You can use these certainty-equivalent returns as ultraconservative safe withdrawal rates. And the Merton share as the optimal allocation to stocks in your portfolio.

Click Calculate to see example calculations using historical averages. Or follow the instructions below the calculator to make your own personalized calculations.

Note: The initial default inputs use U.S. historical real average returns and volatility with one simplification: While the historical volatility of intermediate US treasury bonds equals 5 percent. I set this input to 0 (zero) to match the typical textbook treatment.

How Calculator Works

The super safe withdrawal rate calculator steps through three calculations.

First, it takes the average geometric returns expected from equities and risk-free assets and adjusts these percentages so they approximate arithmetic mean returns. (Mechanically, the formulas add one-half the squared volatility.)

Second, the calculator estimates the equity premium. (If equities return 7 percent and risk-free bonds return 2 percent, the equity premium equals 5 percent.)

Third, finally, the super safe withdrawal rate calculator estimates the certainty-equivalent returns (CERs) as well as the resulting Merton “equity allocation” shares suggested for the standard set of risk tolerances.

Personalized Super Safe Withdrawal Rate

To calculate a personal super safe withdrawal rate, replace the default annual average “geometric mean” returns for equities and risk-free bonds with your forecasted returns. And then also estimate the volatilty, or standard deviation, for both asset classes.

You may aleady have estimates for these inputs. But if you don’t? No problem. The large investment services also provide this information regularly. (See here, for example, for the December 2024 Market Outlook from Vanguard.)

Certainty-equivalent Returns and Merton Shares in a Picture

A simple line chart accurately shows how Merton shares and certainty-equivalent returns work (see below).

The blue line shows the average expected arithmetic returns for portfolios using a variety of stock allocations: 0%, 10%, 20%, 30% and so on. If the portfolio holds only risk-free assets, for example, the expected return equals the risk-free return. If the portfolio holds only equities, the expected return equals the equity return. In between those equity percentages, the expected return reflects a weighted average.

The line chart hints at the portfolio risks using those two dashed grey lines. They show the 25th and the 75th percentile returns. (All of these calculations reflect the historical real returns of US stocks and risk-free assets and their volatility. Also, in this chart to make it make sense, I did set the standard deviation of the risk-free assets to 5%.)

That green line shows the certainty-equivalent returns, or CERs, and graphically shows the utility the investor enjoys at various stock allocations. The green line flattens as the investor increases the allocation to stocks. That visually signals the diminished marginal utility. In effect, the formulas assume there’s a risk penalty diminishing the expected value.

By the way, the green line reflects a good guess as to the utility. The Python script that draws the line chart uses the standard utility function, or formula, economists think does a pretty good job. But the main takeaway here for non-economists? Sure, you and I get larger returns by allocating ever larger percentages to stocks (see the blue line). But risks explode as we do this (see the two grey dashed lines.) The utility we enjoy (see the green line) essentially tops out at the Merton share.

Historical Context Helps

Using the historical default numbers, which is what the line chart does, the Merton share formula suggests a 62.5% allocation to stocks based on a constant relative risk aversion equal to 2. (More on this constant in a minute.) So very close to the orthodox 60-percent stocks and 40-percent bonds asset allocation. Furthermore, the certainty-equivalent return per the formula equals about 3.56%. Which is interestingly close to the cannonical four percent safe withdrawal rate.

Personalizing Your Relative Risk Aversion

For practical purposes, the super safe withdrawal calculator above assumes your personal relative risk aversion equals 1, 2, 3, 4 or 5. The way the Merton share and CER formulas work, those values are sort of the standard “shoe sizes.”

Most people, according to the research, feel a constant relative risk aversion equal to 2 or 3.

A constant relative risk aversion equal to 1 might signal someone comfortable with a leveraged portfolio in many economic scenarios. (In late December 2024, a constant relative risk aversion equal to 1 would mean an investor focusing on only US stocks might invest between 60 and 65 percent of their portfolio in US equities.)

A constant relative risk aversion equal to 4 or 5 would suggest in the current market an allocation to US equities of maybe 10 percent to 15 percent.

Use CER as a Super Safe Withdrawal Rate?

The $64 question: Can you or I really use certainty-equivalent returns as a “safer” safe withdrawal rate? Good question. And one worth chewing over a bit.

Certainty-equivalent returns can provide a good safe withdrawal rate number. As noted, if you make the calculations using historical averages? The resulting Merton shares and CERs mesh with the almost canonical 4 percent rule and popular 60-percent stocks and 40-percent bonds asset allocation. But you need to be careful here.

