A Practical Guide to Monte Carlo Simulation

Mike's Notes

A great introduction to using Microsoft Excel for Monte Carlo simulation by Jon Wittwer. The 4-part series is combined here as one. Vetex42 is an excellent resource.

I recently discovered that Pipi has used Monte Carlo since version 6, but the method was unnamed, as is much of Pipi's internal functioning (A disadvantage of working visually 😇 and finding ideas from anywhere to solve a problem)

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Last Updated

18/12/2025

A Practical Guide to Monte Carlo Simulation

By: Jon Wittwer
Vertex42: 1/06/2004

Jon Wittwer started Vertex42 in 2003 while working on a PhD in mechanical engineering. After finishing his degree, he worked for Sandia National Laboratories where he kept Vertex42.com running on the side. In 2008, he left the labs to work on Vertex42 full time. In addition to his expertise with Excel, Dr. Wittwer is respected in multiple fields for his development of financial tools such as the Debt Reduction Calculator, project management tools like the Gantt Chart Template, statistical tools such as the Monte Carlo Simulator, and a large collection of business productivity and time management tools.

A Monte Carlo method is a technique that involves using random numbers and probability to solve problems. The term Monte Carlo Method was coined by S. Ulam and Nicholas Metropolis in reference to games of chance, a popular attraction in Monte Carlo, Monaco (Hoffman, 1998; Metropolis and Ulam, 1949).

Computer simulation has to do with using computer models to imitate real life or make predictions. When you create a model with a spreadsheet like Excel, you have a certain number of input parameters and a few equations that use those inputs to give you a set of outputs (or response variables).

This type of model is usually deterministic, meaning that you get the same results no matter how many times you re-calculate.

Example 1: A Deterministic Model for Compound Interest

Deterministic Model

Figure 1: A parametric deterministic model maps a set of input variables to a set of output variables.

Monte Carlo simulation is a method for iteratively evaluating a deterministic model using sets of random numbers as inputs. This method is often used when the model is complex, nonlinear, or involves more than just a couple uncertain parameters. A simulation can typically involve over 10,000 evaluations of the model, a task which in the past was only practical using super computers.

By using random inputs, you are essentially turning the deterministic model into a stochastic model. Example 2 demonstrates this concept with a very simple problem.

Example 2: A Stochastic Model for a Hinge Assembly

In Example 2, we used simple uniform random numbers as the inputs to the model. However, a uniform distribution is not the only way to represent uncertainty. Before describing the steps of the general MC simulation in detail, a little word about uncertainty propagation:

The Monte Carlo method is just one of many methods for analyzing uncertainty propagation, where the goal is to determine how random variation, lack of knowledge, or error affects the sensitivity, performance, or reliability of the system that is being modeled.

Monte Carlo simulation is categorized as a sampling method because the inputs are randomly generated from probability distributions to simulate the process of sampling from an actual population. So, we try to choose a distribution for the inputs that most closely matches data we already have, or best represents our current state of knowledge. The data generated from the simulation can be represented as probability distributions (or histograms) or converted to error bars, reliability predictions, tolerance zones, and confidence intervals. (See Figure 2).

Uncertainty Propagation

Monte Carlo Analysis

Figure 2: Schematic showing the principal of stochastic uncertainty propagation. (The basic principle behind Monte Carlo simulation.)

If you have made it this far, congratulations! Now for the fun part! The steps in Monte Carlo simulation corresponding to the uncertainty propagation shown in Figure 2 are fairly simple, and can be easily implemented in Excel for simple models. All we need to do is follow the five simple steps listed below:

  • Step 1: Create a parametric model, y = f(x1, x2, ..., xq).
  • Step 2: Generate a set of random inputs, xi1, xi2, ..., xiq.
  • Step 3: Evaluate the model and store the results as yi.
  • Step 4: Repeat steps 2 and 3 for i = 1 to n.
  • Step 5: Analyze the results using histograms, summary statistics, confidence intervals, etc.

On to an example problem ...

