EFFICIENT POST-RETIREMENT ASSET ALLOCATION

Transcription

1 EFFICIENT POST-RETIREMENT ASSET ALLOCATION Barry Freedman* ABSTRACT To examine pos-reiremen asse allocaion, an exension o he classic Markowiz risk-reurn framework is suggesed. Assuming ha reirees make consan (real dollar) annual wihdrawals from heir porfolios, reward and risk measures are defined o be he mean and sandard deviaion of wealh remaining a end of life. Asse reurns and ime of deah are boh reaed as random variables. Assuming consan lifeime asse allocaion, he risk and reward measures can be evaluaed analyically, and an efficien fronier can be deermined. Life annuiies can be used o exend he lef-hand (low-risk) side of he efficien fronier. The desired level of wealh a end of life can be used o choose a desirable porfolio on he efficien fronier. The desirable porfolio srongly depends on he wihdrawal rae. I is suggesed (alhough no proven) ha asse allocaions sraegies ha vary wih age do no add efficiency in his model, and asse allocaion sraegies ha vary wih wealh can add efficiency. 1. INTRODUCTION As he Baby Boom generaion approaches reiremen, and as employers swich from defined benefi o defined conribuion pension plans, he already imporan issue of pos-reiremen asse allocaion becomes ever more prominen. Unforunaely, as discussed in Bodie (3), The new science of finance has had a profound impac on he pracice of insiuional risk managemen....in comparison, applicaions of his new science o he imporan life-cycle issues households face have been limied. Online financial planning ools and opimizers lag far behind he bes heory. The aim of his paper is o sugges a simple mehod o evaluae pos-reiremen asse allocaion sraegies. I is hoped ha his mehod will be a modes sep forward oward he goal of increasing reiree financial securiy. This paper is organized as follows. Secion lays ou he basic model framework and discusses how his model differs from previous work Secion 3 conains he mahemaical developmen of he model. Secion 4 illusraes he mehodology by analyzing a simple case and deriving conclusions, including a rule of humb for amoun of annuiy purchase. Secion 5 discusses some exensions of he analysis and suggess areas of fuure research.. A FRAMEWORK FOR EVALUATING POST-RETIREMENT ASSET ALLOCATION A sandard mehod of analyzing asse allocaion is he Markowiz Porfolio Selecion Model (see, e.g., chaper 8 in Bodie, Kane, and Marcus 1999). This model is aracive in ha i allows a simple graphical illusraion of any porfolio s risk and reward poenial and allows for he discovery of an efficien fronier. In he Markowiz model he reward and risk of a porfolio are defined o be he expecaion and sandard deviaion of porfolio reurn over a single period. This model has proven iself o be excepionally useful in he asse accumulaion phase of he life cycle; however, he reiremen phase presens an addiional series of issues for consideraion, as illusraed in Table 1. * Barry Freedman, FSA, is a Direcor a Sun Life Assurance of Canada, One Sun Life Execuive Park, Wellesley Hills, MA, 481, Barry.Freedman@sunlife.com. 8

2 EFFICIENT POST-RETIREMENT ASSET ALLOCATION 9 Table 1 Comparison of Accumulaion and Pos-Reiremen Life-Cycle Phases Time horizon Issue Accumulaion Phase Pos-Reiremen Reiremen. Can be regarded as a fixed horizon, or as a source of opionaliy (e.g., in he even of good invesmen experience one can choose o reire early). See Bodie (1). Deah. Should be regarded as a random ime horizon. The random variable represening ime of deah is highly volaile and has no implici opionaliy (see Fig. 1). Desired wealh a horizon Enough wealh o reire Enough asses a he end of life o pay for medical and long-erm-care expenses and o mee beques desires Cash flows (prior o ime horizon) Impac of underperformance Posiive or zero conribuions o asse funds Delay reiremen or change projeced reiremen sandard of living Sysemaic wihdrawals Inabiliy o mee expenses or unplanned reducion in reiremen sandard of living Clearly any pos-reiremen asse allocaion model needs o address he issues raised in Table 1. To accommodae hese differences, a few small changes o he classic Markowiz framework are suggesed. As discussed, he accumulaion phase model implicily describes a porfolio of asses wih a sochasic rae of reurn and a single-year ime horizon. To amend he Markowiz framework o reflec posreiremen realiies, 1. Include a consan (real dollar) rae of wihdrawal. Replace he one-year ime horizon wih a sochasic ime horizon se o ime of deah 3. Quanify risk and reward as he mean and sandard deviaion of remaining (real dollar) wealh a ime of deah: as menioned in Table 1, his wealh will be used o pay for medical and long-ermcare expenses and o mee beques desires (if any). 1 Figure 1 Probabiliy of Deah by Age for a 65-Year-Old Male Noe: The probabiliy disribuion of age of deah is wide and relaively fla, leading o he conclusion ha an individual would be impruden o use life expecancy as he ime horizon for reiremen planning. 1 The desired level of wealh a ime of deah is generally referred o in he lieraure as he beques moive. Alhough here is lile mahemaical disincion beween he desire for a level of beques and he desire for available wealh a (or near) end of life o manage end of life expenses, here is probably a significan difference from a psychological perspecive.

