The perfect withdrawal amount over the historical record

E. Dante Suareza,*

aDepartment of Finance and Decision Sciences, Trinity University School of Business, One Trinity Place,

San Antonio, TX 78212, USA

Abstract

What has been the perfect withdrawal amount (PWA) from retirement savings accounts in long-

term historical data? The PWA is that which, if taken out in the first year of retirement and used again

every year adjusted by inflation, leaves exactly the desired final balance on the account. We present the

formula for obtaining this measure and evaluate the values it has taken in the past under varying combi-

nations of the relevant parameters. We find that safety-minded investors should enter retirement with a

higher stock allocation than what is currently used in most investment funds designed to provide

income during retirement. © 2020 Academy of Financial Services. All rights reserved.

JEL classification: J26; D14; G11

Keywords: Retirement income; Perfect withdrawal amount; Sequencing risk; Sustainable withdrawal rate;

Distribution strategies

1. Introduction

A considerable body of research has explored the question of what is the safe level of

withdrawals from a savings account that must provide income during retirement. A frequent

objective of these studies is to examine the historical record to determine the withdrawal fig-

ure that has “rarely” depleted the funds before the end of the retirement period. However, if

the withdrawal amount chosen does not exhaust the funds in the account, it means that it left

a positive balance at the end. If the retiree had not planned to leave a bequest or posthumous

gift—or if the final balance is larger than what she intended to leave behind—this is money

*Corresponding author. Tel.: +1-210-999-7860; fax: +1-210-999-8134.

E-mail address: esuarez@trinity.edu

1057-0810/20/$ – see front matter © 2020 Academy of Financial Services. All rights reserved.

Financial Services Review 28 (2020) 96–132



that she would rather have consumed herself. In these cases, it can be said that the amount

withdrawn was not perfect; it should have been higher.

Here we use results presented in Suarez, Suarez, and Walz (SSW; 2015) that show that

for any given series of return rates on a portfolio, there is one and only one constant with-

drawal amount that will take the starting balance to the desired ending balance in a given

span of time. If this time span is the number of years for which the retiree wishes to plan,

then this is the perfect withdrawal amount (PWA) for her: it’s constant (no ups-and-downs

in the income stream) and it attains the final goal exactly and just in time (no portfolio “fail-

ure,” or inheritance shortfall, or overaccumulation at the end). If she withdraws more than

her PWA, she will run out of funds—or at least leave behind less money than she intended.

If she withdraws less, she will leave money on the table.

It is important to stress that the PWA is a measure inherent in any sequence of return rates.

A sequence of returns has a PWA just as it has a mean, a standard deviation, or any other con-

ventional statistical measure, and by calculating these values we are not endorsing any particu-

lar viewpoint on retirement withdrawals. Rather, we intend for this paper to be of the same

nature as Ibbotson and Sinquefield’s seminal 1976 article (Ibbotson, R. G., & Sinquefield, R.

A., 1976), where long-run historical rates of return were first presented. We argue that the

retirement withdrawal question can be understood as the search for what the PWA will turn out

to be (eventually) for each retiree’s portfolio and objectives. And a first step in this direction

should be a revision of the values that this measure has taken, historically, for different values

of the relevant parameters. In this sense, the results presented here can significantly assist the

decision-making process in retirement.

We also refer to another result presented in SSW, namely that if the retiree is inter-

ested in using up her retirement savings in stages—for example, withdrawing a certain

amount for a number of periods and then a different, smaller amount as social security

benefits start to come in—the “perfect” set of figures is a function of the “regular” PWA

that is reported here, which is a constant amount throughout the retirement period

(Bridges & Choudury, 2007). The numbers presented here can thus provide guidelines

for many different circumstances and cases.

We present results for the PWA over the period 1926-2014. Using data from Ibbotson

SBBI Classic Yearbook (Morningstar, Inc., 2015), we start by computing PWAs for a base-

case scenario of all-stock portfolios and 30-year long retirement periods, where the invest-

or’s goal is to exhaust the account’s funds completely. The retirement periods are rolled for-

ward one month at a time, with the first one beginning in January 1926 and ending in

December 1955, and the last one running from January 1985 through December 2014. This

one-month-shift approach produces a series of 708 figures that, although not all independent,

provide a bird’s-eye view of how this particular PWA has evolved over time.

We also use a separation result obtained in SSW to look at how the PWA is affected by

the sequential order in the portfolio’s returns. This is known as sequencing risk, and we

examine it by computing the values for the sequencing factor—a measure developed in that

paper for this type of risk—over the same period.

We then compute the values corresponding to other asset allocations for the portfolio. By

increasing the bond allocation in 10 percentage-point increments—and decreasing the correspond-

ing allocation to stocks—the relation between asset allocation and historical PWAs is explored.

E.D. Suarez / Financial Services Review 28 (2020) 96–132 97



Next, we look at how our results change when the retiree’s goal is not to exhaust her savings

entirely, but to leave a portion of her balance as bequest or final gift. This is perhaps a more rele-

vant case, especially since the protected balance can also be considered an emergency reserve

for unforeseen expenditures and, in this sense, it might be a desirable feature in most retirement

plans. This interpretation of a nonzero final balance goal as a safety device suggests an impor-

tant question that, to our knowledge, has not been discussed before: if the retiree wants to “play

it safe,” should she withdraw the amount that rarely exhausts the balance in her account, or the

amount that frequently produces a final balance that can cover some unexpected outlays? Are

these approaches equivalent? The results in this section provide insight for this discussion.

We close with the PWA results for other horizon lengths. By changing the duration of the

retirement period from 30 years to 25, 35, and 40 years we obtain a ballpark estimate of the

level of risk incurred by using a specific endpoint in retirement plans, when in reality this

closing date is of course unknown.

Taken together, the numbers presented in this paper draw the outline of a surface that can

be used for decision-making. The true perfect withdrawal amount awaiting in each retiree’s

future cannot be known in advance, but by perusing the cross-sections of this object we can

make well-educated guesses of what it may end up being in each case. The aim of this paper

is to recruit the assistance of financial history to make this guesswork more robust.

2. Previous research

The research program on optimal retirement withdrawals began in 1994, with the release

of two seminal papers. The first one, Bierwirth (1994), actually computes what we are now

calling PWAs, although he obtains them by trial-and-error. Bierwirth does not seek the with-

drawal amount that exhausts the starting balance, but rather the figure that leaves the account

balance unchanged in nominal terms after 25years (even though the withdrawal amount is

kept constant in real terms). However, the stated goal of Bierwirth’s work—and its lasting

contribution—is to use historical data instead of simplistic assumptions to evaluate any with-

drawal strategy. This key insight is taken up by Bengen (1994) to develop the concept of

“portfolio longevity,” which he uses to arrive at the conclusion that “a first-year withdrawal

of 4%, followed by inflation-adjusted withdrawals in subsequent years, should be safe.”

This “4% rule” was later reinforced by a series of papers by Cooley et al. (1998, 1999,

2003, 2011). These studies used a more exhaustive scope, and varying conditions and

assumptions, to confirm that this strategy provides a high probability (85%–95%) that the re-

tiree will not run out of funds ahead of time.

The unifying feature of these “first-generation” approaches is that the retiree is asked to
select a specific withdrawal amount at the beginning of a very long period, and then she is

supposed to take out that same amount, in either nominal or real terms, each and every year.

This motivated attempts to develop “adaptive” rules, where the withdrawal amount is modi-
fied in some particular way depending on the circumstances faced in every period. In this

direction, the field has veritably exploded in recent years (Mitchell & Blanchett, 2011).

Currently there are, for example, rules that modify the withdrawal amount when the rate

of return is negative (Guyton & Klinger, 2006), or when it deviates too much from an

98 E.D. Suarez / Financial Services Review 28 (2020) 96–132



average value (Frank et al., 2011), or when the withdrawal amount/rate becomes higher

than a predefined number (Pye, 2000; Zolt, 2013). There are rules that change the

amount every five years to meet withdrawal rate targets (Spitzer, 2008), and rules that

change it every year depending on the length of the remaining horizon (Blanchett &

Frank, 2009). Some approaches incorporate mortality tables to the analysis, including

the investor’s age in the calculation of the withdrawal amount (Stout & Mitchell, 2006),

or computing the expected present value of all future withdrawals (Mitchell, 2011;

Stout, 2008).

Furthermore, there are even some approaches that forgo the stability criterion altogether,

letting the withdrawal amount bear the brunt of the variability in the rates of return. Waring

and Siegel (2015), for example, propose a rule in which “the investor will receive 30 years

of payments of varying size, each appropriate to the then-current value of the portfolio and

the interest rate . . . the only risk is to the size of each payment, which will vary . . . with
investment results.” Although these procedures may have some merits, we think that most

retirees prefer a stable flow of income year after year.

