Investor’s Guide to the Monte Carlo Simulation

Did you go through a random phase as a teenager?

You know, colorful hair, eccentric clothes, and maybe even a piercing or two? 🤦‍♂️

Indeed, being random was ‘in’ back in the day—even if you cringe today when looking at those old pictures (at least we didn’t have Facebook then). But, here’s the good news: You can also use “randomness” to help with managing risk—and growing your portfolio.

So how can you use randomness to make meaningful decisions? Financial at that. Isn’t that like gambling?

As strange as it might sound—while bizarrely interwoven with gambling since the method is named after a famous casino resort town—it is much more a game of skill than luck. The Monte Carlo method has a very illustrious history, and a fairly scary formula, but it really isn’t hard to understand—and can help you significantly once you get it.

So, without further ado, let’s dive in and find out what this method is, how it works, and how it can help you make clever investments.

What Exactly is a Monte Carlo Simulation? 📚

The Monte Carlo simulation is a computational algorithm that obtains its results by repeated sampling using semi-random variables. It has numerous applications in fields as wide and varied as engineering, finance, research management, a game of D&D, animation, and physics—with its roots stemming from the research into the atomic bomb.

It is also—clearly—a mouthful. However, this is no reason to fear as what it does isn’t nearly as complicated as its definitions might imply. Essentially, rather than choosing one scenario when predicting an outcome of a decision, the Monte Carlo method picks multiple possibilities and runs the simulation as many times as there are possibilities.

Then, it looks at what possible outcomes each simulation comes up with and provides the most likely ones based on the decision an investor wants to take. For example, if an investor is looking to buy stocks of a certain company, they’ll probably want to know what effect such a purchase would have on their portfolio.

Usually, investors look at the company’s past performance—though it’s ideal to perform proper technical analysis before any financial decision—to figure out how the prices could change. This would, however, probably not yield reliable data.

The Monte Carlo simulation also takes into account a company’s past performance but would run the simulation numerous times with different potential values to give the best idea possible of the likeliest future returns.

The result of the Monte Carlo simulation can align neatly with an investor’s other projections, but can also yield completely different results.

History of the Monte Carlo Simulation 📖

The creation of the Monte Carlo simulation is linked to the card game Solitaire and the illness of a mathematician named Stanislaw Ulam. Reportedly, while Ulam was recovering from his illness, he fought boredom by playing—and constantly losing—solitaire.

At one point, he became frustrated and decided to try and calculate the likelihood of actually winning a game. After a while, Ulam designed a method for calculating probabilities but realized that the math was far too complicated to solve without a computer.

Fortunately for Ulam, he had multiple influential friends among whom was another mathematician—John von Neumann—who had access to the supercomputers of the day—ENIACs.

Ulam and von Neumann went on to conduct the very first Monte Carlo simulation on such a computer in just a few hours and then further developed the method which ultimately led to the creation of the first hydrogen bomb.

Don’t be frightened by the fact it took ENIAC hours to compute the simulation—modern computers are incalculably more powerful than those machines and can usually run all the necessary math in seconds.

Understanding the Monte Carlo Simulation 👨‍🏫

A fairly easy way to understand how the Monte Carlo method works would be a hypothetical study to determine the average height of a human being.

Understandably, the only way to get the exact average height of a human is to measure every living person in the world. This, fortunately, is not how most studies are conducted. Usually, researchers take a limited number of people, measure them, and assume that the average height of the measured group is representative of the entire population.

This represents the first idea behind the Monte Carlo simulation—it takes and measures a large but finite number of possible outcomes to try and get an accurate prediction of the future.

This isn’t the entire picture, however. If we wanted to make that height study worthwhile, we’d have to ensure at least two more conditions are met. First, the sample size should be as large as possible—measuring 1,000,000 people will give results closer to reality than measuring 1,000.

Second, we’d want our sample to be as random as possible. Children are usually shorter than adults, and men are usually taller than women—and people from some parts of the world simply tend to be taller than others. 

