resuls in significan differences in annuiy benefis, which are up by 19 percen compared o he pure deerminisic rend case. 3.4 Welfare Implicaions Our analyses so far have shown ha in many of he scenarios discussed above he individual s opimal porfolio includes a significan fracion of deferred annuiies already early in life. This indicaes ha including hose annuiies in he invesmen menu will have welfare enhancing effecs. To es his hypohesis, we in his secion analyze he uiliy gains aribuable o graning access o markes for deferred annuiies for he various scenarios scruinized above. For individuals aged 20 and 65, we deermine he increase in uiliy aribuable o he availabiliy of deferred annuiies by calculaing he lump-sum paymen required by individuals wihou access o annuiies in order o have he same uiliy level in erms of he value funcion as in a world wih annuiies. Table 5 presens hese lump-sum paymens in percen of he respecive average labor income. Table 5 here Our calculaions sugges ha access o deferred annuiies generaes measurable uiliy gains already for hose aged 20 in mos of our scenarios, wih he low risk aversion and he no labor income risk cases being he excepions. In a world wihou deferred annuiies, our base case invesor would require a lump-sum of 16.2 percen of he average labor income o have he same uiliy as in a world wih annuiies. As could be expeced, his lump-sum rises wih increasing risk aversion and labor income risk and wih decreasing expense loadings and replacemen raes from exogenous pensions. Uiliy gains significanly increase as he individual ages. A reiremen, our base case invesor already requires almos wo imes he average income o be indifferen beween he worlds wih and wihou annuiies. The highly risk averse individual would require a lump-sum of 4.5 imes he average income, while even a reiremen an invesor wih low risk aversion would only ask for a negligible 1.4 percen. Looking a our scenarios wih ime-varying moraliy, we find ha deferred annuiies are 17
18 welfare enhancing paricularly in hose cases where moraliy is sochasic, while heir posiive impac is sharply reduced by moraliy rends. An invesor aged 20 facing non-rending sochasic moraliy will require a lump-sum of 27 percen of he average income, almos wice he amoun required by his counerpar in a world wih consan moraliy. By conras, if moraliy is decreasing deerminisically, our invesor would abandon he opion o inves in annuiies for a mere 10 percen of he average labor income, which is only abou 60 percen he compensaion needed under consan moraliy. In a world where moraliy is sochasic and a he same ime rending downward, hese effecs offse one anoher o some exen bu welfare gains hrough annuiizaion sill exceed hose in he consan moraliy scenario by abou 30 percen. As before, uiliy gains significanly increase wih he age of he invesor. This especially holds for purely sochasic moraliy, where an individual would require almos nine imes he average income o be indifferen. We finally urn o our cases where he deferring age is 85. Again, we find some welfare gains even for individuals aged 20. These are slighly lower han before when moraliy is eiher consan or purely sochasic, while hey are even marginally higher in case moraliy exhibis a rend. 4 Conclusion In a world ha sees he relevance of predicable reiremen income sreams from saeorganized social securiy sysems and occupaional defined benefi pension plans coninuously diminishing, deferred life annuiies offered in he privae marke may provide households wih an adequae means o independenly replicae hese cash flow paerns and he inheren insurance agains longeviy risk. In his sudy, we scruinize he role of deferred annuiies