Retirement income strategies designed in an expected
utility framework

Mark J. Warshawskya,*
aReLIAS LLC, 903 Brentwood Lane, Silver Spring, MD 20902, USA

Abstract

Various classes of retirement income strategies are evaluated and their robustness tested in an
expected utility framework. Fixed percentage systematic withdrawals from an investment portfolio
combined with laddered purchases of immediate life annuities stands out as a superior strategy for
retired defined contribution plan participants and IRA holders, yielding better outcomes than alter-
natives, including longevity insurance. This broad strategy is then applied, step by step, and custom-
ized across a wide range of household risk preferences and situations, with due consideration of
product costs, taxes, and economic risks and returns. © 2017 Academy of Financial Services. All
rights reserved.

JEL classifications: G11; G22; D14

Keywords: Retirement income strategies; Systematic withdrawal; Life annuity; Advanced life delayed annuity
(ALDA)

1. Introduction

Products and strategies are emerging in the market to help retirees invest and draw down
assets from their 401(k) accounts and IRAs during retirement. Given the competing desires
for lifetime income and wealth, for security and flexibility, and for current and future needs,
it is appropriate to compare solutions carefully. This article gives a quantitative analysis of
different strategies for households, including systematic withdrawals, immediate life annu-
ities, mixed strategies of systematic withdrawals with life annuities, and also the use of

* Corresponding author. Tel.: �1-301-908-2846; fax: �1-301-592-9277.
E-mail: markjwarshawsky@gmail.com

Financial Services Review 26 (2017) 113–141

1057-0810/17/$ – see front matter © 2017 Academy of Financial Services. All rights reserved.



longevity insurance (advanced-life delayed annuities, ALDA). The ALDA product has been
featured in other research, and in a recent amendment to the required minimum distribution
(RMD) rules for retirement accounts in the United States.

As realistically as possible, this articles considers fee charges and pricing for various
products, income flows from Social Security and pensions, taxes, and random realizations of
asset returns, interest rates, and mortality (normal and impaired life). Results are computed
from a model using an expected utility framework, which assigns a different utility value to
any small change in consumption or final wealth levels. The robustness of strategies is tested
across households with varying preferences and circumstances.

A household’s chosen strategy is assumed to remain fixed throughout retirement. This
essentially acknowledges the complexity of issues facing older retirees, such as their trust in
financial institutions and their cognitive agility to make midcourse changes, as well as the
governance capacities of financial providers and advisors. The idea of simplifying and easing
the path for investors is also found in the fixed strategies used by the popular target-date
funds and managed accounts during the accumulation phase. At the same time, it is possible
to reoptimize at any time in response to major changes in the retired household’s situation,
such as the death of a spouse. A more dynamic strategy cognizant of the practical complex-
ities of financial and insurance products and taxes would necessitate a much more complex
model which, in turn, would be more opaque to retirees.

Among the many options considered, a new strategy stands out: combining laddered purchases
of nominal immediate life annuities (that is, dollar cost averaging) with systematic withdrawals
(fixed percentage) from a dynamically changing retirement investment portfolio. This combina-
tion strategy outperforms the alternatives using ALDA and inflation-indexed annuities.

2. Evolution of thinking on income strategies

There is growing research literature on retirement income solutions. In a simple loss
aversion framework, Pang and Warshawsky (2009) compared products and strategies for
producing income and managing wealth in retirement accounts, assuming an initial 50–50
equity-bond allocation. The strategies compared ran the spectrum of liquidity and potential
growth.1 At one end, systematic withdrawals as a fixed percentage of the retirement account
provides complete liquidity and growth potential but no guarantees, and significant risk. At
the other end, full annuitization using a straight immediate (nominal) life annuity provides
no liquidity or growth potential but is fully guaranteed for life in nominal terms. Midspec-
trum strategies include a mix of systematic withdrawals and gradual annuitization over ten
years, a deferred variable annuity with a guaranteed minimum withdrawal benefit rider
(“VA�GMWB” with a 5% withdrawal rate), and an immediate variable life annuity invested
in bonds and equities.

In general, among the strategies analyzed, the highest real income and lowest chance of
income shortfalls are achieved by combining systematic withdrawals and gradual but com-
plete annuitization over ten years—that is, the annuity purchase ladder strategy. Only the
purchase of an immediate fixed annuity at retirement, using the entire account balance,
achieved a higher real income, at the median outcome, but that strategy leaves no wealth

114 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



balances ever. The ladder strategy maintains some balances through the first 10 years of
retirement. The other strategies (excluding the immediate variable annuity) offer the advan-
tage of significant real balances (i.e., liquidity). The liquidation value in VA � GMWB tends
to run to zero in later life, owing to withdrawals and fees. A more aggressive portfolio
(70–30 equity-bond split) gives a small relative boost to the VA�GMWB, because the
investor takes greater advantage of the insurer’s guarantee, but the basic results remain. The
relative levels of fees are critical to the analysis. With an overall low level of fees, perhaps
owing to institutional pricing, as long as the fees are reduced proportionally across products,
the advantage of the annuity ladder strategy still holds.

Warshawsky (2012, Chapter 7) and Warshawsky (2016) are the main studies that develop
an analysis of the laddered annuity purchase strategy. Their focus is on optimizing the
ladder strategy of immediate annuity purchases, both because it compared well in earlier
research with other strategies and because it closely resembles the optimum in the theoretical
fully dynamic model of Pang and Warshawsky (2010) described below. The components of
strategy design optimization include the withdrawal rate, and the length and extent of
annuitization.

Pang and Warshawsky (2010) developed a formal theoretical, somewhat stylized, dy-
namic model of expected utility maximization in retirement, where the household optimizes
consumption and allocates wealth across equities, bonds, and life annuities, with consider-
ation of the receipt of lifetime income flows such as Social Security and defined benefit
pensions. In addition to stochastic capital market returns and mortality, households are
exposed in this model to uninsured health care cost risks, which increase with age and
income decile. Absent a bequest motive, the results indicate that retired households should
optimally start annuitizing their wealth around their mid-70s and annuitize fully in their 80s.
Retirees continue purchasing annuities throughout their lives as they save some income for
sequential purchases of higher yielding life annuities to effectively insure for higher health
care expenses later in life. Moreover, with the use of guaranteed annuities, the optimal equity
exposure in the remaining portfolio increases with age, reaching nearly 100% for high-
income households, until nonannuitized wealth is used up. The consumption level is fairly
sustained over the retirement life cycle with the support of annuity income. A modest bequest
motive tempers these results, in particular cutting the ultimate use of life annuities, but does
not overturn them.

In the face of uncertain health care expenses, it seems logical to expect, and some studies
indeed show, immediate life annuities to be a poor investment, owing to their lack of
liquidity. However, uninsured health care spending exposure is highest later in life, espe-
cially for long-term care needs, precisely when newly purchased immediate life annuities (if
available) generate their highest returns, owing to large and growing-with-age mortality
credits. Hence, immediate life annuities are a hedge for long-term care and health spending
in the absence of complete health and long-term care insurance coverage. Note that this point
is orthogonal to the issue of the current health status and, therefore, life expectancy of the
potential insured and the appropriateness of the current purchase of immediate life annuities.

Dus, Maurer, and Mitchell (2005, DMM) analyze various strategies, including an imme-
diate real (that is, inflation-indexed) life annuity, delayed annuitization and systematic
withdrawals. The withdrawal rate is determined according to a fixed dollar benefit level.

