Safe Withdrawal Rates and CertaintyEquivalent Spending
by Druce Vertes
"Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness. Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery." —Mr. Micawber (From Charles Dickens’ “David Copperfield,” 1850)
Financing a safe retirement with a risky portfolio depends on tradeoffs. Increasing spending today is desirable, but it increases the risk of shortfall in the future. High variation in spending is undesirable, but upward adjustments let you increase spending when investments outperform, while downward adjustments after underperformance reduce risk of even deeper future cutbacks. Finally, accepting future spending variation enables higher spending today, if you invest in higherrisk, higherreturn portfolios, or accept a thinner buffer against downward adjustments.
William P. Bengen (1994) pioneered safe withdrawal literature by studying a constant spending strategy. Using this strategy, the retiree determines an appropriate spending amount at retirement, and adjusts it annually for inflation to achieve constant real spending. The model I discuss in this article is a generalization of the Bengen approach to incorporate a variable spending term (a fraction of a smoothed portfolio value, adjusted for remaining life expectancy). If you recalculate a Bengen rule and update spending every few years, or after significant portfolio changes, you approach a smoothed variable spending strategy. Everything that follows will use real, inflationadjusted values.
In this article, I try to accomplish three things:
 Use Bengen’s approach to test strategies incorporating a constant and variable spending amount based on the size of the portfolio.
 Describe the shape of the tradeoff: How much do shortfall risk and severity increase when you increase initial spending?
 Use certaintyequivalent spending to describe how, as you reduce your risk aversion and accept risk and variation in spending, you increase your lifetime spending.
The Withdrawal Rate Model
Bengen’s model had one parameter: constant spending. I added three parameters to determine variable spending that changes based on portfolio size:
 A smoothing parameter (n): I smooth spending variation by basing spending on an exponential moving average of the portfolio value.
 A mortality insensitivity parameter (b): I set the amount available to spend as the smoothed portfolio divided by the remaining life expectancy + b. An 80yearold can safely spend a higher percentage of a $1 million portfolio than a 65yearold, and higher b means spending adjusts more slowly as you age.
 A variable spending parameter (h): How much of the available amount is spent.
The complete spending model is:
s_{i} = K + EMA(P,n)_{i} h
(L_{i} + b)
Where (in addition to parameters above):
 s_{i} is spending in period I,
 K is constant spending,
 EMA(P,n)_{i} is the exponential moving average of portfolio values and
 L_{i} is the retiree’s remaining life expectancy in period i.
If h (variable spending) is 0, this reduces to the Bengen constant spending model. If K (constant spending) is 0, n is 1 (no smoothing) and b is large (minimal mortality updating), it’s similar to a constant percentage model that spends a fixed percentage each year.
Testing Variable Spending in Retirement
I tested about 3.7 million combinations of parameters and portfolios. The test subject is a hypothetical single 65yearold male retiree, who chooses a withdrawal strategy and allocates between a U.S. largecapitalization stock index and a bond index. (The returns are represented by the Ibbotson data for largecap stocks and intermediateterm U.S. government bonds from 1926–2012. For life expectancies, I used Social Security Administration actuarial life tables, which are at www.ssa.gov/oact/STATS/table4c6.html, accessed in May 2013.)
For each portfolio and set of parameters tested, I determined:
 Initial spending;
 Lifetime expected spending;
 Worst shortfall, defined as maximum decline below initial spending (I also calculated shortfall percentiles: 90th percentile = an outcome where 90% of outcomes had better worstcase shortfalls); and
 Shortfall probabilities (probability of a decline below initial spending; probability of 10% decline, etc.).
If a scatter graph with every possible combination of initial spending versus shortfall severity using every parameter value tested is plotted, a cloud is plotted. The top left boundary of the cloud represents the highest initial spending achievable for a given maximum shortfall. Figure 1 shows selected outcomes from this “possibility frontier” or “shortfall severity frontier.”
Probability  
Initial  Worst  Equity  of 10%  Lifetime  
Spending*  Shortfall  Allocation  Constant  Variable  Mortality  Shortfall  Spending  
(%)  (%)  (%)  Spending  Spending  Smoothing  Insensitivity  (%)  (%) 
3.9  0.0  50  3.50  0.1  25.0  9  0  70.68 
4.0  1.1  45  3.75  0.1  25.0  30  0  69.99 
4.1  6.1  45  3.75  0.1  3.0  16  0  72.46 
4.2  14.1  45  3.5  0.4  3.5  40  3  75.18 
4.3  16.4  45  3.5  0.4  1.5  35  7  76.56 
4.4  23.2  45  3.25  0.6  2.5  35  22  79.36 
4.5  25.2  45  3.25  0.7  1.5  40  27  80.22 
4.6  30.4  50  3.00  0.9  1.5  40  32  83.2 
4.7  33.2  50  3.00  0.7  1.0  25  36  85.74 
4.8  38.2  60  2.50  1.3  1.0  40  47  90.08 
4.9  39.4  50  2.75  1.1  1.0  35  47  87.78 
5.0  43.1  50  2.75  1.3  1.0  40  51  89.44 
5.1  44.7  60  1.50  1.5  1.0  25  56  97.34 
5.2  47.1  70  1.25  1.3  1.0  16  54  107.19 
5.3  48.9  65  1.25  1.9  1.5  30  55  101.57 
5.4  51.6  70  1.75  1.7  1.0  30  56  104.08 
*The highest initial spending for a given maximum shortfall.

