Tape Reading By Linda Bradford Raschke_7 pptx

August 22, 2007 Time: 09:50am chapter01.tex Introduction • 11 1.3 References to Actuarial Finance An encyclopedic book of actuarial calculations with different mortality functions is Bowers et al. (1997), published by the Society of Actuaries. Duncan (1952) and Biggs (1969) provide formulas for variable annuities, that is, for annuities with stochastic returns. For an overview of life insurance formulas, see Baldwin (2002). Another useful book with rigorous mathematical derivations is Gerber (1990). Milevsky’s (2006) recent book contains many useful actuarial formu- las for specific mortality functions (such as the Gompertz–Makeham function) that provide a good fit with the data. It also considers the implications of stochastic investment returns for annuity pricing, a topic not discussed in this book. August 18, 2007 Time: 11:25am chapter16.tex CHAPTER 16 Financial Innovation—Refundable Annuities (Annuity Options) 16.1 The Timing of Annuity Purchases In previous chapters (in particular, chapters 8 and 10) we have seen that in the presence of a competitive annuity market, uncertainty with respect to the length of life can be perfectly insured by an optimum policy that invests all individual savings in long-term annuities. The implication of associating annuity purchases with savings is that the bulk of annuities are purchased throughout one’s working life. This stands in stark contrast to empirical evidence that most private annuities are purchased at ages close to retirement (in the United States the average age of annuity purchasers is 62). A recent survey in the United Kingdom (Gardner and Wadsworth, 2004) reports that half of the individuals in the sample would, given the option, never annuitize. This attitude is independent of specific annuity terms and prices. By far, the dominant reason given for the reluctance to annuitize was a preference for flexibility. For those willing to annuitize, the major factors that affected their decisions were health (those in good health were more likely to annuitize), education, household size (less likely to annuitize as household size increases), and income (higher earnings support annuitization). Lack of flexibility in holding annuities was interpreted by the respon- dents as the inability to short-sell (or borrow against) early purchased annuities when personal circumstances make such a sale desirable. A preference for selling annuities arises typically upon the realization of negative information about longevity (disability) or income. In this survey, the reluctance to purchase annuities early in life was hardly affected by the knowledge that annuities purchased later would be more expensive (due to adverse selection). Bodie (2003) also attributes the reluctance to annuitize to uncertain needs for long-term care: “Retired people do not voluntarily annuitize much of their wealth. One reason may be that they believe they need to hold on to assets in case they need nursing home care. Annuities, once bought, tend to be illiquid . ” August 18, 2007 Time: 11:25am chapter16.tex 136 • Chapter 16 Data about the timing of annuity purchases and surveys such as the above suggest a need to develop a model that incorporates uninsurable risks, such as income (or needs such as long-term care) in addition to longevity risk. Further, to respond to the desire of individuals for flexibility, the model should allow for short sales of annuities purchased early or the purchase of additional short-term annuities when so desired. The first part of this chapter builds on a model developed by Brugiavini (1993) with this objective in mind. With uncertainty extending to variables other than longevity, competi- tive annuity markets cannot attain a first-best allocation (which requires income transfers accross states of nature). Sequential annuity market