The perfect withdrawal amount: A methodology for
creating retirement account distribution strategies

E. Dante Suareza,*, Antonio Suarezb, Daniel T. Walzc

aTrinity University, Department of Finance and Decision Sciences, San Antonio, TX 78212, USA
bIndependent Financial Advisor, Monterrey, Mexico

cTrinity University, Department of Finance and Decision Sciences, San Antonio, TX 78212, USA

Abstract

We present a new way to develop withdrawal strategies from retirement portfolios. It is derived
analytically, instead of from empirical testing, and it iterates always in the same manner. It is based
on a new measure we develop, the Perfect Withdrawal Amount, for which we discuss how to construct
a probability distribution and how to apply it sequentially. We also derive a new measure of
sequencing risk. We present new strategies built with this framework. © 2015 Academy of Financial
Services. All rights reserved.

Jel classification: J26; D81; D14

Keywords: Retirement; Optimal withdrawal; Safe withdrawal rate; Withdrawal rule; Sequencing risk

1. Introduction

The question of how much to withdraw from a savings account once an investor has
entered retirement is now more relevant than ever, as current demographic trends make it a
matter of vital importance for a growing number of people every year. The problem itself
arises because retirees generally wish to use their retirement funds to support a standard of
living that is as high as possible, but without depleting their account so quickly that the years
still ahead become difficult to finance—the so-called failure risk. On the other hand,
withdrawing “too little” money might simply translate into excessively high balances in their

* Corresponding author. Tel.: �1-210-999-7860; fax: �1-210-999-8134.
E-mail address: esuarez@trinity.edu (E. D. Suarez)

Financial Services Review 24 (2015) 331–357

1057-0810/15/$ – see front matter © 2015 Academy of Financial Services. All rights reserved.



accounts at the end of the life span horizon—the so-called surplus risk—and the “golden
years” lifestyle would have been restricted unnecessarily.

The goal for the researcher is then to develop formulas or “rules” that dictate withdrawal
amounts in each period based on the retiree’s age, the assets held in the savings account, and
the retiree’s willingness to limit consumption in exchange for future safety. There are, of
course, other factors that affect the optimality of these rules (inflation, tax implications,
covering for emergency withdrawals, etc.), but the three factors mentioned usually have the
largest impact on any recommendation.

The research thus far undertaken has produced a considerable body of knowledge. The
problem is well understood, and many of the pitfalls of the original, simpler rules have been
identified and addressed. The optimal strategies currently available are sophisticated and
address a wide range of different situations and scenarios.

Still, it is our opinion that many of these results are, in a sense, heuristic. These approaches
typically start with an idea that “makes sense” intuitively and then test it to see if improve-
ment was indeed attained. In contrast, we have tried to develop an entirely analytical
treatment of the problem, starting from the functional relationship between the relevant
variables and building up from there before producing any rules or strategies. A central
concept in this effort is a new measure that we have called the Perfect Withdrawal Amount
(PWA). Instead of developing a particular rule for withdrawing, we introduce a fundamen-
tally different way of addressing the problem.

2. Previous research

The origin of the research program on optimal withdrawals in retirement can be traced
back to Bengen’s (1994) pioneering article, in which he presents the basis for what would
come to be known as “the 4% rule.” Specifically, he demonstrates that a 4% withdrawal rate
from a retirement fund, adjusted for inflation, is generally sustainable for normal retirement
periods. A series of studies by Cooley, Hubbard, and Walz (1998, 1999, 2003, and 2011)
then strengthened this conclusion, as they report similar findings using overlapping samples
of historical stock and bond returns.

From a methodological standpoint, the distinguishing feature of these “first generation”
articles is that they rely on a constant withdrawal amount, established from the outset, which
is only adjusted to replenish its purchasing power. This has motivated attempts to develop
“adaptive” rules, aimed at improving the results by applying midcourse corrections. Guyton
and Klinger (2006) develop performance-based rules that decrease or even cancel the
inflationary adjustment when return rates are too low, and that modify the withdrawal
amount when the implied withdrawal rate falls outside ranges they prescribe. Frank, Mitch-
ell, and Blanchett (2011) use adjustment rules that depend on how much the rate of return
deviates from the historical averages. Zolt (2013) proposes curtailing the inflationary ad-
justment to the withdrawal amount to increase the portfolio’s survival rate, and produces
different “rules” by varying the degree to which purchasing power is restored.

Another take on the adaptive theme has been to reassess the situation periodically, taking
into account the shortening of the horizon period. Spitzer (2008) considers this effect and

332 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



resets the withdrawal amount every five years. Blanchett and Frank (2009) annually recal-
culates the probability of depleting the retirement funds too soon. If it becomes higher
(lower) than a set of critical values they posit, the withdrawal amount is decreased (in-
creased) by 3%; otherwise it remains constant.

An additional refinement has been to interpret the planning horizon length as a stochastic
variable instead of a parameter. Under this view, the goal for the planner is to ensure that the
funds in the retirement account “outlive” the retiree (instead of the other way around), no
matter the number of years involved. Stout and Mitchell (2006) use mortality tables to make
sure that a retirement period of uncertain length can be covered. Stout (2008) decreases the
withdrawal amount whenever the account balance falls below a measure of the present value
of the withdrawals yet to be made, and increases it when the balance is above this measure
plus an additional value reserve. Mitchell (2011) uses different thresholds to trigger adjust-
ments and also addresses the risk of superannuation.

A still more recent approach has treated the selection of withdrawal amounts as a
lifetime-utility maximization problem. Milevsky and Huang (2011) posit as the objective
function the total discounted value of the utility derived across the entire retirement period,
in a setting where this length is a stochastic variable and the subjective discount rate is a
measure of “personal impatience.” Williams and Finke (2011) use a similar model with more
realistic portfolio allocations and also consider other sources of income. Blanchett, Kowara,
and Chen (2012) measure the relative efficiency of different withdrawal strategies by
comparing the actual cash flows provided by each strategy to the flows that would have been
feasible under perfect foresight.

The methodology that we present here can also be used to derive adaptive rules and
revisiting schemes, and to perform longevity risk and utility maximization analyses. It is also
well-suited for trying out alternative distributions for the rates of return, such as those
described in Blanchett and Blanchett (2008), Pfau (2012), and Blanchett, Finke, and Pfau
(2014). All of these, however, would now be enhanced by understanding the trajectories that
the resulting strategies trace out in the PWA dimension.

