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What makes a better annuity?

ABSTRACT

Given costly and limited annuity products, we investigate how annuity market innovation could improve participation and increase individual welfare. We find that participation gains are most likely with new annuity products that concentrate on late-life payouts. Our welfare analysis suggests that annuity innovation should focus on adding survival contingencies to assets commonly held by individuals. Finally, in a complete market setting, we find demand only for those annuity contracts with a significant time gap between purchase and payout (a rarity in current contracts). Overall, our analysis indicates ample opportunity for innovation to spur annuity demand and improve individual welfare.

INTRODUCTION

Annuities, unlike most financial assets, require that an investor be alive to collect a payment. In his seminal work, Yaari (1965) shows that a rational investor faced with two distinct choices--financial assets and annuity assets--will only purchase annuities. This result rests on three fundamental assumptions. First, an investor maximizes a personal, time separable, von Neumann--Morgenstern expected utility function, which is subject to exponential discounting. In particular, an investor has no desire to leave bequests to her heirs. Second, the financial and annuity markets are complete; that is, all investments are available with and without a survival contingency. Third, annuities are priced assuming frictionless markets and actuarially fair mortality discounting. Given these assumptions, not only do annuities crowd out all other investments, but they also result in significant welfare gains. In fact, recent research has estimated that an actuarially fair market would result in welfare gains equivalent to a 50 percent increase in investor wealth (Mitchell et al., 1999; Brown and Warshawsky, 2004).

Davidoff, Brown, and Diamond (2005) relax Yaari's (1965) assumptions and show that full annuitization can still occur for utility functions that are not necessarily time separable, exponentially discounted, or satisfy the expected utility axioms. Further, they show that the annuity market does not need to be frictionless--returns on annuity assets only need to exceed the returns available on financial assets. Given these theoretical results, they identify a complete annuity market as the critical factor for full annuitization. Next, they investigate incomplete annuity markets and show that "even for preferences that stretch the bounds of plausible impatience," annuity demand persisted, and "a large fraction of wealth is optimally placed in a constant real annuity." Across all the cases they test, participation in the annuity market was unanimous, annuity allocations were never below 75 percent of wealth, and welfare gains were at least 25 percent. Hence, even in an incomplete annuity market, model investors universally purchase some annuity assets--a result that is in sharp contrast to the observed behavior of real investors. To explain this annuity puzzle, Davidoff, Brown, and Diamond conjectur that "lack of annuity demand may arise from behavioral considerations." In this article, we develop an analytic model of annuity participation that explicitly considers three factors that conspire to limit annuity demand--incomplete annuity markets, annuity costs, and behavioral biases. We model both financial and annuity assets as bundles of Arrow--Debreu, state-contingent claims. This allows us to precisely characterize existing annuity contracts and assess the benefit of introducing new contracts. Annuity costs play a significant role in limiting annuity demand. The 10-100 basis point fees of financial assets are modest compared to the 100-1000 basis point fees typical of annuities (Warshawsky, 1988; Mitchell et al., 1999). Though the sensitivity of annuity demand to costs has been previously investigated, the interaction between costs and incomplete markets on demand has not been explored. For example, the stress tests performed by Davidoff, Brown, and Diamond (2005) assume access to annuities with actuarially fair prices. Adding costs to the analysis could significantly change the resulting annuity demand and welfare gains. Lastly, we consider a class of very general utility functions that includes functions that have a behavioral bias against purchasing an annuity. This model includes cohorts of investors that fully and partially annuitize, as well as cohorts that totally abstain.

We approach the annuity puzzle from a somewhat different perspective. Although the current literature focuses on why individuals fail to purchase existing annuity products, we focus on innovations that are most likely to encourage annuity demand. Every time an insurance company introduces a new annuity bundle, the annuity market moves closer to completeness, and there is a possibility of dramatic increases in participation with substantial welfare gains. We use economic theory to identify those annuity contracts most likely to realize this possibility. In short, we explore what makes a better annuity.

We assume that rational investors choose portfolios that maximize their utility of consumption. Hence, if an investor holding just financial assets can increase her utility by switching a small amount of wealth to an annuity asset, then she will purchase an annuity. We show that her marginal gain is maximized for the annuity product with the largest discount relative to the financial asset that replicates its payouts. Since annuity prices tend to be proportional to survival probabilities, an insurance company hoping to improve annuity sales should focus on annuities with late-life (and highly discounted) payouts. The welfare gained from purchasing an annuity depends on an individual's specific utility function; however, we show that knowing an investor's current portfolio is sufficient to compute a lower bound on this gain--the gain exceeds the discount the annuity offers over the current assets. Hence, by adding a survival contingency to a nonparticipating cohort's representative portfolio, an insurance company can create an annuity product that delivers large welfare gains and entices new participants.

Though ultimately we want to answer questions about participation and welfare, our first task is to specify a robust analytical framework. These preliminary tasks include introducing models for financial and annuity markets, defining annuity market enhancements, and identifying our rational investor's objective. These details are found in the next three sections, and are followed by two sections devoted to participation and welfare.

FINANCIAL AND ANNUITY MARKETS

Throughout our analysis, we call all investments without survival contingencies financial assets and refer to them collectively as the financial market. Similarly, the investments with survival contingencies are called annuity assets and belong to the annuity market. For both markets, we assume that an asset's payouts are easily converted to consumption without additional costs, and therefore payouts and consumption are interchangeable. In this section, we describe and model the investment options that we analyze.

A Complete Financial Market

We assume that the current and all future states of our financial market can be described and collected into discrete sets. In particular, we let St equal the set of possible states at time t [greater than or equal to] 0. The sets [S.sub.0], [S.sub.1], ... are necessarily mutually disjoint. The current state of the world is known, and hence So has only one element. On the other hand, for any future time t > 0, there will generally be multiple states with known probabilities that sum to one. Often, we will work with the union S = [S.sub.0] [union] [S.sub.1] [union] ..., the set of all financial states. Since every state is a member of just one set [S.sub.t], we can define a time function t(s) for every s [member of] S. For simplicity, we will restrict time to discrete, annual intervals. The states describe financial market outcomes, e.g., the cumulative market return equals 80 percent after 5 years is an element of [S.sub.5]. However, no states are allowed to depend on an investor's survival.

Our financial market is complete and defined by a set of Arrow-Debreu contingent claims--one for the current state and one for each of the future financial states. The state-claim on state s c S pays $1 if, and only if, state s occurs. The price of the state-claim, the state-price, is given by the function [psi](s) for each state s [member of] S. Given access to state-claims, an investor can use the financial markets to purchase any tailored package of payouts that she desires--a case of maximal flexibility. A useful package of state-claims is the risk-free, zero-coupon bond that pays $1 at time t, independent of the outcome of the financial market. We assume there are no additional costs associated with purchasing financial market packages; thus, the current price [Z.sub.t] of this zero-coupon bond is simply:

[Z.sub.t] = [[summation over (s[member of][S.sub.t])][psi](s). (1)

In particular, the state-claim on the single state at t = 0 costs $1, and so we have [Z.sub.0] = 1. The set of all zero-coupon bond prices describes the financial market's yield curve.

An Incomplete Annuity Market With Frictions

In an ideal world, the annuity market would mirror the flexibility of the financial market. A complete annuity market is exactly the same as a complete financial market with one exception--the investor must be alive to receive a payout. If we assume that the price for the annuity state-claim is actuarially fair, then its price y (s) is given by

[gamma](s) = [[pi].sub.t(s)] x [psi](s). (2)

In Equation (2), [psi](s) is the price of the corresponding financial state-claim, and [[pi].sub.t] is the probability that an investor is alive at time t, given she is alive at t = 0. (1) The annuity state-claims are only available for the subset [OMEGA] of states for which there is some chance that an investor is alive, that is, [OMEGA] = {s [member of] | [[pi].sub.t] > 0}. We call these states the pool set--the set of financial states an insurer considers when underwriting contingent claims for a cohort. For simplicity, we assume that the survival probabilities are strictly decreasing and satisfy the inequality string: 1 = [[pi].sub.0] > [[pi].sub.1] >, ..., > [[pi].sub.T] > [[pi].sub.T + 1] = 0, where T = max { t | [[pi].sub.t] > 0} is the cohort's maximum possible survival time. In this case, the pool set can be written as the union [OMEGA] = [S.sub.1] [union] ... [union] [S.sub.T]. Further, since [[pi].sub.t] [less than or equal to] 1, we see from Equation (2) that with actuarially fair pricing, an investor in the annuity market never pays more for a payout than an investor in the financial market and, more likely, pays less. Hence, investors with no bequest motives should invest only in annuities--Yaari's (1965) prediction.