In a sense, most of these methods are just different ways of “hunting” for one single, elu-

sive number: the constant withdrawal amount that can be used year after year, over a certain

period in the future, without causing neither early depletion nor overaccumulation. This pa-

per provides the historical series for that number.

3. The perfect withdrawal amount

The perfect withdrawal amount formula can be thought of as a generalization of the payment for-

mula for a fixed-rate loan. In this case, however, the rate is not fixed but changes in every period, so

the simplification allowed by geometric series does not occur. Also, here we are talking about pay-

ments from an investment account, instead of payments into a loan that must be amortized. Finally,

the generalization includes a final balance term that must be non-negative, but not necessarily zero.

As shown in SSW, this constant value is given by:

w ¼ Ks

Qn
i¼1 1þ rið Þ � KE

� �
=on

i¼1
Qn

j¼1 1þ rjð Þ (1)

where w is the yearly withdrawal amount, KS is the starting balance, ri is the rate of return in year i,

n is the planning horizon in years, and KE is the ending balance sought at the end of these n years.

This equation shows that a period of time with a large cumulative return is not necessarily

associated with a high PWA, because the sequential order by which the total return comes

about is also relevant. This effect is seen more clearly by rewriting Eq. (1) as:

w ¼ RnKS � KEð Þ Sn (2)

where Rn ¼
Qn

i¼1 1þ rjð Þ is the return generated by the assets in the portfolio over the entire

period, and [Sn ¼ on
i¼1
Qn

j¼i 1þ rjð Þ]�1 can be interpreted as an “adjustment” applied to this

total return to take into account the specific sequence of period-by-period return rates that

produced it. The expression Sn is what we call sequencing factor.

E.D. Suarez / Financial Services Review 28 (2020) 96–132 99



Ordering matters, of course, because for an investment account that is constantly drawn

down—as is the case in the retirement stage—getting high rates at the beginning and low

rates later on is better than the other way around, because if the high rates come early they

will impact a larger balance. A closer inspection of the sequencing factor formula shows

how this feature is captured by our measure:

Sn ¼ ½on
i¼1
Qn

i¼1 1þ rjð Þ��1 ¼ ½ 1þ r1ð Þ 1þ r2ð Þ 1þ r3ð Þ . . . 1þ rnð Þþ 1þ r2ð Þ 1þ r3ð Þ
. . . 1þ rnð Þþ 1þ r3ð Þ 1þ r4ð Þ . . . 1þ rnð Þþ . . .þ 1þ rn�1ð Þ 1þ rnð Þþ 1þ rnð Þ��1 (3)

We note that the rates that come later in the sequence (higher index number) appear more

times in the summation than the rates that come sooner. Because Sn is the reciprocal of the

summation, this means that if we take a set of return rates (unordered) and arrange it into a

sequence (ordered), we can increase the value of Sn by swapping any pair of rates so that the

larger of the two figures comes before the lower one1. Therefore, as it should, the value of

Sn is higher when the sequence is “better.”

The next step in the PWA approach is just to realize that Eq. (1) defines a stochastic vari-

able. That is, since the rate of return is a random variable, the PWA measure is a random

variable as well. In that sense, the results presented here can be considered as snapshots of a

stochastic process that could have given us other values as observations. The purpose of this

paper is to examine those values—the ones that actually came up and became observable—

to derive insights for the management of retirement accounts.

4. The data

For all the calculations in this paper we use data from Ibbotson SBBI Classic Yearbook, 2015

edition. Specifically, we use the series “Large-Capitalization Stocks: Total Returns” as the

monthly return on stocks, and “Intermediate-Term Government Bonds: Total Returns” as the

monthly return on bonds. All the rates of return are deflated using CPI-U to obtain real-term

yields. Because we use real-term returns throughout, the withdrawal amounts that we obtain are

constant in terms of the purchasing power they provide (the initial withdrawal amount is adjusted

for inflation in every period) and the desired ending balance is attained in real terms as well.

To provide perspective for the results of this study, it is instructive to review these

return figures briefly. The geometric average of the real monthly rate of return for stocks

over the entire period spanning from January 1926 through December 2014

(1,068months) was 0.56% (6.98% annualized), with a minimum of �29.26% (in

September 1931) and a maximum of 42.56% (in April 1933). The arithmetic mean was

0.71%, and the standard deviation was 5.47 percentage points.

In the case of bonds, the geometric average of the real monthly return over the 89-year pe-

riod was 0.18% (2.24% annualized). The minimum value was �7.71% (February 1980), and

the maximum was 10.74% (April 1980). The mean for this series was 0.19%, and the stand-

ard deviation was 1.39 percentage points.

100 E.D. Suarez / Financial Services Review 28 (2020) 96–132



We now ask what the real rate of return has been, for these two financial instruments,

over 30-year periods in the historical record. This 30-year horizon is what we use below as

our “standard scenario,” which in turn corresponds to the set-up that is most frequently used

in the literature on optimal withdrawals.

Fig. 1 shows these 30-year rates, expressed as annual rates, and we notice a major upward

trend in the long-term real returns for bonds. However, although no more recent data can be

included because 30-year periods starting after December 1984 have not yet ended (at least

not in our dataset), using 20- or 15-year periods we see that this series will probably end up

hovering around 3% annual, or even lower, when the next decade of observations becomes

available. In any case, these figures are well known; we show them here just as a reference

for the analysis of the next sections.

5. Results for the standard scenario

The scenario we use as our “base case” is an investor that enters her retirement period

with a portfolio fully invested in stocks, and whose aim is to deplete her balance entirely

over a period of 30 years. In other words, our investor profile in this section is a person of

approximately 65 years of age who wants her money to last until she’s 95 and has no heirs,

or otherwise has no desire to leave behind any inheritance or gift.

When there is no bequest sought, KE in Eq. (2) is zero and the expression simplifies to:

w ¼ RnSnKS (4)

Fig. 1. Real rates of return for stocks and bonds, over 30-year periods beginning on the indicated date. Shown

are yearly geometric averages over the 30-year periods.

E.D. Suarez / Financial Services Review 28 (2020) 96–132 101



Before moving on to the actual results, we make one last modification to express our per-

fect withdrawal amount as a perfect withdrawal rate (PWR), relative to the starting balance

in the account:

w=KS ¼ RnSn (5)

Fig. 2 presents the PWR for the standard scenario as a monthly variable, assigning to

each month the PWR value corresponding to the 30-year period beginning on that month.

This computation assumes that withdrawals are made only once per year, on the very first

day and adjusted by the previous year’s inflation, whereas the returns are credited to the

account on the last day of the year (or, equivalently, at the end of every month). As men-

tioned before, under this convention for dating the data the PWR cannot be computed

beyond December 1984 because the 30-year periods beginning after that date are not yet

complete in our data set.

The first thing one notices in Fig. 2 is the wide range of variation, and how these up-and-

downs do not follow closely the changes in the rate of return for stocks (see Fig. 1). To see

how this can come about, and to further pin down the PWA concept, Fig. 5 shows the

explicit sequence of debits and credits on the account in two retirement periods, beginning

Fig. 2. Perfect withdrawal rate for 100%-stocks portfolio over 30-year period with zero final balance. For the

indicated retirement start dates, these withdrawal rates (applied to the starting balance and updated by inflation

every year) would have exhausted the available funds exactly after 30-years.

102 E.D. Suarez / Financial Services Review 28 (2020) 96–132



in June 1949 and February 1969, respectively, chosen to illustrate some of the points we

make below.

Another salient feature of Fig. 2 is that the 4% rule would seem overly conservative, as

the chart rarely pierces the 4% level. However, in fact, the “90% safe” withdrawal rate is

only slightly higher, at 4.27%. That is, out of the 708 observations in this series, 70 were

lower than 4.27%. Fig. 4 shows the frequencies by PWR range.

Still, the fact that the PWR has been higher than 4% so frequently—and higher by so much,

with more than half of the periods registering over 7%—raises the question of what would have

happened, in the past, to the final balance of an investor who followed the 4% rule. Under the

PWA framework this is easy to answer, as we can rewrite Eq. (2) to solve for the final balance2:

KE ¼ RnKS � w=Sn (6)

Fig. 3. All-stocks portfolio with $1 M starting balance, to be depleted over 30 years, begins retirement at two

different dates. An all-stocks portfolio of $1 million that entered retirement in June 1949 would have been ex-

hausted after 30 annual withdrawals of $139,929 (plus inflation), but this same portfolio would have provided

only $39,612 (plus inflation) per year over 30 years if it had entered retirement in February 1969—even though

the average annual real return over both 30-year periods was practically the same (7.2%).

