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 mid-twentieth century to deal with the uncertainty 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 does 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.

The normal mathematical operations of addition, subtraction, multiplication, and division apply to fuzzy numbers. Exhibit 1 shows how the fuzzy number, the minimum, the midpoint, and the maximum can be factored into a valuation.

When the possible range of values for both the price earnings ratio and the normalized earnings is considered, the value of the company is not simply $1,200,000, the point estimate from traditional math, but rather $1,293,000, the midpoint of the fuzzy number for the overall company value estimate.

Present Value of Future Earnings or Cash Flow

Because all normal mathematical operations apply to them, fuzzy numbers can also be used with present value of future earnings cash flow techniques.

For example, consider ABC Inc., a mature company in a stable industry. Assume a forecast horizon of only three years with a terminal value assumption for the fourth year, consistent with the valuation of a mature company with no anticipated, significant long-term changes.

Assume that current-year free cash flow is $91,000 and is expected to grow 10% annually for the next three years before reverting to the long-term industry growth rate of 5%. The weighted average cost of capital is 8%. The traditional valuation might resemble Exhibit 2, focusing on the value of core operations while ignoring other items that might influence the free cash flow.

This valuation indicates a company value of $3,547,580. Some small changes to the assumed growth rates in the previous assumptions, however, can make a difference. First, assume that the anticipated growth rate for the next three years is a fuzzy number of 10% that ranges from a minimum of 8% to a maximum of 12%. Second, assume that the long-term industry growth rate for Year 4 and beyond is a fuzzy number of 5% that ranges from a minimum of 4% to a maximum of 7%. Changing these two assumptions to fuzzy numbers would result in the valuation in Exhibit 3 and the value of core operations of $5,384,453 is a fuzzy number represented by Sidebar Panel 3.

Panel 3 shows that the value with the highest belief of 1 is a point that is slightly above the $3,500,000 point on the belief graph. This is consistent with the traditional valuation estimate of $3,547,580. The valuation amount using the fuzzy numbers becomes $5,384,453, approximately $1.8 million higher than the traditional valuation of $3,547,580. The higher valuation derives from the conversion of growth rates from traditional point estimates into fuzzy numbers.

The valuations of $3,547,580 and $5,384,453 are both correct according to the assumptions used to produce them. The fuzzy number valuation better reflects the reality that there is greater upside potential to long-term growth than can be expressed by a point estimate. Panel 3 shows that, although the point of highest belief is $3,547,580, there is more upside than downside potential to the valuation. This indicates that the potential value of the company is somewhere between $3,547,580 and $5,384,453. A seller for ABC Inc. should know about the upside potential when negotiating a sale, as should the buyer.