Friday, March 30, 2018

The “Future” of Retirement Planning

When we decide how much money we can spend in the present year of retirement we need to know not only how much spendable wealth we have today but our best guess of how much we will have in the future. Likewise, on the expense side of the ledger, we need to know not only what our expenses were last year but also our best guess of what our expenses will be in the future. We can spend less this year, for example, if we know there is a big expense looming in the future and more if we’re pretty sure we’ll have more money in the future, say a sizable inheritance.

In short, we need a model of our retirement future to properly plan for it and even to determine a safe amount to spend this year.

Of the four inputs to this model I just mentioned, only one, our current wealth, is fairly certain.

If our entire future income will come from annuities, pensions and Social Security retirement benefits then it's relatively predictable, too. To the extent that future income will come from investments, that income is fairly unpredictable.

Expenses are unpredictable, as well. All we know for sure is how much we spent over the past few years. We can’t even be certain of the coming year’s expenses, so they too are uncertain. Retirement spending studies have shown that spending tends to decline as we age [2,3] but it doesn’t for everyone so we can’t assume that ours will. According to David Blanchett, whether or not it declines, annual spending volatility is relatively high (unpredictable from one year to the next).

When we take spending shocks into consideration, the future spending becomes even less certain. I spent $15,000 in the past year for two HVAC systems that I expected would last at least five more years. That has a much greater impact than my cable bill going up 5%.

The primary determinant of retirement cost is longevity. A five-year retirement will be far cheaper than one of 35 years. Our individual life expectancy is completely unpredictable assuming we are healthy.

As researcher Larry Frank keeps telling me, everything in an individual household’s retirement funding is stochastic, i.e., unpredictable.

Following is a graph of 200 randomly-selected portfolio value paths from a simulation of 10,000 scenarios for a retiree with a $1M portfolio from which she plans to spend $45,000 a year. All calculations are in real dollars and life expectancies are randomized using actuarial tables. It assumes a real 5.25% expected market return with a standard deviation of 12%.

Notice that most terminal portfolio values end up lower than the initial $1M portfolio value in real dollars. In this simulation, the median terminal portfolio value was about $860,000. About 75% of the scenarios ended with smaller portfolio values than the $1M they started with. This is typical of simulation results, though spending less would shift the bulk of those blue lines upward and spending more would do the opposite. The central mass of those blue lines would rotate around the starting point like clock hands, farther clockwise with more spending.

I read a comment on a retirement blog this week from a reader who said, “Retirement is uncertain so planning is useless.”

That’s like saying we shouldn’t plan outdoor activities because we can’t know future weather conditions with certainty. It’s like saying that companies shouldn’t bother developing business plans because they can’t know future economic conditions for sure. Of course you can plan where outcomes are uncertain and the best way to do that is with probabilities.

We develop retirement plans using models of the future but some models are much better than others. Nor is the model a plan. If we schedule a picnic for tomorrow and the weather models predict a 20% chance of rain, calculating the 20% is not the plan. The plan is deciding to take an umbrella or planning a backup activity. As in retirement planning, we use the model results to help create the plan.

Another blog suggested that Monte Carlo simulations can generate hundreds of thousands of future scenarios but that using them for planning is a mistake because a retiree can’t know which one she will experience. The first part of that statement is true. Your retirement's future finances might follow one of the blue lines in the chart above — assuming we ignore spending shocks — but it is impossible to know which one is yours.

There is a small chance that yours will follow a better path than any of these and a small chance that it will follow one worse and we can't ignore the latter's risk. (The former would just be sweet – we're OK with things turning out much better than we expected.)

Though it is true that we can't foresee our future path, it is also irrelevant — the purpose of the Monte Carlo model isn’t to predict an individual retiree’s path through the future (that’s impossible) but to explore a broad range of possible scenarios and develop some estimate of the probability of each actually being realized. Simulation is essentially a gigantic "what-if" analysis.

The weather forecasting model is likewise imperfect but the probabilities it provides are extremely useful. If there is a 5% chance of rain tomorrow perhaps we forego the umbrella. With a 95% probability or rain, we might cancel the event altogether. We don’t say, “No use planning based on the weather probabilities because we can’t know for sure.”

