Technical Paper Series 

Congressional Budget Office 

Washington, DC 

Health Shocks and the Demand for Annuities 

Sven H. Sinclair 

Macroeconomic Analysis Division 

Congressional Budget Office 

Washington, DC 

E-mail: sven.sinclair@cbo.gov 

Kent A. Smetters 

University of Pennsylvania and NBER 

Philadelphia, PA 

E-mail: smetters@wharton.upenn.edu 

July 2004 

2004-9 

Technical papers in this series are preliminary and are circulated to stimulate 

discussion and critical comment. These papers are not subject to CBO’s formal 

review and editing processes. The analysis and conclusions expressed in them are 

those of the authors and should not be interpreted as those of the Congressional 

Budget Office. References in publications should be cleared with the authors. 

Papers in this series can be obtained at www.cbo.gov/tech.cfm. 
Abstract 

A new explanation is offered for the thin private market for individual annuities in 

the United States. Individuals face a risk of health shocks which simultaneously cause 

large uninsured expenses and shorten the life expectancy. The value of a life annuity then 

decreases at the same time as the need for cash increases, undermining its effectiveness in 

providing financial security. When the risk of such health shocks is substantial, it is no 

longer optimal for risk-averse individuals with uncertain life spans to hold all of their 

wealth in life annuity form, even if annuity contracts are reversible, and bequest motives, 

transaction costs and adverse selection are absent. 

A dynamic programming model is used to compute the demand for annuities, 

under conditions involving health shocks, in an overlapping-generations setting calibrated 

to resemble the United States economy. The model is used to estimate the demand for life 

annuities and the relative significance of the factors that affect the demand: health shocks, 

Social Security, bequest motives and premium loads. It is useful for understanding 

various modes of drawing down retirement savings, measuring the extent of uninsured 

health-related risks (particularly long-term care expenses) and providing a consistent 

framework for analysis of various approaches to insure such risks. 
1

1. Introduction 

It has become conventional wisdom that life annuities are the “best” retirement 

savings vehicle for most people. Yaari (1965) proved that a risk-averse individual with 

uncertain life spans, intertemporally separable utility and no bequest motives would find 

it optimal to hold all assets in life annuity form if the annuity can be purchased at the 

actuarially fair premium, namely, without transaction costs (policy loading). Moreover, 

more recent work has shown that the more restrictive of Yaari’s assumptions are not 

necessary: Davidoff, Brown and Diamond (2002), for instance, proved that it is sufficient 

to assume that agents have no desire to leave bequests and that premiums are not too far 

from being actuarially fair. 

In contrast with this theoretical result, the actual market for individual life 

annuities in the United States is very thin. Premiums for individual immediate annuities 

totaled $7 billion in 1999. (Brown et al., 2001) Other financial products marketed as 

individual annuities are far more popular, but they are used primarily as tax-efficient 

investment vehicles in the accumulation phase, rather than as a lifetime source of income 

in the payout phase (e.g., Mitchell et al., 1999).1

 

Several explanations have been offered for the discrepancy between the standard 

theoretical result and the observed state of the market. Premium loading is always present 

and is an obvious departure from Yaari’s assumptions, but the observed premiums are not 

so far from actuarially fair that consumers would not want to annuitize at least a 

significant portion of their wealth at the usually assumed levels of risk aversion (Mitchell 

et al., 1999). Social Security and defined-benefit pensions provide a mandatory life 

 

1

 Examples of such investment products are single-premium deferred annuities and variable annuities. 
2

annuity for most Americans, crowding out the market for individual annuities (Townley 

and Boadway, 1988), but this does not explain why wealthier Americans without definedbenefit pensions do not purchase significantly more life annuities, nor why most elderly 

individuals did not hold private annuities prior to the introduction of Social Security 

(Warshawsky 1988). Marriage provides some insurance against the mortality risk 

(Kotlikoff and Spivak, 1981), but the insurance is incomplete, depends on the assumption 

of uncorrelated mortality between spouses, and does not apply at all to unmarried 

individuals. Another common, and powerful, explanation is the bequest motive (Hurd 

1989, Jousten 2001), but it again fails to explain non-purchase of annuities by individuals 

without children. 

Another popular explanation has been that the disciplined, gradual income stream 

of annuities leaves individuals liquidity-constrained. In its simple form, this explanation 

requires the restriction that life annuities, once purchased, may not be sold, and this 

situation is not entirely realistic. Even if law or market failure prevents individuals from 

actually selling annuities on their own life, a negative life annuity can be constructed by 

simultaneously borrowing and purchasing term life insurance. Since it is legal and 

feasible for someone to own both an annuity and a life insurance policy, the assumption 

that annuities are irrevocable may be hard to defend. This paper introduces a closely 

related situation, in which annuity values fall as liquidity needs rise, but no exogenous 

restrictions on annuity purchases and sales are assumed. 

In Yaari’s and all related models, uncertainty is limited to individual mortality, 

but the probabilities of survival for any number of future years are known and 

deterministic. This assumption is not trivial. Anyone who has bought life insurance and 
3

gone through the underwriting process likely knows that future life expectancy depends 

on a number of variables; most of those variables are health-related and many are 

revealed at random times during a person’s life.2

 If, additionally, there is a positive 

correlation between available assets and life expectancy, Yaari’s result does not hold: full 

annuitization is no longer optimal. This paper focuses on the effect of wealth-affecting 

stochastic changes in health (henceforth “health shocks”). 

An idea related to this paper has been analyzed by Brugiavini (1993). The main 

similarity is that both papers use information updating to explain low annuity demand.3

The nature of this updating, as well as the model used, is different. Brugiavini considers 

an adverse selection economy, with ex ante heterogeneous agents who learn their type in 

the early period of life. The present paper assumes ex ante identical agents exposed to 

random shocks, and it does not assume asymmetric information (although it is possible to 

extend the model to include it). Furthermore, Brugiavini limits the analysis to a threeperiod model and derives a prediction with a complete-market flavor: no trades occur 

after the first period. We employ a more realistic computational model which can 

accommodate more flexible assumptions, and the predictions generally turn out to be 

quite sensitive to the parameters used. 

To estimate the magnitude of deviation of optimal asset allocation from full 

annuitization, individual choices are simulated in an overlapping-generation economy 

with realistically calibrated relevant parameters. In this model, health shocks can be 

turned on and off and their magnitude changed; moreover, same can be done to Social 

 

2

 Some of the health information is about genetics (e.g., family history) and doesn’t change through life, 

but other is about personal history and may change (usually for the worse) at any time. Typical questions in 

life insurance applications might be “Have you been diagnosed with cancer/heart disease/AIDS/etc.” 

3

 One further connection between the two papers is that neither poses any exogenous restrictions on 

individuals selling or undoing life annuities. 
4

Security, premium loads and bequest motives, so that the relative sizes of their respective 

effects on annuity demand can be compared. 

2. Basic Framework 

2.1. Health Shocks 

Consider a currently healthy individual whose entire wealth is annuitized, as 

Yaari’s model would predict, and who faces a risk of illness which would require 

expensive treatment during the next year. If that individual becomes ill, two adverse 

effects follow: probability of survival in current and future years decreases, and the need 

for cash increases due to the medical and related expenses. We will refer to health 

changes with those two effects as health shocks. The necessary medical expense — 

required for survival or simply for maintaining a minimally acceptable quality of life — 

can consume a large portion of the individual’s wealth and can easily be larger than the 

annual income provided by the annuity. Whether these expenses are insurable or not, the 

aggregate effect of uninsured expenses is significant: one in four petitioners for personal 

bankruptcy identify an illness or injury as a reason for filing for bankruptcy; among 

petitioners age 65 or higher, the proportion is roughly one in two (Warren, Sullivan and 

Jacoby, 2000). Following a health shock, the affected individual might like to sell (or 

undo) the annuity in order to pay for the health-related expenses. Such a transaction, 

however, would be based on the updated mortality table, which incorporates the 

knowledge of illness and consequently higher probabilities of death.4

 The value of the 

annuity would now be lower than that for a healthy individual of the same age: health 

 

4

 This paper assumes symmetric information throughout. 
5

shocks reduce the value of annuities precisely at the times annuitants would prefer to 

liquidate. If this effect is significant, it is better not to annuitize all wealth, and it may be 

best not to annuitize at all. 

Various health conditions can cause catastrophic health care expenses, but costs 

of most hospital stays and medical treatments are typically well insured for a majority of 

Americans, including virtually all those over age 65, who are covered by Medicare. Many 

people are still exposed to large out-of-pocket expenses, as the cited bankruptcy data 

demonstrate, but the actual exposure to expense due to acute illness is hard to estimate. 

Nursing home stays, however, cost tens of thousands of dollars per year, are usually not 

eligible for Medicare reimbursement, and are rarely insured in the private market. The 

practically relevant, and straightforward to calibrate, interpretation of health shocks is 

thus as onsets of long-term-care (LTC) episodes. In addition, when working-age 

individuals are considered, the financial part of a health shock may take the form of a loss 

of earnings in case of disability. 

2.2. Assets: Bonds and Annuities 

Individual decisions in this model involve the consumption and portfolio choice 

with two available securities: a bequeathable asset and life annuity. No result will depend 

critically on the kind of bequeathable asset used; we will use risk-free bonds as the 

simplest asset to model. There is no loss of generality from this as long as the insurance 

company that provides annuities can hold the same portfolio of assets backing that 

annuity as the individuals would hold themselves. No matter what that portfolio is, the 

expected return on annuity will be higher by the transfer of wealth from those who die to 
6

those who survive.5 Annuities yield a higher expected rate of return, since their payout 

includes a survivorship premium (transfer from decedents to survivors), but they involve 

risk. In the traditional model, with deterministic survival probabilities, annuities are 

effectively riskless because their return, conditional on survival, is certain, and those are 

the only relevant states for individuals not concerned with bequests. With health shocks, 

annuity returns are risky even when restricted to the states in which the individual 

survives (and hence cares about them). 