in opimal porfolios of households facing un-insurable labor income risk, uncerain capial marke reurns, and sochasic moraliy. We se off by analyzing he relevance of annuiies wih a deferring age of 65, which is equal o he reiremen age. Hence, hey serve as a direc supplemen o reiremen income from he social securiy sysem. We show ha invesors wih moderae risk aversion and some exposure o labor income risk will sar purchasing deferred annuiies from heir early fories and will hen gradually shif funds from liquid savings ino annuiies unil heir porfolio share reaches abou 80 percen a reiremen. Wih increasing risk aversion, higher labor income risk, lower replacemen raes from sae pensions, and sochasic moraliy raes, annuiy purchases commence even earlier and he fracion of oal financial wealh invesed in annuiies increases. Consisen wih findings in oher sudies, early in he life-cycle liquid savings are fully invesed in equiies and hen gradually shifed ino annuiies. Liquid bonds in mos cases only play a minor role and are quickly crowded-ou by annuiies. We also find ha households use deferred annuiies o acively hedge agains longeviy risk. Even when payous only commence a he very advanced age of 85, he invesor slowly begins purchasing deferred annuiies a age 42. In case he household faces o sochasic and rending life expecancy, invesmens in deferred annuiies already begin a age 31 and he allocaion o annuiies is significanly higher compared o a consan moraliy scenario. Consisen wih prior sudies on immediae annuiies, we find subsanial welfare gains from including deferred annuiies in he invesmen menu. In a world wih sochasic and rending moraliy, even an invesor aged 20 is willing o give up abou one quarer of average labor 18
19 income o have access o deferred life annuiies wih payous saring a age 85. Consequenly, due o heir comparably low price, deferred annuiies migh be a good insrumen o overcome he relucance o annuiize. 19
20 References Blake, D., A. Cairns, and K. Dowd, 2003, Pension Merics 2: Sochasic Pension Plan Design during he Disribuion Phase, Insurance: Mahemaics and Economics, 33 (1), 29 47. Bodie, Z., R. Meron, and W. Samuelson, 1992, Labor supply flexibiliy and porfolio choice in a life cycle model, Journal of Economic Dynamics and Conrol, 16, 427 449. Brown, J., O. Michell, J. Poerba, and M. Warshawsky, 2001, The Role of Annuiy Markes in Financing Reiremen, MIT Press, Cambridge, M.A. Brugiavini, A., 1993, Uncerainy Resoluion and he Timing of Annuiy Purchases, Journal of Public Economics, 50 (1), 31 62. Cairns, A., D. Blake, and K. Dowd, 2006, A wo-facor model for sochasic moraliy wih parameer uncerainy: Theory and calibraion, Journal of Risk and Insurance 73: 687-718. Cairns, A., D. Blake, and K. Dowd, 2006b, Pricing Deah: Frameworks for he Valuaion and Securiizaion of Moraliy Risk, ASTIN Bullein, 36, 79 120. Cairns, A., D. Blake, and K. Dowd, 2008, Modelling and managemen of moraliy risk: a review, Scandinavian Acuarial Journal 2008(2-3): 79-113. Cocco, J., 2005, Porfolio Choice in he Presence of Housing, The Review of Financial Sudies, 112, 61 156. Cocco, J., F. Gomes, and P. Maenhou, 2005, Consumpion and Porfolio Choice over he Life Cycle, The Review of Financial Sudies, 18, 491 533. Finkelsein, A., and J. Poerba, 2004, Adverse Selecion in Insurance Markes: Pol-icyholder Evidence from he U.K. Annuiy Marke, Journal of Poliical Economy, 112, 183 208. Gomes, F., and A. Michaelides, 2005, Opimal Life-Cycle Asse Allocaion: Undersanding he Empirical Evidence, Journal of Finance, 60, 869 904. Gong, G. and A. Webb, 2009, Evaluaing he Advanced Life Deferred Annuiy An Annuiy 20
