What happens when we stop trying to predict one future and instead simulate thousands of possible futures? Monte Carlo simulation turns randomness into a powerful tool for understanding uncertainty.
Imagine trying to answer a question about the future: how much might an investment be worth in 20 years?
There is no single answer because investment returns are uncertain. Instead of pretending that we know exactly what will happen, we can generate thousands of plausible sequences of returns.
Each sequence represents one possible future. We then look at the distribution of all the outcomes.
Monte Carlo simulation does not predict the future. It generates many plausible futures and examines the distribution of outcomes.
Before looking at investments, consider something much simpler: rolling a fair six-sided die.
In theory, each number has a probability of 1/6. But if we roll the die only a few times, the results can look very uneven.
As the number of rolls increases, the observed frequencies tend to move closer to the theoretical probabilities.
Randomness can look chaotic in a small sample. Repeated trials allow the underlying probabilities to become visible.
One of the most famous Monte Carlo experiments estimates the value of π using random points.
Imagine a square containing a quarter of a circle. Random points are thrown into the square. Some land inside the quarter circle and some outside it.
The proportion of points inside the circle can be used to estimate π.
With only a small number of points the estimate can be quite poor. As the number of random trials grows, the estimate generally becomes more stable.
Now we can apply the same idea to financial markets.
Suppose an investment starts at €100,000. Instead of assuming that it earns exactly the same return every year, we generate a random return for each year.
A simplified model can describe returns using an expected return and a measure of volatility.
This is deliberately simplified. Real financial return distributions are more complicated, which we will explore later.
The path above is simply one possible sequence of returns. Run it again and you will get a different future.
This is where Monte Carlo becomes useful: rather than looking at one path, we can generate thousands.
Now generate thousands of possible investment futures. Each simulation starts with the same initial investment, but receives a different sequence of random annual returns.
After thousands of simulations, we no longer have one answer. We have a distribution of possible final values.
This distribution tells us much more than a single forecast. We can see relatively poor outcomes, typical outcomes, and unusually strong outcomes.
The middle of the distribution represents more typical outcomes. The tails represent less common but potentially important outcomes.
We can go one step further. Instead of looking only at the final value, we can examine how the distribution of possible wealth evolves over time.
The chart below shows several percentiles at every point during the investment period.
The further we look into the future, the greater the range of possible outcomes becomes. Uncertainty compounds over time.
One of the most useful features of Monte Carlo simulation is that we can ask probability questions.
For example: what proportion of the simulated futures finish above a particular target?
Run the Monte Carlo simulation first.
Instead of asking "What will my investment be worth?", we can ask questions such as:
"What is the probability that it will be worth at least €200,000?"
A Monte Carlo estimate itself is uncertain. If we run only a small number of simulations, the result can move around significantly from one run to another.
Increasing the number of simulations generally makes the estimated probabilities and percentiles more stable.
This means that increasing the number of simulations improves precision, but with diminishing returns.
Results can be noticeably affected by random noise.
Estimates become considerably more stable.
The distribution is usually much more clearly defined.
Once results are sufficiently stable, additional simulations provide progressively smaller improvements.
We have moved from random observations, to samples, to distributions of sample statistics, and now to thousands of simulated futures.
The model used on this page is deliberately simple. It assumes that returns can be represented using a stable distribution with constant volatility.
Real financial markets are more complicated.
Extreme market movements can occur more frequently than a Normal distribution suggests.
Financial return distributions are not always symmetrical around their centre.
Market volatility tends to change over time rather than remaining constant.
Markets can move between different environments, such as calm, stressed and highly volatile periods.
More realistic Monte Carlo models can incorporate Student-t distributions, skew-t distributions, GARCH volatility models and Markov regime switching.
We have now built the basic machinery of Monte Carlo:
The next question is one of the most important in financial modelling:
What should the random returns actually look like?
The Normal distribution is a useful starting point, but financial returns often display characteristics that the Normal model cannot capture.