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The 4% rule – At what price?

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The 4% rule is the advice many retirees follow for managing spending and investing. We examine this rule’s inefficiencies—the price paid for funding its unspent surpluses and the overpayments made to purchase its spending policy. We show that a typical rule allocates 10–20% of a retiree’s initial wealth to surpluses and an additional 2–4% to overpayments. Further, we argue that even if retirees were to recoup these costs, the 4% rule’s spending plan remains wasteful, since many retirees actually prefer a different, cheaper spending plan.
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JOURNAL OFINVESTMENT MANAGEMENT, Vol. 7, No. 3, (2009), pp. 31–48
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THE 4% RULE—AT WHAT PRICE?
Jason S. Scott, William F. Sharpeaand John G. Watsonb
The 4% rule is the advice many retirees follow for managing spending and investing. We examine
this rule’s inefficiencies—the price paid for funding its unspent surpluses and the overpayments
made to purchase its spending policy. We show that a typical rule allocates 10–20% of a retiree’s
initial wealth to surpluses and an additional 2–4% to overpayments. Further, we argue that even
if retirees were to recoup these costs, the 4% rule’s spending plan remains wasteful, since many
retirees actually prefer a different, cheaper spending plan.
1 Introduction
Retirees must annually decide how to invest their
retirement portfolios and how much to with-
draw from them to cover their living expenses.
Some retirees turn to financial planners for advice,
whereas others consult brokers, investment pub-
lications, or web sites. Although these sources
are quite different, their spending and investment
advice is consistently the same—the 4% rule. This
rule of thumb originated in the financial planning
Corresponding author. Managing Director, Retiree
Research Center at Financial Engines, Financial Engines,
Inc., 1804 Embarcadero Road, Palo Alto, CA 94303,
USA. Tel.: 650-565-4925; fax: 650-565-4905; e-mail:
jscott@financialengines.com
aSTANCO 25 Professor of Finance, Emeritus Graduate
School of Business, Stanford University.
bFellow, Retiree Research Center at Financial Engines, Finan-
cial Engines, Inc., 1804 Embarcadero Road, Palo Alto, CA
94303, USA.
literature, and it was quickly adopted by many
financial firms to advise their retail customers.
Much of the financial press and many investor web
sites now embrace the rule, and hence now it is the
most endorsed, publicized, and parroted piece of
advice that a retiree is likely to hear. Therefore, it
behooves readers of this journal to be familiar with
the rule’s approach, features, and flaws.
A typical rule of thumb recommends that a retiree
annually spend a fixed, real amount equal to 4% of
his initial wealth, and rebalance the remainder of
his money in a 60–40% mix of stocks and bonds
throughout a 30-year retirement period. For exam-
ple, a retiree with a $1MM portfolio should confi-
dently spend a cost of living adjusted $40K a year for
30 years, independent of stock, bond, and inflation
gyrations. Confidence in the plan is often expressed
as the probability of its success, for example, in nine
of ten scenarios, our retiree will sustain his spend-
ing. Modifications to this basic example include
THIRD QUARTER 2009 31
32 JASON S. SCOTT ET AL.
changing the amount to withdraw, the length of the
plan, the portfolio mix, the rebalancing frequency,
or the confidence level. However, all these variations
have a common theme—they attempt to finance a
constant, non-volatile spending plan using a risky,
volatile investment strategy. For simplicity, we refer
to this entire class of retirement strategies as 4%
rules, the sobriquet of its first and most popular
example.
Supporting a constant spending plan using a volatile
investment policy is fundamentally flawed. A retiree
using a 4% rule faces spending shortfalls when
risky investments underperform, may accumulate
wasted surpluses when they outperform, and in
any case, could likely purchase exactly the same
spending distributions more cheaply. The objec-
tive of this study is to price these inefficiencies—we
want to know how much money a retiree wastes
by adopting a 4% rule. In the next sections, we
review the 4% rule’s history and examine its popu-
larity. We then present a financial parable featuring
two aging boomers and the single spin of a bet-
ting wheel. Our parable illustrates the flaws of
the 4% rule, both qualitatively and quantitatively.
Next, we use standard assumptions about capi-
tal markets and show that the approach of the
4% rule to spending and investing wastes a sig-
nificant portion of a retiree’s savings and is thus
prima facie inefficient. Finally, we argue that a better
solution can be obtained by formulating the retire-
ment problem as one of maximizing the retiree’s
expected utility, an approach advocated by financial
economists.
2 History
Not long ago, many financial planners estimated
a retiree’s annual spending budget using a mort-
gage calculator, an estimate of the average rate of
return on the retiree’s investments, and the retiree’s
horizon—the number of years that a retiree’s invest-
ments had to support his spending.1Further, to
include a cost of living increase, the planner
would adjust the average nominal investment return
downward by an estimate of the average inflation
rate and compute the real spending. This method
is only valid when all the future yearly investment
returns and inflation rates are nearly equal to their
estimated averages, and hence non-volatile.
First Larry Bierwirth (1994), and then William
Bengen (1994) argued that since actual asset returns
and inflation rates were historically quite volatile,
retirement plans based on their averages were unre-
alistic. Bengen proposed an alternative strategy that
retained the basic investment and spending strate-
gies inherent in the mortgage calculation. In partic-
ular, he assumed that a retiree’s assets were invested
in a mix of stocks and bonds and annually rebal-
anced to fixed percentages. Further, he assumed that
in terms of real dollars, a retiree’s annual spending
was constant and financed by a year-end, inflation-
adjusted withdrawal from the portfolio. Hence,
choosing a stock-bond mix and a withdrawal rate—
the ratio of annual, real spending to initial wealth—
specified a retirement plan. Now, for a given
horizon, some of these plans would have historically
performed better than all the other possibilities.
Therefore, Bengen collected scenarios of past asset
returns and inflation rates, simulated a number of
plans under these scenarios, and identified the best
performers.
Although a retiree wants the highest withdrawal
rate possible, he also wants to sustain his spending
throughout his retirement years. Bengen required
that all his recommended plans be historically
sustainable—the investments had to support all
scheduled withdrawals for every historical scenario.
Further, he sought the sustainable plans with the
largest withdrawal rates. For example, using a port-
folio mix consisting of between 50% and 75%
stocks, Bengen (1994, 172) reported:
Assuming a minimum requirement of 30 years of portfolio
longevity, a first-year withdrawal of 4 percent, followed by
inflation-adjusted withdrawals in subsequent years, should
be safe.
JOURNAL OFINVESTMENT MANAGEMENT THIRD QUARTER 2009
THE 4% RULE—ATWHAT PRICE?33
Larger (smaller) withdrawal rates were recom-
mended for shorter (longer) horizons. However,
the horizons of most interest were in the 30–40-
year range and had withdrawal rates that were near
4%. For this reason, Bengen’s approach is now
commonly called the 4% rule.
In his original article, Bengen did not give a confi-
dence level for his rule—he deemed it safe, since it
had never historically failed. However, in a similar
study, Cooley et al. (1998, 20, Table 3) reported a
95% historical success rate for a 30-year horizon, a
4% withdrawal rate, and 50–50% mix of stocks and
bonds. This success rate increased to 98% when the
percentage of stocks was increased to 75%. This
study is often cited as the Trinity Study (all three
authors are finance professors at Trinity University
in San Antonio, Texas).
Bengen (1996, 1997, 2001, 2006a) extended his
approach in a series of articles. For example, Ben-
gen (1996) treats both tax-deferred and taxable
accounts, and recommends that conservative clients
reduce their risk with age by yearly decreasing their
percentage of stocks by 1%. Bengen (1997) extends
the investment choices to include Treasury Bills
and small-cap stocks. For a summary of Bengen’s
work, we recommend his recent review (Bengen,
2006b). Cooley et al. (1999, 2001, 2003a) also
continued to analyze withdrawal rates. Cooley
et al. (1999) focuses on monthly versus annual
withdrawals, and Cooley (2003a) concludes that
investors “would benefit only modestly in the long
run from international diversification.”
When estimates of success rates are based on a small
number of scenarios, they are prone to estimation
error. This is particularly true for estimates that
use overlapping historical scenarios. This problem
led some investigators to develop market models—
stochastic models of asset returns and inflation rate
processes. A model’s parameters are chosen so that
the joint probability distribution of the processes
reflects the average values, variances, and correla-
tions commonly observed. An unlimited supply of
scenarios can be numerically generated from these
models, and then statistics, such as the success
rate, can be computed using Monte Carlo meth-
ods. Further, in a few cases, estimates can be derived
analytically.
George Pye (2000) simulated all-equity portfolios
whose real returns were log-normally distributed
with a mean return of 8% and a standard devia-
tion of 18%. Pye concluded that his modified 4%
rule would be safe for a 35-year horizon. Pye’s strat-
egy increases a portfolio’s longevity by ratcheting
down spending when markets perform poorly. We
include his rule in the 4% class since its “focus is on
sustaining the initial withdrawal” (Pye, 2000, 74)
and invests in a volatile asset.
Most of the withdrawal rate research ignores a
retiree’s mortality and assumes a fixed planning
horizon. An exception is the series of articles by
Ho et al. (1994a, 1994b), Milevsky et al. (1997),
and Milevsky and Robinson (2005). In particular,
Milevsky and Robinson (2005) chose simple mod-
els for both the markets and mortality and devel-
oped estimates of success rates without using simu-
lation. We recommend the article by Milevsky and
Abaimova (2006) for a summary of this approach
and its application to retirement planning.
Cooley et al. (2003b) compared the results obtained
using historical data and market models, but their
study “does not take sides on which methodology is
better.” Finally, in another approach, Spitzer et al.
(2007) generate scenarios using a bootstrap algo-
rithm to resample historical data with replacement.
3 Popularity
The authors conducted an informal survey of retiree
guidance sources and their recommendations on
spending and investment strategies. We were struck
by the universal popularity of the 4% rule—retail
THIRD QUARTER 2009 JOURNAL OFINVESTMENT MANAGEMENT
34 JASON S. SCOTT ET AL.
brokerage firms, mutual fund companies, retire-
ment groups, investor groups, financial websites,
and the popular financial press all recommend it.
Sometimes the guidance explicitly references Ben-
gens work, the Trinity Study, or related research,
but more often, it is presented as the perceived wis-
dom of unnamed experts. In this section, we report
a sample of our findings.
Vice President of Financial Planning Rande Spiegel-
man of the Schwab Center for Financial Research
wrote an article on retirement spending, subtitled
“The 4% Solution”, for the August 17, 2006 issue of
Schwab Investing Insights®, a monthly publication
for Schwab clients. In this article, he recommends
the basic 4% rule that we used in the introduc-
tion to this study (30-year horizon, 4% withdrawal
rate, 60–40% mix of stocks and bonds, and 90%
confidence level).
T. Rowe Price’s web site (2008a) suggests, “If you
anticipate a retirement of approximately 30 years,
consider withdrawing no more than 4% of your
investment balance, pretax, in the first year of retire-
ment. Each year thereafter, you’ll want to increase
that dollar amount 3% every year to maintain
your purchasing power.” A popular feature of this
web site is the “Retirement Income Calculator” (T.
Rowe Price, 2008b) that simulates 500 scenarios
of returns for seven asset classes, where the monthly
returns are assumed to be jointly normal. The calcu-
lator automatically accounts for minimum required
distributions after age 701/2, attempts to decrease
equity exposure every five years, and yearly inflates
withdrawals by 3%. For a single retiree starting
retirement at age 65, having a 30-year horizon,
beginning with a $1MM portfolio, investing ini-
tially in a 60% equity mix (portfolio E), and
withdrawing $3.3 K per month, the withdrawal rate
is 3.96%, and the calculator predicts a 90% success
rate.
The Vanguard Group (2008) advises “making with-
drawals at rates no greater than 3% to 5% at the
outset of your retirement ... They also provide a
tool for retirees to determine how much they can
annually withdraw in real dollars. The tool uses 81
historical scenarios, and “the monthly withdrawal
amount shown by the tool is the highest level of
spending in which 85% of these historical paths
would have left you with a positive balance at the
end of your chosen investment horizon.” A retiree
with a 30-year horizon is advised to withdraw at a
rate of 3.75%, 4.75%, or 5.25%, if he is invested
in a conservative (less than 35% equities), moder-
ate (between 35% and 65% equities), or aggressive
(higher than 65% equities) portfolio, respectively.
AARP (2008) indicates that “most experts” recom-
mend 4% spending and that stocks should be added
to the portfolio to “help your money last.” Jane
Bryant Quinn echoes Bengens recommendations
in the June 2006 AARP Bulletin. John Markese
(2006), President of the American Association of
Individual Investors (AAII), writes, “Most research
and simulation studies conclude that portfolios with
annual withdrawal rates of 4% or less and greater
than 50% in stock are most likely to last throughout
retirement.”
Liz Weston (2008), writing for MSN Money Cen-
tral, refers to Bengen and the T. Rowe Price
Retirement Income Calculator (2008b) and rec-
ommends a 3–4% withdrawal rate for retirees with
horizons of 30 or more years. Scott Burns (2004),
business columnist for the Dallas Morning News,
was an early champion of the Trinity Study. Also,
Walter Updegrave (2007), Money Magazine senior
editor, often fields questions on the 4% rule in his
Ask the Expert” forum. In his CNN Money article,
“Retirement: The 4 Percent Solution”, he recom-
mends that a 65-year-old person withdraw money
at the 4% rate, invest in a 50–50% mix of stocks
and bonds, and shift gradually more to bonds with
age. Further, he adds, “The reason many pros rec-
ommend this rate is because studies show that it
JOURNAL OFINVESTMENT MANAGEMENT THIRD QUARTER 2009
THE 4% RULE—ATWHAT PRICE?35
provides a high (but not absolute) level of assurance
that your savings will last 30 years or more.
The explanation of the 4% rule’s popularity is its
simplicity—it is easy to understand and implement.
Many of today’s retirees use the rule and have bene-
fited from its disciplined approach to spending and
investment. However, as we shall see, that benefit
comes with a price.