equilibrium is characterized by the purchase of long-term annuities, short sale of some of these annuities later on, or the purchase of additional short-term annuities. Since the competitive equilibrium is second best, it is natural to ask whether there are financial instruments that, if available, are welfare- improving. We answer this question in the affirmative, proposing a new type of refundable annuities. These are annuities that can be refunded, if so desired, at a predetermined price. Holding a portfolio of such refundable annuities with varying refund prices allows individuals more flexibility in adjusting their consumption path upon the arrival later in life of information about longevity and income. We show that refundable annuities are equivalent to annuity options. These are options that entitle the holder to purchase annuities at a later date at a predetermined price. Interestingly, annuity options are available in the United Kingdom. It is worth quoting again from a textbook for actuaries Guaranteed Annuity Options. The option may not be exercised until a future date ranging perhaps from 5 to 50 years hence . . . . The mortality and interest assumptions should be conservative . . . . The estimates of future improvement implied by experience from which mortality tables were constructed suggest that there should be differences in rates according to the year in which the option is exercisable . . . . A difference of about 1 4 % in the yield per $100 purchase price could arise between one option and another exercisable ten years later . . [Such] differences in guaranteed annuity rates according to the future date on which they are exercisable do therefore seem to be justified in theory. (Fisher and Young, 1965, p. 421.) Behavioral economics, addressing bounded rationality (see below) seems to provide additional support to the offer of annuity options that involve a small present cost and allow postponement of the decision to purchase annuities. It has been argued (e.g., Thaler and Benartzi, 2004; Laibson, 1997) that these features provide a positive inducement August 18, 2007 Time: 11:25am chapter16.tex Financial Innovation • 137 to purchase annuities for individuals with tendencies to procrastinate or heavily discount the short-run future. 16.2 Sequential Annuity Market Equilibrium Under Survival Uncertainty Individuals live for two or three periods. Their longevity prospects are unknown in period zero. They learn their period 2 survival probability, p (0 ≤ p ≤ 1), at the beginning of period 1. Survival probabilities have a continuous distribution function, F ( p), with support [p , p] ∈ [0, 1]. In period 0, all individuals earn the same income, y 0 , and do not consume. They purchase (long-term) annuities, each of which pays 1 in period 2 if the holder of the annuity is alive (all individuals survive to period 1). Denote the amount of these annuities by a 0 and their price by q 0 . Individuals can also save in nonannuitized assets which, for simplicity, are assumed to carry a zero rate of interest. The amount of savings in period 0 is y 0 − q 0 a 0 . At the beginning of period 1 (the working years), individuals earn an income, y 1 , learn about their survival probability, p, p ≥ p ≥ p, and make decisions about their consumption in period 1, c 1 , and in period 2, c 2 (if alive). They may purchase additional one-period (short-term) annuities, a 1 , a 1 ≥ 0, or short-sell an amount b 1 of period-0 annuities, b 1 ≥ 0. Since some consumption is invaluable, they will never sell all their long-term annuities; that is, a 0 − b 1 > 0. In period 2, annuities’ payout is a 0 + a 1 − b 1 if the holder of the annuities is alive, and 0 if the holder is dead. (a) First Best Suppose that income in period 1, y 1 , is known with certainty so that individuals are distinguished only by their realized survival probabilities