Determining how much you can safely spend this year requires a good model of the future.
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The alternative the blogger suggested was simply to build a spreadsheet for a near-worst-case life expectancy using an expected market return and ignoring stock return variance. Ignoring variance, of course, ignores sequence of returns risk. This is the model of the future you get using a spreadsheet that assumes the same expected market return every year. It's the kind of analysis we did before Bengen. It's the kind of thinking that led Peter Lynch to suggest 7% annual spending from a stock and bond portfolio would be safe.

(When I say "spreadsheet" in this post, I'm referring to any model that assumes zero portfolio volatility and a fixed life expectancy. Actually, you can use a Monte Carlo simulator and set the portfolio volatility to zero and get the same results. On the other hand, you can build a Monte Carlo simulator with market volatility in an Excel spreadsheet – I have the scars to prove it. You can download the Mother of All Monte Carlo Spreadsheets at the Retire Early Home Page[1].)

The path to the median outcome is also in that jumble of blue lines above, so the first question we should ask the blogger is, "If you can't know which of those blue paths to choose, why did you go ahead and pick one (the one with the mean annual return), anyway?"

The following chart shows all those blue paths again with the "spreadsheet" prediction of the future superimposed (red) assuming a fixed 30-year retirement and zero market return variance. That’s effectively what a spreadsheet model produces.

Why is the spreadsheet future a nice, smooth upward curve while all the simulated blue lines are jagged and head off in all directions including ruin? 

The answer is sequence of returns risk. The spreadsheet ignores market volatility and consequently, it ignores sequence risk. The model in the simulation is much more realistic.

And why does the spreadsheet portfolio end up so large compared to most of the blue lines?

The answer is sequence risk plus longevity. Note that life expectancies are simulated to generate the blue lines (they end at different years of retirement), while the spreadsheet model assumes a fixed-length retirement of 30 years. Real-life portfolios and retirees don't often last 30 years so their portfolios most often have less time to grow.

Of all the paths on this chart, the red spreadsheet path is by far the least likely for you to experience. Twenty consecutive years of identical positive portfolio returns is unimaginable.

With 10,000 simulated scenarios, fifteen survive 20 years and end up within 1% of the spreadsheet path value at 20 years. These are fifteen possible paths to reach the spreadsheet value at 20 years and they don't get there in a straight line, as you can see on the following chart. So, the spreadsheet path is in there, but why one would choose it as the representative outcome remains a mystery.

The path that reaches the median terminal portfolio value, among the 10,000 simulated scenarios, is shown on the graph above in orange. It ends at year 17, which is roughly the median life expectancy for this 65-year old. The spreadsheet path presumably uses average historical market returns, so why is its outcome at 20 years ($1.2M) so much higher than that of the median simulated outcome of $860,000?

The chart shows that using the average market return every year in a spreadsheet (the red line) doesn't produce the average outcome (the orange line). You're probably tired of hearing me say "sequence risk and stochastic life expectancies are the difference" but simulations model them and spreadsheets don't.

The spreadsheet path is quite optimistic. If you insist on a spreadsheet model, you should at least reduce the expected return to compensate for sequence risk. In this comparison, you would need to reduce the expected portfolio return in the spreadsheet model from 5.25% to 3.5% to obtain results similar to the simulation's median outcome at 20 years.

Lastly, let's look at a density histogram of all portfolios that survived at least 20 years.

The blue bars show the portfolio values after 20 years for those 956 portfolios and retirees who survived at least that long. It's a probability density histogram, so the total area of the blue bars equals 1.

The orange curve shows the continuous density. As you can see, the distribution is right-skewed and not a symmetrical normal distribution. The median is less than the mean due to all those huge but highly improbable outcomes along the right tail.

We're more interested in the median ($645,000), the value at which half of the portfolios are larger and half smaller. About 57% of the outcomes after 20 years are less than the mean ($702,000) compared to the 50% of outcomes that are less than the median. The median is the more representative statistic with this skewed distribution.

Finally, the red vertical line represents the spreadsheet model's portfolio value after 20 years, $1.17M. At the 20-year mark, that red portfolio value is larger than 88% of the simulated portfolios that survived that long. $645,000 is a much more representative expectation after 20 years than the spreadsheet prediction.

