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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,

prot 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 Reect 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.