Annuitization and asset allocation
aa r X i v : . [ q -f i n . P M ] J un Annuitization and Asset Allocation
Moshe A. Milevsky and Virginia R. Young York University and University of MichiganCurrent Version: 25 August 2006 Milevsky, the contact author, is an Associate Professor of Finance at the Schulich School of Business,York University, Toronto, Ontario, M3J 1P3, Canada, and the Director of the Individual Finance andInsurance Decisions (IFID) Centre at the Fields Institute. He can be reached at Tel: (416) 736-2100 ext66014, Fax: (416) 763-5487, E-mail: [email protected]. This research is partially supported by a grantfrom the Social Sciences and Humanities Research Council of Canada and the Society of Actuaries. Young is a Professor of Mathematics at the University of Michigan, Ann Arbor, Michigan, 48109, USA.She can be reached at Tel: (734) 764-7227, Fax: (734) 764-7048, E-mail: [email protected]. The authorsacknowledge helpful comments from two anonymous JEDC referees, Jeff Brown, Narat Charupat, GlennDaily, Jerry Green, Mike Orszag, Mark Warshawsky, and seminar participants at
York University , Universityof Michigan, University of Wisconsin , University of Cyprus , the
American Economics Association and the
Western Finance Association annual meeting. bstractANNUITIZATION AND ASSET ALLOCATION
This paper examines the optimal annuitization, investment and consumption strategies of autility-maximizing retiree facing a stochastic time of death under a variety of institutional restric-tions. We focus on the impact of aging on the optimal purchase of life annuities which form thebasis of most Defined Benefit pension plans. Due to adverse selection, acquiring a lifetime payoutannuity is an irreversible transaction that creates an incentive to delay. Under the institutionalall-or-nothing arrangement where annuitization must take place at one distinct point in time (i.e.retirement), we derive the optimal age at which to annuitize and develop a metric to capture theloss from annuitizing prematurely. In contrast, under an open-market structure where individualscan annuitize any fraction of their wealth at anytime, we locate a general optimal annuity purchas-ing policy. In this case, we find that an individual will initially annuitize a lump sum and then buyannuities to keep wealth to one side of a separating ray in wealth-annuity space. We believe our pa-per is the first to integrate life annuity products into the portfolio choice literature while taking intoaccount realistic institutional restrictions which are unique to the market for mortality-contingentclaims.
JEL Classification:
J26; G11
Keywords:
Insurance; Mortality; Retirement; Contingent-Claims; Financial Economics
Introduction and Motivation
Asset allocation and consumption decisions towards the end of the human life cycle are complicatedby the uncertainty associated with the length of life. Although this risk can be completely hedged ina perfect market with life annuities – or, more precisely, with continuously-renegotiated tontines –real world frictions and imperfections impede the ability to do so in practice. Indeed, empirical andanecdotal evidence suggests that voluntary annuitization amongst the public is not very common,nor is it well understood even amongst financial advisors. Therefore, in attempt to fill this gap andintegrate mortality-contingent claims into the finance literature, this paper examines the optimalannuitization strategy of a utility-maximizing retiree facing a stochastic time of death under avariety of institutional pension and annuity arrangements. We also examine the usual investmentand consumption dynamics but focus our attention on the impact of aging and the increase in theactuarial force of mortality on the optimal purchase of mortality-contingent annuities, which formthe basis of most Defined Benefit (DB) pension plans. One of our main insights is that due tosevere adverse selection concerns, acquiring a lifetime payout annuity is an irreversible transactionthat, we argue, creates an incentive to delay. We appeal to the analogy of a classical Americanoption which should only be exercised once the value from waiting is no more than the value fromexercising.For the most part of the paper, the focus of our attention is a life annuity that pays a fixed(real or nominal) continuous payout for the duration of the annuitant’s life. (The appendix extendsour basic model to variable immediate annuities.) From a financial perspective, this product isakin to a coupon-bearing bond that defaults upon death of the holder and for which there is nosecondary market. Under the institutional all-or-nothing arrangement, where annuitization (i.e.the purchase) must take place at one distinct point in time (i.e. retirement), we locate the optimalage at which to annuitize and develop a metric to capture the loss from annuitizing prematurely.This optimal age, which is linked to the actuarial force of mortality, occurs within retirement yearsand is obviously gender specific but also depends on the individual’s subjective health status. Allof this will be explained in the body of the paper.In contrast to the restrictive (yet not uncommon) all-or-nothing arrangement, under an open-market structure where individuals can annuitize a fraction of their wealth at distinct points intime, we locate a general optimal annuity purchasing policy. In this case, we find that an individualwill initially annuitize a lump sum – if they do not already have this minimum level in pre-existingDB pensions – and then buy additional life annuities in order to keep wealth to one side of aseparating ray in wealth-annuity space. This is a type of barrier control result that is common inthe literature on asset allocation with transaction costs.We believe our paper is the first to integrate life and pension annuity products into the portfolio The first (working paper) version of this paper was circulated and presented at the January 2001 meeting of the
American Economics Association in New Orleans.
The remainder of this paper is organized as follows. In Section 2, we provide a brief explanation ofthe mechanics of the life annuity market and review the existing literature involving asset allocation,personal pensions, and payout annuities. In Section 3, we present the general model for our financialand annuity markets. In Section 4, we consider the case for which the individual is required toannuitize all her pensionable wealth at one point in time. This is effectively an optimal retirementproblem and is akin to the situation (up until recently) in the United Kingdom, where retireescan drawdown their pension but must annuitize the remaining balance by a certain age, or to thesituation for which individuals have the choice of when to start their retirement (DB) pension butmust do so at one point in time. In fact, most Variable Annuity contracts sold in the United Stateshave an embedded option to annuitize that can only be exercised once. Our analysis would coverthis too. In this restrictive (but common) framework, we locate the optimal age for her to do so, andthen define a so-called option value metric as the gain in utility from annuitizing optimally. Section5 provides a variety of numerical examples for the optimal time to annuitize and also pursues theoption analogy as a way of illustrating the loss from annuitizing pre-maturely.Then, in Section 6, we consider a less restrictive open-market arrangement whereby the indi-vidual may annuitize any portion of her wealth at any time. This arrangement is applicable toindividuals with substantial discretionary wealth who can purchase small (or large) quantities ofannuities on an ongoing basis. In this case, we find that the individual annuitizes a lump sum assoon as possible (the amount might be zero) and then acquires more annuities depending on theperformance of her stochastic wealth process. If her wealth subsequently increases in value, shepurchases more annuities by annuitizing additional wealth; otherwise, she refrains from additionalpurchases and consumes from her originally-purchased annuities, as well as from liquidating invest-ments in her portfolio. Furthermore, we explicitly solve for the optimal annuity purchasing policyunder this less restrictive case when the force of mortality is constant, which implies that the futurelifetime is exponentially distributed. Section 7 provides a variety of numerical examples for theopen-market framework and flushes-out and explores a number of insights. Non-essential proofsand theorems are relegated to an Appendix (Section 9), while Section 8 concludes the paper withour main qualitative insights. 2
The Annuity Market and Literature
Life annuities are purchased directly from insurance companies and form the basis of most DBpension plans. In exchange for a lump-sum premium, which the company invests in its generalaccount, the company guarantees to pay the annuitant a fixed (monthly or quarterly) payout forthe rest of his or her life. This payout rate – which depends on prevailing interest rates and mortalityprojections – is irrevocably determined at the time of purchase (a.k.a. annuitization) and does notchange for the life of the contract. The following chart illustrates some sample quotes which wereprovided by a life-annuity broker (in Canada).Certain \ Age m f m f m f m f m f m f
631 590 686 633 765 694 877 780 1039 911 1259 1096
628 589 681 631 755 689 855 770 989 887 1146 1036
10 yrs
620 584 666 623 726 674 799 741 879 825 940 901
15 yrs
607 578 644 611 687 652 729 699 764 745 774 765
20 yrs
591 569 618 596 643 625 662 651 673 668 665 664
25 yrs
573 559 589 578 601 594 608 605 610 609 N.A. N.A.
Sample monthly payout based on a $100,000 initial premium (purchase)
The term word certain refers to the guarantee period built into the annuity. If the guaranteeperiod is n years, then the individual buying such an annuity (or his or her estate) will receive thestated income for n years; thereafter, the individual will receive the money only if he or she is alive.For example, in exchange for a $100,000 initial premium, a 75-year-old female will receive $911per month for the rest of her life. This life annuity has no guarantee period, which means that ifshe were to die one instant after purchasing the life annuity (technically it would have to be afterthe first payment), her beneficiaries or estate would receive nothing in return. The $911 monthlyincome for those who survive consists of a mix of principal and interest as well as the implicit fundsof those who do not survive. A male would receive slightly more per month, namely $1,039, due tothe lower life expectancy of males.A few things should be obvious from the table. First, the higher the purchase age, all else beingequal, the greater the annuity income. In this case, the future life expectancy is shorter and theinitial premium must be amortized and returned over a shorter time period. Likewise, a longerguarantee period yields a lower annuity income. In fact, an 80-year-old female buying a 20-yearguarantee will receive virtually the same amount ($664) as a male of the same age, since neitheris likely to live past the 20-year certain period; hence, the annuity is essentially a portfolio ofzero-coupon bonds. Some other points are in order, especially for those not familiar with insurancepricing concepts. 3. The law of large numbers and the ability to diversify mortality risk is central to the pricing oflife annuities. The above-mentioned payouts are determined by expected objective annuitantmortality patterns together with prevailing interest rates of corresponding durations. Profits,fees, and commissions are built into these quotes by loading the pure actuarial factor on theorder of 1% to 5%.2. Payout rates fluctuate from week-to-week because most of the insurance companies’ assetsbacking these lifetime guarantees are invested in fixed-income instruments. This can some-times cause quotes to change on a daily basis. It is therefore reasonble to model the evolutionof these prices in continuous time.3. Most of the existing open-purchase (a.k.a retail) annuity market in North America is basedon fixed nominal (and not inflation-adjusted) payouts. Real annuities are quite rare, which isan ongoing puzzle to many economists. Consequently, most of the numerical examples in ourpaper focus on nominal values, although there is nothing in our model that precludes usinginflation-adjusted prices and returns as long as they are not mixed in the same model.4. An additional form of life annuity is the variable payout kind whose periodic income is linkedto the performance of pre-selected equity and bond indices. In this case, the above-mentioned$100,000 premium would go towards purchasing a number of payout units (as opposed todollars) whose value would fluctuate over time. These annuities are the foundation of theUS-based TIAA-CREF’s pension plan for University workers, but are quite rare anywhereelse in the world. As a result, the bulk of our paper addresses the fixed payout kind, but werefer the interested reader to Appendix C (namely, Section 9.3) in which these products areintegrated into our model. This paper merges a variety of distinct strands in the portfolio choice and annuity literature. First,our work sits squarely within the classical Merton (1971) optimal asset allocation and consumptionframework. However, in contrast to extensions of this model by Kim and Omberg (1996), Koo(1998), Sorensen (1999), Wachter (2002), Bodie, Detemple, Otruba, and Walter (2004), or therecent book by Campbell and Viceira (2002), for example – which are concerned with relaxing thedynamics of the underlying state variables and/or investigating the impact of (retirement) timehorizon on portfolio choice – our model attempts to realistically incorporate mortality-contingentpayout annuities within this framework.A life-contingent annuity is the building block of most DB pension plans – see Bodie, Marcus,and Merton (1988) for details – but can also be purchased in the retail market. The irreversibilityof this purchase is due to the well-known adverse selection issues identified by Akerlof (1970) andRothschild and Stiglitz (1976). 4n its own, the topic of payout annuities has been investigated quite extensively within thepublic economics literature. In fact, a so-called annuity puzzle has been identified in this field.The puzzle relates to the incredibly low levels of voluntary annuitization exhibited by retirees whoare given the choice of purchasing a mortality-contingent payout annuity. For example, holders of variable annuity saving We also argue that the decision of when to purchase an irreversible life annuity endows theholder with an incentive to delay that can be heuristically viewed as an option. Indeed, undermany institutional pension arrangements (such as in the United Kingdom up until recently or withregards to the rules concerning variable annuity saving policies in the U.S.), individuals are allowed Recent papers that attempt a portfolio-based model for annuitization along the same lines – most written afterthe first draft of this paper was released – include Kapur and Orszag (1999), Blake, Cairns, and Dowd (2000), Cairns,Blake, and Dowd (2005), Neuberger (2003), Dushi and Webb (2003), Sinclair (2003), Stabile (2003), and Battocchio,Menoncin, and Scaillet (2003), Koijen, Nijman and Werker (2006).