3 3 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 The model suggesed above has a number of posiive feaures: 1. The sandard inuiive risk and reward presenaion of resuls is mainained, bu he model now incorporaes pos-reiremen requiremens of consan annual wihdrawal, higher end-of-life expenses, a beques moive, and a random ime of deah.. The pos-reiremen porfolio is designed around ypical liquid asses invesmen sraegies bu can also incorporae payou annuiies. 3. The suggesed risk and reward measures can be easily calculaed using sochasic modeling mehods for any asse allocaion sraegy. Furhermore, for a consan asse allocaion sraegy (i.e., an asse allocaion sraegy ha is independen of ime and level of wealh), he suggesed risk and reward measures can be handled analyically, as described below. The exisence of analyical soluions allows for he simple and quick calculaion of efficien froniers. On he oher hand, he model has cerain negaive feaures: 1. A major pos-reiremen risk is he probabiliy of running ou of money, which is no explicily used in his model (alhough i can be calculaed when one has developed a suggesed pos-reiremen sraegy). This parameer is, unforunaely, no easily amenable o analyical reamen (alhough see Milevsky and Robinson for an approximaion echnique).. Similarly, a major pos-reiremen reward is he abiliy o spend more freely prior o he end of life. The model, as described, canno easily incorporae addiional discreionary spending ino he reward measure. 3. The use of sandard deviaion as a risk measure suffers from he fac ha he disribuion of wealh a he end of life may no be symmeric. Before proceeding o a mahemaical developmen, i is imporan o consider he place of his model in he conex of prior work in his field. A significan vein of research ino pos-reiremen financial decision making frames he problem as one of maximizing uiliy funcions. Analyses following his approach ofen make use of dynamic programming or sochasic opimal conrol and allow boh he porfolio composiion and he wihdrawal amouns o vary. Examples of his mehod can be found in Meron (199, chs. 4 6) and Gerrard, Haberman, and Vigna (4, 6). The srengh of his ype of analysis is is compleeness and rigor; however, he associaed mahemaical sophisicaion make hese ideas difficul o apprehend for acuarial praciioners (no o menion consumers). Perhaps responding o his issue, a second vein of research has approached he problem using simplifying assumpions and simpler mahemaical analyses. An apparen aim of his research is o produce conceps and illusraions ha are simple enough o be generally usable, bu complex enough o capure he essence of he problem. For example, he general echnique of uiliy maximizaion is used in a simpler fashion o answer he asse allocaion porion of he pos-reiremen decision-making problem by Chen and Milevsky (3) and Charupa and Milevsky (). Similarly Blake, Cairns, and Dowd (3) make use of numerical opimizaion of he uiliy funcion and a limied number of posreiremen wihdrawal sraegies. Simplifying he problem even furher, a number of auhors aemp o reduce complexiies of pos-reiremen asse allocaion o a relaively simple and inuiive se of risk and reward parameers. The model presened in his paper is inended o follow his pah. Oher examples of his approach include ha of Milevsky and Robinson (), who describe a mehod for approximaing he probabiliy of running ou of money prior o deah given a consan asse allocaion. Dus, Maurer, and Michell (4) use sochasic modeling o calculae a variey of risk and reurn saisics for a few differen pos-reiremen asse allocaion and drawdown sraegies. Smih and Gould (7) use sochasic simulaions o plo median bequess agains shorfall probabiliy for various invesmen sraegies and wihdrawal amouns. Finally, merging he risk reurn framework wih he mahemaics of dynamic programming (and assuming a consan force of moraliy), Young (4) develops an analyical formula for he invesmen sraegy ha minimizes he probabiliy of lifeime ruin and compues he resuling disribuion of wealh a ime of deah.