It's true that you can't predict which of those 10,000 blue paths your future will mimic but the spreadsheet outcome is one of those. Pick it and you simply decided to pick a path from the 10,000 choices after saying you couldn't. And, then you picked a very unlikely and optimistic one. The spreadsheet predicts portfolio values in the absence of sequence and longevity risk and tells you nothing about the probability of realizing them.

Furthermore, the purpose of simulation isn't to predict your future but to explore the possibilities, so you aren't intended to choose one.

We need a model of the future to plan retirement. We need it to even calculate the safe amount to spend this year. Simulation is a good starting point.

The takeaway, for now, is that if you have planned your retirement with a spreadsheet model, you should take another look, especially if your plan shows a single straight path to doubling (or more) your initial portfolio. It could happen; it just isn't likely and if it does happen it certainly won't be a smooth path.

If you're using an online calculator, make sure it incorporates simulation. There's a Monte Carlo version of E$Planner, for example, and there are free simulators online.

But simulations have issues, too, so they're only a starting point. I'll discuss the Limts of Simulation next time before describing how to make sense of that jumble of blue lines.


[1] Download The Retire Early Home Page spreadsheet.

[2] Expenditure Patterns of Older Americans, 2001‒2009, Sudipto Banerjee, Employee Benefit Research Institute. (Download PDF.)

[3] Estimating the True Cost of Retirement, David Blanchett. (Download PDF.)


  1. if i understand the monte carlo models correctly, they randomly select a return for each year out of a distribution of annual returns. this implies the return each year is independent of the prior year. but i don't think the stock market works like that. wouldn't it be more meaningful to use rolling histories of actual stock market performance? and wouldn't it also be useful to take into account that we are currently at a very, very high valuation level?

    1. Great questions!

      Yes, that's how Monte Carlo simulations work and, yes, this implies the return each year is independent of the prior year but only if you choose not to build autocorrelation into your model. I'm not convinced it's worth the trouble, so I don't bother but that is a subjective choice on my part and not a limitation of Monte Carlo simulation.

      I guess the more important question is whether you believe you can predict next year's market return from this year's. I don't, so IID seems a reasonable guess.

      A key point of this post is not that the stock market "works like that" but that the stock market doesn't "work like anything." It's uncertain.

      The other key point is that when you can't predict how the market will perform, it's helpful to generate a large number of scenarios of how it might perform and make sure your strategy works across most of them.

      Would it be more meaningful to use rolling histories of actual stock market performance? More meaningful in what way? Would it help you predict your individual retirement outcome? Nothing will — it's uncertain. If you believe that the stock market will perform in the future as it has throughout its limited history, then it would be more meaningful. I'm not very sure of that.

      Would it generate more potential future outcomes to consider? Clearly not, since we only have 200 years or so of actual market data, or about seven unique 30-year periods to study.

      Some studies try to get around that sample size limitation by using rolling periods, as you suggest, but there are only about 170 or so of those (better than seven) and rolling periods aren't used evenly. The returns from the earliest and latest years aren't used as often as those in the middle. (Assuming 200 years of returns data, the first and last year's returns would only be used once, for example.)

      Regardless, if you use the relatively small amount of historical market data available, you end up with a small sample size and a large confidence interval.

      Would it also be useful to take into account that we are currently at a very, very high valuation level? I would say "prudent" instead of "helpful."

      Current market valuations are historically high but if most investors agreed that they are unreasonably high, then the market wouldn't still be climbing. (I think they are.) "Very, very high" then becomes your subjective evaluation assuming that future market returns will follow history.

      When we say "valuations are high", we are typically implying that we believe returns tend to mean-revert, so they are more likely to be lower in the future than in the past. This is also a subjective assessment but I tend to agree. It turns out to be fairly simple to adjust Monte Carlo simulations when you suspect valuations are high and that markets mean-revert — just lower the expected future market returns.

      When Wade Pfau says that the sustainable withdrawal rate may be closer to 3% than 4% due to historically high market valuations and historically low bond yields, he means that those conditions make it more likely that future market returns will be lower than in the past, assuming mean-reversion, resulting in a lower sustainable withdrawal rate in the future. You test that by simulating lower future market expectations than historical market returns would suggest. If you're using an online simulator you can typically enter your own estimate of future returns. Just lower it.