REFERENCES:

  • Hoffman, P., 1998, The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 238-239.
  • Metropolis, N. and Ulam, S., 1949, "The Monte Carlo Method." J. Amer. Stat. Assoc. 44, 335-341.
  • Eric W. Weisstein. "Monte Carlo Method." From MathWorld--A Wolfram Web Resource.
  • Paul Coddington. "Monte Carlo Simulation for Statistical Physics." Northeast Parallel Architectures Center at Syracuse University. http://www.npac.syr.edu/users/paulc/lectures/montecarlo/p_montecarlo.html
  • Decisioneering.com. "What is Monte Carlo Simulation?"" Part of: Risk Analysis Overview - http://www.decisioneering.com/risk-analysis-start.html

Our example of Monte Carlo simulation in Excel will be a simplified sales forecast model. Each step of the analysis will be described in detail.

The Scenario: Company XYZ wants to know how profitable it will be to market their new gadget, realizing there are many uncertainties associated with market size, expenses, and revenue.

The Method: Use a Monte Carlo Simulation to estimate profit and evaluate risk.

You can download the example spreadsheet by following the instructions below. You will probably want to refer to the spreadsheet occasionally as we proceed with this example.

Download the Sales Forecast Example

Step 1: Creating the Model

We are going to use a top-down approach to create the sales forecast model, starting with:

Profit = Income - Expenses

Both income and expenses are uncertain parameters, but we aren't going to stop here, because one of the purposes of developing a model is to try to break the problem down into more fundamental quantities. Ideally, we want all the inputs to be independent. Does income depend on expenses? If so, our model needs to take this into account somehow.

We'll say that Income comes solely from the number of sales (S) multiplied by the profit per sale (P) resulting from an individual purchase of a gadget, so Income = S*P. The profit per sale takes into account the sale price, the initial cost to manufacturer or purchase the product wholesale, and other transaction fees (credit cards, shipping, etc.). For our purposes, we'll say the P may fluctuate between $47 and $53.

We could just leave the number of sales as one of the primary variables, but for this example, Company XYZ generates sales through purchasing leads. The number of sales per month is the number of leads per month (L) multiplied by the conversion rate (R) (the percentage of leads that result in sales). So our final equation for Income is:

Income = L*R*P

We'll consider the Expenses to be a combination of fixed overhead (H) plus the total cost of the leads. For this model, the cost of a single lead (C) varies between $0.20 and $0.80. Based upon some market research, Company XYZ expects the number of leads per month (L) to vary between 1200 and 1800. Our final model for Company XYZ's sales forecast is:

Profit = L*R*P - (H + L*C)
Y = Profits
X1 = L
X2 = C
X3 = R
X4 = P

Notice that H is also part of the equation, but we are going to treat it as a constant in this example. The inputs to the Monte Carlo simulation are just the uncertain parameters (Xi).

This is not a comprehensive treatment of modeling methods, but I used this example to demonstrate an important concept in uncertainty propagation, namely correlation. After breaking Income and Expenses down into more fundamental and measurable quantities, we found that the number of leads (L) affected both income and expenses. Therefore, income and expenses are not independent. We could probably break the problem down even further, but we won't in this example. We'll assume that L, R, P, H, and C are all independent.

Note: In my opinion, it is easier to decompose a model into independent variables (when possible) than to try to mess with correlation between random inputs.

Step 2: Generating Random Inputs

The key to Monte Carlo simulation is generating the set of random inputs. As with any modeling and prediction method, the "garbage in equals garbage out" principle applies. For now, I am going to avoid the questions "How do I know what distribution to use for my inputs?" and "How do I make sure I am using a good random number generator?" and get right to the details of how to implement the method in Excel.

For this example, we're going to use a Uniform Distribution to represent the four uncertain parameters. The inputs are summarized in the table shown below. (If you haven't already, Download the example spreadsheet).

Sales Forecast Input Table

Figure 1: Screen capture from the example sales forecast spreadsheet.

The table above uses "Min" and "Max" to indicate the uncertainty in L, C, R, and P. To generate a random number between "Min" and "Max", we use the following formula in Excel (Replacing "min" and "max" with cell references):

= min + RAND()*(max-min)

You can also use the Random Number Generation tool in Excel's Analysis ToolPak Add-In to kick out a bunch of static random numbers for a few distributions. However, in this example we are going to make use of Excel's RAND() formula so that every time the worksheet recalculates, a new random number is generated.

Let's say we want to run n=5000 evaluations of our model. This is a fairly low number when it comes to Monte Carlo simulation, and you will see why once we begin to analyze the results.

A very convenient way to organize the data in Excel is to make a column for each variable as shown in the screen capture below.

Random Inputs in Column Format

Figure 2: Screen capture from the example sales forecast spreadsheet.