21 People Migh Acually Buy, forhcoming in Insurance: Mahemaics and Economics, doi:10.1016/j.insmaheco.2009.08.010. Gourinchas, P.-O., and J. A. Parker, 2002, Consumpion over he Life Cycle, Economerica, 70, 47 89. Heaon, J., and D. Lucas, 1997, Marke Fricions, Savings and Porfolio Choice, Macroeconomic Dynamics, 1, 76 101. Horneff, W., R. Maurer, O. Michell, and I. Dus, 2008, Following he Rules: Inegraing Asse Allocaion and Annuiizaion in Reiremen Porfolios, Insurance: Mahemaics and Economics, 42, 396 408. Horneff, W., R. Maurer, and M. Samos, 2008a, Life-Cycle Asse Allocaion wih Annuiy Markes, Journal of Economic Dynamics and Conrol, acceped, 2008. Horneff, W., R. Maurer, and M. Samos, 2008b, Opimal Gradual Annuiizaion: Quanifying he Coss of Swiching o Annuiies, The Journal of Risk and Insurance, in press, 2008. Hu, Wei-Yin and Jason S. Sco, 2007, Behavioral obsacles in he annuiy marke, in: Financial Analyss Journal, Vol. 63 (2007), No. 6, 71-82. Koijen, R., T. Nijman, and B. Werker, 2006, Dynamic Asse Allocaion wih Annuiy Risk, Discussion paper, Tilburg Universiy - Cener for Economic Research. Lee, R. and L. Carer, 1992, Modeling and Forecasing U.S. Moraliy, Journal of he American Saisical Associaion 87 (419), 659-671. Maurer, R., O.S. Michell, and R. Rogalla, 2009, The Effec of Uncerain Labor Income and Social Securiy on Life-cycle Porfolios, Pension Research Council Working Paper 2009-06. Milevsky, M., 2005a, Real Longeviy Insurance wih a Deducible: Inroducion o Advanced-Life Delayed Annuiies (ALDA), Norh American Acuarial Journal, Vol. 9, No. 21
22 4 (Ocober), 109 122. Milevsky, M., 2005b, The Implied Longeviy Yield: A Noe on Developing an Index for Life Annuiies, Journal of Risk and Insurance, Vol. 72, No. 2 (June), 301 320. Milevsky, M., K. Moore, and V. Young, 2006, Asse Allocaion and Annuiy-Purchase Sraegies o Minimize The Probabiliy of Financial Ruin, Mahemaical Finance, 16, 647 671. Milevsky and Promislow (2001), Moraliy derivaives and he opion o annuiise, Insurance: Mahemaics and Economics 29, 299 318. Milevsky, M., D. Promislow, and V. Young, 2009, Financial Valuaion of Moraliy Risk via he Insananeous Sharpe Raio, Journal of Economic Dynamics and Conrol 33(3), 676-691. Milevsky, M., and V. Young, 2007, Annuiizaion and Asse Allocaion, Journal of Economic Dynamics and Conrol, 31, 3138 3177. Sco, J., 2008, The Longeviy Annuiy: An Annuiy for Everyone? Financial Analys Journal, 64(1), 40 48. Samos, M. Z., 2008, Opimal Consumpion and Porfolio Choice for Pooled Annuiy Funds, Insurance: Mahemaics and Economics, 43, 56 68. Viceira, L., 2001, Opimal Porfolio Choice for Long-Horizon Invesors wih Nonrad-able Labor Income., Journal of Finance, 56 (2), 433 470. Yaari, M., 1965, Uncerain Lifeime, Life Insurance, and he Theory of he Consumer, Review of Economic Sudies, 32, 137 150. Yagi, T., and Y. Nishigaki, 1993, The inefficiency of Privae Consan Annuiies, The Journal of Risk and Insurance, 60 (3), 385 412. 22
23 Yogo, M., 2008, Porfolio Choice in Reiremen: Healh Risk and he Demand for Annuiies, Housing, and Risky Asses, Discussion paper, Wharon School. 23
24 Table 1: Implied Longeviy Yield of Deferred Annuiies Deferring Age 65 Deferring Age 85 Age Female Male Female Male 20 12 27 84 129 40 21 45 121 185 60 48 94 207 309 80 - - 484 659 Noes: Auhors calculaions. Implied longeviy yields in BP: excess reurn over discoun rae generaed by buying deferred annuiies a various ages compared o posponing annuiizaion unil deferring age. Real discoun rae: 2%. Moraliy assumpion: 2000 US Annuian Basic wih q 100 := 1. 24