      Ultimately, however, market returns are uncertain, so we can't know with any certainty that's how things will play out in our lifetimes. Monte Carlo simulation just provides a much larger set of possible future outcomes to consider and that's its goal — not predicting the future from the past.

      Thanks for writing.

    2. Hi Dirk,

      Intersting research.

      In this case, I agree with Anonymous, however. Monte Carlo simulations are not 10,000 possible outcomes. The bottom and top 5-10% are clearly not possible.

      For example, the lowest line would be 30 2008s in a row. That is certainly less likely than 30 8% years in a row.

      The point is that stock market returns are not random - especially at the extremes. Large gains and large losses typically follow each other.

      For example, 100% of the calandar losses greater than 20% in the S&P500 in the last 150 years were followed by a gain of more than 20% (except for 1929-32).

      After a major decline, a major recovery almost always happens. Monte Carlo does not describe actual stock markets. It just uses statistics to create thousands of possible and impossible scenarios based on mean and standard deviation.

      Whenever you use Monte Carlo, I would suggest to ignore the bottom and top 5-10%. They are impossible.

      Using a random selection of actual stock market histories is far more accurate and informative.


    3. Ed, with all due respect, what you suggest is a near-perfect explanation of how Monte Carlo simulation doesn’t work.

      The reason MC, developed for the Manhattan Project, exists is to generate lots of scenarios that are likely and far fewer that are unlikely, usually based on a normal distribution. The extreme scenarios you suggest are incredibly unlikely to show up in a simulation. The probability of a MC model generating the outcome you imagine as common to simulation (30 consecutive years of losses and 30 consecutive years of gains) is infinitesimal.

      The problem with using actual historical returns is there aren’t enough of them to generate a reasonable confidence interval; otherwise, we would. If they were to be selected randomly, as you suggest, that would remove the momentum you say you want to include. (Autocorrelation can also be simulated.)

      More importantly, you ignore sequence risk. The ten best and worst scenarios of any simulation are unlikely to have the 10 best and worst market returns. They are more likely to have the 10 best and worst sequences of those returns from among the simulated scenarios.

      If Monte Carlo simulation is so flawed, as you suggest, why do you think that virtually all academic research on this topic is based on it? And, why would a Nobel laureate in Economics (William Sharpe) develop Monte Carlo retirement simulation software and distribute it freely?

      Monte Carlo simulation of retirement scenarios is imperfect, as I will describe in my next post, but it is the best alternative available for developing a starting point for a retirement plan.

      Thanks for sharing your thoughts!

  2. Very good article. Did you use R to create the graphs? Have you ever considered posting the code on GitHub for others to use? (Similar to what Nick does over at


    1. Thanks, Matthew. I use R for just about everything, nowadays, including those graphs. I have posted some of my Retirement Cafe code at github (under "Dirk Cotton") but I doubt many of my readers have even heard of R so I try to provide spreadsheets, instead, from time to time.

      The simulation script is large and complex but as soon as I have adequate confidence in it, I will post it at github. I'm currently doing some research with a UNC economics professor and I'll post that R script after the paper is published, as well.

      Thanks for writing and for your kind words.

    2. Thanks, Matthew and Dirk, for bringing up and answering the "where's the code?" issue. Dirk, it may be true that few of your readers know about R. For those of us who do, though, it's wonderful to have you doing the coding and us doing the benefiting. I (and at least one other person; Matthew) look forward to your posting that code on Github.

      And, to reinforce you once again, thanks for this article. You address the strengths and weaknesses of modeling in a fair manner. Presenting both sides of something is uncommon in most recent discourse--on just about any topic.

    3. OK! So, there are at least three of us. The simulator is a work-in-progress but one day. . .

      My philosophy is that if retirees understand the benefits and risks of retirement strategies and retirement tools then they should choose the one that makes them happy. That might not be the one that makes me happy, but my situation is probably different than yours and ultimately you're the one who has to live with the decision, not me. None is perfect. And since the outcomes are stochastic, no one can be certain which bet is best. The goal should be to make a good bet.

      I'll probably misquote Michael Finke here, doing it from memory, but I like what he said and I'll catch the spirit. "It's our job to make retirees happy and sometimes what makes them happy is less risk in their investment portfolio."

      I'll talk about the weaknesses of simulation next time and then move on to how to best use these tools.

      Thanks for the comment!