The red line at the bottom (worst case) intersects the vertical axis at 3.9%, representing a strategy allowing 3.9% initial spending with worstcase decline from initial spending = 0% (no shortfall). The green line second from the bottom (90th percentile) intersects at 5.1%, indicating a strategy with 5.1% initial spending, which has no shortfall in 90% of outcomes. Observing where the 5.1% initial spending level intersects the red line, there is a strategy allowing 5.1% initial spending, which experienced a worstcase decline below initial spending of about 45%.
Probability  
Worst  Equity  Initial  Lifetime  of 10%  
Certainty  Shortfall  Allocation  Constant  Variable  Mortality  Spending  Spending  Shortfall  
Equivalent*  (%)  (%)  Spending  Spending  Smoothing  Insensitivity  (%)  (%)  (%) 
81.3  0  70  0.75  0.5  9.0  2  3.36  100.7  0 
81.4  0  75  0.75  0.5  9.0  2  3.36  103.5  0 
81.6  0.6  70  1.75  0.3  7.0  0  3.50  99.0  0 
81.7  1.1  75  1.75  0.3  7.0  0  3.50  101.6  0 
81.9  1.7  80  1.75  0.3  7.0  0  3.50  104.4  0 
82.2  3.2  80  1.25  0.4  8.0  1  3.45  106.1  0 
82.4  4.1  55  2.00  0.3  5.0  0  3.75  94.5  0 
82.9  4.5  60  2.00  0.3  5.0  0  3.75  96.9  0 
83.3  5.1  65  2.00  0.3  5.0  0  3.75  99.4  0 
83.7  6.0  70  2.00  0.3  5.0  0  3.75  102.1  0 
83.9  9.5  75  1.25  0.4  4.5  0  3.58  112.6  0 
85.1  12.1  65  1.5  0.4  3.5  0  3.83  107.2  2 
85.3  12.7  70  1.5  0.4  3.5  0  3.83  110.6  2 
85.4  13.5  75  1.5  0.4  3.5  0  3.83  114.1  3 
85.7  16.7  85  1.5  0.4  4.0  0  3.83  121.8  3 
85.8  19.1  75  1.25  0.5  3.0  1  4.00  115.4  12 
85.9  20.4  70  1.00  0.5  2.0  0  3.91  115.5  15 
86.1  20.7  75  1.75  0.4  1.5  0  4.08  114.8  19 
86.2  21.1  80  1.75  0.4  1.5  0  4.08  118.6  22 
86.4  21.2  80  1.00  0.5  2.5  0  3.91  124.3  15 
86.6  22.5  75  1.75  0.4  2.0  0  4.08  115.0  15 
86.7  23.2  80  1.75  0.4  2.0  0  4.08  118.9  15 
86.9  24.9  70  1.75  0.4  2.5  0  4.08  111.7  8 
*Certaintyequivalent spending with a risk aversion of 8.

