n. 586 January 2017
ISSN: 0870-8541
The Method of
Market Multiples on the Valuation of Companies:
A Multivariate Approach
1
José Couto
1,2
Paula Brito
1
António Cerqueira
1
FEP-UP, School of Economics and Management, University of Porto
2
LIAAD/INESC TEC
The Method of Market Multiples
on the Valuation of Companies: A Multivariate Approach
José Couto
Faculdade de Economia, Universidade do Porto, Portugal
[email protected]
Paula Brito
Faculdade de Economia & LIAAD INESC TEC, Universidade do Porto, Portugal
[email protected]
António Cerqueira
Faculdade de Economia, Universidade do Porto, Portugal
[email protected]
Dezembro, 2016
Abstract. The main goal of this study is to investigate, using multivariate analysis, the
possibility of defining comparable firms as those with economic and financial characteristics
closest to the company under evaluation, rather than adopting the "same industry" criterion, and
thereby improve the estimation errors when the multiples valuation process is used to estimate
the enterprise value and the market capitalization of a company. The analysis is performed
running formal tests to compare mean values of the distributions of errors.
The results obtained using cluster analysis reveal that considering comparable companies as
those with economic and financial ratios closer to the company under evaluation generally
reduces the mean of the estimation errors for almost all groups of ratios considered. For those
groups for which the improvement is not significant, the results are similar to those obtained
using the industry membership criterion.
Keywords: Cluster Analysis; Estimation Errors; Relative Valuation; Method of Multiples;
Market Multiples Classification-JEL: G32; G12; G14, C38
1
1 Introduction
Multiples are an important tool used by many analysts, investors, researchers and other public
interested in the valuation of assets or generally interested in the stock market. Despite the solid
and extensive literature on valuation methodologies such as the Dividend Cash Model (DDM)
or the Discounted Cash Flow (DCF), multiples are frequently used to translate the results of
such methodologies into intuitive figures (implied multiples), in combination with those
acknowledged methods (on the perpetuity of those models) or as an alternative to estimate the
value of a company in an easier and faster way. Among professionals, multiples are already an
accepted tool, but in the academic world they are still considered a subjective and understudied
approach, which means that their coverage in the financial analysis courses is limited, what
ultimately threatens its credibility (Bhojraj & Lee, 2002, p. 408).
Multiples appear frequently in all kinds of valuation reports, on fairness opinions documents,
on business newspapers and websites - they even appear in some M&A offers. Their widespread
use can be attributed to their simplicity (Schreiner, 2007a, p. 1). A multiple is simply a ratio,
obtained dividing the market or estimated value of an asset by a specific item of the financial
statements or other measure. Multiples are thus easier to explain to clients by the professionals
than the fundamental analysis methods. However, this apparent simplicity is quite illusory, as
all the explicit assumptions needed during the fundamental analysis are still implicitly
synthetized in the multiples, such as the risk, growth, potential cash-flows as well as the market
mood.
The method of multiples, also known as the four-step process, consists in the following: 1)
select a sample of comparable companies; 2) choose and compute a multiple for those
comparables; 3) aggregate those multiples into a single figure using a central statistics, such as
the mean, the median, the harmonic mean or the geometric mean; 4) apply the aggregated
multiple of comparables to the corresponding value of the firm under analysis in order to
estimate its value. Each of these steps raises a complex issue that requires a decision in order
to be implemented.
This study is motivated by the idea that it is possible to rely on the proximity of the economic
and financial characteristics, rather that the “same industry” criterion, in order to select a set of
comparables (1st step). We also study the impact of choosing among different multiples (2nd
step) as well as the impact of the aggregation measure (3rd step).
2
In order to structure our research we address the following questions: Q1: What ratios are
closely associated with each multiple?; Q2: What is the best measure to aggregate the
information of each multiple (mean, median, harmonic mean or geometric mean)?; Q3: Does
the adoption of the closest financial characteristics criterion improve the estimation errors when
compared to the same-industry criterion?
We tackle the first issue studying the correlation coefficients between ratios and multiples. The
second and third issues are approached comparing the valuation errors under the different
calculation procedures.
In the next section we examine the literature review, linking it with the issues related to each of
the four steps mentioned above. The third section explores the methodology and the data
building process. The fourth section presents the empirical results and the fifth and last one
brings together the findings of this work.