  3. A very interesting post Dirk, and thanks for the shout-out.

    This is an interpretation of simulation runs over set time periods. The problem with the data generation is that the cash flows are time sensitive based on age. As one gets older, the time frame for the simulation period gets shorter. Shorter time periods generate different cash flow capability (you can spend more with less time remaining oversimplifying the concept). Different cash flows for those shorter periods (older ages) result in different portfolio balances at each and every subsequent age as a result. The graphs would look different for both income and portfolio balances as a result.

    "Basically the derivation and remainder values of today’s calculations/simulations are interpreted as possible future states. The research on yesterday’s post shows “possible future states” (remainder values) is not a valid interpretation of the data because the subsequent year’s solutions would be different; because the time frame for those solutions are different due to aging and period life table statistics, not fixed as viewed today as a result of calculation methodology." ... quote from my post summarizing research on the rolling calculation solution to this simulations interpretation issue

    Your point about modeling is spot on. They are useful for decision making and planning. My example was hurricane plots that keep changing as time goes on and the possible paths get updated with new data inputs and reviews over time. Thus, interpretation of today's retirement model would also be modified each year the model is rerun with updated age and portfolio characteristic results along with life table adjustments.

    Retirement planning is not a "one and done" model. It is a "rinse and repeat" model.

    Great post Dirk!

    1. Thanks, Larry. I agree with your assessment. I think the issue you raise is making the assumption that one runs this simulation at the beginning of retirement and assumes those probable future states are constant throughout retirement — I don't.

      I believe they need to be re-run every year with revised facts and assumptions, which would obviously create a new set of likely future scenarios every year. "Possible future states" is a valid interpretation of the data if the underlying assumption is "valid until information changes", which is always mine.

      I find a sailing trip representative. When you set sail for a destination you have a plan and a heading and an ETA. Whether you're off course by 5 miles or a hundred at the end of each day, you have to create a revised plan based on a new set of likely future outcomes. It would be irrational to use yesterday's heading without replanning.

      I suppose I could be more explicit and say, "these are likely future scenarios based on what we know and think today but their likelihood will be different once either or both of those things changes significantly."

      I agree with your last paragraph and, frankly, don't see a rational alternative. I'm not sure why anyone does. I hope my readers understand by now that any retirement planning method (including simulation) needs to be reviewed and possibly revised as new information comes available and as our life expectancy inexorably declines with age.

      Thanks for joining in!

  4. Excellent presentation of Monte Carlo and enlightening people to this vs straight line approach. Thanks.

  5. Too complicated for me. can we just KISS?
    Heard a former school teacher put all in dividend paying stocks and never sale any of them. it took her over 20 years to cross 1 million mark at age 69. Now she is 97 years old and receiving over $900K income from those stocks every year. with that one would guess the value is over 30 millions.

  6. Now, THAT is an interesting story. :-)

  7. I enjoy reading your posts as it is clear you take the time to think things through. Yet here is my question as I am one of those spreadsheet guys. Assuming one starts with a decent sized portfolio, lowers expected returns, and adjusts 'the sails' yearly, what is wrong with using a correctly set up spreadsheet?

    1. Jim, the answers to your questions are visualized in the second graph of this post. The information that your approach provides is shown by the red line and you can see that the blue lines provide a great deal more information.

      The shortcoming of your approach is that its (singular) prediction is based on a single factor (expected annual market return), which isn't even the most important consideration.

      This is somewhat like trying to guess someone's weight given just their height. This strategy is probably better than a poke in the eye but you would make a far better guess if you also knew the subject's age, gender and nationality.

      A good model will consider several other factors. The most important is length of retirement and the second most important when you plan to spend from a volatile portfolio is market return variance (the sequence of market returns in this case is more important than the returns, themselves). Your model considers neither of these.

      If you lower your expected return, you bend the red line downward but you still have only one red line that appears to never fail. You still know nothing about the probability of achieving the "red line" outcome or how likely you are to prematurely deplete your portfolio given planned spending. In effect, you are saying that you know your prediction is optimistic so you will lower the return by some guess in hopes of making it less optimistic. You will spend less to be a little safer because you know your model doesn't make good predictions.