4 EFFICIENT POST-RETIREMENT ASSET ALLOCATION MATHEMATICAL DEVELOPMENT OF THE MODEL In he classic Markowiz model a consan asse allocaion is assumed. This is clearly appropriae given he shor ime horizon of he model; however, in he proposed variaion he assumpion of a consan asse allocaion is no necessary and is probably no appropriae. In paricular, asse allocaion may depend on boh age and wealh. As a base case, however, and for mahemaical simpliciy, a consan asse allocaion will be assumed in he developmen below. In he search for an efficien asse allocaion his assumpion (which can be reaed analyically) can be viewed as providing a lower bound, and sochasic modeling mehods can be used o refine he sraegy. The mahemaical developmen is largely inspired by Milevsky and Robinson (). The evoluion of pos-reiremen wealh is governed by dw (W, )W d (W, )W dz kd, (3.1) where W is wealh a ime, and iniial wealh is W k is he annual wihdrawal Z is sandard one-dimensional Brownian moion is he mean (real) asse reurn: depends on asse allocaion decision, which can vary wih wealh and ime is he sandard deviaion of asse reurn. The goal is o calculae E[e T W T ] and Var[e T W T ], where T represens ime of deah and is an ineres rae used for presen value purposes. Sensible choices for include he risk-free rae (assuming ha wealh a end of life will be used for bequess) or he expeced excess of medical or long-ermcare inflaion above price inflaion (assuming ha wealh a end of life will be used for medical expenses or long-erm care). For clariy of developmen, is se equal o zero for he remainder of his paper; however, he equaions below are exended in he Appendix for a nonzero. As described above, his developmen will assume a consan asse allocaion hroughou life, which simplifies equaion (3.1): dw W d W dz kd. (3.) There is a echnical problem wih equaion (3.) if wealh becomes negaive. As wrien, his equaion would imply ha he reiree is borrowing money a a risky rae, which is clearly no realisic. One soluion is o assume ha when wealh becomes zero, wihdrawals (k) immediaely reduce o zero, and wealh unil he end of life would herefore remain a zero, hus compressing he lef-hand side of he probabiliy disribuion of W T. This soluion is realisic bu is difficul o handle analyically. A second soluion is o simply allow equaion (3.) o exend ino negaive wealh. The ne impac of his soluion is o reduce he reward measure and increase he risk measure for any sraegy ha has a significan shorfall probabiliy. Because he goal of his analysis is o help choose a pos-reiremen asse allocaion, adding addiional penalies o disaser scenarios is no inherenly unreasonable. Equaion (3.) is equivalen o (1/ )Z (1/ )szs W e W k e ds. (3.3) The reward measure is defined o be he mean wealh a ime of deah. Le ime of deah be T, and where reward E [W ] d p q E [W ], (3.4) T,Z T x x Z E T,Z [] indicaes expecaion over he random variable T and he Brownian moion pah denoed by Z p x probabiliy of survival from iniial age x o age x and