3. The perfect withdrawal amount

We begin the development of our methodology by assuming a scenario where annual
withdrawals are made on the first day of the year and annual returns accrue on the last
day.1 There’s no inflation (or, alternatively, the rates of return used are in real terms) and
no taxes. This simplifies the setup without removing any crucial element in the rela-
tionship between the main variables and, as will be shown below, any additional pieces
needed to represent a real-world situation are easily added on top of this skeleton
framework.

We now posit that for any given series of annual returns there is one and only one constant
withdrawal amount that will leave the desired final balance on the account after n years (the
planning horizon). This can be verified by solving a problem that is formally equivalent to
that of finding the fixed-amount payment that will fully pay off a variable-rate loan after n
years. In other words, we rederive the traditional PMT() formula found in financial calcu-

333E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



lators, but with three amendments: (1) interest rates are not fixed but change in every period,
(2) the desired ending value is not necessarily zero, and (3) we are dealing with drawdowns
from an asset instead of payments to a liability.2

The basic relationship between account balances in consecutive periods is:

Ki�1 � (Ki � w) (1 � ri) (1)

where Ki is the balance at the beginning of year i, w is the yearly withdrawal amount, and
ri is the rate of return in year i in annual percentage. Applying Eq. (1) chain-wise over the
entire planning horizon (n years), we obtain the relation between the starting balance KS (or
K1) and the ending balance KE (or Kn):

KE � ({[(Ks � w) (1 � r1) � w] (1 � r2) � w} (1 � r3) . . . � w) (1 � rn) (2)

And we solve Eq. 2 for w to get:

w � �Ks � i�1
n (1 � ri) � KE� � �i�1

n �j�i
n (1 � rj) (3)

Eq. (3) provides the constant amount that will draw the account down to the desired final
balance if the investment account provides, for example, a 5% return in the first year, 3%
in the second year, minus 6% in the third year, and so forth, or any other particular
sequence of annual returns. This figure we call the PWA. If one were to know in advance
the sequence of returns that will come up in the planning horizon, one would compute
the PWA, withdraw that amount, and reach the desired final balance exactly and just in
time.3

To provide a concrete example of the relationship between a sequence of returns and its
corresponding PWA, we will use one of the estimation runs involved in the exercise
presented in the next section to provide a numerical illustration. Suppose that the retirement
period is just starting out and that we know in advance that in the next 30 years our
investment account will yield the returns presented in Fig. 1.

Fig. 1. Sequence of returns produced by Monte Carlo engine through random drawing from S&P 500 historical
data.

334 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



Then, assuming that we want to exhaust our account in full (no inheritance) in 30 years,
Eq. (3) indicates that each year we should withdraw exactly 7.2556 cents per dollar of initial
balance. Fig. 2 confirms.

Therefore, with $1 million as starting balance and a final balance goal of zero, $72,556 is
the Perfect Withdrawal Amount for this sequence of returns. We withdraw the same amount
every year and reach our desired final balance with no ups and downs in the income stream,
no portfolio failure, and no surplus.

In this manner we are characterizing every sequence of returns using one particular figure:
the corresponding PWA. Therefore, from here we argue that the retirement withdrawal
question is, at its core, a matter of “guessing” what the PWA will turn out to be (eventually)

Fig. 2. Arithmetical confirmation that the PWA for the sequence in Fig. 1, when operating on $1 million starting
balance and aiming for $0 ending balance, is $72,556.

335E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



for each retiree’s portfolio and objectives. The methodology presented here can then be
understood as a way to go about this guessing in a statistically sound way. Specifically, our
problem now becomes how to estimate the probability distribution for PWAs from the
probability distribution for the returns on the assets held in the retirement account. Before
moving on to explore how to derive the PWAs distribution, we should mention several
results provided by our approach even at this early stage.

First, Eq. (3) can be restated in a particularly useful way. The term �i�1
n (1 � ri) in the

numerator is simply the cumulative return over the entire retirement period, so we’ll now call
it Rn. We keep the subscript n to underscore the fact that this is the total return, not the
average, and thus it depends on the length of the planning horizon.

The denominator, in turn, can be interpreted as a measure of sequencing risk:

�i�1
n � j�i

n (1 � rj) � (1 � r1) (1 � r2) (1 � r3) . . . (1 � rn) � (1 � r2) (1 � r3) . . .

(1 � rn) � (1 � r3) (1 � r4) . . . (1 � rn) � . . . � (1 � rn�1) (1 � rn) � (1 � rn) (4)

We note that, for any given (unordered) set of rates, this expression decreases if “big” rates
show up at the beginning of the retirement period and “small” rates show up at the end,
because the last rates appear more times than the first rates in the summation.

To make the interpretation more natural, we’ll say that it is the reciprocal
�1/�i�1

n �j�i
n (1 � rj)� that captures the sequencing effect, because this item goes up when

the sequence is favorable.4 This we call Sn, and we note that two r-vectors can have the same
Rn and yet different Sn’s—a given cumulative return can come up in different ways,
order-wise.

Now Eq. (3) simplifies to:

w � (RnKS � KE) Sn (5)

This provides us with an expression that captures the impact of sequencing explicitly,
through a factor (Sn) that “hits” the withdrawal amount in a perfectly sensible way.

This reformulation is important because the sequencing issue is precisely what makes the
optimal withdrawal problem unique. In most financial analysis discussions, an understanding
of the total accrued return will suffice, but here it is of the essence to know not just how much
but when this much. In retirement, it makes a huge difference if “good” financial results come
first and “bad” ones later, instead of the other way around, because in this stage of the life
cycle the rates of return apply sequentially to an ever-dwindling capital base—the retiree is
constantly drawing down her savings to support herself.

This is also why previous studies have sought for a way to account for this “sequencing
risk” (Frank and Blanchett, 2010; Frank, Mitchell, and Blanchett, 2011). Indeed, even very
recent discussions such as Pfau (2014) continue to address this problem, developing proxy
variables to measure the correction required because of the sequencing issue. Although it is
reasonable to expect that some of these constructs will capture the general features of the
adjustment factor, our Eq. (5) comes directly from the simplest, most natural interpretation
of the problem—so Sn is not a proxy. Rather, we think Sn is an expression that should be
investigated further because it is a measure of orientation (return rates going up, going down,

336 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



up a little then down a lot, etc.), and this is the crucial element that the adjustment factor
should capture.

Second, we modify Eq. (5) to express the withdrawal rate, instead of the withdrawal
amount:

w/KS � RnSn � Sn (KE/KS) (6)

Additionally, we note how it is dependent on the ratio KE/KS. This means that to derive an
optimal withdrawal rate we need to know what fraction of the initial account balance is to
be bequeathed.