Having all our participants be kindergartners would produce inaccurate results. Ideally, we’d want our sample group to come from all age groups, genders, and from as many geographical locations as possible.

A common place to see this approach is with political polls. Gallup doesn’t survey every American when trying to predict who the next president is going to be. The typical sample size for a Gallup poll, whether it’s a traditional stand-alone poll or just one night’s interviewing, is 1,000 national adults. They often have a margin of error of ±4 percentage points.

Likewise, a Monte Carlo simulation should be conducted using random values and should be repeated as many times as possible to get the results as close to reality as possible. While using random values to get true results might appear strange at first, a pretty big testament to the method’s value is the fact that we can pretty accurately estimate the value of π using the Monte Carlo simulation.

Monte Carlo Simulation: Probability Distributions 📜

Since the Monte Carlo method runs numerous simulations with different variables, it gives many, often vastly different, results. These results tend to come in clusters.

Often we’ll see that while, for example, the highest possible future returns of an asset are 50%, and the lowest -20%, most results are in the area between 10 and 15%.

This result is what we’d most consider when comparing our determined risk tolerance to the risk level of an investment. If the simulation had 1,000,000 iterations—each iteration being a possible future with a different value assigned to the variable—and 1 gave the returns of -20%, and 3 gave 70%, those are the possibilities that can be ignored.

On the other hand, if 500,000 gave the expected return rate of 14, 15, or 16%, those are the numbers we’re likely to get in real life if we go through with the investment. Obviously, this is just one possible probability distribution, and Monte Carlo simulation can give us many others, so let’s take a closer look at what they are.

Normal Distribution ☑

The normal distribution is a probability distribution used to calculate quantitative and qualitative financial decisions. It’s also the most common type of distribution assumed in technical stock market analysis and other variations.

With a normal distribution, there are two parameters: the mean (the peak of the curve) and the standard deviation (the variation around the mean). This distribution shows up as a symmetric curve on a chart, where the mean, median, and mode are equal.

The normal distribution is perhaps the most common outcome of the Monte Carlo simulation. It is also widely known as the bell curve—which we’ve all likely seen many times in our lives, especially in various textbooks. 

In finance, however, normal distribution can be used when conducting a technical analysis of the stock market. This model is best for analyzing the risks and returns of an investment, where the mean represents returns and the standard deviation represents risk. The higher the standard deviation on a chart, the riskier the investment could be, making it more volatile.

The bell curve is also useful as it is a continuous distribution—it is capable of providing key information on whether an investor’s portfolio’s returns will be good or bad, but can also estimate how good or bad they will be.

This distribution works well with stocks that are low in volatility and have a relatively predictable behavioral pattern. Blue-chip stocks tend to fall under this category. So if we wanted to analyze the potential returns for a blue-chip company like IBM or Coca Cola, we can expect to see a bell-curve distribution on our charts.

Lognormal Distribution 🔗

The lognormal distribution occurs when all the outcomes of a Monte Carlo simulation are positive. Most people are familiar with normal distributions, but they may not be as familiar with lognormal distribution. But, the two have a relationship.

Normal distributions can be converted to lognormal distributions using logarithmic mathematics. This is a core fundamental of lognormal distributions – they can only come from a normally distributed set of random variables.

One would opt for lognormal distributions in cases where they strictly need positive values; this is common when assessing asset prices, e.g. stocks, oil, gold, and real estate. 

While investors can use a normal distribution to determine the expected returns of a stock (where positive and negative values are applicable), they would need a lognormal distribution to study potential price changes of a stock.

This distribution can also be used in assessing overall market performance. For example, a chart with a fat tail—when the part of the chart going towards zero appears to be taking a long time to get there—can indicate that a crash could be imminent.

Triangular Distribution 📐

The triangular distribution is considered a continuous probability distribution. It occurs when the potential maximum, minimum, and most likely outcomes are known beforehand and predefined in the simulation.