5 3 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 q x d insananeous probabiliy of dying a beween age x and x d (assuming survival o x ). Inegraing equaion (3.4) by pars, d reward W d px E Z[W ]. (3.5) d In he Appendix (eq. A.3) i is shown ha d E Z[W ] (W k)e. (3.6) d Therefore, reward W (W k) d pxe, (3.7) which can be easily calculaed numerically for any moraliy able. The risk saisic is defined as he sandard deviaion of remaining wealh a ime of deah: risk E T,Z[W T] (E T,Z[W T]). (3.8) E T,Z [W T ] is, of course, he reward parameer and is calculaed above, while E T,Z [ following he mehod above: W T ] can be calculaed d E T,Z[W T] W d px E Z[W ], (3.9) d where a formula for d/d E Z [ W ] is given in he Appendix as equaion (A.4). Developing a formula for he variance is simply a maer of algebra and is given as equaion (A.7). The furher mahemaical exension o e T W T is also given in he Appendix as equaions (A.15) (A.18). The developmen above focuses on radiional asse classes. Incorporaing (inflaion-indexed) annuiies is simple. Purchasing an annuiy a ime reduces iniial wealh and reduces fuure wihdrawals from liquid asses. Therefore he purchase of an annuiy would require a new calculaion wih a new value of k (lowered by he annual annuiy paymen) and a new W (lowered by he price of he annuiy). 4. SAMPLE MODEL RESULTS The model described in Secion 3 allows for he numerical calculaion of a pos-reiremen efficien fronier assuming consan asse allocaion. Using he equaions in he Appendix, he numerical calculaions are simple enough o allow he opimizaion o be easily implemened in a spreadshee program. An illusraive analysis has been developed wih he assumpions provided in Table. Figure shows he efficien fronier for a 5% annual wihdrawal rae. Figure a shows he fronier assuming invesmen only in risky asses (sock and bonds), Figure b includes invesmen risk-free asses, and Figure c includes invesmen in annuiies. Clearly he inroducion of a risk-free asse class exends he fronier, as does he inroducion of an annuiy. Choosing an opimal locaion on an efficien fronier is generally presened as a funcion of a individual s risk olerance. In his case he risk olerance can be illusraed by making some assumpion abou he desired level of W T. For example, one migh assume ha a pruden reiree wih wealh on reiremen of $1,, migh desire o have a leas $5, (real) dollars available for end of life expenses. This desire could be approximaed by suggesing ha he reiree would like E[W T ]obea leas one sandard deviaion above $5,. This level of risk olerance requiremen is illusraed in Figure 3 as a sraigh line on he risk-reward graph. From he inersecion poin in his char one can draw he conclusion ha his reiree should definiely desire some allocaion o risky and risk-free asses and would likely feel he need for an annuiy. This conclusion changes for differen assumed

6 EFFICIENT POST-RETIREMENT ASSET ALLOCATION 33 Table Sample Model Assumpions Available Liquid Asses: Asse Mean Real Reurn () Sandard Deviaion of Reurn () Socks 7% % Bonds 4 7 Risk-free Sock/bond reurn correlaion 3% No shor selling Moraliy Raes: U.S Annuiy Basic Table for a male aged 65 (wih no projeced moraliy improvemen) Annuiy Price: Presen value of expeced moraliy experience a he risk-free rae. For he 65-year-old male his implies ha $1/year could be purchased for $15.6. Annual Wihdrawal Raes: 4%, 5%, and 6% of iniial wealh wihdrawal raes. Figures 4 and 5 show he efficien fronier and risk olerance curve for wihdrawal raes of 4% and 6%, respecively. From Figure 4 (wihdrawal rae of 4%) one can conclude ha an allocaion o annuiies is no desirable. From Figure 5 (wihdrawal rae of 6%) one can conclude ha he reiree canno mee boh he desired wihdrawal rae and he desired level for W T. Table 3 shows he desired porfolios implied by he inersecion poins in Figures 3, 4, and 5. I is imporan o reierae ha he asse allocaions shown would merely provide a sensible saring poin for an acual asse-allocaion decision; however, direcionally hey are likely o provide some useful suggesions o he reiree. The resuls in Table 3 also sugges a rule of humb for annuiy purchase. As is generally assumed, a wihdrawal rae of 4% is relaively safe and requires no annuiizaion. A wihdrawal rae of 5%, on he oher hand, is seen o require he annuiizaion of 39% of iniial asses. Given he annuiy pricing in Table, his implies ha he reiree would receive half of his or her 5% wihdrawal from an annuiy and half from liquid asses. The wihdrawal rae from liquid asses would hen be 4.1% (i.e.,.5% divided by 61%). This suggess he following rule: Given ha a 4% wihdrawal rae is safe, reirees should consider annuiizing enough so ha hey can wihdraw 4% from heir remaining liquid asses. 5. EXTENSIONS OF THE ANALYSIS 5.1 Evaluaing he Impac of Adding Typical Fixed (Non-Inflaion-Indexed) Annuiies o a Porfolio I is convenien mahemaically and heoreically o allow reiree porfolios o include inflaion-indexed annuiies; however, in he U.S. marke inflaion-indexed annuiies are rare. Therefore, i is desirable o undersand how a ypical non-inflaion-indexed fixed annuiy fares in his model. I is simple o evaluae he mean and sandard deviaion of wealh a deah sochasically in he presence of hese annuiies; however, one is obligaed o add a simulaed random variable represening increase in CPI (wih sensible correlaions o he reurn on fixed income asses.) I is also possible o evaluae his porfolio analyically given he (unrealisic) assumpion of a consan rae of inflaion. The basic analysis is essenially idenical o ha given in Secion 3 and in he Appendix (alhough he algebra is messier). Assuming a consan rae of inflaion and assuming ha he pricing of he ypical fixed annuiy is consisen wih he pricing of he inflaion-indexed annuiy (i.e., idenical moraliy wih ineres raes There is no perfec risk-free asse class ha exends from reiremen o deah; however, a Treasury I-bond provides a good approximaion.

