A New Approach to Uncertainty in Business Valuations

CPA Journal, The, Apr 2004 by McKee, Thomas E

The typical business valuation has a significant limitation: the failure to recognize uncertainty. Business valuation specialists try to cope with uncertainty by triangulation of three different valuation techniques, applying a premium or discount to a capitalization rate, or adjusting future revenue and expense projections. These techniques generally can do no better than narrow the valuation range among valuation results to /- 15%, a limitation that users should understand.

Fortunately, "fuzzy math" functions in spreadsheets can formally incorporate uncertainty in business valuations in a way that incorporates significant additional information into valuation reports and helps mitigate the limitations of traditional valuation approaches.

Uncertainty in Valuation Opinions

The typical report-"It is our considered opinion that the Fair Market Value of 100% of the common stock of ABC Inc. as of December 31, 2003, is best expressed as $12,800,000"-would not reveal the possibility that ABC Inc. might be worth as much as $15 million or as little as $10 million. The range of possible values usually is not available under traditional valuation reporting approaches.

Contrast the previous opinion with the following opinion and Panel 1 of the Sidebar: "It is our considered opinion that the Fair Market Value of 100% of the common stock of ABC Inc. as of December 31, 2003, is best expressed as most likely to be $12,800,000, according to the enclosed belief graph."

The belief graph in Panel 1 shows a 40% probability that the company may be worth as little as $10 million. It also indicates the belief that there is 0% probability of the company being worth more than $17 million.

The belief graphs in the Sidebar illustrate possible reporting tools with fuzzy math.

Risk Assessment

Risk is the possibility of an adverse event. For a potential purchaser of ABC Inc., the company in the previous example, an adverse event would be paying $ 12 million for the company only to find out subsequently that its fair market value is only $10 million.

Risk is typically assessed in terms of both the likelihood an adverse event will occur and the monetary impact it would have. A purchaser of ABC Inc. willing to pay $12 million faces a 40% possibility that the company is worth $2 million less.

Risk can be assessed in terms of statistical probabilities determined by sampling from large populations. Further refinement through simulation analyses can provide additional insights. Simulation approaches can be extremely complex and time-consuming, however, leading to a search for alternatives for typical valuation work.

Another approach to risk assessment considers the possibility or likelihood of an outcome. For example, any valuation expert performing a valuation of ABC Inc. would know that it is not absolutely true that the company value is exactly $12,800,000; this value simply represents the single best estimate. Fuzzy math logic provides a means to manage, and disclose, the degree of uncertainty or imprecision in the valuation amount of $12,800,000.

Fuzzy Logic

Fuzzy logic was developed in the midtwentieth century to deal with the uncertainly that arises from ambiguity or vagueness, which differs from the randomness associated with uncertainty in statistical probability. Ambiguity or vagueness may occur because of imprecision in linguistic terms or from an inability to measure an object precisely.

Under classical logic, a statement is either true or false; however, under fuzzy logic, the truth of a statement can be described as anything between 0 (false) and 1 (true). Thus, a statement with a value of .8 would represent a fairly strong belief that the statement is true. Fuzzy logic has become widely accepted by scientists and mathematicians, who use it in a wide array of applications, including weather forecasting.

Fuzzy math allows the simultaneous assignment of possibilities to a number of mutually exclusive outcomes. For example, a valuation of 10 could occur with a belief of 100%, but a valuation of 9 could occur with a belief of 50%. One belief docs not preclude the other. Beliefs about many different valuations over an interval would be possible.

Fuzzy math beliefs are not the same as statistical probabilities. Statistical probabilities for an event typically have to sum to 1, which implies 100% certainty in statistical probability. Fuzzy math beliefs do not need to sum to 1 or any other value.

Implementing Fuzzy Logic in Business Valuations

Fuzzy logic can be implemented in business valuations through spreadsheet software such as Microsoft Excel. FuziCalc, by FuziWare Inc., introduced a practical Windows-based spreadsheet incorporating a variation of fuzzy math over a decade ago.

For example, using the multiple of earnings valuation model, with an earnings multiple of 10, a company with normalized earnings of $120,000 would have an estimated company value of $1,200,000.

Sensitivity analysis using fuzzy math can convert earnings multiples and normalized earnings point estimates to fuzzy amounts by associating possibility beliefs with them. For example, it could be determined that an earnings multiple between 8 and 12 is appropriate, with 10 being the most likely. The multiple could be expressed in a triangular belief graph shaped similar to the one shown in Panel 2. Similarly, it could be determined that normalized earnings of $120,000 are most likely but, based on past variations, earnings could range from slightly above $100,000 to slightly below $160,000, as shown in Panel 2. Note that the midpoint for this belief graph is not the normalized earnings estimate of $ 120,000 but rather $125,900, because the interval is weighted in this direction. The midpoint is the point at which half of the distribution is on either side. By introducing the range of possible values for normalized earnings, new information, such as the midpoint of the belief function, becomes available.