Nevertheless, certaintyequivalent spending has the advantage of being a consistent, objective measure that takes into account risk and variation, and it can identify a schedule of solutions at different levels of initial and lifetime spending and worst shortfall that are locally optimal, and you can pick the ones that look most desirable. For instance, the solution shown in bold on Table 2 has:
 Initial spending of 3.83%,
 2% probability of shortfall greater than 10%,
 Worst shortfall of 13% and
 Lifetime spending greater than 110% of initial portfolio.
Compared to maximizing initial spending for 0% shortfall in Table 1 (3.9% initial spending, 70.68% lifetime spending), a greater than 50% increase in lifetime spending at a price of less than a 2% decrease in initial spending and a worstcase decline of 12.7% from initial spending is achieved.
Final Notes
This analysis assumes that the future will be similar to the past. Paraphrasing Warren Buffett, historical returns should be an aid to thinking, not a substitute for it. In 2013, with Treasuries offering yields below inflation, historical returns appear unobtainable in bonds. The same may be true in equities, if stocks are priced off of Treasury yields and the realized equity risk premium turns out to be in line with historical averages. This analysis also excludes taxes and investment expenses, which may be significant. On the other hand, expanding the investment universe to a wider range of assets—including higher risk, higher return assets as part of a more diversified portfolio, as well as life annuities at the right price—would improve the risk/reward tradeoff.
It also assumes you pick a strategy and stick with it for 30 years. Recomputing the optimal strategy every few years, or when the portfolio changes, makes the constant parameter function a slowadjusting variable parameter, likely introduces variation, increases lifetime spending and mitigates risk of extreme shortfall.
What is more important than the specific initial rate is a strategy with an appropriate degree of spending flexibility, giving you the ability to take advantage of higher longterm equity returns without imposing intolerable risk and volatility, and a clear understanding of the range of potential outcomes.
To the extent possible, retirees should err on the side of moderate initial spending, embrace the volatility they can tolerate as the key to unlocking maximum lifetime spending and accept that their retirement trajectory is ultimately dependent on how the timing of their retirement intersects with longterm economic and market trends.
Vertes’ working paper, “Safe Withdrawal Rates, Optimal Retirement Portfolios, and CertaintyEquivalent Spending,” can be read at http://ssrn.com/abstract=2263998.
Lifetime Spending  Worst Shortfall  Equity %  Constant Spending  Variable Spending  Smoothing  Mortality Insensitivity  Initial Spending  Probability of 10% Shortfall 
118.06

