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Do Not Use the Arithmetic Mean to Average Multiples

Authors:
  • Sutter Securities Financial Services, San Francisco

Abstract

The use of arithmetic mean to average multiples is mathematically incorrect. The multiple is an inverted ratio with price in the numerator. Therefore, the harmonic mean should be used as the appropriate measure of central tendency. As a crosscheck, the median should also be considered.
TIMELY NEWS, ANALYSIS, AND RESOURCES FOR DEFENSIBLE VALUATIONS
Vol. 27, No. 4, April 2021
BUSINESS VALUATION UPDATE
bvresources.com
Reprinted with permissions from Business Valuation Resources, LLC
By Gilbert E. Matthews, CFA,
Sutter Securities Inc. (San Francisco, Calif., USA)
Valuation professionals should not use the arith-
metic mean of multiples. It is mathematically
incorrect because it gives excessive weight to
high multiples. A multiple is an inverted ratio
with price in the numerator. Therefore, the har-
monic mean should be used as the appropriate
measure of central tendency. As a cross-check,
the median should also be considered. Although
abnormally low multiples can overly affect the
harmonic mean, excluding outliers can correct
this.1
The harmonic mean is calculated by: (i) taking
the sum of the reciprocals of each value in a
data series; (ii) dividing the sum by the number
of values in the data series; and (iii) taking the
reciprocal of that number. It is easy to calculate
using Excel with the fx [Insert Function] button
or clicking on Fn+Shift+F3. Select HARMEAN,
scroll over the datapoints to be averaged, and
click. Alternatively, use the Σ [sum] function and
replace SUM with HARMEAN in the formula bar.
The Median
The median is the midpoint of a range of numbers.
It is a commonly used measure of central ten-
dency approach and is a useful supplement to
1 Outlying low multiples can distort the result. Since
excluding a low outlier could be deemed to be
selective, it is best to use a trimmed harmonic mean,
excluding a high multiple for each low multiple
excluded.
the harmonic mean for averaging multiples. In
practice, the median of multiples is usually close
to the harmonic mean.
The author has been using both the harmonic
mean and the median in corporate valuations
since the 1970s. In my experience, the median
multiples are higher than the harmonic mean
more often than they are lower than others;
however, it is common for some medians of mul-
tiples within the same group of guideline compa-
nies to be lower than the harmonic mean while
others are higher.
The median is not useful for small samples; with a
limited number of guideline companies, the har-
monic mean is the only useful measure of central
tendency. The harmonic mean is superior to the
median in another respect—because the median
uses only one datapoint, it does not give any
consideration to skewness in the data.
Support for the Harmonic Mean
Using the harmonic mean of multiples is not a
new concept. Graham and Dodd’s classic book,
Security Analysis, used the harmonic mean to
average P/E ratios in 1951. The use of the har-
monic mean for averaging multiples was ex-
plained in detail in a book chapter on fairness
opinions published in 1990.2 A classic valuation
book, Shannon Pratt’s Valuing a Business,
2 Gilbert E. Matthews and M. Mark Lee, “Fairness
Opinions & Common Stock Valuations,” in The Library
of Investment Banking, Vol. 4, Robert L. Kuhn, ed.
(Dow Jones Irwin, 1990): 381, 405-407.
Do Not Use the Arithmetic Mean to Average Multiples
2 Business Valuation Update April 2021 Business Valuation Resources
DO NOT USE THE ARITHMETIC MEAN TO AVERAGE MULTIPLES
explained in 1996, “The harmonic mean is used
to give equal weight to each guideline company
in summarizing ratios that have stock price or
MVIC [market value of invested capital] in the
numerator.”3
Empirical analyses by Baker and Ruback in a 1999
Harvard working paper demonstrated that the
arithmetic mean was a poor measure of central
tendency for multiples of revenues, EBITDA and
EBIT. They also concluded that the harmonic
mean was somewhat better than the median.4
Liu, Nissim, and Thomas, in a 2002 empirical
study, arrived at the same conclusion.5 Numerous
subsequent studies have arrived at the same con-
clusion based on empirical data.6 A few studies
3 Shannon P. Pratt, Robert F. Reilly, and Robert P.
Schweihs, Valuing a Business: The Analysis and
Appraisal of Closely Held Companies, 3rd edition
(New York: Irwin, 1996): 225.