5o drawdown their pension via discretionary consumption but must eventually annuitize at onepoint in time their remaining wealth. We refer to this system as an all-or-nothing arrangement andargue that this is similar to Stock and Wise’s (1990) option to retire and echoes the framework ofSundaresan and Zapatero (1997) who examine optimal behavior (and valuation) of various pensionbenefits. Other institutional structures allow for annuitization at any time and in small quantitiesas well, and we refer to these systems as anything anytime throughout the paper. We investigatethe optimal annuitization policy in both of these cases and provide extensive numerical examplesthat compare the two.
In this section, we describe our model for financial and annuity markets. We assume that anindividual can invest in a riskless asset whose price at time s , X s , follows the process dX s = rX s ds, X t = X >
0, for some fixed r ≥
0. Also, the individual can invest in a risky asset whoseprice at time s , S s , follows geometric Brownian motion given by ( dS s = µS s ds + σS s dB s , S t = S >
0, (1)in which µ > r, σ >
0, and B is a standard Brownian motion with respect to a filtration {F s } ofthe probability space (Ω , F , P ). Let W s be the wealth at time s of the individual, and let π s bethe amount that the decision maker invests in the risky asset at time s . Also, the decision makerconsumes at a rate of c s at time s . Then, the amount in the riskless asset is W s − π s , and whenthe individual buys no annuities, wealth follows the process dW s = d ( W s − π s ) + dπ s − c s ds = r ( W s − π s ) dt + π s ( µds + σdB s ) − c s ds = [ rW s + ( µ − r ) π s − c s ] ds + σπ s dB s , W t = w >
0. (2)In Sections 4 and 5, we assume that the decision maker seeks to maximize (over admissible { c s , π s } and over times of annuitizing all his or her wealth, τ ) the expected utility of discountedconsumption. Admissible { c s , π s } are those that are measurable with respect to the informationavailable at time s , namely F s , that restrict consumption to be non-negative, and that result in (2)having a unique solution; see Karatzas and Shreve (1998). We also allow the individual to valueexpected utility via a subjective hazard rate (or force of mortality), while the annuity is priced byusing an objective hazard rate; which may or may not be the same.Our financial economy is based on the (simpler) geometric Brownian motion plus risk-free ratemodel originally pioneered by Merton (1971), as opposed to the more recent and richer modelsdeveloped by Kim and Omberg (1996), Sorensen (1999), Wachter (2002), or Campbell and Viceira62002) for example. The reason is that we are primarily interested in the implications of intro-ducing a mortality-contingent claim into the portfolio choice framework, as opposed to studyingthe impact of stochastic interest rates or mean-reverting equity premiums per se . By avoiding thecomputational price of a more complex set-up, we are able to obtain analytical solutions to ourannuitization problems.We now move on to the insurance and actuarial assumptions. We let t p Sx denote the subjectiveconditional probability that an individual aged x believes he or she will survive to age x + t . It isdefined via the subjective hazard function, λ Sx + t , by the formula t p Sx = exp (cid:18) − Z t λ Sx + s ds (cid:19) . (3)We have a similar formula for the objective conditional probability of survival, t p Ox , in terms of theobjective hazard function, λ Ox + t .The actuarial present value of a life annuity that pays $1 per year continuously to an individualwho is age x at the time of purchase is written ¯ a x . It is defined by¯ a x = Z ∞ e − rt t p x dt. (4)We deliberately use the risk-free rate r in our annuity pricing because most of the recent empiricalevidence suggests that the money’s worth of annuities relative to the risk-free Government yieldcurve is relatively close to one. In other words, the expected present value of payouts using therisk-free rate is equal to the premium paid for that benefit. Thus, it appears that the additionalcredit risk that the insurance company might take on by investing in higher risk bonds is offset byany insurance loads and commissions they charge. We refer the interested reader to the paper byMPWB (1999) for a greater discussion of the precise curve that is used for pricing in practice.In terms of notation, if we use the subjective hazard rate to calculate the survival probabilitiesin equation (4), then we write ¯ a Sx , while if we use the objective (pricing) hazard rate to calculatethe survival probabilities, then we write ¯ a Ox . Just to clarify, by objective ¯ a Ox , we mean the actualmarket prices of the annuity net of any insurance loading, whereas ¯ a Sx denotes what the marketprice ‘would have been’ had the insurance company used the individual’s personal and subjectiveassessment of her mortality.We refer the interested reader to Hurd and McGarry (1995, 1997) for a discussion of exper-iments involving “subjective” versus “objective” assessments of survival probabilities. We will Also, more recently, Smith, Taylor, and Sloan (2001) claim that their “findings leave little doubt that subjectiveperceptions of mortality should be taken seriously.” They state that individuals’ “longevity expectations are reason-ably good predictions of future mortality.” Other researchers, such as Bhattacharya, Goldman and Sood (2003) claimthat individuals are biased in their estimates of mortality as evidenced by the viatical and life settlement market.We do not take a position on whether individuals estimate and discount mortality using the same forward curve asthe insurance company and therefore allow for the two functions to be distinct. perfect market
Yaari (1965) framework,subjective survival rates do not play a role in the optimal policy. We will show that if the con-sumer disagrees with the insurance company’s pricing basis regarding her subjective hazard rate -or personal health status - she will delay annuitization in an all-or-nothing environment.In Section 6, we start by assuming that the decision maker maximizes (over admissible { c s , π s , A s } )the expected utility of discounted lifetime consumption as well as bequest, in which A s is the an-nuity purchasing process. A s denotes the non-negative annuity income rate at time s after anyannuity purchases at that time; we assume that A s is right-continuous with left limits. The sourceof this income could be previous annuity purchases or a pre-existing annuity, such as Social Securityor pension income. We assume that the individual can purchase an annuity at the price of ¯ a Ox + s per dollar of annuity income at time s , or equivalently, at age x + s . In that case, the dynamics ofthe wealth process are given by ( dW s = [ rW s − + ( µ − r ) π s − c s + A s − ] ds + σπ s dB s − ¯ a Ox + s dA s , W t − = w >
0. (5)The negative sign on the subscripts for wealth and annuities denotes the left-hand limit of thosequantities before any (lump-sum) annuity purchases.