3. Analytical Aspects 

3.1. A Three-Period Model 

An individual lives for up to three periods, j = 1, 2, 3. The probability of surviving 

to period j+1, conditional on being alive in period j, is pj, where p0 = 1 and p3 = 0. In each 

period the individual earns income wj and consumes cj, while the unconsumed portion of 

wealth can be saved either as a noncontingent bond yielding interest rate r or as a life 

annuity with a rate of return ρ > r conditional on being alive. The individual’s period 

utility u(·) is a concave function of consumption only, and the rate of time preference is β. 

The individual has no desire to bequeath any wealth. At time t the individual has total 

assets At = at + bt, where at is the actuarial value of the annuity and bt is the value of 

bonds. In each period the individual learns his current wealth and income and the 

available returns r and ρ, and decides how to allocate his resources among a, b, and c.

The individual’s problem is: 

 

5

 Furthermore, in an economy with no aggregate uncertainty, a well-diversified market portfolio is also 

risk-free. 
7

 

2

1 1 2 12 3 , ,

1

max ( ) ( ) ( )

s.t. , 1,2

,

, , 0, .

abc

j j j j t j jj

jjj

jj j

EU u c p u c p p u c

A A w c a br j

c Aw j

cbA j

β β

+ ρ

=+ +

= + −+ + =

≤+ ∀

≥ ∀

 (1) 

If u(·) satisfies the Inada conditions, the nonnegativity of consumption constraint 

can be omitted. Obviously for any increasing u(·) the individual consumes all wealth in 

the last period. In period j the uncertain quantities are {ws, s > j}, {ps, s > j}, and {ρs, s ≥

j}. The last two will generally be closely related since the annuity return normally 

consists of a bond return and the annuity premium. The general form is: 

1 1 1 j j

j

j

E π

ρ

π

+ + + = , (2) 

where Ej denotes conditional expectation with respect to the period j information set, and 

πj is the single premium in period j for an annuity paying 1 per period for the rest of the 

life, the first payment to occur at the beginning of period j+1. If the annuity premiums are 

actuarially fair, the return can be expressed as: 

2

1

12 1

2

2

1 1 1

1

1 1

r p r

r Ep p

r

p

ρ

ρ

   ++ +

+ =       + +

+

+ =

 (3) 

An important result follows in the simplest case, where the annuity returns ρs are 

deterministic (and, naturally, greater than the risk-free interest rate): the constraint 

preventing borrowing at the risk-free rate is binding and the individual will hold all 

savings in the annuity form whenever he holds any savings at all. This is a special case of 

Yaari’s (1965) result and is intuitively obvious because annuities dominate bonds as an 

asset in all states the individual cares about. Another result under certainty is that the ratio 
8

of marginal utilities in consecutive period equals the individual’s subjective discount 

factor: u’(cj) / u’(cj+1) = (1+ ρj) β pj, whenever the problem for total savings has an 

interior solution.6

 This is the well-known optimality condition for saving under certainty. 

These obvious results illustrate the view of annuities which will be taken throughout the 

paper: a life annuity is a (generally risky) financial asset, and annuitization decisions are 

essentially portfolio decisions. The only special feature of this asset is that it is a 

derivative of the individual owner’s expected mortality. 

Even when there is uncertainty, the full annuitization result will hold in the 

second-to-last period. This is because p3 = 0 is certain, and hence ρ2 is also certain, 

leading to the first-order condition involving the multiplier of the borrowing constraint 

(λ) that is essentially the same as under certainty, since both returns can be taken outside 

the expectation: 

2 23 23 { [ ][ ]}

22 2 3

(1 ) ( ) (1 ) ( )

( ) ( ).

p E uc E r uc

p r Eu c

λβ ρ

β ρ

= + −+ ′ ′

= − ′ (4) 

The right-hand side is strictly positive, so the borrowing constraint is always binding in 

period 2. This also explains why the simplest model of annuity demand with updating of 

mortality information must have at least three periods. No demand for noncontingent 

assets can arise in the penultimate period even if income is uncertain. In the present 

model, anything interesting can happen only in the first period. 

Consider the first-order condition, analogous to (4), for the first period. It can be 

reduced to: 

1 [ ][ ] 12 1 2 1 E (1 ) ( ) (1 ) ( ) + =+ + ρ λβ uc E ruc p ′ ′ , (5) 

 

6

 If (1+rj) β > 1 — and hence (1+ ρj) β pj > 1 — a sufficient condition for an interior solution, for any 

increasing utility function, is A1 + w1 ≥ w2 and w1 ≥ w2. 
9

where λ is the Lagrange multiplier of the period-1 borrowing constraint. Now the return 

ρ1 is uncertain and the next-period wealth, and hence also consumption c2, will depend on 

it, and will obviously increase as ρ1 increases. Since u(c) is concave, u’(c2) will decrease 

as ρ1 increases, so it will be (perfectly) negatively correlated with ρ1. Therefore: 

[ ] [ ][ ] 12 1 2 E (1 )() 1 () + <+ ρ ρ uc E Euc ′ ′ , (6) 

and it may be possible to satisfy (5) with λ = 0 (borrowing constraint not binding), 

although E[1+ ρ1] > E[1+r]. This simply shows that a life annuity behaves like a risky 

asset. It turns out that, for realistic parameters and an actuarially fair ρ1, the magnitude of 

the correlation is not sufficient to produce an interior solution. If, however, income w2 is 

also positively correlated with the annuity return, the correlation can be much greater and 

the effect will be amplified. The annuity acts as a negative hedge for the risk of changes 

in future income, and this makes it a less desirable asset. 

As an illustration, consider the extreme case where period-2 survival probability 

may take values 0 or 1. Let the probability in period 1 of those events be (1– α) and α, 

respectively. According to (3), the annuity return will be either 

2

1 1 (1 ) (1 ) 1 S ρ α = + ++ − r rp or 1 1 (1 )(2 ) (1 ) 1 H ρ α = + + ++ − rr rp . Also assume that 

utility is CRRA, 1 uc c () 1 σ σ − = − , that β = 1, and income is zero in periods 1 and 3, 

while in period 2 it takes on values w2H = h if “healthy” (low mortality) or w2S = –s if 

“sick” (high mortality), where h, s ≥ 0. Even with such a simple utility function the firstorder conditions are rather cumbersome, but are easily solved numerically for various 

parameter values.7

 For simplicity, in this discussion we will assume h = 0, i.e., no 

 

7

 It may be interesting to note that logarithmic utility, which is easier to solve, cannot give rise to less 

than full annuitization in this (three-period) model. 
10

unforeseen income, and vary only s, the unforeseen expense in the sick state. Some 

results are shown in Tables 1 and 2. 

With relative risk aversion σ = 2, initial wealth A1 normalized to 1, interest rate r

= 0.25, probability of survival from first to second period p1 = 0.9, and probability of 

remaining healthy in the second period α = 0.6, the critical value of s for which the 

borrowing constraint ceases to bind is 0.196 — the potential expense in the unfavorable 

state needs to exceed approximately one-fifth of the initial wealth in order to make it 

worthwhile to use bonds in the portfolio.8

 The critical value is higher for higher interest 

rate and lower period-one survival probability, both of which mean more discounting of 

future shocks. It is lower at higher levels of risk aversion as such individuals assign 

relatively less importance to the higher expected return of annuities; it reaches a 

minimum at the intermediate value of the frequency parameter α, because those levels 

maximize the uncertainty of outcome. 

Table 2 shows that the portfolio composition will generally be quite sensitive to 

the magnitude of the shocks. In most cases where shocks are comparable to one’s entire 

wealth, no long position in annuities is taken; the individual prefers to short a life annuity 

if allowed. 

3.2. Infinitely Many Periods 

For another illustration of the implications of updating mortality and health 

information on the demand for annuities, consider a version of the model of perpetual 

youth. Individuals live for an uncertain, but unbounded, number of periods. In the 

standard perpetual youth model (e.g., Blanchard and Fischer 1989, p.115) individuals 

 

8

 Note that one period corresponds to one-third of a lifetime, so an interest rate of 25% is not high. 
11

have a constant (age-invariant) probability of death, so that their distribution of age at 

death is geometric (or exponential in continuous time). For the present problem, assume 

this probability of death is itself stochastic, following a two-state Markov process. We 

can label the state of lower mortality “healthy” and the state of higher mortality “sick”, 

with respective survival probabilities p

H and pS

 (where 1 > p

H > pS

 > 0). To simplify the 

analysis, let the sick state be absorbing, so that the only nontrivial transition probability is 

q = Pr (ξj+1 = S | ξj = H), which can be interpreted as morbidity. When the transition 

from healthy to sick state occurs, the individual’s wealth is reduced. This outcome can 

represent health care expense or reduction in income in case of disability. 

The problem can be formulated recursively, using the value function V (A): 

{ } , ,

( , ) max ( ) ( , )

s.t. (1 ) (1 )

, 0.

abc

V A w u c pEV A w

A a b r wc

c Aw

b c

β

ρ

= + ′′′

′ = + + ++−

≤ +

≥

 (7) 

where primes denote next-period quantities. 