4 Malcom Baker and Richard S. Ruback, “Estimating
Industry Multiples.” Working Paper. Harvard University
(1999), pp. 4-5, available at hbs.edu/faculty/
Publication%20Files/EstimatingIndustry_b4e64d71-
c8fd-4a5e-b31a-623d3a7d02bc.pdf.
5 Jing Liu, Doron Nissim, and Jacob Thomas, “Equity
Valuation Using Multiples.Journal of Accounting
Research 40 (1) (2002): 135, 137, 157.
6 E.g., Ingolf Dittmann and Ernst G. Maug, “Biases
and Error Measures: How to Compare Valuation
Methods,” ERIM Report Series Reference No. ERS-
2006-011-F&A; Mannheim Finance Working Paper
No. 2006-07 (Aug. 25, 2008), pp. 2, 8, available at
ssrn.com/abstract=947436; Stefan Henschke and
Carsten Homburg, “Equity Valuation Using Multiples:
Controlling for Differences Between Firms” (May
2009), p. 22, available at papers.ssrn.com/sol3/
papers.cfm?abstract_id=1270812; Toby Tatum,
“Harmonic Mean Value: The Appropriate Measure of
Central Tendency,Business Appraisal Practice (3rd
quarter 2011) 28, 31; Georgia Pazarzi, “Comparison
of the Residual Income and the Pricing Multiples
Equity Valuation Models,” II International Journal in
Economics and Business Administration, Issue 3, 88,
102 (2014); Jens Overgaard Knudsen, Simon Kold,
and Thomas Plenborg, “Stick to the Fundamentals and
Discover Your Peers,” 73 Financial Analysts Journal
84, 104 (2017); William H. Black and Lari B. Masten,
“Empirical Investigation of Alternative Measures of
Central Tendency,5 Journal of Forensic Accounting
Research 216 (2020).
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DO NOT USE THE ARITHMETIC MEAN TO AVERAGE MULTIPLES
Reprinted with permissions from Business Valuation Resources, LLC
showed that the median was somewhat better
than the harmonic mean,7 others concluded that
they were similar,8 but all agreed that the arith-
metic mean was an inferior approach.
A Flawed Contrary View
In contrast, a 2015 BVU article argues that “the
harmonic mean should be avoided because it
is inherently biased low.9 The article’s authors
correctly conclude that the median is superior
to the arithmetic mean, but they argue that the
harmonic mean should not be used. They rely
on unsound reasoning in rejecting the harmonic
mean. Their conclusions would be valid if they
were examining datapoints with price in the de-
nominator, such as dividend yields. Their analy-
ses fail because they do not take into account the
basic reason why the harmonic mean is appro-
priate for multiples: the fact that a multiple is an
inverted ratio because price is in the numerator.
The authors acknowledge that the arithme-
tic mean “is most frequently higher than the
median,”10 but they erroneously assert that the
arithmetic mean gives the same weight to each
multiple.11 They fail to recognize that the reason
7 E.g., Volker Herrmann and Frank Richter, “Pricing with
Performance-Controlled Multiples,” 55 Schmalenbach
Business Review 194, 212 (2003); Andreas Schreiner
and Klaus Spremann, “Multiples and Their Valuation
Accuracy in European Equity Markets,” Working Paper
(Aug. 13, 2007), p. 12, fn. 5, available at SSRN:
ssrn.com/abstract=957352.
8 E.g., Mingcherng Deng, Peter D. Easton, and Julian
Yeo, “Another Look at Equity and Enterprise Valuation
Based on Multiples” (April 2010), pp. 15-16, available
at ssrn.com/abstract=1462794; Thomas Plenborg and
Rene Coppe Pimentel, “Best Practices in Applying
Multiples for Valuation Purposes,” 59 The Journal of
Private Equity 55, 59 (Summer 2016); Emanuel Bagnal
and Enrico Cotta Ramusino, “Market Multiples and
the Valuation of Cyclical Companies,” 10 International
Business Research (Issue 12) 246, 252 (2017).