115M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



Under a fixed dollar formula, benefits are paid until the plan participant dies or funds are
exhausted. Alternate variable formulas are based on fixed percentages or relate to remaining
life expectancy. Their evaluative approach is broadly similar to Warshawsky (2012, Chapter
7) using shortfall risk of expected payouts and real balances. If no annuities are available, the
optimal strategy is a fixed percentage withdrawal of 7.3%, and the optimal equity share is
75%—remarkably close to the results mentioned above. DMM compare these results to a
fixed payment systematic withdrawal equal to the immediate real life annuity payout.
Focusing on the expected present value of benefits paid (a concept similar to money’s worth
in the annuity pricing literature—the larger, the better) as the best single evaluative measure,
the fixed percentage approach is shown to be superior to the fixed dollar approach.2

DMM find that annuitization is more appealing to older retirees, as compared with
systematic withdrawals, another result quite similar to those summarized above. They infer
that a strategy of systematic withdrawals followed by full annuitization at age 75 or 85
(delayed annuitization) increases the expected present value of benefits and shrinks the
expected present value of shortfall income. This evokes the annuity ladder. Finally, they
evaluate the immediate purchase of a delayed annuity (to pay at age 75 or 85) at the
beginning of the retirement period—the longevity insurance strategy advocated by some
analysts and market participants. DMM find that for retirees who desire any bequests or
liquidity, these outcomes are generally inferior to delayed annuitization, particularly for
longevity insurance paying at age 85. Moreover, DMM assumed the load on the delayed
annuity is the same as for immediate annuities, whereas empirical evidence finds that
immediate life annuities have lower loads (see below).

A related article by Horneff, Maurer, Mitchell, and Dus (2008) uses a utility-based
framework, with stochastic capital markets (but not inflation) and uncertain lifetimes, to
evaluate phased withdrawal plans and fixed payout annuities. They find that the fixed benefit
rule performs poorly, running out of funds by age 80, is consistent only with very low levels
of risk aversion, and is dominated by other payout rules (the fixed percentage in particular)
and by the life annuity.

Horneff et al., also consider combination strategies, in particular the life annuity and the
life expectancy withdrawal rule. They find that as risk aversion increases above relatively
low levels, the devotion of a significant share of wealth (approaching 90% with no bequest
motive) to the immediate life annuity increases welfare significantly. For moderate risk
aversion, it is better to delay annuitization until around age 80.

Economists refer to the actual dearth of voluntary annuitization—despite the evidence of
economic theory and simulations showing the high utility value of immediate life annuities
as insurance against outliving wealth—as the “annuity puzzle.” Various explanations have
been suggested. Brown and Warshawsky (2004), Dushi and Webb (2004), and Inkmann et
al., (2011), among others, show that the extent of annuitization is affected by the availability
of annuities in retirement plans, the share of wealth represented by Social Security and
defined benefit plans, the load on life annuities arising from adverse selection, extra mar-
keting costs and other factors, levels of financial wealth, life expectancy, education, and
bequest motive.

Behavioral biases are now favored as an explanation of the annuity puzzle, prompting
calls for new strategies and public policies to overcome cognitive blocks. One such proposed

116 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



strategy is longevity insurance, a deeply deferred life annuity, also known as an advanced-
life delayed annuity (ALDA). An ALDA is purchased at retirement but payouts do not begin
until the retiree reaches an advanced age, usually 85. It has been promoted by Milevsky
(2005) and Scott (2008), as well as by the prior, Obama, Administration.3 The ALDA
strategy claims to provide liquidity through most of retirement, that is, partial annuitization
with longevity risk coverage late in life for a premium that might be perceived as a cheap
price. For example, Sexauer et al. (2012) show that in 2010, a 65-year-old retiree would have
needed “only” 12% of the portfolio to purchase a deferred (albeit nominal) annuity such that
the first payout at age 85 would equal the last payout from a self-amortizing (over 20 years)
portfolio of laddered Treasury Inflation Protected Securities (TIPS). Haensly and Pai (2015)
have also examined the TIPS/ALDA strategy. It should be noted though that there is no
upside investment potential with this strategy and it may not therefore be desirable to many
retired investors.

Gong and Webb (2010) compare longevity insurance with complete immediate annuiti-
zation, delaying the complete annuitization to a late age, and systematic withdrawals; they
do not consider ladders of annuity purchases. They use an expected utility model, but with
no risky assets or bequest motive, and assume all annuities are loaded equally. With
retirement at age 65 and moderate risk aversion, Gong and Webb find that complete
immediate annuitization wins out over longevity insurance (although Gong and Webb say
they favor the ALDA).

Gong and Webb also calculate loads on commercial annuity products using the standard
money’s worth methodology of Mitchell et al. (1999). They find that the load is about five
percentage points higher on real annuities than on nominal annuities, and that the load is
fairly constant across ages 60 through 75 but increases thereafter, again by about five
percentage points, based on the best pricing of immediate annuities among three issuers, and
a pricing model using Treasury bond yields and general population mortality. A comparison
of the loads on longevity insurance (paying at age 85) with immediate annuities, on a
nominal basis and issued at age 65, finds the load on the deferred annuity to be about five
percentage points higher than for the immediate annuity at one insurance company, and
about 20 to 30 percentage points higher at another. Although these findings reflect pricing on
one day only (in January 2008), Finkelstein and Poterba (2004) reached broadly similar
conclusions based on data from a large United Kingdom insurance company over a 17-year
issuance period through 1998. They find that the load is from five to 10 percentage points
higher on real than on nominal immediate annuities. We will use these results in our
empirical simulations below. Note that even though there is no real ALDA being sold in the
market, I do model it in a couple of the simulations below, and make an inference on what
its load would be from the above literature.

This literature review has focused on studies bearing directly on retirement income
strategies. Other recent studies, however, are also quite relevant to this article because they
address directly and empirically parameter values for the bequest motive that are only
guessed at in the previous simulation literature. Ameriks et al. (2011), De Nardi et al. (2010),
and Lockwood (2012) carefully estimate preference parameters in structurally similar life-
cycle expected utility models. They find varying degrees of bequest prevalence and strength
of the bequest motive. Below, we use the average and range of their parameter estimates in

117M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



evaluations of income strategies. Similarly, survey-based formal evidence on the parameters
for risk aversion is found in Kimbal et al. (2008). Their risk aversion parameter estimates are
measured precisely, based on answers to hypothetical risk situations. By contrast, the time
preference parameter is less consistently and more widely measured in a large literature, so
we hew more closely here to convention and intuition.

3. A utility-maximizing framework for the evaluation of strategies

3.1. Lifetime utility

I now specify the algorithm or model by which retired households find their optimal
strategy. This stochastic simulation optimization model is in the spirit of the theoretical
model by Pang and Warshawsky (2010) but is more realistic of product and market
conditions. The particular functional forms used here are chosen to (1) enable the use of
empirical results in the literature estimating various parameter values (which assumed these
functional forms), (2) they are more easily manipulated in stochastic simulation work than
other forms, and (3) are fairly common and long-standing in both the theoretical and
empirical literatures.

A household is assumed to seek an income and wealth management strategy that maxi-
mizes its retirement lifetime utility, which is defined as a function of consumption flows and
the bequest amount upon death. Its lifetime utility in retirement is expressed as follows:

V� � �t�0
��1 ��thtu�ct

ht
�� � ��v�b��,

where the realized lifetime utility V� depends on the survival from t � 0 (retirement age 65)
through � (stochastic, maximum age 105), ct is household consumption and ct/ht is on a per
capita basis with ht being the effective number of adults, b� is monetary wealth as bequest
upon death, and � � 0.97 is the discount factor (time preference) initially (later we will vary
and increase the time preference parameter). Ignoring children, we set ht to 1 for single
retirees and 20.5 for couples, taking into account economies of scale in consumption.