True, using CERs as a super safe withdrawal rate delivers some unique benefits. The approach considers the risks of a particular portfolio. It explicitly addresses periods where expected returns going forward will probably be lower. (If you don’t like the idea of forecasting lower expected equity returns, you can surely see it makes sense to forecast lower expected bond returns if interest rates have dropped.) Further for investors with long retirements and who want to preserve their wealth? The ultraconservative nature of CERs mean they’re almost guaranteed not to fail. (If this sounds implausible, consider the CER percentages start lower. And then if portfolios shrink in value, that CER percentage probably increases but it also gets multiplied by the new year’s lower portfolio value.)

However, using the CER as a super safe withdrawal rate may not make sense in many situations. Currently, the formula returns a very low withdrawal rate for investors who limit their equity investments to US stocks. (The certainty-equivalent return in late 2024 might suggest a super safe withdrawal rate of less than 2 percent for an all US stocks investor.) The CER formula would also often result in investors simply not spending much of their retirement nest egg. (That doesn’t really make sure.) And the formula would tend to restrict a retiree’s spending. (That doesn’t sound great.)

Two Final Thoughts

A couple of other thoughts before I end.

First, if you’ve been using the four percent safe withdrawal rate, one way to maybe benefit from the super safe withdrawal rate calculator is to make the calculations for your portfolio. And then think about an average of the CER percentage and that four percent figure. That hybrid approach hedges your bets a bit.

A second idea: Calculating CERs and Merton shares may help you think about diversifying away from US stocks. (Adding more international stocks will make the numbers work better.) And doing the arithmetic may also help you more unemotionally calibrate your portfolio risks. (The CERs and Merton share math help you quantatively adjust your portfolio risk.) Those effects? Arguably pretty good.

Related Resources

This companion calculator and discussion may be interesting as you’re learning about this stuff: Merton Share Estimator.

This related discusion of the variability of portfolio returns may provide nice context: Retirement Plan B: Why You Need One.

I like the insights the Merton share and certainty-equivalent returns provide when thinking about retirement. But personally? I think it makes more sense to do Monte Carlo simulations to think about the risks. That topic is discussed in more detail here: Monte Carlo Safe Withdrawal Rates for Low Expected Returns.

Finally, if you’re struggling with the math and logic of certainty-equivalent returns and how lower-returning bonds can possibly help? Check this blog post: Monte Carlo Simulations Show How Bonds Dampen Retirement Risk. It provides a visual approach to exploring how risk-free assets can mostly dial down your risks during retirement.

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Those low returns raise interesting questions. But one that matters a lot to retirees? How much can you safely draw if, going forward, stocks, bonds and other assets deliver a real return roughly half their historical average. So maybe for stocks something like 3 or 3.5 percent rather than 7 percent.

The Problem with Historical Data

A great question. But a question hard to answer. And for a simple reason. We don’t have much historical data to use for modeling what works safely when investors should expect low returns.

In fact, I count two examples where investors should have expected low returns going forward: Right before Black Tuesday in October 1929. And then in the late 1990s before the dot-com crash.

But that’s not enough to generalize. The 1929 scenario probably did fail for 100 percent stock portfolios if someone started in 1929 and withdrew 4 percent. (A balanced portfolio probably didn’t.)

And for those retirements that start in say 1999 or 2000? We don’t know yet. We haven’t had a full thirty years of data. (Those retirees do look to be in pretty good shape so far.)

Thus, this idea: Maybe we use a Monte Carlo simulation. And calculate not historical safe withdrawal rates. But Monte Carlo safe withdrawal rates.

The attraction there? We need just a couple of pieces of data: The arithmetic mean return expected. And then a standard deviation.

But with those two values, we can get an idea as to how much one might draw in a worst-case situation in a low-expected-returns environment.

Using a Monte Carlo Simulation Given Absence of Good Historical Data

Conveniently, the Red Portfolio Black Portfolio spreadsheet I constructed recently for a blog post last year (see below) will let you do that.

You can download a free copy here: RedPortfolioBlackPortfolioVer1. And then you just follow these steps:

1. Specify the starting balance into cell B4,
2. Enter a negative value into the cell B5 to show the probable starting withdrawal amount.
3. Specify the percent increase in the draw using cell B6. (This is essentially the inflation rate.)
4. Enter the standard deviation for your primary “Red” portfolio into cell B8 and the arithmetic mean return into cell B9. Back of the envelope calculations, which I’ll describe in a minute, suggest an arithmetic mean return of maybe 6.2% for a balanced portfolio of 70% stocks and 30% bonds like you might use for retirement? That portfolio’s standard deviation probably roughly equals 11.3%?
5. The spreadsheet uses a second standard deviation and arithmetic mean as a benchmark “Black” portfolio for comparison. For now, enter the Black Portfolio standard deviation as 6% and its arithmetic mean as 5.4%. These values represent my guess of what a 100% Treasury Inflation Protected Securities (aka TIPS) portfolio would show. But more on that in a few paragraphs.