2 Literature Review
The literature concerning multiples is scarce and very fragmented in its findings. A broad and
consistent over time study has not been done yet. The different focus on different multiples (e.g.
P/E vs PBV) and the different assumptions on the operationalization of the four-step process
(e.g. the choice among different aggregation measures), make the comparison of results
difficult. Therefore an important work, in order to standardize the methodological process of
carrying out these studies, or, alternatively, a theoretical framework that allows understanding
the impact of such changes on the results, is still lacking.
Choosing the multiple: Kim & Ritter (1999), studying multiples on IPOs valuations in the US
between 1992 and 1993, conclude that forward-looking P/E multiples outperform historical P/E
multiples. They also find that estimation errors are smaller for older companies than for young
companies (less than 10 years). Liu, Nissim, & Thomas (2002) find also that forward P/E
multiples perform better than trailing P/E, cash-flows measures (EBITDA, CFO) and PBV are
tied in third place, sales achieves the worst place. The finding that both P/E outperform cashflow measures is contrary to the belief presented in some standard books, CFO (Cash-Flow
from Operations) performing considerably worse than EBITDA.
Herrmann & Richter (2003), for a sample of European and US firms, investigate the accuracy
of a set of multiples concluding that, for the non-financial-services firms, P/E is a much better
multiple than all the other investigated multiples if they are not controlled for growth and
profitability. Controlling comparables for those factors instead of using the same SIC code,
3
improves the accuracy of multiples, which may be ranked as follows: P/E, EV/EBIAT, PBV,
EV/EBIDAAT, EV/TA e EV/S.
Schreiner (2007a) examining a set of companies from the DJ Stoxx 600 (Europe) and the SP&P
500 (US) finds that equity value multiples outperform entity multiples, knowledge multiples
(created by the author) outperform traditional multiples and two-year forward P/E multiple
outperform trailing multiples. He also suggests that the findings regarding the best multiple
depend on the set of companies: for the European companies the two-year forward P/EBT
multiple ranks first and the one-year forward P/E ranks second, while for the US companies the
two-year forward P/E ranks first and the one-year forward P/EBT follows it – this may occur
due to different corporate tax laws in Europe, according to the author.
Choosing comparables: Alford (1992) who is one the first authors to study this subject,
examines the accuracy of the P/E multiple when comparables are chosen on the basis of SIC
codes, size (proxy for risk) and return on equity (proxy for growth). He finds that using a threedigit SIC code to select comparables is preferable to a broader code but no improvement occurs
when the four-digit code is chosen. Choosing comparables based on risk and growth together
perform similarly well but using those variables separately does not perform well. The author
also concludes that further controls on the industry membership such as size, growth or leverage
(using the EV/EBIT) do not improve prediction errors significantly. Kim & Ritter (1999)
conclude that investment bankers are able to improve the valuation accuracy of P/E multiples
selecting comparables than just automatically using the same industry SIC code.
Bhojraj & Lee (2002) study the possibility of selecting comparables using a multiple regression
approach based on underlying economic variables, in order to attribute a warranted multiple to
each company. These warranted multiples are then used to select comparables as those with the
closest warranted multiple. They conclude that this method improves the prediction errors
comparing to the industry and size matches. This technique is used for the EV/S and PBV
multiples but the best set of comparables is not necessarily the same for both multiples, this is
an important finding for our study as we shall see. Dittmann e Weiner (2005) investigate the
comparables selection method when using EV/EBIT multiple to estimate the value of
companies, finding that selecting comparables based on similar return on assets clearly
outperforms a selection based on industry membership (preferably the same four-digit SIC
code) or total assets. These authors study if the set of comparables should be picked from the
same country, region or from all OECD countries, concluding that for most 15 EU countries
4
comparables should be selected from the same region, except for the UK, Denmark, Greece and
the US where comparables should be selected only from the same country.