4 The parable of two boomers2
A simple example often yields insights that extend
to more complicated venues—this is true in our
case. In this section, we present a parable about
two boomers, who share similar spending goals, but
have radically different investment strategies. One
of their strategies wastes money, and we will show
that it can be replicated by a series of less-expensive
strategies. In later sections, we use this same
approach to show that the 4% rule wastes money.
Our two boomers, Eric and Mick, want to hear
Bob Dylan in concert. Dylan’s next concert has
three-ticket tiers: floor pass ($125), general admis-
sion ($75), and pay-per-view ($50). Although both
boomers would prefer a floor pass, unfortunately,
each boomer has only $100 to spend. Fortunately,
the concert promoters have set up a betting wheel
and will allow Eric and Mick to each wager on a sin-
gle spin. The betting wheel has three sectors of equal
area that are equally likely to be selected. If a sector is
selected, a chance on that sector will pay $1; other-
wise the price of the chance is lost. Our boomers can
purchase chances on any or all of the wheel’s sectors.
However, the promoters have contrived to make the
costs of the three sectors differ3—they are 10c, 30c,
and 40c for sectors one, two, and three, respec-
tively, or in vector short hand (10c, 30c, and 40c).
Eric decides to purchase 125 chances on each sec-
tor. He allocates his $100 wager as follows ($12.50,
$37.50, and $50.00) to receive the payouts ($125,
$125, and $125). Eric’s can’t-lose, no-blues strat-
egy is guaranteed to have a 25% return, and he will
be enjoying the concert from the floor. However,
Eric’s friend Mick, who briefly attended the London
School of Economics, likes high-return, high-
risk gambles. Mick makes the purchases ($35.00,
$15.00, and $50.00) and will receive the payouts
($350, $50, and $125). This strategy has a 75%
expected return and a 127% standard deviation.
Mick has focused on his betting strategy and has lost
sight of his quest, a concert ticket. In fact, if you
take his goal into account, Mick is actually wasting
money in three ways. First, Mick is paying for a
surplus he is not planning to spend. If sector one
occurs, Mick will get $350, buy a floor pass for
$125, and be left with a $225 surplus. At the rate
of a dime for a dollar, Mick is wasting $22.50 on
the surplus, or alternatively, he has wagered only
$77.50 of his $100 toward his ticket.
Second, Mick is also overpaying for his ticket
chances—even if the surplus is eliminated. Mick’s
payouts without the surplus are ($125, $50, and
$125). He has a two-third chance of a floor pass and
a third chance of watching on pay-per-view. How-
ever, consider the alternative strategy that wagers
($12.50, $37.50, and $20.00) and costs just $70.
It has the payouts ($125, $125, and $50) and
would give Mick the same ticket chances as his
current strategy and the same concert experience.4
The alternative plan is less expensive because it
pairs payouts and sectors so that the largest payouts
correspond to the cheapest sectors.
Third, suppose that Mick ranks his ticket prefer-
ences using Billboard bullets: a floor-pass gets five
bullets, a general admission ticket gets four bul-
lets, and pay-per-view gets just one bullet. It is clear
that Mick loves a live concert (he often performs
himself). Further, he would prefer (as measured
in expected bullets) a guaranteed general admission
ticket (4 bullets) to a two-third chance at a floor pass
and a third chance for pay-per-view (32/3bullets).
THIRD QUARTER 2009 JOURNAL OFINVESTMENT MANAGEMENT
36 JASON S. SCOTT ET AL.
The betting strategy ($7.50, $22.50, and $30.00)
would pay ($75, $75, and $75) and would guaran-
tee a general admission ticket. Since Mick prefers
the ticket chances of this $60 alternative strategy,
he is wasting an additional $10 by following his
current plan.
Our parable illustrates a number of key points. Eric
wanted a floor pass to the concert, and he placed
bets that paid $125 no matter what the wheel’s
outcome. Eric’s fixed spending was paired with a
risk-free strategy. On the other hand, Mick’s wager
has volatile payouts that overshoot the $125 goal
when sector one is the outcome, but fall short for
the remaining sectors. Generally, fixed spending is
best supported by fixed payouts—volatile payouts
lead to surpluses and deficits. In fact, Mick required
$22.50 of his original stake to fund his surplus—
wasted money. Further, Mick’s particular strategy
has wasted an additional $7.50 by overpaying for
his spending distribution. Generally, when payouts
are volatile, there may be alternative wagers that
generate the same payout distribution, and some
will be cheaper than others. Lastly, Mick arguably
wasted another $10.00 by subscribing to an infe-
rior ticket plan. More generally, when wagers and
preferences are at odds, even more money can be
wasted.
5 An experiment
Unfortunately, if a retiree adopts a 4% rule, he
faces outcomes much like Mick’s—he will waste
money by purchasing surpluses, will overpay for his
spending distribution, and may be saddled with an
inferior spending plan. In this section, we describe
a numerical experiment that demonstrates the first
two problems. For simplicity, we limit our analysis
to the specific, but representative, planning horizon
of 30 years.5
Our experiment uses a simple market model—an
economy with just two basic assets. The first is a
one-year bond that pays a guaranteed annual 2%
real return. A multiyear, risk-free bond can be repli-
cated by purchasing a series of these one-year bonds.
The second asset is a market portfolio consisting
of all other bonds and all stocks, held in mar-
ket proportions. The annual real returns on our
market portfolio are independent and log-normally
distributed with an expected value of 6% and a
standard deviation of 12%. Because the returns
are independent, they are serially uncorrelated. We
note investors can create portfolios with any desired
volatility by purchasing a mix of the two basic assets.
Our market model is similar to those used by
investment consultants for asset allocation and asset
liability studies. A 2% risk-free real rate is broadly
consistent with the historical record for U.S. Trea-
sury STRIPS and TIPS investment returns. In
addition, our market portfolio assumptions imply
a Sharpe ratio of a third, a fairly typical choice.
While the actual market values of bonds and stocks
vary over time, on average, bonds contribute about
40% of the value of the market portfolio and stocks
60%. Thus, a strategy that invests 100% in the
market portfolio can be thought of as a 60% equity
strategy.
An investor can guarantee a real dollar every year
for 30 years by purchasing a series of zero-coupon,
risk-free bonds. The cost of this investment is
the sum of the discounted prices6$1/(1.02) +
$1/(1.02)2+ ··· + $1/(1.02)30, which amounts
to a little less than $22.40. Alternatively, if a retiree
invests in a risk-free bond portfolio, he can safely
withdraw at a yearly rate that is a bit more than
$1.00/$22.40 4.46%. This withdrawal rate—
the guaranteed rate—is the maximum withdrawal
rate that can be guaranteed to never fail. This risk-
free strategy is analogous to Eric’s strategy and is a
special case of the 4% rule—the limit of zero invest-
ment volatility. This version of the 4% rule never has
a surplus, never has a shortfall, and is the cheapest
way to receive a constant, guaranteed payout every
JOURNAL OFINVESTMENT MANAGEMENT THIRD QUARTER 2009
THE 4% RULE—ATWHAT PRICE?37
year. If a cheaper investment were to exist, then
there would be an arbitrage opportunity.