The simplest case involves an individual who has no income, just an initial 

endowment of wealth to be consumed over the remaining life. For tractability and time 

invariance, let the reduction in wealth at transition to the sick state be multiplicative: 

whatever assets Aτ the individual has at the time, the wealth available for future 

consumption becomes λAτ, where 0 < λ < 1.9

 This is essentially the simple portfolio 

problem with two assets (e.g., Sargent 1987), with the simple modifications in the rate of 

time preference (pξ

β, ξ∈{H, S}, instead of β) and the next-period wealth being multiplied 

 

9

 This is obviously not a very realistic model of health expenses if comparison across wealth levels is a 

goal. It also distorts the incentive to save; however, its effect on the portfolio choice for a given level of 

savings is small. We use it here for tractability and to avoid the possibility of the expense exceeding the 

available wealth. In numerical computations we use a more realistic model of expenses. 
12

by the random variable λ

% which takes the value λ if the individual becomes sick 

(probability q) and 0 otherwise. The distribution of asset returns is discrete with two 

states. The bond return, (1+r) is equal in both states, and the annuity return can be shown 

to be: 

0 1 1 1 , S

S

r p

p

ρ

+ −

+ =+ (8) 

if sick in current period, 

1 (1 ) 1( ) 1 1 , 1 (1 )

H S

H

H S

r qp r p

p r qp

ρ

  +− − +−   + =+   +− −  

 (9) 

if healthy now and next period (probability 1–q), and 

2 (1 ) (1 ) (2 )(1 ) 1 1 , (1 ) (1 )H S

S

H S

r p qp q r

p r qp

ρ

+ + − −− +     + =+   +−−  

 (10) 

if healthy now and sick next period (probability q). 

First note that an individual who is already sick (and this is known to the insurer) 

would behave just like in the conventional model and hold only annuity and no bonds, 

since the only uncertainty that is left is that of the time of death. Therefore, in order to 

study the interesting effects of information updating, it suffices to consider the problem 

of a healthy individual. The Bellman equation for the individual’s problem is: 

{ ( ) [ ] 1 } ,

( , ) max ( ) (1 ) (1 ) , . t t

H

tt t t t t t t t a b

V A H u A a b p EV a r b ξ β λρ ξ = = −− + + ++ + % % (11) 

 If we further assume logarithmic utility, u(c) = ln(c), then it is straightforward to 

obtain the expressions for a and b:

10

 

10 For a general CRRA utility, at and bt can also be expressed in closed form. The expressions are more 

complicated and they depend on λ, but the qualitative behavior is similar. 
13

(1 ) (1 ) , ( )( )

(1 ) ( ) . ( )( )

H HS

t t H S

H SH S

t H S t

p rr q q a A

r r

p q qr q r qr r b A

r r

β ρρ

ρ ρ

β ρρ ρ

ρ ρ

+ −− −     = − −

  −− + + −+ −   = − −

Using expressions(8)–(10) for ρ

H and ρS

, we can express at and bt in terms of 

fundamental quantities q, p

H

, p

S

, r and β. The expressions are rather cumbersome and are 

not shown. The optimal annuity share in the portfolio is plotted in Figures 1 and 2 for 

select values of the parameters. There is generally a region in which this optimal share is 

less than one, meaning the borrowing constraint is not binding. As one would expect, this 

region is in the northwest of the (p

S

, q) plane. Comparing the two figures, it is apparent 

that the optimal annuity share is quite sensitive to the parameters and varies rather steeply 

near the curve on which the borrowing constraint is exactly met. While the perpetual 

youth model is far from realistic, it should provide useful insight into the life-cycle 

planning of someone around the retirement age, who may well live for another forty or so 

years and is facing slowly changing average survival probability for at least a decade. 

Under conditions represented in Figure 1, with quite plausible probability of shock of 2% 

and mortality rate in the sick state of 25%, only half the wealth would be annuitized. 

4. The Computational Model 

The models of the previous section provide a useful insight into the mechanism 

by which updating of mortality information reduces individuals’ demand for annuities. 

They are, of course, too stylized to be useful for real-world quantitative estimates. 

Besides the obvious fact that realistic mortality and morbidity tables do not resemble a 

perpetual youth model, there are interesting, but complicated, situations to which 
14

analytical models do not apply. For example, expenses can be so high that they exhaust 

an individual’s entire wealth; there are many people whose total assets do not exceed the 

cost of a year’s stay in a nursing home, and many more whose assets are roughly of that 

order of magnitude.11 Typically, some form of bankruptcy (or limited liability) constraint 

applies in those cases, often in the form of the state as the insurer of last resort. It has 

been observed that people behave rationally taking this into account, as evidenced by the 

“Medicaid spend-down”. Almost half of the current nursing home residents admitted as 

privately paying eventually became Medicaid beneficiaries during their stay (U.S. House 

of Representatives 1998, p. 1062). Obviously, running out of private wealth is a 

possibility that any quantitatively useful model must include, but such a model is almost 

sure to be analytically intractable. 

4.1. Individuals: Age and Health 

The economy is populated by overlapping generations of individuals who live to a 

maximum of J periods (years), with one-period survival probabilities p(j, χj) at age j

dependent on health status χj, which in turn is also uncertain. The state variable χ takes 

values on a discrete set. While the precise definition of “health” is not critical for the 

functioning of the model, the health states are generally distinguished and ordered by the 

respective survival probabilities. Assuming higher value denotes “worse” health: 

χ ξ χ ξ χξ > ⇒ pj pj j ( ) () , ≤ ∀∀ ,,, , . (12) 

 

11 Annual cost of nursing home is roughly $50,000 (e. g., McGarry and Schoeni 2001), about the same as 

the mean wealth of the 3rd decile in the Health and Retirement Study (HRS), excluding Social Security 

wealth which cannot be liquidated (Moore and Mitchell 2000). The median HRS family wealth, excluding 

Social Security, equals approximately the cost of four years in a nursing home. 
15

Another feature of health is that states of bad health can be connected with 

financial losses, consistent with the definition of health shocks in Section 2.1. 

Initially all individuals are “healthy,” but in any period, one’s health can change. 

Accordingly, χ follows an M-state Markov process with an age-dependent transition 

matrix Rmn(j); m, n = 1, ..., M. For our present purposes, which focus on explaining 

annuity demand, M = 3 will suffice; more health states might be required in different 

contexts, such as pricing long-term care insurance or evaluating its money’s worth, as in 

Brown and Finkelstein (2003).12

4.2. Savings: Bonds and Annuities 

Individuals can hold the wealth they do not consume in two forms: noncontingent 

bonds and life annuities. The basic properties of these assets were introduced in Section 

2.2; when there are no transaction costs (premiums are actuarially fair), we can rely fully 

on the concept, described in Section 3, of annuities as “actuarial notes” which mature in 

one period and pay $1 plus the amount equal to the next-year annuity premium for the 

same person. In this case it is sufficient to have one state variable for an individual’s 

wealth because the portfolio weights (bonds vs. annuities) can be changed in each period 

at no cost. Optimal decisions do not depend on the portfolio composition at the beginning 

of the period. 

Actuarially fair annuities are generally not available (Mitchell et al. 1999) and 

premium loading obviously can influence demand, so a realistic model ought to allow for 

 

12 In fact, two health states are sufficient for most purposes. Three states add some additional flexibility 

and dynamics without significant extra computational cost. For example, with three states it is possible to 

treat the first year of illness differently from subsequent years in both its financial impact and its impact on 

mortality. 
16

it. This is not a trivial adaptation since the problem essentially becomes a multi-period 

portfolio choice with transaction costs, which requires an additional state variable. 

Without transaction costs, purchasing a single-premium life annuity and simultaneously 

borrowing the amount equal to the annuity premium and buying life insurance with 

annual premium equal to the annuity payment is equivalent to no action at all. With 

transaction costs, it involves a “round-trip” transaction cost and is clearly inferior. 

Therefore, in every decision, one must consider not just the current total wealth, but also 

the initial portfolio composition. Therefore, the model must keep track of bonds and 

annuities as two separate state variables. To avoid slowing down the computation, it is 

best to use a model without transaction costs whenever those are not the object of 

attention. The precise way in which asset state(s) are modeled is described in the 

Appendix. 

There is no formal restriction on borrowing or short sales of annuities. However, 

it is impossible for an individual, who might die before the loan repayment time, to 

borrow at the risk-free rate without carrying life insurance in the amount of the loan. 

Such a restriction is natural, and would arise in a rational equilibrium even if not modeled 

explicitly. Since borrowing plus buying life insurance is equivalent to selling an annuity 

on one’s own life, we consider all transactions in the insurance market as annuity 

purchases or sales. Some formal restrictions do exist. One cannot trade annuities on other 

people’s lives, consistent with prevalent laws that give life insurance companies the 

exclusive right to sell life annuities. An individual’s net worth A must always remain 

nonnegative. This applies also to the situation when a health shock, and the associated 

expense, would otherwise make the individual’s wealth negative. As the utility would 
17

become –∞ at zero consumption, this restriction is ensured the minimum level of 

consumption (bankruptcy/welfare protection) in each period. 

4.3. Individuals’ Preferences 

Individuals are fully rational and have preferences for consumption and possibly 

for leaving bequests. Preferences for consumption are additively time-separable, with a 

constant relative risk aversion period utility (felicity). This choice is common in lifecycle literature as well as in multi-period finance, it characterizes risk aversion in a 

plausible way, and is easy to use and control in computations. To avoid problems with 

tractability and uniqueness that arise in OLG models with altruism, bequest motives are 

modeled as “joy of giving” and total felicity is additively separable in consumption and 

bequest. The full utility function is then: 

1

1

1 1

( ) 1

J J

j j j

j jj

j j

c

U uc DA

σ

β βζ

σ

−

+

= =

 

== +   −   ∑ ∑ (13) 

Here, j denotes age, β is the rate of time preference, cj is consumption at age j, σ is 

the curvature parameter, Aj is bequeathable wealth at age j, Dj is an indicator that equals 1 

in the year of death and 0 otherwise, and ζ is a parameter that determines the strength of 

the bequest motive. As there are no preferences over leisure, the labor supply is inelastic. 