0.0%

100%

0.00

0.4

10.0

0

2.33

0%

119.88

0.5%

100%

0.00

0.4

9.0

0

2.33

0%

121.80

2.1%

100%

0.00

0.4

8.0

0

2.33

0%

123.76

4.3%

100%

0.00

0.4

7.0

0

2.33

0%

125.08

4.6%

100%

0.25

0.4

7.0

0

2.58

0%

127.20

7.6%

100%

0.00

0.5

7.0

1

2.75

0%

128.11

8.4%

100%

0.50

0.4

6.0

0

2.83

0%

128.94

11.6%

100%

0.00

0.5

6.0

1

2.75

2%

130.06

11.6%

100%

0.25

0.5

6.0

1

3.00

2%

130.77

12.3%

100%

0.75

0.4

5.0

0

3.08

2%

131.82

12.4%

100%

1.00

0.4

5.0

0

3.33

2%

132.43

14.2%

100%

1.00

0.4

4.5

0

3.33

3%

133.41

14.6%

100%

1.25

0.4

4.5

0

3.58

3%

133.89

16.3%

100%

1.25

0.4

4.0

0

3.58

8%

134.30

18.0%

100%

1.25

0.4

3.5

0

3.58

11%

135.15

18.2%

100%

1.50

0.4

3.5

0

3.83

12%

137.21

18.6%

95%

0.25

0.5

4.0

0

3.16

8%

142.69

20.2%

100%

0.25

0.5

4.0

0

3.16

8%

143.30

22.4%

100%

0.50

0.5

3.5

0

3.41

12%

143.95

24.0%

100%

1.00

0.5

3.0

0

3.91

19%

149.68

31.4%

100%

0.00

0.6

2.5

0

3.49

25%

150.14

41.3%

100%

0.00

0.6

3.0

0

3.49

18%

151.72

45.9%

100%

0.00

0.7

1.5

0

4.07

38%

152.64

49.4%

100%

0.00

0.7

2.0

0

4.07

33%

153.09

63.9%

100%

0.00

0.8

1.5

0

4.65

42%

154.10

75.9%

100%

0.00

0.8

2.0

0

4.65

39%

156.09

100.0%

100%

0.00

0.8

4.5

0

4.65

29%

Discussion
Henry Hanau from NY posted about 1 year ago:
What if we take 5% of the portfolio on Dec. 31 and agree to live on that amount for the year? We can carry that a bit further and say that when we've had a good year we don't go overboard on the spending and save a little. When the portfolio didn't do well, maybe we use some of that savings. Further, at 80 with a life expectancy of 8+ years, we might think about the fact that our desire to travel at 85 might be constrained. Thoughts?
Druce Vertes from NY posted about 1 year ago:
Hi Henry  thanks for the comment. If you look at Table 1, you'll see that for our hypothetical 65yearold with a 30year retirement, the 5% initial spending solution with the lowest historical worstcase shortfall has a 43.1% worstcase decline from initial spending, and that solution would have resulted in at least a 10% decline from initial spending 51% of the time.
There are very important caveats in the Final Notes section.
The methodology assumes people prefer constant spending throughout retirement. Some people may have income requirements diminish as they get older. But it's hard to quantify in way that would be right for most people. Everyone's needs are different, and some people could have increasing medical or other expenses.
A key takeaway is that willingness to accept some variability in spending can allow retirees to maximize lifetime spending.
Mike Muhle from TX posted about 1 year ago:
Dear Druce,
Very nice article.
I wanted more details so I also reviewed your paper on SSRN.
I did have a couple of questions I was hoping you could clarify for me.
1. You mention you use the Ibbotson data from 19262012 for returns and 3.2 million spending paths. I was wondering how you handled the return data. Do you split this up into consecutive segments and get the average or just one simulation on all the return data?
2. In calculating the “severity frontier” vs a variable such as the shortfall pct, I assume you use a procedure similar to a portfolio efficient frontier, i.e., specify the constraint variable, such as the shortfall pct and find the value that gives the maximum initial spending or lifetime CE whichever is of interest. I looks like you may have just windowed the data (20^5 points) and determined the maximum rather than a search algorithm?
Druce Vertes from NY posted about 1 year ago:
Hi Michael  thanks for the questions!
The basic methodology works like this: Suppose you want to find the highest noshortfall withdrawal rate. In our example, we're looking at a 30year retirement. To test a set of the 5 parameters:
 first, take the return data from 1926 to 1954, which is what someone who retired on Dec 31, 1925 would have experienced. (Since we withdraw at the beginning of the year, only 29 years of returns are used, the last year wouldn't impact spending.) Using the 5 parameters that define this strategy, determine what the spending path would have looked like over that period, and the worst shortfall, if any.
 Repeat the process for 1927, 1928 up to 1984, which is the last year for which we have 29 years of data.
 If no shortfall occurred in any of those 59 retirement windows, save the starting spending rate.
 Repeat this process for every combination of the 5 parameters. The highest noshortfall spending rate is our solution.