The products in actual annuity markets have limited flexibility--an investor cannot purchase an arbitrary package of annuity state-claims, but only standard packages--the ones offered by the insurance companies. We model an annuity package as simply an arbitrary collection of future payouts. For example, an annuity package might pay an amount G(s) [greater than or equal to] 0 in each state s [member of] S. The actuarially fair price for this annuity package [[gamma].sub.G] is given by

[[gamma].sub.G] = [summation over (s[member of][OMEGA)] G(s) x [gamma](s). (3)

We can construct a comparable package of payouts from the financial market. If we ignore packaging and transaction costs, the financial market price [[psi].sub.G] of a package with payout function G is

[[psi]G = [summation over (s[member of]S)] G(s) x [psi](s). (4)

In addition to limited availability, we assume annuity packages carry prices higher than their actuarially fair price. We will model additional costs using the money's worth percentage. Formally, the money's worth percentage is the ratio of the actuarial fair price of an annuity to its actual cost (Mitchell et al., 1999). Informally, a value of 100 percent means an investor is getting her full money's worth, and any lesser value models annuity market frictions. (2) For an annuity package with payout function G, its actual price [[GAMMA].sub.G] can be written in terms of its money's worth [K.sub.G] and its actuarially fair price [[gamma].sub.G ]as follows: (3)

[[GAMMA].sub.G] = [[gamma].,sub.G]/[K.sub.G]. (5)

Even if annuity markets were complete, survival contingent payouts could still entail additional costs. If we allow for frictions in the pricing of the state-claim annuity described in Equation (2), its price [[GAMMA].sub.s](s) is given by

[[GAMMA].sub.s](s) = [gamma](s)/[K.sub.s] = ([[pi].sub.t](s)/[K.sub.s]) x [psi](s). (6)

The right-hand side of Equation (6) illustrates the important tension between mortality discounts and annuity frictions. When annuity market frictions are nontrivial ([K.sub.s] < 1), an annuity state-claim may no longer sell at a discount to its financial counterpart. A discount exists only for the states s for which the money's worth exceeds the survival probability ([K.sub.s] > [[pi].sub.t]). In fact, investors should clearly avoid purchasing annuities in states for which frictions exceed the benefit from pooling mortality risk ([K.sub.s] < [[pi].sub.t]). Thus, frictions in the annuity market have two important implications. First, full annuitization is no longer necessarily optimal--some states are fundamentally better handled by the financial markets. Second, large frictions imply investors should focus their annuity purchases on states with relatively high mortality.

Sample Fixed Annuity Markets

Although our theoretical analysis considers annuity markets with arbitrary packages, we will illustrate many of our results by analyzing four fixed annuity markets: immediate, delayed purchase, longevity and zero coupon. We refer to these as markets for fixed annuities because the annuity payouts are deterministic (i.e., independent of market states). Although all of these annuity markets allow for deterministic payouts, the markets are not equal. In fact, as we will detail later, each market in this progression represents an enhancement over the previous annuity markets.

Immediate Annuities. A majority of the research on annuities assumes there is but a single option available--the immediate annuity. For an initial premium, an immediate annuity provides equal payments that begin immediately and last for life. The price of an immediate annuity [[GAMMA].sub.I] is the ratio of its actuarial fair price to its money's worth parameter [K.sub.I]. The actuarially fair price of an immediate annuity that pays $1 annually follows from Equations (3) and (1) with G(s) [equivalent to] 1. Combining these results and simplifying yields

[[GAMMA].sub.I] = (1/[K.sub.I]) x [[summation over 0[less than or equal to]t[less than or equal to]T)] [[pi].sub.t] x [Z.sub.t]. (7a)

As expected, the price of an immediate annuity [GAMMA].sub.I] is a function of the annuity market frictions [K.sub.I], cohort survival probability [[pi].sub.t], and the financial term structure [Z.sub.t].

Delayed Purchase Annuities. Many researchers have pointed out that investors typically have an option to delay the purchase of an immediate annuity; that is, they can set aside funds now to purchase an immediate annuity at a future time. (4) We treat an immediate annuity with this delay option as a separate class of annuities--delayed purchase annuities. Because insurance company policies may restrict sales of immediate annuities to investors below a certain maximum age, there is often a time limit on how long an investor can delay a purchase. Further, for convenience, we assume that all delays are annual. The price [[GAMMA].sub.D]([tau]) of the delayed purchase annuity that begins in r years and annually pays $1 is given by the formula:

[[GAMMA].sub.D]([tau]) = (1/[K.sub.D]) x (1/[[pi].sub.[tau]]) x [summation over ([tau][less than or equal to]t[less than or equal to]T)] ([[pi].sub.t] x [Z.sub.t]). (7b)

In Equation (7b), the money's worth [K.sub.D] generally depends on the cohort and the delay [tau], but for simplicity we suppress these dependencies. Since the underlying immediate annuity is purchased if, and only if, the investor is alive, the survival probabilities in Equation (7b) must be conditioned on survival to time [tau], and the factor [[pi].sub.[tau]] appears in the denominator of the price. Strictly speaking, the delayed payout annuity is a hybrid asset--it is an annuity provided an investor lives for r or more years, and it pays a death benefit equal to its purchase price times the risk-free rate if the investor dies before its conversion to an immediate annuity. Thus, its payout function has two components--a survival-contingent payout and a death benefit payout. In particular, the survival payout is G(s) = 1 for all s [member of] [S.sub.[tau]] [union] ... [union] [S.sub.T], and 0 otherwise.

Longevity Annuities. The longevity annuity is a recent innovation in the annuity product space. (5) MetLife introduced the first of these products in September 2004 (MetLife, 2004) and other insurers have followed suit. (6) Simply put, a longevity annuity is an immediate annuity whose first few payments are skipped. In principle, payments could commence at any future date; however, insurance companies often require payments to begin prior to a set maximum age--typically age 85.

The survival payouts of a longevity annuity that begins its payments in r years are exactly the same as an immediate annuity whose purchase is delayed by r years, that is, a delayed purchase annuity. However, the longevity annuity has no death benefit; all of its payouts are survival contingent. The price [[GAMMA].sub.L]([tau]) of the longevity annuity that begins its payments in [tau] years and annually pays $1 is given by the formula:

[[GAMMA].sub.L]([tau]) = (1/[K.sub.L]) x [summation over ([tau][less than or equal to]t[less than or equal to]T)] ([[pi].sub.t] x [Z.sub.t]). (7c)

Equation (7c) is very similar to Equation (7b); however, there is no need for the [[pi].sub.[tau]] factor, since all survival probabilities are conditioned on survival at t = 0 for which [[pi].sub.0] = 1. In this sense, a longevity annuity price dominates the analogous delayed purchase annuity assuming comparable cost factors.

Zero-Coupon Annuities. Next, we introduce the zero-coupon annuity, a zero-coupon bond with a survival contingency. The price [[GAMMA].sub.Z]([tau]) of a zero-coupon annuity that pays $1 in 0 [less than or equal to] [tau] [less than or equal to] T years is

[[GAMMA].sub.Z]([tau]) = [[pi].sub.[tau]] x [Z.sub.[tau]]/[K.sub.Z]. (7d)

The payout function for this annuity is G(s) = 1 for s [member of] St, and 0 otherwise. Though this annuity is not available commercially, it is often a useful benchmark since it allows for any deterministic payout stream to be survival contingent.

To analyze our fixed annuity markets, we will need to make some assumptions regarding the prevailing money's worth percentage. The money's worth percentage for actual annuity products has been estimated to be between 70 percent and 90 percent (Mitchell et al., 1999; Warshawsky, 1988; Poterba and Warshawsky, 2000). Although our theoretical results apply to situations where the money's worth percentage is arbitrary and could vary between products, our examples will assume a single money's worth percentage applies across annuity products. Thus, our examples will focus on the impact of all annuity products becoming more or less expensive. Interestingly, the results reported in Mitchell et al. (1999) and Poterba and Warshawsky (2000) are suggestive of a single money's worth percentage for a given cohort. (7) Dushi and Webb (2004) and Purcal and Piggott (2008) argue for a single money's worth percentage and perform their analysis based on this assumption.

ANNUITY PRODUCT ENHANCEMENTS

In this section, we introduce a method for ranking annuity markets. As we have defined it, an annuity market consists of a set of survival-contingent products with known payouts and prices. Though some of these products may have a death benefit, we will focus entirely on their survival-contingent payouts. Further, we only consider annuity markets that are paired with a complete financial market. A financial asset's payments are not survival contingent but pay either an investor or her heirs. Said another way, a financial asset has both a survival-contingent payout and a death benefit of equal value. If we ignore death benefits, we can think of financial assets as possibly expensive, but very flexible, annuity assets. In fact, investors can purchase any survival-contingent consumption stream they desire, provided they can afford it.

We say that an annuity market A' dominates an annuity market A if the following two conditions are met. First, the minimum price of every survival-contingent payout stream purchased using the products in A' is never greater than the same stream purchased using the products in A. We emphasize that the assets in both annuity markets can be freely packaged with, assets from the financial market. Second, there is at least one stream that is cheaper to purchase in A' than it is in A.

We now formally define annuity market dominance. Let c(s) [greater than or equal to] 0 be any consumption stream and F(c) be the minimum cost of this stream using the products in annuity market A and the financial market. If x(s) is the quantity of each state-contingent financial security purchased, and y(a) is the quantity of each annuity product purchased, then F(c) is the optimal objective value of the following cost minimization problem:

F(c) = min [summation over (s[member of]S)] x(s) x [psi](s) + [summation over (a[member of]A)] y(a) x [[GAMMA].sub.a] (8a)

c(s) [less than or equal to] x(s) + [summation over (a[member of]A)] y(a) x [G.sub.a](s),s [member of] [OMEGA] (8B)

x(s) [greater than or equal to] 0, S [member of] [OMEGA] (8C)

y(a) [greater than or equal to] 0, a [member of] A. (8d)

In Equations (8), [[GAMMA].sub.a] is the price and [G.sub.a](s) [greater than or equal to] 0 is the survival-contingent payout function for the annuity product a [member of] A. Similarly, we let F'(c) be the minimum cost function for annuity market A'. We say that the annuity market A' dominates A if (1) F'(c) [less than or equal to] F(c) for all streams c, and (2) F'(c) < F(c) for at least one stream c. Condition (1) ensures that all consumption streams are no more expensive, and condition (2) requires that at least one stream is cheaper.