E.D. Suarez / Financial Services Review 28 (2020) 96–132 103



If we express the final balance as a fraction of the starting balance and define the with-

drawal rate r=w/KS, we get:

KE=KS ¼ Rn � r=Sn (7)

Fig. 5 presents the results of Eq. (7), over 30-year periods, for an all-stocks portfolio using

r = 4%—the 4% rule. This chart brings forward a point that was made in SSW, namely, that

choosing a withdrawal amount that provides a high safety level is tantamount to accepting a

high probability of ending up with an account balance much larger than what was intended.

We find that in 81% of the periods, following the 4% rule would have left the investor

with more money—in real terms—than what she had at the beginning (instead of exhausting

the balance). In almost two-thirds of the historical instances the final balance would have

been more than twice the starting balance. An in fully one-third of the cases the account

would have closed with five times or more the original money!

Since overaccumulation means that the retiree’s standard of living was unnecessarily re-

stricted, the fact that it has happened so frequently under the 4% rule (and that it is so large)

is in itself a significant drawback for this common recommendation. Additionally, when we

consider that retirement savings accounts are rarely the only source of income for

retirees3—so that even if withdrawals exhaust the balance, there is usually another source of

funds—a likely culprit for this excess balance comes to the fore: using 90% as the chosen

safety level is probably an inadequate requirement for an adaptive plan.

Fig. 4. Frequency distribution of PWR in standard scenario. In the standard scenario the portfolio is invested

entirely in stocks and is completely depleted (no bequest) after 30 years. The first decile cutoff is at 4.27%, so

this is the “90%-safe” withdrawal rate in this case.

104 E.D. Suarez / Financial Services Review 28 (2020) 96–132



Returning to Fig. 2, we see basically three “times” when the PWR dipped below the 4%

level. First, there are the periods starting in December 1928 through November 1929, or in

March 1930 through June 1930. Anybody who started retirement in these specific months,

with stocks as their savings vehicle, would have met the brunt of the Great Depression at a

very early stage of their retirement period, which is when the retiree’s financial situation is

most vulnerable. In the first period mentioned, for example, they suffered the huge crashes

of October and November 1929 (-19.7% and �12.5%, respectively, in real terms) during

their very first year of retirement. Everybody in this group bore the losses of June and

September 1930 (-15.8%, �13.3%) and also took, only one year later, the largest one-month

drop in the history of the stock market (-29.3%, September 1931). Furthermore, a few

months after that, stocks fell an additional 43.2% in the three-month period of March

through May 1932.

All this would seem obvious if not for the fact that the PWR turned out to be much higher

(in the 6%–8% range) for people who retired only a few months earlier. Indeed, for people

who retired in July 1927 (only 17months before the first of our ill-fated groups) the PWR

was 6.2%. If the retirement period had started in August 1931—taking the “mother of all

crashes” head on, and then going through the ordeal of the spring of ‘32—the PWR would

still have been 5.9%. And if the retirement period had started only a little after this turmoil,

in July 1932, the retiree would have obtained the highest PWR ever recorded—an eye-pop-

ping 15.3%—even though the awful returns of 1937 would have hit her while still in the first

stage of the retirement period.

Fig. 5. Final balance for 100%-stocks portfolio under 4% rule after 30 years, relative to starting balance. The

4% rule would have led to large (possibly unwanted) final balances in the historical record.

E.D. Suarez / Financial Services Review 28 (2020) 96–132 105



The examination of this first segment in the PWR series throws into sharp relief the com-

plex relationship between rates of return and optimal withdrawal rates. Although this volatile

behavior could be attributed to the general market turbulence of the time, the sensitivity of

the results to what might seem like minor disturbances is still remarkable. In particular, we

see how market rebounds sometimes manage to “save” the portfolio, but they have to hap-

pen at a specific time to produce a meaningful impact—sequencing is crucial.

The sequencing effect is crucial because it determines the perfect withdrawal amount. We

stress the point that a set of return rates that might be considered “good”—in the sense that

upon compounding it produces a high total return for the entire period—will nonetheless

lead to a low PWA if the particular ordering of the set is unfavorable (from low returns to

high returns, in general). In particular, a “bad” ordering will be associated with a relatively

low PWA, and this cannot be changed by adjusting the withdrawal amount from year to year

as it would only zig-zag around, hover about or glide towards an inevitably low value.4

Next in our analysis of very low PWR periods is the six-month stretch beginning in

October 1965. This instance of “bad times to start retirement” is puzzling because not much

happened market-wise in that period. Rather, it would seem that it was just a few instances

of negative monthly returns (March, May, August 1966) that, though none terribly bad, com-

bined with a generally lackluster market in the first few years to produce a very deleterious

effect.

The most recent period with a very low PWR is the one spanning from August 1967 to

June 1969. In this 23-month period, 13months had a PWR of less than 4%, with the remain-

ing 10months barely managing to surpass this benchmark. As in the previous case, this

instance is hard to understand except as a case of peculiar timing. It would seem that a series

of early punches to the investment account (October 1967, Jan-Feb 1968) combined with the

bear market of 1969-1970 to create the kind of “perfect storm” that hurts new retirees very

much.

It would seem appropriate to also discuss periods when the PWR was particularly high,

but by now it is clear that the relationship between rates of return and PWRs is too complex

to allow any definite conclusions. For example, the lowest 30-year real rate of return on re-

cord for the all-stocks portfolio corresponds to the period starting in October 1955 (4.1%

annualized), but the PWR for the cohort that began retirement on that month is 6.2%—not

particularly low. The so-called Kennedy Slide of 1962 produced very low returns at the be-

ginning of these retirement periods, yet the PWR ended up in the 4.5%–5.5% range—higher

than at other, less remarkable times. Although the results for 30-year periods starting just

before the crash of 1987 are not yet available, Fig. 2 shows that retiring at the end of 1984—

only three years before that big hit—nonetheless provided an extremely high PWR (10.7%).

The complicated relationship between rates of return and PWRs is illustrated in Fig. 6.

Panel (a) superimposes the rate of return for stocks shown in Fig. 1 with the PWR presented

in Fig. 2. Panel (b) shows the corresponding scatter plot, where a linear regression between

these two variables attains a coefficient of determination of just 14%.

As can be seen, it is hard to establish a clear-cut connection between these two variables

over the historical record. We notice, for example, a long stretch of time spanning from late

1946 until mid-1954 when the PWR was much higher than what the return rates at the time

would seem to warrant. And the opposite holds true from mid-1967 to mid-1972: high return

106 E.D. Suarez / Financial Services Review 28 (2020) 96–132



Fig. 6. Real rate of return (annualized) and standard-scenario PWR. Panel (a) compares the all-stocks average

rate of return over 30-year periods with the corresponding PWR over the historical record. Panel (b) plots the

same-date values of these two variables against each other. The X’s in panel (b) mark Jun ‘49 and Feb ‘69, the

two dates used to construct Fig. 3.

E.D. Suarez / Financial Services Review 28 (2020) 96–132 107



rates, but low PWRs. As shown in Fig. 3, the annualized real rate of return for the 30-year

period beginning in June 1949 was 7.2% and the PWR was 14.0%. For the period beginning

in February 1969 the rate of return was also 7.2%, but the PWR was 4.0%—10 percentage

points lower!5 The period starting in August 1952 obtained an average rate of return of just

4.6%, yet managed to attain a PWR of 10.0%. The period starting in January 1933 got a

much higher rate of return, at 10.1%, but ended up with the same PWR of 10.0%!

It is now that the explanatory power of Eq. (5) becomes truly useful, as all these puzzling

cases become just specific instances of one single relationship. This expression tells us con-

cisely and unambiguously that the gist lies in the sequencing part of the equation, and it also

tells us how to explore this further. Therefore, even if a specific sequence of return rates will,

in all likelihood, never be repeated, the problem of forecasting the PWR in any particular case

is no longer the problem of forecasting the whole series of rates. Rather, as the derivations pre-

sented here show, what the investor needs to do is to obtain an estimate of the average annual

return for the portfolio she holds (for which there is abundant literature that can help her), and

then address the problem of the sequencing factor value that she must use to adjust it. The next

section focuses on how can the historical evidence shed light on this latter problem.

