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# The Superiority of the Harmonic Mean as a Method for Averaging Multiples of Guideline Companies and Acquisitions

Authors:
• Sutter Securities Financial Services, San Francisco

## Abstract

This article discusses why harmonic means are statistically superior to arithmetic means in averaging data with price in the numerator.
FINANCIAL VALUATION - The Market Approach to Value
two numbers is 0.06667; the reciprocal
of 0.6667 is 15x. We will show later in
of 20x would be misleadingly high for
valuation purposes.
The formula is:
H = n ÷ ((1 ÷ M1) + (1 ÷ M2) + (1 ÷ M3) +
… + (1 ÷ Mn))
Where:H = the harmonic mean;
n = the number of companies for
which multiples are computed; and
M1, M2, etc. = the guideline
companies' multiples.
HOW CAN THE HARMONIC
MEAN BE EASILY CALCULATED?
Before computers, the harmonic mean
Fortunately, Microsoft Excel® now
makes it quite easy to calculate the
harmonic mean of a set of multiples:
1) Click on either "Insert/Function" or the
"function" (fx) button on the toolbar.
2) Select the statistical function "HARMEAN."
3) Click "OK."
4) Highlight (or type in) the range of data
points (e.g., multiples) to be included in the
calculation.2
5) Click "OK" to get the harmonic mean of the
selected multiples.
For example, the formula in Excel® for the har-
monic mean of numbers in cell B3 and cells B6
through B13 is:
=HARMEAN(B3,B6:B13)
WHY IS THE HARMONIC MEAN
PREFERABLE AND WHEN IS IT
APPLICABLE TO VALUATIONS?
The harmonic mean calculates the av-
erage of the inverse of data points. All
multiples, such as a P/E ratio, have
price in the numerator. The basic
strength of the harmonic mean for av-
eraging multiples is that it produces a
value giving an equal weight to each
multiple. In contrast, an arithmetic
mean gives triple weight to a multiple
of 30x as compared to a multiple of
10x. The arithmetic mean over-
weights high multiples. Thus, when a
wide range of multiples exists, the di-
rect consequence will be an overvalu-
ation.
A median gives predominant
weight to the central data point. It
avoids the problem of overweighting
high multiples but, as shown below, it
ignores the dispersion of values in the
sample. Therefore it does not reflect
some of the information supplied by
the sample.
The harmonic mean should al-
ways be used instead of the arithmetic
mean when price is in the numerator,
as it is for multiples. However, when
GILBERT E. MATTHEWS, CFA
Sutter Securities Incorporated
San Francisco, CA
gil@suttersf.com
The Superiority of the Harmonic Mean
as a Method for Averaging Multiples of
Guideline Companies and Acquisitions
Average multiples are an important
input in valuations using guideline
companies and guideline acquisitions.
Most analysts use the arithmetic mean
and/or the median for determining
average multiples in their guideline
company and guideline transaction
valuations. The analyst, however,
should be aware that both of these
measures can give distorted results.
The arithmetic mean necessarily over-
states the central tendency of multi-
ples because it gives greater weight to
higher multiples.1This upward bias
can lead the analyst to overvalue com-
panies. The median does not have an
upward or downward bias, but it re-
lies on a single data point and thus
discards the information offered by all
of both of these measures can be
avoided by using a readily accessible
and superior measure of central ten-
dency: the harmonic mean.
that the harmonic mean is the most
accurate method for averaging multi-
ples, and we provide a simple method
by which to calculate and use this
measure. We have used the harmonic
mean for more than 30 years and have
written and spoken for more than 20
years about the preferability of using
we continue to believe that the valua-
tion community should adopt the har-
monic mean for averaging multiples.
WHAT IS THE HARMONIC
MEAN?
The harmonic mean is a measure of
central tendency that is calculated by
averaging the reciprocals of the data
points. For example, the harmonic
mean of multiples of 10x and 30x is
15x. It is calculated by averaging 0.1
(the reciprocal of 10) and 0.03333 (the
reciprocal of 30). The average of these
Continued on next page
Page 12 • June/July 2008 • FVLE Issue 13
expert
TIP
The harmonic mean is statisti-
cally superior to the arithmetic
mean for averaging multiples,
for the data set than does the
median.