We seldom compare two distinct annuity markets; rather, we are more interested in the effect new products have on an existing market. If, after adding products to a market, the new market At dominates the old market A, we say that the new products enhance the old market. Further, since the new market includes all of the old market products, we only need to show that condition (2) holds. If a new set of products replicates the existing set of products, but with higher money's worth, then they enhance the market. However, increasing money's worth is not the only way to enhance a market. Even an annuity product that sets a new low for money's worth percentage could enhance the market if it covers payouts in states that previously could not be annuitized.

To illustrate market enhancements, we first consider a market of just financial state claims, and introduce immediate annuities. Since the payouts from an immediate annuity are constant, this annuity is an enhancement if, and only if, its price [[GAMMA].sub.I] is less than the price of a constant payout, risk-free bond ladder; that is, the money's worth [K.sub.I] must exceed the bound:

[K.sub.I] > [summation over (0[less than or equal to]t[less than or equal to]T)] [[pi].sub.t] x [Z.sub.t]/ [summation over (0[less than or equal to]t[less than or equal to]T)] [Z.sub.t]. (9a)

We note that the critical value of the money's worth depends on the cohort's survival probability and the current term structure. Next, we reintroduce immediate annuities, but this time we give investors an option to delay their purchases. If we compare the costs of a delayed purchase annuity and a bond ladder for a constant payout stream starting in r years, we find that the delayed purchase annuity is an enhancement whenever its money's worth [K.sub.D] satisfies the following inequality:

[K.sub.D] > 1/[[pi].sub.[tau]] x ([summation over (0[less than or equal to]t[less than or equal to]T)][[pi].sub.t] x [Z.sub.t]/ [summation over (0[less than or equal to]t[less than or equal to]T)] [Z.sub.t]). (9b)

As long as this inequality is satisfied for at least one value of [tau], the option to delay the purchase of an immediate annuity enhances the annuity market. For typical values of the cohort survival probability and the term structure, the graph of the right side of Equation (9b) is a U-shaped function of the delay [tau]. Hence, even though the current purchase of an immediate annuity ([tau] = 0) may not enhance the market, the flexibility of delaying its purchase ([tau] > 0) may enhance the market.

Suppose we add longevity annuities to an annuity market of delayed purchase annuities. These products will enhance the annuity market if at least one of them is cheaper than both a bond ladder and a delayed purchase annuity. In short, the following condition must hold for at least one value of [tau]:

[K.sub.L] ? max {[[pi].sub.[tau]] x [K.sub.D], [summation over (0[less than or equal to]t[less than or equal to]T)][[pi].sub.t] x [Z.sub.t]/[summation over (0[less than or equal to]t[less than or equal to]T)] [Z.sub.t]}. (9c)

Finally, we can add zero-coupon annuities to our market. Since these annuities only have financial market analogues, they necessarily enhance the market provided at least one offers a cheaper way to purchase a riskless payout. To be cheaper, the money's worth for a zero-coupon annuity must exceed the survival probability for the given payout horizon.

Utility Maximization

We model an investor's preferences with a utility function U(c), whose arguments are the state specific levels of consumption c(s). We assume that our investor has no desire to leave a legacy--this has two consequences. First, the consumption stream c(s) for U(c) only includes the states in the pool set [OMEGA]. Second, since all consumption utility is survival contingent, we can ignore any asset's death-benefit payout. Further, we assume that the partial derivative of U with respect to c(s), [U.sub.s], is continuous and positive--more is always better. Also, we require that as c(s) approaches zero, the marginal utility becomes unbounded (see Davidoff, Brown, and Diamond, 2005). This requirement prevents optimal solutions with zero-consumption states (i.e., starvation states). Finally, we will assume that U(c) is concave for c(s) > O.

In spite of all these restrictions, the class of utility functions that we consider is quite large. It includes functions based on expected utility, for example, the expected value of a time-separable, power utility function, as well as functions that do not satisfy the axioms of expected utility. Further, our class simplifies the task of maximizing utility subject to linear constraints (e.g., budget constraints). In our case, the first-order conditions, the Karush-Kuhn-Tucker conditions, are both necessary and sufficient conditions for the existence of a global utility maximum with positive consumption (see Luenberger, 1995).

As an example, suppose an investor with an initial budget of [W.sub.0] invests only in financial assets. She can solve the following mathematical program to find her maximum utility [U.sub.*] and optimal consumption stream [c.sub.*]:

[U.sub.*] = U([c.sub.*]) = max U(c) (10a)

c(s) = x(s),s [member of] [OMEGA] (10b)

[W.sub.0] = [summation over (s[member of][OMEGA])] x(s) x [psi](s). (10c)

The decision variable x(s) is the number of state-claims purchased for state s [member of] [OMEGA], and since each state-claim pays $1, c(s) = x(s). Equation (10c) is the budget constraint--the initial endowment [W.sub.0] equals the total cost of all state claims. Typically, the total cost only needs to be less than the endowment; however, since the marginal utilities are all positive, the full endowment will be spent. The first-order conditions for this problem simplify to

[U.sub.s]([c.sub.*]) = [[lambda].sub.*] x [psi](s), s [member of] [OMEGA] (11a)

[W.sub.0] = [summation over (s[member of][OMEGA])] [c.sub.*](s) x [psi](s). (11b)

Here, the Lagrange multiplier for the budget constraint, [[lambda].sub.*], > 0, measures the marginal utility with respect to the initial endowment [W.sub.0].

More generally, an investor with access to annuity products will solve the following program:

[U.sub.*] = U([c.sub.*]) = max U(c) (12a)

c(s) = x(s) + [summation over (a [member of] A)] y(a) x [G.sub.a](s),s [member of] [Omega] (12b)

[W.sub.0] = [summation over (s[member of][OMEGA])] x(s) x [psi](s) x [psi](s) + [summation over (a[member of]A)] y(a) x [[GAMMA].sub.a] (12c)

x(s) [greater than or equal to] 0,s [member of] [OMEGA] (12d)

y(a) [greater than or equal to] 0, a [member of] A. (12e)

In Equations (12), the decision variable x(s) is the number of financial state-claims purchased for the state s [member of] [OMEGA], and the variable y(a) is the number of shares of the annuity product a [member of] A. The survival-contingent consumption c(s) is the sum of the payouts from the state-claims x(s) and all annuities--the contribution from the annuity a is y(a) * [G.sub.a](s). Since no future trading is allowed, the current choices for x and y determine all current and future consumption c. Equation (12c) is the budget constraint; the full endowment [W.sub.0] is used to purchase assets. Equations (12d) and (12e) are no short sale constraints for financial assets and annuity products, respectively. Allowing short positions in annuity products is problematic due to the ability to fund arbitrary current consumption by issuing future survival-contingent liabilities. However, short positions in financial assets are often reasonable, for example, if the present value of all financial assets is always nonnegative, then short positions have collateral if the investor dies. Still, we opt for the more restrictive, but simpler, no short sale condition. In the Appendix, we identify and solve the first-order conditions for the mathematical program described by Equations (12). The solution prescribes the number of shares of financial assets [x.sub.*] and annuity assets [y.sub.*] that an investor should purchase to maximize her utility. Generally, the optimal utility U, and consumption [c.sub.*] will depend on which products are available in her annuity market.

ANNUITY MARKET PARTICIPATION

The original annuity puzzle centered on Yaari's (1965) prediction that a rational investor without a bequest motive would only purchase annuity assets. Since the potential benefits are large, why do so few investors actually adopt the strategy? Some economists argue that actual benefits have been overestimated, whereas others argue that actual costs have been underestimated. Still, both camps consistently conclude that individuals should invest a significant portion of their wealth in annuities (Brown and Warshawsky, 2004; Davidoff, Brown, and Diamond, 2005). Since Yaari's analysis, the puzzle has evolved--economic theory predicts almost everyone should participate somewhat in the annuity market, but surprisingly, virtually no one does. Davidoff, Brown, and Diamond (2005) suggest that the reasons are behavioral, and Brown et al. (2008) and Hu and Scott (2007) have investigate this hypothesis. In this section, we focus on what drives annuity market participation and which products would increase it. We examine the predictions for both rational investors and for investors with a behavioral bias against annuity purchases.

Participation for Rational Investors

Fundamentally, annuity assets are purchased because they offer a discount on consumption. For our analysis, we measure this discount as the spending-improvement quotient q identified in Scott (2008). The q for a particular annuity asset is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13a)

The spending-improvement quotient reflects the savings (or additional spending) one could achieve by replicating the payouts of a financial asset with a dollar invested in an annuity asset. We define an annuity market's spending-improvement quotient Q as the maximum q available across all annuity products:

[alpha] = [summation over (a[member of]A)] y(a) x [[GAMMA].sub.a]. (13b)

An annuity market participant is any investor who owns an annuity asset, whereas nonparticipants own only financial assets. To analyze the participation decision, we must investigate an investor's marginal utility as a function of her wealth [alpha] allocated to annuities:

[alpha] = [summation over (a[member of]A] y(a) * [[GAMMA].sub.a]. (13c)

A nonparticipant's investments are all in the financial market, and so [alpha] = 0. We can identify participants in the annuity market by first assuming they only have access to the financial market, and then assessing the marginal utility associated with the first dollar annuitized. Participants will find they get positive marginal utility from the first dollar annuitized. The marginal utility for initially swapping money out of financial assets and into annuity assets is given by (see the Appendix for details):

d[U.sub.*]/d[alpha][|.sub.[alpha]=0] = [[lambda].sub.*] * Q. (13d)

where [[lambda].sub.*] is the Lagrange multiplier of the budget constraint that appears in Equation (10c). Since [[lambda].sub.*] is strictly positive from our assumption that more consumption is preferred to less, Equation (13d) implies that investors participate in the annuity market whenever Q > 0, or equivalently, whenever at least one annuity product is offered at a discount, [[GAMMA].sub.G] < [[psi].sub.G]. Moreover, we see that marginal utility from participation is proportional to the annuity market's spending-improvement quotient Q. Surprisingly, all investors of a given cohort agree on which annuity product delivers the maximum marginal benefit from participation, even though they will likely disagree on the magnitude of the benefit, since the latter depends on an investor's utility function via the multiplier [[lambda].sub.*]

Given Equation (13d), it is not surprising that previous research consistently finds at least partial allocations to the annuity market uniformly optimal. Even though we allow for flexible annuity bundles and a general utility function, we find participation is unanimous provided at least one annuity product is offered at a discount. For our rational investors, the only explanation for a dearth of participation is if the market Q is negative. In other words, if annuitization costs are high enough to overwhelm the mortality discount for all available annuity products, then no rational investor should participate.