The risk-free strategy supports constant spending
using the constant-return, risk-free bond—a perfect
match. However, a typical 4% rule advises support-
ing constant spending using a volatile-return, stock,
and bond mix—a mismatch. Our first set of exper-
iments will analyze constant mix portfolios with six
volatility levels, evenly spaced from 0% to 15%.
These levels correspond to market exposures from
0% to 125%, or alternatively, equity exposures of
from 0% to 75%.7We pair each of these portfolios
with a range of withdrawal rates that brackets the
guaranteed rate—4.0%, 4.25%, 4.46%, 4.75%,
and 5.0%. Except for the pair that corresponds to
Eric’s strategy, our benchmark, these pairs will all
exhibit at least one of Mick’s problems.
We will report three metrics for each pair of invest-
ment and spending strategies: the failure rate,
the cost of the surplus, and the overpayment for
the spending distribution. A strategy’s failure rate
is the probability that the actual spending in the
last year is less than the spending goal; in other
words, the probability of a shortage. Failure rates
(and success rates or confidence levels) have been
the main focus of previous investigators, and we
include them here for comparison. The final port-
folio value, the surplus, is a function of random
market returns. Hence, we can use the machinery
of derivative pricing to find its fair price, the cost of
funding the surplus. Similarly, we can compute the
present value of the actual spending and the price
of an alternative investment that delivers the same
spending distribution, but at the cheapest price.
The difference in these prices is the overpayment
for the spending strategy. We refer the reader to
Appendix A for more details.
For each pair of investment and spending strategies,
we numerically simulated many equally probable,
30-year-long, scenarios. The portfolio value for
each scenario was initially set to $100, without loss
of generality, and the annual spending goal was set
to the withdrawal rate times $100. We began each
scenario by drawing a random real market return
from its lognormal probability distribution, and
recorded its value. Given the investment strategy
and the returns on the risk-free bond and the mar-
ket, we determined the real value of the portfolio
at the end of the first year. Next, we computed
the first year’s actual real spending—the smaller of
the portfolio’s value and the spending goal—and
deducted it from the portfolio. The actual spending
amount and the portfolio’s post-withdrawal value
were recorded. This process was then repeated for
the remaining 29 years. We then computed the met-
rics of our experiment using Monte Carlo estimates,
for example, the failure rate was approximated by
the percentage of scenarios for which there was a
spending shortage. Again, we refer the reader to
Appendix A for more details.
5.1 Failure rates
Table 1 summarizes the failure rates for our first set
of experiments. The columns in Table 1 correspond
to constant mix investment strategies (labeled with
the strategy’s volatility), and its rows to constant real
spending strategies (labeled with the strategy’s with-
drawal rate). The investment portfolios are annually
rebalanced to maintain a constant risk level. The
numbers in the body of the table are the estimated
failure rates, the percent of scenarios that fell short of
the spending goal. For example, with 12% portfo-
lio volatility and a 4.25% withdrawal rate, spending
had a shortfall in 8.1% of cases. The standard errors
of the estimates reported in this table are less than
0.01%. The failure rates reported in Table 1 are
consistent with the previous studies.
Table 1 illustrates several key features of the 4% rule.
First, the influence of portfolio risk depends criti-
cally on the spending level. For rates less than or
equal to the guaranteed rate, adding risk necessarily
introduces the possibility of failure. Moreover, the
chance of failure increases with portfolio volatility.
THIRD QUARTER 2009 JOURNAL OFINVESTMENT MANAGEMENT
38 JASON S. SCOTT ET AL.
Table 1 Failure rates for constant-mix portfolios.
Constant-mix volatility
Withdrawal
rate 0% 3% 6% 9% 12% 15%
4.00% 0.0% 0.3% 1.9% 3.9% 5.7% 7.6%
4.25% 0.0% 1.9% 4.4% 6.3% 8.1% 9.9%
4.46% 0.0% 6.8% 7.9% 9.2% 10.6% 12.1%
4.75% 100.0% 22.5% 15.0% 14.0% 14.5% 15.4%
5.00% 100.0% 44.2% 23.4% 19.2% 18.4% 18.7%
Table 2 Cost of surpluses as a percentage of initial wealth for constant-mix portfolios.
Constant-mix volatility
Withdrawal
rate 0% 3% 6% 9% 12% 15%
4.00% 10.4% 10.8% 13.0% 15.8% 18.8% 21.8%
4.25% 4.8% 6.3% 9.3% 12.5% 15.7% 19.0%
4.46% 0.0% 3.4% 6.8% 10.1% 13.5% 16.8%
4.75% 0.0% 1.2% 4.2% 7.5% 10.8% 14.2%
5.00% 0.0% 0.4% 2.7% 5.7% 8.9% 12.2%
However, for withdrawal rates above the guaranteed
rate, investing in the risk-free asset is guaranteed to
fail, so adding risk to the portfolio is the only pos-
sible hope for success. For example, by increasing
the portfolio volatility from 0% to 3%, the 4.75%
spending plans failure rate drops from 100% to
22.5%. However, beyond a certain level, additional
risk begins to increase the chances of failure. For
example, with the 4.75% withdrawal rate, a port-
folio volatility of 9% has the smallest failure rate,
that is, 14.0%.8It is this pattern that led some
researchers to devise strategies that minimize the
failure rate and the risk of ruin.
5.2 Cost of surpluses
Retirees who follow a 4% rule will often generate
portfolio surpluses that waste money. In Table 2,
we report the results for our numeric experiment.
The layout of Table 2 is the same as Table 1; how-
ever, the values in its body are estimated percentages
of initial wealth that are spent on funding sur-
pluses. The standard errors of all estimates are
uniformly less than 0.05%. Clearly, withdrawing
less (more) than the guaranteed rate and invest-
ing in the risk-free bond always (never) generates
surpluses. However, for all rates, volatility adds
significantly to the surplus and shifts money away
from retirement spending. In particular, an investor
withdrawing at the guaranteed rate (4.46%) and
investing in the market portfolio (12% volatility)
has allocated 13.5% of his portfolio to surpluses,
and only 86.5% to actual spending. Strikingly,
the surplus from this strategy could be used to
increase every retirement payout by nearly 16%
(13.5%/86.5%).
5.3 Spending overpayments
In Table 3, we present our third metric—the spend-
ing overpayment. This table’s structure is the same
as its two predecessors, but the values in its body are
JOURNAL OFINVESTMENT MANAGEMENT THIRD QUARTER 2009
THE 4% RULE—ATWHAT PRICE?39
Table 3 Spending overpayments as a percentage of initial wealth for constant-mix portfolios.
Constant-mix volatility
Withdrawal
rate 0% 3% 6% 9% 12% 15%
4.00% 0.0% 0.2% 1.1% 1.9% 2.5% 3.0%
4.25% 0.0% 0.7% 1.6% 2.4% 3.0% 3.5%
4.46% 0.0% 1.2% 2.1% 2.8% 3.4% 3.8%
4.75% 0.0% 1.7% 2.6% 3.3% 3.8% 4.2%
5.00% 0.0% 1.9% 2.9% 3.6% 4.1% 4.5%
the estimated percentages of initial wealth wasted
by obtaining the spending distribution using the
4% rule versus a more cost-effective strategy. The
standard errors of all estimates are uniformly less
than 0.05%. The waste increases with increasing
withdrawal rate, since relatively more money goes
toward spending, and also increases with portfolio
volatility. With a market portfolio investment (12%
volatility), spending overpayments claim an addi-
tional 2–4% of a retiree’s initial portfolio wealth.