Can I also throw out a tangential comment? I think given the purpose of this Monte Carlo simulation, you use rounded whole percentages. Like maybe 11 percent and 6 percent from the Red and Black Portfolio’s standard deviations. And then maybe 6 percent and 5 percent for the arithmetic averages. Those rounded percentages might be good reminders that what we’re doing here? A learning thing. Not hardcore research for some academic study.

Interpreting Monte Carlo Safe Withdrawal Rates Line Chart

Once you get the inputs entered, Excel recalculates the workbook and plots 100 simulation results for the Red Portfolio using red lines. It also plots the best and worse scenarios from another 100 simulation results for the Black Portfolio using thick, black dashed lines.

A line chart that uses the inputs above appears below. And what the line chart suggests? A 4 percent withdrawal looks risky. Those red lines dropping down to zero? Those signal failure scenarios. You can experiment a bit. I think you’ll find your Monte Carlo safe withdrawal rate looks more like 3% to 3.5% if you’re talking a three-decade retirement funded by a balanced portfolio. So, making the math easy, with a $1,000,000 to start, a starting draw of $30,000 to $35,000.

Benchmarking Against TIPS

I designed the Red Portfolio Black Portfolio spreadsheet so someone can compare one portfolio’s simulations (the Red Portfolio) to the best and worst outcomes of another portfolio (the Black Portfolio).

That begs a question, which is what do you use as the alternative when you’re thinking about safe withdrawal rates in a low expected returns environment? I wondered about this bit. And then realized the Bogleheads, or some of them, already have a go-to idea. Worried about low returns requiring a low safe withdrawal rate? Just use Treasury Inflation Protected Securities or TIPS.

That makes sense as a hunch. So for fun, I estimated the nominal arithmetic mean return on TIPS as equal 5.4% based on the the fixed interest rate (2.2%) for TIPs the day I drafted this article and a 3% inflation rate, and then the historical standard deviation (6%) . Note that the spreadsheet calculates and works with nominal returns which means you need to adjust the real return TIPS deliver (so the 2.2% in the earlier example) for the inflation you anticipate.

The worst and best case scenarios for this TIPS portfolio show up in those thick black dashed lines in the chart shown earlier. Interestingly, as the line chart shown illustrates, you can fail with a 4% withdrawal rate. But a 100% TIPS portfolio also might not fail even after 40 years.

Again, I used inputs that suggest more precision than is probably appropriate. I’d say go with rounded whole percentages if that makes you feel better.

Calculating Arithmetic Means for Monte Carlo Safe Withdrawal Rates

Obviously, the trick here is coming up with good arithmetic returns and standard deviations. And for portfolios like the ones you want to understand. Wade Pfau provides an online resource you can use for some of this information (see here.)

For a balanced portfolio’s expected returns, you need to go to a little bit of effort to get the numbers you need. But here’s how I’d approach the work. I think you use the 3.5% geometric mean Illmanen expects for US stocks as your expected return for that part of your portfolio. That number, he calculates as the sum of the 1.5 percent growth in earnings per share plus the 2 percent-ish dividend yield. If one adds two to three percent for inflation, you’re at 5.5 to 6.5 percent as a nominal geometric return. So maybe we split the difference and say a 6 percent nominal return on stocks.

Ten-year treasuries return (roughly) 4.7 percent as I write this and maybe represent a guess for that part of your portfolio? Thus, if you go with 70 percent in stocks generating 6 percent and 30 percent in bonds generating 4.7 percent, your portfolio earns a geometric return of roughly 5.6%. To roughly convert that geometric mean to an arithmetic mean, add one half of the squared standard deviation, which Portfolio Visualizer suggests is around 11.3%, and the adjustment equals .6%. That gives you basically 6.2% arithmetic return.

For the 100% TIPS portfolio, I’d look at today’s real yield (roughly 2.2% as I write this), add a 3% inflation guess to this to get a geometric return of 5.2.%-ish and again bump up this geometric return by one half of the 6-percent-ish squared standard deviation. That’s about a .18% adjustment. Which means to model TIPS, you use a roughly 5.4% arithmetic mean and a 6%-ish standard deviation.