in period 1. Expected lifetime utility, V, is V = E [u(c 1 ) + pu(c 2 )], (16.1) where u  (c) > 0, u  (c) < 0 and the expectation is over p ∈ [ p, p]. The economy’s resource constraint is E [c 1 + pc 2 ] = y 0 + y 1 . (16.2) Optimum consumption, the solution to maximization of (16.1) subject to (16.2), may depend on p,(c ∗ 1 (p), c ∗ 2 (p)). However, the concavity of V August 18, 2007 Time: 11:25am chapter16.tex 138 • Chapter 16 and the linear constraint yield a first-best allocation that is independent of p: c ∗ 1 (p) = c ∗ 2 (p) = c ∗ , where c ∗ = y 0 + y 1 1 + E(p) , (16.3) and E(p) =  ¯ p p pdF(p) (16.4) is the expected lifetime. We shall now show that a competitive long-term annuity market attains the first-best allocation. (b) Annuity Market Equilibrium: No Late Transactions In period 1, the issuers of annuities can distinguish between those who purchase additional annuities (lenders) and those who short-sell period-0 annuities (borrowers). Since borrowing and lending activities are distin- guishable, their prices may be different. Denote the lending price by q 1 1 and the borrowing price by q 2 1 . The individual’s maximization is solved backward: Given a 0 , p, q 1 1 , and q 2 1 , individuals in period 1 maximize utility, max a 1 ≥0, b 1 ≥0 [u(c 1 ) + pu(c 2 )], (16.5) where c 1 = y 0 + y 1 − q 0 a 0 − q 1 1 a 1 + q 2 1 b 1 , c 2 = a 0 + a 1 − b 1 . (16.6) The first-order conditions are −u  (c 1 )q 1 1 + pu  (c 2 ) ≤ 0 (16.7) and u  (c 1 )q 2 1 − pu  (c 2 ) ≤ 0. (16.8) Denote the solutions to (16.6)–(16.8) by ˆ a 1 (p), ˆ b 1 (p), ˆ c 1 (p), and ˆ c 2 (p), where we suppress the dependence on y 0 − q 0 a 0 , q 1 1 , q 2 1 , and y 1 . It can be shown (see the appendix) that when ˆ a 1 (p) > 0, so (16.7) holds with equality, ∂ ˆ a 1 /∂p > 0, and that when ˆ b 1 (p) > 0, so (16.8) holds with equality, ∂ ˆ b 1 /∂p < 0. A higher survival probability increases the amount of lending and decreases the amount of borrowing whenever these are positive. Assume that optimum consumption is strictly positive, ˆ c i (p) > 0, i = 1, 2, for all p ≤ p ≤ ¯ p (a sufficient condition is that u  (0) =∞). August 18, 2007 Time: 11:25am chapter16.tex Financial Innovation • 139 When q 2 1 < q 1 1 , then by (16.7) and (16.8), individuals are either lenders ( ˆ a 1 > 0) or borrowers ( ˆ b 1 > 0) but not both. It is shown below that this condition always holds in equilibrium. In period 0, individuals choose an amount a 0 that maximizes expected utility, anticipating optimum behavior in period 1: max a 0 ≥0 E[u( ˆ c 1 ) + pu( ˆ c 2 )] (16.9) subject to (16.6). By the envelope theorem, the first-order condition is −E [u  ( ˆ c 1 )]q 0 + E [pu  ( ˆ c 2 )] = 0. (16.10) Denote the optimum amount of period 0 annuities by ˆ a 0 . Since in period 0 all individuals are alike and purchase the same amount of annuities, the equilibrium price, ˆ q 0 , is equal to expected lifetime, (16.4), ˆ q 0 = E( p). (16.11) The equilibrium prices of a 1 and of b 1 are determined as follows. When (16.7) holds with equality at the “kink,” ˆ a 1 = ˆ b 1 = 0, this determines a survival probability, p a , p a = λq 1 1 , where λ = u  (y 0 + y 1 − E( p) ˆ a 0 ) u  ( ˆ a 0 ) , (16.12) with ˆ a 0 determined by (16.10) and (16.11): − E [u  (y 0 + y 1 − E( p) ˆ a 0 − q 1 1 ˆ a 1 (p) + q 2 1 ˆ b 1 (p))]E ( p) + E [pu  ( ˆ a 0 + ˆ a 1 (p) − ˆ b 1 (p))] = 0. (16.13) When ˆ a 1 (p) = ˆ b 1 (p) = 0 for all p, ¯ p ≥ p ≥ p, then, from (16.13), λ = 1 (because marginal utilities are independent of p). When p a < ¯ p, then, by (16.7), ˆ a 1 (p) > 0for ¯ p ≥ p ≥ p a and ˆ a 1 (p) = 0forp a ≥ p ≥ p. Using a similar argument for short sales, define p b = λq 2 1 . The condition q 2 1 < q 1 1 implies that p b < p a . It can be seen from (16.8) that if p b > p, then ˆ b 1 > 0for p ≤ p < p b and ˆ b 1 = 0for ¯ p ≥ p ≥ p b . Summarizing, ˆ a 1 > 0, ˆ b 1 = 0, p a < p ≤ ¯ p, ˆ a 1 = ˆ b 1 = 0, p b ≤ p ≤ p a , ˆ b 1 > 0, ˆ a 1 = 0, p ≤ p < p b . (16.14) August 18, 2007 Time: 11:25am chapter16.tex 140 • Chapter 16 The equilibrium prices ˆ q 1 1 and ˆ q 2 1 are determined by zero expected profits conditions  p p a ( ˆ q 1 1 − p) ˆ a 1 (p) dF( p) = 0 (16.15) and  p b p ( ˆ q 2 1 − p) ˆ b 1 (p) dF( p) = 0. (16.16) Note that the bounds of integration, p a and p b , depend on the equilibrium values ˆ q 1 1 and ˆ q 2 1 . As shown in chapter 8 and first stated by Brugiavini (1993), equilibrium prices that satisfy (16.15) and (16.16) are q 1 1 = p and q 2 1 = p, which implies that ˆ a 1 = ˆ b 1 = 0 for all p. Under a certain condition, this solution is unique. Proof is provided in the appendix to this chapter. This solution entails that ˆ c 1 (p) and ˆ c 2 (p) are independent of p and, by (16.13), equal to the first-best allocation, ˆ c i (p) = c ∗ , i = 1, 2, given by (16.3). Conclusion: When uncertainty is confined to future survival proba- bilities, consumers purchase early in life an amount of annuities that generates zero demand for annuities in older ages, ensuring a consum- ption path that is independent of the state of nature ( ˆ c 1 and ˆ c 2 independent of p). Consequently, there will be no annuity transactions late in life. This conclusion is in stark contrast to overwhelming empirical evi- dence showing that private annuities are purchased by individuals at advanced ages. 1 Indeed, we shall now show that the above conclusion does not carry over to more realistic cases with uncertainty about (uninsurable) future variables, such as income, in addition to survival probabilities. 16.3 Uncertain Future Incomes: Existence of a Separating Equilibrium Suppose that in period 0, the probability of survival to period 2 and the level of income in period 1, y 1 , are both uncertain, the realizations occurring at the beginning of period 1. 2 The realized levels of p and y 1 1 See Brown et al. (2001). 2 An alternative formulation is to make utility in period 1 depend on a parameter needs, whose value is unknown in period 0 and realized at the beginning of period 1. This formulation yields the same results as those shown below. August 18, 2007 Time: 11:25am chapter16.tex Financial Innovation • 141 are assumed to be private information unknown to the issuers of annuities. For simplicity, assume that y 1 is distributed independently of p. Its distribution, denoted by G(y 1 ), has a support (y 1 , y 1 ). (a) First Best As before, the first-best allocation maximizes expected utility, (16.1), subject to the resource constraint E [c 1 + pc 2 ] = y 0 + E(y 1 ). Again, the solution is independent of p: c ∗ 1 = c ∗ 2 = y 0 + E(y 1 ) 1 + E(p) . (16.17) However, unlike the previous case where the early purchase of annu- ities could fully insure against survival uncertainty and, consequently, implement the first-best allocation, it is seen from (16.17) that the first-best solution with income uncertainty requires income transfers, providing the expected level of income to everyone. Indeed, income insurance would enable such transfers. However, for obvious reasons, the level of realized income is assumed to be private information, and this precludes insurance contingent on the level of income. Conse- quently, the annuity market cannot, in general, attain the first-best allocation. (b) Sequential Annuity Market Equilibrium As before, maximization is done backward. In period 1, utility maximiza- tion with respect to a 1 yields the first-order condition −u  1 ( ˆ c 1 )q 1 1 + pu  ( ˆ c 2 ) ≤ 0, (16.18) with equality when ˆ a 1 > 0. Setting ˆ a 1 = ˆ b 1 = 0, (16.18), with equality −u  (y 0 − q 0 a 0 + ˜ y 1 1 (p)) q 1 1 + pu  (a 0 ) = 0, (16.19) defines for each p a critical level of income, ˜ y 1 1 (p). Since −u  (y 0 − q 0 a 0 + y 1 + q 2 1 b 1 )q 1 1 + pu  (a 0 − b 1 ) > 0 for all y 1 > ˜ y 1 1 (p) and b 1 ≥ 0, it follows