In the next section, we shall look at least cost
strategies in greater detail and see how to eliminate
overpayments.
6 Least cost spending strategies
We have identified and quantified the 4% rule’s
first two sources of waste—funding unspent sur-
pluses and overpaying for spending. However, what
investment strategy can a retiree use to recoup these
losses? In this section, we answer that question and
also demonstrate a useful diagnostic tool for discov-
ering spending overpayments. For concreteness, we
consider a retiree that has a 30-year horizon, starts
with $100, withdraws at the 4.46% guaranteed rate,
and invests in the market portfolio.
Figure 1 is a scatter plot of our retiree’s last year
of spending versus the cumulative market return
over the 30-year period since his retirement began.
For convenience, the cumulative market return has
been annualized. Each point on the plot represents
$0.00
$1.00
$2.00
$3.00
$4.00
$5.00
4.0% 2.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0%
Cumulative market return (annualized)
Spending
A
B
Figure 1 Spending versus cumulative market return in
year 30 for a 4.46% withdrawal rate and the market
portfolio.
a scenario. As shown in Table 1, our retiree spent
less than his $4.46 goal in 10.6% of the scenarios,
and in fact, he spent nothing at all in 9.6% of them.
The remaining 89.4% of the scenarios had sufficient
funds to fully support the $4.46 spending goal. The
Monte Carlo estimate of the cost of purchasing the
last year’s payout is 96c. Note that if our retiree
had invested in the risk-free bond, he would have
been guaranteed a $4.46 payout in the last year.
The cost of the bond investment would have been
$4.46/1.0230 $2.46. Hence what happened to
the $1.50 saved by allowing deficits? It was wasted
on funding surpluses.
Our retiree spent 96c to fund the last year of his
plan, but he could have obtained its spending distri-
bution more cheaply.9To see why, consider points
A and B in Figure 1. Point A is a scenario for
which spending was $4.46 and the market averaged
a mere 21 basis point increase over 30 years—times
THIRD QUARTER 2009 JOURNAL OFINVESTMENT MANAGEMENT
40 JASON S. SCOTT ET AL.
$0.00
$1.00
$2.00
$3.00
$4.00
$5.00
4.0% 2.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0%
Cumulative market return (annualized)
Spending
Figure 2 Spending versus cumulative market return in
year 30 for the least cost-spending strategy.
were tough. In contrast, point B is a scenario for
which spending completely disappeared, and yet
the market averaged a hefty 6.92% real, annual-
ized increase. Since the cost of money is cheaper
in good times than it is in bad times, our retiree
could have saved money by finding an alternative
investment that paid nothing in scenario A, $4.46
in scenario B, and kept the payouts in all the remain-
ing scenarios unchanged. If our retiree exploited all
these money-saving swaps, he would have a least
cost strategy (Dybvig, 1988a, 1988b). Figure 2 is a
scatter plot of the least cost-spending strategy ver-
sus the cumulative market return. The spending
distributions of Figures 1 and 2 are the same—
the spending amounts have the same values and
frequency—however, the spending amounts have
been reassigned to different market returns and have
different underlying investment strategies. Our esti-
mated price to implement the least cost strategy is
69c, a savings of 27c over the 4% rule.10
Generally, scatter plots of spending versus the
cumulative market return can identify overspend-
ing. A least cost strategy will plot as a smooth
non-decreasing curve (see Figure 2), and spending
will be higher when prices are cheaper. For exam-
ple, the plot of constant, risk-free spending versus
the market is a horizontal line, is non-decreasing,
and is a least cost strategy. Inefficient strategies will
either exhibit scatter (see Figure 1) or, if they are
smooth, have a region where spending decreases
with cumulative market return.
What investments should our retiree make to get the
least cost? Because the shape of Figure 2 is approxi-
mately flat–ramp–flat, a financial engineer would
suggest a strategy of buying and selling 30-year,
European call options on the market portfolio. Our
retiree would purchase calls with an exercise price
equal to the rightmost value for which the payouts
are zero. Simultaneously, he would sell an equal
number of calls with an exercise price equal to the
leftmost value for which the payouts equal $4.46.
By adjusting the numbers of calls purchased and
sold, a payoff diagram very similar to that shown
in Figure 2 would be obtained. Specifically, sup-
pose that Ncalls are bought at the strike price
Kb=F(P0) and sold at Ks=F(PF), where P0
and PF(the failure rate) are the probabilities of a
zero withdrawal and a shortage, respectively, and
the function F(·) is the inverse cumulative proba-
bility distribution for the (log-normal) cumulative
market return. The number of calls to buy and sell
is N=δ·W0/(KsKb), where δis the withdrawal
rate and W0is the initial portfolio value. The prices
of the calls are easily determined from the Black–
Scholes formula. For our example, P09.56%,
PF10.58%, and δ·W0=$4.46, and we
get the values Kb$2.12 (2.53% annualized),
Kb$2.19 (2.65% annualized), and N=57.97.
The net cost of the replicating call options is 68.5c,
which equals our Monte Carlo price to three digits.
The above analysis could be repeated to cover all 30
years of spending. Efficient strategies for each year
would entail the purchase and sale of call options
with different exercise prices. The total cost would
equal the cost of all the options purchased minus
the proceeds from selling all the options written.
For this particular investment and spending com-
bination, our retiree would save 3.4% of his initial
portfolio by adopting the least cost strategy, where
0.27% of the savings can be attributed to the last
year.
It is, of course, possible that no counterparty would
offer the multiyear options necessary for a retiree
JOURNAL OFINVESTMENT MANAGEMENT THIRD QUARTER 2009
THE 4% RULE—ATWHAT PRICE?41
to replicate the 4% rule’s spending distribution. If
not, it is still possible to replicate the least cost
strategy by following a dynamic strategy using the
riskless bond and the market portfolio. Further,
if enough retirees wanted the 4% rule’s spending
distribution and the least cost-investment strategy
was sufficiently below other readily available invest-
ment options, then financial institutions would be
motivated to offer new products and services that
replicated the spending distribution at a cheaper
price for the retiree, but at a profitable price for the
institution.
The key to recouping surpluses is to just not fund
them—just fund spending. The key to recouping
overpayments is to spend relatively large amounts
when payouts are cheap and to spend relatively small
amounts when payouts are expensive. A retiree can
get the 4% rule’s spending distribution at a much
cheaper cost by buying and selling call options or by
implementing a dynamic strategy. However, does he
really want this spending distribution or is there a
cheaper distribution he would prefer?
7 Spending preferences and expected utility
Mick wagered $100 and had a two-third chance of
winning a floor pass and a third chance of watching
Dylan on pay-per-view (32/3bullets). Mick’s gam-
ble can be replicated for just $70 by eliminating a
surplus and using a least cost strategy, but this is still
wasteful, since Mick would have preferred a guar-
anteed general admission ticket (4 bullets), which
he could have had by wagering just $60. In our
parable, the money spent on surpluses and over-
payments is only a lower bound on waste; choosing
a strategy whose payouts are inconsistent with pref-
erences wastes even more. In this section, we argue
that this result is also true for a retiree who follows
a 4% rule, but prefers a different spending plan.