6. The sequencing factor over the historical record

The sequencing factor can be interpreted as an “adjustment” that has to be applied to the

rate of return to express the effect of the specific sequence of results that produced that total

return. It can be “good” or “bad”; in general, a series of returns that goes from high to low is

better than one that goes from low to high. The periods discussed at the end of the previous

section show that the magnitude of the sequencing impact can be downright stunning.

Eq. (3) shows the explicit formula for the sequencing factor which, as discussed in SSW,

is not a proxy measure. It comes directly from an analytic dissection of the problem and it is

a measure of orientation, in the same sense that variance is a measure of dispersion—and

the orientation of return rates (going up, going down, up a little then down a lot, etc.) is the

crucial element that the adjustment factor should capture. Fig. 7 displays the values that this

formula has taken over the historical record in the standard scenario, with the corresponding

frequency distribution shown in Fig. 8.

The first salient feature in Fig. 7 is that the sequencing factor is low (unfavorable order-

ing) and relatively stable for a very long time at the beginning of the chart. For the first

173months (almost 14 and a half years!) the sequencing factor stays inside the lowest two

brackets of the frequency distribution in Fig. 8, not surpassing the 0.0075 mark until June

1940. The reason for this is, of course, the general instability that affected equity markets

during most of the 1930s—the Great Depression.

This “shaky ground” situation meant that if an investment portfolio entered the with-

drawal stage at any time during this period, it didn’t take long for it to reach one or more

months when the stock market crashed violently. For the 30-year periods beginning during

1926, 1927, or 1928, the crash of October 1929 came along very quickly, hurting the

sequencing factor irreparably even though the average rate of return ended up in the 6%–8%

108 E.D. Suarez / Financial Services Review 28 (2020) 96–132



range. The periods beginning shortly after Black Tuesday did not fare much better,

sequence-wise, since “just around the corner” they would meet terrible results in June 1930,

September 1931, and the March-April-May stretch of 1932. Later periods in this era would

be similarly crippled by major downturns in September 1937, March 1938, and May 1940.

Fig. 7. Sequencing factor for 100%-stocks portfolio over 30-year periods. The sequencing factor is a measure

of the orientation of an ordered set of numbers. If it is large it means that the set goes from high to low, instead

of the other way around—which is good, if the numbers in the set are rates of return.

Fig. 8. Frequency distribution for sequencing factor in standard scenario. The sequencing factor is a measure of ori-

entation in return rates (“going up” vs. “going down”). In general, a low value of the sequencing factor means that

the sequence of returns went mostly downwards, which in turn is correspondingly worse for the retiree.

E.D. Suarez / Financial Services Review 28 (2020) 96–132 109



Therefore, we could say that in these years the sequencing factor was low because the return

rates could not sustain a normal level for long, instead crashing constantly. The cumulative

rate of return over 30 years was very acceptable, but the initial years were bad and the dam-

age was done.

Next in Fig. 7 we come across a stretch of rising values culminating in August 1952,

when the sequencing factor reached its maximum value of 0.0261. Indeed, the sequencing

factor was over 0.02 for 17months—between September 1951 and January 1953—which

produced PWRs around 10% even though the average return rate was quite low. And none

of these 17months produced spectacularly high returns (the maximum was 5.7% in

November 1952); rather, it was a period when returns followed more or less an increasing

pattern: from slightly negative in its first two months to moderately strong at the end of

1952. The return sequences then managed to stay clear of major crashes for a number of

years, but otherwise these moderately favorable starts were all it took for the sequencing fac-

tor to attain very high values. It would seem that a favorable return sequence is not so much

one that starts with a bang, but one that doesn’t get its legs cut off too quickly.

Finally, we comment on a period of nearly two years—from January 1969 to November

1970—when the sequencing factor again dropped below 0.005. This case would seem to be

different from the Great Depression period discussed above, because now the low values of

the sequencing factor were not caused by “crashes” in the early stages of the 30-year peri-

ods. Rather, this was a somewhat peculiar time that registered negative (and positive, but

small) returns in the first months, and then rose to strong returns in the second half of 1970.

This unfavorable ordering caused the PWR to hover in the 4%–6% range, even though the

average return was between 7% and 9% for these periods.

7. Results for other portfolio asset mixes

We now examine the PWR results for portfolio mixes that include bonds as well as

stocks. We retain the parameters from the standard scenario regarding horizon length

(30 years) and bequest goal (zero), but now allow bonds to take up a constant, positive frac-

tion of the asset mix. We identify each asset mix by its percentage of stocks/bonds—as in

90/10 for 90% stocks and 10% bonds—and consider different mixes by varying the propor-

tions in 10-point steps. Thus, we examine the 90/10 portfolio, the 80/20, the 70/30, and so

on, all the way to the 0/100 (100% bonds) portfolio.

To maintain the same asset mix in the portfolio at all times, we assume that a rebalancing

of asset classes takes place every month. That is, after the account obtains the return for each

asset class at the end of every month, the investor buys and sells each instrument as needed

to restore the corresponding proportions to the asset mix. We clarify this assumption because

our set-up would also be consistent with an annual rebalancing scheme (withdrawals are

made only once a year), and the two rebalancing methods are not equivalent. We chose

monthly rebalancing because our retirement horizons are being rolled over month-by-month,

and this procedure facilitates comparisons across periods that overlap. It seems to us that

both methods produce very similar results, but in any case, the choice is arbitrary.

110 E.D. Suarez / Financial Services Review 28 (2020) 96–132



We begin by adding the minimal amount of bonds to the mix (10%), and present the

PWR results for this 90/10 portfolio in Fig. 9 The results for the all-stocks portfolio are also

shown for comparison. Fig. 10 presents the differential between the two series, as well as

the variation in the actual withdrawal amount resulting from the change of mix. Variation

charts similar to Fig. 10 could be produced for all the alternate model specifications pre-

sented in this paper; we present only this one instance, to convey the idea of the type of anal-

ysis that can be conducted.

These charts showcase the enhanced clarity provided by the PWR approach. An examina-

tion of Figs. 9 and 10 may lead us to conclude that the all-stocks portfolio is the best alterna-

tive, as the instances when the 90/10 mix provides better results are less frequent and less

significant. By changing the all-stocks mix to 90/10, the PWR improves in 173 of the periods

considered (24% of the 708 total), but in 104 of these instances the increase is less than one-

tenth of a point—presumably negligible. The 69 periods where the PWR increased by more

than 0.1 points represent just 10% of the total, and are all in the Great Depression era. Also,

these increases never surpass 0.42 points, whereas the PWR decreases resulting from the

change reach as high as 1.43 points.

However, the periods that were benefited the most by the change in the mix also had a

low PWR in the standard scenario, so the impact on the perfect withdrawal amount is much

more significant. The PWA variation is a more appropriate measure because it presents the

situation as the retiree perceives it: her retirement income becoming x% higher or lower

because of the asset mix change. This is the dashed line in Fig. 10 which, for example, at its

highest point informs us that if a person entering retirement in September 1929 had switched

Fig. 9. PWR for all-stocks portfolio versus 90/10 portfolio. When the asset mix is changed from all-stocks to

90% stocks and 10% bonds, the PWR changes accordingly.

E.D. Suarez / Financial Services Review 28 (2020) 96–132 111



10% of her retirement funds to bonds instead of stocks, her annual income over the next

30 years would have been 12.5% higher. This is a significant impact but, overall, the chart

shows unfavorable PWA variations, so the case against including bonds seems to stand—at

least in the marginal proportion examined so far.

What happens if we allot a larger proportion to bonds? For the sake of clarity in the

charts, the next results are presented in sets of three portfolio mixes: 100/0 versus 70/30 ver-

sus 50/50, and 100/0 versus 30/70 versus 0/100. The summary results for the entire set of 11

mixes (from 100/0 to 0/100, in 10-point increments) are given later in this section.

Now our previous conclusion that bonds should not be included in the portfolio becomes

less clear-cut, but the arguments in favor and against are easily grasped from our charts. If

we want to make the withdrawal profile more stable, the 30/70 mix seems appealing (see

Fig. 12). If what the investor wants is to minimize the probability of a catastrophic outcome,

50/50 looks like the way to go (Fig. 11). The grid shown in Fig. 13 is particularly useful for

this kind of decisions.

