Monte Carlo Methods encompass a variety of mathematical problem-solving techniques based on randomly generated numbers and probability distributions. The term was coined by Von Newman, after the famous casino in Monaco. Monte Carlo Methods were central to some of the simulations used in the Manhattan Project.

An easy way to understand this approach is by using random numbers to estimate a surface area. Consider a one square foot area subdivided into four equal parts. A randomly generated pair of numbers between zero and one will have a probability of one fourth to be in upper corner sub-square.

Alternatively, imagine the same square hanging on a wall with the upper right sub-square painted in red and a group of drunken people aimlessly throwing darts at it. Of the total amount of darts that hit the square, about one fourth will hit inside the red area. The larger the number of darts, the closer the proportion will be to one quarter.

Take this one step further. Instead of squares let’s draw a complex figure within the square, one whose surface area would be really difficult to calculate, say a Mickey Mouse silhouette. After the drunken crew aimlessly throws enough darts so that 100 hit the square, the number of darts inside the figure will be proportional to its surface area, say 75. Hence, three-quarters of a square foot will be a close approximation of the area of Mickey Mouse!

### Random Sampling in Business and Finance

Managing uncertainty is an inherent part of the business, where the future heavily influences the present and the reliability of an outcome is a function of the combined reliabilities of its inputs. One way to cope with this uncertainty is using Sensitivity Analysis to understand the behavior of a given system better. With more than a few variables, it can become a complex and often frustrating task.

### Modeling Net Present Value

To take an example from finance, consider a Net Present Value (NPV) calculation. A simple model will likely include variables such as market demand, sales volume by period, unit price, materials cost, inventory levels, etc. Some of these variables can be considered fixed or determined by other variables in our model; others would be expected to vary according to a certain probability distribution.

For example, unit price could be modeled to be a function of expected demand, given its elasticity, but demand itself would be uncertain. Once the probability distribution of each of the relevant variables is determined, a simulation software (well-known packages are @Risk and Crystal Ball) can randomly sample each variable and calculate the NPV many times. The resulting values will constitute an estimate of the probability distribution of the NPV itself.

Understanding the probability distribution of the NPV of a project, allows us not only to understand its variability in a quantitative way (via its standard deviation), but to make reasonable estimates of the likelihood of a given result. For instance, the probability that NPV is greater than zero.

The same ideas can be expanded to virtually any area of business that faces uncertainty, from inventory management to marketing.

### A Word of Caution

As with any other modeling method, the validity of simulation results obtained using a Monte Carlo approach is determined by the quality of the model itself and its inputs. While modern software allows the construct fairly sophisticated models with little effort this is no substitute for understanding the underlying concepts. Impressively looking results based on an allocation model can, undeservedly, create the illusion of detail.