To assess costs as an explanation for participation, we analyze our fixed annuity markets as viewed by a cohort of women aged 65. (8) For example, consider an immediate annuity with $1 payouts that begin at age 65. Assuming actuarially fair pricing, the price for this annuity is $17.96. Purchasing the same payout stream in the financial market costs $26.00. However, if instead of actuarially fair pricing, this immediate annuity is priced assuming a money's worth value of 69 percent, then the annuity and the financial price for this payout stream would be equal. A money's worth higher than 69 percent implies the immediate annuity offers a discount on consumption, and any value below 69 percent indicates the annuity is inferior to the pricing in the financial market. At this breakeven level of costs, q takes on a value of zero. Since the reported range for money's worth percentages is 70 percent to 90 percent, perhaps costs are a key driver explaining the dearth of annuitization. However, Equation (13d) requires all annuities have inferior pricing if we are to explain an absence of participation.

Figure 1 reports the breakeven money's worth percentages for a range of fixed annuity products. The horizontal axis in Figure 1 identifies the age of the first survival contingent payout for the given annuity package. The vertical axis corresponds to the "breakeven" money's worth value that equates the annuity and financial market costs for the given payout bundle. For example, an immediate annuity, a delayed purchase annuity, and a longevity annuity are equivalent assuming their respective payouts all begin at age 65. As illustrated in Figure 1, the breakeven money's worth for this annuity bundle is 69 percent. Data for delayed purchase and longevity annuities are included out to age 85 since this corresponds to the latest age a typical policy allows one to start payments.

[FIGURE 1 OMITTED]

Breakeven values for delayed purchase annuities decline from 69 percent at age 65 to 55 percent at age 85. Thus, in a high-cost environment, it could be optimal to delay the purchase of an immediate annuity. For costs to preclude the use of delayed purchase annuities, money's worth percentages would have to drop below 55 percent, a value lower than any reported in the literature. Longevity annuities also exhibit declining breakeven money's worth values as the payout age increases. Since longevity annuities price dominate delayed purchase annuities, their breakeven money's worth percentages are lower. By age 85, the breakeven money's worth for a longevity annuity is 35 percent. In other words, with a longevity annuity, insurance companies must charge a price roughly triple the actuarially fair price to eliminate the discount.

Given our model, a cohort of rational investors uniformly decides whether or not to participate based on the sign of Q. A negative Q is possible given sufficiently high fees. If writing annuity contracts involve high costs which necessitate high fees, then insurance companies should introduce annuities with payouts concentrated in low survival states. This class of annuity packages would allow both a high insurance fee while still providing a discount relative to the financial market. By separating the purchase date from the payout date for all payouts, longevity annuities significantly lower the survival probability for each payout. The result is a contract that offers a discount even in extremely high cost environments. If costs were the main barrier to demand, the introduction of longevity annuities should significantly increase participation. However, since delayed purchase annuities also have a relatively low breakeven money's worth and have been readily available for years, costs alone seem insufficient to explain low annuity market participation.

Participation for Investors With a Behavioral Bias

If costs considerations are insufficient to preclude annuitization, perhaps some other factor is involved. In this section, we consider a simple model for investors with a behavioral bias against purchasing annuities. We assume that their utility functions have one additional argument--the amount of wealth invested in annuities [alpha] given by Equation (13c). In particular, we let V(c, [alpha]) represent the utility function of our behavioral investor. Further, we assume that any increase in [alpha] decreases utility, that is, [partial derivative] V / [partial derivative] [alpha] < 0. To identify an annuity market participant, we first constrain an individual to only use the financial markets and then evaluate the marginal utility of the first dollar annuitized. Annuity market participants will find their initial allocation to the annuity market generates positive marginal utility. As we demonstrate in the Appendix, the marginal utility from initial participation is given by

d[U.sub.*]/d[alpha][|.sub.[alpha]=0] = [[lambda].sub.*] * (Q - P) (14a)

P [equivalent to] [partial derivative]V([c.sub.*], 0)/[partial derivative][alpha]/[[lambda].sub.*] (14b)

The annuity penalty P is dimensionless, positive, and depends on an investor's utility function; that is, it varies within a cohort. We see from Equation (14a) that an individual will increase her utility and purchase annuities provided the market's Q exceeds her value of P. Although no member of a behavioral cohort will participate unless Q exceeds the cohort's minimum P, all members will participate if Q exceeds its maximum P. However, it is more likely that some members will participate, and the remaining will not. In short, participation now depends on the magnitude of Q rather than just its sign.

Table 1 reports the magnitude of Q for various combinations of fixed annuity markets and money's worth percentages. For example, an actuarially fair immediate annuity has a q-value of q = ($26.00 - $17.96)/$17.96 = 0.45. If this is the only annuity available, then this is also the market Q as reported in Table 1. Now, suppose our cohort can also purchase an actuarially fair longevity annuity with $1 payments that begin at age 85 at a price of $3.26. The cost of this annuity's replicating bond ladder is $9.32, and so q = ($9.32 - $3.26)/$3.26 = 1.86. If age 85 is the maximum age payouts can begin, then this will also be the annuity market Q.

Assuming a money's worth value of 80 percent, then historically light participation in the delayed purchase annuity market suggests that most investors have a value of P that exceeds this market's Q, a value of 0.45 (see Table 1). In other words, the behavioral disutility of annuitizing the first dollar exceeds 0.45 times the utility gained from a dollar increase in initial wealth endowment. For the same money's worth of 80 percent, a smaller behavioral distortion can explain the unpopularity of immediate annuities purchased at age 65: the disutility of annuitizing only needs to exceed 0.16 times the marginal utility of wealth. Although introducing new annuities does not affect an individual's P, the marginal utility of participation can increase if the introduction increases the market Q. The introduction of longevity annuities nearly tripled the market's Q to 1.29. If this simple behavioral model is correct, then we should see a boost in participation. The disutility of annuitizing would have to exceed the marginal utility of wealth to suppress participation in this market. Moreover, underwriters could increase participation further by introducing new products that increase the market's Q.

Although Yaari (1965) predicted individuals would spend every last dollar on annuities, participation merely requires some annuity allocation. Surprisingly, we find that the marginal utility from participation is governed for everyone by the same quantity, the market spending improvement quotient Q. This suggests that insurance companies looking to improve participation with new annuity products should focus on increasing Q. These new products are not necessarily low cost, high money's worth products. Indeed, a high Q is most easily attained by offering annuity products with payout bundles focused on high mortality states. Even a relatively high cost longevity annuity market, with a money's worth value of 70 percent, offers double the Q compared to an actuarially fair immediate annuity.

INDIVIDUAL WELFARE

Whereas our analysis of the participation decision focused on the marginal utility from an initial allocation to annuities, in this section we investigate the total welfare an investor gains from access to an annuity market and which market enhancements potentially provide the largest gains. We first introduce an investor's expenditure function E([U.sub.0]), the minimum cost to achieve the utility level [U.sub.0] using the annuity and financial markets:

E ([U.sub.0]) = min W (15a)

W = [summation. over (s[member of][OMEGA])] x(s) * [psi](s) + [summation. over (a[member of]A)] y(a) * [[GAMMA].sub.a] (15b)

c(s) = x(s) + [summation. over (a[member of]A)] y(a) * [G.sub.a](s) (15c)

U(c) [greater than or equal to] [U.sub.0] (15d)

x(s) [greater than or equal to] 0, s [member of] [OMEGA] (15e)

y(a) [greater than or equal to] 0, a [member of] A. (15f)

For example, consider an investor with an initial endowment W0 and investments in the annuity market A. Her maximum utility [U.sub.*] is the solution to the program given by Equations (12), and the value of her expenditure function for this level is E([U.sub.*]) = W0. Suppose an investor chooses products from an annuity market A' that enhances the market A. We expect that she may pay less, but never more, and still achieve the same level of utility. The money she saves is her welfare gain. Formally, if we let E and E' be the expenditure functions for the annuity markets A and A', respectively, then the welfare gain [DELTA]W is defined as

[DELTA]W [equivalent to] E'([U.sub.*]) - E'([U.sub.*]) = [W.sub.0] - E'([U.sub.*]). (16)

Informally, an investor's welfare gain is the money she can save by using the products in the enhanced annuity market and still maintain her original level of utility. Given an investor's utility function, we can easily calculate welfare gains.

Portfolio Decisions and Welfare Bounds

Unfortunately, an individual's utility function is not directly observable. Utility functions are individual specifi[c.sub.*] and although our examples can illustrate effects, welfare calculations based on a particular utility function have limited practical application. In the following, we get around this difficulty by recasting the problem in terms of an investor's observed actions (she makes investments to support her future spending) and away from her unobserved decision process (she maximizes her personal utility function). Our method produces some very general results about how annuity enhancements influence welfare.

