Computer Simulation Example

As a computer simulation example I will use the comprehensive example involving the sale of the Pennypinch CPA practice. Click here for details of this example. Recall that based on the analysis of the Maximum Competitive Advantage it was decided that the optimal buyer was the owner’s current employee Leona. The key computation for Leona is differential compensation: the difference between what she could derive in discretionary earnings (DE) if she buys the practice versus the amount of compensation she could expect if she remains an employee. The actual bid is based on what Leona required in the form of additional compensation in return for assuming the burdens of managing and servicing the practice.

Here are the key forecast parameters subject to variation and risk in the computation of differential compensation:

  • the transfer rate                                                                                  the attrition rate                                                                                  the growth rate                                                                                   the DE per customer per year                                                                                                                                          

In modeling these variables, our primary concern will be ordinary variation and possible extraordinary uninsurable risk. Are there any such extraordinary risks? The loss of a lease will not have a significant impact on the size of the client base as convenience is not a big factor in patronization decisions. If we assume that there is no one big employer whose sudden loss would cause a dramatic decrease in population and then a decline in client base, we need not include this factor.

So what’s left? One possibility is that Congress overhauls the tax code and greatly simplifies the compliance burden to the point where the demand for tax preparation services declines to next to nothing. While any such event is probably unlikely, its catastrophic impact leads us to include it in our forecasts of future DE. In the simulation we will run we will assess this risk to be 1%.

There is one more uninsurable risk that should be reflected in the forecast, the loss of a key employee. It is understood that there is a challenge to replace Peter. It is understood that Leona as the senior trained accountant will be able to service many of Peter’s clients. However, she will have to find someone to replace her. Someone of hopefully equal skill, diligence and charm.

As anyone who has run a small business knows hiring, training and retaining the kinds of employees that can provide the level of services the clients expect is hardly ever easy. The potential financial impact from being unable to hire and retain employees with the right skills can be significant. The impact is felt in higher than normal attrition experienced due to higher than expected levels of client dissatisfaction. The impact from employee problems is also cumulative.

Recall from the earlier example that Peter has 750 current clients. Suppose she nets 90% of these clients initially on the transfer. Suppose Leona’s first hire after acquiring the practice is less than optimal. Problems arise in the timeliness of return processing, increased errors on the returns discovered by the clients and personality clashes. Many of the clients previously serviced by Peter and Leona are now dissatisfied and many of these do not return in the second year after the transfer. Some but not all of the clients lost may be replaced by new clients but if the level of dissatisfaction caused by the bad hire is high enough it may take many years to return to a total client base that would have prevailed had the hire been a good one. The risk of a bad hire does not exist simply in the first year. Suppose Leona finds a good hire and that hire stays for say five years and then leaves. Now there is a potential crisis two years later when the impact of a bad replacement hire is fully realized.

This entails that in the simulation model the number of clients served during the year has to be sequentially connected. Stated as an equation: 

 Ct = Ct-1 – Ca +Cg 

Where Ct = Clients served in time t.

 Ct-1 = Clients served in time t-1.

 Ca = Clients lost after being served in time t-1.

 Cg = Clients gained during time t. 

A simulation of 1,000 trials basically outlines 1,000 possible different future paths of the firm’s client base. We also assume in this example that there is an 850 client capacity that once reached will not be exceeded. The clients served in year 1 after the transfer will be a function of the transfer rate. This transfer rate will be modeled not as a fixed rate but as a probability function. The number of new clients gained in year 2 and beyond will be modeled as a rate based on the prior year ending balance and will also be based on a probability function. These new clients are derived from referrals or inexpensive positional advertising.

Finally, the attrition experienced will consist of two components: normal and abnormal. Normal attrition will be modeled based on a constant percentage. Abnormal attrition will be added only if one of two events occur: dramatic change in the tax code and/or some form of significant employee problem. The occurrence or non-occurrence of these events will be modeled on a Bernoulli probability function. The expected degree of client loss in the case of a significant employee problem will also be modeled probabilistically. In the event of a major overhaul of the tax code it will be assumed that the practice will simply cease to exist.

Computer Simulation of DE

There are many factors that drive the DE of a firm all of which are sources of variation that can be modeled separately. These factors include the mix of services offered at different prices, the degree to which the owner relies on employee labor to provide services, the cost of those employees and the fixed cost structure of the firm. However, I believe the cost and effort to model these factors separately is not worth the benefit in more accurate forecasting. It is better to simply model the variation in the end measure of DE per client per year. Recall that this DE rate is computed before taking into account the cost of acquiring the practice.

Summary of Computer Simulation Example Model

The following tables summarize the simulation model assumptions. We begin with those variables that we treat as constants:

Here are the variables that will be modeled utilizing triangular probability distributions:

Here are the two major risk events modeled on a Bernoulli distribution:

Here for purposes of illustrating the structure of the simulation model is the Excel spreadsheet for one of 1,000 future paths simulation of the first two of the twenty five years forecast:

The transfer rate cell is computed by drawing from the probability function described above and is multiplied by the fixed 750 clients available at time of transfer. No net growth or attrition is assumed in the first year. The second year begins with the balance of clients from the end of the first year. A fixed normal attrition rate is used as a subtraction to the ending balance of clients from the previous year.