FVLE Issue 12 • April/May 2008 • www.valuationproducts.com • Page 13
FINANCIAL VALUATION - The Market Approach to Value
Continued on next page
averaging yields or any other ratio
where price is the denominator, the
arithmetic mean is appropriate and
the harmonic mean should not be
used. A dividend yield is a divi-
dend/price ratio. If we used the
“earnings yield”3(the earnings/price
ratio), we would calculate the average
earnings yield with the arithmetic
mean. The harmonic mean of P/E ra-
tios (i.e., price in the numerator) is an
equivalent to an arithmetic mean of
the earnings/price ratios.
HIGH MULTIPLES INFLATE THE
ARITHMETIC MEAN
Arithmetic means of multiples such as
P/E ratios and ratios of total capital-
ization to EBITDA give greater weight
to higher multiples. Mathematically,
each guideline company's market
multiple is weighted by the magni-
tude of the multiple itself, so that the
arithmetic mean causes overvalua-
tions. The harmonic mean, which
gives equal weight to each guideline
company's multiple, is statistically su-
perior as a measure of central tenden-
cy of multiples. This can be demon-
strated by the following calculations.
Let us assume that the sample
consists of five similar companies
with EPS of \$2.50, \$2.00, \$1.60, \$1.00
and \$0.80, respectively, each having a
market price of \$20 per share. Exhibit
A calculates the arithmetic mean of
the five P/E ratios (15.1x) and Exhibit
B calculates the harmonic mean of the
same P/E ratios (12.7x). Given the
higher weight of the higher multiples,
the arithmetic mean is higher.
Why is the harmonic mean a
better measure of central tendency for
valuations? The superiority of the
harmonic mean over
the arithmetic mean can
be shown by hypotheti-
cal calculations com-
paring two investment
approaches: (1) an in-
of shares that would
give him/her an equal
amount of earnings in
each company and (2)
an investor makes an
equal dollar investment in each com-
pany. We assume a group of five com-
panies, with EPS of \$0.80, \$1.00, \$1.60,
\$2.00 and \$2.50, respectively, each sell-
ing at \$20 per share (same as prior ex-
ample). Exhibit C summarizes the
EXHIBIT B
P/E ratio Reciprocal
8.0x 0.125
10.0x 0.100
12.5x 0.080
20.0x 0.050
25.0x 0.040
Mean of reciprocals 0.079
Harmonic mean =
1 ÷ 0.079 = 12.7x
purchases in the hypothetical situa-
tion where the investor buys the num-
ber of shares that represent \$1,000 of
earnings.
In this scenario, the portfolio
cost would be \$75,500 and the P/E
ratio of this portfolio (\$75,500 ÷
EXHIBIT A
Price EPS P/E ratio
\$20.00 \$2.50 8.0x
\$20.00 \$2.00 10.0x
\$20.00 \$1.60 12.5x
\$20.00 \$1.00 20.0x
\$20.00 \$0.80 25.0x
Arithmetic mean 15.1x
\$5,000) would be 15.1x. This is the
same 15.1x that is the arithmetic mean
of the P/E ratios in Exhibit A.
It is unlikely, however, that an
investor would weight a portfolio in
this manner. The more rational ap-
proach— investing an equal amount
in each company— is shown in Exhib-
it D. It shows the result of 20 percent
of the same \$75,500 in shares of each
of the five companies.The P/E ratio of
this portfolio (\$75,500 ÷ \$5,964) would
be 12.7x. This is the same 12.7x calcu-
lated as the harmonic mean in Table B.
This confirms that the harmonic mean
gives an equal weight to an equal dol-
lar investment in each company.
These two examples show that the
harmonic mean more accurately re-
flects reasonable portfolio investment
practice. No rational investor would
have a policy of investing proportion-
ally larger amounts in higher-multi-
ple companies. These ex-
amples demonstrate why
the harmonic mean is a
better measure of central
tendency than the arith-
metic mean. The median
omits useful information
by giving all the weight
to the midpoint.
Many analysts prefer
the median to the arith-
metic mean. The median
(the middle value of a set
of data points) is useful for large sets
of multiples, but is less reliable for
small sets. The writer has found that
medians of large sets of multiples are
lower than the arithmetic mean in
most cases, and that the medians tend
EXHIBIT D
Market price EPS Shares Cost Earnings
\$20.00 \$2.50 755 \$15,100 \$1,888
\$20.00 \$2.00 755 \$15,100 \$1,510
\$20.00 \$1.60 755 \$15,100 \$1,208
\$20.00 \$1.00 755 \$15,100 \$755
\$20.00 \$0.80 755 \$15,100 \$604
3,775 \$75,500 \$5,964
EXHIBIT C
Market price EPS Shares Cost Earnings
\$20.00 \$2.50 400 \$8,000 \$1,000
\$20.00 \$2.00 500 \$10,000 \$1,000
\$20.00 \$1.60 625 \$12,500 \$1,000
\$20.00 \$1.00 1000 \$20,000 \$1,000
\$20.00 \$0.80 1,250 \$25,000 \$1,000
3,775 \$75,500 \$5,000
FINANCIAL VALUATION - The Market Approach to Value
to be much closer to harmonic means.