Our first step is to rewrite welfare as a function of consumption. If [c.sub.*] is the optimal consumption computed from minimizing expenses in annuity market A, then E ([U.sub.*]) = F([c.sub.*]), where F is the consumption based cost function defined by Equations (8). Similarly, for the annuity market enhancement A', we have the optimal consumption [[c'.sub.*].sub.*], and expenditure E'([U.sub.*]) = F'([[c'.sub.*].sub.*],). Hence, using these equalities and Equation (16), we can rewrite welfare in terms of the optimal consumptions as follows:

[DELTA]W = F([c.sub.*]) - F'([[c'.sub.*].sub.*]) = [F([c.sub.*]) - F'([c.sub.*])] + [F'([c.sub.*]) - F'([[c'.sub.*].sub.*])]. (17)

The second equality in Equation (17) splits the welfare gain into two components. The first term in brackets is the money saved by purchasing the original consumption in the enhanced market. We call this the welfare from savings, and since we assume that A' enhances A, it is always nonnegative. The second term in brackets is the welfare from shifting consumption streams. Both [c.sub.*] and [[c'.sub.*].sub.*] have the same utility [U.sub.*]; however, in the enhanced market A', the latter is optimal and will never be more expensive than the former. (9) The second term is called the welfare from consumption shifting, and it is also nonnegative. Since both terms are nonnegative, there is never a welfare loss if an annuity market A' enhances a market A.

Equation (17) relates welfare to consumption--it depends on both the current market consumption [c.sub.*] and the hypothetical, enhanced market consumption [[c'.sub.*].sub.*]. However, at the cost of some precision, we can eliminate the latter dependence. Since the welfare from shifting is nonnegative, a lower bound on total welfare is just the welfare from savings, which depends only on the current consumption [c.sub.*]. Hence, we have

[DELTA]W [greater than or equal to] F([c.sub.*]) - F'([c.sub.*]). (18)

The sharpness of this welfare bound depends on the size of the welfare from shifting. In general, welfare from shifting depends on the degree to which prices for consumption change and the substitutability of the arguments of the utility function. Since consumption across states is generally modeled with a low degree of substitutability--gorging 1 year does not make up for starvation the next--the welfare from shifting tends to be small resulting in a sharp bound on the welfare gain. We can make a stronger argument for the quality of this bound in the limit as price changes become small. For small price changes, welfare from shifting vanishes and the total welfare gained is approximately the welfare from savings:

[DELTA]W [approximately equal to] F([c.sub.*]) - F'([c.sub.*]). (19)

In this context, a price change refers to the price differential between an annuity product in the enhanced market and the minimum cost replicating package of financial and annuity products in the original annuity market. (10)

Generally, the bound described by Equation (18) requires full knowledge of the optimal consumption bundle [c.sub.*]; however, we can obtain a weaker bound using partial consumption information. First, consider the investment shares [x.sub.*] and [y.sub.*] that are purchased to form the least cost portfolio for [c.sub.*]. These shares can be arbitrarily parceled out to multiple, virtual accounts. We call these virtual accounts partial portfolios, and we call their payouts partial consumption streams. If we create M partial portfolios, there will be M partial consumptions [[xi].sub.1], [[xi].sub.2],..., [[xi].sub.M],. and their sum will equal or exceed [c.sub.*]. Partial consumption streams have the property that the sum of the welfare savings from each stream is a bound on the welfare savings of their sum, the total consumption. (11) Using this property, we get the following weaker bound on welfare:

[DELTA]W [greater than or equal to] F[c.sub.*]) - F'([c.sub.*]) [greater than or equal to][[summation over 1[less than or equal to]m[less than or equal to]M] [F([[xi].sub.m]) - F'([[xi].sub.m])]. (20)

Each partial portfolio's contribution to the above bound is its welfare from savings and is nonnegative. Hence, if any one of the partial consumption streams is unknown, its term can be bounded by zero and dropped from the sum. Given Equation (20), we no longer need full knowledge of an individual's portfolio to calculate a bound on welfare gained. Instead, if we can identify any part of an individual's investment portfolio, we can bound the welfare based on the savings the enhanced annuity market offers on replicating that partial portfolio. As our knowledge of the partial portfolio becomes more complete, the welfare bound from Equation (20) approaches the bound from Equation (18).

The welfare bounds identified in Equation (18) and Equation (20) have important implications for annuity product introductions. Since welfare gains are primarily savings gains, annuity providers can use a cohort's representative portfolios to design new products. Rather than designing products based on individual utility, underwriters can focus on products that match the payouts of currently held investments. For example, a house represents the single largest asset owned by many individuals. As such, the preceding analysis suggests the nascent market for reverse mortgages could provide large welfare gains for many individuals. Whether the asset is a stock, bond, or piece of real estate, the fundamental insight is individual portfolio decisions provide important clues as to desirable annuity product innovations. We next illustrate these ideas with two examples. In the first example, we argue that investments providing constant risk-free consumption should receive special attention since this likely makes up a sizable fraction of many individual portfolios. In the second example, we analyze a representative utility specification to illustrate the quality of the welfare bounds.

Example 1: Welfare Bounds and Minimum Consumption

Equation (20) suggests we focus our attention on any portfolio that is held, either in part or in full, by a large number of people since reducing the cost of this portfolio yields widespread welfare gains. Our utility framework identifies one such portfolio. In our framework, starvation states are explicitly ruled out. Hence, every optimal solution will have a minimum positive consumption value (i.e., a consumption floor). Although our framework requires at least some wealth be allocated to minimum consumption, the fraction of wealth allocated could be arbitrarily small. However, a review of the literature suggests wealth allocated to minimum consumption could actually be quite large. For example, in traditional, time-separable utility models, high levels of minimum consumption often result from intertemporal risk aversion or are explicitly imposed with a large subsistence requirement. More recently, habit formation models have developed to explain the surprisingly high levels of consumption persistence. Constantinides (1990) and Sundaresan (1989) pioneered this approach. In addition to the academic literature, consumption smoothing is also a ubiquitous theme in the financial planning literature. Arguably, the most popular rule of thumb for retirement consumption is Bengen's (1994) 4 percent rule. This rule pegs retirement spending to a real, constant value--4 percent of initial wealth. Bengen states that his clients "wanted to spend as much as possible each year from their retirement accounts, while maintaining a consistent lifestyle throughout retirement." (12) In short, there is substantial evidence that many investors want a sizable floor--a constant, risk-free, consumption stream. Hence, an annuity market enhancement which specifically lowers the cost of floor consumption will likely improve welfare across a broad population.

We can estimate the welfare gain derived from using annuities to support floor consumption using an investor from our sample cohort. Initially, suppose that she invests $100 in a zero-coupon bond ladder to provide her consumption floor. Table 2 reports her savings for various annuity market enhancements (the columns) and money's worth values (the rows). If all annuities are actuarially fair (K = 100 percent), then the savings is the same across all markets--the floor purchased for $100 can now be purchased for $69.10, a savings of $30.90. From Equation (20), we see that our investor's total welfare gain is at least $30.90. (13) As the money's worth decreases, early annuity payouts are no longer cheaper than early zero-coupon bond payouts. A longevity annuity selectively avoids these expensive annuity payouts, and in combination with bonds, it saves more money on a floor than an immediate annuity. Delayed purchase annuities are superior to immediate annuities, but inferior to longevity annuities. Further, the welfare gains of longevity annuities are robust to annuity market frictions; for example, at the 70 percent money's worth level, where the welfare gain from an immediate annuity is negligible, the gain from a longevity annuity still exceeds half of its actuarially fair level. Enhancing the market further by providing zero-coupon annuities does not improve upon longevity annuities at these cost levels. Money's worth levels would have to dip below the age 85 survival probability (~63 percent) for zero-coupon annuities to provide a cheaper path to minimum spending.

Example 2: Welfare Bounds and Habit Formation Utility

We have presented two bounds on the welfare gain--the savings welfare bound and the floor welfare bound. The floor bound depends only on an investor's floor consumption; however, if we know all her portfolio's payouts, we can calculate the welfare from savings and improve that bound. Finally, if we know an investor's utility function, we can calculate her actual welfare gain. The bounds are often good estimates of the actual welfare, and in this section, we illustrate their accuracy with a numerical experiment.

Consider an investor from our sample cohort with the following utility U:

U(c) = [T.summation over (t=0)][[pi].sub.t] * [(1 + [delta]).sup.-t][([c.sub.t]/[v.sub.t]).sub.1-[phi]]/(1 + [beta] ) (21a)

[v.sub.t] + 1 = ([v.sub.t] + [beta] * [c.sub.t])/(1 + [beta]). (21b)

[FIGURE 2 OMITTED]

This habit formation utility was introduced by Diamond and Mirrlees (2000) and later analyzed by Davidoff, Brown, and Diamond (2005). Note that consumption [c.sub.t] at time t is independent of state, that is, risk free. In Equation (21a), [[pi].sub.t] is the cohort's survival probability, [delta] is a time-discount parameter, [phi] is a risk-aversion parameter, and [v.sub.t] is the habit level. The habit level changes with time following Equation (21b) and depends on its previous level, the previous consumption, and the habit persistence [beta]. For our numerical examples, we use [phi]) = 2 and [beta] = 1 (Davidoff's values) and a real interest rate of [delta] = 2 percent. We set the initial habit level [v.sub.0] to $3.85, the floor level our cohort can purchase with a bond ladder. (14)

In Figure 2, we plot two optimal consumption streams versus age for an investor with the above habit utility and parameters. The solid bar at age 65 corresponds to the initial habit level [v.sub.0] = $3.85. The first stream [c.sub.*] (solid diamonds) is optimal for an initial endowment of $100 and when our investor has no access to annuities. The second stream [c'.sub.*], (empty squares) is optimal for an initial endowment of $75.3 and when our investor has access to a longevity annuity market with K = 80 percent. The endowment for [c'.sub.*], was chosen so that the optimal utility for both streams is the same, that is, U([c.sub.*]) = U([c'.sub.*],). Hence, the endowment difference, $24.7, is the welfare gain from introducing the longevity annuity market.

