Adam M. Grossman

**IN JANUARY 1946,** a man named Stanislaw Ulam found himself confined to a hospital bed, having suffered an encephalitis attack. A brilliant scientist and a veteran of the Manhattan Project, Ulam wasn’t the type to sit idly while he recuperated. Instead, after playing innumerable games of solitaire to pass the time, Ulam began to examine the statistical aspects of the game.

Among the questions he asked: How can you accurately estimate the probability of winning a game? To answer this question, Ulam ended up devising a novel statistical technique that he dubbed Monte Carlo analysis. Today, this approach is broadly accepted and widely used in everything from engineering to biology to financial planning to meteorology—and even basketball.

How exactly does Monte Carlo analysis work? The idea is this: As long as you know the basic dynamics of how something works—whether it’s weather patterns, card games or anything else—you can use a computer to simulate an experiment thousands of times and then simply count up the frequency of various outcomes.

How is Ulam’s technique applied to financial planning? Knowing how the stock market has performed in the past, and the degree to which it varies from year to year, you can simulate the market’s future performance. But instead of focusing on the stock market’s average annual return, Monte Carlo analysis focuses on average multi-year returns. This allows financial advisors to reassure their clients with confident-sounding statements such as: “I’ve tested 10,000 scenarios and can tell you that your retirement plan has a 90% probability of success.” To see what a Monte Carlo simulation looks like, check out Vanguard Group’s retirement nest egg calculator.

While Monte Carlo analysis is widely used in financial planning, I would advise caution, for two reasons:

**1. Subjective inputs**. Monte Carlo simulation works well when forecasting physical or mechanical processes—things that act in predictable ways or, at least, within known limits. A card game, for example, can develop in numerous ways. Still, it’s always played with a fixed set of rules and an identical deck of cards. While you don’t know which card will come up next, you do know there will never be five aces in a deck. As a result, the set of possible outcomes is necessarily limited.

When it comes to the stock market, though, the opposite is true: An infinite combination of political and economic events—coupled with human emotions—can drive the market in unpredictable ways. In a card game, it’s difficult to know what will happen next. In the stock market, it’s impossible to know. And it’s not just the stock market. Lots of other variables impact the success of a retirement plan, including inflation, tax rates, interest rates and potential changes to Social Security and Medicare.

Try this thought experiment: Tomorrow morning, pick up the newspaper and ask yourself, “How many of these stories could I have predicted five or 10 years ago?” Not many, I suspect—and yet that’s what we are doing when we expect Monte Carlo analysis to help us forecast multi-decade retirement scenarios. To be sure, the past serves as a guide to the future, but it’s just a guide. That’s why any simulation of the stock market rests on shaky ground, simplistically assuming that the future will mirror the past.

**2. Less-than-useful output**. If the inputs to a Monte Carlo analysis are subjective, the outputs are even more troubling.

First, Monte Carlo output is normally expressed as a percentage—a 90% probability of success, for example—but what exactly does “success” mean? It means simply that you won’t run out of money. That sounds logical. Problem is, it’s defined very literally. If a Monte Carlo simulation determines that there will be even $1 left at the end of someone’s life, that is defined as success. For a computer, that may make sense. But for a real person seeing their funds rapidly approach zero late in life, that hardly sounds like a pleasant or successful outcome. But because this is the way Monte Carlo simulations work, they may lull people into a false sense of security or, alternatively, scare the daylights out of them, when neither reaction may be warranted.

The second problem with Monte Carlo output is that, if you dig into the results, it provides a range of potential outcomes so wide that it borders on the absurd. Use a typical Monte Carlo program to simulate the growth of a $1 million portfolio over a 30-year retirement, for example, and it will unhelpfully project that your assets will fall somewhere between zero and $29 million. While this may be statistically accurate, personally I don’t find it very useful.

To be sure, Monte Carlo analysis has its place in scientific disciplines. But when it comes to retirement planning, I would urge you to be skeptical. Always look at the numbers behind the numbers—and never let any one analysis drive your financial decisions.