From the Fig. we see that the 50/50 mix is the one that has breached the 4% PWR level

the least number of times, so this could be considered the ideal portfolio for a safety-minded

investor. Indeed, regarding the 4% rule it is interesting to note that in none of the 708 histori-

cal periods covered here has it been the case that the PWR was over 4% for the 100/0 portfo-

lio, but less than this benchmark for the 50/50 portfolio. However, the opposite has indeed

happened, as in 32 historical periods the 100/0 PWR was less than 4% but the 50/50 PWR

was more than 4%. Therefore, an investor who is satisfied with a 4% withdrawal rate should

Fig. 10. Change in PWR and variation in PWA when switching from all-stocks to 90/10 mix. PWR differential

(how many points is the withdrawal rate affected) is in percentage points, solid line, left axis. PWA variation

(how large is the modification in the withdrawal amount) is in percentage, dashed line, right axis.

112 E.D. Suarez / Financial Services Review 28 (2020) 96–132



never hold the all-stocks portfolio: 50/50 is better. By the same token, the idea that an inves-

tor who cares a lot about safety should hold more bonds, seems unwise. In particular, if the

funds’ owner is overly concerned that her withdrawal amount might crash below some sub-

sistence level, she should avoid asset mixes such as 20/80, 10/90, or 0/100.6

However, another interpretation of “safe” is that the retiree should pick a withdrawal

amount that has a high probability of being sustainable. Under this version of safety, the

task of selecting a portfolio mix then consists of finding the proportions that maximize,

say, the 90%–safe rate. We have included these values at the end of Fig. 13, where we

find that the 70/30 mix is now ideal because the upper bound of the first decile in this

PWR distribution is almost 4.5%. Indeed, we can say that by selecting an appropriate

asset mix, the 4% rule’s withdrawal levels can be improved by over 10% without incur-

ring additional risk.

Using the upper bound of the first decile as the watermark for separating safe from risky

is, of course, an arbitrary criterion. Especially if, as discussed in SSW, the procedure for

selecting the withdrawal amount calls for periodic reassessments—or if, as argued above,

the retiree has other, reliable sources of income—the retiree may be willing to take a less

conservative approach. After all, under a periodic revision scheme the retiree’s main concern

is not so much running out of funds, but having to decrease the withdrawal amount signifi-

cantly in the future. Be it for those reasons, or simply because the retiree has a higher toler-

ance for risk, it is useful to review the figures corresponding to other “safety levels” in the

historical record. Fig. 14 presents these milestone values by asset mix.

Fig. 11. PWR for 100/0 (all-stocks) portfolio versus 70/30 and 50/50 portfolios.

E.D. Suarez / Financial Services Review 28 (2020) 96–132 113



8. Results for bequest goals other than zero

In this section we restore the asset mix from the standard scenario (100% stocks) to look

at the behavior of the PWA when the “perfection” of the measure does not mean that it will

Fig. 12. PWR for 100/0 (all-stocks) portfolio versus 30/70 and 0/100 (all-bonds) portfolios.

Fig. 13. Frequency distribution for historical PWR, by portfolio mix, for 30-year periods and zero bequest goal.

Over the historical record, the PWR was lower than 4% in the least number of periods using the 50/50 mix. But

the 90%-safe rate was highest when the 70/30 mix was used.

114 E.D. Suarez / Financial Services Review 28 (2020) 96–132



exhaust the account balance in a given number of periods but, rather, that it will make that

balance reach a certain figure by the end of the retirement period. As in the previous sec-

tions, we assume a horizon length of 30 years and all figures are in real terms, so the bequest

goal is attained in the sense that this is the purchasing power that the ending balance

provides.

This investigation is important not only because many prospective retirees are actually

interested in bequeathing or giving away some of their funds, but also because of the pos-

sibility of introducing positive-valued bequest goals as safety devices in retirement plans.

That is, if the investor is concerned about getting low returns, or that perhaps a large

unforeseen expenditure may arise, she could protect against these contingencies by plan-

ning as if she intended to leave an inheritance. This maneuver provides the funds’ owner

with three adjustment levers to react to new developments: the withdrawal amount, the

safety level, and the bequest goal. Thus, for example, in a scenario of poor portfolio per-

formance the retiree can either reduce her withdrawals, accept a lower safety level, or

reduce her bequest target/protection balance—or combine these adjustments to spread

the impact.

Before presenting our results, we discuss a change in the role that the sequencing factor

plays when the bequest goal is nonzero. The zero final balance specification used in the

Fig. 14. Cutoff points of the first five deciles in historical PWR distributions, by asset mix. If the retiree wants

to be 90%-safe she should use the 70/30 mix, but for lower safety levels (higher risk tolerance) she should add

more stocks.

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standard scenario produces the very succinct expression shown in Eq. (5), but the corre-

sponding equation for the general case is:

w=KS ¼ RnSn � Sn KE=KSð Þ (8)

This complete version reveals a peculiar feature of the relationship between the PWR and

the sequencing factor. Although a favorable ordering in the sequence of return rates will pro-

vide a higher PWR, it will also make the result more sensitive to any change in the bequest

goal (KE/KS).

As we did with the asset mix, we change the bequest goal parameter in 10-point incre-

ments from zero to 100%, going from the case where the account’s funds are exhausted

completely (no bequest) to the case where the balance on the account is unchanged in real

terms at the end of the planning horizon (100% bequest). The intermediate points correspond

to bequeathing 10% of the starting balance, 20%, 30%, and so forth, always in constant pur-

chasing power terms. We start by comparing, in Fig. 15, the PWR for the standard scenario

against the results for 10% and 30% bequest goals.

We will argue that the main thing to see in Fig. 15 is, ironically, that there is not much to

see in it. That is, the three variables shown are so close to each other that they “fudge” the

chart, making it difficult to distinguish one line from the others. What this means is that the

reduction in the PWR necessary to protect or salvage a non-negligible fraction of the

Fig. 15. PWR for standard scenario (no bequest) versus 10% bequest goal and 30% bequest goal. The adjust-

ment in the withdrawal rate needed to accommodate significant bequest intentions is surprisingly small (see

also Fig. 16).

116 E.D. Suarez / Financial Services Review 28 (2020) 96–132



investment balance is surprisingly small. One would think, for example, that if a person who

enters retirement with $1 million in her account wants to leave a $300,000 estate, she would

have to reduce her withdrawal amount very significantly with respect to what she could take

out if she didn’t want to leave any funds at all. However, the evidence in the historical record

indicates that the PWR differential in this case (all-stocks portfolio over 30-year periods)

has never been larger than 0.8% points, it has been as low as 0.1 points, and its median value

was 0.3 points—that’s all she would have to shave off.

Again, it may be more relevant to discuss the required variation in the PWA, that is, how

much of her annual income would the retiree have to sacrifice to reach a given bequest goal.

Our results for the 30% bequest goal say that the most she would have had to cut back was

8.9%, and that in some periods the necessary decrease in her withdrawal amount was just

0.1%. The median PWA variation was �3.7%.

However, it would be wrong to conclude that by committing to reductions as small as

these the retiree would immediately have 30% of the balance available for emergencies. The

figures presented here leave the desired remainder in the account, yes, but at the end of the

planning horizon, and they depend crucially on the length of this timespan. Emergencies and

unforeseen expenditures can arise at any time, and if they come up in the first few years of

retirement we would be “cashing our bequest early” to deal with them. This would render

the results in this section inapplicable by, in effect, shortening the retirement horizon. Still,

by following these guidelines an investor will come progressively closer to having a free-

disposition fund available—if she manages to navigate a number of years in retirement with-

out major mishaps.

The next chart presents the results for bequest goals of 50% and 100% of the starting

balance. The results for all the bequest-goal levels considered are presented at the end

of this section.

With these two bequest-goal levels we may be moving into unrealistic territory, as per-

haps very few people will want to leave behind 50% of their starting balance (let alone

100%) when they could have used it for consumption during retirement. But 50% is psycho-

logically appealing (the “half for me, half for the kids” approach), whereas there is some-

thing very realistic about the 100% bequest goal: endowment funds. An institutional

endowment fund is, essentially, a retirement investment account that must last forever. The

“infinitely-lived retiree” (the institution) is interested in withdrawal strategies that provide a

relatively stable flow of funds, but leave the principal value unchanged in the long run, so

that the withdrawing process can go on indefinitely. This is similar to leaving a bequest of

100% after a number of years, so we comment on the results for this specification at some

length below.

As with the smaller bequest goal levels shown in Fig. 15, in Fig. 16 we find it remarkable

that the lines in the chart are not further apart, even though we are now dealing with very

ambitious bequest intentions. To leave behind a full 50% of the starting balance as departure

gift, there have been periods when the retiree only had to decrease her withdrawal rate by

0.2 percentage points. For the period beginning in June 1935, for example, the PWR for zero

bequest was 8.0% and the PWR for 50% bequest was 7.8%. Of course, in some other periods

the half-and-half approach has been a much more costly endeavor. For the period beginning

in August 1952 (a peculiar period that we have already singled out) the PWR with no

E.D. Suarez / Financial Services Review 28 (2020) 96–132 117



bequest was 10.0%, while for the 50% bequest goal the PWR was 8.7%—a 1.3-point differ-

ence. The median reduction in the PWR, when going from zero bequest to 50% bequest,

was just 0.4 points.