The period utility function of consumption takes the constant relative risk aversion
(CRRA) form:

u�c� �
c1��

1��
,

where a higher value of � indicates greater risk aversion. The utility from bequest has a
similar functional form, as follows:

v�b� �

��� �
b

�
�1��

1��
,

118 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



where � indicates the strength of the bequest motive and � imposes a threshold of consump-
tion (in thousands of dollars) above which the bequest motive becomes operative. As the
value of � rises, bequests are increasingly considered a luxury. Using rounded averages of
parameter values estimated by De Nardi et al., (2010) and Lockwood (2012),4 we initially set
� � 4, � � 30, and � � 10. These parameters depict a modestly conservative attitude of
retirees toward risk and a middling motive for a bequest. Bequests are feasible for house-
holds, but are not necessarily easy to manage with the consumption threshold of $30,000 a
year (� � 30). We consider alternative values later in sensitivity tests. The objective of
households is to maximize the expected V� over all possible life outcomes of �. The
numerical result of V� is a ranking of utility and does not conveniently measure the
magnitude of welfare gain or loss of one strategy versus another. Therefore, we calculate
the so-called certainty equivalent (CE) consumption that would generate the same level of
value V� for a life path �. This constant CE consumption, on a per capita basis, is determined
such that:

�t�0
� ��thtu�CE�� � V�.

Further, over N possible life paths (i.e., N series of simulations from t � 0 through �), we
calculate the average certainty equivalent consumption (ACE) as:

ACE � �j�1
N CEj

N
.

The objective of households is now transformed to search for an income distribution and
wealth strategy that achieves the highest ACE. When evaluating strategies, both across broad
categories and for specific implementations within a category, the highest ACE should be
chosen, for a particular set of preference and economic condition parameters.

3.2. Parameterizations and stochastics

Households (singles or couples) are initially assumed to start with $250,000 in accounts
upon retirement. Household consumption ct is equal to income that is the sum of systematic
withdrawals, annuity payouts, and Social Security benefits. The Social Security benefit is
initially set to be 12 � $1150for singles or survivors and 12 � $1150 � 1.5 for couples (a
50% spousal benefit). These amounts are based on the 50th percentile of the new Social
Security awards to retired workers in 2010. The bequest b� is set as the fund balance upon
death, if greater than zero, plus one month’s Social Security benefit.

Households are initially assumed to have an initial 50 –50 equity-bond portfolio mix.
This simple allocation, common in many target-date funds at the point of retirement, is
chosen so that we can focus on the implications of various withdrawal and annuitization
strategies; we leave the analysis of more complex asset allocations, such as international,
developing market, real assets, and so on, to future work. As wealth is annuitized, the
equity share increases in the remaining assets (up to 100%) so as to maintain roughly the
same overall risk exposure.5 This dynamic asset allocation strategy is an approximation
to the optimal one resulting from the theoretical model of Pang and Warshawsky (2010)

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mentioned above. Although some retirees might initially balk at this high allocation to
equities, its riskiness is only apparent and not real because of the substitution of fixed life
annuities for bonds; this logic could be explained to retired households by financial
advisors.

Variable returns on equities and bonds are proxied by the S&P 500 and the United States
10-year Government Bonds Total Return indexes, respectively. Inflation is measured by the
change in the CPI-U index. The stochastic dynamics of asset returns and inflation are
modeled as a vector autoregressive (VAR) process, following Campbell and Viceira (2005).
The VAR coefficients and variance-covariance matrix, estimated on 1962–2011 quarterly
data, are embedded in the simulations to generate a large number of multiyear series of rates
and returns. This approach captures the serial correlations among variables and the contem-
poraneous correlations of market shocks. Summary statistics are reported in Table 1 and
simulation details are in the Appendix, section A. Investment management fees are sub-
tracted from returns. We also allow for insurer bankruptcies, with small probabilities, and
stochastic partial policyholder recoveries.

The management fee for investments is initially assumed to be a relatively low 25 basis
points, which is consistent with a portfolio composed mainly, but not entirely, of indexed
equity and bond funds in employer-sponsored retirement accounts or discount IRAs. For
annuity purchases before age 75, a 10% load is assumed for nominal immediate life
annuities, a 15% load for real immediate life annuities, a 15% load for the nominal ALDA,
and a 20% load for the real ALDA (as mentioned above, this latter product does not exist in
the market, so it should be regarded as hypothetical here). For annuity purchases after age
75, loads are further assumed to increase linearly by up to 5% until age 85 and flatten off
thereafter. These load differentials are consistent with the empirical evidence of age-varying
actuarial fairness of annuities reported in Gong and Webb (2010).6 Single households
purchase single life annuities, while married couples purchase joint and survivor (J&S)
annuities with 75% survivor benefit. The process of annuitization can be gradual for
immediate annuities, taking as long as 30 or more years, although in actual practice, it would
be appropriate to limit the process up to age 90, in line with current U.S. market practices
for the maximum age for immediate annuity sales. For the ALDA, it is a one-time purchase
upon retirement at age 65, with payment commencing at 85—the approach advocated by
some analysts, market makers, and policymakers.

Table 1 Summary statistics of simulated annual rates and returns

Equity return Bond return Bond yield Inflation

Real (%)
Mean 4.9 2.8 2.5 —
Standard
deviation

17.8 9.7 2.4 —

Nominal (%)
Mean 8.9 6.9 6.5 4.1
Standard
deviation

17.3 9.0 2.5 2.7

Source: Author’s simulations based on 1962–2011 data.

120 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



The underlying assets for life annuities are assumed to be invested in nominal bonds. The
calculation of annuity price/factor uses the 10-year government bond yield, which is sto-
chastic through time, and life tables for annuitants rather than those for the general popu-
lation, which reflects adverse selection in the voluntary annuity market. Note that this annuity
pricing module, including the loads mentioned above, is quite conservative and biases the
optimal strategies somewhat away from annuitization. The survivals of households are
simulated in the model based on general population mortality rates.

3.3. Implementations and strategies considered

I now consider several broad strategies, including systematic withdrawals, immediate
annuities, and deeply deferred annuities, nominal and real. I model retired investors search-
ing among the implementations for the best strategies, first simple ones and then more
complex combinations.

3.3.1. Systematic withdrawals: Fixed real dollars or fixed percentage of balances
Retirees can continue to hold investment funds and take periodic systematic with-

drawals, which can be constant in real dollars. That is, investors withdraw a certain
amount in the first period and adjust the amount for inflation in the following periods.
This strategy provides retirees with the same purchasing power over time, but they risk
outliving resources at older ages. This approach is related closely to the so-called Bengen
rule, discussed below.

Alternatively, withdrawals can be a fixed percentage of the portfolio balance in each
period, which will not exhaust the retiree’s wealth and implicitly assumes some self-
discipline or flexibility on consumption because withdrawals can be very low in adverse
investment climates. This approach provides liquidity to investors and bequest potential
to their heirs. It allows investors to consume more when funds perform well, but also
exposes them to possibly painful declines in consumption when investments fare
poorly.7

3.3.2. Immediate life annuity: Nominal or real
Life annuities address longevity risk and offer a steady flow of income. Lacking an

annuity, retirees’ income flow and consumption hinge on how quickly they draw down
wealth, how long they live, and their investment outcomes. Retirees who consume too
quickly may outlive their financial resources, especially given ever-increasing life expec-
tancy, while the overly cautious may consume well below their means. We consider the most
widely available nominal immediate life annuities, whose payouts are constant in nominal
terms, as well as real life annuities, whose payouts are indexed to inflation, with extra cost
loads, as observed in the market.

3.3.3. Advanced-life delayed annuity
As a relatively recent innovation to insure against longevity, the ALDA purchased at

retirement begins payouts at an advanced age, such as 85, to surviving investors. The ALDA
premium is typically a fraction of the premium for an immediate annuity with the same

121M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



payout. The annuity load is nonetheless higher because ALDA providers face higher risk in
guaranteeing the interest rate during the interval between purchase and payouts and also
possibly from a greater extent of adverse selection among purchasers. Determining the
desired level of ALDA payouts and smoothly managing income before benefits begin
remains practically challenging for investors. Our simulations assume that investors purchase
the ALDA at age 65, make systematic withdrawals before 85, and lower withdrawals by the
amount of ALDA payouts at 85. The lifetime income flow can be volatile because of the
uncertainty in investment outcomes.