A triangular distribution is also useful for people who want to simplify their results—making them easier to read. For example, the triangular distribution can highlight the outcome of a portfolio assessment if one isn’t interested in the nitty-gritty, but are simply looking to find the worst, best, and most likely outcomes of one’s investment strategy.

Discrete Distribution 📊

The discrete distribution represents the outcome of a simulation where there is a finite number of possible outcomes. A good example would be the distribution of a Monte Carlo simulation estimating the probabilities of throwing a pair of dice 100 times. The number of outcomes is finite because it is impossible for a throw to yield a number smaller than two, or greater than 12.

A simple example of finding this result in the area of finance might be with some binary options contracts. This would be most likely with some simple forms of binary options like betting on whether the prices of a commodity will go up, down, or remain the same—thus making only three outcomes relevant in that assessment.

Uniform Distribution 📈

Uniform distribution is a rather interesting case. Visually, it looks like a rectangle and it represents a simulation in which every possible outcome is equally probable. It’s most likely to get such a distribution when the choice is binary—like whether the ball will rest on red or black in roulette. This, however, isn’t the only case where a uniform distribution is possible.

Once again, binary options are the most likely sphere to come across this distribution. However, this would be relatively rare as—following from our previous example—it would require the likelihood of the prices staying the same, rising, and falling to be the same.

PERT Distribution 📉

PERT distribution is similar to the triangular distribution in that it has minimum, maximum, and expected outcomes predetermined. However, it adds additional weight to the expected outcome making it more likely to occur in simulations and giving the distribution a shape more akin to a wave than a triangle. 

Additionally, this means that fringe results are going to pop up in a smaller percentage of cases encouraging you to ignore them.

For example, Sam runs a Monte Carlo simulation on Disney’s stock. It is fairly safe to assume that as long as he won’t let the simulation run ad infinitum, any outcome in which the company goes bankrupt can be completely ignored. 

Additionally, considering Disney is a huge, established blue-chip company, it is unlikely its value will quadruple in the foreseeable future and such outcomes can also be safely ignored—no controversy over bills from Florida will kill the company, and there are no more Lucasfilms to be acquired.

Thus, in a case like this, Sam might set his Monte Carlo simulation towards a PERT distribution.

Calculating a Monte Carlo Simulation 🧮

While there are multiple companies that offer running Monte Carlo simulations on behalf of investors, it is always useful to understand how various methods of stocks analysis, be they technical or fundamental, work.

Furthermore, finding a  reliable stock analysis software helps prevent taking unfeasible amounts of time to do the math manually. Alternatively, investors can also look into using robo advisors as they often come with the capability to do Monte Carlo simulations.

But if an investor would like to take matters into their own hands, then they’ll need to be comfortable with spreadsheets as this is the main tool for conducting Monte Carlo simulations.

Generally, the ideal place to begin calculating is analyzing the past performance of a company to determine its drift, variance, and standard deviation. These variables shouldn’t be too difficult to come by even without doing the math as they are commonly used for other risk-assessing methods such as the Sharpe Ratio.

First, we start setting up our simulation by finding a number of periodic daily returns using the formula given and subsequently using these returns to determine the average daily return.

Another factor that needs to be determined is drift. Drift can often be simply set to 0—which is likely to have negligible consequences if the simulation is run for short periods, but can have a significant impact if the aim is to project farther into the future. Drift can be found using a relatively simple formula though.

Since the Monte Carlo simulation is based around using random variations to estimate future outcomes, we also need to determine a random input.

The last equation we’ll need is for calculating the next day’s price (also known as tomorrow’s price). This is where our previous calculations come into play.

It’s advisable to repeat this calculation multiple times in order to get as many likely outcomes as possible. This increases the accuracy of the Monte Carlo simulations estimates.

Using a Monte Carlo Simulation for Portfolio Management 📝

While the Monte Carlo simulation does have numerous applications in a plethora of fields, when it comes to personal finances, this simulation is most used for portfolio management. 