Age 85 Age 75 Age 65 Age 55 Age 45 Base Case Low Risk Aversion Very High Risk Aversion Med. Rep. Rae Low Rep. Rae No Labor Income Risk High Labor Income Risk Low Loadings High Loadings 25 Table 2: Impac of Cenral Risk Parameers Age 1 s Ann Purchase 40 63 28 39 38 53 24 40 40 Equiies (%) 92.1 100.0 48.2 90.5 88.3 100.0 46.0 92.2 92.3 Bonds (%) 4.9 0.0 12.1 5.1 5.3 0.0 31.3 4.9 4.9 Annuiies (%) 3.0 0.0 39.7 4.3 6.4 0.0 22.7 2.9 2.7 Savings 10.6 4.0 9.3 10.9 11.2 2.8 19.0 10.6 10.7 Equiies (%) 71.2 100.0 36.9 67.0 61.8 99.0 46.9 71.8 73.0 Bonds (%) 3.3 0.0 6.2 2.8 2.4 0.2 6.6 3.4 3.6 Annuiies (%) 25.5 0.0 56.9 30.2 35.8 0.8 46.5 24.8 23.4 Savings 12.8 7.1 9.4 12.7 12.5 6.4 21.3 12.9 13.2 Equiies (%) 21.8 99.2 6.6 19.4 17.0 31.3 18.0 25.4 32.8 Bonds (%) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Annuiies (%) 78.2 0.8 93.4 80.6 83.0 68.7 82.0 74.6 67.2 Savings 4.5 9.2 1.7 4.4 4.2 2.9 9.3 5.2 6.7 Equiies (%) 26.2 98.9 14.8 23.5 20.9 36.0 22.0 29.6 36.6 Bonds (%) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.3 Annuiies (%) 73.8 1.1 85.2 76.4 79.1 63.9 78.0 70.3 63.1 Savings 4.2 4.7 2.8 4.1 3.9 2.7 8.6 4.8 6.0 Equiies (%) 23.6 95.4 21.1 21.3 18.9 32.3 19.9 27.0 33.9 Bonds (%) 0.2 0.0 0.2 0.2 0.2 0.2 0.2 0.3 0.8 Annuiies (%) 76.2 4.6 78.6 78.5 80.9 67.5 80.0 72.7 65.3 Savings 2.3 0.8 2.6 2.3 2.2 1.5 4.8 2.7 3.6 Noes: Auhors calculaion. Base Case: Reiremen Age: 65, Replacemen Rae: 68%, Risk Aversion: = 5, Med. Labor Income Risk: Vola of Transiory Income Shock = 15%, Vola of Permanen Income Shock = 10%, No Loadings. Low Risk Aversion: Base Case wih = 2. Very High Risk Aversion: Base Case wih = 8. Medium Replacemen Rae: Base Case wih 60% rep. rae. Low Replacemen Rae: Base Case wih 50% rep. rae. No Labor Income Risk: Base Case wih vola of Transiory Income Shock = 0%, vola of Permanen Income Shock = 0%. Low Labor Income Risk: Base Case wih Vola of Transiory Income Shock = 7.5%, Vola of Permanen Income Shock = 5%. Low Loadings: Base Case + Loadings of 2.38%. High Loadings: Base Case + Loadings of 7.2%. Asse weighs in percen of oal wealh (liquid savings + presen value of annuiy claims); Savings as a muliple of iniial labor income. 25
26 Table 3: Impac of Time-Varying Moraliy Asse Allocaion Age 45 55 65 75 85 Age 1 s Ann. Purchase Consan Moraliy 40 Equiies (%) 92.1 71.2 21.8 26.2 23.6 Bonds (%) 4.9 3.3 0.0 0.0 0.2 Annuiies (%) 3.0 25.5 78.2 73.8 76.2 Savings 10.6 12.8 4.5 4.2 2.3 Sochasic Moraliy w/o Trend 37 Equiies (%) 88.7 66.3 18.0 24.6 24.4 Bonds (%) 5.0 3.9 0.0 0.0 0.3 Annuiies (%) 6.3 29.8 82.0 75.4 75.2 Savings 11.5 13.7 4.2 4.6 3.1 Deerminisic Moraliy Trend 35 Equiies (%) 88.8 67.2 35.9 39.2 41.0 Bonds (%) 0.8 1.3 0.0 0.8 2.6 Annuiies (%) 10.4 31.6 64.1 60.0 56.3 Savings 9.9 12.2 8.0 8.0 6.6 Sochasic Moraliy wih Trend 34 Equiies (%) 85.4 63.8 32.0 37.1 40.2 Bonds (%) 1.2 1.7 0.0 1.1 3.2 Annuiies (%) 13.4 34.5 68.0 61.8 56.6 Savings 10.8 13.1 7.9 8.6 7.7 Noes: Auhors calculaion. Consan Moraliy (equal o Base Case in secion 3.1): Reiremen Age and Deferring Age: 65. Replacemen Rae: 68%. Risk Aversion: = 5. Med. Labor Income Risk: Vola of Transiory Income Shock = 15%. Vola of Permanen Income Shock = 10%. Sochasic Moraliy (for boh. annuian s & individual moraliy) follows a bounded, symmeric 9-level rinomial ree wih p u =p d =0.1 and jump size equal o 5.77 percen of he original moraliy rae (oher parameers equal o Consan Moraliy case). Trend: moraliy raes decline geomerically by 1.42 percen per year (oher parameers equal o sochasic moraliy case). Asse weighs in percen of oal wealh (liquid savings + presen value of annuiy claims); Savings as a muliple of iniial labor income. 26