9 Robert M. Dohmeyer, Herbert Kierulff, and Janae
Castell, “Mean, Median, Harmonic Mean: Which Is
Best?” Business Valuation Update, Jan. 2015, p. 1.
10 Id. at 4.
11 Id.
that the arithmetic mean is almost always higher
is that it is upwardly biased by high multiples.
Moreover, the distribution of multiples is almost
always positively skewed.
They attempt to “prove” that the harmonic mean
is biased using this statistically unsound analysis:
To further test the bias of the harmonic mean,
we used the random number generator in
Excel. In Excel, if you type “rand()” into a cell,
it will generate a random number between 0
and 1. Since the central tendency—median and
average—of Excel’s random generator is 0.50,
we know in advance the true unbiased result.12
They then conclude that the mean and median
in their test of “random numbers” were both 0.50
and the harmonic mean was 0.26. Based on this
determination, they claim, “[W]e demonstrated
that the harmonic mean is a biased low estimator
of central tendency when data are distributed
normally.” They mistakenly assume that the ap-
propriate measure of central tendency does not
depend on the nature of the underlying data.
They concede the obvious fact (knowable even
without using Excel) that the arithmetic mean
and the median of numbers from 0.00 to 1.00 (or
0.01 to 0.99, or 0.10 to 0.90) is 0.50. If we take the
reciprocals of numbers from 0.01 to 0.99 (analo-
gous to ratios with price in the numerator), the
harmonic mean and the median are 0.50 and the
arithmetic mean is 0.19. This shows that, for recip-
rocals, it is the arithmetic mean that is a biased
measure of central tendency. The question is
when it is appropriate to average raw numbers
or to average their reciprocals.
Is it more reasonable to average multiples using
reciprocals? The exhibit shows that: (a) if an in-
vestor invested equal amounts in a $100,000
portfolio of four companies with P/E ratios of
50x, 25x, 15x, and 10x, the portfolio would have
earnings of $5,667 and a multiple of 17.6x (the
12 Id.
4 Business Valuation Update April 2021 Business Valuation Resources
DO NOT USE THE ARITHMETIC MEAN TO AVERAGE MULTIPLES
Reprinted with permissions from Business Valuation Resources, LLC
harmonic mean); and (b) if the investor bought
equal amounts of earnings in each company (i.e.,
investing ve times as much at a 50x multiple as
in a 10x multiple), the portfolio’s earnings would
be $4,000 and its multiple would be 25x (the
arithmetic mean).
Equal Investment vs. Equal Earnings
Equal investment: Equal earnings:
Invested P/E Earnings Invested P/E Earnings
$25,000 50.0x $500 $50,000 50.0x $1,000
$25,000 25.0x $1,000 $25,000 25.0x $1,000
$25,000 15.0x $1,667 $15,000 15.0 x $1,000
$25,000 10.0x $2,500 $10,000 10.0x $1,000
$100,000 17. 6x $5,6 67 $100,000 25.0x $4,000
To give equal weight to each ratio with price in
the numerator, it is necessary to use the harmonic
mean—that is why it is the best measure of central
tendency for multiples. The arithmetic mean gives
ve times as much weight to a 50x multiple com-
pared to a 10x multiple, demonstrating that there
is upward bias to an arithmetic mean of multiples.
The harmonic mean of datapoints is always lower
than the arithmetic mean. Because the arithmetic
mean of multiples gives excessive weight to high
multiples, it necessarily results in an overvaluation.
Weighted Harmonic Mean
Some valuation experts favor the use of a weight-
ed harmonic mean.13 This method is appropri-
ate for calculating the multiple of a weighted
index. However, use of a weighted mean requires
a subjective judgment as to what factor to use
for weighting the multiples. Several alternatives
could be chosen, such as market capitalization,
revenues, free cash ow, and net income. One
study has concluded that the accuracy of the
13 E.g., Toby Tatum, “In Defense of Tatum’s Law of Market
Multiples,” Business Valuation Update, April 2018,
Special Supplement, p. 18.
harmonic mean can be improved by weighting
the harmonic mean by the growth rate but not
by other factors.14
Importantly, any weighting based on size nec-
essarily gives more weight to larger guideline
companies than to smaller ones. Why should
larger companies be given greater weight? A
weighted mean devalues the input from smaller
guideline companies, even though the company
being valued is commonly closer in size to them
than to the larger ones.