      Since you plan to "adjust the sails" yearly, you have a variable-spending strategy. That's good — it makes it harder to deplete your portfolio than fixed spending would. It doesn't, however, guarantee that the safe amount that you calculate to spend each year will meet your expenses. You probably won't go broke but you may well lose your standard of living to a poor sequence of market returns.

      But I suspect your real question is "what could go wrong if I use the spreadsheet approach?" There are two big concerns. The first is that your intrinsically optimistic view of the future will repeatedly suggest a higher current spending amount and possibly leave you with less available spending later in retirement.

      The second concern is that your approach provides no insight to guide planning for bad outcomes. (It never has a bad outcome.)

      Essentially, the spreadsheet approach ignores the most important risks of retirement.

      I hope that further explanation helps but let me add that my goal is to inform you, objectively, of the pros and cons of each strategy. Once you understand them, then I firmly believe you should choose the strategy that makes you happy.

      Thanks for writing!

  8. I recently discovered your blog, and am having a great time working my way through your posts. I particularly like your engaging and enlightening “Unraveling Retirement” posts, which have really helped me clarify my thinking on retirement planning and spending.

    I completely agree with this post’s message that it’s vital to take account of uncertainty in retirement planning. Still, I was struck by the fact that the (no-variance) spreadsheet model end result is higher than the great majority of your Monte Carlo results. This seemed (to me, at least) pretty counterintuitive. After pondering it a bit, I believe the explanation lies in the distinction between the geometric and arithmetic means for your assumed rate of return.

    Let me illustrate. Using an assumed 5% investment return (rounded from 5.25% to simplify the math), two years’ worth of returns using the no-variance spreadsheet would cumulatively result in an increase of just over 10%. That is, $100 on day 1 would become $110 (and change) after two years of returns – in line with what we would expect for 5% return. If, on the other hand, we use returns that are one standard deviation away from the average – the first year up at 117%, the second year down at 93% -- we end up with slightly less than a 9% increase after 2 years, i.e., a little under $109 --close to the no-variance return, but a bit lower. If we look at two years of more extreme returns, this time two standard deviations away from the average (129%, then 81%), we’re quite a bit lower -- $104.50 at the end of two years. In all three cases, the “average” annual return is 5%, but the realized return varies – in the last case, it has dropped to a little over 2%! The results are similar if you carry this comparison out to, say, four years. Interestingly, the order of the returns (high, then low, or the reverse) doesn’t matter.

    What this example illustrates is the difference between the arithmetic mean (add up the annual returns and divide by the number of years) and the geometric mean (the constant annual return that would result in the final amount). In your post, the 5.25% spreadsheet return is both the arithmetic and the geometric mean. In your Monte Carlo simulation, on the other hand, I believe the 5.25% is the arithmetic mean around which your scenarios cluster -- which, if I’m reading your results correctly, results in something like a 3.5% realized annual mean return. With a withdrawal rate of 4.5% and a (geometric) return of 3.5%, it’s not surprising that, on average, the portfolio ends up lower than it started!

    Burton Malkiel (2007), using Ibbotson data, noted something similar: large company stocks returned 10.4% (7.4% real) between 1926 and 2005 when calculated as a geometric mean, but the same return was 12.3% (9.3% real) when calculated as an arithmetic mean. So – is your Monte Carlo simulation model using the mean return you really intend (e.g., 9% arithmetic average return on stocks, if you’re using history as a guide)?

    Sorry for the blog-length (book-length?!) comment. Keep up the great work!

    1. Retired Keith, thanks for the question. The short answer to your long question is yes, the model used the expected return I really intended.

      I'm not sure how you calculated 3.5%. The model parameter was 5.25% annual compound growth with 12% annual standard deviation. The mean annual growth rate after 10,000 scenarios was 1.05293, or about 5.29%, as expected. If I ran it many times, it would converge to 5.25%.

      You may be confusing arithmetic and geometric returns of historical returns with expected returns. The Malkiel numbers you quote are inflated; I use real returns and I don't expect future returns to be as high as past.

      Each scenario (blue line) has its own geometric mean that can't be calculated until after the simulation. Each year's results for each scenario are the product of the previous year's ending portfolio balance (minus annual spending) and a growth rate, not an arithmetic average. I'm not sure it's possible to make the error you suggest.