*Adam M. Grossman’s previous articles include Get a Life, Higher Taxes and *Moving Target*. Adam is the founder of **Mayport Wealth Management**, a fixed-fee financial planning firm in Boston. He’s an advocate of evidence-based investing and is on a mission to lower the cost of investment advice for consumers. Follow Adam on Twitter **@AdamMGrossman**.*

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DI came back to this article from one that Adam just published on the blog. I like that it explains two things: 1) Where Monte Carlo simulations started (an early version of the kinds of probability theories that have become quite popular with the massive power of modern computers (from my point of view.)) 2) A few reasons why the seemingly powerful modeling approach might fail to provide a good guide to practical solutions.

The goal of any modeling, computer or otherwise, is ultimately to guide problem solving. Models people don’t understand often lead to outcomes that may not be justified.

I would add another thought: The behavior of financial markets at any time may not be random, but rather have significant relationship to what has happened up until then. Perhaps it’s only in the very short term that the behavior appears mostly random. If this were not true, there would be no “technical analysts” or “algorithm-based trading.” Of course, some might argue that both technical analysis and algorithmic trading are fictional constructs, so there’s that!

I couldn’t agree more with this article. I am an engineer and the rule of any calculation or software is bad data in bad data out. As every financial institution is required to say, “past performance is not an indicator of future results”. Past performance of the stock market is not an indicator of future results, even if you run thousands of historical scenarios. There is a bias in the historical data – it is historical.

Monte Carlo simulations may also fail to take into account one large impact in the standard 401K plan – RMD. You will not be free to remove only what you want to remove to optimize longevity. The government will tell you what you must withdraw. The only optimal way around this is Roth conversions from age 55 to age 70.

Everyone’s best opportunity for success is to save as much as you reasonably can. Then plan a budget to live on less than that. Live debt free. In good years park the excess in savings. In bad years, live on less and if necessary access savings.

I created my own Monte Carlo simulation for my retirement planning and am targeting a 95% success rate, where success is defined as never running out of money. I feel it’s a much more sophisticated method that can test against long bear runs during your retired years better than a simple rule of thumb can like the 4% safe withdrawal rate rule can.

In school I was required to do single variable linear regression problems with a basic calculator. Afterwards we were allowed to use fancy calculators and computers for multivariate stuff (punch-card fed mainframes!). This forced me to understand the weaknesses of this very useful tool that could lie like the devil himself if you fed it autocorrelated data. At the suggestion of a prof I proved with near mathematical certainty that the mid-60’s bull market was inversely correlated to (i.e. caused by) the length of women’s skirts. The shorter the skirts, the higher the market. Go-Go baby!

In other news, a hobby in electronics led to the use of Monte Carlo analysis to predict filter behavior with user specified component tolerances (resistors, capacitors, etc). Plots of the extremes and everything in between are drawn as the math varies component values. This process is dead accurate and incredibly helpful. You save a bunch of time and money using math and a minimum of prototypes to predict maximum permitted tolerances for production purchases. While electronic component ranges can be known, how far you can go with this in a domain heavily influenced by human behavior is unknown.

Math is wonderful, you just have to know what’s going on under the hood if you’re going to use it for important decisions. Otherwise, consider the well-chosen title of this article.

The risk of using Monte Carlo are the normal distributions of its random inputs. Normal distributions are Gaussian in nature, but variations are more power law than Gaussian in nature. What does this imply? Power law distributions may have a median, but not a well defined average. Standard deviation is even less defined. Power law distributions are commonly observed in biology and physics. Examples are average salary and average city size. The 1987 crash was 20 standard deviations outsize the average, which is pretty much physically impossible. Use Monte Carlo as a rough guideline, but don’t believe it entirely, but do believe in “black swans”, which is what not Gaussian distributions are about.

I would suggest using not ruin but the original amount in real terms. That is leaving money to your heirs, but the rate is not that much different though more reassuring. Historical data is always problematic in a changing world, but behavior can work for you if you can adjust.