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When Averaging Multiples, Apply the Harmonic Mean

Authors:
  • Sutter Securities Financial Services, San Francisco
Article

When Averaging Multiples, Apply the Harmonic Mean

Abstract

This article discusses why harmonic means are statistically superior to arithmetic means in averaging data with price in the numerator.
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When Averaging Multiples,
Apply the Harmonic Mean
By Gilbert E. Matthews, CFA*
Editor’s note: This article lls a potential gap in
methods for averaging multiples. It responds to
an abstracted article by Jim Hitchner (see BVU,
April 2006), which considered the median and
the arithmetic mean—but did not address the
harmonic mean.
When using the guideline company and guideline
transaction approaches, it is critical to select an appro-
priate method for averaging multiples. The arithmetic
mean and the median may be the most common mea-
sures that analysts use in their valuations—but the
harmonic mean is a better measure. The arithmetic
mean has a built-in upward bias that overstates the
central tendency when averaging multiples, and the
median can often discard useful information.
What is the harmonic mean?
To calculate the harmonic mean, average the re-
ciprocals of ratios, as follows:
n
H
=
n
(1/mi)
i=1
Where:
H = the harmonic mean;
n = the number of companies for which ratios are
computed; and
m = the multiple of a guideline company.
* Gil Matthews is currently Chairman and Senior Managing Direc-
tor at Sutter Securities Incorporated. (San Francisco); from 1970
– 1995, he directed the nationwide fairness opinion practice at
Bear Stearns, where he initiated the use of the harmonic mean.
With respect to multiples of earnings, the harmonic
mean uses the inverse of the price/earnings (P/E)
ratio: It calculates an average based on an earnings/
price ratio. In the 1960’s, the Financial Times called
this ratio the “earnings yield,” and included it in its
daily London Stock Exchange tables. However, when
price is in the denominator—as in a dividend yield,
the arithmetic mean is appropriate; dividend yield is
simply the dividend/price ratio. The harmonic mean
is applicable to multiples, not to yields, and should be
used when price is in the numerator.
Arithmetic mean is biased upward
Valuators commonly calculate the average of mul-
tiples such as P/E ratios or the ratios of aggregate
market value (AMV) to EBITDA or EBIT by using the
arithmetic mean, the median, or both. The arithmetic
mean of ratios with prices in the numerator always
gives greater weight to higher multiples in the sample
than to lower multiples. It mathematically weights
each guideline company’s market multiple in propor-
tion to the magnitude of the multiple. A company with
a P/E ratio of 20x has twice the weight in the average
than a company with a P/E of 10x. Statistically, the
harmonic mean is a superior measure because it gives
equal weight to each of the guideline companies.
The following example illustrates the upward bias
of the arithmetic mean and the advantage of the
harmonic mean. Assume that four companies each
have a stock price of $30 per share and have earnings
per share (EPS) of $3.00, $2.50, $1.20, and $0.50,
respectively. Table 1 shows that the arithmetic mean
of the P/E ratios of the four companies is 26.8x.
Reprinted with permission from Business Valuation Resources, LLC.
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FMV Opinions, Inc.—Irvine, Calif.
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CPA/ABV, ASA
The Financial Valuation Group—Atlanta,
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McDermott, Will & Emery—Chicago, Ill.
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Maurice Jay Kutner & Associates,
P.A.—Miami, Fla.
GILBERT E. MATTHEWS,
CFA
Sutter Securities Incorporated—San
Francisco, Calif.
JOHN W. PORTER
Baker & Botts, LLP—Houston, Texas
JAMES S. RIGBY, ASA,
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Financial Valuation Group—Los Angeles,
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School of Accountancy, DePaul
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Continued to next page...
Table 2
P/E ratio Reciprocal
10.0x 0.1000
12.0x 0.0833
25.0x 0.0400
60.0x 0.0167
Mean of reciprocals 0.0600
Harmonic mean (1÷0.06) 16.7x
Ass ume that an investor ele cted to purchase shares
representing $600 of earnings of each company.
Table 3 summarizes the purchases.
Table 3
Market
price EPS Shares
bought Cost Earnings
bought
$30.00 $3.00 200 $6,000 $600
$30.00 $2.50 240 $7,200 $600
$30.00 $1.20 500 $15,000 $600
$30.00 $0.50 1,200 $36,000 $600
2,140 $64,200 $2,400
The total cost of the portfolio would be $64,200 for
shares with aggregate earnings of $2,400. The P/E
multiple of this portfolio ($64,200 ÷ $2,400) would be
26.8x, i.e., the arithmetic mean of the P/E ratios. The
arithmetic mean awards a 60x multiple six times the
weight awarded to a multiple of 10x. However, no
rational investor would weight a portfolio by allocating
six times more dollars to a high-multiple stock than to
a low-multiple stock.
Alternatively, assume that an investor elected to
invest $64,200 by buying an equal dollar value of
shares of each company, a more rational investment
strategy. Table 4 summarizes this portfolio.
Table 1
Market price EPS P/E ratio
$30.00 $3.00 10.0x
$30.00 $2.50 12.0x
$30.00 $1.20 25.0x
$30.00 $0.50 60.0x
Arithmetic mean 26.8x
Table 2 calculates the reciprocals of the multiples,
showing that the harmonic mean is 16.7x.
When Averaging Multiples
...continued from previous page
Reprinted with permission from
Business Valuation Resources, LLC.
june 2006 Business Valuation Update 3
Table 4
Market
price EPS Shares