For example, in Exhibits A and B pre-
viously, the median of 12.5x is close to
the harmonic mean of 12.7x and far-
ther from the arithmetic mean of
15.1x.
There is no question that the
median is a better measure of the cen-
tral tendency of multiples than the
arithmetic mean if there are enough
guideline companies to give a mean-
ingful result. However, the median
suffers from excluding information
from the remaining multiples. There-
fore, the median may not be fully re-
flective of the underlying data and
can be misleading as a measure of
central tendency. The harmonic mean
more accurately captures the disper-
sion of the multiples in a sample and
therefore can lead to more reliable
valuations.
The following three tables
demonstrate this weakness in the me-
dian. Exhibit E shows that three sam-
ples with the same median can con-
tain different data points, and there-
fore have different harmonic (and
arithmetic) means.
Exhibit F uses the same sets of
multiples except that the midpoints
are 15.0x instead of 12.0x. If we com-
pare Exhibit F to Exhibit E, we ob-
serve that the median goes up by 3.0x,
while the means go up approximate-
ly 0.6x. The only information in the
median is that half the data points are
higher and half are lower; the amount
by which the other multiples
are higher or lower has no ef-
fect on the median.
Exhibit G shows that if
we take only the data in Set 1A
and change the third line from
12.0x to 11.0x, 14.0x and 17.0x,
and calculate columns 1C, 1D
and 1E, the median goes up by
3.0x in each column, while the
Again, we get no information
from the higher and lower
multiples.
THE HARMONIC
MEAN’S PREFERABILI-
TY HAS BEEN SUP-
PORTED BY VARIOUS
STUDIES
bility of the harmonic
mean for averaging multi-
ples in valuations was pre-
sented in a 1990 publica-
tion,4valuation literature
seldom discussed the har-
monic mean until recently.
have now highlighted the
merits of using the har-
monic mean for averaging
multiples.5A 1999 work-
ing paper by Malcolm
Baker and Richard Ruback
School compared applica-
tions of the arithmetic
mean, the harmonic mean,
and the median to multi-
ples. After reviewing
multiples in 22 industries,
they found that the har-
monic mean had the smallest mini-
mum variance. They concluded that
the harmonic mean was the best way
to average multiples and that it was
superior to the median for that pur-
pose. They also confirmed that values
were consistently overestimated
when the arithmetic mean was used.
In 2002, UCLA and Columbia
scholars performed a series of analy-
ses and determined that the harmonic
mean was a superior measure.6They
stated that “performance improves
when [average] multiples are comput-
ed using the harmonic mean, relative
to the [arithmetic] mean or median
ratio of price to value driver for com-
parable firms.”7They also concluded
that the “performance of median mul-
tiples is worse than for harmonic
mean multiples.”8The same authors
again used harmonic means in a 2007
article, pointing out that the harmonic
mean mitigates the problem caused
by the impact of high multiples on
arithmetic means.9A European study
in 2005 also supported and utilized
the harmonic mean, stating that "the
harmonic mean leads to more accu-
rate forecasts [of value] than the arith-
metic mean or the median.”10
CONCLUSION
Some experts have said that the har-
monic mean is not used more fre-
quently because most valuators, as
well as the parties who engage them,
are unfamiliar with it and its merits.
