What the Heck is Monte Carlo Analysis?

In reading about personal finance, you may have come across references, often hushed and worshipful, to a mysterious thing called Monte Carlo analysis.  Perhaps you wondered:  What is it?  Is it really the gold standard for financial planning?  Are there any alternatives?  If this sounds like you, read on!  

What is it?

Monte Carlo analysis, developed by mathematician and nuclear scientist Stanislaw Ulam while working at Los Alamos in the 1940s, is an important tool for modeling a complex system when a key variable is unpredictable, but whose possible values can be described with a distribution (such as a bell curve).  Monte Carlo essentially substitutes an entire probability distribution for the unknown variable. (Here is a readable explanation of the general method.)

Because we have a good understanding of historical stock and bond returns (but admittedly can’t predict the future), Monte Carlo analysis is often used to model possible financial outcomes – usually under the assumption that the future range of outcomes will be similar to what we’ve seen in the past.  MC does this through a rather brute force method in which hundreds or even thousands of scenarios are run, each one drawing a single, random value from the distribution the modeler has defined (say, historical annual returns of the S&P 500).  By doing this, it is possible to look at all the outcomes of a retirement model and notice, for example, that in 90% of them your retirement funds last until you’re one hundred. 

Why use it?

Projecting possible financial outcomes in a world of uncertain and often volatile returns is tricky.  Say you want to know how your nest egg, invested 50/50 in stocks and bonds with some living expenses withdrawn every year, will do over the course of your retirement – till you’re 95 or 100, to be safe.  One approach would be to simply assume that you’ll receive a fixed, historically-based rate of return such as 5% real (after inflation) annual return.  (This is about the average return of a mixed 50/50 portfolio over time.)  This approach might be adequate when you’re younger and accumulating a nest egg over a period of decades, because over a long period of time you very likely will receive something close to the long-term average return (if you stay invested and don’t panic at the wrong times). 

Unfortunately, this simple approach is not good enough for modeling what might happen in retirement.  While stocks have returned about 7% after inflation over the past 100 years, there have been long stretches, decades or longer, in which they gained nothing or even lost value – such as the 1910s, the 1970s and the 2000s.  (Interestingly, the 1930s had a slight positive real return, due to deflation.)  If you’re unlucky enough to be hit by low returns for a number of years early in retirement, your portfolio, beset by the double whammy of bad returns and withdrawals, may shrink enough that it can’t recover even if returns improve later on.   Assuming you’ll receive a constant 5% real (not counting inflation) return – or even a more conservative 4% or 3% — might turn out to be too optimistic. 

Monte Carlo analysis models how this sequence of returns risk might play out by simulating a large number of retirement scenarios with different returns randomly drawn from a distribution of possible values that represents actual historical returns.  Instead of making a single assumption about returns – which might be a best guess or something more conservative – you’re essentially playing out almost every conceivable possibility, including a few quite improbable strings of positive and negative returns that are actually more extreme than anything in the historical record.  By looking at the results collectively, you can project how robust your retirement plan might be in withstanding the slings and arrows the markets may send its way – does it work 75% of the time?  95%? 

Sounds great – are there any drawbacks?

Despite its power, Monte Carlo analysis is not a magic bullet.  It is, after all, based on history.  Even if a modeler runs 10,000 simulations, the underlying rate of return data are based on about a hundred years of actual history in the stock and bond markets.  More simulations do not create any more data. 

Historical data are all we’ve got, but we don’t know for sure that they’re a good predictor of the future.  A number of financial analysts argue that future stock and/or bond returns will be lower than in the past.  (See this article, for example.)  They, of course, don’t really know.  Other planners believe that basing future return estimates on past experience is the best – really the only – approach available. 

More technical criticisms of using Monte Carlo analysis for financial modeling point to the common assumption that returns are distributed in a normal (bell curve) distribution.  In fact, historical stock returns are not distributed normally; the actual distribution of returns has “fatter tails” than a normal distribution.  Using a normal distribution as an approximation of the ‘real’ underlying returns might lead to an underestimate of very bad (as well as very good) return years – not so helpful if you’re interested in the probability of a string of those bad years occurring in your retirement. 

