# Monte Carlo Simulation

# Blog: Monday Project Management Blog

Every decision you make involves risk, from choosing what to eat for lunch to deciding whether to sell your scrappy startup to a big corporation. Risk analysis is at the forefront of many businesses. Although we’re well into the information age, seeing into the future with clarity is impossible. But there are ways to peek ahead and make better decisions.

With a computational algorithm known as the Monte Carlo simulation, you can generate every possible decision outcome and determine each one’s probability. While the method doesn’t show you the future, it supercharges risk analysis for businesses across every industry.

Below, you’ll learn about the Monte Carlo simulation, how it works, and why it’s a powerful predictive method for analyzing risk and uncertainty. We can’t promise you’ll acquire the ability to predict the future, but you’ll likely have a much better understanding of probability.

**What is the Monte Carlo simulation?**

The Monte Carlo simulation is a mathematical algorithm that estimates uncertain outcomes and their probability. It analyzes risk in stock markets, artificial intelligence, sales forecasts, and even project management.

Given its wide-ranging applications, the Monte Carlo simulation is an excellent tool in risk analysis. Since the method relies on random variables and valid data, it’s invaluable for predicting results when facing uncertainties. While powerful modern computing has brought Monte Carlo methods forward, the technique is far from new.

**A short history of the Monte Carlo simulation**

Created by Manhattan Project scientist Stanislaw Ulam and colleague John Von Neumann, Monte Carlo Simulation was developed after World War II to enhance decision-making in uncertain conditions. The technique is named after Monaco, the famous casino town, since a tenant of the model is the element of chance, not mathematically unlike playing a game of dice or roulette. It was during Ulam’s recovery from brain surgery while playing countless games of solitaire that he began thinking deeply about probability determination.

**How does the Monte Carlo simulation work?**

The Monte Carlo simulation predicts outcomes based on a range of values, each value a variable with uncertain outcomes. The algorithm uses these variables to build a model of possible results using a probability distribution for each variable. The algorithm then calculates all possible results repeatedly using every random value from highest to lowest. In a typical simulation, the calculation can occur tens of thousands of times.

Since the entire method is built on generating random values repeatedly, its accuracy varies depending on the input samples provided. That said, when provided with a high number of reliable values, Monte Carlo methods are remarkably accurate. Aside from regular applications in science and engineering, the methods have practical applications in predicting price fluctuations in financial markets and even cost overruns in project management.

Let’s use a large container of jelly beans as an example. You need to know how many jelly beans the container holds, but counting them one by one is not an appealing task. With Monte Carlo methods, you add variable inputs such as the height, width, and circumference of a few jelly beans. You then add fixed inputs based on the measurements of the container.

The Monte Carlo simulation would essentially generate randomly sized objects between the smallest and largest jelly bean you measured and fill the virtual jar with them. Then it would do it a few more thousand times to produce a very close approximation to the number of jelly beans in the jar.

While counting jelly beans is a simplistic example, they’re equally effective in complex systems. Monte Carlo simulations are even used for far-reaching predictions. The more input you have for the methods, the higher the accuracy. One of the key elements, however, is using the right probability distribution.

**Probability distributions and the Monte Carlo simulation**

The Monte Carlo simulation generates a probability model based on fixed inputs, such as historical data or even educated guesses, and a probability distribution model. In simple terms, this model determines the extent that each input value is randomized to determine probability.

Some of the most common probability distributions are:

**Normal distribution:**A normal probability distribution model or, more commonly, a bell curve, defines the mean input value and how much that value deviates from the mean. A simple example of a normal distribution model would be the height of a given population.**Lognormal:**In a lognormal distribution model, input values represent non-negative numbers with unlimited positive potential. These models are typically used in areas related to finance, such as the stock market and real estate.**Uniform:**In a uniform model, every possible value has an equal chance of occurring. Uniform distribution is used when there’s little or no previous data on which values are more or less likely to occur.

Apart from these probability distributions, typically used in mathematics, statistics, and engineering, Monte Carlo methods are increasingly common in project management.

**The Monte Carlo simulation in project management and finance**

The Beta function, for example, lets businesses analyze risks associated with project budgets and schedules. PERT, another project management distribution, is an advanced method that follows a three-point distribution, with a minimum, maximum, and most likely value. In PERT, the most likely value is weighted more than the minimum and maximum values. It also identifies uncertainties when an activity’s duration changes by modifying the deadline into a randomized variable.

The Monte Carlo simulation also touches multiple areas of the finance world. It can help identify and predict risks across areas such as options valuation and equity options pricing, and can simulate fluctuation in share valuations. Monte Carlo methods can even help predict portfolio valuations by simulating all possible outcomes based on simulated portfolios.

**How to use Monte Carlo methods**

To the layman, the Monte Carlo simulation seems complex. Mathematically, however, the methods are elegant. Despite this fact, accurate predictions — especially when a large number of uncertain values are present — require immense processing power. You can certainly run simple simulations on a laptop, but serious predictions need more computational capacity.

That said, the Monte Carlo simulation is a three-step process:

**Setup up the predictive model:**Identify and specify the independent variables that require prediction as well as the input variables to guide the algorithm.**Specify the probability distribution:**Define the probability distribution of the independent variables using available data and assign likelihood for each variable.**Run simulations repeatedly:**Run the simulation repeatedly, generating random values for the independent variables until there’s enough data to build a probability sample of all probabilities.

Put simply, the Monte Carlo simulation is an algorithm for calculating possible outcomes and the probability of each outcome occurring by processing random samples repeatedly. It’s a process that’s useful for any project, in any field, and across every industry for predicting uncertain outcomes and avoiding unnecessary risk.

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