4. Results

The strategies may use nominal or real (inflation-adjusted) product components in their
implementations, but reported outcomes are always adjusted for stochastically realized
inflations, that is, reported incomes and balances are in real dollars. To initially speed up the
simulation computations, which involve searching across a grid of expected utility values to
find the optimum, the fixed percentage withdrawal has a one percentage point increment, and
the fixed real dollar (annual) withdrawal has a $5,000 increment. The annuitization process
can take up to 30 or more years, with a 5-year time period increment, and the purchase can
start and end in the range of 0–100% of expected wealth, with a five percentage point
increment. Later simulations do use a more refined search grid. I consider both singles and
couples as retired households.

4.1. Singles with normal life expectancy

Although the at-least-partial use of annuities is indicated by the literature, and the ACE
values (for like preference parameters) reported below are higher than found in strategies
using systematic withdrawals alone, I start with systematic withdrawals alone to gain some
insight into the best form and level of systematic withdrawals. Among fixed-dollar inflation-
indexed withdrawals, the simulations find that a $15,000 withdrawal (that is, 6% of the initial
balance, inflation-indexed in subsequent years) generates the highest lifetime utility. Real
consumption remains constant, as long as the accounts are not exhausted. If the distribution
strategy is based on a fixed percentage of asset balances with a varying dollar amount, the
optimal withdrawal is 9% of the portfolio balance each period. Relative to the fixed
real-dollar withdrawals, this strategy tends to generate higher consumption in earlier years
but lower consumption in later years, owing to a higher volatility of incomes. Overall, the
ACE is higher here than with the fixed dollars implementation, indicating the superiority of
the fixed percentage approach. See Table A-1 in the Appendix, section B, for the consump-
tion and wealth outcomes for single retirees.

Financial planners often advise retirees to draw down their assets using a 4% rule, as
suggested by Bengen (1994). By this rule, households in our model would withdraw $10,000
a year in real dollars, which is lower than the optimal withdrawal amount given above and
produces lower expected lifetime utility (by an ACE measure of 23.0, not reported, vs. 25.1
in Table A-1). One important factor is that my analysis includes a safe and inflation-indexed

122 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



lifetime income flow from Social Security, which supports a higher withdrawal and reduces
the risk to income of wealth running out. Scott et al. (2009) argue that the 4% rule is
inefficient because it leaves a significant portion of resources unspent. Also, most financial
advisors deal with clients with above-average wealth; their spending needs and desires may
be relatively modest compared with their assets.

Let individuals now consider adding (laddered) purchases of immediate life annuities
(Table 2). In scenario 2a, with both fixed dollar withdrawals and annuity payouts indexed to
inflation, singles annuitize 40% of wealth initially and 45% over 25 years.8 They also
simultaneously withdraw $10,000 a year in real terms. This combination foregoes some
liquidity (lower balances) but generates a higher and more sustainable income flow than
systematic withdrawals alone, implying a greater lifetime utility (ACE of 25.6 vs. 25.1 in
Table A-1).

When fixed percentage withdrawals and nominal life annuities are considered in scenario
2b, it is optimal to not annuitize initially (zero percentage), but purchases of immediate
annuities reach 45% of wealth by 20 years. The postponed and laddered purchase of nominal
annuities helps maintain the real purchasing power of payouts and reduces the timing risk of
purchases. For early years in retirement, income is achieved primarily through withdrawals
(8% of balance), which are nontrivial amounts given the size of account balances. The ACE
is highest here across the strategies/implementations analyzed, at 26.3.

With withdrawals fixed in real dollars along with a ladder of nominal annuities
considered in scenario 2c, singles withdraw $10,000 a year and also generate income
through significant annuity purchases—initially 30% of wealth and reaching 55% by 25
years. The ACE is lower than in the above “nominal” strategy, because the rigid fixed
dollar withdrawal imposes greater risk of running out of funds. Overall, the strategy of
fixed percentage withdrawals combined with laddered purchases of nominal immediate
annuities wins.9

Table 2 Search for optimal strategies among systematic withdrawals and laddered purchases of immediate
life annuities—singles

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Options: Fixed real dollars � real annuity
Optimal strategy: Withdrawal $10, annuity initial 40% of wealth, ending 45% by 25 years, ACE 25.6

Income 32.6 29.5 18.6 28.0 4.4
Balance 199.8 89.4 0.0 88.9 72.5

b. Options: Fixed percentage � nominal annuity
Optimal strategy: Withdrawal 8%, annuity initial 0% of wealth, ending 45% by 20 years, ACE 26.3

Income 35.6 28.2 21.2 28.4 4.6
Balance 250.0 105.3 0.0 113.2 87.9

c. Options: Fixed real dollars � nominal annuity
Optimal strategy: Withdrawal $10, annuity initial 30% of wealth, ending 55% by 25 years, ACE 25.8

Income 33.3 30.1 18.0 28.4 4.9
Balance 219.2 101.3 0.0 99.4 82.1

Source: Author’s simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

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In Table 3, an ALDA is considered instead of laddered purchases of immediate annuities.
In scenarios 3a and 3c, the retiree would optimally withdraw $15,000 systematically,
inflation-adjusted, and use 5% of initial wealth to purchase an ALDA, with payouts indexed
to inflation or nominal, respectively. The simulated level of demand for the ALDA seems
trivial. Recall, however, that the ALDA payout commences 20 years after purchase. The ages
covered by an ALDA have higher mortality rates than earlier years, which actuarially reduces
the cost for any life annuity payout. Further, with discounting for time (interest) for the
20-year waiting period, the ALDA premium is substantially lower. Discounting would lower
the premium by about 65% for a real ALDA and 85% for a nominal ALDA, assuming the
average interest and inflation rates reported in Table 1. Put differently, the age-85 payout
from an ALDA that is purchased with 5% of wealth is equivalent to the payout from an
immediate annuity that is purchased with 15–30% of initial wealth, depending on the
contract terms.

Nonetheless, the ALDA has significant shortcomings. First, achieving a smooth connec-
tion between withdrawals and commencement of payout is difficult. Consumption may
experience cliff changes by the time the ALDA begins payouts because of declines in wealth.
Notably in scenario 3b, the ALDA loses its appeal entirely when a fixed percentage
withdrawal is used. And second, households are likely to be better off using immediate annuities
rather than an ALDA, as indicated by higher ACEs in Table 2 versus Table 3, in part owing to
the lower loads on immediate annuities. More important, the immediate annuities perform better
in managing a steady income flow and avoiding extremely low incomes.

4.2. Singles with a short life expectancy

I examine now how the strategies/implementations would vary for single retirees with an
impaired life expectancy. I model impaired mortality as equivalent to those with a spinal cord
injury, based on the estimates of Strauss et al. (2005). Their life expectancy is about seven

Table 3 Search for optimal strategies among systematic withdrawals and age-85 ALDA—singles

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Options: Fixed real dollars � real ALDA
Optimal strategy: Withdrawal $15, ALDA 5% of wealth, ACE 25.2

Income 28.8 28.8 15.4 27.2 4.3
Balance 295.9 161.4 0.0 151.1 100.6

b. Options: Fixed percentage � nominal ALDA
Optimal strategy: Withdrawal 9%, ALDA 0%, ACE 25.7

Income 36.7 26.7 17.6 27.2 6.4
Balance 254.1 140.3 41.0 146.3 71.4

c. Options: Fixed real dollars � nominal ALDA
Optimal strategy: Withdrawal $15, ALDA 5% of wealth, ACE 25.2

Income 28.8 28.8 15.2 27.2 4.3
Balance 293.7 160.0 0.0 150.7 100.2

Source: Author’s simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

124 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



years shorter than that for the general population. As shown in Table 4, these households would
make more aggressive withdrawals—$5,000 more a year as a fixed dollar withdrawal strategy or
three percentage points higher as a fixed percentage strategy (compared with cases 2a and 2b
above, respectively). Being less likely to reach advanced ages, these retirees would also generally
reduce their purchases of immediate life annuities. Here mortality is simulated based on the
impaired population life table but assumes that annuity pricing uses the regular annuitant life
table, because annuity underwriting is uncommon and expensive in the United States.