In this section, we examine the institutional arrangement where the individual is required to an-nuitize all her wealth in a lump sum at some (retirement) time τ . If the volatility of investmentreturn σ = 0, and assuming the objective hazard rate increases over time, then we show that theindividual annuitizes all her wealth at a time T for which µ = r + λ Ox + T , which is the time at whichthe hazard rate (a.k.a. mortality credits) plus the risk-free rate is equal to the expected return fromthe asset. Furthermore, if λ Sx + t = λ Ox + t for all t >
0, then the individual will optimally consumeexactly the annuity income after time T . Therefore, for small values of σ and for λ Sx + t ≈ λ Ox + t for all t ≥
0, the individual will consume approximately the annuity income after annuitizing herwealth, at least for time soon after the annuitization time. In fact, the classical annuity results, forexample Yaari (1965), prove that consuming the entire annuity income is optimal in the absence ofbequest motives. This is why, to simplify our work we assume that at some time τ , the individualannuitizes all her wealth W τ and thereafter consumes at a rate of W τ ¯ a Ox + τ , the annuity income.It follows that the associated value function of this problem is given by8 ( w, t )= sup { c s ,π s ,τ } E w,t (cid:20)Z τt e − r ( s − t ) s − t p Sx + t u ( c s ) ds + Z ∞ τ e − r ( s − t ) s − t p Sx + t u (cid:18) W τ ¯ a Ox + τ (cid:19) ds (cid:21) = sup { c s ,π s ,τ } E w,t (cid:20)Z τt e − r ( s − t ) s − t p Sx + t u ( c s ) ds + e − r ( τ − t ) τ − t p Sx + t u (cid:18) W τ ¯ a Ox + τ (cid:19) ¯ a Sx + τ (cid:21) , (6)in which u is an increasing, concave utility function of consumption, and E w,t denotes the expecta-tion conditional on W t = w . Note that the individual discounts consumption at the riskless rate r .If we were to model with a subjective discount rate of say ρ , then this is equivalent to using r as in(6) and adding ρ − r to the subjective hazard rate. Thus, there is no effective loss of generality insetting the subjective discount rate equal to the riskless rate r . In other words, while some life-cyclemodels in the literature adjust the discount rate for perceived risk and other subjective factors, weremind the reader that our underlying hazard rate λ Sx + t effectively adjusts the discount rate for theprobability of survival and, thus, takes these risks into account implicitly.Note that in this section, we do not account for pre-existing annuities, such as state and cor-porate pensions for example. We anticipate that such annuities will change the optimal time ofannuitization, but we defer this problem to Section 6. Also, note that we take the annuity prices asexogenously given. We are not creating an equilibrium (positive) model of pricing as in the adverseselection literature of Akerlof (1970) or Rothschild and Stiglitz (1976), but rather a normativemodel of how people should behave in the presence of these given market prices. Expanding toequilibrium considerations is beyond the scope of this (normative) paper, although we do makesome statements regarding equilibrium pricing of annuities in the concluding remarks.We also restrict our attention to the case in which the utility function exhibits constant relativerisk aversion (CRRA), γ = − cu ′′ ( c ) /u ′ ( c ). That is, u is given by u ( c ) = c − γ − γ , γ > , γ = 1 . (7)For this utility function, the relative risk aversion equals γ , a constant. The utility function thatcorresponds to relative risk aversion 1 is logarithmic utility.We next show that for CRRA utility, solving the problem in (6) is equivalent to assuming thatthe optimal stopping annuitization time is some fixed time in the future, say T . Based on that valueof T , one finds the optimal consumption and investment policies. Finally, one finds the optimalvalue of T ≥ τ is not random, but rather deterministic. For other utilityfunctions, τ will be a random stopping time that depends on stochastic (state variable of) wealth,as is the case for the exercise time of American options.To show that the optimal stopping time is deterministic for CRRA utility, we use the fact thatif we can find a smooth solution V to the following variational inequality, then that smooth solution9quals the value function U in (6); see Øksendal (1998, Chapter 10), for example. (cid:0) r + λ Sx + t (cid:1) V ≥ V t + rwV w + max c [ u ( c ) − cV w ] + max π (cid:20) ( µ − r ) πV w + 12 σ π V ww (cid:21) , (8)and V ( w, t ) ≥ ¯ a Sx + t u (cid:18) w ¯ a Ox + t (cid:19) , (9)with equality in at least one of (8) and (9), and with u given by (7).We look for a solution of this variational inequality of the form V ( w, t ) = − γ w − γ ψ γ ( t ). If V is of this form, then ψ necessarily solves11 − γ (cid:0) r + λ Sx + t (cid:1) ψ ≥ γ − γ ψ ′ + δψ + γ − γ , (10)and 11 − γ ψ γ ( t ) ≥ − γ ¯ a Sx + t (cid:0) ¯ a Ox + t (cid:1) − γ , (11)with equality in at least one of (10) and (11). So, if we find a smooth solution ψ to this variationalinequality in time t , then we are done, and we can assert that U ( w, t ) = − γ w − γ ψ γ ( t ). The keyfeature to note is that wealth w and time t are multiplicatively separable in U ; thus, the optimaltime to annuitize one’s wealth is independent of wealth and is, therefore, deterministic.To solve the variational inequality for ψ , we hypothesize that the “continuation region” (i.e.the time when one does not annuitize) is of the form (0 , T ). In other words, one does not annuitizeone’s wealth until time T . If our hypothesis is correct, then (10) holds with equality on (0 , T ),while (11) holds with inequality on (0 , T ) and with equality at t = T . Given any value of T , we cansolve this boundary-value problem; write φ = φ ( t ; T ) as the solution of this problem. For t < T ,the function φ is given by φ ( t ; T ) = ¯ a Sx + T (cid:0) ¯ a Ox + T (cid:1) − γ ! γ e − r − δ (1 − γ ) γ ( T − t ) (cid:0) T − t p Sx + t (cid:1) γ + Z Tt e − r − δ (1 − γ ) γ ( s − t ) (cid:0) s − t p Sx + t (cid:1) γ ds, (12)with δ = r + γ (cid:0) µ − rσ (cid:1) . For t ≥ T , we have φ ( t ; T ) = ¯ a Sx + T (cid:0) ¯ a Ox + T (cid:1) − γ ! γ . (13)The function ψ is the smooth solution of the variational inequality in (10) and (11), for whichthere will be a unique T . (That is, φ will have a continuous first derivative at t = T ≥ T ≥ − γ φ γ ( t ; T ) is maximized. Note that it is also possiblethat the optimal time to annuitize is right now (i.e. T = 0).10efine the function ˜ U by ˜ U ( w, t ; T ) = − γ w − γ φ γ ( t ; T ). Then, the value function U in (6) isgiven by U ( w, t ) = max T ≥ ˜ U ( w, t ; T ), in which the optimal value of T is independent of wealth w because w factors from the expression for ˜ U . In other words, the optimal time to annuitize is deterministic at time t .One can obtain the optimal consumption and investment policies from the first-order necessaryconditions in (8); they are given in feedback form by C ∗ t = c ∗ ( W ∗ t , t ) = W ∗ t ψ ( t ) , (14)and Π ∗ t = π ∗ ( W ∗ t , t ) = µ − rσ γ W ∗ t , (15)respectively, in which W ∗ t is the optimally controlled wealth before annuitization (time T ). If weare in the case of logarithmic utility one can show that the optimal consumption rate is C ∗ t = W ∗ t ¯ a Sx + t . It is interesting to note that if r = 0, in which case the denominator of the optimal consumption, ψ , collapses to a (subjective) life expectancy, then the consumption rate is precisely the minimumrate mandated by the U.S. Internal Revenue Service for annual consumption withdrawals fromIRAs after age 71. Specifically, the proportion required to be withdrawn from one’s annuity eachyear equals the start-of-year balance divided by the future expectation of life. Because r > U = ˜ U ( w, t ; T ) with respect to T , whileassuming t < T . One can show that δ ˜ UδT ∝ γ − γ ¯ a Sx + T ¯ a Ox + T ! − − γγ − − γ + ¯ a Sx + T ¯ a Ox + T + ¯ a Sx + T (cid:2) δ − (cid:0) r + λ Ox + T (cid:1)(cid:3) . (16)Thus, if the expression on the right-hand side of (16) is negative for all T ≥
0, then it is optimalto annuitize one’s wealth immediately, and we have U ( w, t ) = ˜ U ( w, t ; 0). However, if there existsa value T ∗ > ≤ T < T ∗ and isnegative for all T > T ∗ , then it is optimal to annuitize one’s wealth at time T ∗ , and we have U ( w, t ) = ˜ U ( w, t ; T ∗ ). In all the examples we present below, one of these two conditions holds. Werepeat that the decision to annuitize is independent of one’s wealth, an artifact of CRRA utility.It is straightforward to show that ˜ U ( w, t ; T ) having a continuous derivative at t = T > T is a critical point of ˜ U ; that is, the right-hand side of (16) is zero at that value of T .Moreover, if that critical point T = T ∗ maximizes ˜ U ( w, t ; T ), then (11) holds with strict inequalityon (0 , T ∗ ), as desired. See the earlier working paper version of this paper, Milevsky and Young (2002a) for a proof and exploration ofthis fact.
11f the subjective and objective forces of mortality are equal, then we have δ ˜ UδT ∝ [ δ − ( r + λ x + T )] . (17)In this case, if the hazard rate λ x + t is increasing with respect to time t , then either δ ≤ ( r + λ x ),from which it follows that it is optimal to annuitize one’s wealth immediately, or δ > ( r + λ x ), fromwhich it follows that there exists a time T in the future (possibly infinity) at which it is optimalto annuitize one’s wealth. The optimal age to purchase a fixed life annuity is when the force ofmortality λ x is greater than a constant (reminiscent of Merton’s constant) defined by: M := 12 γ (cid:18) µ − rσ (cid:19) . (18)One can then think of the hazard rate as a form of excess return on the annuity due to the embeddedmortality credits and the fact that liquid wealth reverts to the insurance company when the buyerof the annuity dies. This annuity purchase condition leads to a number of appealing insights.Namely, higher levels of risk aversion ( γ ) and higher levels of investment volatility ( σ ) lead to lowerannuitization ages, since the constant M decreases under larger γ, σ and increases under higherlevels of µ .We observe that if the subjective force of mortality is different than the objective force ofmortality, then the optimal time of annuitization increases from the T given by the zero of theright-hand side of (16). We can show mathematically that this is true if the subjective force ofmortality varies from the objective force to the extent that ¯ a Sx < a Ox for all ages x (see AppendixA), and we conjecture that it is true in general. Note that this inequality is automatically true forpeople who are less healthy because in this case ¯ a Sx < ¯ a Ox for all x . For an individual who is lesshealthy than the average person, the annuity will be too expensive, and the person will want todelay annuitizing her wealth.On the other hand, for an individual who is healthier than the average person, the annuitywill be relatively cheap. However, such a healthy person will live longer on average and will beinterested in receiving a larger annuity benefit by consuming less now and by waiting to buy theannuity later in life. Therefore, a healthy person is also willing to delay annuitizing her wealth inexchange for a larger annuity benefit (for a longer time). Note that this result is likely driven bythe fact that in this model, we force the investor to consume the totality of the income from theannuity. If the investor were allowed to save some of that annuity income and invest it in the stockmarket, this result would not necessarily hold.Of course, by following the optimal policies of investment, consumption, and annuitizing one’swealth, an individual runs the risk of being able to consume less after annuitizing wealth than ifshe had annuitized wealth immediately at time t = 0. Naturally, there is the chance of the exactopposite, namely that the lifetime annuity stream will be higher. Therefore, to quantify this risk,we calculate the probability associated with various consumption outcomes. See Appendix B for12he formula of this probability. We include calculations of it in a numerical example below.Finally, we define a metric for measuring the loss in value from annuitizing prematurely bycomputing the additional wealth that would be required to compensate the utility maximizer forforced annuitization. This is akin to the annuity equivalent wealth used by MPWB (1999), whichwe prefer to label a subjective option value. Technically, it is defined to be the least amount ofmoney h that when added to current wealth w makes the person indifferent between annuitizingnow (with the extra wealth) and annuitizing at time T (without the extra wealth). Thus, h is givenby U ( w, t ; T ) = U ( w + h, t ; 0) , (19)in which T is the optimal time of annuitization. In the examples in the next section, we express h as a percentage of wealth w . This is appropriate because U exhibits CRRA with respect to w . In this section, we present two numerical examples to illustrate the results from the previous section.To start, although most mortality tables are discretized, we require a continuous-time mortalitylaw. We use a Gompertz force of mortality, which is common in the actuarial literature for annuitypricing. See Frees, Carriere, and Valdez (1996) for examples of this model in annuity pricing. Thismodel for mortality has also been employed in the economics literature for pricing insurance; seeJohansson (1996), for example. The force of mortality is written λ x = exp (( x − m ) /b ) /b in which m is a modal value and b is a scale parameter. Note how the force of mortality itself increasesexponentially with age.In this paper, we fit the parameters of the Gompertz, namely m and b , to the IndividualAnnuity Mortality 2000 (basic) Table with projection scale G. For males, we fit parameters ( m, b ) =(88 . , . . , . . Initially, we assume that the subjective and objective forcesof mortality are equal. Throughout this section, we assume that the seller of the annuity uses thefemale hazard rate to price annuities for women; similarly, for men. Figure 1 shows the graph ofthe probability density function of the future-lifetime random variable under a Gompertz hazardrate that is fitted to the discrete mortality table. Figure 1 about here.
As for the capital market parameters, in both our examples, the risky stock is assumed to havedrift µ = 0 .
12 and volatility σ = 0 .
20. This is roughly in line with numbers provided by IbbotsonAssociates (2001), which are widely used by practitioners when simulating long-term investment We actually fit a Makeham hazard rate, or force of mortality, namely λ + exp (( x − m ) /b ) /b in which λ ≥ λ was 0, so the effective form of the hazard rateis Gompertz (Bowers et al., 1997). r = 0 .