The variation in the PWA, when moving from a zero-bequest goal to a 50% bequest goal,

was also rather small. The maximum decrease in the withdrawal amount was 14.9%, and the

minimum was 1.8%. The median variation was �6.2%, which means that, by reducing the

amount withdrawn by only that much, the retiree would have ended up with a final balance

equal to 50% or more of the starting balance in one half of the historical periods covered

here—instead of zero dollars at the end.

As for the 100% bequest level, the necessary decrease in the PWR to achieve this goal

ranged from 0.4 to 2.6 points with respect to the standard scenario, with a median of �0.9

points. The PWA variation, in turn, was as large as �29.8% in some periods, and as small as

�3.5% in others. Not surprisingly, these extreme values correspond to retirement periods for

which the average rate of return was also close to the endpoints of its own range. The small-

est required decrease in the withdrawal amount—to leave the real balance unchanged after

30 years instead of exhausting it—occurs in the period beginning in June 1932 (11.8% aver-

age real return); the largest, in October 1955 (4.1% average real return, lowest ever, see Fig.

1). The median PWA variation was �12.4%.

It could even be argued that the inclusion of a high bequest goal should be the hallmark

of a safe retirement plan. The 4% rule, for example, is considered a safe strategy because

Fig. 16. PWR for standard scenario (no bequest) versus 50% bequest and 100% bequest.

118 E.D. Suarez / Financial Services Review 28 (2020) 96–132



it lowers the probability of “failing,” but it defines failure as running out of funds alto-

gether! Even in the presence of other sources of income, would it not be more prudent to

plan so that the worst-case scenario—or at least the most unlikely ones—still provides

means for sustenance, or some amount of supplementary funds? A promising idea would

be to develop retirement plans that, at the outset, use a very high bequest goal—perhaps

100%. This feature would provide the plan with an “embedded safety” that would sup-

port the selection of an otherwise risky withdrawal rate—a 50%–safe level could even

be used. Then, if future developments indicate that our case is in the unfavorable range,

the bequest goal can be adjusted gradually until a safe situation is reestablished. It

should be noted that if we select the median value from the correct PWR distribution, in

one half of the cases the withdrawal amount would be revised up, not down.7

With respect to endowment funds, the manager’s problem is to determine the maximum

amount that can be withdrawn consistently each year without jeopardizing the long-term via-

bility of the fund. This is equivalent, to some extent, to the problem of sustaining a long

retirement period as comfortably as possible and yet bequeath a large portion (close to

100%) of the starting balance in the account.8 For example, an all-stocks fund manager that

would like to protect the entirety of the fund’s assets, but who considers acceptable to end

up with a 10% decrease in the balance after a 30-year period, may use the values shown in

Fig. 17 as a reasonable range for the withdrawal rate.

From Fig. 17 we see that a withdrawal rate between 3.5% and 4.1% would seem to be

appropriate in this case. If our manager takes out more than 4.1%, the “probability” that

the fund’s balance after 30 years will be lower than 90% of the starting balance would

be in excess of 20%—too much risk, perhaps. On the other hand, if she were to withdraw

Fig. 17. Milestone safety levels in historical PWR distributions with 100% bequest goal and 90% bequest goal.

Possible range of withdrawal rates for balance protection. Portfolios are all-stocks and horizon length is

30 years.

E.D. Suarez / Financial Services Review 28 (2020) 96–132 119



less than 3.5%, the likelihood that the balance in the distant future will be larger than

the current balance would be more than 90%, which seems overly conservative. Because

the actual PWR is likely to be higher than these values—at least under our interpretation

of past frequency as probability—perhaps the ideal procedure would be to “hunt down”

the correct withdrawal rate. Instead of taking out the same amount (in real terms) every

year, the fund administrator would use this range of rates to see if they translate into

withdrawal amounts with the same purchasing power as what was taken out in the previ-

ous year. In case of significant discrepancy, any adjustment would be pondered against

the benefits of a stable income flow.

It is interesting to note that a more simplified approach to the endowment fund problem

would identify the average real return as the 100%–bequest PWR. However, these two con-

cepts are not the same and, in general, do not coincide. They would be strictly equal only if

the rate of return was the same in every period, but otherwise the sequencing effect makes

them differ. Fig. 18 superimposes these two variables, and we see that the difference has

been very substantial. Fig. 19 displays the relative frequency of the perfect withdrawal rates

according to the range of bequest goals.

Finally, our discussion about how it is not so difficult to bequeath a large portion of the

retirement funds is just the flip side of what we mentioned in a previous section: if the retiree

undershoots her PWR (takes out less than the perfect amount) even by just a fraction of a

point, she can end up with huge unwanted balances at the end. These realizations can lead us

to conclude that the whole business of determining an optimal withdrawal amount is too

Fig. 18. Comparison of average real return versus perfect withdrawal rate for all-stocks portfolio with 100%

bequest and 30-year horizon. Although it might be argued that the applicable withdrawal rate for endowment

funds is simply the average real return on the investment account, in the historical record the correct rate has

turned out to be a very different figure.

120 E.D. Suarez / Financial Services Review 28 (2020) 96–132



fragile, subject to some kind of “butterfly effect.” However, for us they only underscore the

need to develop algorithms, instead of rules of thumb, to navigate this stage of life with

some degree of stability. We think that should be the research program for this field and,

although we do not present any proposals here, SSW can provide sketches of what a PWA-

based algorithm would look like. We close this section with the PWR frequency distribution

under different bequest goals, computed using a 100%–stocks portfolio and with a planning

horizon of 30 years.

9. Results for other planning horizon lengths

The last item in our research list is to see what happens to the PWR when the length of

the planning horizon changes. As in the previous two sections, we reset the values of the

other two parameters in Eq. (1) to what they are in the standard scenario. Thus, in this sec-

tion the rates of return ri correspond to an all-stocks portfolio and the ending balance KE is

set at zero, but we use different values for n.

Exploring the impact of horizon length changes is important for at least two reasons.

First, the horizon length used when putting together a retirement plan will be either an edu-

cated guess of the person’s remaining life span (based on age, life expectancy data, health

outlook, family history, etc.) or an intentional “overshoot” figure to accommodate the possi-

bility of an unusually long life. If she is using the realistic figure, she wants to know what

happens if she misses; in particular, she would like to know how large would the

Fig. 19. Frequency distribution for historical PWR, by bequest goal level, for all-stocks portfolio and 30-year

horizon. For any given period in the historical record, the reduction in the withdrawal rate required to accom-

modate a positive final balance is equal to the sequencing factor for that period times the bequest goal sought.

This means that periods that registered a high sequencing factor (good for the retiree) are also more sensitive to

the introduction of a bequest goal.

E.D. Suarez / Financial Services Review 28 (2020) 96–132 121



adjustments be if she ends up living longer than she thought. If she is “padding up” the hori-

zon length, she wants to know the price she is paying for doing this, that is, how much is the

withdrawal amount lowered by this arbitrary extension of the planning horizon.

The second reason is that the retiree will probably want to change this figure along the

way. Perhaps a new medical breakthrough will benefit her, increasing her life expectancy.

Or if she adopts healthier habits—exercise, better nutrition, quitting smoking—what hap-

pens then? How bad would it be if she suddenly needed to make her funds last a longer num-

ber of years? And we must also be ready for the opposite case, when health developments

are such that a reduction in the planning horizon seems to be in order.

Indeed, the number of years remaining in the horizon is a different sort of uncertainty that

needs to be addressed, perhaps stochastically. In this paper we are treating it as a purely sub-

jective choice, but even a change in the perception of the numbers of years still ahead for

the retiree (or a change in the current age of the retiree under consideration) is a drastic mod-

ification of the estimation problem. Even though the return rates going forward remain of

course the same, a change in the planning horizon is a change in our choice of cutoff point

for that future sequence. We start by extending the planning horizon to 35 and 40 years. Fig.

20 presents the values that the PWR has taken, over the historical record, for these alternate

specifications.

Once again, as in the case of bequest goals, we find that the decrease in the PWR pro-

duced by the extension of the planning horizon is remarkably small. For the 40-year specifi-

cation, the difference in the PWR with respect to the standard scenario is at most 1.5

Fig. 20. PWR for standard scenario (30 years) versus 35-year long and 40-year long horizons. All cases com-

puted using 100%-stocks and zero bequest (total exhaustion). The adjustments required to accommodate longer

lifespans are smaller than might be expected.

