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From the developers of Pratt’s Stats™

Timely news, analysis, and resources for defensible valuations Excerpt from Vol. 12, No. 6, June 2006

BUSINESS

VA

LUATION

UPDATE™

BVR

What It’s Worth

Continued to next page...

When Averaging Multiples,

Apply the Harmonic Mean

By Gilbert E. Matthews, CFA*

Editor’s note: This article ﬁ 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.

Business Valuation Update june 2006

2

Managing Editor: Sherrye Henry, Esq.

Associate Editors: Paul Heidt & Adam Manson

Publisher: Doug Twitchell

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President: David Foster

Customer Service: Pam Pittock

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Publisher Emeritus:

Shannon Pratt

RONALD D. AUCUTT, Esq.

McGuireWoods LLP—McLean, Va.

JOHN A. BOGDANSKI, JD

Lewis & Clark Law School—Portland, Ore.

HON. WILLIAM A.

CHRISTIAN

N.C. 11th Judicial District Court—Sanford,

N.C.

S. STACY EASTLAND, Esq.

The Goldman Sachs Group,

Inc.—Houston, Texas

BARNES H. ELLIS, Esq.

Stoel Rives LLP—Portland, Ore.

NANCY J. FANNON, ASA,

CPA/ABV, MCBA

Fannon Valuation Group—Portland, Me.

JAY E. FISHMAN, ASA, CBA

Financial Research Associates—

Philadelphia, Pa.

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newsletter may be reproduced without express written consent from BVR.

BUSINESS VALUATION UPDATE

LARRY WELDON GIBBS, JD

Gibbs Professional Corporation—San

Antonio, Texas

LYNNE Z. GOLD-BIKIN, Esq.

Wolf, Block, Schorr & Solis-Cohen,

LLP—Norristown, Pa.

LANCE S. HALL

FMV Opinions, Inc.—Irvine, Calif.

JAMES R. HITCHNER,

CPA/ABV, ASA

The Financial Valuation Group—Atlanta,

Ga.

JARED KAPLAN, Esq.

McDermott, Will & Emery—Chicago, Ill.

MAURICE KUTNER, Esq.

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,

CPA/ABV

Financial Valuation Group—Los Angeles,

Calif.

ARTHUR D. SEDERBAUM,

Esq.

Patterson, Belknap, Webb & Tyler—New

York, N.Y.

DONALD S. SHANNON,

Ph.D., CPA

School of Accountancy, DePaul

University—Chicago, Ill.

BRUCE SILVERSTEIN, Esq.

Young, Conaway, Stargatt & Taylor,

LLP—Wilmington, Del.

GEORGE S. STERN, Esq.

Stern & Edlin—Atlanta, Ga.

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?

Difﬁ 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 ﬂ 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 ﬁ 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.