that ˆ a 1 (p, y 1 ) > 0 for all ¯ y 1 ≥ y 1 > ˜ y 1 1 (p) and ˆ a 1 (p, y 1 ) = 0 for all y 1 ≤ y 1 < ˜ y 1 1 (p) (see figure 16.1). August 18, 2007 Time: 11:25am chapter16.tex 142 • Chapter 16 Figure 16.1. Pattern of period-1 annuity purchases. Similarly, the first-order condition with respect to b 1 is u  ( ˆ c 1 )q 2 1 − pu  ( ˆ c 2 ) ≤ 0, (16.20) with equality when ˆ b 1 > 0. Again, setting ˆ a 1 = ˆ b 1 = 0, (16.20) with equality defines for each p a critical level of income, ˜ y 2 1 (p). Since u  (y 0 − q 0 a 0 + y 1 )q 2 1 − pu  (a 0 ) > 0 for all y 1 ≤ y 1 < ˜ y 2 1 (p) and ˆ a 1 ≥ 0, it follows that ˆ b 1 (p, y 1 ) > 0 for all y 1 ≤ y 1 < ˜ y 2 1 (p) and ˆ b 1 (p, y 1 ) = 0for all ¯ y 1 ≥ y 1 > ˜ y 2 1 (p). To make the pattern displayed in figure (16.1) consistent, it is nece- ssary that ˜ y 2 1 (p) < ˜ y 1 1 (p) for all p, which is equivalent to the condition that q 2 1 < q 1 1 . That is, the borrowing price is lower than the lend- ing price. 3 We shall show that this condition is always satisfied in equilibrium. 3 For a 2 × 2 case, Brugiavini (1993) shows that the condition is that income varia- bility be large relative to the variability of survival probabilities. This ensures that all individuals with a high income and with any survival probability purchase annuities, and vice versa. August 18, 2007 Time: 11:25am chapter16.tex Financial Innovation • 143 Equilibrium prices, ( ˆ q 1 1 , ˆ q 2 1 ), are defined by zero expected profits conditions  ¯ p p ( ˆ q 1 1 − p) ˆ a 1 (p, ·) dF( p) = 0 (16.21) and  ¯ p p ( ˆ q 2 1 − p) ˆ b 1 (p, ·) dF( p) = 0, (16.22) where ˆ a 1 (p, ·) =  ¯ y 1 ˜ y 1 1 (p) ˆ a 1 (p, y 1 ) dG(y 1 ) and ˆ b 1 (p, ·) =  ˜ y 2 1 (p) y 1 ˆ b 1 (p, y 1 ) dG(y 1 ) are total demands for a 1 and b 1 , respectively, by all relevant income recipients with a given p. Recall that ˆ a 1 and ˆ b 1 depend implicitly on q 1 1 and q 2 1 and on ˜ y 1 1 (p) and ˜ y 2 1 (p), defined above. Thus, the existence and uniqueness of ( ˆ q 1 1 , ˆ q 2 1 ), defined by (16.20) and (16.21), requires certain conditions. From (16.21) and (16.22), ˆ q 1 1 − ˆ q 2 1 =  p p pϕ( p) dF( p), (16.23) where ϕ( p) = ˆ a 1 (p, ·)  ¯ p p ˆ a 1 (p, ·) dF( p) − ˆ b 1 (p, ·)  ¯ p p ˆ b 1 (p, ·) dF( p) . (16.24) Clearly,  p p ϕ( p) dF( p) = 0. Hence, ϕ(p) changes sign at least once over [p , ¯ p]. Since ˆ a 1 (p, ·) strictly increases and ˆ b 1 (p, ·) strictly decreases in p, ϕ( p) strictly increases in p. This implies that there exists a unique ˜ p, 0 < ˜ p < 1, such that ϕ(p)  0asp  ˜ p. It follows that ˆ q 1 1 − ˆ q 2 1 =  ¯ p p pϕ( p) dF( p) > ˜ p  ¯ p p ϕ( p) dF( p) = 0. (16.25) Thus, the condition for an equilibrium with active lending and borrowing in period 1 is satisfied. As before, the equilibrium price for period-0 annuities is equal to life expectancy: ˆ q 0 = E( p) =  ¯ p p pdF(p). (16.26) Of course, 0 < ˆ q 0 < 1. Notice that since ˆ a 1 (p, ·) strictly increases and ˆ b 1 (p, ·) strictly decreases in p,1> ˆ q 1 1 > ˆ q 0 , while ˆ q 2 1 < ˆ q 0 , reflecting the adverse selection in period 1. [...]... improve the market allocation in terms of expected utility?4 We answer this question in the affirmative by proposing a new financial instrument that may achieve this goal The proposal is to have a new class of annuities, each carrying a guaranteed commitment by the issuer to refund the annuity, when presented by the holder, at a (pre) specified price Call these (guaranteed) refundable annuities As shown below,... refund of ri when presented by the holder in period 1 There are k types of such refundable annuities, ranked from the highest refund down, r1 > r2 > · · · > rk ≥ 0 Denote the price and the amount of type i i ri annuities purchased by q0 and a0 , respectively The amount of type ri i i i annuities refunded in period 1 is denoted b1 , a0 ≥ b1 ≥ 0 Individuals’ consumption is now given by k c1 = y0 + y1 − k i... compared to the purchase by rational individuals By deliberately delaying implementation of the purchase of annuities, this plan may accommodate hyperbolic discounters (Laibson, August 18, 2007 Time: 11:25am chapter16.tex Financial Innovation • 149 1997) who put a high discount rate on short-run saving Thaler and Benartzi report that their plan has been successfully implemented by a number of firms There... 