4.3. Couples

Tables 5 and 6 report the results for two-person (same age) retired households when both
annuities and systematic withdrawals are considered. The optimal strategies for couples are

Table 4 Search for optimal strategies among systematic withdrawals and immediate annuities—singles with
an impaired life expectancy

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Options: Fixed real dollars � real annuities
Optimal strategy: Withdrawal $15, annuity, initial and ending, 25% of wealth, ACE 27.2

Income 35.7 34.0 18.7 32.6 4.6
Balance 220.5 138.8 0.0 126.0 72.0

b. Options: Fixed percentage � nominal annuity
Optimal strategy: Withdrawal 11%, annuity, initial 0% of wealth, ending 30% by 20 years, ACE 27.9

Income 41.3 32.5 23.3 32.8 6.3
Balance 250.0 138.5 6.8 137.7 79.6

Source: Author’s simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

Table 5 Search for optimal strategies among systematic withdrawals and laddered purchases of immediate
life annuities—couples

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Options: Fixed real dollars � real annuity
Optimal strategy: Withdrawal $10, annuity initial 35% of wealth, ending 45% by 25 years, ACE 23.5

Income 37.8 33.3 17.4 30.4 6.7
Balance 213.6 87.3 0.0 90.5 80.0

b. Options: Fixed percentage � nominal annuity
Optimal strategy: Withdrawal 8%, annuity initial 0% of wealth, ending 50% by 25 years, ACE 24.3

Income 41.3 31.3 19.9 31.1 7.0
Balance 250.0 88.9 0.0 102.6 87.2

c. Options: Fixed real dollars � nominal annuity
Optimal strategy: Withdrawal $10, annuity initial 25% of wealth, ending 65% by 25 years, ACE 23.8

Income 39.3 34.2 17.2 31.0 7.4
Balance 223.3 77.3 0.0 88.8 86.1

Source: Author’s simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

125M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



similar to those for singles. Using systematic withdrawals alone, it is optimal for couples to
withdraw $15,000 in real dollars, or alternatively 9% of balances each year (Table A-2 in
Appendix, section B). When the option of laddered purchases of immediate life annuities is
considered, couples should eventually annuitize from 45 to 65% of wealth, varying with
the combinations of nominal and real products (Table 5). Initial annuitization is gener-
ally lower for couples than for singles. Also note that the ACE is lower for couples than
for singles because the same resources have to be shared for a couple even when there
are some economies of scale in living. For retired couples, the optimal strategy is a
program of systematic withdrawals of 8% of balances combined with a laddered pur-
chase of immediate nominal life annuities over 25 years, rising from zero percentage to
50% of wealth. When an ALDA is considered, a small fraction (5%) of wealth on ALDA
purchase is optimal and improves welfare relative to the systematic withdrawals alone,
according to the measure of ACE (Table 6). Overall, households benefit from adding
annuities to their retirement portfolios. Immediate life annuities again serve better than
ALDAs.

4.4. Other household preferences and situations using the combined withdrawal-laddered
strategy

I now consider alternative preferences and household situations. Given the prior
results indicating the overall superiority of combinations of fixed percentage withdrawals
with laddered purchases of nominal life annuities, we search for optimal strategies just
within that class. Table 7 shows the specific optimal strategies and range of results in
several cases.

With greater wealth (here $600,000), there is more room for the operation of the bequest
motive (the minimum consumption threshold is easily met), and therefore the withdrawal
rate is lower as is the ultimate extent of annuitization (compare Table 7a with Table 2b).

Table 6 Search for optimal strategies among systematic withdrawals and age-85 ALDA—couples

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Options: Fixed real dollars � real ALDA
Optimal strategy: Withdrawal $15, ALDA 5% of wealth, ACE 23.6

Income 35.7 35.7 16.2 30.7 6.5
Balance 296.5 142.1 0.0 139.4 105.5

b. Options: Fixed percentage � nominal ALDA
Optimal strategy: Withdrawal 9%, ALDA 0%, ACE 23.9

Income 43.1 30.0 17.5 30.2 8.5
Balance 250.3 126.8 37.0 136.8 71.6

c. Options: Fixed real dollars � nominal ALDA
Optimal strategy: Withdrawal $15, ALDA 5% of wealth, ACE 23.7

Income 35.7 35.7 16.1 30.7 6.6
Balance 296.3 142.8 0.0 139.6 104.4

Source: Authors’ simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

126 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



Hence, the balance holdings across possible outcomes are all proportionately higher here. If
risk aversion is higher (� � 6 vs. 4 in the baseline), the single-member household would
boost eventual annuitization by 10 percentage points across the life cycle (compare Table 7b
with Table 7a) owing to its greater preference for security, in particular about income; indeed
income outcomes are uniformly larger and balances lower. In situation 7c, I change a number
of preferences and situations simultaneously: the household is wealthier ($1,000,000), more
risk averse (� � 7.6), more desirous of leaving a bequest (� � 20 vs. 10 in the baseline),
more forward-looking/patient (� � 1.10), while our search grid across solutions to find the
optimal strategy is more refined both for withdrawal rates, and the extent and timing of
annuitization. These changes drop the fixed percentage withdrawal rate substantially, while
the annuitization pattern is largely unchanged. Wealth outcomes are uniformly and relatively
higher here because the optimal strategy consistent with the preferences and situation gives
more emphasis to retaining balances.

In the final frame of Table 7, I move in the opposite direction—preferences remain the
same, but wealth is lower and retirement for the couple is at age 62. The optimal strategy
places a higher emphasis on income, as evident in the much greater extent of annuiti-
zation, albeit the purchase laddering takes place over a longer horizon because the
younger couple has a much higher probability of at least one member surviving to old
age. The longer horizon for the household also explains the lower rate of withdrawals.
The specific optimal strategies vary widely with household preferences and demographic
situations.

Table 7 Search for optimal strategies among fixed percentage systematic withdrawals and laddered
purchases of nominal immediate life annuities—alternative situations and preferences

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Situation: Initial wealth is $600,000
Optimal strategy: Withdrawal 7%, annuity initial 0% of wealth, ending 20% by 20 years, ACE 41.49

Income 61.92 45.39 30.04 45.74
Balance 638.12 359.16 75.66 363.56

b. Situation: As in a. above, but with higher risk aversion � � 6
Optimal strategy: Withdrawal 7%, annuity initial 10% of wealth, ending 30% by 20 years, ACE 43.40

Income 62.78 46.14 30.79 46.51
Balance 579.88 319.14 57.62 322.98

c. Situation: Initial wealth is $1,000,000, high risk aversion � � 7.6 stronger bequest motive � � 20 more
forward-looking, patient � � 1.10, refined search grids

Optimal strategy: Withdrawal 4%, annuity initial 7.5% of wealth, ending 20% by 21 years, ACE 49.1

Income 79.6 55.9 39.5 56.9 12.5
Balance 1314.7 806.5 348.2 806.4 301.4

d. Situation: As in a. above, except couple both age 62, initial wealth is $150,000, and � � 1.0
Optimal strategy: Withdrawal 6%, annuity initial 20% of wealth, ending 75% by 30 years, ACE 19.76

Income 31.69 26.56 17.36 25.31 4.83
Balance 127.57 48.77 0.0 54.32 45.98

Source: Author’s simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

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4.5. Add consideration of household taxes

I now add consideration of personal taxes on the retired household to the model. In particular,
I add parameters for the payment of taxes (aggregating federal, state and local government
impositions) on income and bequests made at the average effective rates of the household. For
most households, estate tax rates are low or zero, but income tax rates, even for middle-class
retirees, can be 15% or higher. I also add to the model the minimum distribution requirements on
tax-deferred account balances which can force distributions in excess of optimal levels deter-
mined, particularly at older ages; for example, beyond the ages of 90 and older, distributions of
10% and more are required. If the legally required distribution in a year is higher than the optimal,
then we place the distribution in excess of the optimal in a taxable investment account that will
eventually be used for liquidity or bequest purposes, that is, to support consumption if all other
retirement wealth has been used up, but otherwise held in reserve.