We also see that for this situation, the two welfare bounds provide a good estimate of total welfare. The introduction of longevity annuities significantly changes the price of consumption across the various ages. The result is fairly substantial shifts in consumption. Consumption prior to age 84 declines on average by 12.5 percent. Consumption at age 84 and beyond increases by an average of 31 percent. Even with relatively large shifts in consumption, the welfare from savings is still a good bound for the total welfare gain. With longevity annuities, the original consumption stream can be replicated at a cost savings of $20.9; thus, the welfare from savings captures about 85 percent of the total welfare gain. Prior to the introduction of longevity annuities, the minimum consumption level was $3.19. The cost savings on this floor consumption was $17.7, capturing approximately 72 percent of the total welfare gain.

Table 3 reports the welfare gain for four annuity markets (column groups) and various values of the money's worth (rows). Welfare triplets are reported for each combination--actual welfare, savings welfare, and floor welfare. For all welfare calculations, an annuity market is considered as an enhancement over the financial market. For example, the actual, savings, and floor welfare gains for an actuarially fair (K = 100 percent) immediate annuity are $30.4, $28.4, and $25.6, respectively. Similarly, the triplet for the longevity annuity (K = 80 percent) is $24.7, $20.9, and $17.7--these values correspond to the consumption streams discussed above and plotted in Figure 2.

Consider the top row of Table 3--the row corresponding to actuarially fair markets. We see that as annuity markets become more flexible, moving from left to right on the chart, welfare increases. For example, as we move from an immediate annuity market to a longevity annuity market, welfare increases from $30.4 to $34.7. Moreover, the value of flexibility becomes more dramatic in the face of higher costs. For the money's worth value K = 70 percent, as we move from an immediate annuity to a longevity annuity market the welfare increases from $1.2 to $20.6. Both welfare values decrease, as they must, but the percentage decrease of the longevity annuity is much less than that of the immediate annuity.

The second value in each welfare triplet is the welfare from savings. The actual welfare is the sum of the welfares from savings and consumption shifting, and we observe that former dominates the latter. Across all annuity enhancements and all cost levels, the savings welfare captures at least three-quarters of the actual welfare, and often captures 90 percent or more. (15) The welfare from shifting tends to be larger for longevity and zero-coupon annuity markets because these markets allow for more flexible spending patterns. The floor welfare, the third value in the triplet, is a bound on savings welfare. The quality of this bound depends critically on the utility function selected and the resulting taste for floor consumption. However, Table 3 does illustrate how, for this common class of utility specifications, welfare gains are reasonably well estimated by cost savings on floor consumption.

COMPLETING THE ANNUITY MARKET

In the preceding sections, we assumed the annuity market was characterized by bundling and frictions and investigated the types of annuity market enhancements likely to improve participation and generate welfare gains. In this section, we take a different approach. Here, we consider the types of annuity products demanded when annuity markets are complete. Characterizing annuity products that people want, given access to a complete market, could identify desirable annuity packages that effectively substitute for a complete market. To give an extreme example, suppose, when given access to infinite choice in the annuity market, everyone decides they want the same exact bundle of annuity payouts. This would be good news indeed for annuity providers. Instead of having to go to the expense of creating every possible annuity package, they need only create a single package. Access to that single package would provide all the benefits of a complete market.

To investigate this possibility, we assume our annuity market is complete in the sense that every state has an available survival-contingent asset. However, our annuity market is not frictionless. Annuity prices are described by Equation (6). Since each state has a financial and an annuity asset available, we can define a state-specific spending improvement quotient, q(s). Equation (6) and the definition of q imply:

q(s) = [K.sub.s] - [[pi].sub.s]/[[pi].sub.s] (22)

Financial and Annuity Market Separation Theorem

When an annuity market is incomplete, only specific annuity packages are available. In this case, costs are not the sole factor, and suitability depends on whether a package's payouts complement an investor's desired payouts. However, when an annuity market is complete, we can say a good deal about which annuities are purchased, independent of an investor's utility function. In particular, for both a rational and behavioral investor, there will exist a critical [q.sub.*] that separates financial and annuity market purchases. Here, an optimal investment plan will only purchase financial assets in the states s [member of] [OMEGA] with a spending improvement level q(s) less than [q.sub.*] and only annuity assets in all states with a q(s) greater than the same [q.sub.*] (see the Appendix for a demonstration of this result). We call this result the Separation Theorem for complete annuity markets. (16) Informally, annuities are purchased exclusively in high q states, whereas financial assets are exclusively purchased in low q states.

For a purely rational investor, [q.sub.*] equals zero. In other words, in any state where the money's worth exceeds the survival probability the investor will rely on annuity assets for consumption. Similarly, in any state with a negative q the investor relies solely on financial market assets. This recovers Yaari's (1965) result of full annuitization since actuarially fair pricing implies all values of q are at least nonnegative. It extends Yaari's analysis by describing the solution for complete annuity markets with frictions. Now, only states that offer a discount in the annuity market are supported with annuity investments. The result for a behavioral investor with a distaste of annuities is similar except now [q.sub.*] > 0.

The Separation Theorem has significant implications for the annuity market. In general, optimal annuity bundles should never have payouts that occur in states with a survival probability too high to cover the costs of creating the annuity contract. For example, suppose an annuity market is characterized by a money's worth percentage of 85 percent. For our sample cohort, survival rates do not drop below 85 percent until age 77. Given access to a complete market, our investor would fund all consumption between the ages of 65 and 76 with financial market securities and all later life consumption with annuity market assets. In addition, if her preferences include a distaste for annuity investments, then the critical age would be even older.

The package of annuity products that all individuals want when markets are complete all share one common property. They all create sufficient time between the purchase date and the payout date to allow the survival discount to overcome any annuity costs. Surprisingly, this property characterizes few existing annuity products. With the exception of the recently introduced longevity annuity, annuity packages uniformly include payouts with little to no potential for a survival discount. The alignment of the longevity annuity payouts with those desired given a complete market provide an important insight into why longevity annuities are a significant annuity market enhancement with the potential to increase participation and create welfare gains.

CONCLUSION

The disparity between economic prediction and observed annuity market demand is well documented. However, the disparity between ideal and actual annuity markets is equally large. The difficulty associated with writing and monitoring a survival-contingent contract suggests that annuity markets will never offer actuarially fair pricing nor will they likely approach financial markets in terms of flexibility. Given this reality, what types of annuity product innovations might close the gap between current and ideal annuity markets? In short, what makes a better annuity?

To answer this question, we must first identify the goal of annuity market innovation. We considered the annuity market implications of two separate objectives: increased participation and improved individual welfare. With increased participation as the objective, we suggested insurance companies focus on maximizing the spending improvement quotient, q. We demonstrated that annuity product introductions that do not alter the maximum q cannot influence participation. Similarly, introductions that significantly increase the maximum q could spur large increases in participation. Although q is related to both mortality and costs, the most promising area for large increases in q is not lowering annuity costs but rather offering annuity products focused on late-life payouts. If our objective instead is to increase welfare, we suggested insurance companies pay close attention to the assets held by their potential customers. These assets offer important clues as to the types of payouts these individuals find desirable. In fact, the extent to which an annuity product innovation helps individuals replicate their existing portfolio payouts at a lower cost often provides a good estimate for total welfare gains. Thus, innovation focused on minimizing the cost of existing asset allocations likely garners a significant fraction of the potential welfare gain.

Finally, we considered which annuity packages are essential to completing the annuity market. Given complete annuity markets, annuity purchases should only occur when the mortality discount is sufficient to overcome any annuity costs. With positive costs, an annuity purchase thus requires a gap between the purchase date and the payout date. Surprisingly, the vast majority of annuity products available today bundle at least some payouts without a gap. This fundamental inefficiency suggests innovative new products may ultimately supplant many incumbent annuity products. Although annuity demand is low, our research suggests that existing annuity markets offer ample opportunities for improvement.

We show how individuals who maximize utility (with perhaps a bias against annuity contracts) allocate their wealth between financial and annuity investments assuming annuity contracts are both costly and limited. Although we view this analysis as creating an important foundation, there are a number of additional areas for future extensions. For example, our analysis does not consider the impact of a bequest motive. Allowing for some individuals to garner utility from wealth remaining after death would be a significant extension. In principle, a bequest motive would tend to dampen annuity demand, although as Davidoff, Brown, and Diamond (2005) point out, annuity contracts can effectively manage the risks (as perceived by an heir) associated with a bequest that depends on an uncertain date of death. Our analysis considers a robust financial market where individuals can purchase payouts contingent on any future market state, and a more limited insurance market where individuals can purchase payouts contingent on both the market state and a personal state--their survival. Moreover, there are a number of other personal states that could impact decision making. In particular, our framework could be extended to consider health insurance questions. Much like annuity contracts, health insurance should also cover outcomes sufficiently unlikely that the probability discount exceeds the cost of creating the contract. The introduction of multiple personal states also allows an investigation into the impact of correlations. For example, the fact that health shocks and mortality are correlated could have implications for insurance demand. (17) Finally, we underscore that our analysis presumed access to a complete financial market. Modeling an incomplete financial and insurance market allows an analysis of numerous interesting issues such as the demand for liquidity and the impact of volatile interest rates. Although many areas of investigation remain, our hope is that the fundamental relationships between participation, welfare, costs, and bundling identified in this research results in annuity market innovation that ultimately delivers on the benefits Yaari (1965) identified over four decades ago.