While this is a departure from realism in working years and especially around normal 

retirement age, which now must be exogenously determined, it does help focus on the 

problem of annuity demand, for several reasons. Firstly, annuity demand mainly exists 

due to uncertain consumption needs late in life; secondly, few people return to work at a 

very old age (which would be a significant issue if it were common) and it is unclear how 
18

utility for leisure should be described at those ages. Thirdly, and perhaps most 

importantly, it would be potentially difficult to sort out the general equilibrium effects of 

annuities if a change in savings level also caused a change in the labor supply. 

When a “healthy” individual becomes “sick,” his or her probability of survival 

decreases and a health-related expenditure may become necessary. This expenditure does 

not contribute to the individual’s utility for consumption; it is assumed to be necessary 

for any chance of survival, and hence perfectly inelastic. In effect, individuals have 

lexicographic preferences in which one’s own survival trumps any other form of 

consumption — probably a slight exaggeration, but a reasonably plausible one.13 

4.4. Production and Labor Productivity 

The output Yt of the economy in period t is determined by the constant returns-toscale technology with a Cobb-Douglas production function Y = θt Kα

 L1–α, where K

denotes capital and L labor, in efficiency units as discussed below. 

During the individuals’ working years, their earning capacity (productivity) 

changes with age in a predictable pattern, as well as randomly due to individual 

(idiosyncratic) productivity shocks. This idiosyncratic productivity is quantified by a 

Markov state variable η with a transition matrix Qkl(j); k, l = 1, ..., L. While this is an 

additional, and realistic, source of financial risk, its main purpose in the present model is 

as a source of wealth heterogeneity. All individuals are a priori identical and all 

differences among them arise due to random events during their lifetimes (possibly 

 

13 Equivalently, preferences can be described using state-dependent utility, where sick individuals who 

do not receive adequate treatment derive no (or sufficiently little) utility from any consumption or bequests. 

Also, as we do not deal with moral hazard in the present paper, the fact that health expenses are 

exogenously fixed presents no significant restriction on the rational behavior of individuals. 
19

including being born into different productivity and health states). Productivity is 

generally zero beyond normal retirement age and in the highest (sickest) health state, 

regardless of the value of η.

14 An individual’s wage is a product of age-related 

productivity εj, individual productivity η, an indicator of the health status, and the market 

wage rate per unit of labor. 

Individuals do not know what their productivity and health state will be in the 

next period, but know their respective distributions, so they make their decisions to 

maximize the expected utility over the set of next-period states. With a given 

consumption and saving decision, next-period wealth is uncertain, but depends only on η

and χ, so it is possible to optimize over consumption, saving in bonds, and saving in 

annuities using standard constrained-optimization techniques. 

Individuals spend the first part of their lives working and earning, and the later 

years in retirement. They can satisfy their consumption needs in retirement by 

accumulating some wealth through private savings during their working years, by relying 

on government transfers, or both. We model government-provided retirement scheme as 

unfunded (pay-as-you-go) transfer from workers to retirees in each period, so that it is 

effectively a mandatory life annuity to a retiree. The “Social Security” transfer tax is 

determined from the equilibrium distribution of individuals and the desired average 

income replacement rate. Government also acts as the insurer of last resort for long-term 

care, providing transfers to those individuals whose nursing home costs exceed their total 

available assets. As with Social Security, this transfer is financed through a balancedbudget tax. 

 

14 Age-related zero productivity reflects inelastic labor supply, as discussed later. 
20

In the absence of Social Security and transaction costs, an individual is fully 

described by four state variables: age j, health χ, idiosyncratic productivity η, and wealth 

(assets) A. The economy, as a collection of individuals, is then described by the measure 

Φ(j, χ, η, A) of individuals by state, and by the values of market wage w, interest rate r, 

capital stock K, and labor supply L. Presence of transaction costs requires an extra 

variable, so A must be replaced by two variables (A for annuities and B for bonds), and 

otherwise the structure is unchanged. Social Security has a peripheral role in this model, 

so it is simplified to avoid adding a state variable for earnings history.15

4.5. The Dynamic Programming Problem 

To complete the description formally, an individual characterized by age j, wealth 

A, productivity state η and health state χ solves the following problem at time t, taking the 

prices w, r, ρ and initial conditions j, A, η, χ as given: 

1 [ ] , ,

(,, ,)

max ( ) ( , ) ( , ; ), , , 1 ( , ) ( , )

t

t j abc

VA j

uc p j V A ab j Q d R d

χ η

η χ

βχ χ + η χ η η χ χ

′ ′

=

      + + ′ ′ ′′ ′ ′

    ∫ ∫ (14) 

subject to: 

( ) ( ) ( )( )

( )( )

11 1

0

0 1,

j

j

A a b r I H w T c B Tr L

b

c A I H w T B Tr L

ρ εη χ

εη χ

′ = + + + + = − −+ + −

≥

≤≤ + = − ++ −

where εj is the (deterministic) average productivity of a worker of age j, B is the 

amount received from the distribution of unintended bequests, L is the financial loss in 

the sick state, Tr is the sum of all government transfers received (Social Security and 

welfare), and T is the total tax rate required to finance those transfers. Here A∈R+ , 

 

15 More about modeling Social Security is discussed in Section 5, Calibration. 
21

1 2 { , ,..., } η ∈ = H ηη ηn , 1 2 { , ,..., } χ χχ χ ∈ = X m , j J ∈ = J {1,2,..., } , and the functions 

{ } 1 ,,, : t ttt t Vabc ∞

→ + = S R are measurable with respect to F , where S R HXJ = ××× + , 

( ) () () ( ) F , and = ××× B PPP R HXJ + P( )⋅ denote power sets and ( ) B R+ is the Borel σalgebra of R+ . 

5. Calibration 

Mortality data for the population as a whole are based on the U. S. Social Security 

Administration tables (Faber and Wade 1982; Bell, Wade and Goss 1992). Since the 

“sick state” in the context of this paper corresponds most closely with long-term 

disability during working years and with a need for long-term care for retirees, the 

sources of mortality data for “sick” individuals are RP-2000 disabled life mortality tables 

(Society of Actuaries 2000) for ages 21–65 and nursing home discharge and mortality 

data from the 1998 Green Book and the National Nursing Home Survey (NNHS) for ages 

over 65. The latter data are in coarse age groups, so the rates by age by year were 

obtained using the relative rates from population mortality and from mortality in 

continuing-care retirement communities (Barney 1998). Survival probabilities by age and 

health state are shown in Table 3. 

Three health states are generally enough to capture all information available from 

the existing data. Besides the healthy and sick states with by now obvious interpretations, 

an intermediate state is useful as representing impaired health with significantly higher 

mortality, but without actually being institutionalized and suffering the associated 

expense. For working years, we use the intermediate state to represent the first year of 

disability so that we can vary recovery rates with the duration of disability. Transition 
22

probabilities between health states are shown in Table 4. Probabilities for the workingage population are based on long-term disability morbidity tables (Society of Actuaries 

1982) and Social Security data regarding the number of persons receiving benefits. For 

older ages we used nursing home population and admission data from the Census Bureau, 

National Nursing Home Survey, Medicare, and the 1998 Green Book; we refined these 

transition probabilities based on the actuarial model of Robinson (1996). The expenditure 

connected with poor health for retired population is based on average nursing home costs 

as reported in the Green Book and NNHS. The information in LTC data is necessarily 

incomplete: on one hand, some expenditures are higher than necessary and in fact 

represent voluntary (and hence elastic) consumption; on the other hand, expenditures 

from loss of income of a family member caring for a non-institutionalized person 

requiring LTC are not captured. 

Individual productivity states and transition probabilities are taken from 

Nishiyama and Smetters (2003), with the original eight states combined into two for most 

computations. When the two states represent the bottom 60% and the top 40% of the 

workers for a given age, idiosyncratic productivity factors are given in Table 5 and the 

transition probabilities in each period are R11 = 0.9422, R12 = 0.0578, R21 = 0.0867, and 

R22 = 0.9133. Macroeconomic variables are also calibrated consistent with Nishiyama 

and Smetters (2002): the capital share of output is α = 0.32, the depreciation rate of 

physical capital is δ = 0.046, and the capital-to-output ratio is 2.8. 

Individuals are followed from the beginning of their working lives, and it is 

assumed that all start out healthy. This obviously does not capture congenital or earlyonset disabilities, but those would deprive the individual of earning power before he or 
23

she could make any decisions on saving; such cases are thus irrelevant for the present 

model. Disabled children would, however, have an effect on their parents, so it would 

perhaps be ideal to increase effective disability rates during working years (and introduce 

another state, since the parents’ earning power would remain, perhaps diminished). 

Nevertheless, as the most affected ages precede significantly the ages typical for major 

annuity purchases, there is little harm in ignoring such cases at this stage. 

The rate of population growth is assumed to be a constant 1%. This roughly 

approximates population growth in the United States, but the particular value is not 

critical. The rate of productivity increase in the production function is also not critical 

and, for the sake of simplicity, productivity is assumed to be unchanging. 