122 E.D. Suarez / Financial Services Review 28 (2020) 96–132



percentage points, and sometimes it is as low as 0.1 points. The median differential is �0.4

points.

If we use the PWA variation as measure, the results remain striking. In the historical re-

cord, the cutback to the withdrawal amount required to make the funds last 40 years instead

of 30 has never been larger than 11.4%. As for the minimum, an investor who entered retire-

ment in April 1932 with $1 million could have taken out $94,143 for 30 years, or $92,634

for 40 years—a variation of just 3.5%!

The median value for the PWA variation is �6.3%, which means that if a retiree

decreases her withdrawal from the outset by just 6.3%, she will have a 50% chance of being

able to cover 10 more years.9 We believe that a majority of investors and retirees would think

that an increase of one-third in the number of periods that must be covered with the funds avail-

able would have a much larger impact on the amount of money they can withdraw.

It is certainly possible that some 35- or 40-year periods had very high returns in the last

five or 10 years, so the horizon extension managed to carry its own weight, so to speak, and

did not affect the withdrawal amount very much. But this would be a rare occurrence, and

Fig. 20 shows that a small decrease in the PWR is the norm, not the exception. Rather, this

feature seems to be embedded in the intrinsic dynamics of the drawdown process. This is an

important result, as it seems that prospective retirees perhaps do not have to worry so much

about underestimating their life expectancy: there will be plenty of time to react and, in all

likelihood, the reaction will not have to be severe.

It is important to stress, however, that for this result to hold the horizon extension needs

to be acknowledged and applied in the early stages of the retirement period. Thus, for exam-

ple, if at the very beginning of retirement, the investor is unsure whether to use a 30-year

long or a 35-year long planning length, it is useful to know that the respective PWA distribu-

tions are not very different. However, this is not the same as traversing, say, one half of a

30-year retirement horizon and then decide at that point that we would like to cover five

more years. In that case we would need to craft a 20-year plan—what we need to cover from

that point on—to substitute for a 15-year plan—the remaining horizon according to the orig-

inal plan. This is a different situation, although it can be assessed in a similar fashion. To

provide an idea of what these “midterm” adjustments might entail, we present summary

results for 20-, 15-, 10-, and 5-year long horizons later in this section.

Although the literature in this field has settled on 30 years as the typical horizon length

for planning purposes, there are reasons to consider it excessively long. The full retirement

age to receive social security benefits is currently 66 years and, according to data compiled

by the Social Security Administration (SSA), an American man reaching age 65 today can

expect to live, on average, until age 84.3. An American woman turning 65 today can expect

to live, on average, until age 86.6. Only about one out of every four 65-year-olds today will

live past age 90, and one out of 10 will live past age 95.10 This means that the standard sce-

nario has a large additional safety margin, as this horizon length will actually be required by

just 10% of all retirees.

Our position here is that it is advisable to use a withdrawal amount with a good chance of

being sustainable, but this must be taken from a PWA distribution with a realistic horizon

length. Otherwise, the sustainability of the chosen amount would be incorrectly estimated,

and the true safety level will likely be much higher than what was intended. Under this

E.D. Suarez / Financial Services Review 28 (2020) 96–132 123



view, the SSA’s life expectancy data indicate that 25 years would be more appropriate than

30 years for first-approximation calculations. Fig. 21 presents the PWR results under this ho-

rizon length,11 keeping in mind that about 25% of all retirees—particularly women—will

need to apply corrections along the way to cover additional years.

We see that the PWA for 30-year periods has been between 2.9% and 13.1% lower than

for the 25-year periods beginning on the same month, with a median of �5.3%. This means

that an intervention in the early years of a 25-year plan, aiming to turn it into a 30-year plan

instead, should call for reductions of the withdrawal amount in the 3% to 10% range to real-

istically expect to cover the additional longevity.

The relatively small size of these adjustments supports our notion that it is a good idea to

build retirement plans using 25-year horizons, and then operate on these plans using prudent

withdrawal strategies. The 90%-safe rate in 25-year horizons with no bequest is maximal in

the 70/30 mix, reaching 4.80%. The 50%-safe rate in 25-year horizons with 100% bequest—

the version of conservative approach that we advocate in this paper—is highest in the all-

stocks allocation, at 6.6%. Per life expectancy data, this timespan should be appropriate for

most retirees and, if this is not the case, then the conservative withdrawal program will prob-

ably provide sufficient leverage to cover the extra years. Furthermore, even if the perfect

storm materializes in the form of a longer-than-usual life span and a terrible sequence of

returns, the option of curtailing the withdrawal amount does not look catastrophic because

the required reduction is probably not so large.

As for the timing of the actual adjustments to be made, we advocate in favor of the proce-

dure proposed in SSW. As time moves forward—and either the planning horizon shortens or

the retiree decides to choose an endpoint farther into the future—the withdrawal amount

Fig. 21. PWR for standard scenario (30 years) versus 25-year planning horizon. The 25-year planning horizon is

more realistic for people retiring at the age of 66.

124 E.D. Suarez / Financial Services Review 28 (2020) 96–132



under consideration must be assessed in terms of the percentage it represents of the account

balance at that point (i.e., that is, the implied withdrawal rate). The withdrawal amount is to

be modified when the location of this withdrawal rate in the corresponding PWR distribution

falls outside the confidence range (probability interval) with which the retiree feels more

comfortable: if the confidence level becomes too high, it is advisable to increase the

Fig. 22. Frequency distribution for historical PWR, by horizon length, for 100%-stocks portfolio and zero

bequest goal. Panel (a) shows bins in integer percentage points. Panel (b) shows cutoff points for first five

deciles.

Fig. 23. PWR for standard scenario (30 years) versus 20-, 15-, 10-, and 5-year planning horizon. These shorter

timeframes may be used for midcourse adjustments, once inside the retirement stage.

E.D. Suarez / Financial Services Review 28 (2020) 96–132 125



withdrawal amount; if it gets too low, the amount withdrawn should be reduced. The magni-

tude of the adjustments, in turn, may conceivably be based on the results presented in this

paper. Thus, for example, if the investment account is all-stocks, the retiree is trying to use

all her funds for consumption, and the appropriate planning horizon length for the current

year is deemed to be, say, 15 years, then the third column on the right side of Fig. 24 is appli-

cable. There we find that she should take out at least 6.1% of her balance, lest she takes on

too much risk of running up an excess balance, and perhaps no more than 10.5% of the bal-

ance—unless she is not troubled with being more than 50% likely of having to reduce her

withdrawal amount in the future. This is what the evidence in the historical record tells us.

Fig. 22 summarizes the main features of the historical PWR distributions for the four dif-

ferent horizon lengths that we discussed in detail in this section. Fig. 23 presents the corre-

sponding values for shorter horizons, and Fig. 24 shows the historical evolution of the PWR

for these shorter timeframes.12

As for how does the sequencing factor change when the length of the planning horizon

is modified, an algebraic analysis indicates that it should vary inversely with the horizon

length parameter. First, in general, a longer horizon should lead to lower withdrawal

rates simply because the starting balance would need to cover a larger number of periods

(a longer life span). Second, the total compounded return generated by the investment

portfolio over the entire period should become larger as the planning horizon lengthens

—again, in general—because there will now be more periods over which the return rates

will compound to provide the total return. Finally, from the decomposition of the PWR

Fig. 24. Same as Fig. 22, but with shorter planning horizons.

126 E.D. Suarez / Financial Services Review 28 (2020) 96–132



provided by Eq. (5) (w/KS = RnSn) we see that a lower withdrawal rate (left-hand side of

the equation) may be reconciled with a larger total return (Rn, in the right-hand side of

the equation), only if the corresponding sequencing factor decreases, because Sn is the

sole remaining term in the right-hand side of the equation.

Fig. 25. Sequencing factor for 100%-stocks portfolio over different horizon lengths. In general, longer time-

frames should be associated with lower values for the sequencing factor but, although true in general in the his-

torical record, we see that this has not always been the case.

Fig. 26. Historical 90%-safe PWR for zero bequest goal, as asset mix and horizon length vary. This is a particu-

lar 3-day section of the historical PWR hypercube; similar sections can be produced for other safety levels or

bequest goals. Also, any of these two control parameters may be swapped for one of the three variables in the

axes of this chart, so as to explore different situations (see, for comparison, the work of Frank et al. (2011)).