I illustrate in Table 8 the impact of taxes on optimal strategies for four cases shown above,
in panels 2b, 5b, 7b, and 7c. I assume a 15% effective income tax rate and a zero effective

Table 8 Search for optimal strategies among fixed percentage systematic withdrawals and laddered
purchases of nominal immediate life annuities—alternative situations and preferences, with additional
consideration of taxes and minimum distribution requirements

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Situation: Panel 2b
Optimal strategy: Withdrawal 9%, annuity initial 0% of wealth, ending 55% by 20 years, ACE 23.2

After-tax income 31.5 24.8 17.4 24.4 4.6
Tax-deferred balance 250.0 80.9 0.0 97.5 88.9
Taxable balance

At age 95 0.5 0.0 0.0 0.1 0.5

b. Situation: Panel 5b
Optimal strategy: Withdrawal 8%, annuity initial 0% of wealth, ending 60% by 25 years, ACE 21.03

After-tax income 35.2 26.6 16.7 26.5 6.0
Tax-deferred balance 250.0 77.8 0.0 95.8 88.2

Taxable balance
At age 95 0.1 0.0 0.0 0.1 0.9

c. Situation: Panel 7b
Optimal strategy: Withdrawal 7%, annuity initial 15% of wealth, ending 40% by 20 years, ACE 38.3

After-tax income 54.2 40.1 26.9 40.4 8.8
Tax-deferred balance 545.9 285.9 25.2 290.1 172.8
Taxable balance

At age 95 40.7 9.9 0.0 13.8 14.0

d. Situation: Panel 7c
Optimal strategy: Withdrawal 5%, annuity initial 5% of wealth, ending 20% by 30 years, ACE 44.30

After-tax income 70.3 50.6 33.7 50.8 11.4
Tax-deferred balance 1195.5 746.6 250.7 734.5 296.8
Taxable balance

At age 95 331.8 138.0 42.6 154.9 94.5

Source: Author’s simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

128 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



estate tax rate. In general, because a significant portion of retirement resources are sent to the
government, to produce income levels (and utility) roughly comparable (but inevitably a bit
lower) to the situation before the consideration of taxes, higher withdrawals and greater
annuity purchases must be made. The remaining balances will be lower. In addition, with the
addition of minimum distribution requirements, for higher wealth levels or favorable invest-
ment outcomes, some tax-favored retirement assets will eventually be placed in taxable
investment accounts, particularly at older ages.

4.6. Add pensions

Finally, I add defined benefit pensions to the model. The pension benefits can be either
indexed to inflation or not, and can be either for the individual alone or for both members of the
couple (assumed as a joint-and-two-thirds-to-survivor annuity). Here I also only use a refined
search grid to find the optimal strategy and limit the laddering of annuities up to age 90. With the
addition of pension income but no subtraction of other retirement resources, we naturally will
expect a higher ACE, everything else equal. Because the pension is paid as a life annuity, one
would also expect less need for purchasing immediate life annuities as well as lower withdrawals.
These are indeed the model outcomes shown below in Table 9 when I add pensions (at the same
level as Social Security, but unindexed and split among both members of the couple), on top of
taxation and minimum distribution requirements, to cases 7c (8d) and 7d.

4.7. The full model treatment of three disparate example cases

Here I show three example cases, with divergent preferences and household demographic
and economic situations, their optimal strategies determined by the full model, and the range

Table 9 Search for optimal strategies among fixed percentage systematic withdrawals and laddered
purchases of nominal immediate life annuities—alternative situations and preferences, with additional
consideration of taxes and minimum distribution requirements, and defined benefit pensions added

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Situation: Panel 7c (8d)
Optimal strategy: Withdrawal 4.5%, annuity initial 0% of wealth, ending 15% by 24 years, ACE 49.0

After-tax income 76.4 56.8 39.1 56.8 11.5
Tax-deferred balance 1301.8 813.4 276.5 798.5 316.9
Taxable balance

At Age 95 527.1 202.2 67.7 238.6 149.9

b. Situation: Panel 7d
Optimal strategy: Withdrawal 4.0%, annuity initial 0% of wealth, ending 80% by 27 years, ACE 21.9

After-tax income 35.3 29.9 18.9 28.4 5.6
Tax-deferred balance 167.3 73.6 0.0 75.7 60.2
Taxable balance

At Age 95 31.6 1.3 0.0 4.6 9.6

Source: Author’s simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

129M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



of possible outcomes over the households’ retirement lifetimes. Table 10 gives the full
specifications of the cases in terms of preference parameters, demographic situation, wealth,
and so on for each household.

Now I show in Table 11 the optimal strategies and range of outcomes for the three
households. The first household has fairly moderate retirement means, but a large bequest
motive, and is somewhat risk averse. Their solution is a quite modest withdrawal rate,
and significant annuitization over time, which produces fairly steady income and,
relative to their means, significant asset holdings, both in the tax-deferred and taxable

Table 10 Description of three different retired households

Household Age(s) Wealth Equity
allocation

Investment
expense

Tax
rate

Health
status

Social
security

Pension

A 68, 63 $400K 70% 77bps 20% Good $18K $12K both
B 70 $1.5M 75% 47bps 25% Poor $18K $12K
C 66, 64 $250K 40% 33bps 10% Good $15K $6K both, COLA

Household � � �

A 1.075 30 7.6
B 1.075 30 5.1
C 1.15 3 10.4

Source: Author.

Table 11 Search for optimal strategies among fixed percentage systematic withdrawals and laddered purchases
of nominal immediate life annuities—three households, full model

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Household: 10A
Optimal strategy: Withdrawal 4%, annuity initial 0% of wealth, ending 35% by 21 years, ACE 27.6

After-tax income 48.3 38.8 26.1 37.9 7.2
Tax-deferred balance 505.5 257.1 20.4 258.2 157.7
Taxable balance

At age 95 211.8 64.3 12.1 103.1 94.0

b. Household: 10B
Optimal strategy: Withdrawal 6%, no annuity, ACE 43.4

After-tax income 107.4 82.7 49.7 80.3 17.9
Tax-deferred balance 1927.5 1354.7 658.3 1313.8 393.8
Taxable balance

At age 95 250.0 124.5 NA 154.2 67.6

c. Household: 10C
Optimal strategy: Withdrawal 4%, annuity initial 0% of wealth, ending 80% by 24 years, ACE 27.4

After-tax income 41.8 34.6 24.8 33.7 5.3
Tax-deferred balance 265.5 119.7 0.0 122.5 96.0
Taxable balance

At age 95 43.8 4.2 0.1 10.9 19.7

Source: Author’s simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

130 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



accounts. The second household is composed of an older sick person with significant
asset holdings. Because of his large bequest motive and his poor health, no annuitization
is optimal while the moderate withdrawal rate, combined with Social Security and
pensions, gives plenty of income and preserves most of the tax-deferred and taxable
assets to grow and to leave as an inheritance. The third household has even more modest
retirement means than household A, but is more risk-averse, more desirous of future than
current spending, and much less interested in leaving a bequest. Therefore, the optimal
strategy for household C places a much higher allocation to annuitization, laddered over
an extended period, which in turn produces a less volatile income flow over the
household’s retirement lifetime.

Figs. 1 through 4 show some of the outcomes graphically for household C, using the
optimal strategy indicated above.

Fig. 1 shows the mean of stochastic outcomes under the optimal strategy in terms of pretax
income, by source and by age. Social Security and, in this case, pensions are inflation-
indexed, so the income flow from them is steady, although they do decline, on average, with
age, because of the possibility that one member of the couple dies. Income from the series
of nominal life annuities being purchased over time grows through age 90, reaching about
$12,000 annual income, in real terms, but then declines because purchases of the

Fig. 1. Sources of total annual income, mean, by age ($ real). Source: Author’s simulations, for household C in
Table 11.

