In finance, there is a fair amount of uncertainty and risk involved with estimating the future value of figures or amounts due to the wide variety of potential outcomes. Monte Carlo simulation (MCS) is one technique that helps to reduce the uncertainty involved in estimating future outcomes. MCS can be applied to complex, non-linear models or used to evaluate the accuracy and performance of other models. It can also be implemented in risk management, portfolio management, pricing derivatives, strategic planning, project planning, cost modeling and other fields.

## Definition

MCS is a technique that converts uncertainties in input variables of a model into probability distributions. By combining the distributions and randomly selecting values from them, it recalculates the simulated model many times and brings out the probability of the output.

## Basic Characteristics

• MCS allows several inputs to be used at the same time to create the probability distribution of one or more outputs.
• Different types of probability distributions can be assigned to the inputs of the model. When the distribution is unknown, the one that represents the best fit could be chosen.
• The use of random numbers characterizes MCS as a stochastic method. The random numbers have to be independent; no correlation should exist between them.
• MCS generates the output as a range instead of a fixed value and shows how likely the output value is to occur in the range.

## Some Frequently Used Probability Distributions in MCS

Normal/Gaussian Distribution – Continuous distribution applied in situations where the mean and the standard deviation are given and the mean represents the most probable value of the variable. It is symmetrical around the mean and is not bounded.

Lognormal Distribution – Continuous distribution specified by mean and standard deviation. This is appropriate for a variable ranging from zero to infinity, with positive skewness and with normally distributed natural logarithm.

Triangular Distribution – Continuous distribution with fixed minimum and maximum values. It is bounded by the minimum and maximum values and can be either symmetrical (the most probable value = mean = median) or asymmetrical.

Uniform Distribution – Continuous distribution bounded by known minimum and maximum values. In contrast to the triangular distribution, the likelihood of occurrence of the values between the minimum and maximum is the same.

Exponential Distribution – Continuous distribution used to illustrate the time between independent occurrences, provided the rate of occurrences is known.

## The Math Behind MCS

Consider that we have a real-valued function g(X) with probability frequency function P(x) (if X is discrete), or probability density function f(x) (if X is continuous). Then we can define the expected value of g(X) in discrete and continuous terms respectively:

﻿\begin{aligned}&E(g(X))=\sum^{+\infty}_{-\infty}g(x)P(x),\\&\qquad\qquad\qquad\qquad\qquad\text{ where }P(x)>0\text{ and} \sum^{+\infty}_{-\infty}P(x)=1\\&E(g(X))=\int^{+\infty}_{-\infty}g(x)f(x)\,dx,\\&\qquad\qquad\qquad\qquad\text{ where }f(x)>0\text{ and }\int^{+\infty}_{-\infty}f(x)\,dx=1\\&\text{Next, make n random drawings of X (x_1,\ldots,x_n), called}\\&\text{trial runs or simulation runs, calculate g(x_1),\ldots,g(x_n)}\\&\text{and find the mean of g(x) of the sample:}\end{aligned}﻿

﻿\begin{aligned}&g^\mu_n(x)=\frac{1}{n}\sum^n_{i=1}g(x_i),\text{ which represents the final simulated}\\&\text{value of }E(g(X)).\\\\&\text{Therefore }g^\mu_n(X)=\frac{1}{n}\sum^n_{i=1}g(X)\text{ will be the Monte Carlo}\\&\text{estimator of }E(g(X)).\\\\&\text{As }n\to\infty, g^\mu_n(X)\to E(g(X)), \text{thus we are now able to}\\&\text{compute the dispersion around the estimated mean with}\\&\text{the unbiased variance of }g^\mu_n(X)\text{:}\\&Var(g^\mu_n(X))=\frac{1}{n-1}\sum^n_{i=1}(g(x_i)-g^\mu_n(x))^2.\end{aligned}﻿

Simple Example

How will the uncertainty in unit price, unit sales and variable costs affect the EBITD?

Copyright Unit Sales)-(Variable Costs + Fixed Costs)

Let us explain the uncertainty in the inputs – unit price, unit sales and variable costs – using triangular distribution, specified by the respective minimum and maximum values of the inputs from the table.