E.D. Suarez / Financial Services Review 28 (2020) 96–132 127



Fig. 25 shows the actual values of the sequencing factor, over the historical record, for

the four horizon lengths that we examined more closely in this section. It is interesting to

note that, the arguments of the previous paragraph notwithstanding, there are several periods

in the historical record where the sequencing factor for shorter horizons was lower than the

factors for longer horizons beginning on the same date. In general, however, the arguments

presented above are confirmed by our results.

It would also be useful to understand the PWR interplay between horizon lengths and

asset mixes in the historical record, as this would provide guidelines for the construction of

glide path strategies and for the management of funds that are to provide income during

retirement. Of course, all the algorithms and rules currently used by serious practices have a

strong statistical foundation, but it would still be instructive to know what these programs

look like when placed under the lens of historical experience. We offer the results shown in

Fig. 26 as a point of entry for this discussion.

This chart shows that for the 40, 35, and 30-year horizon lengths, the highest 90%-

safe rate is attained with the 70/30 mix, whereas for the 25-year horizon this rate is

slightly higher with the 60/40 mix. From the point of view of mutual funds designed to

provide income during retirement, the 25-year and 30-year timespans can be considered

reasonable or even short, and even so the “very safe” withdrawal rate is highest at equity

allocations that are larger than what is normally used in these funds.13 Also, a strategy

that switches the portfolio mix away from stocks when a certain accumulation goal is

attained (Basu et al., 2011) may also be called into question, because presumably the

retiree’s goal is not to reach the payout phase with a large balance in her account, but to

make a series of large withdrawals from it. Keeping a high stocks allocation can help

her achieve this, even after the phase transition point is passed.

10. Conclusions

The examination of historical data using the PWA approach yields useful results for the

management of retirement investment accounts. The historical record would recommend the

use of relatively stock-heavy portfolios, with about 70% of total funds allocated to equity.

Even with horizons of just 25 years—that would be well inside the retirement stage accord-

ing to the assumptions of most models—the maximization of the 90%-safe withdrawal rate

calls for stock allocations of at least 60%.

For retirement plans that intend to exhaust the available funds after 30 years—the most

frequent setup in optimal withdrawal studies—portfolios containing 70% stocks and 30%

bonds produced a PWR higher than 4.5% in 90% of the historical periods covered here, and

in none of the 708 periods examined did it drop below 3.9%. In our opinion, this calls for a

reexamination of the 4% rule because “safe” withdrawal amounts can be increased by more

than 10% through a judicious selection of asset allocation. The 4% rule would seem to be

more appropriate for managing endowment funds, instead.

We argue, however, that safe withdrawal strategies should include a bequest goal—

even if the retiree has no such intentions—for at least three reasons. The first is that

128 E.D. Suarez / Financial Services Review 28 (2020) 96–132



retirement plans should strive to avoid undesirable outcomes, but running entirely out of

funds is too drastic as worst-case scenario. The second reason is that the retiree needs to

have a recourse in case that unexpected large outlays arise for any reason. And the third

is that the presence of a bequest goal in the plan creates an additional adjustment lever

for midcourse corrections, should these become necessary in case of poor portfolio

performance.

Under this interpretation, safety would be provided by the bequest goal embedded in

the plan, not by the location of the withdrawal amount in a histogram. For an all-stocks

portfolio with a 100% bequest goal and a 30-year planning horizon, the PWR historical

series produced a median of 6.4%. This could then be used as a safe withdrawal guide-

line, with the bequest goal as an adjustment lever that will have to be lowered in about

50% of all cases. If an adjustment via the bequest goal becomes insufficient and the

withdrawal amount has to be reduced, our results suggest that the impact on the PWA

for the remaining horizon length will not be very large, if the correction is applied in the

first five to 10 years.

Indeed, the sensitivity of PWA distributions to changes in parameter values—be it the ho-

rizon length or the bequest goal level—is surprisingly small. Upon changing the planning

horizon from 30 to 35 years, the median decrease in the zero-bequest PWR for all-stocks

portfolios was just 0.3 percentage points. If the horizon is extended to 40 years, the median

PWR decrease was only slightly higher at 0.4 points. This would indicate that the adjust-

ments demanded by longer lifespans are not very large, if detected and implemented in the

early stages of retirement.

In turn, if the horizon length is kept at 30 years but the bequest goal is increased to

50%—one-half of the starting balance’s purchasing power remaining in the account at

the end—the median PWR decrease for all-stocks portfolios was 0.4 points. Compared

with the zero-bequest case, a reduction of just 6.2% in the amount withdrawn would be

sufficient to achieve (or surpass) the 50% bequest goal in one-half of the historical peri-

ods examined.

Researchers are already applying the PWA methodology with some profit, with Clare et

al. (2016a) using it to implement a trend-following overlay that improves the 90%-safe rate

by 2.2 percentage points in 20-year horizons. In another paper, these authors employ the

PWA concept to discuss ways to address the problem of sequence of returns in decumulation

portfolios (Clare et al., 2016b). In turn, Estrada (2018) takes the methodology to an interna-

tional context and computes PWAs for 21 different countries.

The PWA framework was conceived as a toolkit for constructing withdrawal strategies.

The results presented here should be considered guidelines, provided by financial history, to

assist in the design and calibration of these strategies.

Notes

1 This characteristic of non-commutativeness (changing the order in the sequence

does change the result) lies at the heart of the sequencing factor formula. It is thus

E.D. Suarez / Financial Services Review 28 (2020) 96–132 129



fundamentally different from the “volatility drag” formula (½ s2), and we thank an

anonymous reviewer for comments that prompted this clarification.

2 This expression is valid only when it produces non-negative values. If it is negative

it must have happened that the withdrawal amount was larger than the available bal-

ance at some point, which is either not allowed or implies a change in the rate of

return (to a rate of interest).

3 Bridges and Choudhury (2007) report that 89% of the population aged 65 and older

receive Social Security benefits. In turn, Coile and Milligan (2009) present data from the

Health and Retirement Study that show that for households where the older member of

the couple is in the 65-69 age bracket, 39.1% of their financial assets are not in their

IRAs, nor as stocks, bonds, or bills, but in checking, savings, and money market

accounts.

4 We thank an anonymous referee for bringing to our attention the need to establish

this point clearly.

5 This example, borne out of actual historical data, illustrates the extent to which

sequencing can be crucial indeed. The return rates in both periods produced the

same total return, yet the corresponding PWAs were very different solely by the

effect of sequencing. If the corresponding retirees had made periodic adjustments to

the withdrawal amount it would not have made any difference: the one retiring in

’49 would have fared well, the one retiring in ’69 would have done poorly.

6 This is perhaps a very relevant observation. Studies of actual asset allocation choices

have found more than 55% of people of retirement age having no equity at all in

their IRA portfolios (Waggle and Englis, 2000).

7 The median for the 100% bequest all-stocks PWR with 30-year horizons over the

historical record is 6.4%. With the caveats mentioned regarding mid-course correc-

tions, this could then be called the “6.4% rule”.

8 Strictly speaking, the problem should be posed as the limit convergence of Eq. (1)

with KS = KE as n increases. We use the results for n=30 as an approximation, with

the next section discussing their sensitivity to extensions of this horizon. We also

simplify by ignoring additional future contributions.

9 The number of PWR figures that can be produced with our dataset is 648 for the 35-

year horizon length and 588 for the 40-year length. This is less than the 708 data

points we had in the standard scenario, so the validity of these findings for inference

purposes is correspondingly weaker.

10 Social Security Administration (2016).

11 For the 25-year horizon length, the number of PWR figures that can be produced

with our dataset is 768. The dashed line in Fig. 21 (25-year long horizon) is thus lon-

ger than the solid line (30-year long horizon).

12 The number of data points for these horizon lengths is 828 (20 years), 888 (15

years), 948 (10 years), and 1,008 (5 years). So the lines in Fig. 24 become longer as

the planning horizon shortens.

13 John Hancock Investments’ Retirement Living Portfolios enter the retirement stage

with a 50% stock allocation. Fidelity Investments’ Income Replacement Funds offers

information that implies allocations of 53%, 34%, and 13% to stocks, bonds, and cash

130 E.D. Suarez / Financial Services Review 28 (2020) 96–132



in Year 1 of retirement. Vanguard’s Managed Payout Fund invests 57.5% in stocks,

17.9% in bonds, and 24.6% in “other”. Charles Schwab’s Monthly Income Fund-

Enhanced Payout allocates between 10% and 40% to equities, and the Monthly

Income Fund-Maximum Payout uses 0% to 25% as its equity allocation range.

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