Social Security benefits are modeled with the primary goal of matching the 

average level. A worker who earns the average wage (according to SSA) throughout his 

or her career and retires at the normal retirement age, will get a replacement ratio of 

approximately 45%.16 Social Security formula is redistributive and hence the worker who 

remains in the high productivity state throughout his or her career will have only 2.14 

times higher benefits than the worker who always remains in the low-productivity state, 

although the former worker will have 3.5 times the career earnings of the latter.17 These 

ratios are based on extreme cases, comprising just the top 7.29% and the bottom 1.85% of 

the earnings distribution in the two-state model; most workers will not have spent their 

entire career in one productivity state. 

 

16 Author’s calculation based on benefit information published on the SSA web site, http://www.ssa.gov. 

17 Based on Nishiyama productivity factors; using Conesa-Krueger factors, the Social Security ratio is 

1.49 and the earnings ratio 1.74. These examples ignore any additional differences that may arise from 

spousal benefits. 
24

When the accuracy of Social Security benefits is highly important in a dynamic 

programming model, a good approach is to use an additional state variable representing 

the accrued benefits. In our model, however, there are already so many state variables 

and states that this approach would be computationally very costly, while the benefits 

would be minimal for at least two reasons. First, we are primarily interested in the effect 

on aggregate demand for annuities, as that can be compared to observations. Individual 

differences in benefit levels, within realistic limits, are not likely to have much effect on 

the aggregate demand. Second, such differences in benefits as do exist are the most 

significant among relatively low earners, who are likely to have low wealth and little 

influence on the annuity market. On the other hand, the difference in benefits between 

average and high earners is small and that between high and extremely high earners is 

zero due to the earnings and benefit limits.18 To avoid using an additional state variable 

while using as much information as possible with the existing set of state variables, we 

use the conditional expectation of benefits by final productivity state: an individual who 

reaches the last working year in state i is assigned the expected benefit amount 

conditional on this (last working year state) information only. Using a simple Monte 

Carlo simulation, the ratio of the two levels of benefits is calculated to be approximately 

1.197. 

Size and distribution of bequests can potentially have a significant impact on 

savings behavior, but no consensus seems to exist regarding the strength of bequest 

motives or even the fraction of aggregate wealth that is bequeathed. For this reason, the 

 

18 According to the examples on the SSA web site, http://www.ssa.gov/OACT/COLA/exampleAvg.html, 

for workers retiring in 2003 at age 65, one with average earnings throughout the career will receive a 

monthly benefit of $1,158, while one whose earnings equaled or exceeded the earnings limit in every year 

will receive $1,721 monthly, or 1.486 times more. 
25

strength of bequest motives cannot be truly calibrated. Instead, several different levels 

and distribution modes are used for sensitivity analysis.19 

6. Results 

6.1. Basic model 

In the absence of health shocks, we know that symmetric information, actuarially 

fair premiums, lack of bequest motives, zero transaction costs, and no exogenous 

obstacles to annuity market would be a sufficient set of conditions for full annuitization 

to be optimal for every individual. It is therefore the most natural to start by asking what 

the effect of health shocks will be if all other potential impediments to annuity markets 

are absent. At this point we also turn off any source of retirement income other than 

private savings. It is true that the mere existence of mandatory life annuities, holding 

everything else constant, would not alter the optimality of full annuitization, but would at 

most reduce the portion of the portfolio an individual can control. When other factors 

affect the dominance of annuities over bonds, however, existence of mandatory annuities 

can compound the effect. As we want to isolate the effect of health shocks, we consider 

the counterfactual world with no Social Security. 

An example of individual choice is shown in Figure 3. We chose a healthy 

individual age 65 as perhaps the most typical potential purchaser of a life annuity. Here 

and in the rest of this section, the magnitude of a health shock (about $50,000) is used as 

a unit of wealth. In this diagram, the coefficient of relative risk aversion σ equals 2, and 

 

19 Perhaps the most reliable indicator of the strength of bequest motives is the observation that the assets 

of retirees do not tend to decrease with time, even at very old ages. PSID data show that households whose 

head was aged 75+ in 1999 had higher net worth than households whose head was aged 65-74 in 1989 (i.e., 

same cohort, 10 years earlier). 
26

the interest rate and time preference parameter are set to 5% and 0.95, respectively. 

Optimal portfolio deviates significantly from full annuitization for a fairly wide range of 

wealth levels; the effect is the most pronounced for average to slightly below-average 

wealth. The poorest, who would exhaust all of their assets if ever hit by a health shock 

even for one period, annuitize fully; the wealthy, who can afford annuities with periodic 

payout that leaves sufficient funds even in the event of a shock, have little or no need to 

avoid annuities. Individuals most averse to holding annuities are those who are not poor, 

but would quickly become poor if hit by a health shock.20 Individuals with assets between 

1 and 4 (corresponding to approximately $50,000 to $200,000) want to annuitize, on 

average, less than 20% of their wealth. This fraction gradually increases with wealth, but 

it does not reach one until wealth is about 15 times the magnitude of the shock.21 A 

majority of people of this age have assets in the range of 1–5 times the cost of a year’s 

stay in a nursing home (Moore and Mitchell 2000), so the aggregate effect on annuity 

demand, in terms of the number of policies, is substantial. 

One of the goals of this paper is to analyze the factors of aggregate demand for 

annuities, so the fraction of aggregate wealth held in annuity form is one of the most 

interesting results of each simulation of a steady-state economy. The aggregate departure 

from full annuitization is not as dramatic as Figure 3 might have suggested, but are 

certainly not negligible: 4.9% of total wealth will be held as bonds.22 If we consider only 

the wealth of retirees, 6% of it will be in bequeathable form—despite a complete lack of 

 

20 Departures from the comparable result in Essay 2 include a somewhat smaller magnitude of the effect 

and a shift to lower wealth levels. 

21 The apparently non-smooth behavior of the portfolio composition as a function for wealth for wealth 

levels of the same order of magnitude as the shock is due to the discrete nature of the health states and to 

nonconcavity of the value function arising from limited liability. 

22 Besides the lack of significant reduction in annuity holding at high wealth levels, it also turns out that 

at most ages the effects are not as strong as at age 65. 
27

desire to leave bequests. The aggregate annuitization percentage results are summarized 

in Tables 4 and 5. 

Figure 4 shows the same decision, under equal conditions, for a more risk-averse 

individual (σ = 4). The effect is dramatic: many people — those whose assets are one to 

ten times the size of the shock — up to about half a million dollars — would not want to 

hold any life annuities. The aggregate effect is also much higher now: 17.3% of all 

wealth, and 13.4% of retirees’ wealth, is held as bonds. On the other hand, we do not 

show the diagram for σ = 1 as it looks rather boring: there is no effect at all. Individuals 

with high risk tolerance will still want to annuitize fully, at any level of wealth. 

6.2. Social Security 

As mentioned in the previous subsection, Social Security cannot, by itself, render 

full annuitization suboptimal in a world in which annuities dominate bonds. It can, 

however, interact with other factors of annuity demand and can alter annuitization 

decisions of some individuals, depending on their individual circumstances. Besides the 

obvious direct effect of by imposing some existing annuitized fraction of wealth on the 

individual, there are also indirect, general equilibrium effects. We have seen in the last 

subsection that annuity demand is sensitive to the interest rate, which is in turn higher 

with pay-as-you-go Social Security than without it. It is thus appropriate to evaluate the 

effect of Social Security with constant factor prices as well as in the general equilibrium. 

We focus on the former in this paper. 

When factor prices are kept constant, the aggregate effect of Social Security on 

the level of annuitization of wealth is small and ambiguous. At moderate levels of risk 
28

aversion (σ = 2), less wealth is annuitized with Social Security than without it, but the 

difference is less than 3% for retiree wealth and even less in total. It should be noted that 

the denominator (total wealth) in these results does not include the “Social Security 

wealth” (i.e., the expected present value of future Social Security benefits) so the relative 

reliance on annuity income actually increases. For higher levels of risk aversion, the 

additional effect of Social Security is even less significant, adding only about 2% to the 

bond portfolio weight at σ = 4. It does not appear that Social Security has a higher 

crowdout effect on annuities than on savings in general. 

It is interesting, however, that an individual’s optimal portfolio, as a function of 

wealth, is noticeably altered when Social Security exists. As Figures 5 and 6 show, there 

is no full-annuitization region at low wealth levels now. The overall effect on low-tomedium wealth 65-year-olds is more pronounced, leading to no annuity holding at all 

(actually, they prefer to hold negative private annuities to offset the Social Security 

annuity in part). 

6.3. Bequest Motives and Transaction Costs 

Bequest motives, if they are present, are certain to diminish the demand for 

annuities. We find indeed that, by varying the altruism parameter ζ (the strength of the 

bequest motive) in the utility function, we can reduce the annuitized fraction of wealth to 

an arbitrary degree. For example if ζ = 1.0, only 34% of total wealth and only 19% of the 

retirees’ wealth will be annuitized. However, this is an implausible value of altruism; it 

would mean that consumption, no matter how high one’s wealth, would be limited to 

about $50,000 per year. In terms of macroeconomic variables, capital-to-output ratio 
29

would be at least 4.5 and interest rate by about three percentage points lower than 

calibrated. We would also find that intergenerational bequests exceed 10% of GDP and 

14% of consumption; all this is incompatible with empirical data. A high but still 

somewhat plausible value of ζ = 0.5 (resulting in K/Y = 3.6 and total bequests consistent 

with observation) would reduce total annuitization (at σ = 2) to 72.6%, and that of retirees 

to 66.5%. This is approximately a quadrupling of the bond portfolio weight, relative to 

the no-altruism case. On the other hand, lower values of altruism, which do not alter 

macroeconomic quantities significantly, have a very small effect on annuitization. With ζ

= 0.25, over 90% of wealth would still be held in annuity form.23

Premium loading is another obvious deterrent from annuity purchases, but its 

importance in deviations from Yaari have been questioned recently (Mitchell et al. 1999; 

Davidoff et al. 2002). In the framework of our model, we are not limited to evaluating the 

effect of the higher price on money’s worth of an annuity; we can also look at the more 

subtle transaction-cost nature of the loading. Transaction costs limit the usefulness of the 

annuity as a short-term investment and, if they are high enough, can make annuities an 

inferior investment for those who cannot commit to them for a long term or indefinitely. 

