# Monte Carlo Simulation

## Calculate Many Different Outcomes and Their Probabilities of Occurrence with Monte Carlo Simulation Software ## Resources ### Monte Carlo Analysis Probabilistically Assesses the Impact of Risk

Risk and forecast analysis is part of every decision you make. We are constantly faced with uncertainty, ambiguity, and variability. And even though we have unprecedented access to information, we can’t accurately forecast the future. Monte Carlo simulation (also known as the Monte Carlo method) lets you see all possible outcomes of your decisions, including the actual probabilities each will occur. This lets you quantitatively assess the impact of risk, allowing for more accurate forecasting and, ultimately, better decision-making under uncertainty.

## What is Monte Carlo Simulation?

The Monte Carlo method is a computerized mathematical technique that allows people to quantitatively account for risk in forecasting and decision-making. At its core, the Monte Carlo method is a way to use random samples of parameters to explore the behavior of a complex system. A Monte Carlo simulation is used to handle an extensive range of problems in a variety of different fields to understand the impact of risk and uncertainty.

### A Way to Account for Risk

Monte Carlo Simulations have assessed the impact of risk in stock prices, project management, AI, and many other real-life scenarios.The Monte Carlo method provides a number of advantages over predictive models with fixed inputs, such as the ability to conduct sensitivity analysis or calculate the correlation of inputs.

A Forecast Analysis Tool That Works in Many Fields

The technique is used for forecasting, which takes into account risk, uncertainty and variability. Project managers and decision-makers use the Monte Carlo Simulation tool to estimate the impacts of various risks on the project cost and project timeline. Using this method, one can easily find out what will happen to the project schedule and cost in case any risk occurs. The Monte Carlo Simulation is used in many different fields, including:

Use cases run the gamut and include cash flow analysis, capital investments, reserves estimation, pricing, cost estimation, project management, product pipeline analysis, portfolio optimization, supply chain risk, and more.

### A Range of Outcomes

Monte Carlo simulation furnishes the decision-maker with a range of possible outcomes and the probabilities they will occur for any choice of action. It shows:

• the extreme possibilities
• the outcomes of going for broke and for the most conservative decision
• along with all possible consequences for middle-of-the-road decisions

### History of Monte Carlo Simulation

The technique was first used by scientists working on the atom bomb; it was named for Monte Carlo, the Monaco resort town renowned for its casinos. Since its introduction in World War II, Monte Carlo simulation has been used to model a variety of physical and conceptual systems.

# How Monte Carlo Simulation Works

Monte Carlo simulation performs risk analysis by building models of possible results by substituting a range of values—called a probability distribution—for any factor that has inherent uncertainty. It then calculates results over and over, each time using a different set of random values from the input probability distributions. Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands or tens of thousands of recalculations before it is complete. The result of a Monte Carlo simulation is a range – or distribution – of possible outcome values. This data on possible results enables you to calculate the probabilities of different outcomes in your forecasts, as well as perform a wide range of additional analyses. Monte Carlo simulation software builds a spreadsheet model that lets you evaluate your plan numerically, allowing you to change the numbers, ask ‘what if’ and see the results.

By using probability distributions for uncertain inputs, you can represent the different possible values for these variables, along with their likelihood of occurrence. Probability distributions are a much more realistic way of describing uncertainty in variables of a risk analysis, making Monte Carlo simulation far superior to common “best guess” or “best/worst/most likely” analyses. ### Common Probability Distributions Include #### Normal

Or “bell curve.” The user simply defines the mean or expected value and a standard deviation to describe the variation about the mean. Values in the middle near the mean are most likely to occur. It is symmetric and describes many natural phenomena such as people’s heights. Examples of variables described by normal distributions include inflation rates and energy prices. #### Lognormal

Values are positively skewed, not symmetric like a normal distribution. It is used to represent values that don’t go below zero but have unlimited positive potential. Examples of variables described by lognormal distributions include real estate property values, stock prices, and oil reserves. #### Uniform

All values have an equal chance of occurring, and the user simply defines the minimum and maximum because they have no knowledge of which values are more likely than others. Examples of variables that could be uniformly distributed include manufacturing costs or future sales revenues for a new product. #### Triangular

The user defines the minimum, most likely, and maximum values. Values around the most likely are more likely to occur. Variables that could be described by a triangular distribution include past sales history per unit of time and inventory levels. #### PERT

The user defines the minimum, most likely, and maximum values, just like the triangular distribution. Values around the most likely are more likely to occur. However values between the most likely and extremes are more likely to occur than the triangular; that is, the extremes are not as emphasized. An example of the use of a PERT distribution is to describe the duration of a task in a project management model. #### Discrete

The user defines specific values that may occur and the likelihood of each. An example might be the results of a lawsuit: 20% chance of positive verdict, 30% change of negative verdict, 40% chance of settlement, and 10% chance of mistrial.

## Random Sampling Versus Best Guess

During a Monte Carlo simulation, values are sampled at random from the input probability distributions. Each set of samples is called an iteration, and the resulting outcome from that sample is recorded. Monte Carlo simulation does this hundreds or thousands of times, and the result is a probability distribution of possible outcomes. In this way, Monte Carlo simulation provides a much more comprehensive view of what may happen. It tells you not only what could happen, but how likely it is to happen.

Monte Carlo simulation provides a number of advantages over deterministic, or “single-point estimate” analysis:

• Probabilistic Results. Results show not only what could happen, but how likely each outcome is.
• Graphical Results. Because of the data a Monte Carlo simulation generates, it’s easy to create graphs of different outcomes and their chances of occurrence. This is important for communicating findings to other stakeholders.
• Sensitivity Analysis. Deterministic analysis makes it difficult to see which variables impact the outcome the most. In Monte Carlo simulation, it’s easy to see which inputs had the biggest effect on bottom-line results. This allows you to identify and mitigate factors which cause the most risk.
• Scenario Analysis: In deterministic models, it’s very difficult to model different combinations of values for different inputs to see the effects of truly different scenarios. Using Monte Carlo simulation, analysts can see exactly which inputs had which values together when certain outcomes occurred. This is invaluable for pursuing further analysis.
• Correlation of Inputs. In Monte Carlo simulation, it’s possible to model interdependent relationships between input variables. It’s important for accuracy to represent how, in reality, when some factors go up or down, others go up or down accordingly.

An enhancement to Monte Carlo simulation is the use of Latin Hypercube sampling, which samples more accurately from the full range of values within distribution functions and produces results more quickly.

### Monte Carlo Simulation with Palisade

Palisade’s @RISK software puts this powerful technique within reach for any Excel user faced with uncertainty in their analyses. @RISK makes it easy to graphically define risk models, run simulations, and analyze the results, all with the click of a mouse. @RISK is 100% integrated with Excel, adding hundreds of new functions to Excel so that users can quickly understand their risks without learning a new application. First introduced in 1987 for Lotus 1-2-3, @RISK has a long-established reputation for computational accuracy, modeling flexibility, and ease of use, making it the dominant Monte Carlo simulation software in the market today.