27 Table 4: Asse Allocaion wih Deferring Age 85 Asse Allocaion Age 45 55 65 75 85 Age 1 s Ann. Purchase Consan Moraliy 42 Equiies (%) 92.9 75.4 61.3 46.8 0.1 Bonds (%) 6.3 18.7 26.4 23.4 0.0 Annuiies (%) 0.8 6.0 12.3 29.9 99.9 Savings 10.9 16.7 20.0 19.5 0.0 Sochasic Moraliy w/o Trend 37 Equiies (%) 89.5 71.1 57.9 44.1 0.2 Bonds (%) 7.6 21.1 28.5 25.2 0.0 Annuiies (%) 2.8 7.8 13.6 30.7 99.8 Savings 12.1 18.5 22.2 22.7 0.1 Deerminisic Moraliy Trend 33 Equiies (%) 86.6 71.5 59.2 47.0 0.4 Bonds (%) 3.8 14.1 20.5 14.4 0.0 Annuiies (%) 9.5 14.4 20.3 38.6 99.6 Savings 10.1 15.7 19.4 18.9 0.1 Sochasic Moraliy wih Trend 31 Equiies (%) 84.4 68.4 56.3 44.9 5.1 Bonds (%) 5.3 16.4 22.5 16.4 0.0 Annuiies (%) 10.3 15.1 21.1 38.7 94.9 Savings 11.4 17.5 21.6 22.1 1.7 Noes: Auhors calculaion. Consan Moraliy (equal o Base Case in secion 3.1): Reiremen Age: 65. Deferring Age: 85. Replacemen Rae: 68%. Risk Aversion: = 5. Med. Labor Income Risk: Vola of Transiory Income Shock = 15%. Vola of Permanen Income Shock = 10%. Sochasic Moraliy (for boh. annuian s & individual moraliy) follows a bounded, symmeric 9-level rinomial ree wih p u =p d =0.1 and jump size equal o 5.77 percen of he original moraliy rae (oher parameers equal o Consan Moraliy case). Trend: moraliy raes decline geomerically by 1.42 percen per year (oher parameers equal o sochasic moraliy case). Asse weighs in percen of oal wealh (liquid savings + presen value of annuiy claims); Savings as a muliple of iniial labor income. 27
28 Table 5: Welfare Analysis Welfare Gains (in % of Avg. Labor Income) Age 20 65 Consan Moraliy Base Case 16.2 198.4 Low Risk Aversion 0.0 1.4 High Risk Aversion 66.6 453.0 Medium Replacemen Rae 21.1 250.1 Low Replacemen Rae 28.5 341.1 No Labor Income Risk 1.1 126.3 High Labor Income Risk 46.0 346.4 Low Coss 14.7 177.8 High Coss 11.8 140.9 Time-Varying Moraliy Sochasic Moraliy w/o Trend 27.0 898.6 Deerminisic Moraliy Trend 10.4 82.6 Sochasic Moraliy wih Trend 21.1 103.2 Deferring Age 85 Consan Moraliy 13.9 205.3 Sochasic Moraliy w/o Trend 24.8 238.1 Deerminisic Moraliy Trend 11.5 134.1 Sochasic Moraliy wih Trend 23.0 164.3 Noes: Auhors calculaion. Base Case: Reiremen Age: 65, Replacemen Rae: 68%, Risk Aversion: = 5, Med. Labor Income Risk: Vola of Transiory Income Shock = 15%, Vola of Permanen Income Shock = 10%, No Loadings. Low Risk Aversion: Base Case wih = 2. Very High Risk Aversion: Base Case wih = 8. Medium Replacemen Rae: Base Case wih 60% rep. rae. Low Replacemen Rae: Base Case wih 50% rep. rae. No Labor Income Risk: Base Case wih vola of Transiory Income Shock = 0%, vola of Permanen Income Shock = 0%. Low Labor Income Risk: Base Case wih Vola of Transiory Income Shock = 7.5%, Vola of Permanen Income Shock = 5%. Low Loadings: Base Case + Loadings of 2.38%. High Loadings: Base Case + Loadings of 7.2%. 28
29 Figure 1: Opimal Asse Allocaion Policies: Base Case Noes: Auhors calculaion. Base Case: Reiremen Age: 65. Replacemen Rae: 68%. Risk Aversion: = 5. Med. Labor Income Risk: Vola of Transiory Income Shock = 15%. Vola of Permanen Income Shock = 10%. 29
20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 Asse Allocaion 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 30 Figure 2: Expeced Opimal Life-Cycle Profile and Asse Allocaion: Base Case A) Expeced Life-cycle Profile 14 12 10 8 6 Labor Income Consumpion Ann. Income Ann. Purchases Savings 4 2 0 Age B) Expeced Asse Allocaion 100% 90% 80% 70% 60% Annuiies Bonds Equiies 50% 40% 30% 20% 10% 0% Noes: Auhors calculaion. Base Case: Reiremen Age: 65. Replacemen Rae: 68%. Risk Aversion: = 5. Med. Labor Income Risk: Vola of Transiory Income Shock = 15%. Vola of Permanen Income Shock = 10%. Asse weighs in percen of oal wealh (liquid savings + presen value of annuiy claims); Savings. Consumpion. Labor & Annuiy Income as well as Annuiy Purchases as a muliple of iniial labor income. Age 30