Regression Analysis
Rather than using a measure of central tendency
for multiples, some writers have used a regres-
sion analysis for valuation purposes.15 Regression
analyses are not useful unless there is a large
number of observations.16 This approach, like
the weighted mean, brings in an element of sub-
jective judgment: Which factors (e.g., revenues,
prot margins, growth rate, market value, and
payout ratio) should be considered in the regres-
sion analysis?
Some studies have found that the regression ap-
proach fails in empirical tests.
Finally, a regression-based approach to di-
rectly estimating valuation multiples does
14 Ian Cooper and Neophytos Lambertides, “Is There
a Limit to the Accuracy of Equity Valuation Using
Multiples?” (2014), p. 22, available at papers.ssrn
.com/sol3/papers.cfm?abstract_id=2291869.
15 See, e.g. Mark Filler, “Letter to the Editor,Business
Valuation Update, August. 2006, p. 20. For a dis-
cussion of the use of regression analysis (which is
outside the scope of this article), see, e.g. Aswath
Damodaran, Investment Valuation, 3rd edition (Wiley,
2012), pp..464-66, 562-69; Henschke and Homburg,
pp. 3-18; Sanjeev Bhojraj, Charles Lee, and David
Ng, “International Valuation Using Smart Multiples,
working paper, Cornell University 2003, available at
semanticscholar.org/paper/International-Valuation-
Using-Smart-Multiples-Bhojraj-Lee/0e8ce3d2ffdc87fe
905a3b25bb9193f19326f2c6.
16 Baker and Ruback, p. 2.
bvresources.com April 2021 Business Valuation Update 5
DO NOT USE THE ARITHMETIC MEAN TO AVERAGE MULTIPLES
Reprinted with permissions from Business Valuation Resources, LLC
not necessarily improve valuation accuracy....
Further analysis reveals that the relationship
between the nancial ratios and ... multiples is
nonlinear and hence, a linear regression model
leads to suboptimal results.17
Despite Its Merits, the Harmonic Mean
Is Not Widely Used
In 2016, Hitchner asked a group of valuation pro-
fessionals which averages they typically use for
multiples. The replies: 72% said the median, 31%
said the arithmetic mean, and only 16% said the
harmonic mean.18
Although average multiples are used in most
fairness opinions, a review of fairness opinions
on EDGAR shows that the median is used far
more often than the arithmetic mean and that
the harmonic mean of multiples is rarely used
(other than in fairness opinions by Bear Stearns,
which had used the harmonic mean since the
1970s).19
The harmonic mean for averaging multiples has
rarely appeared in published court decisions,
most likely because the expert witnesses did
not discuss the subject. The author’s Westlaw
search found only ve relevant cases. The har-
monic mean was accepted twice and rejected
thrice. A 1999 study that addressed Tax Court
cases posited that arithmetic means were the
17 Henschke and Homburg at 17. See also, e.g., Volker
Herrmann, Marktpreisschätzung mit kontrollierten
Multiplikatoren (Cologne: Josef Eul Verlag, 2002):
233.
18 James R. Hitcher, “Poll Results Reect Current Trends
in Business Valuation,Financial Valuation and
Litigation Expert (February-March 2017) at 6. Poll
taken Feb. 3, 2016.
19 The author was chairman of Bear Stearns’ Valuation
Committee, which was responsible for all fairness
opinions it issued from 1970 through 1995.
poorest method for averaging multiples and that
using reciprocals was the preferable method for
averaging multiples. 20 However, the Tax Court
has never discussed the harmonic mean.
Pratt wrote in 2001: “Although the harmonic
mean is not used frequently, probably because it
is unfamiliar to most readers of valuation reports,
it is conceptually a very attractive alternative
measure of central tendency.”21
Despite numerous academic studies since then
that demonstrate the harmonic mean’s superiori-
ty, valuation professionals, investment bankers, or
courts still do not widely use the harmonic mean.