In other words, they reduce the liquidity of annuities, compounding the problem of 

possible high cash need as caused by health shocks.24 

 

23 This is not a small value for altruism. In our model it means no individual ever spends more than about 

$100,000 per year, even with unlimited wealth. 

24 Our preliminary finding is that a premium loading of 10% reduces annuitization of retirees’ wealth to 

62%, having a greater effect than a moderately high bequest motive. 
30

7. Conclusion 

This paper provides the insight into the effect of mortality information updating 

and related changes in wealth on the demand for life annuities. It shows that, under 

realistic mortality and morbidity assumptions and expense levels, many individuals do 

not find it optimal to hold all of their wealth in life annuity form, even in the absence of 

premium loading, adverse selection, and bequest motives. Effectively, a life annuity 

ceases to be a dominant asset and becomes a risky asset with an expected, but not 

guaranteed, premium over the riskless return. 

While this mechanism does not depend on the other, better known, factors 

affecting demand for annuities, it is not mutually exclusive with any of those. In fact, a 

typical individual will have a variety of factors influencing his or her decision to 

annuitize: transaction costs (premium loading and obstacles to undoing an annuity), a 

desire to bequeath some wealth, private information, presence of mandatory annuities, 

availability of family help or social welfare in case of outliving one’s wealth, and so 

forth. The relative effects of each factor on annuity demand are quite sensitive to the 

precise choice of parameters, but the effect of health shocks is significant in most cases. 

There are numerous potential applications of this model. Long-term care has been 

mentioned repeatedly in the paper as a source of wealth shocks of the type assumed in the 

model. It is thus natural to try to use this framework to study the joint effects of long-term 

care and mortality risks and optimal insurance approaches. In that respect this model can 

complement and extend the work of Murtaugh, Spillman and Warshawsky (2001). The 

issue of payout options from defined-contribution pensions or other individual account 

retirement schemes can also be made clearer with this framework. This paper shows that 
31

annuity payouts do not necessarily provide the most financial security to the beneficiary, 

especially if retirement programs are designed in isolation from health insurance 

programs. The issue of the optimal, welfare-maximizing, level of mandatory 

annuitization of retirement wealth is a nontrivial one and requires a sound analytical and 

computational framework. On the other hand, government transfers such as income floor 

protection and insurance of the last resort may provide incentives for some individuals to 

draw down their savings early. Mandatory annuitization can prevent that and help control 

the cost of such programs, but that there is no general guarantee of a welfare gain from 

such mandates. 
32

APPENDIX: Discretization of State Space 

A1. No Transaction Costs 

Total wealth at age j, Aj, is represented as one of 101 points of the wealth grid, Ajk, 

k = 0, 1, ... , 100. We fix point Aj0 = 0, Aj100 equals the assumed maximum wealth, and 

the value of Ajk increases with k. Ideally, for best interpolation during optimization and 

evaluation, the spacing between adjacent grid points should be narrower where the value 

function’s curvature is the most pronounced. We chose equidistant points at the low end 

of the wealth distribution, and geometrically increasing values for intermediate to high 

wealth. For example, if Aj100 = 100 (in units where 1 is the average annual wage), then: 

 Aj,k = Aj,k–1 + 0.05, k = 1, ... , 20 (Aj20 = 1) 

 Aj,k = 1.05 Aj,k–1, k = 21, ... , 75 (Aj75 = 14.63563) 

 Aj,k = 1.0799 Aj,k–1, k = 76, ... , 100. 

As most people’s wealth increases during the early part of life, the maximum 

wealth Aj100 does not have to be the same for all ages; we also allow the grid to be 

expanded during the computation if the maximum wealth is actually reached by a 

positive measure of agents. 

When the optimal policy (consumption, bond saving and annuity saving) is 

computed for an agent at the node (j, h, i, k), where the indices represent age, health, 

productivity, and wealth, respectively, the wealth A’ next period (age j+1) is allowed to 

take any positive value, rather than be limited to the values of the grid points. The value 

function V(j+1, h’, i’, A’) corresponding to that wealth is determined by interpolation 

between the two grid points bracketing it, for the given final health and productivity state 

(h’, i’) and age j+1. 
33

The simplest way to interpolate V is by piecewise linear base functions: V is 

assumed to be linear in A between Ak and Ak+1 such that Ak < A < Ak+1. The disadvantage 

of this approach is that is alters the shape (curvature) of the value function. In 

optimization problems, concavity is often very important and it is desirable to keep 

concave functions concave not just globally but also locally, on any scale. We thus use 

interpolation by Schumaker’s shape-preserving quadratic splines (Judd 1999, pp. 231–

234). They preserve local concavity/convexity of V as a function of wealth, as well as its 

smoothness, avoiding artificial kinks at grid points. 

With respect to the curvature of the value function, it is important to keep in mind 

that this is not a standard, well-behaved concave problem. Limited liability introduces 

nonconcavity in wealth, and there are further complications due to the discrete nature of 

health. Risk involved in annuity returns arises from the difference in the next-period 

present value of the annuity in healthy and sick states, as well as the transition 

probabilities for the respective states, so this is the main determinant of the optimal 

portfolio weights. The objective function can, and frequently does, have multiple local 

optima. For this reason, it is necessary to perform optimization at each node starting from 

several different initial guesses chosen by randomized search techniques. 

In most cases we find insignificant difference between results of runs using linear 

and spline interpolation. Greater difference is normally a sign that the grid spacing is too 

wide, and it is then generally best to rerun the computation using a finer grid. 

When the measure of agents is computed, it is also represented as having a 

discrete distribution, with probability masses at the nodes of the grid. For this purpose, a 

value from the continuum must be apportioned to the nearest two grid points. This 
34

process must preserve expected utility as well as the total measure, so the weights given 

to the two points will be inversely proportional to the distance to them (i.e., they will be 

based on a linear interpolation scheme). 

The number of nodes in the full dynamic-programming tree is J × m × n ×

(k

max+1), where J is the maximum age (actually the age span between the minimum and 

the maximum), m is the number of health states, n the number of productivity states, and 

k

max the highest index of the wealth grid. We use ages from 21 to 120, so J = 100; as 

defined above, k

max = 100, and as discussed in the paper, m = 3 and n = 8 (or less in some 

versions). Therefore, J × m × n × (k

max+1) = 100 × 3 × 8 × 101 = 242,400. (The smallest 

version, with n = 2, has 60,600 nodes.) 

A1.2. With Transaction Costs 

When transaction costs are present, portfolio composition is part of the 

description of the state, so there will not be one, but two continuous state variables, and 

each of them must be discretized in computation. We use liquid wealth (bond) holding 

and annual payout of the annuity policy owned as the two variables. Annual payout 

remains constant if the annuitant enters no annuity transactions during a period, and is 

hence a better modeling choice than the present value of the annuity, which changes with 

age and health. The resulting two-dimensional grid for each age-health-productivity triad 

has a large number of nodes, slowing the computation down considerably.25 Second, twodimensional interpolation is needed instead of one-dimensional as described in the 

previous section. While shape-preserving bivariate interpolation schemes do exist, they 

 

25 With 40 grid points for bond holding and 40 for annuities — the bare minimum for a reasonable 

spacing — and 2 productivity states, there are 960,000 nodes to evaluate in each pass in this problem. 
35

are more complicated to implement (and the problem is already computationally 

intensive), so we limit the interpolation to a simple bilinear procedure. 

While the transaction cost model has an additional state variable, its optimization 

space is essentially the same as for the model without transaction costs, and the 

associated nonconcavity problems are similar. Interpolation leads to potential new 

problems, however. If an individual’s optimum annuity holding (measured by annual 

payout) is constant over a number of years, but not exactly on a node of the annuity grid, 

this point may never be reached if the transaction costs are high enough. Since the initial 

state is always at a grid node, and it is advantageous to avoid annuity transactions, 

solutions will also have a tendency to “stick” to the nodes. This tendency may lead to 

understatement of the dynamics of annuity transactions over time. The safest way to 

minimize this problem is to use as dense a grid for annuities as possible, but of course, 

there is a tradeoff between accuracy and computational speed. 
36

Table 1 

Critical Magnitude of Shocks in the Three-Period Model 

Relative Risk 

Aversion 

(σ) 

Interest Rate 

(r) 

Probability of 

Remaining 

Healthy (α) 

Survival 

Probability 

(p1) 

Critical Value 

of Shock 

(s) 

2 0.25 0.6 0.9 0.196 

2 0.25 0.6 0.75 0.533 

2 0.25 0.8 0.9 0.824 

2 0.25 0.4 0.9 0.222 

2 0.1 0.6 0.9 0.150 

2 0.5 0.6 0.9 0.278 

1.5 0.25 0.6 0.9 0.256 

4 0.25 0.6 0.9 0.100 

Note: The rightmost column shows the magnitude of second-period shock (s), as a 

fraction of first-period wealth, at which the borrowing constraint ceases to be binding, for 

some values of the parameters (shown in the first four columns) in the three-period model 

of Section 3.1. At higher levels of s, it is optimal to include bonds in the portfolio. 
37