20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 Annuiy Weigh (in %) 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 Annuiy Weigh (in %) Annuiy Weigh (in %) 31 Figure 3: Impacs on Allocaion o Deferred Annuiies over he Life-Cycle Relaive Risk Aversion Replacemen Rae 100% 100% 90% Base Case 90% Base Case 80% Rho = 2 80% RR = 60% 70% Rho = 8 70% RR = 50% 60% 60% 50% 50% 40% 40% 30% 30% 20% 20% 10% 10% 0% 0% Age Age Labor Income Risk Annuiy Loadings 100% 90% 80% 70% Base Case No LIR High LIR 60% 50% 40% 30% 20% 10% 0% Age Noes: Auhors calculaion. Base Case: Reiremen Age: 65. Replacemen Rae: 68%. Risk Aversion: = 5. Med. Labor Income Risk: Vola of Transiory Income Shock = 15%. Vola of Permanen Income Shock = 10%. No Labor Income Risk: Base Case wih vola of Transiory Income Shock = 0%. vola of Permanen Income Shock = 0%. High Labor Income Risk: Base Case wih vola of Transiory Income Shock = 30%. vola of Permanen Income Shock = 20%. Asse weighs in percen of oal wealh (liquid savings + presen value of annuiy claims). 31
20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 Annuiy Weigh 32 Figure 4: Impac of Time-Varying Moraliy on Annuiy Allocaion 100% 90% Consan Moraliy 80% Soch. Mor. w/o Trend 70% Deerminisic Mor. Trend 60% Soch. Mor. wih Trend 50% 40% 30% 20% 10% 0% Noes: Auhors calculaion. Consan Moraliy (equal o Base Case in secion 3.1): Reiremen Age and Deferring Age: 65. Replacemen Rae: 68%. Risk Aversion: = 5. Med. Labor Income Risk: Vola of Transiory Income Shock = 15%. Vola of Permanen Income Shock = 10%. Sochasic Moraliy (for boh. annuian s & individual moraliy) follows a bounded, symmeric 9-level rinomial ree wih p u =p d =0.1 and jump size equal o 5.77 percen of he original moraliy rae (oher parameers equal o Consan Moraliy case). Trend: moraliy raes decline geomerically by 1.42 percen per year (oher parameers equal o sochasic moraliy case). Asse weighs in percen of oal wealh (liquid savings + presen value of annuiy claims). Age 32
20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 Annuiy Weigh 33 120% 100% 80% 60% Figure 5: Annuiy Allocaion wih Deferring Age 85 Consan Moraliy Soch. Mor. w/o Trend Deerminisic Mor. Trend Soch. Mor. wih Trend 40% 20% 0% Noes: Auhors calculaion. Consan Moraliy (equal o Base Case in secion 3.1): Reiremen Age: 65. Deferring Age: 85. Replacemen Rae: 68%. Risk Aversion: = 5. Med. Labor Income Risk: Vola of Transiory Income Shock = 15%. Vola of Permanen Income Shock = 10%. Sochasic Moraliy (for boh. annuian s & individual moraliy) follows a bounded, symmeric 9-level rinomial ree wih p u =p d =0.1 and jump size equal o 5.77 percen of he original moraliy rae (oher parameers equal o Consan Moraliy case). Trend: moraliy raes decline geomerically by 1.42 percen per year (oher parameers equal o sochasic moraliy case). Asse weighs in percen of oal wealh (liquid savings + presen value of annuiy claims). Age 33
34 Noes i See for insance Blake, Cairns, and Dowd (2003), Horneff, Maurer, and Samos (2008a, b), Horneff, Maurer, Michell, and Dus (2008), Koijen, Nijman, and Werker (2006), Milevsky and Young (2007), Milevsky, Moore, and Young (2006), and Yogo (2008). ii Oher work abou deferred annuiies includes Sco, Wason, and Hu (2008), who recommend purchasing deferred annuiies in case he exen o which annuiies can be bough is limied. Gong and Webb (2009) evaluae deferred annuiies using he annuiy equivalen wealh concep inroduced by Michell e al. (1999). iii This can be jusified ha he household pu only ha par of financial wealh ino he annuiy which is no inended for a beques, i.e. we only consider ha par of wealh which is required for consumpion purchases. Also see Samos (2008) on his poin. 34