“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.
in Valuation Opinions
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.
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
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.
belief graphs in the Sidebar
illustrate possible reporting tools with fuzzy math.
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.
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
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
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
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.
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.
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.
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
Fuzzy Logic in Business Valuations
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.
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
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,
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.
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.
Value of Future Earnings or Cash Flow
all normal mathematical operations apply to them, fuzzy
numbers can also be used with present value of future earnings
cash flow techniques.
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.
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.
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.
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.
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.