Many valuators are unfamiliar with the concept.
Importantly, the harmonic mean is hardly ever
discussed or even mentioned in books on cor-
porate valuation.
Valuation practitioners should reject the use of
the arithmetic mean for averaging multiples.
Those who do not use the harmonic mean
should review the available literature and decide
whether they concur that it is the optimum
measure of central tendency for ratios with price
in the numerator.
Gilbert E. Matthews, CFA, is chairman emeritus
and a senior managing director of Sutter Securi-
ties Inc. (San Francisco). He has more than 50
years of experience in investment banking and
has spoken and written extensively on fairness
opinions, corporate valuations, and litigation re-
lating to valuations.
20 Randolph Beatty, Susan M. Riffe, and Rex
Thompson “The Method of Comparables and Tax
Court Valuations of Private Firms: An Empirical
Investigation,” 13 Accounting Horizons (Vol. 2) 177,
188-189 (1999).
21 Shannon P. Pratt, The Market Approach to Valuing a
Business (New York; John Wiley & Sons, 2001): 133.
... 27 However, this article bases its conclusion on questionable reasoning and a statistically flawed analysis. 28 Several other studies determined that the harmonic mean gave superior results. Pazarzi's 2014 study of multiples of UK companies concluded that the harmonic mean was more reliable than the median and far better than the arithmetic mean: [ 29 A 2017 article by Knudsen, Kold, and Plenborg reviewed multiples for the S&P 1500 and observed that ''we confirmed that averages based on the harmonic mean yield more accurate valuation estimates than do averages based on the [arithmetic] mean or median.'' ...
Article
Full-text available
This article posits that using the arithmetic mean to average multiples is mathematically inferior. A multiple is an inverted ratio with price in the numerator. The harmonic mean is a statistically sound method for averaging inverted ratios. It should be used as a measure of central tendency for multiples, along with the median. Empirically, the harmonic mean and the median of a set of multiples are usually similar. Because the harmonic mean can be overly affected by abnormally low multiples, the valuator must use judgment to exclude outliers.
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There is ongoing controversy in the business valuation literature regarding the preferability of the arithmetic mean or the harmonic mean when estimating ratios for use in business valuation. This research conducts a simulation using data reported from actual market transactions. Successive random samples were taken from data on valuation multiples and alternative measures of central tendency were calculated, accumulating more than 3.7 million data points. The measures (arithmetic mean, median, harmonic mean, geometric mean) were compared using hold-out sampling to identify which measure provided the closest approximation to actual results, evaluated in terms of least squares differences. Results indicated the harmonic mean delivered superior predictions to the other measures of central tendency, with less overstatement. Further, differences in sample size from 5 to 50 observations were evaluated to assess their impact on predictive performance. Results showed substantial improvements up to sample sizes of 20 or 25, with diminished improvements thereafter.
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We investigate biases of valuation methods and document that these depend largely on the choice of error measure (percentage vs. logarithmic errors) used to compare valuation procedures. We analyze four multiple valuation methods (averaging with the arithmetic mean, harmonic mean, median, and the geometric mean) and three present value approaches (dividend discount model, discounted cash flow model, residual income model). Percentage errors generate a positive bias for most multiples, and they imply that setting company values equal to their book values dominates many established valuation methods. Logarithmic errors imply that the median and the geometric mean are unbiased while the arithmetic mean is biased upward as much as the harmonic mean is biased downward. The dividend discount model dominates the discounted cash flow model only for percentage errors, while the opposite is true for logarithmic errors. The residual income model is optimal for both error measures.
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Comparison of the Residual Income and the Pricing Multiples Equity Valuation Models
  • Toby Tatum
Toby Tatum, "Harmonic Mean Value: The Appropriate Measure of Central Tendency," Business Appraisal Practice (3rd quarter 2011) 28, 31; Georgia Pazarzi, "Comparison of the Residual Income and the Pricing Multiples Equity Valuation Models," II International Journal in Economics and Business Administration, Issue 3, 88, 102 (2014);