Cell A2 contains the formula:

=Model!$F$14+RAND()*(Model!$G$14-Model!$F$14)

Note that the reference Model!$F$14 refers to the corresponding Min value for the variable L on the Model worksheet, as shown in Figure 1. (Hopefully you have downloaded the example spreadsheet and are following along.)

To generate 5000 random numbers for L, you simply copy the formula down 5000 rows. You repeat the process for the other variables (except for H, which is constant).

Step 3: Evaluating the Model

Our model is very simple, so to evaluate the output of our model (the Profit) for each run of the simulation, we just put the equation in another column next to the inputs, as shown in Figure 2.

Cell G2 contains the formula:

=A2*C2*D2-(E2+A2*B2)

Step 4: Running the Simulation

To iteratively evaluate our model, we don't need to write a fancy macro for this example. We simply copy the formula for profit down 5000 rows, making sure that we use relative references in the formula (no $ signs). Each row represents a single evaluation of the model, with columns A-E as inputs and the Profit as the output.

Re-run the Simulation: F9

Although we still need to analyze the data, we have essentially completed a Monte Carlo simulation. We have used the volatile RAND() function. So, to re-run the entire simulation all we have to do is recalculate the worksheet (F9 is the shortcut).

This may seem like a strange way to implement Monte Carlo simulation, but think about what is going on behind the scenes every time the Worksheet recalculates: (1) 5000 sets of random inputs are generated (2) The model is evaluated for all 5000 sets. Excel is handling all of the iteration.

If your model is not simple enough to include in a single formula, you can create your own custom Excel function (see my article on user-defined functions), or you can create a macro to iteratively evaluate your model and dump the data into a worksheet in a similar format to this example (Update 9/8/2014: See my new Monte Carlo Simulation template).

In practice, it is usually more convenient to buy an add-on for Excel than to do a Monte Carlo analysis from scratch every time. But not everyone has the money to spend, and hopefully the skills you will learn from this example will aid in future data analysis and modeling.

A Few Other Distributions

My new Monte Carlo Simulation template includes a worksheet that calculates inputs sampled from a variety of distributions. Some of the formulas are listed below.

Normal (Gaussian) distribution

To generate a random number from a Normal distribution you would use the following formula in Excel:

=NORMINV(rand(),mean,standard_dev)
Ex: =NORMINV(RAND(),$D$4,$D$5)
Excel 2010+: =NORM.INV(RAND(),$D$4,$D$5)

Lognormal distribution

To generate a random number from a Lognormal distribution with median = exp(meanlog), and shape = sdlog, you would use the following formula in Excel:

=LOGINV(RAND(),meanlog,sdlog)
Ex: =LOGINV(RAND(),$D$6,$D$5)
Excel 2010+: =LOGNORM.INV(RAND(),$D$4,$D$5)

Weibull distribution

There isn't an inverse Weibull function in Excel, but the formula is quite simple, so to generate a random number from a (2-parameter) Weibull distribution with scale = c, and shape = m, you would use the following formula in Excel:

=c*(-LN(1-RAND()))^(1/m)
Ex: $C$5*(-LN(1-RAND()))^(1/$C$6)

Beta distribution

This distribution can be used for variables with finite bounds (A,B). It uses two shape parameters, alpha and beta. When alpha=beta=1, you get a Uniform distribution. When alpha=beta=2, you get a dome-shaped distribution which is often used in place of the Triangular distribution. When alpha=beta=5 (or higher), you get a bell-shaped distribution. When alpha<>beta (not equal), you get a variety of skewed shapes.

Excel 2010+: =BETA.INV(RAND(),alpha,beta,A,B)

MORE Distribution Functions: Dr. Roger Myerson provides a free downloadable Excel add-in, Simtools.xla, that includes many other distribution functions for generating random numbers in Excel.

Creating a histogram is an essential part of doing a statistical analysis because it provides a visual representation of data.

In Part 3 of this Monte Carlo Simulation example, we iteratively ran a stochastic sales forecast model to end up with 5000 possible values (observations) for our single response variable, profit. If you have not already, download the Sales Forecast Example Spreadsheet.

The last step is to analyze the results to figure out how much the profit might be expected to vary based on our uncertainty in the values used as inputs for our model. We will start off by creating a histogram in Excel. The image below shows the end result. Keep reading below to learn how to make the histogram.

