A New Approach to Uncertainty in Business Valuations

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

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 longterm 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.