06. We displayvalues for the option to delay annuitization h , for three different levels of risk aversion, γ = 1(logarithmic utility) and γ = 2 and γ = 5. A variety of studies have estimated the value of γ to lie between 1 and 2. See, the paper by Friend and Blume (1975) that provides an empiricaljustification for constant relative risk aversion, as well as the more recent MPWB (1999) paper inwhich the CRRA value is taken between 1 and 2. In the context of estimating the present valueof a variable annuity for Social Security, Feldstein and Ranguelova (2001) provide some qualitativearguments that the value of CRRA is less than 3 and probably even less than 2. On the otherhand, some of the equity premium literature, see Campbell and Viceira (2002) suggests that riskaversion levels might be much higher, which is why we have also displayed results for γ = 5. Table 1 provides the optimal age of annuitization – and what we have labeled the value of theoption to delay as a percentage of initial wealth – as well as the probability of consuming less atthe optimal time of annuitization than if one had annuitized one’s wealth immediately. We referto this as the probability of a deferral failure. We provide numerical results for both males andfemales under very low ( γ = 1), low ( γ = 2) and high ( γ = 5) coefficients of relative risk aversion. Table 1a about here.
Note that females annuitize at older ages compared to males because the mortality rate offemales is lower at each given age. Also, note that more risk averse individuals wish to annuitizesooner, an intuitively pleasing result. However, notice that even at relatively high ( γ = 5) levels ofrisk aversion, males do not annuitize prior to age 63 and females do not annuitize prior to age 70.Finally, the value of the option to delay annuitization – which is effectively the certainty equivalentof the welfare loss from annuitizing immediately – decreases as one gets closer to the optimal ageof annuitization, as one expects. Table 1b about here.
The probability of deferral failure reported in Table 1a, although seemingly high, is balancedby the probability of ending up with more than, say, 20% of the original annuity amount. Forexample, for a 70-year-old female with γ = 2, the probability of consuming at least 20% more atthe optimal age of annuitization than if she were to annuitize immediately is 0.474. Obviously,on a utility-adjusted basis this is a worthwhile trade-off as evidenced by the behavior of the valuefunction. See Table 2 for tabulations of the probability that the individual consumes at least 20%more at the optimal age of annuitization than if he or she were to annuitize immediately, for variousages and for γ = 1 and 2. 14 able 2 about here. These “upside” probabilities decrease as the optimal age of annuitization approaches. Also, fora given age, they decrease as the CRRA increases. This makes sense because a less risk-averseperson is less willing to face a distribution with a higher variance.
We continue the assumptions in the previous example as to the financial market. We have amale aged 60 with γ = 2, whose objective mortality follows that from the previous example;that is, annuity prices are determined based on the hazard rate given there. For this example,suppose that the subjective force of mortality is a multiple of the objective force of mortality;specifically, λ Sx = (1 + f ) λ Ox , in which f ranges from − proportional hazard transformation in actuarial science introducedby Wang (1996), and it is similar to the transformation examined by Johansson (1996) in theeconomic context of the economic value of increasing one’s life expectancy.In Table 3, we present the imputed value of the option to delay annuitization, the optimal ageof annuitization, the optimal rate of consumption before annuitization (as a percentage of currentwealth), and the rate of consumption after annuitization (also, as a percentage of current wealth).For comparison, if the male were to annuitize his wealth at age 60, the rate of consumption wouldbe 8.34%. Also, the optimal proportion invested in the risky stock before annuitization is 75%. Table 3 about here.
Note that as the 60-year-old male’s subjective mortality gets closer to the objective (pricing)mortality, then the optimal age of annuitization decreases. It seems that the optimal age of annu-itization will be a minimum when the subjective and objective forces of mortality equal, at leastfor increasing forces of mortality. We conjecture that this result is true in general, but we onlyhave a proof of it when ¯ a Sx < a Ox ; see Appendix A. Also, note that the consumption rate beforeannuitization increases as the person becomes less healthy, as expected.Compare these rates of consumption with 8.34%, the rate of consumption if the male were toannuitize his wealth immediately. We see that if the male is healthy relative to the pricing forceof mortality, then he is willing to forego current consumption in exchange for greater consumptionwhen he annuitizes, at least up to f = − . . Past that point, the optimal rate of consumption beforeannuitization is greater than 8.34%. For a 60-year-old male with f = − . initial wealth. We also graph the 25 th and 75 th percentiles of his random consumption. This individualexpects to live to age 84.4. Note that the annuitant has roughly a 70% chance of consuming morethroughout the remaining life compared to annuitizing at age 60.15 igure 2 about here. In this section, we consider the optimal annuity-purchasing problem for an individual who seeksto maximize her expected utility of lifetime consumption and bequest. In Section 6.1, we allowthe individual to have rather general preferences, while in Section 6.2, we specialize to the case forwhich preferences exhibit constant relative risk aversion. We allow the individual to buy annuitiesin lump sums or continuously, whichever is optimal. Our results are similar to those of Dixit andPindyck (1994, pp 359ff). They consider the problem of a firm’s irreversible capacity expansion.For our individual, annuity purchases are also irreversible, and this leads to the similarity in results.Specifically, a discrete jump in wealth can only occur at the initial instant (in our case, with a lump-sum purchase of an annuity; in their case, with an initial investment of capital); thereafter, theannuity income remains constant or increases incrementally to keep wealth below a given barrier(for Dixit and Pindyck, capital stock was either constant or changed incrementally). In other words,the optimal control is a “barrier control” policy.In Section 6.2.1, we continue with CRRA preferences and linearize the HJB equation in theregion of no-annuity purchasing via a convex dual transformation in the case for which there isno bequest motive. In Section 6.2.1.1, we provide an implicit analytical solution to the optimalannuity purchasing problem developed in Section 6.2.1 in the case for which the force of mortalityis constant. This leads us to Section 7, which provides a full set of numerical results.
In this section, we show that the individual’s optimal annuity purchasing is given by a barrierpolicy in that she will annuitize just enough of her wealth to stay on one side of the barrier inwealth-annuity space. In equation (5), we described the dynamics of the wealth for this individual.Denote the random time of death of our individual by τ d . We assume that τ d is independent of therandomness in the financial market, namely the Brownian motion B driving the stock price. Thus,her value function at time t , or at age x + t , is given by U ( w, A, t )= sup { c s ,π s ,A s } E w,A,t (cid:20)Z ∞ t e − r ( s − t ) s − t p Sx + t u ( c s ) ds + e − r ( τ d − t ) u ( W τ d ) (cid:21) = sup { c s ,π s ,A s } E w,A,t (cid:20)Z ∞ t e − r ( s − t ) s − t p Sx + t (cid:8) u ( c s ) + λ Sx + s u ( W s − ) (cid:9) ds (cid:21) , (20)in which u and u are strictly increasing, concave utility functions of consumption and bequest,respectively. Also, E w,A,t denotes the expectation conditional on W t − = w and A t − = A . In the last16quality, we used the independence of τ d from the Brownian motion B to simplify the expressionfor U . Note that we assume the individual discounts future consumption at the riskless rate r sincethe mortality discounting – which increases the effective discount rate – is incorporated separately.The value function U is jointly concave in w and A .We continue with a formal discussion of the derivation of the associated HJB equation. Supposethat at the point ( w, A, t ), it is optimal not to purchase any annuities. It follows from Itˆo’s lemmathat U satisfies the equation at ( w, A, t ) given by( r + λ Sx + t ) U = U t + ( rw + A ) U w + max π (cid:20) σ π U ww + ( µ − r ) πU w (cid:21) + max c ≥ [ − cU w + u ( c )] + λ Sx + t u ( w ) . (21)Because the above policy is in general suboptimal, (21) holds as an inequality; that is, for all( w, A, t ),( r + λ Sx + t ) U ≥ U t + ( rw + A ) U w + max π (cid:20) σ π U ww + ( µ − r ) πU w (cid:21) + max c ≥ [ − cU w + u ( c )] + λ Sx + t u ( w ) . (22)Next, assume that at the point ( w, A, t ) it is optimal to buy an annuity instantaneously. Inother words, assume that the investor moves instantly from ( w, A, t ) to ( w − ¯ a Ox + t ∆ A, A + ∆
A, t ).Then, the optimality of this decision implies that U ( w, A, t ) = U ( w − ¯ a Ox + t ∆ A, A + ∆
A, t ) , (23)which in turns yields U A ( w, A, t ) − ¯ a Ox + t U w ( w, A, t ) = 0 . (24)Note that the lump-sum purchase is such that the marginal utility of annuity income equals theadjusted marginal utility of wealth, in which we adjust the marginal utility of wealth by multiplyingby the cost of $1 of annuity income. This result parallels many such in economics. Indeed, themarginal utility of annuity income is the marginal utility of the benefit, while the adjusted marginalutility of wealth is the marginal utility of the cost. Thus, the lump-sum purchase is such that themarginal utilities are equated. Due to space constraints, we refer the interested reader to the earlier working paper version by Milevsky andYoung (2002b) for a detailed discussion of this and other properties of U . a Ox + t U w ( w, A, t ) − U A ( w, A, t ) ≥ . (25)By combining (22) and (25), we obtain the HJB equation (26) below associated with the valuefunction U given in (20). The following result can be proved as in Zariphopoulou (1992), forexample. Proposition 6.1:
The value function U is a constrained viscosity solution of the Hamilton-Jacobi-Belman equation min (cid:2) ( r + λ Sx + t ) U − U t − ( rw + A ) U w − max π (cid:0) σ π U ww + ( µ − r ) πU w (cid:1) − max c ≥ ( − cU w + u ( c )) − λ Sx + t u ( w ) , ¯ a Ox + t U w − U A (cid:3) = 0 . (26)Equation (24) defines a “barrier” in wealth-annuity income space. If wealth and annuity incomelie to the right of the barrier at time t , then the individual will immediately spend a lump sumof wealth to move diagonally to the barrier (up and to the left). The move is diagonal because aswealth decreases to purchase more annuities, annuity income increases. Thereafter, annuity incomeis either constant if wealth is low enough to keep to the left of the barrier, or annuity incomeresponds continuously to infinitesimally small changes of wealth at the barrier.Thus, as in Dixit and Pindyck (1994, pp 359ff) or in Zariphopoulou (1992), we have discoveredthat the optimal annuity-purchasing scheme is a type of barrier control. Other barrier controlpolicies appear in finance and insurance. In finance, Zariphopoulou (1999, 2001) reviews the roleof barrier policies in optimal investment in the presence of transaction costs; also see the referenceswithin her two articles. See Gerber (1979) for a classic text on risk theory in which he includes asection on optimal dividend payout and shows that it follows a type of barrier control. In this subsection, we specialize the results of the previous subsection to the case for which theindividual’s preferences exhibit CRRA. For this case, we can reduce the problem by one dimension,and we show that the barrier given in the previous section is a ray emanating from the origin inwealth-annuity space. Let u ( c ) = c − γ − γ , and u ( w ) = ku ( w ) , γ > , γ = 1 , k ≥ . (27)The parameter k ≥ U is a solutionof its HJB equation in the classical sense, not just in the viscosity sense. Generally, if the force18f mortality is “eventually” large enough to make the value function well-defined, then this resultholds for our problem, too.For the utility functions in (27), it turns out that the value function U is homogeneous of degree1 − γ with respect to wealth w and annuity income A . That is, U ( bw, bA, t ) = b − γ U ( w, A, t ) for b >
0. Thus, if we define V by V ( z, t ) = U ( z, , t ), then we can recover U from V by U ( w, A, t ) = A − γ V ( w/A, t ) , for A > . (28)It follows that the HJB equation for U from Proposition 6.1 becomes the following equation for V :min [ ( r + λ Sx + t ) V − V t − ( rz + 1) V z − max ˆ π (cid:18) σ ˆ π V zz + ( µ − r )ˆ πV z (cid:19) − max ˆ c ≥ (cid:18) − ˆ cV z + ˆ c − γ − γ (cid:19) − kλ Sx + t z − γ − γ , ( z + a Ox + t ) V z − (1 − γ ) V ] = 0 , (29)in which ˆ c = c/A , and ˆ π = π/A . Davis and Norman (1990) and Shreve and Soner (1994) use thesame transformation in the problem of consumption and investment in the presence of transactioncosts. Also, Duffie and Zariphopoulou (1993) and Koo (1998) use this transformation to studyoptimal consumption and investment with stochastic income.Due to space considerations we simply refer to the working paper version by Milevsky andYoung (2002b) which study properties of the optimal consumption and investment policies. Pleaserefer to that work for details on the proof of the following proposition that describes the actions ofthe individual. Proposition 6.2:
For each value of t ≥ , there exists a value of the wealth-to-income ratio z ( t ) that solves ( z ( t ) + ¯ a Ox + t ) V z ( z ( t ) , t ) = (1 − γ ) V ( z ( t ) , t ) , (30) such that (i) If z = w/A > z ( t ) , then the individual immediately buys an annuity so that w − ∆ A ¯ a Ox + t A + ∆ A = z ( t ); (31) Thus, V ( z, t ) = V ( z ( t ) , t ) in this case. (ii) If z = w/A < z ( t ) , then the individual buys no annuity; i.e., she is in the region ofinaction. Thus, in this case, V solves ( r + λ Sx + t ) V (32)= V t + ( rz + 1) V z + max ˆ π (cid:18) σ ˆ π V zz + ( µ − r )ˆ πV z (cid:19) + max ˆ c ≥ (cid:18) − ˆ cV z + ˆ c − γ − γ (cid:19) + kλ Sx + t z − γ − γ . It follows that at each time point, the barrier w = z ( t ) A is a ray emanating from the origin andlying in the first quadrant of ( w, A ) space. z ( t ) < ∞ , then it is optimal for the individual to have positive annuity incomebecause the positive w axis lies in the region { ( w, A, t ) : w/A < z ( t ) } .Davis and Norman (1990) and Shreve and Soner (1994) find results similar to those in Propo-sition 6.2 for the problem of optimal consumption and investment in the presence of proportionaltransaction costs. In the next subsection, we show how to linearize the HJB equation of theindividual who has no bequest motive. Up until now we have assumed both utility of bequest and consumption in our specification. In thissubsection, we linearize the nonlinear partial differential equation for V in the region of inactiongiven by equation (32) with no bequest motive ( k = 0). To this end, we consider the convex dualof V defined by ˜ V ( y, t ) = max z> [ V ( z, t ) − zy ] . (33)The critical value z ∗ solves the equation 0 = V z ( z, t ) − y ; thus, z ∗ = I ( y, t ), in which I is the inverseof V z with respect to z . Note that one can retrieve the function V from ˜ V by the relationship V ( z, t ) = min y> h ˜ V ( y, t ) + zy i . (34)Indeed, the critical value y ∗ solves the equation 0 = ˜ V y ( y, t ) + z = − I ( y, t ) + z ; thus, y ∗ = V z ( z, t ).In the partial differential equation for V with no bequest motive ( k = 0), let z = I ( y, t ) andrewrite the equation in terms of ˜ V to obtain˜ V t − ( r + λ Sx + t ) ˜ V + λ Sx + t y ˜ V y + my ˜ V yy = − y − γ − γ y − γ , (35)in which m = (cid:0) µ − rσ (cid:1) . Note that (35) is a linear partial differential equation.Next, consider the boundary condition U A ( w, A, t ) = ¯ a Ox + t U w ( w, A, t ) from equation (24). Interms of V , this condition can be written as in equation (27), and we repeat it here for convenience − (1 − γ ) V ( z ( t ) , t ) + ( z ( t ) + ¯ a Ox + t ) V z ( z ( t ) , t ) = 0 . (36)Smooth pasting at the boundary implies that the derivative of this boundary condition with respectto z evaluated at z = z ( t ) holds and is given by γV z ( z ( t ) , t ) + ( z ( t ) + ¯ a Ox + t ) V zz ( z ( t ) , t ) = 0 . (37)We also have a boundary condition at z = 0 because at that point, the individual has no wealthto invest in the risky asset. Write ˆ π ∗ in terms of ˜ V : ˆ π ∗ ( y, t ) = µ − rσ y ˜ V yy . Thus, for z = 0 (with thecorresponding value for y written y a ( t )), we have that either y a ( t ) = 0 or ˜ V yy ( y a ( t ) , t ) = 0.20ecause V z > z , we have y a ( t ) > y ( t ) ≥ t ≥ y a ( t ) and y ( t ) are defined by y a ( t ) = V z (0 , t ) , and y ( t ) = V z ( z ( t ) , t ) . (38)Thus, because y a ( t ) >
0, in terms of ˜ V , the boundary conditions become˜ V y ( y a ( t ) , t ) = 0 , (39)for ˜ V yy ( y a ( t ) , t ) = 0 , (40)and (1 − γ ) ˜ V ( y ( t ) , t ) + γy ( t ) ˜ V y ( y ( t ) , t ) = ¯ a Ox + t y ( t ) , (41)for ˜ V y ( y ( t ) , t ) + γy ( t ) ˜ V yy ( y ( t ) , t ) = ¯ a Ox + t . (42) While still operating within the zero bequest world, if we assume that the forces of mortalityare constant, that is, λ Sx + t ≡ λ S and λ Ox + t ≡ λ O for all t ≥
0, then we can obtain an “implicit”analytical solution of the value function V via the boundary-value problem given by (35) and (39)- (42). See Neuberger (2003) for recent and related work. In this case, V , ˜ V , y a , and y areindependent of time, so (35) becomes the ordinary differential equation − ( r + λ S ) ˜ V ( y ) + λ S y ˜ V ′ ( y ) + y ˜ V ′′ ( y ) = − y − γ − γ y − γ , (43)with boundary conditions ˜ V ′′ ( y a ) = 0 , (44)for ˜ V ′ ( y a ) = 0 , (45)and (1 − γ ) ˜ V ( y ) + γy ˜ V ′ ( y ) = y r + λ O , (46)for 21 V ′ ( y ) + γy ˜ V ′′ ( y ) = 1 r + λ O . (47)The general solution of (43) is˜ V ( y ) = D y B + D y B + yr + C y − γ , (48)with D and D constants to be determined by the boundary conditions, with C given by C = r + λ S γ − m − γγ , (49)with B and B given by B = 12 m (cid:20) ( m − λ S ) + q ( m − λ S ) + 4 m ( r + λ S ) (cid:21) > , (50)and B = 12 m (cid:20) ( m − λ S ) − q ( m − λ S ) + 4 m ( r + λ S ) (cid:21) < . (51)The boundary conditions at y give us D { γ ( B − } y B + D { γ ( B − } y B + y r = y r + λ O , (52)and D B { γ ( B − } y B + D B { γ ( B − } y B + y r = y r + λ O . (53)Solve equations (52) and (53) to get D and D in terms of y : D = − λ O r ( r + λ O ) 1 − B B − B y − B γ ( B − , (54)and D = − λ O r ( r + λ O ) B − B − B y − B γ ( B − . (55)Next, substitute for D and D in ˜ V ′ ( y a ) + γy a ˜ V ′′ ( y a ) = 0 from (44) and (45) to get λ O r + λ O B (1 − B ) B − B (cid:18) y a y (cid:19) B − + λ O r + λ O B ( B − B − B (cid:18) y a y (cid:19) B − = 1 . (56)(56) gives us an equation for the ratio y a /y >
1. To check that (56) has a unique solution greaterthan 1, note that the left-hand side (i) equals λ O / ( r + λ O ) < y a /y = 1, (ii) goes toinfinity as y a /y goes to infinity, and (iii) is strictly increasing with respect to y a /y .Next, substitute for D and D in ˜ V ′ ( y a ) = 0 from (44) to get22 λ O r ( r + λ O ) B (1 − B ) B − B ( y a /y ) B − γ ( B − − λ O r ( r + λ O ) B ( B − B − B ( y a /y ) B − γ ( B − r + C (cid:18) − γ (cid:19) y − γ a = 0 . (57)Substitute for y a /y in equation (57), and solve for y a . Finally, we can get y from y = y a y a /y , (58)and D and D from equations (54) and (55), respectively.Once we have the solution for ˜ V , we can recover V from V ( z ) = max y> h ˜ V ( y ) + zy i = max y> h D y B + D y B + yr + C y − γ + zy i , (59)in which the critical value y ∗ solves D B y B − + D B y B − + 1 r + C (cid:18) − γ (cid:19) y − γ + z = 0 . (60)Thus, for a given value of z = w/A , solve (60) for y and substitute that value of y into (59) toget U ( w, A ) = V ( z ). Perhaps more importantly, we are interested in the critical value z abovewhich an individual spends a lump sum to purchase more annuity income. We pursue this in theexamples in the next section. In this section, we provide a variety of numerical examples to illustrate the results of our anythinganytime model. We focus attention on the impact of risk aversion, investment volatility, andinsurance fees on the optimal amount annuitized.In the first set of results, we assumed the following values for the hazard rate parameters: λ S = λ O = 0 .
04. That is, the force of mortality is constant and therefore the expected futurelifetime is: 1 /λ = 25 years. Furthermore, we set the risk-free interest rate to be r = 0 .
04, the driftof the risky asset is µ = 0 .
08, and its volatility is σ = 0 .