APPENDIX

Rational investors will maximize their utility and choose the consumption stream given by the mathematical program described by Equations (12). In this appendix, we consider the more general utility function V(c, [alpha]) that depends on the consumption stream c(s) for s [member of] [OMEGA] and allows for a potential dependency on the amount [alpha] invested in annuities:

[V.sub.*] = V([c.sub.*], [[alpha].sub.*],) = max V(c,[alpha]) (A1a)

c(s) = x(s) + [summation over (a[member of]A])] y(a) * [G.sub.a](s), s [member of] [omega] (A1b)

[alpha] = [summation over (a[member of]A])] y(a) * [[GAMMA].sub.a] (A1c)

[W.sub.0] = [summation over (s[member of][OMEGA]]) x(s) * [psi](s) + [summation over (a[member of]A]) y(a) * [[GAMMA].sub.a] (A1d)

x(s) [greater than or equal to] 0, s [member of] [OMEGA] (A1e)

y(a) [greater than or equal to] 0, a [member of]A. (A1f)

Here, the decision variables are c(s) and x(s) for s [member of] [OMEGA], y(a) for a [member of] A, and [alpha]. The Karush-Kuhn-Tucker conditions (Luenberger, 1995, eqs. C.4, pp. 460-461) simplify to

[c.sub.*](s) = [x.sub.*](s) + [summation over (a[member of]A])][y.sub.*](a) * [G.sub.a](s), s [member of][OMEGA] (A2a)

[[alpha].sub.*] = [summation over (a[member of]A])][y.sub.*](a) * [G.sub.a](s) (A2b)

[W.sub.0] = [summation over (s[member of][OMEGA]])] [x.sub.*](s) * [psi](s) + [summation over (a[member of]A])][y.sub.*](a) * [[GAMMA].sub.a] (A2c)

[mu](s) = [[lambda].sub.*] * [psi](s) - [V.sub.s]([c.sub.*],[[alpha].sub.*]),s [member of][OMEGA] (A2d)

[x.sub.*](s) * [mu](s) = 0, [x.sub.*](s) [greater than or equal to] 0, [mu](s) [greater than or equal to] 0, s [member of] [OMEGA] (A2e)

v(a) = [[GAMMA].sub.a] * [[[lambda].sub.*] - [V.sub.[alpha]]([c.sub.*], [[alpha].sub.*])] - [summation over (s[member of][OMEGA]])][V.sub.s]([c.sub.*],[[alpha].sub.*]) * [G.sub.a](s),a [member of]A] (A2f)

[y.sub.*](a) * v(a) = 0, [y.sub.*](a) [greater than or equal to] 0, v(a) [greater than or equal to] 0, a [member of] A. (A2g)

We assume that V is concave and that these conditions are both necessary and sufficient for a maximum. Further, we assume that the budget constraint, Equation (A2c), is binding and that its Lagrange multiplier [lambda] is positive.

We use Equations (A2) to demonstrate a number of useful theorems. Before proceeding, we derive an alternative form of Equation (A2f) that we use extensively in these proofs. If we eliminate the partial derivate Vs from Equation (A2f) using Equation (A2d), we obtain

v(a) = [summation over (s[member of][OMEGA]])[mu](s) * [G.sub.a](s) - [lambda] * [[GAMMA].sub.a] * [q(a) - [P.sub.*]. (A3a)

In the above equation, q(a) refers to an annuity's spending-improvement quotient:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A3b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A3c)

The price [[psi].sub.G] is the price of replicating the annuity's payouts with financial assets. Also in Equation (A3a), we have introduced [P.sub.*], the penalty for purchasing annuities:

[P.sub.*] [equivalent to] [V.sub.[alpha]] ([c.sub.*], [[alpha].sub.*])/[[lambda].sub.*]. (A3d)

Since we assume that utility decreases as the wealth invested in annuities increases, the penalty [P.sub.*] is positive for behavioral investors (and zero for purely rational investors).

Theorem: The spending-improvement quotient of any purchased annuity must be greater than or equal to the behavioral penalty, a nonnegative number.

Proof: Suppose that y(a) > 0 for some annuity a [member of] A. Equation (A2g) implies v(a) is zero, and hence the right side of Equation (A3a) is zero:

[lambda] * [[GAMMA].sub.a] * [q(a) - [P.sub.*]] = [summation over (s[member of][OMEGA]])[mu](s) * [G.sub.a](s). (A4a)

We have [mu](s) [greater than or equal to] 0 by Equation (A2e), and since [G.sub.a](s) [greater than or equal to] 0, the right side of the above equation is nonnegative. Further, since [lambda] is positive, we must have q(a) [greater than or equal to] [P.sub.*]. Q.E.D.

Corollary: Iran annuity is purchased, then its price is no greater than the price of the bundle of financial assets that replicates its payouts.

Proof: From the previous theorem, the spending-improvement quotient q(a) must be nonnegative. It follows from its definition that if q [greater than or equal to] 0, then [[psi].sub.G] [greater than or equal to] [[GAMMA].sub.a]. Q.E.D.

Theorem: If the spending-improvement quotient of any annuity is greater than the behavioral penalty, then at least one annuity product is purchased.

Proof: For any annuity product a, the Lagrange multiplier for its no short-sale constraint must be nonnegative, i.e., v(a) [greater than or equal to] O. Using Equation (A3a), we get the following inequality for every a [member of] A:

[summation over (s[member of][OMEGA]])[mu](s) * [G.sub.a](s) [greater than or equal to] [lambda] * [[GAMMA].sub.a] * [q(a) - [P.sub.*]]. (A5a)

If for some annuity a, its quotient q(a) exceeds [P.sub.*], then the sum on the left side of Equation (A5a) is strictly positive. Hence, there must be at least one state s'[member of] [OMEGA] where both [G.sub.a](s') and [mu](s') are simultaneously greater than zero. However, if [mu](s') > 0, then it follows from Equation (A2e) that x(st) = 0. We know from our no-starvation assumption that every state must receive either a financial or annuity payout. Since the payout in state s' is not financial, it must be from an annuity, say a'. Though a' and a are not necessarily the same annuity, they will have at least one payout state in common. Q.E.D.

Corollary: For nonbehavioral investors, if the price of any annuity is strictly less than the price of the bundle of financial assets that replicates its payouts, then at least one annuity product is purchased.

This theorem follows from the previous theorem when [P.sub.*] is zero. Q.E.D.

Separation Theorem: When the annuity market is complete, annuity state-claims are purchased on the states for which q(s) > [P.sub.*] and financial state claims are purchased on the states for which q(s) < [P.sub.*].

Proof: For a complete annuity market, we have an annuity state-claim corresponding to each financial state-claim, that is, A [equivalent to] S. Further, for each annuity state-claim a, the payout function [G.sub.a](s) equals one when s equals a, and equals zero otherwise. The sum in Equation (A3a) simplifies, and the equation reduces to

[lambda] * [[GAMMA].sub.s] * [q(s) - [P.sub.*]] = [mu](s) - v(s). (A6)

When q(s) exceeds [P.sub.*] [mu](s) must be positive, and from Equation (A2e), it follows that [x.sub.*](s) = 0, that is, no financial state-claims are purchased. Since starvation states are not allowed, we have [y.sub.*](s) > 0. Similarly, q(s) < [P.sub.*] implies [x.sub.*](s) > 0 and [y.sub.*](s) = O. Q.E.D.

Theorem: For an investor with all her wealth in financial assets, the marginal utility of switching money out of financial assets and into annuity assets is proportional to Q - [P.sub.*], where Q is the market's spending-improvement quotient and [P.sub.*] is her behavioral penalty.

An investor who chooses not to purchase annuities will solve Equations (Al), but with the no-short constraints (y [greater than or equal to] 0) changed to no-purchase constraints (y = 0). In this case, Equations (A2) remain valid, except that we drop the nonnegativity constraints on the Lagrange multipliers v--they now can have either sign. If we were to relax one of the no-purchase constraints, then the marginal increase in maximum utility for an increase in shares of this annuity is proportional to the constraint's multiplier, that is, d[V.sub.*]/dy = -v, where v is given by Equation (A3a). Note that if v is positive, then there is no advantage to purchasing this annuity. Since all consumption is funded with financial assets, we have x(s) > 0 and [mu](s) = 0 for all s [member of] [OMEGA]. In this case, the marginal utility at y = 0 simplifies to

d[V.sub.*]/dy[|.sub.y=0] = [lambda].sub.*] * [[GAMMA].sub.a] * [q(a) - [P.sub.*]]. (A7a)

Alternatively, we can compute the marginal utility of investing an amount [alpha] in this same annuity, and since the investment equals the product of shares and price, we have

d[V.sub.*]/d[alpha][|.sub.[alpha]=0] = [lambda].sub.*] * [q(a) - [P.sub.*]].( A7b)

Equations (A7) are valid for relaxing the no-purchase constraint on any one annuity, while enforcing it on all the remaining products. If we simultaneously open the contest to all annuities, the product(s) with the largest q will win the investment, and so for an unrestricted investment a, we have

d[V.sub.*]/dy[|.sub.y=0] = [lambda].sub.*] * [[GAMMA].sub.a] * [q(a) - [P.sub.*]]. (A7a)

d[V.sub.*]/d[alpha][|.sub.[alpha]=0] = [lambda].sub.*] * [Q - [P.sub.*]].(A8a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A8b)

In Equations (A8), the market's spending-improvement quotient Q is the largest quotient of any of its products. Q.E.D.