To trace the efficient frontier:
 We run the process above for every combination of the 5 parameters and save each starting spending rate and worst shortfall in the database.
 First, select the highest withdrawal rate where max shortfall = 0  the solution we found above. This is the leftmost point of the frontier where it intersects the yaxis. Set that return as the starting hurdle.
 To find the 2nd point, select the lowest shortfall where return > hurdle return. That return is the new hurdle.
 Repeatedly select the lowest shortfall where return > than previous hurdle until none is found (100% shortfall)
As you mention, it's a brute force sampling process. We test all combinations sampled over a range, ie corresponding to constant spending of 0%, 0.1%, etc.
It's likely that if 3.9% is a noshortfall solution, then 3.901% is also going to work, possibly with a small adjustment to equity % etc. Using an optimizer would work too. Iteratively find the lowest feasible shortfall subject to a higher and higher return constraint. It might have been a little tricky because the solution space is piecewise differentiable. As you move to the right on the frontier increasing the sustainable rate, you might be gradually increasing for instance the equity percentage, then you might hit a point where you move discontinuously to, for instance, a higher equity percentage and a higher smoothing factor. So if you're using a gradient descent optimizer you have to be careful to make sure it converges. And ultimately for implementation purposes, we don't necessarily care too much if the maximum sustainable rate turns out to be 3.9% or 3.94% ie a 1% difference in spending for the retiree. As an estimate of outcomes going forward and the best strategy for the retiree the error is probably going to be larger than that. And the brute force method gives us a database of outcomes we can slice in various ways to see how parameters change the outcomes.
thanks for the questions, glad to see you took the time to understand it, and it was more or less understandable. It's a challenge to explain clearly, especially with space constraints.
Harold Marrett from NY posted about 1 year ago:
A big shortfall of this and many more articles is the lack of an example from which the reader can calibrate what they think the author is trying to say.
Every constant $$$ payout be it Social Security, fixed pensions or constant withdrawal amounts have a Taleb function (a Black Swan) that can make them blow up. From my nearly 20 years of retirement with all the blow ups in the market one needs to assume ever few years they are reretiring and should recalculate their actual money situation. This more intelligently keeps their spending under contol than using a cost of living number.
Hal Marrett
RDV from FL posted about 1 year ago:
The full paper would likely include some definitions missing from the abridged article that difficult for me to follow (didn't want to sign up for yet another site to get it).
As it is I'm not clear/sure as to the Table 1 meaning (for say the 4.0% line) of 1.1% Worst Shortfall (less than initial or running out or ?) 3.75 constant spending (portion of the 4.0% ?), 0.1 Variable spending (of 4.0% ?) 25.0 smoothing, 30 mortality insensitivity, 70.68% lifetime spending (relative to ?).
How would this line play out for a $100,000 portfolio?
Reading a response above raised a secondary question: if the 30 year tests stop with 1984, is it fair to conclude that only a handful of test years include the 2008 market collapse? Could that alter the outcomes for upcoming test years?
I don't dare delve into certainty equivalents and Table 2, lol. The certainty I gathered is that the commonly cited 4% rule of thumb seems safe.
H Dornbush from CA posted about 1 year ago:
I really tried to slog it through and understand this article, but I failed. Please consider a rewrite that speaks in laymen terms. This felt like professorspeak.
Bruce Mac Naughton from CA posted about 1 year ago:
I am not as sophisticated as most of you but my approach has worked very well for me.
Each quarter, I calculate the value of 4% of my retirement assets, divide by 12 and then withdraw that amount each month during the next quarter. I also round the distribution amount down to the nearest $100 or $1000 to leave a little bit more in the kitty. Obviously, my income has gone both up and down over the last eight years.
My allocations have ranged between 5070% equities and 5030% cash and bonds for the last eight years.
My total assets have grown over that period.