20. We have selected these numbers –which are lower than those used in the earlier examples – to better capture a real (after-inflation)case in which Social Security benefits would be considered as part of the pre-existing annuity.In Table 4a, for various values of γ , we give the critical value of the ratio of wealth to annuityincome z = w/A above which the individual will spend a lump sum of wealth to increase herannuity income. We also include the amount that the individual will spend on annuities for a given(pre-existing) annuity income of A = $25 , w − z A ) / (1 + ( r + λ O ) z ).23 able 4a about here. For example, a retiree with $1,000,000 in liquid investable assets and with $25,000 in pre-existingannuity income would immediately (and irreversibly) annuitize between $727,620 and $914,176depending on the coefficient of relative risk aversion. Notice from the same Table 4a that theamount spent on annuities increases for a given level of wealth as the individual becomes more riskaverse, which is an intuitively pleasing result. Also, for a given level of risk aversion, the amountspent on annuities decreases as wealth decreases. In Table 4b, we present the results for the casewhen A = $50 , Table 4b about here.
In fact, in this case someone with $100,000 of wealth, or less, will not annuitize any additionalwealth when their coefficient of relative risk aversion is γ = 2 or less. We emphasize – once again –that these numerical results assume that there is absolutely no bequest motive and no importanceplaced in inheritance, a spouse or any other survivors. In the presence of some weight on bequest(which would be expected in the real world) the amount annuitized would not be any higher. Table 4c about here.
Table 4c looks at a different aspect of the problem, namely the impact of investment volatility σ on the optimal amount annuitized in the anything anytime environment. In this case, we assumethat same λ O = λ S = 0 .
04 hazard rate, r = 0 .
05 risk free rate and µ = 0 .
12 drift for the riskyasset. We provide results under two differing levels of risk aversion: high ( γ = 5) and low ( γ = 2).In both of these cases, we assume a retiree/investor with $1,000,000 of liquid investable wealth and$40,000 in pre-existing annuity income.Notice from Table 4c that as the investment volatility σ increases from 0 .
12 to 0 .
20, the amountannuitized declines under both (and we can show, all) levels of risk aversion. In fact, under highlevels of volatility the retiree/investor annuitizes $472,871 for γ = 2 and $768,568 for γ = 5. Thesenumbers drop to $12,692 and $469,789, respectively, when the investment volatility is reduced from0.20 to 0.12. The economic intuition for this result is quite clear. As the relative risk of investingin the high-return alternative declines, it becomes much less appealing, on a risk-adjusted basis, toannuitize one’s wealth. Table 4d about here.
Table 4d investigates the impact of subjective vs. objective health status on the amount an-nuitized in the same anything anytime framework. In this case we assume an objective (annuity24ricing) hazard rate of λ O = 0 .
04, but vary the subjective λ S from a value of 0.03 to 0.055. Thus,the insurance company believes the annuitant has a life expectancy of 1 / .
04 = 25 . / .
03 = 33 . / .
055 = 18 . r = 0 .
05 for the risk free rate, µ = 0 .
10 drift of the risky asset and σ = 0 .
16 for the investment volatility.In this case, and in contrast to the results from the all-or-nothing results, an increasing hazardrate has a uniform impact on the amount annuitized. Namely, people who consider themselves tobe in worse health annuitize less, all else being equal. More specifically, a γ = 2 retiree/investorwho believes he has a life expectancy of (only) 18.2 years will annuitize $514,496 in contrast to$574,840 when he believes his life expectancy is a higher 33.3 years. The same result is observed athigher levels of risk aversion, although it is less extreme in nominal (and marginal) terms. Noticethat a 15-year difference in perceived life expectancy reduces the amount annuitized by $60,000under low ( γ = 2) levels of risk aversion. But, at higher levels of ( γ = 5) risk aversion, the differenceis a mere $28,000.We remind the reader that these results have been obtained under a variety of assumptions,namely a constant hazard rate (exponential future lifetime distribution) and zero bequest motive,as well as constant risk-free rate and risk-premia. To eliminate these restrictions is the subject ofongoing research. Our paper locates the optimal dynamic policy of a utility-maximizing individual who is interested inincorporating lifetime payout annuities (or defined benefit pension income) into his or her retirementportfolio. We investigated a variety of institutional arrangements and market structures withdiffering restrictions and constraints. Our main results can be stated as follows. • An individual who is faced with the irreversible decision of when to start a fixed (nominalor real) lifetime pension annuity – with the proviso that annuitization must take place inan ‘all-or-nothing’ format – is endowed with an incentive to delay that is quite valuable atyounger ages. • There is an incentive to delay annuitization even in the absence of bequest motives. This isbecause market imperfections do not allow retirees to purchase payout annuities that offercomplete asset allocation flexibility to match ones subjective consumption preferences vis avis their health status. Stated differently, a market in which instantaneously-renegotiatedlife-contingent tontines are available would not give rise to our ‘option to delay’ result.25
Under an all-or-nothing framework, which is a feature of many public and private pensionsystems, the optimal age to annuitize is the age at which the option to delay has zero timevalue. This value is defined equal to the loss in utility that comes from not being able tobehave optimally. This value depends on a person’s coefficient of relative risk aversion, aswell as their subjective health status. • Using historical market parameters and realistic mortality estimates, we conclude that in thisall-or-nothing framework the optimal age to purchase a pure life-contingent annuity does notoccur prior to age 70. This result is consistent with a variety of probabilistic-based modelswhich are based on the relationship between mortality credits and the returns from competingasset classes. • In the event that complete asset allocation flexibility is available within the payout annuity,which is akin to variable immediate annuities that are available in some countries and ju-risdictions, the optimal age to annuitize is indeed earlier. Of course, high management feesand expenses relative to non-annuitized wealth can have a strong mitigating impact on thebenefits from annuitization. • When we move towards an open institutional system in which annuitization can take placein small portions and at anytime, we find that utility-maximizing investors should acquirea base amount of annuity income (i.e. Social Security or a DB pension) and then annuitizeadditional amounts if and when their wealth-to-income ratio exceeds a certain level. In thiscase which we label anything anytime , individuals annuitize a fraction of wealth as soon asthey have opportunity to do so – i.e. they do not wait – and they then purchase moreannuities as they become wealthier. • Thus, in contrast to the all-or-nothing pension structure, in the case of an open systemwhere annuities can be purchased on an ongoing basis we find that individuals prior to age70 should have a minimal amount of annuity income and should immediately annuitize afraction of wealth to create this base level of lifetime income if they do not already havethis from pre-existing DB pensions. We reiterate that individuals should always hold someannuities, even in the presence of a bequest motive, as long as z ( t ) in Proposition 6.2 is lessthan infinity. • Finally, our anything anytime model indicates that ceteris paribus, a larger amount of wealthrelative to pre-existing annuity income, higher levels of risk aversion, greater investmentvolatility σ and better health status (i.e. lower subjective mortality rates) will all serve toincrease the amount that is voluntarily annuitized. And, although the magnitude of these This after-age-70 result has also been advocated in a variety of popular-press article, such as a recent piece inthe
Wall Street Journal , 3 September 2003, page C1, “It Pays to Delay: The Longer You Wait To Buy an Annuity,the More You Get,” by Jonathan Clements.
That our paper raises a number of questions that should be considered in any future research onthe topic, which we will now elaborate on.First, by far the strongest assumption we have made in our modeling for both the restricted all-or-nothing market and the anything-anytime environment, is that the risk-free rate and the marketrisk premium are assumed constant. Thus, we have abstracted from any term structure effects,or predictability in the evolution of interest rates as well as stochasticity of investment volatility σ and the market risk premium. Yet, recent models of asset allocation and portfolio choice havegone well beyond the classic Mertonian framework, which was the foundation of our analysis. Thenext step, from our perspective, is to enhance the financial chassis of our model and allow for morecomplicated market dynamics.Indeed, a number of practitioners have advocated that people refrain from annuitizing wheninterest rates are low and that annuitization is more appealing when interest rates are high. Othercommentators have advocated a dollar-cost averaging approach to annuitization to smooth out theinterest rate risk, which resonates with the results from our anything-anytime analysis. Obviously,one might question the precise definition of high and low interest rates in this context, but it wouldcertainly be interesting to investigate the impact of assuming a mean-reverting process for interestrates and possibly the yield curve as a whole. This would necessitate modeling the evolution ofnominal versus real interest rates as well as the behavior of inflation. We, thus, envision an advancedmodel in which real (inflation-adjusted) and nominal life annuities co-exist in the optimal portfolio.This complicates our basic model by introducing at least one more state variable in the PDE/ODE,which is why we leave for future research.Likewise, a recent innovation in the U.S. retirement income market is the introduction of guar-antee living benefits, which are essentially staggered put options on a portfolio that promise aminimal level of income for as long the annuitant lives. These products which fall under the indus-try label of Guaranteed Minimum Withdrawal Benefits (GMWB), have some longevity insurancefeatures and some derivative securities features. These products which are part of the trillion dollarVariable Annuity (VA) industry in the U.S. are growing in popularity and might actually competewith conventional life annuities as a way of generating a sustainable retirement income. Furtherresearch would examine the optimal demand and asset allocation including these hybrid GMWBproducts.Furthermore, given the normative focus on this paper, we have ignored the positive equilibriumimplications. The main question in this case would be how annuity prices would be affected byindividuals’ desire to annuitize at an optimal time. Indeed, the payoff from conventional financial27ssets are not age- or mortality-dependent and thus do not depend on the demographic structure orhealth status of the marginal investor. Under the discretionary and voluntary life annuities that wehave analyzed, one can envision a situation in which very few people purchase a life annuity at age50, which thus reduces the pool of individuals across which mortality risk can be diversified via thelaw of large numbers. This will have immediate implication on the pricing curve and the objectivehazard rate, which we have taken as given. A detailed equilibrium analysis would attempt to derivethe market’s objective hazard as a function of the heterogeneous mix of participant’s subjectivehazard rates. But, this is far beyond the scope of the current paper.On a related note, our model implicitly assumes that mortality rates are fully predictable inthe future and that we are able to specify a survival function and annuity pricing equation duringthe entire horizon, conditional on the value of interest rates. In other words, we assume that theobjective hazard rate is deterministic. However, there is a growing body of empirical and theoreticalliterature that argues that mortality risk is priced in equilibrium. In the extreme, this would implythat if one delays annuitization one runs the risk that annuity prices will actually increase, eventhough the individual has aged. This, of course, would introduce yet another variable in the decisionand further complicate the analysis of the optimal age at which to annuitize. That said, with theimpending retirement of American baby boomers and the industrial shift from Defined Benefit (DB)to Defined Contribution (DC) pension plans, we believe these issues will demand further academicattention as they take on greater practical importance.28 Appendix