      Even had I mistakenly used an arithmetic mean of historical returns instead of a geometric mean as my expected return (I didn't), the same return would have been used and applied as a growth rate in both models and wouldn't explain the difference.

      The red line won't always have the same relationship to the blue lines that this graph shows, so wondering why it is so much higher in this particular example won't be helpful. It's acceleration will depend on the withdrawal rate assumption. Regardless, it won't be representative, which is the point.

      The reason the red line is so different is due to — as I might have mentioned in the post, he said with tongue in cheek — longevity risk and sequence of returns risk. When I set the MC model's standard deviation of market returns to 0% and life expectancy to 30 years (as the spreadsheet models assume) all of those blue lines converge into the red one.

      Long questions are fine, by the way. I often have to figure out what short questions are really about. And keep checking my math. We all make mistakes.

      Thanks for writing!

    2. Dirk –

      Thanks for the quick and thoughtful reply. To follow up just a bit: the 3.5% realized return number I mentioned was my interpretation of your statement that you had to drive the spreadsheet return down to 3.5% to come up with the median simulation result. Perhaps I misunderstood, but if 3.5% was the constant annual return, compounded over time, that led to your 50th percentile result, it sounds to me as though it is indeed the compound annual growth rate (CAGR), aka geometric mean, of that outcome. In other words, 20 years of returns, averaging 5.25% and with a 12% standard deviation, averaged over 10,000 simulations, seem to center on a CAGR of 3.5% -- even as you have confirmed that “the mean annual growth rate” of your scenarios is almost exactly 5.25%. This discrepancy is analogous to the observed return on stocks from 1926 to the present: 7% real CAGR, or geometric average annual return, but 9% real annual return as an arithmetic average.

      Much as it pains me to say this (since I also think the geometric mean is the superior measure of return!) I submit that the correct value to input as the mean of your Monte Carlo distribution is, in fact, the arithmetic mean of the portfolio return distribution you’re trying to model. Take a look at this 12/27/2017 article by Michael Kitces. (Not sure if the link will translate, but hopefully you can find it by date and author.)

      Apologies for taking up more space on this rather abstruse point. Keep up the great work!

  9. "if 3.5% was the constant annual return, compounded over time, that led to your 50th percentile result, it sounds to me as though it is indeed the compound annual growth rate (CAGR), aka geometric mean, of that outcome."

    It is not. It is merely one among countless CGRs that could result in that median terminal portfolio value.

    Your argument seems to assume that the median terminal portfolio value results from the (singular) median compound growth rate. While that is correct with a simple compounding portfolio and will appear correct from a spreadsheet model that ignores volatility, that relationship does not hold when spending from a volatile portfolio.

    When spending from a volatile portfolio, the terminal portfolio value is more a function of the sequence of returns than of the returns themselves, so there is no one-to-one relationship between a single CGR and a TPV. There will be countless CGRs that can produce that same TPV, given the right sequence of those returns.

    This is visualized in the third chart above that shows 15 paths (scenarios) that lead to the median terminal portfolio. In reality, there are countless more paths. Because each scenario consists of randomly-generated returns, the CGRs for each of those paths is different, yet they all end at the median TPV. You cannot work backward from the median terminal portfolio value to determine the median compound growth rate for the simulations, which, as I previously explained, is 5.29%, not 3.25%.

    Don't let that arithmetic/geometric-mean concern cause you too much pain — it's coincidental in this example and irrelevant.


  10. Dirk, you mention running 10,000 scenarios in your simulations. How many runs are required to make a valid conclusion? Some retirement models out there use fewer runs. Would say, 500 runs, be OK?

    1. Good question.

      More scenarios is better but with diminishing returns. In Chapter 4 of William Sharpe’s paper on RISMAT (easy to Google), he provides a statistical argument that 100,000 scenarios will provide a good analysis, so I don’t bother with more.

      I can run 100,000 scenarios in a few seconds but rarely see a significantly different answer than with 10,000, given all the uncertainty of predicting the future.

      Is 500 scenarios “enough”? It depends on how you define “enough.” It’s certainly a lot better than the one scenario you get with the spreadsheet approach.

      10,000 scenarios would provide more information than 500 but will it be different enough to cause you to change strategies? Hard to say.

      There are many, many free simulators on the web, though, so I’d try to find one that runs more scenarios.

      Thanks for the question.