Due to its ability to account for multiple variables in a vast array of possible sequences of events, this method is commonly used for determining “the chance of success” of a person’s investment strategy.

The simulation can be used to predict a portfolio’s anticipated return, adjust the data for factors like inflation, additional investments, and unexpected expenses and compare the results with the desired outcome of one’s stock market activity.

The Monte Carlo simulation then repeats these predictions while constantly readjusting all these factors over time, further mixing and matching to generate more potential outcomes.

 Finally, the simulation can see how often an investor failed to stay afloat with their current investment strategy, and how often they succeeded. For this reason, many experts believe that the Monte Carlo simulation is an important method for predicting whether you’ll have enough retirement money decades before actually leaving the workforce.

The insight gained from running the simulation can then be used by financial analysts and advisors to inform investors on what changes they’d have to make to ensure long-term success. 🗓

On the other hand, this isn’t the only use the simulation has for portfolio management. It is a very useful risk-assessment tool that can be used on individual assets. The foresight gained from running the Monte Carlo method on both stocks we already own and are looking to buy can help us distinguish the opportunities from the liabilities. Essentially, it can be an invaluable tool for detecting undervalued stocks while avoiding probable major future losses.

For those who are very cautious and like to plan as much as possible in advance, the Monte Carlo simulation can help with picking the composition of a portfolio before buying a single asset. 

Basically, we can run the simulation using various potential portfolio compositions before finding one that has an outcome most in line with our respective lifestyles and goals.

Think of such an approach as a forward-looking variant of the dragon portfolio experiment.

The Benefits of the Monte Carlo Method 🌟

The Monte Carlo method has two most obvious benefits: it helps investors clearly visualize the long-term consequences of complex decisions, and it helps remove some of the bias inherent in decision-making.

Analyzing Potential Long-Term Outcomes 🔍

The Monte Carlo method’s ability to consider numerous possible outcomes and test them by making alterations to the variables in multiple iterations is invaluable. It really helps to account for the factors and possibilities we wouldn’t have even thought about.

Since it is computerized, this simulation helps alleviate the issue of wishful thinking. For example, if an investor is really hoping that a new meme stock will take flight and make them a lot of money overnight, they might be inclined to ignore all the warning signs of how risky such an investment really is.

The many futures the Monte Carlo method considers can go a long way in snapping one back to reality. Likewise, a properly set-up simulation could predict the likelihood that assets otherwise considered excellent have a higher-than-expected risk associated with them.

Just think of how perfect the ARK ETF appeared almost right up to the moment Morningstar downgraded it in the spring of 2022. The many iterations employed by the Monte Carlo simulation could have pointed towards such an outcome even when most investors were simply star-struck by the golden age of exchange traded funds.

Escaping Your Biases 🎰

Another bias the Monte Carlo simulation is very good at dispelling is the so-called gambler’s fallacy—which is one of the big reasons gamblers go hot and lose a lot of money in the casinos. Everyone venturing money in any way is certainly susceptible to a fallacy like this, and, likewise, isn’t immune to other gambler’s problems like the fear of missing out, and the sunk cost fallacy.

The gambler’s fallacy is a common belief that any uncanny set of events will be followed by a reversal of fortunes. We can liken it to believing that we’re about to have a spectacular weekend after a terrible week. Or believing our favorite sport’s teams are going to achieve an amazing victory after suffering defeat after defeat throughout the entire season.

The same line of thinking can apply to investing during bull and bear markets in two distinct ways. On the one hand, investors might become overly secure during a long bull market—it’s been going strong for 10 years, why should it stop now? On the other hand, a good thing lasting too long can lead to an anticipation of impending doom.

Many people have an inherent feeling that anything that’s “too good” must balance itself out with something equally bad. A major bull market is likely to lead to irrational fears of a terrible crash, and a horrible bear market can make someone feel like their fortunes are about to reverse, and they are going to hit it big. Both lines of thought are profoundly flawed.