7 34 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 Figure Pos-Reiremen Efficien Fronier Real Annual Wihdrawal of 5% of Iniial Wealh

8 EFFICIENT POST-RETIREMENT ASSET ALLOCATION 35 Figure 3 Pos-Reiremen Efficien Fronier Real Annual Wihdrawal 5.% of Iniial Wealh Figure 4 Pos-Reiremen Efficien Fronier Real Annual Wihdrawal 4.% of Iniial Wealh

9 36 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 Figure 5 Pos-Reiremen Efficien Fronier Real Annual Wihdrawal 6.% of Iniial Wealh differing only by he inflaion rae), one can reach he following conclusion: The locaion of he efficien fronier does no change when one assumes ha inflaion-indexed annuiies are unavailable, and ypical fixed annuiies mus be used as a subsiue. This is no surprising because inflaion is assumed o be consan and predicable, and wo forms of annuiy are priced consisenly. In realiy, inflaion is volaile, leading (presumably) o a preference for an inflaion-indexed annuiy; on he oher hand, an inflaion-indexed annuiy would likely be priced more conservaively han a ypical annuiy (because of he addiional insurer risk and he newness of he produc). The combined impac of hese wo facors meris furher sudy. 5. Examining he Impac of Allowing Liquid Asse Allocaion o Vary wih Age I is well esablished in he economics lieraure ha, conrary o popular belief, a longer invesmen ime horizon need no necessarily lead o a higher allocaion o socks. For example, Meron (199, chs. 4 5) calculaes he opimal allocaion o equiy under a variey of uiliy funcions for an individual making wihdrawals from a fund unil a fixed ime horizon. Alhough some uiliy funcions resul in ime-dependen equiy allocaions, here are also classes of uiliy funcions leading o equiy allocaions ha remain consan wih ime. Using one of hese classes of uiliy funcions, Charupa and Table 3 Efficien Pos-Reiremen Asse Allocaion Efficien Asse Allocaion Wihdrawal Rae Annuiy (CPI-Indexed) Risk-Free Asse Bonds Socks 4% (Fig. 4) % % 66% 34% 5 (Fig. 3) (Fig. 5) No soluion given consrains