In this appendix, we show that if the subjective force of mortality varies slightly from the objectiveforce to the extent that ¯ a Sx < a Ox for all x , then the optimal time of annuitization increases fromthe T given by the zero of the right-hand side of (16). In particular, if the individual is less healthythan normal ( λ Sx > λ Ox for all x ), then ¯ a Sx < ¯ a Ox for all x , from which it follows that the optimaltime of annuitization is delayed. Also, if the individual is more healthy than normal but only tothe extent that ¯ a Sx < a Ox for all x , then the optimal time of annuitization is delayed.Suppose ¯ a Sx + T = ¯ a Ox + T + ε for some small ε , not necessarily positive. Then, equation (16) at thecritical value T becomes0 = γ − γ ¯ a Ox + T + ε ¯ a Ox + T ! − − γγ − − γ + ¯ a Ox + T + ε ¯ a Ox + T + (cid:0) ¯ a Ox + T + ε (cid:1) (cid:2) δ − (cid:0) r + λ Ox + T (cid:1)(cid:3) . (61)We can simplify this equation to0 = (cid:0) ¯ a Ox + T + ε (cid:1) (cid:2) δ − (cid:0) r + λ Ox + T (cid:1)(cid:3) − (cid:18) − − γγ − (cid:19) ε ¯ a Ox + T ! − (cid:18) − − γγ − (cid:19) (cid:18) − − γγ − (cid:19) ε ¯ a Ox + T ! + . . . , (62)if ε ¯ a Ox + T lies between − ε ∗ between 0 and ε ¯ a Ox + T such that 0 = (cid:0) ¯ a Ox + T + ε (cid:1) (cid:2) δ − (cid:0) r + λ Ox + T (cid:1)(cid:3) + 12 γ ( ε ∗ ) , or equivalently, 0 = (cid:2) δ − (cid:0) r + λ Ox + T (cid:1)(cid:3) + ( ε ∗ ) γ a Ox + T (1 + ε ¯ a Ox + T ) . (63)The second term of the above equation is positive (and small) regardless of the sign of ε . Thus, T is determined by setting (cid:2) δ − (cid:0) r + λ Ox + T (cid:1)(cid:3) equal to a negative number. It follows that, for λ Ox increasing with respect to x , this value of T will be larger than the zero of (17).Note that a sufficient condition for the above result is that ε ¯ a Ox + T lie between − a Sx + T < a Ox + T . For less healthypeople ( λ Sx > λ Ox for all x ), we have ¯ a Sx < ¯ a Ox for all x, so ¯ a Sx + T < a Ox + T holds automatically. Also,there is some leeway in this inequality, so that even healthier people might have that ¯ a Sx + T < a Ox + T .Even when this inequality does not hold, we conjecture that we still have a delay in the time of29nnuitization beyond that given by the zero of the right-hand side of (17), as we see in one of theexamples in Section 5. The probability that consumption (as a percentage of initial wealth) at optimal time of annuitizationis p % less than the consumption if one annuitizes immediately equals P W ∗ T ¯ a Ox + T < (1 − . p ) w ¯ a Ox | W = w ! = P we (cid:18) δ − r ( µ − r )22 σ γ (cid:19) T − R T k ( s ) ds + µ − rσγ B T < (1 − . p ) w ¯ a Ox + T ¯ a Ox ! = P e µ − rσγ B T < (1 − . p ) ¯ a Ox + T ¯ a Ox e − δ − r − ( µ − r )22 σ γ ! T + R T k ( s ) ds = P B T < ln (cid:18) (1 − . p )¯ a Ox + T ¯ a Ox (cid:19) − (cid:16) δ − r − ( µ − r ) σ γ (cid:17) T + R T k ( s ) ds µ − rσγ = Φ ln (cid:18) (1 − . p )¯ a Ox + T ¯ a Ox (cid:19) − (cid:16) δ − r − ( µ − r ) σ γ (cid:17) T + R T k ( s ) ds µ − rσγ √ T . (64)Here, Φ denotes the cumulative distribution function of the standard normal. In the body of the paper, we assumed that the only life annuities available upon annuitizationprovided a fixed payout. In this appendix, we examine a market in which annuity ‘wrappers’are available on all asset classes with full asset allocation mobility. We investigate the optimalconsumption, investment, and annuitization policies in this market. We introduce the symbol β to represent the proportion of wealth at the time of annuitization that is invested in the variableannuity, so that 1 − β is the proportion invested in the fixed annuity. We assume that the mixbetween the variable and fixed annuities, namely β versus 1 − β , stays fixed throughout the remaininglife of the annuitant, which is a so-called money mix plan and has certain optimality features asshown by Charupat and Milevsky (2002). Again, we consider CRRA utility and provide formulasfor the power utility case. One can easily deal with the logarithmic case by letting λ approach 1 inthe consumption, investment, and annuitization policies under power utility u ( c ) = − γ c − γ , γ > , γ = 1 . To further capture the salient features of this product, we assume that the provider of theannuity has a nonzero insurance load on the fixed annuity such that the effective “return” on the30xed annuity is r ′ , with r ′ ≤ r . Similarly, there is a nonzero insurance load on the variable annuitysuch that the drift on the variable annuity is µ ′ with µ ′ ≤ µ and µ ′ − r ′ ≤ µ − r .For the mixture of a variable and a fixed annuity, define the value function V by V ( w, t ; T ) = sup { c s, π s, β } E w,t "Z Tt e − r ( s − t ) s − t p Sx + t c − γs − γ ds (65)+ Z ∞ T e − r ( s − t ) s − t p Sx + t − γ W T ¯ a O,r ′ x + T e β (cid:18) µ ′ − r ′ − βσ (cid:19) ( s − T )+ βσ ( B s − B T ) ! − γ ds, in which the second superscript on ¯ a O,r ′ x + T , namely r ′ , is the rate of discount used to calculate theactuarial present value of the annuity.We can deal with the choice of β by noting that (Bj¨ork, 1998) E w,t " W − γT e β (1 − γ ) (cid:18) µ ′ − r ′ − βσ (cid:19) ( s − T )+ β (1 − γ ) σ ( B s − B T ) (66)= E w,t h W − γT i e β (1 − γ ) (cid:18) µ ′ − r ′ − βγσ (cid:19) ( s − T ) . Thus, the expectation is maximized if we maximize β (cid:16) µ ′ − r ′ − βγσ (cid:17) . It follows that the optimalvalue of β equals β ∗ = µ ′ − r ′ σ . (67)Note that the optimal choice of β is independent of the optimal time T of annuitization. Naturally,if β ∗ is greater than one, the holdings are truncated by the seller of the annuity at 100% of therisky stock.It follows that V solves the HJB equation (cid:0) r + λ Sx + t (cid:1) V = V t + max π (cid:2) σ p V ww + ( µ − r ) πV w (cid:3) + rwV w + max c ≥ h − cV w + − γ c − γ i ,V ( w, T ; T ) = − γ (cid:18) w ¯ a O,r ′ x + T (cid:19) − γ ¯ a S,r − (1 − γ ) ( µ ′− r ′ ) sσ y x + T . (68)Note that this is the same as the previous HJB equation, except that the boundary conditionreflects the mixture of the variable and fixed annuities. The second superscript on the actuarialpresent value denotes the rate at which the annuity payments are discounted. It follows that V hasthe form as that given in equation (12), except that ¯ a Sx + T = ¯ a S,rx + T is replaced with¯ a S,r − (1 − γ ) ( µ ′− r ′ ) sσ y x + T , (69)31nd ¯ a Ox + T = ¯ a O,rx + T is replaced with ¯ a O,r ′ x + T . Also, note that the optimal investment policy is similar inform to the one given in the body of the paper for the fixed-only payout case. If there are no loads onthe fixed and variable annuities, that is, if r ′ = r and µ ′ = µ , then the proportion of wealth investedin the risky asset from before annuitization equals the proportion after annuitization; however, inthis case, the optimal strategy of the individual is to annuitize immediately. This latter resultfollows from the work of Yaari (1965). For a CRRA investor (with no bequest motives) with noinsurance loads on the annuities, the optimal mixture between risky and risk-free assets is invariantto whether the portfolio is annuitized or not.The derivative of V with respect to T is proportional to δVδT ∝ γ − γ ¯ a S,r − (1 − γ ) ( µ ′− r ′ ) sσ y x + T ¯ a O,r ′ x + T − γγ − − γ + ¯ a S,r − (1 − γ ) ( µ ′− r ′ ) sσ y x + T ¯ a O,r ′ x + T (70)+ ¯ a S,r − (1 − γ ) ( µ ′− r ′ ) sσ y x + T (cid:2) δ − δ ′ − λ Ox + T (cid:3) , in which δ ′ = r ′ + ( µ ′ − r ′ ) σ γ . We can use this equation to determine the optimal value of T in anygiven situation. In Table 5, we compare the optimal ages of annuitization and the imputed valueof delaying when the individual can only buy a fixed annuity (compare with Table 1) and when theindividual can buy a money mix of variable and fixed annuities. Table 5a about here.
We assume that the financial market and mortality are as described in Section 5, except thatfor the variable annuity, the insurer has a 100-basis-point Mortality and Expense Risk Charge loadon the return, so that the modified drift is µ ′ = 0 .
11, and for the fixed annuity, the insurer has a50-basis-point spread on the return, so that the modified rate of return is r ′ = 0 . Table 5b about here.
Assume that the individual has a CRRA of γ = 2, from which it follows that the individualwill invest 75.0% in the risky stock before annuitization and 68.7% in the variable annuity afterannuitization. Suppose that the individual can buy an escalating annuity. An escalating annuity is one for whichthe payments increase at a (constant) rate g . These are known as COLA (Cost Of Living Adjust-ment) annuities and are available from vendors that sell fixed annuities. These escalating annuities32re popular as a hedge against (expected) inflation, since it is virtually impossible to obtain trueinflation-linked annuities in the U.S. The actuarial present value of an escalating annuity can bewritten ¯ a r − gx ; that is, the rate of discount r is reduced by the rate of increase of the payments g . Asin the previous two sections, we consider CRRA utility and provide formulas for the power utilitycase. Thus, we can define the corresponding value function by V ( w, t ; T ) = sup { c s ,π s ,g } (cid:20) E Z Tt e − r ( s − t ) s − t p Sx + t − γ c − γs ds (71)+ Z ∞ T e − r ( s − t ) s − t p Sx + t − γ W T ¯ a O,r − gx + T e g ( s − T ) ! − γ ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W t = w This expression is maximized with respect to g if11 − γ R ∞ e − ( r − (1 − γ ) g ) s (cid:2)R ∞ e − ( r − g ) s s p Ox + T ds (cid:3) − γ (72)is maximized. The derivative of this expression with respect to g is proportional to R ∞ se − ( r − (1 − γ ) g ) s s p Sx + T ds R ∞ e − ( r − (1 − γ ) g ) s s p Sx + T ds − R ∞ se − ( r − g ) s s p Ox + T ds R ∞ e − ( r − g ) s s p Ox + T ds . (73)Note that this is a difference of expectations of “ s ” with respect to two probability distributions.Also, note that if λ Sx = λ Ox − c for all x and for some constant c , then the optimal value of g is g ∗ = cy . For example, if the individual is healthier to the extent that the subjective hazard rateis 0.01 less than the objective (pricing) hazard rate, then optimal rate of increase of the annuitypayments (once the individual annuitizes his or her wealth) is . γ . Note that in general, healthier people will want to buy escalating annuities with a positive g , while sicker people will want to buyescalating annuities with a negative g . This makes sense because healthier people anticipate livinglonger than normal, so they will be able to enjoy those larger annuity payments in the future. Onthe other hand, less healthy people will not live as long, so they demand higher payments now.Table 6 provides a numerical example for an individual who believes he or she is healthier thannormal with f = − .