I also found that keeping enough cash to cover the disbursements substantially reduces your income tax exposure.
Henry Hanau from NY posted about 1 year ago:
Bruce
In my opinion, you've got it right.
Edward Campbell from NJ posted about 1 year ago:
Ditto Dornbush and Cabler, this article was painful and not to the point.
Larry Keefe from NY posted about 1 year ago:
Granted, I'm no mathematical genius. Still, I have read this article three times and have trouble understanding it, unlike most articles in the AAII Journal. It appears that the definition of terminology is not clear enough from the beginning, so it's hard to evaluate the over all intent of the author. And this, in fact, leads to my main point. While the normal 65 year old investor may be able after much study to perform the needed mathematical calculations, can we be sure that at an age of 80 or 85 their surviving spouse would be able to perform them? Or evenheaven forfendthe AAII member him or herself? The constant percentage withdrawal rate with income variations acceptedmaybe calculated annually, maybe quarterlyseems like an excellent alternative.
Druce Vertes from NY posted about 1 year ago:
Hi RDV  what the first line means is, the rule that gave no shortfall and the highest initial spending used the parameters described  3.75% of initial portfolio fixed spending and a small variable amount using the parameters given, adding up to 3.9% initial spending.
no shortfall means there was never any decline from the initial amount. Since there is a variable spending term, spending would have gone up as the portfolio grew in value.
The 4% line means these are the parameters with the lowest recorded decline from initial spending, which was 1.1%. This means that at some point the value of the portfolio would have declined, and the variable spending term would have resulted in a 1.1% decline in total spending from the initial level.
Since the data go up to 2012, they include paths for the financial crisis. But in those paths, the crisis comes at the tail end of retirement. The really bad years to retire were in late 60s and early 70s and those account for a lot of worstcase scenarios.
I wouldn't get too hung up on the math. The point is, historically a 4% fixed rule worked. But since that rule, as tested by Bengen, never goes up as the portfolio goes up, it doesn't lead to high lifetime spending. If you incorporate a variable term, you get higher lifetime spending, but you have to accept some risk of shortfall.
It's interesting to chart that relationship, and see how much additional lifetime spending you get by accepting different degrees of shortfall risk.
'Certaintyequivalent spending' is a concept that 1) lets you compare different strategies based on your level of risk aversion and 2) shows us that if you're highly riskaverse, you'll stick with something like the fixed 4% rule, and if you're riskneutral, you'll just try to maximize lifetime spending with a strategy that varies based on the size of the portfolio.
Druce Vertes from NY posted about 1 year ago:
and thanks for all the comments! I wish I could have explained more clearly.
Robert Dellavedova from FL posted about 1 year ago:
Druce, thanks for the thorough response.
Your fifth paragraph puts it into fairly clear perspective for me. The concept is simple, but elusive because my inclination would be to avoid higher lifetime spending (barring unexpected pressing necessity).
I don't want to run out or be forced to curtail drastically in late years, should I get there. I'll aim at a lifestyle that works within the 4% rule (maybe a little splurge or two if the portfolio builds enough) and hopefully leave some behind for the family.
G Alcott from TX posted about 1 year ago:
Druce,
I am getting close to retirement and have been reading the articles on withdrawal strategy in the AAII Journal with interest. I have read this article several times and am having trouble understanding the equations in the Withdrawal Rate Model. Please provide an example of how the exponential moving average and s(i)are calculated.
My trouble with s(i) is that s(i) is a percentage, K appears to be a percentage (although the article doesn't state this), EMA is in $, h is dimensionless, and L(i) and b are in years. The second term in the equation then appears to have the units $/year rather than %. What am I missing?
Druce Vertes from NY posted about 1 year ago:
As an example, take the first line in table 2
K (constant spending) = 0.75%
h (variable spending) = 0.5
b (mortality insensitivity) = 2
for our example 65 yearold male, life expectancy in first year of retirement is 17.19 years
0.0075 + 0.5 / (17.19 + 2) = 0.033555 =~ 3.36%