Does a retiree really want the spending distribution
of a 4% rule strategy? Consider a retiree who is
planning on spending $40K a year for the next 30
years. Now, suppose that after the first year, his port-
folio suffers a 20% loss. Without knowing much
about our retiree, what would you expect him to do?
Would he continue spending $40K, cut his spend-
ing by 20% to $32K, or do something in between?
We might reasonably expect him to spread the pain
of his loss across all his remaining years, and adopt
one of the latter two options. However, if our retiree
were a strict follower of the 4% rule, he will con-
tinue spending $40K a year, until his portfolio is
exhausted.11 It seems unlikely that a retiree would
prefer fixed dependable spending in his early retire-
ment years and the volatile spending of Figures 1
and 2 in his late retirement years. Our retiree could
have addressed this issue much earlier, when he cre-
ated his plan, either by investing in a riskless bond
or by relaxing his requirement for fixed spending.
However, without knowing more about his prefer-
ences, we do not know which option would have
been more appropriate.
Financial economists advocate a very different
approach toward retirement spending and invest-
ing. Building on the seminal work of Hakansson
(1970), they suggest that a retiree’s preferences con-
cerning different amounts of income in different
years be represented with a set of utility functions.
Then they determine an integrated investment and
spending policy that can be funded with the retiree’s
initial wealth and that will maximize the expected
value of his overall utility.
Figure 3 places our earlier analysis in the context
of retiree preferences and utility. Each point on
the plot is a retirement strategy. The vertical axis
plots the expected utility of a strategy, and the
horizontal axis plots the cost of the strategy. The
strategies that maximize the expected utility plot
on the solid curve. The strategy labeled A represents
the traditional 4% rule—an initial portfolio value
of $100, withdrawal rate of 4.0%, and 60–40%
stock-bond mix (i.e., the market portfolio). This
strategy does not maximize expected utility—it has
THIRD QUARTER 2009 JOURNAL OFINVESTMENT MANAGEMENT
42 JASON S. SCOTT ET AL.
0
20
40
60
80
100
120
$0 $100
Cost
Expected Utility
4% Rule
B
C
C
B
A
Figure 3 Expected utility versus cost for various retire-
ment strategies.
unspent surpluses and overpays for spending—and
must plot below the curve. The least cost-spending
strategy has the same spending distribution as the
4% rule, and therefore the same expected utility.
However, this strategy recoups the $18.80 spent on
surpluses (Table 2) and the $2.50 spent on overpay-
ments (Table 3), and can be purchased for $78.70.
The least cost-spending strategy (labeled B) plots
to the left of the 4% rule strategy (labeled A) on
Figure 3.
Generally, we do not expect strategy B to maximize
a retiree’s expected utility, and therefore, there may
be many strategies that deliver the same benefit as A,
but at a lower cost than B’s. For example, suppose a
retiree truly prefers constant spending—even a 1%
chance of failure makes him extremely uncomfort-
able. He might be equally happy if he traded his
current strategy that targets a $4 withdrawal, but
has a chance of shortfalls, for a new strategy, which
pays half as much, but is guaranteed. In Figure 3,
we have labeled this case as C—it invests in the
risk-free bond and costs $44.80. By knowing this
retiree’s preferences, we can make him just as happy
as he was with the 4% rule, but at less than half
the cost.
Of course, no advisor would recommend strat-
egy B or C, since they both cost less than the
available $100 and would leave money on the table.
One strategy, labeled B, fully funds the least cost-
spending strategy, but uses the savings on surpluses
and overpayments to increase all the payouts. This
strategy costs $100, increases all payouts by more
than 27%, and plots directly above the 4% rule in
Figure 3. It is unlikely that this strategy would max-
imize the expected utility, unless our retiree enjoys
the 4% rule’s occasional spending shortages and his
only mistake is failing to find the least cost way of
achieving them. Hence, typically, there are many
strategies with higher expected utility than Bthat
cost the same. If Cis one of these preferred strate-
gies, it will plot above Bin Figure 3. As an example,
consider a retiree who has no tolerance for risk and
prefers investing in the risk-free bond. He would
use his full $100 to get a guaranteed $4.46 annual
payout. This strategy’s payout is at a minimum 10%
higher than any payout of the 4% rule, and never
fails to pay. It clearly dominates the 4% rule for
all retirees, will not dominate the least cost strat-
egy for a few retirees, and will be likely dominated
itself for retirees who have a modest tolerance for
risk.12
To actually reach the efficient curve on our figure,
we need to know a retirees utility function. How-
ever, much of the gap between the efficient curve
and the 4% rule can be closed with a less for-
mal description of a retiree’s preferences. Basically,
retirees need to understand that the cost of provid-
ing spending when markets perform poorly is often
substantially greater than when they perform well.13
If this point is well understood, then most retirees
are likely to choose to spend less in bad scenarios in
order to be able to spend more in good scenarios,
versus spending the same amount in all scenarios.
The willingness to make these spending tradeoffs
and to assume the investment volatility needed to
implement them will differ among investors, and
therefore so should their retirement strategies.
JOURNAL OFINVESTMENT MANAGEMENT THIRD QUARTER 2009
THE 4% RULE—ATWHAT PRICE?43
8 Glide-path investment strategies
Our first set of experiments focused on constant-
mix investment strategies—portfolios are annually
rebalanced to maintain a constant volatility level.
This was the original approach of Bengen (1994),
and it is often recommended. Recently, however,
glide-path investment strategies have experienced a
surge in popularity. These strategies systematically
reduce portfolio volatility as a retiree gets older,
and several authors, including Bengen, have sug-
gested that retirees adopt this alternative. Two years
after his original article, Bengen (1996) concluded
that, “All things considered, I recommend that you
adopt a phase-down of one percent of your stock
allocation each year ...” Jennings and Reichenstein
(2007) analyzed a number of lifecycle mutual funds
and found that these funds, even when intended for
post-retirement use, generally follow an investment
glide path, averaging an equity percentage equal to
approximately 120 less the target investor’s age.
We have argued that the major flaw of the 4%
rule is its attempt to support nonvolatile spending
with volatile investing. Perhaps a glide path that
systematically reduces risk could cause less ineffi-
ciency and reduce failure rates? Using our previous
withdrawal rates and our previous volatility levels,
we ran a numeric experiment to test this hypothe-
sis. However, this time the volatility level was only
used as the initial level for the first year’s invest-
ments, and thereafter, the level was decreased each
Table 4 Failure rates for glide-path portfolios.
Initial-mix volatility
Withdrawal
rate 0% 3% 6% 9% 12% 15%
4.00% 0.0% 0.2% 2.0% 4.1% 6.0% 7.7%
4.25% 0.0% 2.1% 5.3% 7.4% 9.1% 10.6%
4.46% 0.0% 9.8% 10.6% 11.5% 12.5% 13.5%
4.75% 100.0% 35.0% 21.6% 18.6% 17.9% 17.9%
5.00% 100.0% 63.9% 34.2% 26.2% 23.3% 22.3%
year so that in the 30th year the volatility was zero,
and the portfolio was invested in just the risk-free
bond.