Herrmann & Richter (2003) consider comparables as those that deviate less than 30% from
certain control factors concluding that this approach is a better method instead of using the SIC
classification. Those factors are derived from valuation models for the following multiples: P/E
(factors: roe and earnings growth), EV/EBIAT (factors: roic and earnings growth), P/B (factors:
roe and earnings growth), EV/TA (factors: roic and earnings growth), EV/S (factors: EBIAT/S,
S/IC and earnings growth) and EV/EBIDAAT (EBIAT/EBIDAAT and EBIDAAT/IC). This
finding suggests the SIC code approach does not contain superior information to that controlled
using derived factors. An alternative regression approach to P/E and PBV multiples using the
above factors does not improve the accuracy.
Cooper & Cordeiro (2008) investigate the effect of increasing the number of comparables on
the accuracy of the forward P/E multiple. They discover that using a selection rule based on the
proximity of the expected earnings growth, ten companies are enough on average to deliver the
same accuracy as using the entire set from the same industry. They suggest that it is better to
use a small number of comparables with closest growth rates than to use the entire set; more
firms introduce on average more noise.
The aggregation measure: Studies performed by Liu, Nissim, & Thomas (2002) and Baker &
Ruback (1999) suggest that the harmonic mean is the best central tendency measure to adopt
on valuation multiples. However, Herrmann & Richter (2003) disagree with this view
suggesting the median as the best aggregation measure, mainly when we deal with a
heterogeneous sample. These latter authors argue that in homogeneous samples the harmonic
mean leads to similar results than the median but in heterogeneous samples the harmonic mean
regularly underestimates the company’s value. The arithmetic mean is presented as a poor
aggregation measure in all examined studies, leading consistently to the overestimation of
firm’s value due to the right skewed nature of multiples distributions.
Combination of multiples: Cheng & McNamara (2000) examine the accuracy of P/E and PBV
multiples separately and a combination of both. They find that for both multiples using the same
SIC classification combined with the ROE is the best method to select comparables but if a
combined P/E-PBV is computed, then the same industry membership is enough. This P/E-PBV
method (computed using equal weights) performs better than P/E and PBV alone, but
comparing both multiples alone P/E performs better.
5
Yoo (2006) examines the possibility of combining several multiples valuations to improve the
accuracy of the simple valuation technique. He finds that using a combination of historical
multiples reduces the valuation errors but that combination should not include the forward P/E.
This means that historical multiples do not increment information to a forward P/E valuation
but that combination improves historical multiples, so it should be performed when forwardlooking information is not available. To calculate the weight of each multiple valuation Yoo
(2006) conducts a linear regression approach, obtaining the following overall rank of weights:
P/E, PBV, P/EBITDA and P/S. Schreiner (2007a) finding support for the existence of industrypreferred multiples, seeks a combination of those with the PBV multiple for five European key
industries. This two-factor model approach delivers different weights for each multiple
depending on the analysed industry. The proposed weights are determined minimizing
valuation errors for each industry. The results suggest that the two-factor model adds value to
the “oils & gas”, “health care” and “banks” industries but no value is added to the “industrial
goods & services” and “telecommunications” industries because the PBV proposed weight
equals zero.
Determinants of multiples: Damodaran (2002) deduces analytically the determinants of various
multiples, relying on valuation models, and promotes the use of regression analysis to determine
a firm’s value. However, that approach fails empirical tests since it faces multicollinearity
issues and a non-Normal distribution of regression residuals (Schreiner, 2007a, pp. 75-76).
Herrmann & Richter (2003) and Schreiner (2007a) also deduce similar factors from models
such as the DDM, the DCF and the RIV model.
It can be inferred, from the above, that an intrinsic relationship between all multiples and a set
of determinants, more or less popular (e.g. Herrmann & Richter’s EV/EBIAT multiple), can be
determined. It also becomes clear that those determinants depend on the model we are dealing
with, thus different determinants arising from different models for the same multiple can hardly
be put together from a theoretical point of view. Besides, those derivations are laborious and
give no guarantee of empirical success. As we want to study a large set of multiples we chose
an empirical approach to identify the relations between valuation multiples and economic and
financial ratios. That’s what we conduct over the next sections: in Section 3 we formulate the
methodology of that work, in Section 4 we present the data to which it will be applied, so that
over Section 5 we present the empirical results.
6
3 Methodology
To investigate the empirical relationships between 17 valuation multiples and a large set of
popular economic and financial ratios we analysed the corresponding correlation coefficients.