A growth rate is computed based on the probability distribution described above and this rate is multiplied by the ending balance of the previous year clients to reflect the new clients served in the next year. A row is dedicated to the probability of an employee problem that leads to above normal attrition. If based on the probability distribution described above a “1” is drawn this entails that an employee problem occurred. I assume a one year lag on the effect of such an event. The degree of above normal loss is also based on the probability distribution described above. In the above case since there was an occurrence of an employee problem in year 1 a client loss was computed at 65 (1 x 650 x 10%). The clients at the end of the year is also the total of clients served during the year, 593 (650-33+417-65).

The DE rate per client was drawn from the probability distribution described above. This rate times the clients served represents the forecast DE for the year. Note the line for a major tax code revision. This is drawn from the probability distribution described above. Had this resulted in a “1” representing the occurrence of this event then the end of the year client total would have been set to 0 resulting in the end of the practice.

As the reader can see, the basic causal structure of the DE process and all critical variables subject to ordinary variation and extraordinary risk have been captured in a twenty line spreadsheet. The full model extends an additional 23 years for a total of 25 years. We can use this model for multiple purposes. First we can compute the differential compensation.

The Differential Compensation Forecast

The reader will recall that this forecast measures the difference between what Leona will realize in DE if she acquires the firm versus what she can earn in compensation if she remains an employee. For this computation I assumed that Leona’s base salary was $85,000 as opposed to the $75,000 she currently earns. A 2% annual inflation factor was applied to all monetary parameters. The model also assumes that the firm under Leona’s ownership could grow to a maximum capacity of 850.

In developing this model I assume that if the firm goes out of business due to the elimination of the tax code that Leona would no longer be able to earn a salary as a tax preparer. Also recall that this computation is made before taking into account the cost of acquiring the firm. The simulation of 1,000 runs shows the net present value of the annual difference over twenty five years discounted at 4%. Inflation is assumed to be 2% annually. Here are the summarized results:

Because of the extreme low values caused by the remote possibility of the elimination of the income tax, the median rather than the mean should be used as an overall measure of central tendency. It should be noted that our dynamic simulation model shows a differential compensation of about $100,000 less than our static model.

The question facing Leona is whether the additional expected DE from owning and operating the practice offsets the burdens. I believe that stating the differential compensation measure in a lump sum present value does not clarify the answer sufficiently. A better measure might be the average additional DE over the expected annual salary. Here are the simulation results:

Here Leona can see that before reflecting the cost of acquiring the practice that on average she would take home about $61,000 more income by buying the firm versus remaining an employee. Of course focusing on the measures of central tendency helps, but Leona should also have a sense of the potential variation of DE. After all the point of the simulation is to highlight the risks and variation in possible outcomes. One useful measure of variation is known as the inter-quartile range. This measure shows the range results that are expected to occur between 25% and 75% of the time. On this simulation run the inter-quartile range is between about $48,000 and $74,000.

Since the average annual differential salary is a measure of the responsibility premium Leona might also find another simulation statistic useful: the average or median number of clients served in a tax season. Here are the results:

The inter-quartile range is 646 to 730. These are, or should be, eye opening results for Leona. Essentially what these forecasts predicts is zero growth in the client base over a twenty five year cycle! Can this be right? Perhaps not. But where is the model incorrect? The client base is a function of only three variables; the transfer rate, the attrition rate and the growth rate. It seem to me that our assumptions are quite reasonable on the first two variables, although perhaps we have exaggerated both the frequency of employee problems and their impact. I think more likely we may have understated the growth potential in new clients. In any event our simulation software allows us to easily change our assumptions in a transparent manner.

The overall results of this modeling should be reassuring to Leona if she uses Peter’s DE as the target DE for her responsibility premium. Recall that Peter’s current DE level is about $140,000 per year. The simulation shows that before taking into account the cost of the equity this is the level Leona can expect to realize (her base salary of $85,000 + $61,000) under the most frequently expected results. The remaining issue is negotiating how long she is willing to wait to achieve this level of DE. As we saw earlier the most likely outcome is a bid of between $133 and $150K.

Interpreting Simulation Results  

The simulation shows the range of possible DE outcomes over 1,000 possible future paths for the PA. The model attempts to incorporate all possible material sources of risk and variation. This stands in marked contrast to standard appraisal practice that attempts to develop one static forecast of cash flow discounted by one combined risk adjusted discount factor.

As we review the results of this example, or any other case in which simulation is used to guide the appraisal and negotiation process, it is well to ask how much we can rely on the accuracy of these multiple forecasts. The answer is that the accuracy of these forecasts will depend on how completely we have identified the sources of risk and variation and how accurately our chosen probability distributions reflect the frequency of risky events and ranges of normal variation. The principle of “garbage in and garbage out” applies to all simulation models.

For more computer simulation resources check out the Probability Management Organization web site.

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Copyright 2018 Michael Sack Elmaleh