Note: If you have questions about the spreadsheet including questions about how to convert geometric means to arithmetic means, refer to its FAQ: Red Portfolio Black Portfolio FAQ.

Closing Remarks about Monte Carlo Safe Withdrawal Rates

Some closing remarks. First, simple Monte Carlo simulations suggest that the old four percent rule of thumb won’t work 95 percent of the time with a balanced portfolio. You and I are probably looking at something closer to three percent if we invest on the basis of a 95 percent success rate. (That would be especially true for members of the FIRE community who want to fund not three decades of retirement but four or more decades.)

Second, even in a world of low expected returns, most people probably can draw more than whatever the Monte Carlo simulation shows as the lowest rate that doesn’t fail more than about 5 percent of the time. (This was always the case with the 4 percent safe withdrawal rate too, right?) So keep that in mind. The theoretical very worst-case scenarios are something you and I need to plan for a little bit. But those outcomes are unlikely.

A third comment straight out of from Ilmanen’s book. His obvious conclusion regarding low expected returns? We may want to save more if we can. Maybe work longer if that’s an option. Or possibly we will need to spend less if we experience a batch patch of returns in retirement. But mostly? We just need to understand returns going forward probably won’t be as good as they’ve been. And forewarned? Yeah, we can make this work.

And, finally, a fourth comment: The TIPS option seems pretty interesting.

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You hear people offer bromides. Like “bonds provide ballast.” Or “bonds smooth returns.” But those truisms don’t really help you or me think objectively.

So this idea: Try plotting investment outcomes in a line chart that compares a portfolio that holds 100% stocks… to a portfolio that holds a balance of stocks and bonds. Say 70% stocks and 30% bonds. You can then visually see if and how much difference bonds make.

If you’re interested, I’ve got a free, downloadable Microsoft Excel spreadsheet that lets you this. But let me describe the approach I think works. And then I’ll walk you through the steps for using the spreadsheet for your own specific situation.

Plotting Monte Carlo Simulation Results

A problem to mention first, however. We don’t really have enough data to plot hundreds or thousands of investment outcomes comparing a 100% stocks portfolio to a 70% stocks and 30% bonds portfolio. If you want to look at 40-year accumulations or 40-year withdrawals? Well, the earliest usable stock and bond return histories for US investors start in 1871. That gives you or me less than four unique 40-year histories.

Thus, this idea: One can use average stock market returns and volatility to run a Monte Carlo simulation that plots likely scenarios. A bunch of likely scenarios. And then we can compare those. For example, you can run a 100 100% stock simulations. And 100 70% stock and 30% bonds simulations. Compare them using a line chart. And see how bonds affect the outcomes.

Charting How Bonds Dampen Investment Risk

The chart below does exactly this, plotting two simulations, one I’m calling the Red Portfolio and the other the Black Portfolio. The Red Portfolio results show as red lines and depict one hundred simulated “100% stocks” portfolios over 40 years. The Black Portfolio results aren’t fully plotted. To keep the chart legible, it plots two dashed black lines. The lines show the best-case and worst-case investment scenarios of a balanced portfolio that combines 70% stocks and 30% bonds. (You can click the chart to see a larger image.)

You may not even need me to explain this. But the dashed black lines show the benefit of adding bonds. You probably avoid those investment returns that show up as red lines that fall beneath the bottom black dashed line. All of those 100% stocks outcomes? Worse than the worst balanced portfolio outcome. But the other thing to note of course? You also probably lose upside when you add bonds. All those red lines that float off above the top black dashed line? All of those 100% stocks outcomes beat the very best balanced portfolio outcome. And that’s the way to visualize the trade-off bonds offer you and me. We probably dodge some downside. And probably give up some upside.

By the way, to keep the chart legible? It logarithmically scales the value axis for legibility. Thus, pay close attention to the scaling so the chart doesn’t mislead you. For example, while the upside risk and downside risk of using a 100% stocks portfolio visually look similar? Sort of a finger’s width? The logarithmic scaling means a 100% stocks portfolio might deliver way, way more upside risk than downside risk. (We’ll look at the actual numbers next section.)

A tangential comment? Note another reality suggested by the line chart. Over time, the range of returns for both the Red Portfolio and the Black Portfolio widen. That widening visually shows the passage of time does not remove risk. It increases the risk. (If time removed risk, the best and worst case scenarios would get closer and closer together… finally converging at some point in the future.) That’s interesting and something hard to understand until you actually “see” it.

The Good and Bad by Numbers

Take a peek at the green, red and charcoal spreadsheet fragments shown below.  The green cells shown the input values: Starting balance, annual addition, growth in additions and then the two portfolio’s returns and standard deviations. The red and charcoal cells summarize the Red and Black Portfolio simulation results plotted in the chart just shown.