131M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



immediate life annuities stop and owing to the force of inflation. Even though this
household prefers future spending over current spending, which keeps optimal income
high and steady over time, eventually income will decline with age because the large
force of mortality makes current spending more salient, and, for a couple, one member
is likely to have passed away and spending for him/her inevitably drops. Withdrawals
from the retirement investment portfolio, although fixed in percentage terms, decline and
eventually disappear as the retirement investments are withdrawn and also transferred,
because of the minimum distribution requirements, to a taxable portfolio, and life
annuities are purchased.

Minimum distributions (shown in Fig. 2) will be made and transferred to a taxable
portfolio (see Fig. 3) only if the withdrawal rate and annuity purchases are less than the
minimum legally required (which initially is less than 4% at age 71 but increases with age),
and, obviously, if there is value left in the retirement investment portfolio, which in turn
depends on return performance. Here minimum distributions are made only in the upper
percentiles and at upper ages.

Finally, Fig. 4 illustrates the formula for asset allocation given in Footnote 5 above, as
applied to the optimal strategy for this particular household. Here, because annuitization is
quite full, so is the allocation to equity in the remaining investment portfolios (both
tax-deferred and taxable).

Fig. 2. Stochastic range of required minimum distributions, by age ($ real). Source: Author’s simulations, for
household C in Table 11.

132 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



4.8. Different views on investment returns and interest and inflation rates

Thus far, I have used a model (from Campbell and Viceira, 2005) of investment returns
and interest and inflation rates estimated on quarterly data from 1962 through 2011. This is
a reasonable approach for, essentially, forecasting the range of possible experience in the
future. Nevertheless, some might want to put a greater emphasis on current conditions (in
2013), particularly as interest and inflation rates have declined so dramatically recently, as
presumably have overall expected investment returns. It is possible to have the simulations
adjusted to reflect this presumed different environment, without, however, completely
discarding the longer view, based on more distant past experience. Going a step further, it is
even possible to fix interest and inflation rates for the current year, allowing a smaller
dispersion in the second year, slowly spreading thereafter; equity returns would continue to
be as random as modeled earlier. This latter approach may indeed be the best way to express
uncertainty for a household currently retired and planning to implement the produced
strategy immediately. For households that are not yet retired, however, the former approach
(whether emphasizing current conditions or completely historical), reflecting a wider range
of uncertainty, is more appropriate.

In Table 12 below, I give the optimal strategies and range of outcomes for cases 10A and
10C. The first simulation, “Current conditions,” gives lower expected interest and inflation
rates and overall investment returns than the completely historical approach; indeed the real

Fig. 3. Stochastic range of after-tax investment portfolio values, by age ($ real). Source: Author’s simulations,
for household C in Table 11.

133M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



interest rate is reduced more than 100 basis points here. This may reduce the attractiveness
of annuitization somewhat, as it certainly will reduce the ACE. The second simulation,
“Current conditions with fixed initial interest rates,” uses, somewhat arbitrarily, the average
interest rate from the simulation as the fixed initial rate. It should be noted that a fixed initial
interest rate reduces considerably the uncertainty around purchasing a life annuity in the first
couple of years of the plan horizon; this should have the effect of increasing the attractive-
ness of initial annuitization, compared with either historical approach.

The expected changes occurred. Of further note, given the poorer investment environment,
the model favors income over assets, everything else being equal.

5. Policy discussion and conclusions

This analysis is based on a model of rational decision-making by retired individuals and
couples, using currently available investment and insurance products. Yet people are not
likely to use such strategies without a more formalized structure from which to obtain them.
At the least, plan sponsors, plan record keepers, or financial providers or advisors must make
them available to participants.

Fig. 4. Optimal equity allocations of investment portfolios, by age, percent. Source: Author’s simulations, for
household C in Table 11.

134 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



To encourage the use of life annuities, as strongly recommended by theoretical and
simulation models, efforts must go beyond designing the best strategy. As Brown et al.
(2008) show, it is critical to frame the issue so that plan participants understand that their
account balances—although seemingly a large amount of money in an investment frame—
may seem less adequate in a consumption frame as a lifetime annual income flow. Indeed,
based on a survey of people ages 50 and older, Brown et al. find that when the alternatives
are presented in a consumption rather than investment frame, the vast majority prefer an
annuity. Indeed James, Martinez and Iglesias (2006) make a related convincing argument
that the high annuitization rate (more than two-thirds) in the Chilean individual account
retirement system is because of its regulatory structure. They ascribe the high annuitization
rate to a limited range of payout options (effectively only life annuities or systematic
withdrawals), and to the absence of a public defined benefit plan except for a minimum
pension guarantee.

Given these results, proposed regulations by the Obama Administration, also reflected in
proposed bipartisan legislation, could have significantly changed retiree behavior. The
Department of Labor proposed guidelines in 2013 to require defined contribution plan

Table 12 Search for optimal strategies among fixed percentage systematic withdrawals and laddered purchases
of nominal immediate life annuities—two households, full model, alternative investment views

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Household: 10A, current conditions
Optimal strategy: Withdrawal 4.5%, annuity initial 0% of wealth, ending 35% by 21 years, ACE 26.0

After-tax income 47.8 39.9 25.1 38.9 6.8
Tax-deferred balance 428.2 193.3 1.6 205.4 143.1
Taxable balance

At age 95 90.0 23.1 3.7 32.1 30.4

b. Household: 10A, current conditions with fixed initial interest rate
Optimal strategy: Withdrawal 3%, annuity initial 21.5% of wealth, ending 52.5% by 21 years, ACE 26.0

After-tax income 45.6 37.2 24.5 35.9 7.2
Tax-deferred balance 380.0 151.7 0 168.0 130.0
Taxable balance

At age 95 205.7 54.3 12.3 74.2 66.8

c. Household: 10C, current conditions
Optimal strategy: Withdrawal 4%, annuity initial 0% of wealth, ending 67.5% by 24 years, ACE 25.2

After-tax income 37.7 32.5 22.6 31.0 5.0
Tax-deferred balance 250.0 84.1 0.0 100.6 90.4
Taxable balance

At age 95 20.7 3.3 0.2 5.7 8.2

d. Household: 10C, current conditions with fixed initial interest rate
Optimal strategy: Withdrawal 3.5%, annuity initial 31.5% of wealth, ending 80% by 24 years, ACE 25.4

After-tax income 38.0 33.9 22.9 31.6 5.4
Tax-deferred balance 175.9 64.4 0.0 73.5 64.2
Taxable balance

At age 95 34.3 6.7 1.4 10.9 13.1

Source: Author’s simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

135M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



sponsors to give participants annual income illustrations. These are attempts to achieve more
realistic framing and to provide some incentives for partial annuitization. Given strongly
ingrained behavior and market conditions in the U.S. leaning in the opposite direction, the
government might have to take an active role to even the playing field with pure asset
strategies. Certainly the private sector, especially including life insurers, financial companies,
and plan sponsors, need to be more aggressive and creative in their product design and
marketing activities in the retirement field, to emphasize the need for lifetime retirement
income.

Based on surveys of hypothetical choices, Beshears et al. (2014) find that allowing
individuals to annuitize a fraction of their wealth increases annuitization relative to an “all
or nothing” decision. The empirical simulation analysis here indicates that some life annu-
ities should indeed be part of the portfolios for many retirees. Life annuities work to establish
minimum necessary consumption and a certain level of hedging against longevity risk. While
this insurance is being lost with the decline of defined benefit pension plans, it can be
restored.

Some have proposed advanced life delayed annuities as providing the essence of
insurance at lower premiums. Maintaining a sustainable income flow before ALDA
payments begin, however, is no easy task, and the ALDA has higher loads. Welfare
measures using ALDA strategies are lower. A cheaper, less risky and more transparent
strategy is the combination of systematic withdrawals (fixed percentage) with laddered
purchases of nominal immediate life annuities. This strategy is supported by the ana-
lytical work shown here and in the literature, and is robust across the spectrum of
household preferences and situations.

