Monte Carlo Simulation Basics



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Monte Carlo Simulation Basics


http://www.vertex42.com/ExcelArticles/mc/MonteCarloSimulation.html

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 ]


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.

Example 2: A Stochastic Model


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

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 ...


Sales Forecasting Example

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 Example Spreadsheet


MCExample_SalesForecast.zip
File Size: ~420 kB
Requirements: Excel 97 or Later
No Macros Used

Download Instructions:

  1. Download: Click on the download link or image to the left and save the file on your computer. Depending on your browser and operating system, you may need to "right-click" on the link and select "Save Target As..."

  2. Unzip: You will need to "unzip" the file using a program such as Winzip? You can download a free evaluation version of WinZip?at www.winzip.com.

  3. Open in Excel: After extracting the spreadsheet from the .zip file, open it in Excel. If calculation is too slow on your machine, you may need to remove or disable the COUNTIF formulas (used in the cumulative probability calculations).

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.

Generating Random Numbers using Excel


Sales Forecast Example - Part II

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 applys. 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).




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.




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

Cell A2 contains the formula:


=Model!$F$17+RAND()*(Model!$G$17-Model!$F$17)

Note that the reference Model!$F$17 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


Since our model is very simple, all we need to do to evaluate the model for each run of the simulation is put the equation in another column next to the inputs, as shown in Figure 2 (the Profit column).

Cell G2 contains the formula:


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

Step 4: Run the Simulation


We don't need to write a fancy macro for this example in order to iteratively evaluate our model. We simply copy the formula for profit down 5000 rows, making sure that we use relative references in the formula (no $ signs).

Rerun the Simulation: F9


Although we still need to analyze the data, we have essentially completed a Monte Carlo simulation. Because we have used the volatile RAND() formula, to re-run the 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.

Until I get around to providing another example that uses macros, let me just say that 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.

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.

Creating a Histogram in Excel


Sales Forecast Example - Part III

In Part II of this Monte Carlo Simulation example, we completed the actual simulation. (If you haven't already, Download the example spreadsheet). We ended up with a column of 5000 possible values (observations) for our single response variable, profit. The last step is to analyze the results. We will start off by creating a histogram in Excel, a graphical method for visualizing the results.




Figure 1: A Histogram in Excel 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.


Creating a Histogram in Excel


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


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


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 is actually 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.




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

Summary Statistics


Sales Forecast Example - Part IV of V
In Part III of this Monte Carlo Simulation example, we plotted the results as a histogram in order to visualize the uncertainty in profit. In order to provide a concise summary of the results, it is customary to report the mean, median, standard deviation, standard error, and a few other summary statistics to describe the resulting distribution. The screenshot below shows these statistics calculated using simple Excel formulas.

Download the Sales Forecast Example

Figure 1: Summary statistics for the sales forecast example.

Statistics Formulas in Excel


Sample Size (n): =COUNT(G:G)
Sample Mean: =AVERAGE(G:G)
Median: =MEDIAN(G:G)
Sample Standard Deviation (): =STDEV(G:G)
Maximum: =MAX(G:G)
Mininum: =MIN(G:G)
Q(.75): =QUARTILE(G:G,3)
Q(.25): =QUARTILE(G:G,1)
Skewness: =SKEW(G:G)
Kurtosis: =KURT(G:G)

Note: These Excel functions ignore text within the data set.


Sample Size (n)


The sample size, n, is the number of observations or data points from a single MC simulation. For this example, we obtained n = 5000 simulated observations. Because the Monte Carlo method is stochastic, if we repeat the simulation, we will end up calculating a different set of summary statistics. The larger the sample size, the smaller the difference will be between the repeated simulations. (See standard error below).

Central Tendancy: Mean and Median


The sample mean and median statistics describe the central tendancy or "location" of the distribution. The arithmetic mean is simply the average value of the observations.

The mean is also known as the "First Moment" of the distribution. In relation to physics, if the probability distribution represented mass, then the mean would be the balancing point, or the center of mass.