Implementation of the method: To perform the method of market multiples, we divided our
sample randomly into two sets: the Training Group (with 70% of the entire sample) and the
Test Group (containing the remaining 30% of the sample). The Training Group was meant to
provide the set of comparable firms. The Test Group was meant to be the group of firms whose
value is estimated relying on the comparable firms (Training Group). These estimated multiples
will then be used to compute the valuation errors.
To identify the comparable firms from the Training Group to match with the Test Group we
adopted the criterion of proximity of certain ratios. These ratios were previously grouped
according to their correlation intensity with the valuation multiples. Then, using those groups
of ratios, we performed clustering analysis on the Training Group to identify the natural
clusters. For each cluster we computed the mean, median, harmonic mean and geometric mean
of all studied multiples. The matching of each company from the Test Group to each cluster of
the Training Group was made according to the proximity of the economic and financial ratios.
We also matched each company from the Test Group to the Training Group according to the
same-industry criterion. Then the measures of central tendency of each multiple from the
Training Group were attributed to the firms of the Test Group, this was made using all of the
four ICB levels.
Definition of the estimation errors: To decide which of the strategies better suits the purpose of
the method of market multiples, we computed the absolute valuation errors for each firm using
the following formula:
𝐸𝑟𝑟𝑜𝑟𝑦,𝑖𝑡 = |
𝑚
̂ 𝑦,𝑖𝑡 − 𝑚𝑦,𝑖𝑡
𝑚
̂ 𝑦,𝑖𝑡
|=|
− 1|
𝑚𝑦,𝑖𝑡
𝑚𝑦,𝑖𝑡
(3.1)
where m
̂ y,it is the estimated market multiple, my,it is the observed market multiple, y is the
multiple we are dealing with (e.g. P/S, PBV,…), 𝑖 indicates the firm and 𝑡 is the year.
The study of the distributions of the absolute valuation errors, running formal tests, allows
deciding which strategy delivers better results. We compared all the equity multiples among
themselves but separately from the entity multiples because their underlying variable is
different. Absolute valuation errors of equity multiples compares the deviation on the equity
7
variable but absolute valuation errors of entity multiples compares the deviation on the entity
variable. This may be proven by noticing that
̂
̂
(𝐸𝑉⁄𝑆)
𝐸𝑉
|
− 1| = |
− 1|
𝐸𝑉
(𝐸𝑉⁄𝑆)
(3.2)
We can hence understand that to compare the estimated EV/S of a firm with its observed EV/S
multiple is the same as to compare the estimated entity value with its observed market value.
This distribution may be compared with the EV/EBITDA distribution errors as formula (3.3)
suggests:
̂
̂
(𝐸𝑉⁄𝐸𝐵𝐼𝑇𝐷𝐴)
𝐸𝑉
|
− 1| = |
− 1|
𝐸𝑉
(𝐸𝑉⁄𝐸𝐵𝐼𝑇𝐷𝐴)
(3.3)
The same analogy is applicable to the equity multiples. However, we should not compare entity
multiples with equity multiples unless we transform entity values into equity values beforehand,
deducting the net debt and the preferred stock. This is not done in this study, so an estimation
of equity using the P/S multiple differs from an estimation using the EV/S multiple, since the
transformation of the entity estimation delivered by the EV/S multiple into equity would lead
to two different values, and vice-versa.
We should also mention that the valuation error calculation performed, using formula (3.1), is
not ubiquitous among studies. That’s another reason why results across different studies are
difficult to compare, even when a simple approach as the comparison of central tendency
measures of errors is performed.
4 Data
The sample we used consists of the constituents of three merged indices, the World Index, the
Alternext Allshare and the FTSE AIM All-Share, at the end of the first semester of 2012. To
the World Index, containing 6.625 firms from 54 countries1, we added the small and medium
size firms from the NYSE Euronext stock exchange encompassing 181 companies, and the
1
Argentina, Australia, Germany, Belgium, Bulgaria, Brazil, Colombia, Hong Kong, China, Chile, Canada, Cyprus,
Sri Lanka, Czech Republic, Denmark, Spain, Egypt, Finland, France, Greece, Hungary, Indonesia, India, Ireland,
Israel, Italy, Japan, South Korea, Luxembourg, Malta, Mexico, Malaysia, Netherlands, Norway, New Zealand,
Austria, Peru, Philippines, Pakistan, Poland, Portugal, Romania, Russian Federation, South Africa, Sweden,
Singapore, Slovenia, Switzerland, Taiwan, Thailand, Turkey, United Kingdom, United States and Venezuela.