11:25am chapter16.tex 146 • Chapter 16 ˆr ˆr Denote the solution to (16.16) by a0 The equilibrium price, q0 , satisfies a zero expected profits condition: ˆr ˆr q0 a0 = ¯ p p ˆr ˆr p(a0 − b1 ( p; ·) dF ( p) + r ¯ p p ˆr b1 ( p; ·) dF ( p) or ˆr q0 = E( p) + ¯ p 1 ˆr a0 p ˆr (r − p)b1 ( p; ·) dF ( p), (16.31) ˆ while q1 is determined by (16.8) Two observations are in place First, a condition for an active... (r = 0) As before, individuals may borrow against these annuities at the market price for borrowing, described in the previous section Denote r r the amount of type r annuities by a0 , a0 ≥ 0, and the amount refunded r r r by b1 , a0 ≥ b1 ≥ 0 Consider first only one type of refundable annuity For any realization of y1 , consumption in periods 1 and 2 is r r r c1 = y0 + y1 − q0 a0 + r b1 − q1 a1 , r... (16.34) and (16.35) by a1 and b1 , i = 1, 2, , k These are functions k 1 2 ¯ of r = (r1 , r2 , , rk), q0 = (q0 , q0 , , q0 ), and q1 ˆ ˆ i > 0, then bi = a i for all 1 ≥ i > j It is seen from (16.35) that if b1 1 0 That is, all higher-ranked annuities (compared to marginally refunded annuities) are fully refunded The amount of type ri annuities purchased in period 0 is determined by maximization... flexibility offered by holding a portfolio of options to annuitize, there may be additional behavioral reasons in favor of such options A vast economic literature reports experimental and empirical evidence of the bounded rationality and shortsightedness of individuals (e.g., Rabin, 1998, 1999; Mitchell and Utkus, 2004) Of particular relevance to our case seems to be the plan designed by Thaler and Benartzi... conditions u (c1 )r − pu (c2 ) ≤ 0 (16.28) −u (c1 )q1 + pu (c2 ) ≤ 0 (16.29) and ˆr ˆ Denote the solutions to these equations by b1 ( p, y1 ), and a1 ( p, y1 ) r r Again, these functions implicitly depend on y0 − q0 a0 , r , and q1 The optimum level of period-0 annuities is determined by maximization of expected utility, (16.3), assuming an optimum choice, (ˆ 1 , c2 ), in period 1 c ˆ The first-order condition... + u (ˆ 1 ) 1 c u (ˆ 2 ) c (16A.2) Consider the zero expected profits condition (16.5): p pa 1 ˆ (q1 − p)a1 ( p)dF ( p) = 0 (16A.3) 1 Where pa = λq1 , λ is given by (16.12), λ= ˆ u (y0 + y1 − E( p)a0 ) , ˆ 0) u (a (16A.4) ˆ and a0 is determined by (16.13), 1 2ˆ ˆ ˆ −E [u (y0 + y1 − E( p)a0 − q1 a1 ( p) + q1 b1 ( p))E( p) ˆ ˆ ˆ +E [ pu (a0 + a1 ( p) − b1 ( p))] = 0 (16A.5) ˆ ˆ When a1 ( p) = b1 ( p) =... changes a0 , and hence λ, compared to the previous case August 18, 2007 Time: 11:25am chapter16.tex Chapter 16 • 151 Denote by ϕ expected profits in the period-1 market for annuities, 1 ϕ(q1 ) = p pa 1 ˆ (q1 − p)a1 ( p) dF ( p) (16A.6) ˆ1 ˆ1 ¯ An equilibrium price, q1 , is defined by ϕ(q1 ) = 0 Since pa = p when 1 ˆ ˆ = p (because a1 ( p) = 0 and λ = 1), q1 = p is an equilibrium price, implying no purchase . Thaler and Benartzi, 2004; Laibson, 19 97) that these features provide a positive inducement August 18, 20 07 Time: 11:25am chapter16.tex Financial Innovation • 1 37 to purchase annuities for individuals. sufficient condition is that u  (0) =∞). August 18, 20 07 Time: 11:25am chapter16.tex Financial Innovation • 139 When q 2 1 < q 1 1 , then by (16 .7) and (16.8), individuals are either lenders ( ˆ a 1 >. the purchase by rational individuals. By deliberately delaying implementation of the purchase of annuities, this plan may accommodate hyperbolic discounters (Laibson, August 18, 20 07 Time: 11:25am