5; that is, the person has one-half the mortality rate of the average person.Suppose that the CRRA is γ = 1 .
5. We compare these numbers with those when the individual canbuy only a fixed annuity. It turns out that the optimal rate of escalation g = 2% (to two decimalplaces) for all ages and for both genders. Table 6 about here.
Note that the individual is willing to annuitize earlier if there is a 2% escalating annuity available;however, there is still an advantage to wait. 33 able 1a: Optimal Age to Annuitize and Value of Delay under Low Risk Aversion γ = 1, FEMALE (MALE) γ = 2, FEMALE (MALE) Age Optimal Value Prob. Lower Optimal Value Prob. LowerAge of Delay Annuity Age of Delay Annuity60 Now (Now) neg. (neg.) N/a (N/a) Now (Now) neg. (neg.) N/a (N/a)Notes: All-or-Nothing Market: The value of the option to delay annuitization for males andfemales with a coefficient of relative risk aversion of γ = 1 and γ = 2. We assume the non-annuitized funds are invested in a risky asset with drift µ = 0 .
12 and volatility σ = 0 .
20. Therisk-free rate is r = 0 .
06. The mortality is assumed to be Gompertz-Makeham fit to the IAM2000table with projection Scale G. For example, a 70-year-old female with a coefficient of relative riskaversion of γ = 2, will effectively suffer a utility-equivalent 5.2% loss of her wealth if she choosesto annuitize immediately. The optimal time for her to annuitize is at age 78.4. The table alsogives the probability of deferral failure, namely the probability that the annuity purchased at theoptimal age will provide less income that an annuity bought right now.34 able 1b: Optimal Age to Annuitize and Value of Delay under Higher Risk Aversion Coefficient of Relative Risk Aversion γ = 5MALE FEMALE Current Age Optimal Age Value of Delay Optimal Age Value of Delay60 Now neg. 70.4 1.04% Now neg. 70.4 0.01% Now neg. Now neg.Notes: All-or-nothing market. The risk-free rate is r = 0 .
06, the drift is µ = 0 .
12, and thevolatility is σ = 0 .
20. We use Gompertz-Makeham mortality with m = 88 . b = 10 . m = 92 . b = 8 .
78 for females. Notice that even at higher levels of risk aversion, males (andespecially females) do not annuitize prior to age 60.35 able 2: Assuming Optimal Behavior, What is the Probability of HigherConsumption Upon Delay? γ = 1 , FEMALE (MALE) γ = 2, FEMALE (MALE) Age Prob. of consuming at least Prob. of consuming at least20% more than initial annuity 20% more than initial annuity60 N/a (N/a) N/a (N/a)Notes: All-or-nothing market. Assuming the individual self-annuitizes and defers the purchaseof a life annuity to the optimal age, this table indicates the probability of consuming at least 20%more at the time of annuitization, compared to if one annuitizes immediately. Thus, for example,a female (male) at age 65 with a coefficient of relative risk aversion of =1 has a 64.4% (59.6%)chance of creating a 20% larger annuity flow. 36 able 3: How Does Subjective Health Status Impact Optimal Behavior? f Optimal Age of Value Consumption Rate Consumption Rate
More/Less Healthy annuitization of Delay Before Annuitization After Annuitization -1.0 78.28 13.79% 7.55% 13.38%-0.8 74.58 10.54 7.95 11.79-0.6 73.71 9.68 8.18 11.47-0.4 73.29 9.23 8.37 11.33-0.2 73.09 8.99 8.54 11.260.0 73.03 8.87 8.70 11.240.2 73.08 8.84 8.85 11.260.5 73.31 8.93 9.06 11.331.0 74.04 9.34 9.38 11.591.5 75.21 10.00 9.68 12.032.0 76.96 10.89 9.98 12.762.5 79.71 12.01 10.26 14.123.0 85.38 13.38 10.55 18.01Notes: All-or-Nothing Market: The imputed value of delaying annuitization for a male, aged60 with CRRA of γ = 2. We assume the funds are invested in a risky asset with drift µ = 0 . σ = 0 .
20. The risk-free rate is r = 0 .
06. The mortality is assumed to be Gompertz-Makeham fit to the IAM2000 table with projection Scale G. Thus, for example, if the individual’ssubjective hazard rate is 20% higher (i.e. less healthy) than the mortality table used by theinsurance company to price annuities, the optimal annuitization point is at age 73.1, and the valueof the option is 8.84% of his wealth at age 60. While the 60-year-old male waits to annuitize, heconsumes at the rate of 8.85% of assets, and once he purchases the fixed annuity, his consumptionrate - and standard of living - will increase to 11.26% of assets.37 able 4a: How Does Wealth and Risk Aversion Affect Annuitization?
Amount of Money Spent on Annuities for Various Levelsof Wealth and Risk Aversion ( A = $25,000) γ = 1 . γ = 2 . γ = 2 . γ = 3 . γ = 5 . z = 3 . z = 2 . z = 1 . z = 1 . z = 0 . γ , the critical value of the ratio of liquid wealth to pre-existing annuity income z = w/A abovewhich the individual will spend a lump sum to increase her annuity income. We also include theamount the individual will spend on annuities, namely ( w − z A ) / (1 + ( r + λ O ) z ). We assume thefollowing parameter values: The force of mortality λ S = λ O = 0 .
04, which implies a life expectancyof 25 years, the riskless rate of return is r = 0 .
04, the risky asset’s drift is µ = 0 .
08, and the riskyasset’s volatility is σ = 0 .
20. Note that in contrast to the restricted all-or-nothing market, theindividual immediately annuitizes a base level of income and then gradually annuitizes more as hiswealth breaches higher levels. 38 able 4b: How Does Wealth and Risk Aversion Affect Annuitization?
Amount of Money Spent on Annuities for Various Levelsof Wealth and Risk Aversion ( A = $50,000) γ = 1 . γ = 2 . γ = 2 . γ = 3 . γ = 5 . z = 3 . z = 2 . z = 1 . z = 1 . z = 0 . A equals$25,000, while in Table 4b, A is doubled to $50,000. Intuitively, the greater the level of pre-existing annuity income the less liquid wealth must be annuitized to provide the optimal annuitizedconsumption stream. 39 able 4c: How Does Investment Volatility Affect Annuitization? Wealth of $1,000,000 and Initial Annuity Income of $40,000
Investment Volatility
Low Risk Aversion ( γ = 2 ) High Risk Aversion ( γ = 5) σ = 0 .
12 $12,692 $496,789 σ = 0 .
14 $164,292 $598,755 σ = 0 .
16 $289,253 $672,235 σ = 0 .
18 $390,628 $726,853 σ = 0 .
20 $472,871 $768,568Notes: r = 0 .
05 and µ = 0 .
12, with λ O = λ S = 0 .
04. At higher levels of investment volatility(risk), a greater amount is annuitized immediately.40 able 4d: How Does Subjective Health Status Affect Annuitization?
Wealth of $1,000,000 and Initial Annuity Income of $40,000
Subjective Hazard Rate
Low Risk Aversion ( γ = 2 ) High Risk Aversion ( γ = 5) λ S = 0 .
030 $574,840 $817,383 λ S = 0 .
035 $563,603 $812,222 λ S = 0 .
040 $551,941 $806,842 λ S = 0 .
045 $539,862 $801,242 λ S = 0 .
050 $527,375 $795,423 λ S = 0 .
055 $514,496 $789,388Notes: r = 0 . µ = 0 .
10, and σ = 0 .
16, with λ O = 0 .
04. The higher (less healthy) hazard rateleads to reduced levels of annuitization. 41 able 5a: All-or-Nothing Decision with Variable and Fixed Immediate Annuitiesunder Low Risk Aversion
Fixed Annuity, Mixture of Variable (68.7%) andFEMALE (MALE) Fixed Annuities (31.3%),FEMALE (MALE)Age Optimal age of Value of Optimal age of Value ofannuitization delay Annuitization delay60 80.2 (75.2) 21.0 (13.4)% 70.8 (64.1) 3.4 (0.6)%65 80.2 (75.2) 14.8 (7.5) 70.8 (Now) 1.3 (neg.)70 80.2 (75.2) 8.5 (2.5) 70.8 (Now) 0.04 (neg.)75 0.2 (75.2) 2.9 (0.003) Now (Now) neg. (neg.)Notes: In an all-or-nothing annuitization environment – where both fixed and variable annuitiesare available with complete asset allocation flexibility – this table illustrates the imputed value ofdelaying annuitization for males and females with a CRRA of γ = 2. We assume that the non-annuitized funds are invested in a risky asset with drift µ = 0 .
12 and volatility σ = 0 .
20. Therisk-free rate is r = 0 .
06. We introduce insurance loads on the variable and fixed annuities onthe order of 100 basis points and 50 basis points, respectively. The mortality is assumed to beGompertz fit to the IAM2000 table with projection Scale G42 able 5b: All-or-Nothing Decision under Various Levels of Insurance Fees
Low Risk Aversion ( γ = 2) High Risk Aversion ( γ = 5)Insurance Fees Deducted FEMALE MALE FEMALE MALE50 b.p. 67.2 60.0 66.9 60.0100 b.p. 70.8 64.0 68.4 61.2125 b.p. 72.1 65.6 69.1 61.9150 b.p. 73.2 66.9 69.6 62.6200 b.p. 74.9 69.0 70.6 63.8Notes: Assumes the same parameter values as Table 5a.43 able 6: All-or-Nothing Decision with Variable and Fixed Immediate EscalatingAnnuities Fixed Annuity 2% Escalating AnnuityFEMALE (MALE) FEMALE (MALE)Age Optimal Age of Value of Optimal Age of Value ofAnnuitization Delay Annuitization Delay60 80.9 (76.1) 23.68 (15.59)% 78.5 (73.2) 17.41 (9.61)%65 80.9 (76.1) 17.05 (9.24) 78.5 (73.2) 11.50 (4.88)70 80.9 (76.1) 10.22 (3.57) 78.5 (73.2) 5.80 (0.96)75 80.9 (76.1) 3.96 (0.15) 78.5 (Now) 1.29 (0.00)Notes: In an all-or-nothing annuitization environment with escalating annuities available, thistable illustrates the value of delay for males and females with a CRRA of γ = 1 .
5. We assume theliquid funds are invested in a risky asset with drift µ = 0 .
12 and volatility σ = 0 .
20. The risk-freerate is r = 0 .
06. The mortality is Gompertz-Makeham fit to the IAM2000 table with projectionScale G, while the individual has subjective mortality beliefs equal to one-half of the objectivemortality. Note that the availability of an increasing annuity – which better matches the desiredconsumption profile – accelerates the optimal age of annuitization and reduces the option value towait. 44 igure
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