10 EFFICIENT POST-RETIREMENT ASSET ALLOCATION 37 Milevsky () exend Meron s argumen o asse allocaion wihin a variable payou annuiy yielding he same opimal (non-ime-dependen) allocaion o equiies. Furhermore, Bodie (1995) explicily argues ha socks are more risky in he long run han he shor run. Based on hese argumens, one migh expec ha allowing asse allocaion o vary wih age would no necessarily be more efficien han a consan allocaion sraegy. Experimenally his seems o be rue. Afer choosing a poin on he consan asse allocaion efficien fronier for a 65-year-old, sochasic simulaions were used o examine he impac of significanly decreasing asse risk a age 7. This shif does no succeed in producing a risk and reurn saisic ha is above he consan-allocaion efficien fronier (alhough i does reduce risk). 3 This resul, if generally rue, is imporan bu is currenly widely disregarded. 5.3 Examining he Impac of Allowing Liquid Asse Allocaion o Vary wih Wealh In conras wih he probable inefficiency of age-dependen asse-allocaion sraegies, wealhdependen asse-allocaion sraegies can easily be shown o produce risk-reward oupus ha are more efficien han consan-allocaion sraegies. For example, sochasic simulaions in which he allocaion o risk-free asses was increased wih wealh (i.e., locking in gains) produced risk-reurn saisics above he consan-allocaion efficien fronier. This resul is no surprising, corresponding boh o inuiion and o exising work (e.g., Young 4). This is clearly an area ha meris furher sudy. 5.4 Examining he Impac of Purchasing Differen Types of Annuiies or of Annuiizing a Some Laer Time In Secion 4 i was assumed ha an immediae life annuiy would be purchased a he incepion of he modeling period or no a all. In realiy, he annuiizaion decision can be deferred o some laer dae, and he decision can be based on age or on wealh-relaed crieria. The problem of when o annuiize has been discussed a some lengh in he lieraure (for a summary of mehods and conclusions see Table 4 in Blake, Cairns, and Dowd 3). Alernaively he reiree migh choose o purchase a (much cheaper) deferred payou annuiy scheduled o begin paymens on he aainmen of some higher age (e.g., 85; see Milevsky 5 for a discussion of such annuiies). The reiree could also choose o acquire oher insurance producs, such as a variable payou annuiy, a whole life policy, long-erm-care insurance, or criical illness insurance, which would change he cash flow paerns prior o end of life or he required wealh a end of life. An addiional alernaive is he purchase of a convenional variable annuiy wih a GMWB-for-life rider. 4 I is clear ha he risk reward analysis described above can be exended o he analysis of hese alernaives, and his will be he subjec of a fuure paper. APPENDIX If Z is sandard one-dimensional Brownian moion, hen i can be shown ha f()dz 1/ d[ f()] E[e ] e, (A.1) where f() is any (nonpahological) funcion. Equaion (A.1) is ypically proved as a sep in he developmen of he Girsanov heorem, by using Iô s Lemma o show ha e T T T f() dz 1/ d[ f()] 3 Tha is, alhough he sraegy of suddenly lowering equiy allocaion a age 7 reduces risk, an alernaive, consan allocaion sraegy also has he same (lower) risk bu a higher reward. 4 The GMWB-for-life rider is an opion ha can be purchased by a policyholder guaraneeing ha as long as he or she limis heir wihdrawals o a cerain dollar amoun, in he even ha he accoun is depleed, he insurer will coninue paymen of an annual dollar amoun.

11 38 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 is a maringale (e.g., see Nefci 1996). Equaion (A.1) can also be simply (alhough no rigorously) proved by considering n independen normally disribued random variables: Z 1...Z n, each wih a mean of and a variance of. For any se of n inegers f 1,...,f n, one can show ha As approaches, his becomes equaion (A.1). Using equaion (A.1): So f Z fz 1/( f ) 1/( f ) i i i i i i E[e ] E[e ] e e. i i (Z Z ) dz 1/ d 1/ s s s E[e ] E[e ] e e (s). (1/ )Z (1/ )szs z E [W ] E e W k e ds (1/ )Z (1/ )(s)(z Z s ) E W e k e ds (1/ )1/ (1/ )(s)1/ (s) We k e ds ()(s) We k e ds k We (e 1). (A.) Based on equaion (A.) i is clear ha d E Z[W ] (W k)e. (A.3) d A similar, alhough more algebraically complex, calculaion can be used o calculae he derivaive of he second momen: d k e E Z[W ] (W k) d (1/ ) e W k W. (A.4) In he model described in Secions and 3, reward and risk are defined as he mean and sandard deviaion of W T, where W T is wealh a end of life. As described above (eqs. 3.5 and 3.9), d E[W T] W d px E Z[W ], d d E[W T] W d px E Z[W ]. d Using equaions (A.3) and (A.4) and simplifying he noaion by choosing (wih no loss of generaliy) o se W $1 (which implies ha k is now he percen of iniial wealh ha is being wihdrawn annually), hen using algebra, one can derive he risk and reward formulas:

12 EFFICIENT POST-RETIREMENT ASSET ALLOCATION 39 and herefore E[W ] 1 ( k)g(), T k( k) E[W T] 1 g() (A.5) k g( ), (A.6) Var[W T] k g( ) where g(r) is defined by ( k)( k)g() ( k) [g()], r g(r) d pxe. (A.7) (A.8) Realisic values for p x can be easily generaed by means of published moraliy ables. These ables ypically provide annual moraliy raes. Assuming, for example, consan force of moraliy wihin a year, g(r) can be easily exacly calculaed in a spreadshee program. As described in Secion 3, i may be desirable o be able o calculae he momens of he presen value of W T. Forunaely, his calculaion is a relaively simple exension of he work above. Equaions (A.5) and (A.6) were derived beginning wih and clearly n n n T,Z T x x Z x Z E [W ] d p q E [W ] d p E [W ], (A.9) So by defining a new funcion f(x, ) such ha nt n n n T,Z T x x Z E [e W ] d p q e E [W ]. (A.1) and (n) n x x f (x, ) pq e (A.11) (n) lim f (x, ), (A.1) equaion (A.1) can be rewrien nt n (n) n E T,Z[e W T] d f (x, ) E Z[W ] n (n) (n) n Z W f (x, ) d f (x, )E [W ]. (A.13) Comparing equaion (A.13) wih equaions (3.5) and (3.9), i is clear ha virually all of he previous work can be reused. In paricular, equaions (A.5) and (A.6) become