The results of our glide-path experiments are
reported in Tables 4–6, which correspond to
Tables 1–3 of our constant-mix experiments. Com-
paring the failure rate tables, we notice that except
possibly for the lowest volatilities, the glide-path
rates are worse—even though the glide-path port-
folios were on average less risky than the corre-
sponding constant mix portfolios. The reason for
this behavior is that glide-path strategies are just as
likely as constant-mix strategies to have a series of
poor early returns and run out of money. However,
the glide-path strategy tends to lock-in these poor
returns, decreasing the likelihood that future good
returns allow the portfolio to recover and sustain
spending.
A comparison of Tables 5 and 2, the cost of sur-
pluses tables, shows that glide-path strategies do
a bit better than constant-mix strategies in this
category. Basically, a glide-path strategy generates
a smaller surplus. For the market portfolio, the
glide-path strategy spent approximately 2.5% less
on surpluses than the corresponding constant-mix
portfolio. However, a comparison of Tables 6 and 3,
the overpayment tables, shows that the savings
on surpluses are offset by higher overpayments.
For the market portfolio, the glide-path strategy
spent approximately 1–2% more on overpayments
THIRD QUARTER 2009 JOURNAL OFINVESTMENT MANAGEMENT
44 JASON S. SCOTT ET AL.
Table 5 Cost of surpluses as a percentage of initial wealth for glide-path portfolios.
Initial-mix volatility
Withdrawal
rate 0% 3% 6% 9% 12% 15%
4.00% 10.4% 10.6% 11.9% 14.0% 16.3% 18.7%
4.25% 4.8% 5.7% 8.1% 10.6% 13.2% 15.8%
4.46% 0.0% 2.7% 5.4% 8.1% 10.8% 13.5%
4.75% 0.0% 0.7% 3.0% 5.6% 8.2% 10.9%
5.00% 0.0% 0.2% 1.6% 3.9% 6.4% 9.0%
Table 6 Spending overpayments as a percentage of initial wealth for glide-path portfolios.
Initial-mix volatility
Withdrawal
rate 0% 3% 6% 9% 12% 15%
4.00% 0.0% 0.2% 1.5% 2.7% 3.7% 4.5%
4.25% 0.0% 0.9% 2.4% 3.5% 4.4% 5.1%
4.46% 0.0% 1.8% 3.1% 4.1% 5.0% 5.6%
4.75% 0.0% 2.5% 3.8% 4.8% 5.6% 6.2%
5.00% 0.0% 2.6% 4.2% 5.3% 6.0% 6.7%
than the corresponding constant mix strategy. In
total, the surpluses and overpayments of glide-path
strategies are comparable to constant mix strategies.
Further, since we expect that the spending distribu-
tions generated by glide-path strategies are likely at
odds with a retiree’s true preferences, the costs of
surpluses and overpayments are again only a lower
bound on the total money wasted by adopting these
strategies.
9 Conclusion
The 4% rule and its variants finance a con-
stant, nonvolatile spending plan using a risky,
volatile investment strategy. Two of the rule’s
inefficiencies—the price paid for funding its
unspent surpluses and the overpayments for its
spending distribution—apply to all retirees, inde-
pendent of their preferences. For a typical rule, we
used a market model to estimate that between 10%
and 20% of a portfolio’s initial wealth is being
allocated to surpluses, and an additional 2–4% is
going toward overpayments. If the spending dis-
tribution of the 4% rule were inconsistent with a
retiree’s preferences, then the costs can be much
higher. All in all, any retiree who adopts a 4% rule
pays a high price.
Our approach can be easily extended to investigate
other retirement rules of thumb and to use alterna-
tive market models. If a retirement plan generates
unspent surpluses then our approach can price the
surplus. A scatter plot of spending amount ver-
sus cumulative market return will quickly reveal
whether a strategy is of least cost type. Strategies
with overpayments will often generate a cloud of
points (Figure 1), whereas least cost strategies will
generate a non-decreasing curve (Figure 2).
Many practical issues remain to be addressed before
advisors can hope to create individualized retire-
ment financial plans that maximize expected utility
JOURNAL OFINVESTMENT MANAGEMENT THIRD QUARTER 2009
THE 4% RULE—ATWHAT PRICE?45
for investors with diverse circumstances, other
sources of income, and preferences. While we still
may be far away from such an ideal, there appears
to be no doubt that a better approach can be found
than that offered by combinations of desired con-
stant real spending and risky investment. Despite
its ubiquity, it is time to replace the 4% rule
with approaches better grounded in fundamental
economic analysis.
Appendix A. Pricing
Our simple two-asset economy is complete—any
series of payouts, volatile or nonvolatile, can be
dynamically replicated using just the risk-free bond
and the market portfolio. Further, we can assign
a fair, no-arbitrage price to the payouts using the
economy’s state price density function or pricing
kernel (refer to Cochrane (2005) and Sharpe (2007)
for a full discussion). This approach is often used
to price derivative securities, and we use it in this
study to price portfolio withdrawals for spending
and surpluses. In this framework, the current price
Pof an asset that pays a random amount Ctin t
years is equal to the expected value:
P=E[Ct·Mt](A.1)
where the random variable Mtis the pricing ker-
nel. Generally, both Ctand Mtare functions of
the random market returns R1,R2,...,Rt, and the
expected value E[·] in the above equation is with
respect to the joint-density distribution of these
returns.
Equation (A.1) can be used to price the surplus at
the horizon t=Tand to price each annual spend-
ing withdrawal at years t=1, 2, ...,T. The price
of all withdrawals is equal to the sum of the prices
for these annual withdrawals. Note that in our anal-
ysis, the total of all prices (withdrawals and surplus)
will equal the initial portfolio value.
Usually, the pricing kernel Mtdepends on the
annual market returns via the cumulative market
return Vt. In this Appendix, we assume that all
returns are gross returns (ratios of ending value to
beginning value) so that in particular, the cumula-
tive market return is just the product of consecutive
annual market returns (i.e., Vt=R1·R2···Rt).
Hence, whereas the payout often depends on the
order of the returns, the pricing kernel’s value is
independent of order—a low return followed by
a high return is equivalent to the high return fol-
lowed by the low return. When the annual market
returns have identical log-normal distributions and
are independent, the pricing kernel is simply:
Mt=At/Vb
t(A.2)
The positive parameters Aand bare given by the
formulas:
A=(Em·Rf)b1(A.3a)
b=ln (Em/Rf)
ln (1 +S2
m/E2
m)(A.3b)
In the above equations, Rfis the total risk-free
return, Em=E[Rt]is the yearly expected total
market return, and Sm=(Var[Rt])1/2is the annual
market volatility. For our experiments, we use the
values Rf=1.02, Em=1.06, and Sm=0.12,
and so we get the approximate values A1.08 and
b3.02 for the two kernel parameters.
Because the exponent on the cumulative market
return in Eq. (A.2) is positive, payouts are relatively
cheap to secure when the market has high returns.