Page 14 • June/July 2008 • FVLE Issue 13
Continued on page 20
EXHIBIT G Set 1C Set 1D Set 1E
9.0x 9.0x 9.0x
11.0x 11.0x 11.0x
11.0x 14.0x 17.0x
17.0x 17.0x 17.0x
22.0x 22.0x 22.0x
Median 11.0x 14.0x 17.0x
Harmonic mean 12.6x 13.2x 13.7x
Arithmetic mean 14.0x 14.6x 15.2x
EXHIBIT F Set 1B Set 2B Set 3B
9.0x 8.0x 11.0x
11.0x 11.0x 12.0x
15.0x 15.0x 15.0x
17.0x 16.0x 20.0x
22.0x 19.0x 25.0x
Median 15.0x 15.0x 15.0x
Harmonic mean 13.4x 12.3x 15.1x
Arithmetic mean 14.8x 13.6x 16.6x
EXHIBIT E Set 1A Set 2A Set 3A
9.0x 8.0x 11.0x
11.0x 11.0x 12.0x
12.0x 12.0x 12.0x
17.0x 16.0x 20.0x
22.0x 19.0x 25.0x
Median 12.0x 12.0x 12.0x
Harmonic mean 12.8x 11.8x 14.4x
Arithmetic mean 14.2x 13.0x 16.0x
MATTHEWS - Continued
Page 20 • June/July 2008 • FVLE Issue 13
Shannon Pratt wrote, “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 alterna-
tive measure of central tendency.”11
Unfamiliarity is an unfortunate
but understandable reason for the
harmonic mean’s failure to become
more widely accepted As experi-
enced valuators know, the discounted
cash flow method was unfamiliar
three decades ago, but it has since be-
come now widely accepted. Similarly,
the “levels of value” concept was un-
familiar a generation ago and is now
accepted. As discussed above, the
harmonic mean is statistically superi-
or to the arithmetic mean for averag-
mation for the data set than does the
median. It is a more accurate forecast-
er of value. By setting forth the ad-
vantages of the harmonic mean in val-
uations, we are hopeful that valuation
analysts will be encouraged to adopt
it and the courts to expect it. F
1See Shannon P. Pratt, Valuing a Business, Fifth Edition
(McGraw Hill, 2008), p. 292.
2You can insert numbers from an additional group of
data points by typing a comma and then highlighting or
typing in the additional data points.
3This term was commonly used in the U.K. prior to the
1970s. Earnings yields for London Stock Exchange
companies were included in the Financial Times’ daily
stock tables.
4 Gilbert E. Matthews and M. Mark Lee, “Fairness Opin-
ions & Common Stock Valuations,” in The Library of
Investment Banking, R. Kuhn, ed. (Dow Jones Irwin,
1990), pp. 405-407.
5Malcolm Baker and Richard Ruback, “Estimating In-
dustry Multiples,” Working paper, Harvard Business
School (6/11/99), http:///www.people.hbs.hbs.edu/
mbaker/cv/papers/Multiple.pdf. This paper also exam-
ined the value-weighted mean (an approach that is not
appropriate for most business valuations) and found it
inferior to the harmonic mean.
6Jing Liu, Doron Nissim and Jacob Thomas, “Equity Val-
uation Using Multiples,” Journal of Accounting Re-
search, March 2002, pp. 137-172.
7Id., p. 137. “Performance” refers to the ability of an av-
erage to predict market price given a particular value
driver. The “value drivers” they used included, among
others, revenues, EBITDA, free cash flow, and histori-
cal and projected EPS.
8Id., p. 160.
9Liu, Nissim and Thomas, “Is Cash Flow King in Valua-
tions?” Financial Analysts Journal, March/April 2007, p.
2.
10 Ingolf Dittmann and Christian Weiner, “Selecting Com-
parables for the Valuation of European Firms.” SFB
649 Discussion Paper 2005, Humboldt University,
Berlin, p. 2.
11 Pratt, The Market Approach to Valuing a Business
(Wiley, 2001), p. 133.
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Chapter
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This paper also examined the value-weighted mean (an approach that is not appropriate for most business valuations) and found it inferior to the harmonic mean
• Malcolm Baker
• Richard Ruback
Malcolm Baker and Richard Ruback, "Estimating Industry Multiples," Working paper, Harvard Business School (6/11/99), http:///www.people.hbs.hbs.edu/ mbaker/cv/papers/Multiple.pdf. This paper also examined the value-weighted mean (an approach that is not appropriate for most business valuations) and found it inferior to the harmonic mean.
Performance" refers to the ability of an average to predict market price given a particular value driver. The "value drivers" they used included, among others, revenues, EBITDA, free cash flow, and historical and projected EPS
• Id
Id., p. 137. "Performance" refers to the ability of an average to predict market price given a particular value driver. The "value drivers" they used included, among others, revenues, EBITDA, free cash flow, and historical and projected EPS.
The Market Approach to Valuing a Business
• Pratt
Pratt, The Market Approach to Valuing a Business (Wiley, 2001), p. 133.