13 4 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 and where T (1) (1) E[e W T] f (x, ) ( k)g () (A.14) k( k) T () () E[e W T] f (x, ) g () () k g ( ), (A.15) (n) (n) r g (r) d f (x, )e. (A.16) The evaluaion of equaions (A.14) (A.16) can be compleed by showing ha and (n) f (x, ) 1 ng(n) (A.17) n n (n) g (r) 1 g(r n) g(n), (A.18) r r where g(r) is as defined in equaion (A.8). ACKNOWLEDGMENTS I would like o hank Jon Robers and Markus Malik for heir commens on early drafs of his paper, and Alex Scheilin for many useful conversaions. In addiion I would like o hank Moshe Milevsky for poining ou cerain inconsisencies in my argumen and for providing me wih addiional references. Finally I would like o hank Jonahan Treussard for poining me o he many relevan secions in Meron (199). REFERENCES BLAKE, D., A. J. G. CAIRNS, AND K. DOWD. 3. Pensionmerics II: Sochasic Pension Plan Design during he Disribuion Phase. Insurance: Mahemaics and Economics 33: BODIE, Z On he Risk of Socks in he Long Run. Financial Analyss Journal 51(3): Reiremen Invesing: A New Approach. Boson Universiy School of Managemen Working Paper No Life Cycle Invesing in Theory and in Pracice. Financial Analyss Journal 59(1): 4 9. BRODIE, Z., A. KANE, AND A. J. MARCUS Invesmens. 4h ed. Boson: McGraw-Hill. CHARUPAT, N., AND M. A. MILEVSKY.. Opimal Asse Allocaion in Life Annuiies: A Noe. Insurance: Mahemaics and Economics 3: CHEN, P., AND M. MILEVSKY. 3. Merging Asse Allocaion and Longeviy Insurance: An Opimal Perspecive on Payou Annuiies. Journal of Financial Planning 16(6): DUS, I., R. MAURER, AND O. S. MITCHELL. 4. Being on Deah and Capial Markes in Reiremen: A Shorfall Risk Analysis of Life Annuiies versus Phased Wihdrawal Plans. Proceedings of he 14h Annual AFIR Colloquium 1: GERRARD, R., S. HABERMAN, AND E. VIGNA. 4. Opimal Invesmen Choices Pos-Reiremen in a Defined Conribuion Pension Scheme. Insurance: Mahemaics and Economics 35(): The Managemen of Decumulaion Risks in a Defined Conribuion Pension Plan. Norh American Acuarial Journal 1(1): MERTON, R. C Coninuous-Time Finance. Rev. ed. Oxford: Basil Blackwell. MILEVESKY, M. A. 5. Real Longeviy Insurance wih a Deducible: Inroducion o Advanced-Life Delayed Annuiies (ALDA). Norh American Acuarial Journal 9(4): 19. MILEVSKY, M. A., AND C. ROBINSON.. Self-Annuiizaion and Ruin in Reiremen. Norh American Acuarial Journal 4(4): NEFTCI, S. N An Inroducion o he Mahemaics of Financial Derivaives. San Diego: Academic Press. SMITH, G., AND D. P. GOULD. 7. Measuring and Conrolling Shorfall Risk in Reiremen. Journal of Invesing (spring).

14 EFFICIENT POST-RETIREMENT ASSET ALLOCATION 41 YOUNG, V. R. 4. Opimal Invesmen Sraegy o Minimize he Probabiliy of Lifeime Ruin. Norh American Acuarial Journal 8(4): Discussions on his paper can be submied unil January 1, 9. The auhor reserves he righ o reply o any discussion. Please see he Submission Guidelines for Auhors on he inside back cover for insrucions on he submission of discussions.