bought Cost Earnings
bought
$30.00 $3.00 535 $16,050 $1,605.00
$30.00 $2.50 535 $16,050 $1,337.50
$30.00 $1.20 535 $16,050 $642.00
$30.00 $0.50 535 $16,050 $267.50
2,140 $64,200 $3,852.00
The total cost of the investments would be $64,200
for $3,852 of aggregate earnings. The P/E multiple
of this portfolio ($64,200 ÷ $3,852) would be 16.7x,
precisely equal to the harmonic mean. The harmonic
mean gives an equal weight to an equal dollar invest-
ment in each company. The harmonic mean of 16.7x
is clearly a better measure of the average P/E ratio of
this group of guideline companies than the arithmetic
mean of 26.8x.
Median omits useful data
In recent years, many valuators have preferred
using the median to measure the central tendency
of multiples. The median, which is the middle value
of a sample, is useful for large samples, but is less
reliable for small samples. Experience reveals that
medians of large samples of multiples are almost
always lower than the arithmetic mean and tend to
be closer to the harmonic mean. For example, in the
Table 1, the median is 18.5x, the arithmetic mean is
26.8x, and the harmonic mean is 16.7x.
Although the median is a better measure of central
tendency than the arithmetic mean, it effectively elimi-
nates the information in the remaining multiples. Table
5 shows that two samples with the same median can
contain different values, and therefore have different
ha rmoni c means. The harmo nic mean bet ter capt ure s
the variety of the multiples in the sample.
Table 5
Sample A Sample B
9.0x 13.0x
10.0x 14.0x
13.0x 15.0x
15.0x 15.0x
18.0x 20.0x
20.0x 23.0x
22.0x 26.0x
Median
15.0x 15.0x
Harmonic mean
13.8x 16.9x
Why is the harmonic mean not widely used?
Dif culty of computation is a reason analysts often
cite for not using the harmonic mean. Prior to personal
computers, the harmonic mean was cumbersome
to calculate. Now, however, Microsoft Excel helps
calculate the harmonic mean. After clicking either on
the fx button on the toolbar or on “Insert/Function,”
you select the statistical function “HARMEAN,” and
then select the range of data points to include in the
calculation. The resulting number is the harmonic
mean of the selected data points.
Another reason that analysts may not use the har-
monic mean more frequently is that most valuators
(and recipients of valuations) are not familiar with the
measure and its merits. In The Market Approach
to Valuing a Business (2001), 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
alternative measure of central tendency.” But unfa-
miliarity is no reason to disregard a methodology;
30 years ago, the discounted cash ow method was
Continued to next page...
When Averaging Multiples
...continued
Reprinted with permission from Business Valuation Resources, LLC.
Business Valuation Update june 2006
4
unfamiliar—and yet now most analysts widely accept
it, as do the courts.
Until 1999, valuation literature seldom discussed
the harmonic mean.1 Recently, however, several aca-
demic studies have highlighted the merits of using the
harmonic mean for averaging multiples. The landmark
study is a 1999 working paper by Malcolm Baker and
Richard Ruback which compared applications of the
arithmetic mean, the harmonic mean, and the median
to multiples.2 (It also examined the value-weighted
mean, an approach that is not appropriate for most
business valuations.) After reviewing multiples in
22 industries, Baker and Ruback concluded that the
harmonic mean: (a) had the smallest minimum vari-
ance; (b) was the best way to average multiples; and
(c) was superior to the median; they also found that
the arithmetic mean consistently overestimated value.
In addition, several later studies have buttressed the
use of the harmonic mean for averaging multiples.3
Use of harmonic means in practice
Investment advisors such as Van Kampen Invest-
ments (Oakbrook Terrace, IL), apply harmonic means
in their analyses, as do investment bankers in arriving
at fairness opinions. A recent search of investor’s
methodologies on the SEC’s EDGAR database using
When Averaging Multiples
...continued from previous page
10k Wizard (www.tenkwizard.com) found numerous
fairness opinions which used the harmonic mean.
For example, Bear Stearns & Co., Inc. consistently
relied on harmonic means in its fairness opinions,
and at least ten other rms utilized harmonic means
in their analyses.
In sum, the harmonic mean is statistically more
accurate than the arithmetic mean to measure the
central tendency of ratios with price in the numerator,
and it is more informative than the median. The valu-
ation community should adopt the practice of using
the harmonic mean for averaging multiples.
1 But see, Gilbert Matthews and Mark Lee, “Fairness Opinions &
Common Stock Valuations,” in The Library of Investment Banking,
R. Kuhn, ed. (Dow Jones Irwin, 1990), pp. 405-407.
2 Baker and Ruback, “Estimating Industry Multiples,” Working
paper, Harvard Business School (6/11/99); www.people.hbs.hbs.
edu/mbaker/cv/papers/Multiple.pdf
3 See, e.g., Ingolf Dittmann and Christian Weiner, “Selecting
Comparables for the Valuation of European Firms.” SFB 649 Dis-
cussion Paper 2005-002 (2/10/2005), http://141.20.100.9/papers/
pdf-/SFB649DP2005-002.pdf; Jing Liu, Doron Nissim and Jacob
Thomas, “Equity Valuation Using Multiples,” Journal of Accounting
Research, March 2002, pp. 137-172; Randolph Beatty, Susan
Riffe and Rex Thompson, “The Method of Comparables in Tax
Court Valuations of Privately-Held Firms: An Empirical Investiga-
tion,” Accounting Horizons, September 1999, p. 177-199.
Reprinted with permission from Business Valuation Resources, LLC.
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  • Ruback Baker
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