As you might expect, on average the higher risk Red Portfolio delivers a significantly higher average return (cells H5 and K5). Nearly one percent a year. That’s huge. And with that average annual return, an investor on average ends up with about $500,000 more money at the end of the four decades (cells G5 and J5). The higher average return of equities like stocks? The reason we all want to invest as much as we can bear the risk for.

Another tangential point: Online retirement planning tools like FireCalc and cFireSim don’t make it easy to see the downsize risk investors avoid by adding bonds. Or the upside reward bond investors lose. But as we’ve discussed in other blog posts ( Why Bonds Matter for Your Portfolio and  Myth of the Long run Stock Market Return Chart), their historical data and calculations paint a similar picture.

Monte Carlo Simulations for Accumulations

If you’re ready to experiment with the free Monte Carlo simulations spreadsheet, download the spreadsheet (available here: RedPortfolioBlackPortfolio), and then follow these steps:

1. Enter the starting balance into cell B4.
2. Specify any additional annual amounts saved using cell B5.
3. If you plan to increase the annual saving amount, enter your annual percentage adjustment into cell B6.
4. Provide standard deviations for both Red and Black Portfolios into cells B8 and D8.
5. Estimate the average return by entering percentages into cells B9 and D9 for both Red and Black Portfolios.

The workbook automatically recalculates as you enter the data. But you can and should press F9 several times in a row once you enter all the inputs to see additional simulations.

Some quick notes too. First, the logarithmic line chart can’t display negative values. Thus, if a particular simulation results in negative value (signaling a loss), Excel ends the line. (I try to protect against this outcome in most cases.)

Second, Excel may not be able to calculate rates of return for all simulations in which case the outputs will show the #NUM error. (The error occurs typically when the simulation produces wildly extreme results.)

Third, on some computers and with some display property settings? Excel doesn’t always finish updating the chart for every simulation. (If you encounter this, save the workbook to force Excel to redraw the line chart. Or display another window and then redisplay the Excel program window.)

Monte Carlo Simulations for Withdrawals

That line chart show in the beginning? It tries to help you visualize accumulation scenarios with differently-risked portfolios. But you can also use the Red Portfolio Black Portfolio spreadsheet to simulate withdrawal situations, too. To try this, follow these steps:

1. Enter the savings at the start of retirement into cell B4.
2. Specify the annual withdrawal as a negative value using cell B5.
3. If you plan to increase the annual withdrawal—such as for inflation—enter your annual percentage adjustment into cell B6.
4. Provide standard deviations for both Red and Black Portfolios into cells B8 and D8.
5. Estimate the average return into cells B9 and D9 for both Red and Black Portfolios.

The spreadsheet fragment below shows how the inputs look for someone starting retirement with $1,000,000, planning to initially draw $40,000 but bumping this amount by 3% annually. The standard deviation and arithmetic mean inputs reflect a Red Portfolio invested 100% in stocks and a Black Portfolio invested 70% in stocks and 30% in bonds.

The line chart below shows a “withdrawals” Monte Carlo simulation. (Again, click the chart to see a larger image.) When a red or black line drops to zero or close to zero, that reflects a portfolio failure.

Note that nine or ten Red Portfolio lines drop to zero or nearly zero and so represent failures. That single black line doesn’t mean the Black Portfolio fails only once. That’s just the worst case Black Portfolio outcome. (The significant thing to notice here? The Black Portfolio maybe fails for the first time farther into the future. But you’d want to run several simulations.) Also note that three or four of the Red Portfolio failures occur nearly forty years into the accumulation. (That may not matter much.) The thirty year failure rate show above is 6% (6 out of 100 failures). Which roughly matches the conventional wisdom.

Three Final Notes

First of all, if you need help with the Red Portfolio Black Portfolio spreadsheet? Refer to the FAQ I created: Red Portfolio Black Portfolio Frequently Asked Questions. That resource answers a handful of questions and addresses some common issues (including how you come up with inputs to use for your modeling.)

Second, and related to the first point, if you get into this Monte Carlo stuff, know that you can get a good starter set of arithmetic means and standard deviations from Wade Pfau’s excellent resource: Historical Market Returns.

Third, we get so much content plagiarized at our blog, I put a password on the spreadsheet to make the theft a little harder. But if you’re interested in seeing the Microsoft Excel formula that calculates the returns? Just take a peek at this earlier blog post: Stock Market Monte Carlo simulation spreadsheet.

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