If you sort the results from lowest to highest, the median is the "middle" value or the 50th Percentile, meaning that 50% of the results from the simulation are less than the median. If there is an even number of data points, then the median is the average of the middle two points.

Extreme values can have a large impact on the mean, but the median only depends upon the middle point(s). This property makes the median useful for describing the center of skewed distributions such as the Lognormal distribution. If the distribution is symmetric (like the Normal distribution), then the mean and median will be identical.


Spread: Standard Deviation, Range, Quartiles


The standard deviation and range describe the spread of the data or observations. The standard deviation is calculated using the STDEV function in Excel.

The range is also a helpful statistic, and it is simply the maximum value minus the minimum value. Extreme values have a large effect on the range, so another measure of spread is something called the Interquartile Range.


The Interquartile Range represents the central 50% of the data. If you sorted the data from lowest to highest, and divided the data points into 4 sets, you would have 4 Quartiles:



Q0 is the Minimum value: =QUARTILE(G:G,0) or just =MIN(G:G),
Q1 or Q(0.25) is the First quartile or 25th percentile: =QUARTILE(G:G,1),
Q2 or Q(0.5) is the Median value or 50th percentile: =QUARTILE(G:G,2) or =MEDIAN(G:G),
Q3 or Q(0.75) is the Third quartile or 75th percentile: =QUARTILE(G:G,3),
Q4 is the Maximum value: =QUARTILE(G:G,4) or just MAX(G:G).

In Excel, the Interquartile Range is calculated as Q3-Q1 or:



=QUARTILE(G:G,3)-QUARTILE(G:G,1)

Shape: Skewness and Kurtosis

Skewness


Skewness describes the asymmetry of the distribution relative to the mean. A positive skewness indicates that the distribution has a longer right-hand tail (skewed towards more positive values). A negative skewness indicates that the distribution is skewed to the left.

Kurtosis


Kurtosis describes the peakedness or flatness of a distribution relative to the Normal distribution. Positive kurtosis indicates a more peaked distribution. Negative kurtosis indicates a flatter distribution.

Confidence Intervals for the True Population Mean


The sample mean is just an estimate of the true population mean. How accurate is the estimate? You can see by repeating the simulation (using F9 in this Excel example) that the mean is not the same for each simulation.

Standard Error


If you repeated the Monte Carlo simulation and recorded the sample mean each time, the distribution of the sample mean would end up following a Normal distribution (based upon the Central Limit Theorem). The standard error is a good estimate of the standard deviation of this distribution, assuming that the sample is sufficiently large (n >= 30).

The standard error is calculated using the following formula:




In Excel: =STDEV(G:G)/SQRT(COUNT(G:G))

95% Confidence Interval


The standard error can be used to calculate confidence intervals for the true population mean. For a 95% 2-sided confidence interval, the Upper Confidence Limit (UCL) and Lower Confidence Limit (LCL) are calculated as:


To get a 90% or 99% confidence interval, you would change the value 1.96 to 1.645 or 2.575, respectively. The value 1.96 represents the 97.5 percentile of the standard normal distribution. (You may often see this number rounded to 2). To calculate a different percentile of the standard normal distribution, you can use the NORMSINV() function in Excel.



Example: 1.96 = NORMSINV(1-(1-.95)/2)

Commentary


Keep in mind that confidence intervals make no sense (except to statisticians), but they tend to make people feel good. The correct interpretation: "We can be 95% confident that the true mean of the population falls somewhere between the lower and upper limits." What population? The population we artificially created! Lest we forget, the results depend completely on the assumptions that we made in creating the model and choosing input distributions. "Garbage in ... Garbage out ..." So, I generally just stick to using the standard error as a measure of the uncertainty in the mean. Since I tend to use Monte Carlo simulation for prediction purposes, I often don't even worry about the mean. I am more concerned with the overall uncertainty (i.e. the spread).

-End-

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