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# Lost Profits for New Business Using Survivability Statistics and Monte Carlo Simulation

The purpose of this article is to show how business survivability statistics and Monte Carlo simulation can be incorporated into a lost profits analysis for a new business, to assist in meeting the reasonable degree of certainty standard. In order to primarily focus on these aspects, many factors (such as discounting, industry/market share analysis, capital adequacy, management experience, economic cycles in the data, etc.) will be simplified, ignored and/or accepted as adequate.

Here's the scenario:

A new business is claiming lost profits due to breach of contract. The new business never got off the ground due to the alleged breach. The financial forensics expert is tasked with determining, with a reasonable degree of certainty, the lost profits due to the alleged breach. The loss period is five years.

Projected financials:

Management provides projected financials which the financial forensics expert has analyzed and accepted as reasonable projections of profits but for the alleged breach. The financials project profits of \$100,000 per year during the five-year loss period. The financial forensics experts determines that had the business survived during the five-year loss period, it would have generated profits of \$500,000.

Survivability Statistics:

The following table summarizes statistics regarding the survival of U.S. private sector establishments by opening year, for those establishments originated during the year ending March 2013 (www.bls.gov/bdm/us_age_naics_00_table7.txt):

Here are a few examples of how to interpret the data:

• 2nd column: Of the 629,078 establishments originated during the year ending March 2013, 319,206 of those survived through the year ending March 2018.

• 3rd column: Of the 629,078 establishments originated during the year ending March 2013, 50.7% of those survived through the year ending March 2018.

• 4th column: Of the 347,789 establishments that survived through the year ending March 2017 (their 4th year), 91.8% of those survived through the following year, the year ending March 2018 (their 5th year).

So what is the most appropriate way to apply the survivability statistics to the lost profits analysis? Using Monte Carlo simulation to model the possible outcomes will shed more light on this issue.

Monte Carlo Simulation:

I set up the model for lost profits analysis using the survival rates of previous year's survivors (4th column above). I utilized Bernouli distributions for each year's survival outcome (1 = survival; 0 = no survival). If the business sequentially survives each year given its respective survival rate during that year, the lost profits during that year are accumulated to the total.

For instance, the business has a 79.6% probability of survival during its first year. If it doesn't survive its first year, lost profits are \$0 for the first year and the four following years. Accordingly, total lost profits equals \$0. If the business survives its first year, lost profits are \$100,000 for the first year, bringing total lost profits to \$100,000, and it has a 86.6% probability of survival during its second year. If it doesn't survive its second year, lost profits are \$0 for the second year and the three following year. Total lost profits then remain at \$100,000. If the business does survive its second year, lost profits are \$100,000 for the second year, bringing total lost profits to \$200,000, and it has an 89.0% of survival during its third year. This pattern continues through the fifth year.

In this model, the minimum value of lost profits is \$0 which occurs if the business does not survive its first year. The maximum value of lost profits is \$500,000 which occurs if the business survives through all five years.

The program @RISK was utilized to perform the Monte Carlo simulation on this model. The simulation performed 1,000,000 iterations of the model. The following output was obtained from the simulation:

The median value for total lost profits was \$500,000. This outcome is expected given that just slightly over 50% of businesses (50.7% to be exact) survive through their fifth year. The average value for total lost profits was \$316,078, with a 90% confidence interval of +/- \$344.58. This outcome is a little less intuitive. To obtain this value only using the raw statistics without Monte Carlo simulation, the potential lost profits during each year (\$100,000 per year) are multiplied by the survival rates since birth (3rd column above). The result is \$315,900, which is approximately equivalent to \$316,078 and falls within the 90% confidence interval.

Conclusion:

So what is the appropriate value for lost profits? Some may believe that the median, \$500,000, is appropriate, arguing that the new business has a (slightly) greater than 50% chance of surviving through its fifth year. Therefore, its profits but for the breach are \$100,000 for each of the five years, on a more probable than not basis. But does this meet the reasonable degree of certainty standard? I would argue that under these circumstances, it does not.

The most appropriate choice for the value of lost profits under these circumstances, is the mean, \$315,900. Using the mean effectively risk-adjusts each year's potential lost profits by the new business's probability of survival during each respective year, and accumulates the risk-adjusted lost profits for all five years. The risks associated with new business survival are captured using this methodology in lost profits analyses.

If you'd like to discuss this further or obtain a copy of the underlying analysis with the Monte Carlo simulation, please contact me at david@solisff.com.