Although this might seem like a trivial adjustment to the existing strategies, which may
focus on the zero-bequest case only for the sake of clarity, the specific functional form by
which inheritance goals affect the optimal recommendation has not been established before5

(Bengen, 2006; Bernard, 2011; Spitzer, 2008). It is the proportion of the desired final balance
to the starting balance that matters and not, as might be assumed, the amount by which the
initial balance exceeds the target.

For example, a natural way to deal with the complication of a positive bequest goal would
be to “put aside” the inheritance money—effectively removing it from the analysis—and
work with the excess balance as if it were a zero-bequest case (cf. “Two-Bucket” strategy in
Bernard, 2011). However, the separated funds will accumulate returns too, and there is no
reason for these returns not to be available for consumption (at least the real part). Should
the planner separate the funds to be inherited but then plow back the real returns into the
“consumable” balance? Our approach renders these efforts unnecessary by including a
bequest term in the equations, making the no-bequest scenario just a specific instance of the
general case.

4. Construction of a PWA probability distribution

We now proceed to construct an actual probability distribution for PWAs, assuming
a 100% equity portfolio. The decision to use an all-equity scenario is not crucial because
we are only illustrating how the methodology works. This allocation was chosen because
it maximizes the variance of the returns on the account’s assets, and thus produces the
PWA distribution with the largest dispersion possible. By examining this “worst case scenario”
we can get a clearer idea of what PWA distributions look like in general, keeping in mind that
for “normal” portfolio allocations they would be more clustered than what is shown here.

To apply our model we calculated the monthly returns on the S&P500 for the period
January 1957 through April 2013 and then used a Monte Carlo engine to draw 360 values
at random from this set (one at a time, so it’s “with replacement”). These were then
interpreted as the monthly returns in a 30-year planning horizon, in the same order as they
were drawn, so the first 12 values were compounded to get the annual return for year 1, the
next 12 values became year 2, and so on.6

The engine repeated this process 20,000 times and computed the cumulative return (Rn)
and sequencing factor (Sn) for each series of returns obtained. This provided us with 20,000
(Rn,Sn) pairs, each pair standing in for a realistic vector of return rates with 30 entries.7 Fig.

337E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



3 presents the frequency distribution of the PWA formula (Eq. (5)) evaluated at each of these
20,000 (Rn,Sn) pairs, using $1 million as starting balance and with $0 as desired ending
balance. Fig. 4 is the same but with cutoff points for deciles indicated.

Some of the advantages provided by our model can be seen in these charts. For example,
the tabulations used to produce the charts can be used to locate any withdrawal amount, not
just milestone values and not only amounts in the lower end of the range. So we can readily
advise on the consequences of withdrawing, say, $62,000 every year; from the underlying
tabulation we read that 64% of the Monte Carlo runs produced PWAs higher than $62,000,
so failure risk for that withdrawal amount is 36%.

This distribution was calculated using $0 as desired ending balance, so the failure risk
figure estimates the probability of total ruin. However, if the calculations had included a
bequest target, the cumulative areas under the PWA curve would represent probabilities of
not reaching this target instead of total depletion. The retiree might be interested in separating
these two numbers, and the framework provides a way to do this: just compare the failure risk
levels, with and without bequest, at that withdrawal amount. Fig. 5 shows this graphically.

Surplus risk, in turn, can be estimated by inverting the roles of the withdrawal amount and
the ending balance as dependent/independent variables in the analysis. For the example
above, one would use the same set of (Rn,Sn) pairs to evaluate KE � RnKS � 62,000/Sn, and
then tabulate the resulting figures.8

We used this feature to estimate surplus risk in this same scenario using $43,000 as
withdrawal amount (which is approximately the cutoff point for the first decile of the PWA
distribution). As expected, close to 90% of the runs left a positive balance behind—90.9%
“didn’t fail”—but other points along the distribution of ending balances deserve mentioning.

Fig. 3. Frequency distribution of annual PWAs in 20,000 Monte Carlo runs, with the specific parameters shown
in the header. This distribution is then interpreted as a probability distribution for the optimal withdrawal amount.

338 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



Fig. 4. Same as Fig. 3, but now indicating the figures that delimit 10% areas under the curve. These would be
the “milestone” annual withdrawal amounts for the parameter values given in the header.

Fig. 5. The continuous line is Fig. 3. The broken line is the PWA distribution when the desired ending balance
is $500K. Failure risk at $62K rises from 36% in the no-bequest graph, to 46% in the $500K bequest-goal graph.
This means that the probability of a final balance between $0 and $500K is 10%.

339E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



For example, 74% of the runs ended up with more money than they began with, that is, the
final balance was larger than the starting balance (and the desired ending balance was zero).
The final balance is two times or more the starting balance in 58% of the runs. If one were
to take this “4.3% rule” at face value, a 100% equity portfolio would have a 12% probability
of ending up with 10 times or more what it had at the beginning!

Similar results have been found before and have been fully acknowledged (Cooley,
Hubbard, and Walz, 1998), but even so we feel that they call for a reassessment of the “safe
withdrawal rate” approach because of the large size of the ending balances associated with
it. We think this approach could be used as a conservative guideline to start the series of
withdrawals, but understanding that it is very likely that we will be able to increase the
withdrawal amount significantly and yet stay inside the desired failure-risk range. Precisely,
we now move on to the mechanics of how is the withdrawal amount adjusted each year using
this methodology.

5. Sequential application of the PWA formula

Per our assumptions, we make a withdrawal at the beginning of retirement year 1 and live
off this money for the entire period until, at the very end, the yield accrued during the
previous 12 months is actually credited to our account. Now our PWA distribution needs to
be recalculated using Eq. (5), with the balance actually showing as the new starting balance
and shortening the time horizon by one period. The shortening of the horizon is attained by
substituting in a new set of 20,000 (Rn,Sn) pairs, obtained with the same procedure described
in the previous section, but this time drawing return rate sequences that are shorter than before
by 12 data (they represent horizons with one year less to go). From this new distribution we
choose the withdrawal amount for year 2, presumably—but not necessarily—using the same
risk-tolerance profile as in year 1. Furthermore, this process is simply repeated every year.

We must stress that in PWA the process by which the withdrawal amount is selected
is always the same, but this does not mean that the withdrawal amount itself will not
change. Actually, it is rare for a withdrawal strategy built with this framework to
recommend constant withdrawals beyond a small number of periods. PWA incorporates
all new information into the set of data available for the next analyses, and this updating
will create adjustment pressures that, in all likelihood, will end up modifying the
withdrawal amount at some point.

