Risk Analysis Using Monte Carlo
Simulation

Here we present a simple hypothetical budgeting problem for a business startup to demonstrate the key elements
of Monte Carlo simulation. This table shows the setup:
Cost/Budget
Item

Input Values

Min.
Allowed

10%
Likely

Expected
Values

90%
Likely

Max.
Allowed

Administrative

$70,000

$65,000

$67,709

$71,667

$76,097

$80,000

Cost of
Sales

$125,000

$120,000

$122,709

$126,667

$131,097

$135,000

Personnel

$335,000

$325,000

$329,950

$336,667

$343,826

$350,000

Professional
Fees

$15,000

$12,000

$13,533

$15,667

$17,984

$20,000

Sales &
Marketing

$50,000

$45,000

$47,216

$50,000

$52,744

$55,000

Technology

$20,000

$17,000

$18,533

$20,667

$22,984

$25,000

The randomvariable probability distribution we have chosen for each of the input
budget elements is trangular:
This seems like a reasonable distribution to choose because it's peak represents
the initial input values for the budget element variables, that we feel initially are the most likely values; and
then it tails off in a linear fashion as we move left toward the "Minimum Allowed Values" and right toward the
"Maximum Allowed Values" we initially chose to bracket the range of uncertainty around the initial input values.
This triangular distribution results in the "Expected Values," the "Values 10% Likely" and the "Values 90% Likely"
shown in the table above.
Selecting probability distributions for the input variables in a risk analysis can
be tricky. If valid, reliable historical data are available, these should be used as a guide (but it might require
doing a bit of research). Otherwise, a combination of common sense and intuition usually works well. Careful
inspection of the table above reveals that for most of the input variables, we chose maximum allowed values that
are farther above the initial input values than the minimum allowed values are below the initial input values. This
is a somewhat conservative approach that helps us to avoid or minimize bad surprises later on due to
underbudgeting.
This approach is shown graphically below for each individual budget input
variable:
Based on the above distributions, we can now generate a Sensitivity Report that shows us the relative influence
of each input variable on the Total Budget output variable:
In the graph above, the horizontal line represents the Total
Budget number expressed as the sum of the initial budget input variables. A vertical line represents the output
range between an input variable's 10% likely and 90% likely values. Inputs with long vertical lines have the
most effect on the output variable (Total Budget). Below is a tabular Total Budget sensitivity report:
Again we can see that the Personnel Budget has by far the largest influence
on the Total Budget (48%), and the Technology Budget has the smallest influence (5%).
Now we run 1,000 Monte Carlo simulation trials and examine the results:
Minimum result: $600,651
$615,000 Initial Total Budget value
Maximum result: $649,455
20.6% of results are equal or lower
Expected value:
$621,314
79.4% of results are equal or greater
Std Deviation: $7,484
Here is a table showing the percent of simulation values falling below various Total Budget estimates:
5% <= $609,295
10% <= $612,067
15% <= $613,616
20% <= $614,847
25% <= $615,996
30% <= $617,038
35% <= $618,179
40% <= $619,176
45% <= $620,050
50% <= $621,175 (Simulated expected value = $621,314)
55% <= $621,942
60% <= $622,992
65% <= $623,871
70% <= $624,964
75% <= $626,117
80% <= $627,575
85% <= $629,159
90% <= $631,155
95% <= $634,251
100% <= $649,455
If Total Budget is $615,000:
 20.6% of results are equal or lower
 79.4% of results are equal or greater
Here is a histogram of Total Budget simulated values:
Here is the cumulative Total Budget frequency chart:
And here is the tabular version of these distributions:
Here is a summary of the simulations in terms of the amount of change occurring from start to finish for
estimated Total Budget values and the standard deviation of the estimates:
We usually stop simulating when the % change drops below 1.0; otherwise we continue. This simulation exercise
stopped after 1,000 runs. In our case we could have stopped after 500 runs, but by going longer we were able to
reduce the standard deviation slightly.
And finally, here is a calculation of the 95% statistical confidence interval around our simulated expected mean
of $321,314:
Thus, based on our sample of 1,000 runs, if we were to repeat the simulation many more times using a different
random number seed for each simulation, then we would expect 95% of those samples to result in an Expected Total
Budget value between $620,849.99 and $621,778.77.
Conclusion
Based on the results of the simulation, we have learned a lot, and we can be more confident in our budget
planning. First of all, we learned that we would have only about a 20.6% chance of achieving our initially
specified Total Budget of $615,000. If we want to have at least a 50/50 chance of achieving our budget, we should
set the budget level at about $621,314. If we want at least a 75% chance of achieving our budget, we should set the
budget level at about $626,117. And a budget of about $631,155 would give us an even more comfortable 90% chance of
success.
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