Since the Monte Carlo method doesn’t cut corners when doing its math, it is particularly good at dispelling any form of a gambler’s fallacy.

Shortcomings of the Monte Carlo Simulation

While the Monte Carlo simulation is good at what it does, it is still a tool for making estimates and predictions. Its complex and scientific nature can cause a whole other type of bias—forgetting that what it gives are predictions, not truths.

The fact that the method can predict that an investment strategy has a 95% of succeeding still doesn’t mean that in reality, an investor won’t go broke a few years after they retire.

Downsides of Relying on Technology ⚠

Furthermore, the Monte Carlo simulation suffers from a similar issue to many other tools used in technical analysis—it is mathematically elegant and assumes a perfectly efficient market. That is to say that the way asset values are determined is perfectly reflective of the real state of affairs.

Needless to say, this is seldom true and a lot can go awry during a trading day leading both to minor inefficiencies and major disturbances like the flash crash of 2010. This issue is compounded by the fact that the Monte Carlo simulation can be very hardware intensive.

Having mentioned the Flash Crash, and knowing that there is a company in Taiwan whose computer chips basically enable the world economy, this idea of needing powerful machines to run successful simulations might appear like a deal-breaker at first—but it genuinely isn’t.

Very few simulations needed for market activities will need a lot of processing power—the ENIAC took hours to resolve Ulam’s conundrum, but a normal investor can probably get all their answers in seconds, using their laptop. 💻

The really intense problems are more probable when it comes to corporate finance, but a phenomenon known as the embarrassingly parallel problem ensures that corporations won’t be having too many headaches either.

While the Monte Carlo simulation can work with many different moving parts, it tends to look at each of these moving parts separately. For example, when giving an overall estimate of how an investor’s portfolio may fare in the long run, it can consider both their future investments, and unexpected expenses, but it can’t really consider the link between the two.

So, if Sophie, for example, suddenly must pay a large sum of money to cover medical bills after an accident, she won’t have as much money to invest. The simulation might add these expenses but forget to subtract them from her future investments.

The Chance for Human Error 👨‍💻

A common feature of algorithms like the Monte Carlo method is the phenomenon known as “garbage in, garbage out.” This is to say, the quality of the results of the simulation is only going to be as good as the data it was given to work with. Unfortunately, humans are the ones who have to set up and run the method, and humans are certainly fallible.

Additionally, an investor still has to rely on a person – a friend, an advisor, or even just themselves —to interpret the data given by the simulation and plan any necessary course corrections in their investment strategy. These readings are themselves prone to biases, mistakes in general, and sometimes malicious and intentional manipulation.

Lastly, the Monte Carlo method can only work with the information it was given—it can’t take into account elements that the person that has set up the simulation forgot. A big and painful example of an event that can throw a wrench into any Monte Carlo simulation is the Russian invasion of Ukraine.

It would be reasonable not to include a war-related spike in oil prices as there hasn’t been a major land war in Europe since the end of WW2, but such an event can have a profound effect on anybody’s savings and investments.

Another, less grim example of potentially huge market changes that can’t really be predicted using the Monte Carlo method is the various shenanigans of Elon Musk. These can include anything from him recently becoming the biggest shareholder of Twitter, his poll-based selling of Tesla shares, or his pushing of certain meme assets through social media posts.

Conclusion 🏁

In many ways, the Monte Carlo simulation is a double-edged sword. While it offers unique insight among its risk-analyzing peers, it also suffers from some important pitfalls—made worse by the fact it is such a widely used and important tool as it often garners disproportionate and undeserved trust. However, being aware of how it works, and what its strengths and weaknesses are, makes it—in practice—come as close to a flawless tool as it can be.

Still, much like with investing itself, it is important to incorporate as many elements and sources as you can into your decision process. While no tool is perfect, if you combine the results of the Monte Carlo method, with the Calmar ratio, with Warren Buffett’s annual letter, and they all point in the same direction, you might have just found a winning investment.

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