Also, although a year has gone by, the planning horizon may be adjusted differently. The
framework allows the retiree to judge whether the expected number of years still ahead for her
has indeed decreased by one. If, for example, the retiree has stayed particularly healthy, she may
decide to use the same horizon length once more. Or she may even use a longer horizon, if she
has just overcome serious illness or made beneficial lifestyle changes. And, sadly, there will be
opposite cases where the appropriate length reduction is larger than just one period.

However, other than these discretionary length adjustments, which can be considered part
of the information update process, the PWA approach is consistent. It calls for the application
of the same procedure to produce the withdrawal menu every year, without triggering
different schemes if certain conditions are met.

340 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



This consistency allows us to compute the optimal withdrawal rates in the no-bequest case
for the entire length of the planning horizon, and for all confidence ranges, at the same time.
If no bequest is being sought, Eq. (6) becomes:

w/KS � RnSn (7)

Fig. 6 presents the results of evaluating this equation with the sets of (Rn,Sn) pairs corre-
sponding to different horizon lengths (obtained as described above) for the 100% equity case
considered here.

For example, let us consider a retiree that starts out with $1 million, has no desire to leave
behind any money, and thinks that 30 years is a reasonable length for her planning horizon. She
has assets indexed to the S&P 500 and wants to be 90% certain that she will not have to lower
her withdrawal amount later on. Our subject only needs to read from the chart to obtain the
corresponding rate. She does this and, following its advice, pulls out $43,000 (4.3%). Her first
year turns out badly; the stock market goes down 10%, so her remaining balance of $957,000
(after withdrawing the $43K) shrinks to $861,000. Should she withdraw $43,000 again in the
second year? Well, $43K represents 5.0% of her now-current balance and the chart tells us that,
with 29 years remaining, that would take her out of the 90% confidence range and into the
80–90% region. Those are the factors that need to be considered in the decision—nothing else
matters.

Fig. 6. Milestone values in the optimal withdrawal rate distribution for the all-equity, no-bequest case. The rates
are with respect to the actual balance faced each year, not the balance at the beginning of the retirement period.
If, for example, the investor wishes to be 90% safe, each year she should run down the last row and take out the
corresponding fraction of her balance at that point, whatever it may be.

341E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



We make this point to highlight the simplicity of our approach, which cuts through the
Gordian knot of some of the adaptive rules found in the previous literature (Bernard, 2011;
Blanchett and Frank, 2009; Guyton, 2004; Mitchell, 2011; Pye, 2000; Robinson, 2007; Stout
and Mitchell, 2006). For example, in Guyton and Klinger (2006) we find that “withdrawals
are to increase from year to year to make up for inflation, except that there is no increase after
a year where the portfolio’s total return is negative and when that year’s withdrawal rate
would be greater than the initial withdrawal rate” (p. 5), or “when a current year’s withdrawal
rate has risen more than 20% above the initial withdrawal rate, the current year’s withdrawal
is reduced by 10%; this rule expires 15 years before the maximum age to which the retiree
wishes to plan” (p. 7). One can find more rules like these in other studies.

The PWA approach would call this type of adjustments into question, claiming instead
that the statistically appropriate thing to do is to calculate the probability distribution of
PWAs for the current situation and choose from there—every year. All the relevant
information currently available is embedded in that distribution and all new information is
captured by the way the distribution changes as time elapses. An adaptive rule is simply a
movement into different confidence ranges of the corresponding distribution and this, unless
properly understood and consciously chosen, is an arbitrary call.

It must be stressed that these simplified results apply only if no bequest is being sought.
When the retiree wishes to leave a certain amount behind, the withdrawal rate is given again
by Eq. (6), which is the general-case expression:

w/KS � RnSn � Sn (KE/KS) (6)

This can be read as “the withdrawal rate for the no-bequest case, reduced by the sequencing
factor times the fraction of the current balance that is to be bequeathed.” The performance
of the investment portfolio in previous years then comes back into play, but it is fully
captured by the balance actually showing in each period, which is relevant only insofar as it
affects the bequest-to-balance ratio.

6. What confidence ranges really tell us

At this point it becomes important to clarify what is meant by “confidence range” in this
framework. Because the framework is adaptive, now an expression such as “90% safe” no
longer means that there is a 90% chance that the portfolio will not run out of funds. What
it means is that there is a 90% chance that you won’t have to lower your withdrawal amount
in the future, to achieve your bequest target.

We think this interpretation is more informative for the investor because normally in a real
life situation a strategy will not be allowed to fail—the withdrawal amount will be reduced
before that happens. So the truly pressing question is then how much would the withdrawal
amount need to be lowered if we go into the red zone. The model can produce these
quantitative assessments by computing and analyzing alternate PWA distributions, which are
relatively easy to obtain because they can be derived from the original distribution.

For example, the distribution of the PWA after withdrawing an arbitrary amount w* this
year can be obtained using a modified version of Eq. (3):9

342 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



w� � �Rn (KS � w*) � KE� � �i�2
n �j�i

n (1 � rj) (8)

where w’ is the new (modified) PWA for the subsequent years, w* is the withdrawal amount
being considered for this year, and we leave the expression in the denominator in its explicit
form to note that the summation now starts at period 2. We see that the denominator is now
“missing” the product of all the returns from year 1 to year n (the summation starts with the

product of returns from year 2 to year n), which is Rn, so we can express it as
1

Sn
� Rn. And

we now rewrite Eq. (8) as:

w� � �Rn (KS � w*) � KE� Sn/(1 � RnSn) (9)

Finally, we can identify some of these terms as our original PWA, w � (RnKS � KE) Sn, to
see that the modified PWA is a linear transformation of the original:

w� � (1 � RnSn)�1 w � RnSn (1 � RnSn)�1 w* (10)

We can use this expression to compute different PWA distributions using the set of (Rn,Sn)
pairs that we already have. Fig. 7 presents two of these alternate distributions, using
w*�$20,000 and w*�$100,000.

These alternate distributions are then the answer that the framework provides to our
pressing question above. Where most other approaches simply provide the change in the
failure-risk figure, here we can compute the withdrawal amounts corresponding to different

Fig. 7. PWA distributions immediately after making the first withdrawal, but before the first yield accrues. The
thicker line is Fig. 3 (the original distribution). These charts answer the question “what happens if I withdraw x
dollars this year,” and are the PWA alternative to failure-risk figures.