DOI: 10.1111/j.1539-6975.2010.01381.x

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(1) In reality, there are multiple cohorts of individuals, all with different survival probabilities, and all of these cohorts comprise the total annuity market. However, since investors are mostly members of a single cohort, and can only purchase annuities offered to their cohort, we will simplify our presentation by fixing on a single cohort at a time, and one probability of survival. We leave for a future paper some interesting cases that are not covered by this simplification; for example, a couple may want to explore purchasing individual annuity products versus a joint annuity product.

(2) There are a variety of annuity market frictions: (1) asymmetric information on unobserved survival probabilities lead to adverse selection costs in these markets, (2) insurance companies incur administrative and capital costs to create an annuity market, (3) brokerage fees are typically in the range of 3 percent to 10 percent of the contract value, and (4) some states tax annuity premiums, and for these states, the average tax is about 1.5 percent of the contract value (Mitchell et al., 1999).

(3) Equation (5) puts very few restrictions on the annuity market but does implicitly require that annuity pricing is scale independent--purchasing two units of a given annuity package costs twice as much.

(4) The following researchers analyze the strategy of delaying the purchase of an immediate annuity: Kapur and Orszag (1999), Dushi and Webb (2004), Milevsky and Young (2003), Dus, Maurer, and Mitchell (2005), and Horneff et al. (2006). The last reference has an excellent summary of this literature.

(5) The industry sometimes refers to this type of contract as "longevity insurance." We have adopted the term "longevity annuity" to be somewhat consistent with this nomenclature.

(6) The Hartford Financial Services Group introduced its longevity annuity in March 2006.

(7) Using 1995 data, annuitant mortality and a corporate yield curve, Mitchell et al. (1999) report money's worth percentages that range from 84 percent to 86 percent for men and women between the ages of 55 and 75. With 1998 data, Poterba and Warshawsky (2000) report a range of 82.5 percent to 86.1 percent.

(8) All survival probabilities are calculated using the GAM94 mortality table adjusted to 2006. The term structure of interest rates is assumed to be a flat 2 percent.

(9) We can easily show that the consumption shift welfare is nonnegative. First, the consumption streams [c'.sub.*], and [c.sub.*] have the same utility [U.sub.*]. The cost of [c'.sub.*], is F'([c'.sub.*],) = E'([U.sub.*]), but this is the least cost way of getting the utility level [U.sub.*]. Thus, any other consumption stream with the same utility is at least as expensive, and in particular, we have F'([c.sub.*]) [greater than or equal to] F'([c'.sub.*]), or [F'([c.sub.*]) - F'([c'.sub.*])] [greater than or equal to] 0.

(10) Consider a simplified situation where two annuity markets, A and A', offer the same products. Annuity market A' is considered an enhancement because it has lower prices. Let p and p' be the price vectors in A and A', respectively. The welfare gain for a fixed utility level [U.sub.0] is [DELTA]W = e(p, [U.sub.0]) - e(p', [U.sub.0]), where e is the expenditure function. For small price changes, we can approximate the welfare gain with its differential: [DELTA]W [approximately equal to] dW = [summation] ([p.sub.a] - [p'.sub.a]). [partial derivative]e(p, [U.sub.0])/[partial derivative][p.sub.a]. However, from the Envelope Theorem, we have [y.sub.a] = [partial derivative]e(p, [U.sub.0])/[partial derivative][p.sub.a], where [y.sub.a] is the number of product shares purchased in market A for utility level [U.sub.0]. Hence, [DELTA]W [approximately equal to] [summation] ([p.sub.a] * [y.sub.a]) - [summation] ([p'.sub.a] * [y.sub.a]) = F(c) - F'(c), where c is the consumption generated by the portfolio, and F is the cost function. Surprisingly, we can extend this result to any annuity market A and its enhancement A'. To see this, recognize that we can repackage linear combinations of existing products in A to create new products without altering the expenditure function. In particular, in A we can replicate, at minimum cost, the payouts of any product in A'. We let market B correspond to the union of A and all replicated products of A'. Note that B simply offers some redundant packages, but does not offer any economic advantage relative to A. We can similarly construct B' as the union of A' and least cost linear combinations of products in A' that replicate the products in A. Since economic opportunities are unchanged, welfare gained from B to B' is

equivalent to welfare gained from A to A'. However, B and B' offer the same set of products only at different prices, demonstrating that the approximation holds in general.

(11) For annuity market A, suppose we split a minimum cost portfolio for the consumption stream c into two partial portfolios with partial consumption streams [[xi].sub.1] and [[xi].sub.2]. We can show from Equations (12) and the definition of partial portfolios that F([[xi].sub.1] + [[xi].sub.2]) = F([[xi].sub.1]) + F([[xi].sub.2]). In fact, the purpose of partial portfolios is to obtain this equality. Generally, the relation is an inequality, and this is also true for partial consumptions when the costs are computed with a different annuity market, say A', that is, F'([[xi].sub.1]t+ [[xi].sub.1]) [less than or equal to] F'([[xi].sub.1]) + F'([[xi].sub.2]). From these two properties we get the bound:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This result generalizes for any finite number of partial portfolios and consumptions.

(12) See Scott, Sharpe, and Watson (2009) for more detail on the financial planning literature and the 4 percent rule.

(13) In this literature, welfare gains are often reported in terms of Annuity Equivalent Wealth (AEW). AEW assumes an optimal utility is achieved with annuities and then calculates the additional wealth required to compensate for the loss of annuity access. This approach relies on optimizations in the enhanced market which does not allow a welfare calculation based on observable portfolios. We avoid this problem by returning to the fundamental definition of welfare--Equation (16). There is generally not a translation from the reported welfare gain and AEW. However, if the utility function is CRRA and separable, then AEW = $100/($100-[DELTA]W). Thus for a $30 welfare gain, AEW is $100/$70 [approximately equal to] $143.

(14) We use a bond-funded floor to fix the initial habit level, whereas Davidoff, Brown, and Diamond (2005) use levels of 0.5, 1.0, and 2.0 times an immediate annuity funded floor. We also investigated a range of values, and the results of those experiments are very similar to those reported here.

(15) Although not reported in the text, this result held true across a variety of habit utility specifications. Additional data are available from the authors upon request.

(16) This generalizes the separation results presented in Scott, Watson, and Hu (2007).

(17) See Davidoff (2009) for an analysis of the impact of correlated health and mortality shocks.

Jason S. Scott, John G. Watson, and Wei-Yin Hu are with Financial Engines, Inc., 1804 Embarcadero Road, Palo Alto, CA 94303. The Authors can be contacted via e-mail: jscott@financialengines.com, jwatson@financialengines.com, and whu@financialengines.com. They thank William Sharpe, Geert Bekaert, Steve Grenadier, and Jim Shearer for many excellent comments and suggestions. Any remaining errors or omissions are the authors' responsibility. The views expressed herein are those of the authors and not necessarily those of Financial Engines.
TABLE 1
Annuity Market Q

Money's Worth (K)   Immediate   Delayed Purchase   Longevity

100%                  0.45            0.82           1.86
90%                   0.30            0.63           1.57
80%                   0.16            0.45           1.29
70%                   0.01            0.27           1.00

Notes: Interest rates are 2%. Survival probabilities based on age
65 females using GAM 94 mortality tables with adjustments to
2006. The maximum age is assumed to be 100. Delayed purchase and
annuities have a maximum payout start age of 85.

TABLE 2
Welfare Bound per $100 Allocation to Minimum Consumption

                                       Annuity Market

Money's Worth (K)  Immediate  Delayed Purchase  Longevity  Zero Coupon

100%                 $30.9         $30.9          $30.9       $30.9
90%                  $23.3         $23.3          $25.4       $25.4
80%                  $13.7         $14.7          $21.4       $21.4
70%                   $1.3          $8.4          $18.0       $18.0

Notes: Access to the financial market alone is the base case in
all situations. Interest rates are 2%. Survival probabilities
based on age 65 female using GAM 94 mortality tables with
adjustments to 2006. The maximum age is assumed to be 100.
Delayed purchase and longevity annuities have a maximum payout
start age of 85, whereas zero-coupon annuity payout are available
up to age 100.

TABLE 3
Welfare and Welfare Bounds With Habit Formation Utility

                                  Annuity Market

                          Immediate            Delayed Purchase

Money's Worth (K)    U     Savings   Floor    U     Savings   Floor

100%                30.4    28.4     25.6    31.7    29.7     25.6
901%                22.6    21.2     19.3    24.1    22.0     19.3
80%                 13.1    12.2     11.3    15.6    13.5     12.2
70%                  1.2     1.1      1.1     8.8     7.2      6.9

                                     Annuity Market

                          Longevity               Zero Coupon

Money's Worth (K)    U     Savings   Floor    U     Savings   Floor

100%                34.7    30.8     25.6    36.9    31.3     25.6
901%                29.2    25.2     21.1    31.7    25.8     21.1
80%                 24.7    20.9     17.7    27.3    21.5     17.7
70%                 20.6    17.2     14.9    23.2    17.9     14.9

Notes: Access to the financial market alone is the base case
in all situations. For each annuity regime/money's worth
combination, the welfare gain was calculated based on the utility
function, U, described in the text. The welfare from savings
bound (Savings) and the welfare bound based on minimum
consumption (Floor) are also reported for each combination.
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