This is a standard result of asset pricing theory: mar-
ket prices must adjust until the total demand for
claims paying off in states of scarcity will be less than
that for claims paying off in times of plenty. To the
extent that cumulative market returns serve as ade-
quate proxies for overall consumption (scarcity or
plenty), we expect the pricing kernel to be a mono-
tonically decreasing function of cumulative market
return.
Our Monte Carlo simulation generates a number
of equally likely random paths, where each path is
THIRD QUARTER 2009 JOURNAL OFINVESTMENT MANAGEMENT
46 JASON S. SCOTT ET AL.
a scenario of annual market returns of length T.
For each path, we compute sample values for the
payout and the pricing kernel. In particular, let C(i)
t
and M(i)
tbe the payout and kernel values for the ith
path. Using Nof these paths, we can estimate the
expected value of Eq. (A.1) with a sample mean and
get the following approximate price:
P1
N
N
i=1
C(i)
t·M(i)
t(A.4)
The error of this approximation can be made as
small as we want by choosing the number of paths
Nto be sufficiently large.
For complete markets, Dybvig (1988a, 1988b)
derived an elegant formula for computing the min-
imum price for receiving any payout distribution.
Dybvig creates a dynamic investment strategy that
replicates the payout distribution, but purchases
payouts most efficiently—the highest payout is pur-
chased at the cheapest price (when markets perform
the best), the next highest is purchased at the
next cheapest price, and so on. More formally, let
Ft(c)=P(Ctc) and Gt(v)=P(Vtv)be
the cumulative probability densities for the random
payout Ctand market value Vt, respectively. We can
implicitly define the random payout Xtas a func-
tion of the cumulative market value Vtusing the
following equation:
Ft(Xt)=Gt(Vt) (A.5)
For any percentile, let us say the 75th, the 75th
percentile of the market value will be paired with
the 75th percentile of the payout distribution of Ct,
which becomes the 75th percentile of the payout Xt.
Hence, Xtincreases with market value, decreases
with pricing kernel, and has the same distribution
as Ct. Finally, if we replace Ctby Xtin Eq. (A.1),
we obtain the formula for the minimum price.
The Monte Carlo simulation is easily modified to
estimate the minimum price. After simulating N
pairs of payouts and pricing kernels, each variable is
separately sorted. The payouts are sorted in ascend-
ing order, the kernel values in descending order.The
sorted values are re-paired and fed into Eq. (A.4).
The result is an estimated minimum price. In prac-
tice, we generated a large number of paths (N=
50,000) for our estimates, and then further refined
the results by averaging a large number (500) of
independent estimates. For the results we present
in this study, the standard deviation of the esti-
mated prices was uniformly less than 5c for an
initial wealth of $100.
Acknowledgments
The authors thank Robert L. Young, CFA of Finan-
cial Engines for suggesting the title of this study and
Wei-Yin Hu, Ph.D. of Financial Engines for care-
ful reviews of earlier drafts and a number of helpful
comments.
Notes
1For brevity, we will refer to the typical retiree in the singular
and use the male pronoun. However, all our arguments
apply equally well to single females, a married couple, and
partnerships.
2The author who is actually a member of the baby boom
generation wrote this section. He assures his younger and
older colleagues that the parable will have deeper meaning
for readers of his cohort.
3The betting wheel’s sectors have the same $1 payout, have
the same chance of occurrence, but have different costs.
Similarly, in an economy, a boom and a recession may
have the same chance of occurring, but the cost of a $1
in the former state is typically much cheaper than in the
latter state.
4We assume that Mick only cares about the distribution of
tickets and not on the sector chosen by the wheel. More
formally, Mick’s utility is not state dependent.
5Modeling mortality is a very important aspect of retire-
ment planning, and by assuming a fixed retirement period,
we have made an extreme simplification—retirees drop
dead at a specific, deterministic, future date. Further, we
are implicitly assuming that any remaining portfolio value
at that date is wasted. That said, any serious attempt to
include mortality would introduce at a minimum annu-
ities and bequest motives—two complexities that we defer
JOURNAL OFINVESTMENT MANAGEMENT THIRD QUARTER 2009
THE 4% RULE—ATWHAT PRICE?47
to another investigation. Annuities allow retirees to pool
mortality risk, and bequests allow retirees to transfer
surpluses to heirs.
6This is the formula for the present value of an ordinary
simple annuity paying a dollar at the end of each year for
30 consecutive years.
7We included the leveraged market portfolio (125%) in the
analysis to match the volatility of the oft-recommended
75% equity strategy. This implicitly assumes that investors
can borrow at the risk-free rate. If borrowing costs are
higher, the expected return for this portfolio can be
adjusted downward.
8More simulations could be run to refine the estimate of
the volatility that minimizes the failure rate for this, and
any other, withdrawal rate.
9We assume that our retiree does not have state-dependent
utility, cares only about the probability distribution of
spending in each year, and does not care about his spending
sequences.
10 An alternative way to view the inefficiency with the payout
plan shown in Figure 1 is to recognize that our retiree is
exposed to two sources of risk. He does not know where
on the horizontal axis the market’s cumulative return will
plot. This is market risk, for which an investor can expect
to be rewarded in an efficient capital market. But even if
the market’s cumulative return were known, there may still
be uncertainty concerning his payout, since there are mul-
tiple payouts for many of the vertical slices. Our retiree’s
payouts are not only dependent on the cumulative return
on the market, but also on the order of the returns; that
is, the market path. It is this path dependency that causes
the observed multiplicity and creates additional risk. This
additional risk, called path risk or sequence risk, is not
rewarded in a capital market conforming to the standard
assumptions of most asset pricing theories.
11 Many devotees of the 4% rule recommend adjusting
spending and investment after a dramatic market move,
but some are strict adherents. For example, Guyton
(2004) describes a retired couple, who after the bear mar-
ket of 2000–2002, wonder if they “should reduce their
withdrawals to keep their rate at roughly four percent”
(of their depressed 2002 portfolio values). He advised
them “that—on the contrary—it was fine if their cur-
rent withdrawal rates approached six percent or even seven
percent!”
12 We can also plot Mick’s gambles on Figure 3. The points
A, B, and C, correspond to his original gamble ($100),
the equivalent least cost gamble ($70), and the preferred
guaranteed general admission ticket ($60), respectively.
In Mick’s case, the point Bmay correspond to a strategy
that has a third chance of a general admission ticket and a
two-third chance of a floor pass versus his original gamble
that has a third chance of pay-for-view and a two-third
chance of a floor pass. For Mick, the point Cwould
be Eric’s gamble, the guaranteed floor pass, and since
this gamble is optimal for Mick, Cwould plot on the
curve.
13 In Section 6, we estimated that the price of the least cost
strategy for the 4% rule’s spending distribution in year 30
was 69c. This spending distribution pays $4.46 in over
89% of the top performing market scenarios. A risk-free
bond would cover all market scenarios, pay $4.46, and cost
$2.46. Hence, it costs another $1.97 to cover the bottom
11% of market scenarios; that is, almost three times as
much to cover one-ninth of the scenarios.
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Keywords: Retirement economics; expected utility;
fixed withdrawals
JOURNAL OFINVESTMENT MANAGEMENT THIRD QUARTER 2009
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