Notes

1 Ameriks, Veres, and Warshawsky (2001) also use a shortfall framework, but with
historical, not stochastic, simulated data.

2 These are unpublished results, which have been kindly provided by DMM, based on
United States financial and mortality data over the 1967–2002 period for a 65-year-old
single man.

3 The Obama Administration enacted a special dispensation, effective in 2014, from the
retirement account minimum distribution requirements for longevity insurance with
deferred payment up to age 85; the dispensation removes the premium for longevity
insurance from the calculation base used in determining required distributions subject
to income tax; the delayed payments, however, are taxed when paid. Reportedly, some
individuals in high tax brackets are using this dispensation to reduce their tax bill
rather than to manage risks.

4 To be consistent with our functional form, we have rearranged the specifications of De
Nardi et al. (2010) and Lockwood (2012) and recalculated the parameter values.
Ameriks et al. (2011) find their benchmark estimates of � � 7.28 and � � 47.6. These
parameters would lead to little consumption and substantial bequest for households in

136 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



our model because the consumption threshold $7,280 is easily exceeded ($13,800
from Social Security alone for a single retiree, see assumptions below) and the bequest
motive is particularly strong. Nevertheless, we do consider the effect of larger bequest
motive parameters than DeNardi’s but maintain the consumption threshold, so that
lower-income retired households will likely depend more on life annuities, ceteris
paribus.

5 Let e denote the equity share in the remaining non-annuitized wealth and a denote the
degree of annuitization (i.e., a � aw/w, annuitized wealth divided by total wealth). For
the desired 50–50 risk exposure, e*(w-aw)/w � 0.5. Rearranging the equations gives
e � 0.5/(1-a).

6 Using some six months of pricing data from August 2011 through April 2012
provided to me by a large, highly rated, United States insurance company and our
own computations of fair value using government bond yields and general popu-
lation mortality, we calculated that the load differential between a nominal ALDA
commencing payment at age 85 and a single-premium immediate straight fixed-
payout life annuity, both issued at age 65, indeed averaged about five percentage
points.

7 There is some rigidity in these withdrawal options. The fixed dollar strategy does not
respond to realizations of asset returns and the fixed percentage option does not speed
up distribution toward the end of life, as might be desired to avoid leaving too large
a bequest. Rather, they are easy to implement in old age, easy to explain, and require
minimal governance from product issuers. Households could re-optimize their port-
folios any time, perhaps with the help of financial advisors, but to avoid a lot of extra
costs and governance requirements, the rerunning of the algorithm should be done
sparingly, perhaps only at major life events such as the death of a spouse. Sun and
Webb (2012) show that households could spend according to the IRS table for
required minimum distributions (RMD), plus interest and dividends, and get better
utility than from alternatives in the class of systematic withdrawals. Collins and Lam
(2011) use a case study approach to discuss how financial advisors can utilize a
credible simulation model to help investors make informed retirement planning
decisions.

8 The degree of annuitization at any age is expressed as a percentage of the accumulated
value of initial wealth, which reflects the increased with interest (at the average rate).
One could have alternatively expressed it as a percentage of the account balance, but
this would have been misleading. An extreme example: 100% annuitization of the
remaining $1 balance appears as a strong preference for the life annuity but may be
trivial if $99 has been withdrawn over prior years.

9 While the strategy of fixed dollar withdrawals and a real annuity gives completely
steady real income—a goal perhaps desired by the highly risk averse, the combination
of fixed percentage and a real annuity is not examined here because the nominal
annuity has a lower load than the real annuity and the increase in income to cover
inflation can be accommodated through the laddering purchase of smaller amounts of
immediate annuities over time.

137M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



Appendix

Appendix A: Simulations of rates and returns

Asset returns are simulated as a vector autoregressive process (VAR). The VAR coeffi-
cients and variance matrix are first empirically estimated and then embedded in the simu-
lations with stochastic shocks. Technically, let V be a vector containing the variables. The
vector evolves in an autoregressive pattern:

Vt � �0 � �
k�1

K

�kVt�k � �t

where � �(0, �) denotes a vector of serially uncorrelated normal errors with E�t�s � 0, t �
s. The contemporaneous correlations of shocks are incorporated via the variance-covariance
matrix � and serial correlations of the variables via the coefficients �. The econometric
regression on historical data yields estimates of the coefficients, �̂�s and the variance-

covariance matrix �̂. The simulations follow several steps: First, a Cholesky factorization

Table A-1 Search for optimal strategies among systematic withdrawals—singles

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Options: Fixed dollars, inflation adjusted
Optimal strategy: Withdrawal $15, ACE 25.1

Income 28.8 28.8 13.8 27.0 4.8
Balance 315.5 174.2 0.0 163.0 106.3

b. Options: Fixed percentage of balance, nominal
Optimal strategy: Withdrawal 9%, ACE 25.7

Income 36.7 26.7 17.6 27.2 6.4
Balance 254.1 140.3 41.0 146.3 71.4

Source: Author’s simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

Table A-2 Search for optimal strategies among systematic withdrawals—couples

Real $000 95th percentile 50th percentile 5th percentile Mean Standard deviation

a. Options: Fixed dollars, inflation adjusted
Optimal strategy: Withdrawal $15, ACE 23.2

Income 35.7 35.7 13.8 30.4 7.2
Balance 314.0 158.3 0.0 151.7 109.4

b. Options: Fixed percentage of balance, nominal
Optimal strategy: Withdrawal 9%, ACE 23.9

Income 43.1 30.0 17.5 30.2 8.5
Balance 250.3 126.8 37.0 136.8 71.6

Source: Author’s simulations. Higher ACEs (average certainty equivalent consumption) indicate better out-
comes for households, for a given set of preference parameters.

138 M.J. Warshawsky / Financial Services Review 26 (2017) 113–141



decomposes the variance-covariance matrix to a triangle matrix. That is, the factorization

finds a triangle matrix W so that W�W � �̂. Second, a vector of random values are generated
according to IID N(0, 1). Multiplying this vector by the Cholesky factor matrix generates
correlated shocks to rates and returns. Third, multiplying the VAR coefficients by previous
period returns, plus the shocks, gives current period returns. The procedure is repeated
forward until the end-of-time horizon under consideration.

Following the specification in Campbell and Viceira (2005), asset classes include money
market (90-day T-bills), stocks (proxied by the S&P 500 Total Return index), and bonds
(proxied by the 10-year U.S. Government Bond Total Return index). These rates and returns
in the VAR estimation are expressed in logarithm real terms (after adjusting for inflation
measured by the change in the CPI-U index), using quarterly data. Additionally, three
forecasting variables (state variables), which help form expectations of future rates and
returns, include short-term nominal interest rate (nominal T-bills), equity dividend yield, and
the slope of the yield curve (yield spread as the difference between U.S. 10-year T-note
zero-coupon yield and the yield on 90-day T-bills). The entire system is estimated on
1962–2011 quarterly data.

Bankruptcy of insurance companies may occur. It is assumed that an insurance provider
fails with a probability of 0.15% per annum (uniform distribution), based on Moody’s global
analysis of default probability for corporate bonds rated A for 1970–2005. The size of the
loss of insurance contract value is simulated, within the empirical range of economic
contractions estimated by Barro (2006).

Appendix B: Results for additional scenarios

Table A-1 shows the optimal results for single retirees considering systematic withdrawals
alone. For the optimal fixed-dollar inflation-indexed withdrawals, the median level of real
consumption is $28,800, including $13,800 from Social Security. The median real balance
is $174,200 among survivors.

Table A-2 shows the optimal results for married retirees who are considering systematic
withdrawals alone, either fixed real dollars or a fixed percentage of balances.

Acknowledgments

I thank Professors Jing Ai, Raimond Maurer, and Anthony Webb for helpful comments
and access to unpublished empirical findings, the three anonymous reviewers for this journal,
and, especially, Gaobo Pang, for his considerable assistance and advice on this article.

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