343E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



safety levels, with one year less to go, after making a withdrawal of a given amount in the
current period. From the point of view of the retiree, this is the probability that she will have
to settle for a future withdrawal of x dollars or less, after making a withdrawal of y dollars
in the current period. Fig. 8, which is also made with Eq. (10) but using a table form instead
of a chart, shows these results.10

These are examples of a more general feature made available by the PWA framework: the
possibility of optimizing withdrawal paths meeting a specific profile request, and not just
constant streams.11 This can be achieved by introducing the “shape” of the request into the
optimization process.

Consider an investor that is already in retirement but has not yet reached her Social
Security full retirement age. She might prefer to avoid reductions in her benefits by delaying
the time of claiming and use her investment account as her sole source of income for a
number of years. This would then require a two-stage withdrawal plan, taking out relatively
high amounts at the beginning and then smaller amounts after the benefits start to come in.

Fig. 8. Probability that the PWA will drop below a certain amount after a withdrawal of a given size is made in
year 1. Assumptions as in Fig. 7 header. For example, the highlighted cell shows that if a withdrawal amount of
$40,000 (4%) is acceptable for the retiree, then she can take out $110,000 in year 1 and still be “90% safe” with
respect to staying clear of this self-defined “red zone.”

344 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



A plan like this can be optimized by rederiving Eq. (3), with the withdrawals after the switch
year T equal to the first years’ withdrawals minus the benefits B, so that w1 � w2 � … �
wT�1 � wT � B � wT�1 � B � … � wn � B. This scheme produces withdrawal
recommendations aimed at providing constant total income throughout; the path of with-
drawals from the investment account would have to be “kinked,” but this can be handled by
the framework without much difficulty.

We close this section by discussing another feature related to confidence levels which
enhances the model’s flexibility even more. When the retirement plan includes a bequest,
the model provides the investor with an additional adjustment lever: the value of that
bequest.

By Eq. (6), the safety level of a given withdrawal amount depends in part on the
relation between the current balance and the desired ending balance. This means that by
modifying the bequest goal we can attain different confidence levels without changing
the withdrawal amount. For example, with $1 million starting balance and a 30-year
planning horizon, if the retiree intends to bequeath $500,000 the 90% confidence level
is at $34,600. But if she decided to lower that goal to $400,000 the 90%-safe withdrawal
amount would now be $36,157—a 4.5% increase. Settling for an inheritance of just
$200,000 increases this amount an additional 10.4%, to $39,921. Doing away with the
bequest goal altogether takes us to Fig. 4, where we see that the 90%-safe withdrawal
amount is then $43,316. Fig. 9 shows these bequest-adjustment profiles for several
confidence levels.

Of course, modifying the bequest goal is also an option for midcourse corrections, and it
can be combined with other adjustment levers. For example, if the account gets high rates of
return for a number of years, the retiree may decide to split the benefits of the good run by
increasing the withdrawal amount, the confidence level, and the bequest goal. The PWA
framework will provide the relevant terms-of-trade.

Fig. 9. Withdrawal amounts corresponding to different confidence levels for an all-equity portfolio with $1
M starting balance and 30-year horizon. The amounts depend on the investor’s intended bequest. The
highlighted cells show that an investor who wants an annual income close to $52K and wishes to bequest
$500K will have to accept 70% safety. However, if she does away with her bequest intentions her safety
level rises to 80%.

345E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



7. Development and testing of a withdrawal rule inside the PWA framework

Once more, the purpose of this article is not to derive a particular withdrawal strategy but
to present new tools for constructing and assessing strategies in general. Still, by way of
illustration we now present strategies built with this toolkit.

Suppose that a major concern for a strategy under consideration is whether the withdrawal
paths obtained from it will be too “jumpy”—that is, we are worried that the recommended
withdrawal amount might turn out to be very different between one year and the next. This
may arise if, for example, the strategy demands us to stay inside a confidence range that is
very narrow, so that the withdrawal amount has to keep moving up and down to fall back into
the mandated confidence range.

We may address this concern by proposing a two-tiered strategy that starts by selecting a
certain amount of failure risk, but then “tolerates” some degree of variation around this value
for the sake of steadiness in the income flow. Suppose we choose 50% as the acceptable
initial value of failure risk. Anything “safer” than that is not agreeable to the retiree because
she feels it creates too much surplus risk—she has no heirs and wants to consume her savings
down to the last penny. We begin by withdrawing the median amount in the 30-year PWA
distribution for an all-equity portfolio (because that is what she happens to have).

After making this first withdrawal and getting the return in our account at the end of the
year, the withdrawal amount for year 2 is chosen as follows. If the amount withdrawn in the
previous year is still inside the 15%-to-85% confidence range (a rather wide 70-point
interval) once the PWA distribution is recalculated starting from the now-current balance
and with 1 year less remaining in the planning horizon, that same amount is withdrawn once
more. Otherwise the strategy is “recentered” by taking the median value of the recalculated
distribution. This process is then repeated every year.

After setting up our Monte Carlo engine with these parameters, we have run simulations
of the withdrawal paths resulting from this strategy using the now customary $1 million
starting balance and $0 desired ending balance. Figs. 10, 11, and 12 present three of these
simulated paths, each one corresponding to return sequences that give rise to (somewhat) flat,
rising, and decreasing withdrawal profiles.12

Barring perfect foresight, nothing strictly better can be provided for a retiree with these
demands and holding these assets. If we want to improve the results of the strategy along
either one of the three relevant dimensions (safety, income level, stability), we would have
to change either the base safety level, the recentering point or the width of the tolerance
range.

This last point is an important realization about the nature of the problem at hand. By the
structural relationship between the variables, when the parameters of a strategy are modified
to enhance one specific property, it is inevitable for one or more of the other properties to
suffer. This in turn means that the design of optimal withdrawal strategies is not a search for
the dominant scheme; it is a search for the strategy that best suits the needs of the investor
being served.

The next section closes the discussion by looking more closely at this “planner-as-tailor”
aspect of the framework.

346 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



8. A tale of three strategies

We will now use the sequence of returns shown in Fig. 1 to present three different
strategies, once again starting from $1 million and ending at $0. As can be seen in Fig. 1, the
dispersion in that sequence is significant (standard deviation 14.5 points) and it is quite
jumpy, with the values ranging from �12.0% in year 29 to 44.2% in year 8. Therefore, this
particular sequence may be considered a worthy challenge for any strategy to handle, even
though it’s rather “normal” (Fig. 2 informs us that its PWA in this case is $72,556; Figs. 3
and 4 shows this amount is quite close to the median value, and in the peak region of the
chart).