On the other hand, the Monte Carlo approach typically draws return data randomly from its distribution, while many people believe that stock market returns revert to a mean.  Historically, a string of bad years has typically been followed by years with positive returns – and vice versa.  In this respect, Monte Carlo analysis might tend to be overly pessimistic.  (This article makes this case.)

Finally, remember when you run a Monte Carlo analysis, in most cases you’re using a model that contains myriad assumptions (on probability distribution, but also on taxes, inflation, longevity, etc.) you aren’t even aware of. It’s very much a black box.

How do these factors net out?  No one knows.  So, while Monte Carlo analysis is undoubtedly a cool modeling technique, remember to take the results with a grain of salt. 

Are there alternatives?

The simple alternative to Monte Carlo analysis is to use a constant rate-of-return factor appropriate for your portfolio, but supplement it with sensitivity analysis of different return assumptions.  So, if your baseline assumption is that your 50/50 portfolio will return 5% on average after inflation, try out lower rates of return, too – 4%, 3%, maybe even down to 0%.   If changing the assumption to 4% or 3% is enough to cause your plan to fail, you might be vulnerable to sequence of returns risk. 

While better than basing your planning on a single rate of return, sensitivity analysis does not really capture the dynamic variability that Monte Carlo modeling does and that we see in historical returns. Many analysts prefer historical testing, using actual 30-year periods over the last century, to Monte Carlo analysis.  History has the advantage that it is realistic, since it uses actual past returns in a sequence that (unlike Monte Carlo) we know can occur.  By using 30-year rolling periods (1926 to 1955, 1927 to 1956, and so on) from 1926 to 2018, historical testing gives us quite a few periods to test (64, if we begin with 1926).  Note, however, that only three of these are truly independent data points, since the periods overlap so much.  Also, returns from the middle of the historical period (1950s to 1980s) are counted much more frequently than earlier and later years; they may show up in up to 30 rolling periods, while 1926 and 2018 rates of return are used only once.  So, while we may be utilizing all the historical returns from 1926 (or 1871), they’re not all given equal weight.

One practical advantage of historical testing is that the work has already been done for you.  Historical analysis by Bengen and his intellectual heirs has resulted in the 4% rule – if you withdraw 4% of your retirement savings in the first year, then increase that each year for inflation, you would have had sufficient savings in all 30-year periods since 1926.  (See this article arguing that historical analysis and the 4% Rule are still sound.) 

What’s the bottom line?

It’s not possible to predict the future, even with a powerful probabilistic approach such as Monte Carlo analysis.  Nevertheless, given the variability of market returns and, you owe it to yourself to do some analysis of the potential impact of sequence of returns risk on your retirement.   There’s no reason to frighten and depress yourself with extremely improbable, sky-is-falling scenarios, but it does make sense to evaluate some reasonable but pessimistic possibilities. 

Since no one approach is foolproof, I recommend you try all of the approaches outlined above.  First, if you’re using your own financial spreadsheet or an online calculator that uses a constant rate-of-return assumption, experiment with some more conservative (worse) return assumptions, in addition to your base case or best guess, to see how your plan holds up.  Next, review how your portfolio would have done historically by using a calculator such as this one (or simply use the 4% withdrawal rule).  Finally, go to one of the many online calculators (I like this one and this one) and test your financial plan using Monte Carlo analysis.   

Now compare results.  If your retirement financial plan can survive your pessimistic assumptions and succeeds 90% or more of the time in historical and Monte Carlo analyses, it has passed your stress test. Sleep well!

References

Finke, Michael; Pfau, Wade D.; Blanchett, David M.  (2013). The 4 Percent Rule is Not Safe in a Low-Yield World, Journal of Financial Planning 26 (6): 46-55.

Kitces, Michael.  (2012, August 15).  What Returns Are Safe Withdrawals REALLY Based Upon?  Nerd’s Eye View.

Pease, Christopher. (2018, Sept. 6). An Overview of Monte Carlo Methods.

Tharp, Derek.  (2017, July5).  Does Monte Carlo Analysis Actually Overstate Tail Risk In Retirement Projections?, Nerd’s Eye View at Kitces.com