Table 2 

Optimal Portfolio in the Three-Period Model 

Annuity Share of Portfolio for Various Probabilities of Period-1 Survival 

(p1) and Favorable Period-2 State (α) 

Shock Size 

as a Fraction 

of Period-1 

Wealth (s) p1 = 0.95,

α = 0.6 

p1 = 0.9,

α = 0.4 

p1 = 0.9,

α = 0.6 

p1 = 0.9,

α = 0.8 

p1 = 0.75,

α = 0.6 

0.05 1.000 1.000 1.000 1.000 1.000 

0.10 1.000 1.000 1.000 1.000 1.000 

0.15 0.878 1.000 1.000 1.000 1.000 

0.20 0.724 1.000 0.985 1.000 1.000 

0.30 0.429 0.735 0.668 0.758 1.000 

0.40 0.150 0.424 0.370 0.460 1.000 

0.50 0.000 0.139 0.091 0.175 1.000 

0.60 0.000 0.000 0.000 0.000 0.731 

0.80 0.000 0.000 0.000 0.000 1.000 

Note: Entries show proportion of annuities in the savings portfolio for the first period in 

the three-period model of Section 3.1., for selected values of the magnitude of shocks 

(rows) and mortality and morbidity parameters (columns). The interest rate is 0.25 per 

period throughout, and the coefficient of relative risk aversion equals 2. 
38

Table 3 

Survival Probabilities 

Age Healthy Disabled Age Healthy Impaired, 

Not in LTC 

Impaired, 

In LTC 

21 0.999643 0.998394 65 0.987263 0.948234 0.885833

22 0.999634 0.998353 66 0.985591 0.945612 0.884889

23 0.999627 0.998322 67 0.983925 0.942856 0.883937

24 0.999624 0.998308 68 0.982129 0.939960 0.882977

25 0.999624 0.998308 69 0.980198 0.936918 0.882009

26 0.999622 0.998299 70 0.977794 0.933722 0.881033

27 0.999618 0.998281 71 0.975430 0.930364 0.880049

28 0.999607 0.998232 72 0.972719 0.926835 0.879057

29 0.999588 0.998146 73 0.969613 0.923128 0.878056

30 0.999556 0.998002 74 0.966100 0.919233 0.877047

31 0.999501 0.997755 75 0.962166 0.915141 0.876030

32 0.999438 0.997471 76 0.957831 0.910841 0.875005

33 0.999369 0.997161 77 0.953094 0.906324 0.873971

34 0.999298 0.996841 78 0.947877 0.901577 0.872929

35 0.999227 0.996522 79 0.942073 0.898280 0.868672

36 0.999159 0.996216 80 0.935632 0.894872 0.864272

37 0.999096 0.995932 81 0.927959 0.891351 0.859725

38 0.999036 0.995662 82 0.919514 0.887711 0.855026

39 0.998979 0.995406 83 0.910282 0.883949 0.850169

40 0.998921 0.995145 84 0.900221 0.880062 0.845150

41 0.998858 0.994861 85 0.889243 0.876044 0.839963

42 0.998785 0.994533 86 0.877203 0.870000 0.829061

43 0.998701 0.994155 87 0.863957 0.855000 0.817416

44 0.998603 0.993714 88 0.849410 0.840000 0.804978

45 0.998492 0.993214 89 0.833580 0.825000 0.791692

46 0.998384 0.992728 90 0.816592 0.810000 0.777502

47 0.998266 0.992197 91 0.800231 0.790000 0.762345

48 0.998140 0.991630 92 0.783395 0.770000 0.746155

49 0.998005 0.991023 93 0.766338 0.750000 0.728863

50 0.997862 0.990379 94 0.749307 0.740000 0.710392

51 0.997551 0.988980 95 0.732509 0.732509 0.690664

52 0.997333 0.987999 96 0.716095 0.716095 0.669591

53 0.997084 0.986878 97 0.700148 0.700148 0.647083

54 0.996804 0.985618 98 0.684704 0.684704 0.623041

55 0.996376 0.983692 99 0.669793 0.669793 0.597362

56 0.995800 0.981100 100 0.655444 0.655444 0.569933

57 0.995307 0.978882 101 0.641372 0.641372 0.540000

58 0.994727 0.976272 102 0.628315 0.628315 0.520000

59 0.994055 0.973248 103 0.616960 0.616960 0.500000

60 0.993253 0.969639 104 0.607997 0.607997 0.480000

61 0.992324 0.965458 105 0.602114 0.602114 0.460000

62 0.991243 0.960594 106 0.600000 0.600000 0.440000

63 0.989988 0.954946 107 0.600000 0.600000 0.420000

64 0.988720 0.949240 108–119 0.600000 0.600000 0.400000

 120 0.000000 0.000000 0.000000

Source: Author’s calculations based on references cited in Section 3. 
39

Table 4 

Health Transition Probabilities 

Age Q11 Q12 Q13 Q21 Q22 Q23 Q31 Q32 Q33

21 0.999000 0.001000 0 0.600000 0 0.400000 0.300000 0 0.700000

22 0.998500 0.001500 0 0.596000 0 0.404000 0.295000 0 0.705000

23 0.998000 0.002000 0 0.592000 0 0.408000 0.290000 0 0.710000

24 0.997500 0.002500 0 0.588000 0 0.412000 0.285000 0 0.715000

25 0.997500 0.002500 0 0.584000 0 0.416000 0.280000 0 0.720000

26 0.997000 0.003000 0 0.580000 0 0.420000 0.275000 0 0.725000

27 0.997000 0.003000 0 0.576000 0 0.424000 0.270000 0 0.730000

28 0.996500 0.003500 0 0.572000 0 0.428000 0.265000 0 0.735000

29 0.996000 0.004000 0 0.568000 0 0.432000 0.260000 0 0.740000

30 0.996000 0.004000 0 0.564000 0 0.436000 0.255000 0 0.745000

31 0.995500 0.004500 0 0.560000 0 0.440000 0.250000 0 0.750000

32 0.995000 0.005000 0 0.556000 0 0.444000 0.245000 0 0.755000

33 0.994500 0.005500 0 0.552000 0 0.448000 0.240000 0 0.760000

34 0.994000 0.006000 0 0.548000 0 0.452000 0.235000 0 0.765000

35 0.993500 0.006500 0 0.544000 0 0.456000 0.230000 0 0.770000

36 0.993000 0.007000 0 0.540000 0 0.460000 0.225000 0 0.775000

37 0.992500 0.007500 0 0.536000 0 0.464000 0.220000 0 0.780000

38 0.992000 0.008000 0 0.532000 0 0.468000 0.215000 0 0.785000

39 0.991500 0.008500 0 0.528000 0 0.472000 0.210000 0 0.790000

40 0.991000 0.009000 0 0.524000 0 0.476000 0.205000 0 0.795000

41 0.990500 0.009500 0 0.520000 0 0.480000 0.200000 0 0.800000

42 0.990000 0.010000 0 0.516000 0 0.484000 0.195000 0 0.805000

43 0.989500 0.010500 0 0.512000 0 0.488000 0.190000 0 0.810000

44 0.989000 0.011000 0 0.508000 0 0.492000 0.185000 0 0.815000

45 0.989000 0.011000 0 0.504000 0 0.496000 0.180000 0 0.820000

46 0.988500 0.011500 0 0.500000 0 0.500000 0.175000 0 0.825000

47 0.988000 0.012000 0 0.496000 0 0.504000 0.170000 0 0.830000

48 0.987500 0.012500 0 0.492000 0 0.508000 0.165000 0 0.835000

49 0.987000 0.013000 0 0.488000 0 0.512000 0.160000 0 0.840000

50 0.986000 0.014000 0 0.484000 0 0.516000 0.155000 0 0.845000

51 0.985000 0.015000 0 0.480000 0 0.520000 0.150000 0 0.850000

52 0.984000 0.016000 0 0.476000 0 0.524000 0.145000 0 0.855000

53 0.982500 0.017500 0 0.472000 0 0.528000 0.140000 0 0.860000

54 0.981000 0.019000 0 0.468000 0 0.532000 0.135000 0 0.865000

55 0.979500 0.020500 0 0.464000 0 0.536000 0.130000 0 0.870000

56 0.978000 0.022000 0 0.460000 0 0.540000 0.125000 0 0.875000

57 0.976500 0.023500 0 0.456000 0 0.544000 0.120000 0 0.880000

58 0.975000 0.025000 0 0.452000 0 0.548000 0.115000 0 0.885000

59 0.975000 0.025000 0 0.448000 0 0.552000 0.110000 0 0.890000

60 0.975000 0.025000 0 0.444000 0 0.556000 0.105000 0 0.895000

61 0.975000 0.025000 0 0.440000 0 0.560000 0.100000 0 0.900000

62 0.975000 0.025000 0 0.436000 0 0.564000 0.095000 0 0.905000

63 0.972500 0.027500 0 0.432000 0 0.568000 0.090000 0 0.910000

64 0.970000 0.030000 0 0.428000 0 0.572000 0.085000 0 0.915000

65 0.965038 0.034621 0.000341 0.060312 0.929934 0.009754 0.147410 0.399893 0.452697

66 0.962417 0.037170 0.000413 0.059067 0.929877 0.011056 0.128059 0.381839 0.490102

67 0.959573 0.039928 0.000499 0.057856 0.929611 0.012534 0.111249 0.364603 0.524148

68 0.956480 0.042915 0.000604 0.056679 0.929111 0.014211 0.096647 0.348149 0.555204

69 0.953112 0.046157 0.000731 0.055535 0.928351 0.016115 0.083962 0.332441 0.583597
40