The purpose of this exercise is not to establish which strategy is better. The results shown
represent not just the strategy followed, but the specific return sequence used as well. To
compare strategies one would have to average their respective performance statistics over a
large number of randomized runs to determine the frequency with which they produce
“good” or “bad” outcomes, where good and bad are defined by the retiree.

One can compare the situation faced by the retiree to that of a shopper in a clothing
store right after a big sale. The items in the tables and bins are still misplaced because
a horde of customers sorted through them haphazardly the day before. But the most
efficient way to find an item of her size and liking is, still, to look in the places where
it would normally be; there is a higher probability of finding it there than elsewhere. The

Fig. 10. This strategy kicks off by withdrawing the median value in the 30-year PWA distribution. Then it
becomes “sticky,” withdrawing the same amount unless it is “too safe” (more than 85% confidence) or “too risky”
(less than 15% confidence). When this happens, the strategy again takes the median value of the PWA distribution
applicable at that point in time. The PWA for this particular sequence of returns was $84,792 which, by Fig. 4,
is somewhat on the high side.

347E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



PWA framework would then be the knowledgeable clerk who will ask you what you
want and then will tell you where it is more likely that you may find it. Following this
metaphor, this section is a sampler of some of the “items carried” in an all-equity
portfolio facing a 30-year planning horizon.

The first strategy considered is the 4% rule, which withdraws $40,000 every year and is
presented in Fig. 13. We already know that the PWA for this sequence is $72,556, and
therefore that this strategy will be too cautious and end up with surplus balance. The balance
reaches $3,013,737 in year 30, which shows in the figure as the convergence point of the
tolerance range boundaries (only reference points now). These boundaries always come
together at the end because in this framework there is only one possible recommendation at
that point: take out all you have, because this is the last year. After continuing to withdraw
$40,000 (one last time) and accruing the final return of 9.3%, the account bequeaths
$3,250,295.

We see that the overaccumulation trend is evident early on, as the confidence level reaches
98% in year 3 after starting out at 93%—perhaps overly high even from the outset. By year
15 the confidence level goes “off the charts,” as there is not a single run in our simulation
for that horizon length that produces a PWA of $40,000 or lower starting from $1,832,363
(the balance in year 15). These confidence levels “above 100%” persist until the end of the
retirement period.

The second strategy is the one we used in the previous section, which we now show in Fig.
14 as “Sticky Median with 70-point Tolerance and Recentering.” As discussed above, this
strategy uses the central value (the median) in the 30-year PWA distribution as the initial

Fig. 11. Same as Fig. 10, but this time the sequence of returns to which the strategy was applied had a PWA of
$118,498. Fig. 4 shows this is unusually high, so the strategy is “surprised” and reacts by increasing the
withdrawal amount markedly and frequently.

348 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



withdrawal amount. Then it focuses on stability and withdraws the same amount every year,
but only if it is “reasonable” to do so. When taking out the same amount implies falling
outside the 70-point confidence interval centered at 50% (between 85% and 15%), we adjust
the withdrawal and go back to the center of the current PWA distribution. Therefore, the
strategy effectively starts over or “recenters” when it deems that it is no longer prudent to
favor steadiness over safety (failure-wise or surplus-wise).

The median of the 30-year PWA chart is $70,465 (see Fig. 4), so the strategy kicks off
with that, and then it keeps the amount steady for a number of years. We see how the upper
bound gets close to the withdrawal amount in year 7, but the strategy stays the course because
the confidence level is 25% (low, but still allowed). The opposite happens in year 20, when
the lower bound closes in, but again the reins are held tight because the confidence level only
reaches 75%. The tolerance range is not “pierced” until year 23, when holding on to the
previous withdrawal amount would put the account at 91% confidence—an unacceptably
high risk of overaccumulation. We use the balance at that point ($615,189) to compute the
PWA distribution with 8 years to go (year 23 is only starting out), and take out the median
value: $94,648. This new withdrawal amount is almost decreased only two years later, when
the confidence level drops to 16%, but the tolerance range is only grazed and the strategy
holds steady.

The balance at the beginning of year 30 is $71,908, which is taken out in full as final
withdrawal and creates the steep drop at the end of the chart (�24%). The drop is of course
caused by the terrible return rate in year 29 (�12.0%), but in a sense it can also be attributed
to the high volatility of this portfolio allocation. Still, this final amount is higher than what

Fig. 12. Same as Figs. 10 and 11, but here the sequence of returns had a PWA of $53,691. Per Fig. 4, this value
is in the lower half of the distribution, so the strategy is unable to support the median withdrawal amount and
adjusts to a lower level.

349E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



the retiree got during the first two decades and is also higher than what would have been
provided by the 4% rule.

The third strategy is a variation of the second one. It is somewhat exotic, as we modify
the three main parameters of Sticky Median to illustrate the flexibility in strategy design
provided by the PWA framework. This version kicks off by withdrawing the amount at
the 75% safety level in the PWA distribution, so it is more conservative than Sticky
Median, which starts at 50%. Then we narrow the tolerance range around the base safety
level to 30 points, so it now spans the interval from 60% to 90%. This means it stays
closer to the level of risk with which the retiree feels most comfortable (compared with
Sticky Median), but it demands more frequent adjustments to the withdrawal amount
because it allows less “straying.” Therefore, the third change is in how are the adjust-
ments applied; instead of going back to the original safety level (75%), this time the
strategy “pushes” the withdrawal amount (up or down) by the minimum amount neces-
sary to get back into the tolerance range. This adjustment procedure is more subtle than
recentering, but it asks for the retiree to live at the very edge of his risk preference in
some years. These changes produce a peculiar but very reasonable strategy, which we
have called “Sticky First Quartile with 30 point Tolerance and Nudging” and is shown
in Fig. 15.13

The most noticeable change in the resulting withdrawal path is its upward trend, which is
a result of the higher safety level used. The initial withdrawal amount is $55,025 (75% safe)

Fig. 13. The 4% rule is used with the sequence of returns in Fig. 1. As the planning horizon shortens and the
account balance rises, it becomes less likely that $40,000 will end up being the PWA. This increases the
confidence level, even though the withdrawal amount is always the same. After 30 years, the account bequeaths
more than $3 million.