Histogram With Excel

Figure 1: A Histogram in Excel for the response variable Profit, created using a Bar Chart.

(From a Monte Carlo simulation using n = 5000 points and 40 bins).

We can glean a lot of information from this histogram:

  • It looks like profit will be positive, most of the time.
  • The uncertainty is quite large, varying between -1000 to 3400.
  • The distribution does not look like a perfect Normal distribution.
  • There doesn't appear to be outliers, truncation, multiple modes, etc.

The histogram tells a good story, but in many cases, we want to estimate the probability of being below or above some value, or between a set of specification limits. To skip ahead to the next step in our analysis, move on to Summary Statistics, or continue reading below to learn how to create the histogram in Excel.

Creating a Histogram in Excel

Update 7/2/15: A Histogram chart is one of the new built-in chart types in Excel 2016, finally! (Read about it).

Method 1: Using the Histogram Tool in the Analysis Tool-Pak.

This is probably the easiest method, but you have to re-run the tool each to you do a new simulation. AND, you still need to create an array of bins (which will be discussed below).

Method 2: Using the FREQUENCY function in Excel.

This is the method used in the spreadsheet for the sales forecast example. One of the reasons I like this method is that you can make the histogram dynamic, meaning that every time you re-run the MC simulation, the chart will automatically update. This is how you do it:

Step 1: Create an array of bins

The figure below shows how to easily create a dynamic array of bins. This is a basic technique for creating an array of N evenly spaced numbers.

To create the dynamic array, enter the following formulas:

  B6 = $B$2
  B7 = B6+($B$3-$B$2)/5

Then, copy cell B7 down to B11

Array of Bins in Excel

Figure 2: A dynamic array of 5 bins.

After you create the array of bins, you can go ahead and use the Histogram tool, or you can proceed with the next step.

Step 2: Use Excel's FREQUENCY formula

The next figure is a screen shot from the example Monte Carlo simulation. I'm not going to explain the FREQUENCY function in detail since you can look it up in the Excel's help file. But, one thing to remember is that it is an array function, and after you enter the formula, you will need to press Ctrl+Shift+Enter. Note that the simulation results (Profit) are in column G and there are 5000 data points ( Points: J5=COUNT(G:G) ).

The Formula for the Count column:

  FREQUENCY(data_array,bins_array)

  a) Select cells J8:J48
  b) Enter the array formula: =FREQUENCY(G:G,I8:I48)
  c) Press Ctrl+Shift+Enter

Layout for Creating a Scaled Histogram

Figure 3: Layout in Excel for Creating a Dynamic Scaled Histogram.

Creating a Scaled Histogram

If you want to compare your histogram with a probability distribution, you will need to scale the histogram so that the area under the curve is equal to 1 (one of the properties of probability distributions). Histograms normally include the count of the data points that fall into each bin on the y-axis, but after scaling, the y-axis will be the frequency (a not-so-easy-to-interpret number that in all practicality you can just not worry about). The frequency doesn't represent probability!

To scale the histogram, use the following method:

  Scaled = (Count/Points) / (BinSize)

  a) K8 = (J8/$J$5)/($I$9-$I$8)
  b) Copy cell K8 down to K48
  c) Press F9 to force a recalculation (may take a while)

Step 3: Create the Histogram Chart

Bar Chart, Line Chart, or Area Chart:

To create the histogram, just create a bar chart using the Bins column for the Labels and the Count or Scaled column as the Values. Tip: To reduce the spacing between the bars, right-click on the bars and select "Format Data Series...". Then go to the Options tab and reduce the Gap. Figure 1 above was created this way.

A More Flexible Histogram Chart

One of the problems with using bar charts and area charts is that the numbers on the x-axis are just labels. This can make it very difficult to overlay data that uses a different number of points or to show the proper scale when bins are not all the same size. However, you CAN use a scatter plot to create a histogram. After creating a line using the Bins column for the X Values and Count or Scaled column for the Y Values, add Y Error Bars to the line that extend down to the x-axis (by setting the Percentage to 100%). You can right-click on these error bars to change the line widths, color, etc.

Histogram Via Error Bars

Figure 4: Example Histogram Created Using a Scatter Plot and Error Bars.

REFERENCES:

CITE THIS PAGE AS:

Wittwer, J.W., "Creating a Histogram In Excel" From Vertex42.com, June 1, 2004

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