70 0.949435 0.049679 0.000885 0.054423 0.927300 0.018277 0.072943 0.317445 0.609612

71 0.945414 0.053514 0.001072 0.053344 0.925922 0.020734 0.063370 0.303129 0.633501

72 0.941006 0.057697 0.001297 0.052296 0.924179 0.023525 0.055055 0.289461 0.655484

73 0.936160 0.062269 0.001570 0.051280 0.922022 0.026698 0.047831 0.276412 0.675757

74 0.930820 0.067279 0.001901 0.050295 0.919400 0.030305 0.041555 0.263955 0.694490

75 0.924916 0.072782 0.002302 0.049340 0.916253 0.034408 0.036104 0.252061 0.711835

76 0.918371 0.078843 0.002787 0.048415 0.912510 0.039075 0.031367 0.240707 0.727926

77 0.911089 0.085536 0.003374 0.047520 0.908092 0.044388 0.027253 0.229866 0.742881

78 0.902962 0.092952 0.004086 0.046654 0.902909 0.050437 0.023678 0.219516 0.756806

79 0.893155 0.102346 0.004499 0.044664 0.903140 0.052197 0.022696 0.210408 0.766896

80 0.882349 0.112698 0.004954 0.042764 0.903211 0.054025 0.021758 0.201716 0.776526

81 0.870441 0.124104 0.005455 0.040951 0.903124 0.055925 0.020863 0.193421 0.785715

82 0.857318 0.136674 0.006008 0.039221 0.902879 0.057901 0.020010 0.185506 0.794485

83 0.842854 0.150529 0.006617 0.037569 0.902475 0.059956 0.019195 0.177953 0.802853

84 0.826910 0.165802 0.007288 0.035993 0.901914 0.062093 0.018417 0.170745 0.810837

85 0.809333 0.182639 0.008028 0.034489 0.901193 0.064318 0.017676 0.163869 0.818456

86 0.789952 0.201204 0.008844 0.033053 0.900313 0.066634 0.017081 0.158359 0.824560

87 0.768580 0.221676 0.009744 0.031683 0.899270 0.069046 0.016525 0.153200 0.830275

88 0.745007 0.244256 0.010737 0.030376 0.898065 0.071559 0.016006 0.148385 0.835609

89 0.740209 0.247960 0.011831 0.029128 0.896693 0.074179 0.015523 0.143910 0.840567

90 0.734934 0.252027 0.013039 0.029290 0.896120 0.074591 0.015076 0.139772 0.845152

91 0.734350 0.252582 0.013068 0.029459 0.895520 0.075021 0.015017 0.139221 0.845762

92 0.733731 0.253170 0.013098 0.029636 0.894894 0.075471 0.014985 0.138920 0.846096

93 0.733077 0.253793 0.013131 0.029820 0.894238 0.075941 0.014982 0.138894 0.846124

94 0.732383 0.254452 0.013165 0.030014 0.893552 0.076434 0.015012 0.139177 0.845810

95 0.731649 0.255150 0.013201 0.030216 0.892834 0.076950 0.015081 0.139809 0.845110

96 0.730871 0.255890 0.013239 0.030429 0.892081 0.077490 0.015192 0.140841 0.843967

97 0.730046 0.256675 0.013280 0.030651 0.891292 0.078057 0.015353 0.142336 0.842310

98 0.729171 0.257506 0.013323 0.030885 0.890464 0.078651 0.015573 0.144376 0.840051

99 0.728243 0.258389 0.013368 0.031130 0.889595 0.079275 0.015863 0.147066 0.837071

100 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

101 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

102 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

103 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

104 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

105 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

106 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

107 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

108 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

109 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

110 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

111 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

112 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

113 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

114 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

115 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

116 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

117 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

118 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

119 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

120 0.727258 0.259325 0.013417 0.031387 0.888683 0.079931 0.016238 0.150543 0.833218

Source: Author’s calculations based on references cited in Section 3. 
41

Table 5

Productivity Factors by Age and State: 2-State Version 

Age Age-related 

(deterministic) 

productivity 

Idiosyncratic 

productivity, low 

state 

Idiosyncratic 

productivity, high 

state 

21–24 9.8703 0.6612 1.5081 

25–29 12.4385 0.6692 1.4963 

30–34 17.5824 0.5430 1.6856 

35–39 18.7831 0.5895 1.6157 

40–44 21.8091 0.5054 1.7418 

45–49 23.7994 0.5114 1.7330 

50–54 22.9768 0.4812 1.7782 

55–59 22.3280 0.4156 1.8766 

60–64 21.3683 0.3001 2.0498 

Source: Authors’ calculations based on Nishiyama and Smetters (2003). 
42

Table 6 

Annuitization of Aggregate Wealth 

Relative risk aversion σ 1 2 4 

Annuitized fraction of total wealth (no OASI) 99.9% 95.1% 82.7%

Annuitized fraction of total wealth (with OASI) 99.9% 93.7% 81.6%

 

Annuitized fraction of retiree wealth (no OASI) 99.9% 94.0% 86.6%

Annuitized fraction of retiree wealth (with OASI) 99.9% 91.3% 84.1%

Note: Entries show annuitized proportion of aggregate wealth, for various levels of risk 

aversion, in partial equilibrium with bond interest rate and the subjective discount rate 

equal to 5%. Annuity premiums are assumed to be actuarially fair and individuals 

completely selfish, so the existence of health shocks is the only deviation from Yaari’s 

assumptions. 
43

Table 7 

Effect of Bequest Motives on Annuitization 

Annuitized fraction of aggregate wealth Altruism parameter ζ Total Retirees only 

K/Y 

0.0 93.7% 91.3% 3.3 

0.1 93.3% 90.9% 3.4 

0.25 92.1% 89.8% 3.5 

0.5 72.6% 66.5% 3.6 

1.0 34.1% 19.2% 4.5 

Note: Entries in the middle two columns show annuitized proportion of aggregate wealth, 

for various levels of altruism, in partial equilibrium with bond interest rate and the 

subjective discount rate equal to 5%. Annuity premiums are assumed to be actuarially fair 

and Social Security is present. Coefficient of relative risk aversion is assumed to be 2. 

Capital/output (K/Y) ratio is given to illustrate the overall effect on savings of the given 

value of ζ. 
44

Figure 1 

Optimal Portfolio in the Infinitely-Lived Model 

q 0.25 0.5 0.75 1.0 

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

0

0.02

0.04

0.06

0.08

0.1

pS

Note: Contour plot of the annuity share of the portfolio in the infinite-period 

model. Vertical axis is the probability of becoming sick next period, and the 

horizontal axis is the probability of one-period survival in the sick state. The 

lowest curve is the locus of points where the borrowing constraint becomes 

binding; the full annuitization region is below and to the right of it. β = 0.97, 

pH = 0.99, r = 0.04 for this figure. 
45

Figure 2 

Optimal Portfolio in the Infinitely-Lived Model 

q 0.5 0.75 1.0 

0.5 0.6 0.7 0.8 0.9

0

0.05

0.1

0.15

0.2

pS

Note: Contour plot of the annuity share of the portfolio in the infinite-period 

model for β = 0.97, pH = 0.97, r = 0.07. 
46

Figure 3 

Annuitized Fraction of Wealth at Age 65: 

No Social Security, CRRA = 2 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25

Wealth

Annuity (%)

Note: Optimal fraction of wealth held in life annuity form by a healthy person age 

65 with a coefficient of relative risk aversion (CRRA) equal to 2.0, plotted as a 

function of total wealth. The unit of wealth is average loss in the sick state. Social 

Security, transaction costs, and bequest motives are absent. Bond interest rate and 

the subjective discount rate are both 5%. 
47

Figure 4 

Annuitized Fraction of Wealth at Age 65: 

No Social Security, CRRA = 4 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25

Wealth

Annuity (%)

Note: Optimal fraction of wealth held in life annuity form by a healthy person age 

65 with a coefficient of relative risk aversion (CRRA) equal to 4.0, plotted as a 

function of total wealth. The unit of wealth is average loss in the sick state. Social 

Security, transaction costs, and bequest motives are absent. Bond interest rate and 

the subjective discount rate are both 5%. 
48

Figure 5 

Annuitized Fraction of Wealth at Age 65: 

With Social Security, CRRA = 2 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25

Wealth

Annuity (%)

Note: Optimal fraction of wealth held in life annuity form by a healthy person age 

65 with a coefficient of relative risk aversion (CRRA) equal to 2.0, plotted as a 

function of total wealth. The unit of wealth is average loss in the sick state. Social 

Security exists, but transaction costs and bequest motives are absent. Bond 

interest rate and the subjective discount rate are both 5%. 
49

Figure 6 

Annuitized Fraction of Wealth at Age 65: 

With Social Security, CRRA = 4 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25

Wealth

Annuity (%)

Note: Optimal fraction of wealth held in life annuity form by a healthy person age 

65 with a coefficient of relative risk aversion (CRRA) equal to 4.0, plotted as a 

function of total wealth. The unit of wealth is average loss in the sick state. Social 

Security exists, but transaction costs and bequest motives are absent. Bond 

interest rate and the subjective discount rate are both 5%. 
50

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Bell, Felicitie C., Alice H. Wade and Stephen C. Goss (1992). Life Tables for the United 

States Social Security Area: 1900–2080. Social Security Administration Actuarial 

Study #107. 

Brown, J. R. (2000). Differential mortality and the value of individual account retirement 

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Brown, Jeffrey R. and Amy Finkelstein (2003). Why is the market for private long-term 

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paper. 

Brown, J. R. and M. J. Warshawsky (2001). Longevity-insured retirement distributions 

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