350 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



and from there the lower bound of the tolerance range keeps nudging the amount higher. The
only downward adjustment comes at the very end, and is very slight. After withdrawing
$199,620 in year 29 and taking the decrease of �12.0%, the balance reaches $199,197; this
is the last withdrawal.

These three sets of results shed some light on the consequences of pursuing different
goals. An all-out desire to provide withdrawal stability would follow something akin to the
4% rule, even if it means bequeathing an amount much larger than what was intended. An
approach that attempts to maximize income without going out on a limb would prefer
something like the Sticky Median strategy, even if the withdrawal amount can suffer major
jolts in either direction. If the primary concern is safety, and cash flow steadiness is not
essential, a strategy similar to Sticky First Quartile may be in order even though it is likely
to result in a markedly rising withdrawal path. Fig. 16 superimposes the three withdrawal
paths for a more direct comparison.

Again, this exercise must not be taken as evidence in favor or against either strategy. If
instead of the sequence in Fig. 1 we had used the one underlying Fig. 12 (the “decreasing”
path), the interpretation might have been different. Fig. 17 shows the values in that “bad”
sequence, which has a PWA of just $53,691 starting from $1 million and aiming for $0. Fig.
18 is the chart of the resulting withdrawal paths, which may be compared with Fig. 16 for
contrast.

Fig. 14. Same as Fig. 13, but now the strategy followed is “Sticky Median with 70-point Tolerance and
Re-Centering.” The withdrawal amount is much higher than with the 4% rule and yet retains some degree of
stability. Here the entire remaining balance is withdrawn in year 30 (no bequest is desired), but this final
withdrawal is still “inside the ballpark” set by the previous amounts.

351E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



9. Conclusions and next steps

A sequence of return rates, together with a starting balance and a desired ending
balance, determines a “perfect” withdrawal amount. If the retiree’s goal is to obtain a
constant stream of income from her investment account, this is the amount she should
take out in every period. If she withdraws more than the PWA, she will run out of funds
(or leave behind less money than she intended). If she withdraws less, she will leave
money on the table.

The problem, of course, is that we do not know in advance what the PWA will turn out
to be in each case. To make inferences and take decisions one must construct probability
distributions, and the methodology presented here is a way to do this soundly. Here we use
the historic distribution of return rates as the probability distribution of future returns,
assuming it to be independent and identical in all periods. However, any other assumption
may be used, as our framework only discusses how to “process” the assumptions (or
expectations) held by the researcher.

At this stage, we can only make preliminary recommendations. However, one that
seems to take shape intuitively is a major departure from conventional wisdom: the best
level of failure risk is 50%. To be more precise, we should use the mode of the PWA
distribution— because that is the most likely value that the PWA will end up taking in
our particular case. For symmetric or moderately skewed distributions, failure risk at the
mode will be close to 50%. If the assumptions used for the distribution of return rates

Fig. 15. Same as Figs. 13 and 14, but now the strategy followed is “Sticky First Quartile with 30-point Tolerance
and Nudging.” Compared with Fig, 14, this strategy is more cautious (it drifts around 75% safety instead of 50%),
more strict (tolerance 30 points instead of 70), and more subtle (upon piercing the tolerance the amount is nudged
back in instead of recentered).

352 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



are sound, then close to half of the retirees who follow this policy will be able to keep
withdrawing this amount (or more!) for the entire retirement period. For the other half,
the procedure outlined here will steer them clear of funds’ depletion or bequest collapse
through timely warnings that the amount needs to be decreased—and probably not by

Fig. 16. Strategy comparison. The withdrawal paths in Figs. 13, 14, and 15 are superimposed, with the tolerance
range lines removed for clarity. The 4% rule provides a low cash flow and leaves a sizable bequest, so the standard
of living in retirement is overly restricted. Sticky Median provides more money, but failure risk reaches very high
levels along the way. Sticky First Quartile plays it safer, but at the expense of frequent adjustments to the amount
withdrawn.

Fig. 17. Another random sequence of returns produced by Monte Carlo engine. This is the sequence that gives
rise to Fig. 12.

353E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



much. If ours is indeed the case when the sequence of returns awaiting in our future is
a terrible one, then the procedure will keep signaling for downward adjustments, taking
the withdrawal path asymptotically to a very low value, but probably not too far below
what it would have been under perfect foresight.

The obvious next step for the framework’s future development is to use it to derive new
“standard” rules—à la Bengen’s 4%—for different portfolio compositions and retirement
objectives, and to look at the rules currently in favor through the lens of PWA. Also, it would
be useful to add longevity risk into the framework by making the planning horizon a
stochastic variable instead of a parameter. And it would seem possible to adapt the model to
the accumulation phase of the life-cycle, deriving a Perfect Contribution Amount concept
along the lines of PWA.

Once we know more about the PWA for different portfolio compositions, we can proceed
to the derivation of optimal adaptive strategies that produce specific results along any of the
relevant dimensions. Here we would be interested in exploring “gliding” strategies that
change the asset-mix over time, as well.

Finally, we see an opportunity to improve the computation of the PWA probability
distributions by using more advanced methods to construct the probability profiles for the
financial assets’ returns. The bootstrapping procedure used here is very practical, but we are
aware of its numerous drawbacks.

Fig. 18. Comparison of the same three strategies as in Fig. 16, but now the withdrawal paths are the ones produced
by the sequence in Fig. 17 instead of Fig. 1. The 4% rule is closer to the other two in this case, but it still results
in a significant bequest of $735,059 when none was sought.

354 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



Notes

1 Although not entirely realistic, the assumption of annual withdrawals has been the
norm in withdrawal rate studies since the “founding” of the field by Bengen (1994).

2 The fact that this is a mirror image of the paying-off-a-loan situation means that the
variable “r” in the typical formula is now a return rate instead of an interest rate,
the ending value is available capital instead of unpaid balance, and the result of the
formula is a payment from the account instead of a payment into the loan.

3 Blanchett, Kowara, and Chen (2012) presents a measure similar to PWA called
Sustainable Spending Rate (SSR). Our PWA was developed independently, but it
turns out to be a generalization of SSR, SSR being the PWA when the starting balance
is $1 and the desired ending balance is zero.

4 The sequencing factor formula is the same as the sinking fund factor under variable
rates of return, but the interpretation and application is thoroughly different in each
case.

5 Actually, considering only formal, mathematically explicit models, the authors were
unable to find a single study where the desired bequest is an input in the calculations.

6 Similar “bootstrapping” procedures are used, for example, in Blanchett (2007);
Blanchett and Frank (2009); Spitzer (2008); Spitzer, Strieter, and Singh (2007); and
Zolt (2013).

7 In the previous section we said that we characterize each sequence of returns by its
corresponding PWA, but this is only an intuitive description. The formally correct
statement is that we characterize each sequence by its corresponding (Rn,Sn) pair. To
obtain an actual PWA for a specific sequence of returns, the current balance and the
bequest target must be provided.

8 For the (Rn,Sn) pairs that produce a negative ending balance, the numerical value of
this formula would be wrong because the yields cease to apply once the balance moves
into the red. But the tabulation can show the size distribution of the positive ending
balances, with the rest marked as “failures.”

9 Eq. (8) is obtained simply by changing the standpoint of the evaluation: from before
the current year’s withdrawal is made, to after it has been made but before the current
year’s return hits the balance. In other words, we perform the evaluation assuming we
are in the middle of the year, instead of at the very beginning. This changes the starting
balance from Ks to Ks-w*, but leaves the sequence of yield accruals unchanged.

10 We thank an anonymous reviewer for comments that led to the development of the
tables in Figs. 8 and 9.

11 Approaching the problem from a different angle, Robinson and Tahani (2010) also
produce non-constant withdrawal profiles.

12 The three return sequences involved in the figures were produced randomly by the
Monte Carlo engine, but they are not equally likely. We selected them intentionally,
out of the set of 20,000 runs, to illustrate different possibilities for the resulting
withdrawal path.

13 This strategy developed from suggestions by an anonymous reviewer, which we
thankfully acknowledge.

355E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



Acknowledgment

We thank David Blanchett, Wade Pfau, and conference participants at the 2014 Annual
Meeting of the Academy of Financial Services at Nashville, for helpful comments to an
earlier version of this article.

References

Bengen, W. P. (1994). Determining withdrawal rates using historical data. Journal of Financial Planning, 7,
171–180.

Bengen, W. P. (2006). Baking a withdrawal plan ‘layer cake’ for your retirement clients. Journal of Financial
Planning, 19, 44–51.

Bernard, G. A. (2011). Systematic withdrawal programs: unsafe at any speed. Journal of Financial Service
Professionals, 65, 44–61.

Blanchett, D. M. (2007). Dynamic allocation strategies for distribution portfolios: determining the optimal
distribution glide path. Journal of Financial Planning, 20, 68–81.

Blanchett, D. M., & Blanchett, B. C. (2008). Joint life expectancy and the retirement distribution period. Journal
of Financial Planning, 21, 54–60.

Blanchett, D., Finke, M., & Pfau, W. D. (2014). Asset valuations and safe portfolio withdrawal rates. The
Retirement Management Journal, 4, 23–36.

Blanchett, D. M., & Frank, L. R. (2009). A dynamic and adaptive approach to distribution planning and
monitoring. Journal of Financial Planning, 22, 52–66.

Blanchett, D. M., Kowara, M., & Chen, P. (2012). Optimal withdrawal strategy for retirement income portfolios.
The Retirement Management Journal, 2, 7–20.

Cooley, P. L., Hubbard, C. M., & Walz, D. T. (1998). Retirement savings: choosing a withdrawal rate that is
sustainable. Journal of the American Association of Individual Investors, 20, 16–21.

Cooley, P. L., Hubbard, C. M., & Walz, D. T. (1999). Sustainable withdrawal rates from your retirement
portfolio. Financial Counseling and Planning, 10, 39–47.

Cooley, P. L., Hubbard, C. M., & Walz, D. T. (2003). A comparative analysis of retirement portfolio success
rates: simulation versus overlapping periods. Financial Services Review, 12, 115–129.

Cooley, P. L., Hubbard, C. M., & Walz, D. T. (2011). Portfolio success rates: where to draw the line. Journal
of Financial Planning, 24, 48–60.

Frank Sr., L. R., & Blanchett, D. M. (2010). The dynamic implications of sequence risk on a distribution portfolio.
Journal of Financial Planning, 23, 52–61.

Frank Sr., L. R., Mitchell, J. B., & Blanchett, D. M. (2011). Probability of failure-based decision rules to manage
sequence risk in retirement. Journal of Financial Planning, 24, 46–55.

Guyton, J. T. (2004). Decision rules and portfolio management for retirees: is the ‘safe’ initial withdrawal rate
too safe? Journal of Financial Planning, 17, 54–62.

Guyton, J. T., & Klinger, W. J. (2006). Decision rules and maximum initial withdrawal rates. Journal of Financial
Planning, 19, 49–57.

Milevsky, M., & Huang, H. (2011). Spending retirement on planet Vulcan: the impact of longevity risk aversion
on optimal withdrawal rates. Risk Management, 23, 24–38.

Mitchell, J. B. (2011). Retirement withdrawals: preventive reductions and risk management. Financial Services
Review, 20, 45–59.

Pfau, W. D. (2012). Capital market expectations, asset allocation and safe withdrawal rates. Journal of Financial
Planning, 25, 36–43.

Pfau, W. D. (2014). New research on how to choose portfolio return assumptions. Advisor Perspectives, 8, 1–7.
Pye, G. B. (2000). Sustainable investment withdrawals. Journal of Portfolio Management, 26, 73–83.

356 E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357



Robinson, C. D. (2007). A phased-income approach to retirement withdrawals: a new paradigm for a more
affluent retirement. Journal of Financial Planning, 20, 44–56.

Robinson, C., & Tahani, N. (2010). Sustainable retirement income for the socialite, the gardener, and the
uninsured. Financial Services Review, 19, 187–202.

Spitzer, J. J. (2008). Retirement withdrawals: an analysis of the benefits of periodic ‘midcourse’ adjustments.
Financial Services Review, 17, 17–29.

Spitzer, J. J., Strieter, J. C., & Singh, S. (2007). Guidelines for withdrawal rates and portfolio safety during
retirement. Journal of Financial Planning, 20, 52–59.

Stout, R. G. (2008). Stochastic optimization of retirement portfolio asset allocations and withdrawals. Financial
Services Review, 17, 1–15.

Stout, R. G., & Mitchell, J. B. (2006). Dynamic retirement withdrawal planning. Financial Services Review, 15,
117–131.

Williams, D., & Finke, M. S. (2011). Determining optimal withdrawal rates: an economic approach. The
Retirement Management Journal, 1, 35–46.

Zolt, D. M. (2013). Achieving a higher safe withdrawal rate with the target percentage adjustment. Journal of
Financial Planning, 26, 51–59.

357E.D. Suarez et al. / Financial Services Review 24 (2015) 331–357