8
London Stock Exchange, containing 784 companies. The potential size of the sample is then
7.590 companies.
The data was obtained from the Thomson Reuters Datastream database, and several variables
were constructed by us, adopting an economic balance sheet perspective (Fernández, 2007, p.
14). The variables containing missing values were ignored in the construction of the ratios and
we eliminated the severe outliers of all multiples and some ratios. All market multiples were
calculated dividing the market capitalization and the entity value by the accounting information,
both provided by Datastream. The other variables were constructed using the same source. All
the information regarding the stock exchange prices is the one observed at the end of the year
and the accounting information is the one reported in the audited annual accounts. The adopted
industry classification system is the Industry Classification Benchmark (ICB) because it is the
one Datastream uses to categorize companies, that’s not true for other available systems in the
database such as the SIC system (Standard Industrial Classification).
The reference year for the analysis we perform is 2011. We did not mix information from
different moments in time, as some authors do, because they may vary through time as
consequence of the market moods influenced by the economic cycle.
14,0
12,0
10,0
8,0
6,0
4,0
2,0
0,0
2000
2001
2002
EV/S
2003
EV/GI
2004
2005
2006
2007
EV/EBITDA
2008
2009
EV/EBIT
2010
2011
EV/TA
Figure 4.1: Evolution of the median of the entity multiples during the period 2000-2011
Source: Own elaboration
9
18
16
14
12
10
8
6
4
2
0
2000
2001
2002
2003
P/S
P/EBT
2004
P/GI
PER
2005
2006
2007
2008
P/EBITDA
P/B
2009
2010
2011
P/EBIT
P/TA
Figure 4.2: Evolution of the median of the equity multiples during the period 2000-2011
Source: Own elaboration
As we can clearly see in Figure 4.1 and Figure 4.2, market multiples vary across time. The
decrease of all multiples in 2008, when the financial crisis erupted, is evident. Further
investigation on this topic may be of academic interest.
5 Empirical Results
5.1
Univariate Analysis
We report the descriptive statistics of the 17 studied multiples in Table 5.1 (mean, minimum
(Min.), percentile 25 (χ25) or 1st quartile (Q1), median (χ50), percentile 75 (χ75) or 3rd quartile
(Q3), maximum (Max), standard deviation (S.D.), coefficient of variation (C.V.), sample size
or number of valid observations (n), Skewness value (Skew.) and kurtosis (Kurt.)). Those
statistics were obtained for the Training Group, as previously explained.
10
Table 5.1: Descriptive statistics of the market multiples in 2011
Mean
Min.
χ25
χ50
χ75
Max.
S.D.
C.V.
n
Skew.
Kurt.
Entity market multiples:
EV/S
2,0
EV/GI
4,8
EV/EBITDA
8,7
EV/EBIT
12,0
EV/TA
1,5
EV/OCF
11,4
EV/FCFF
15,3
0,0
0,0
0,1
0,1
0,0
0,1
0,0
0,6
2,2
5,2
7,2
0,9
6,4
6,1
1,3
3,8
7,6
10,6
1,2
9,6
11,5
2,6
6,3
10,9
14,9
1,8
14,6
20,4
9,4
17,4
25,5
35,8
4,9
36,0
63,4
2,0
3,7
4,8
6,9
0,9
7,1
13,1
1,0
0,8
0,6
0,6
0,6
0,6
0,9
6.088
5.319
5.700
5.476
6.224
5.795
3.855
1,7
1,3
1,1
1,1
1,5
1,2
1,5
2,4
1,4
1,0
1,2
1,9
1,2
2,0
Equity market multiples:
P/S
1,5
P/GI
3,8
P/EBITDA
6,8
P/EBIT
9,3
P/EBT
10,8
P/E
14,9
P/B
1,7
P/TA
1,4
P/OCF
8,9
P/FCFF
12,1
0,0
0,0
0,0
0,0
0,2
0,2
0,0
0,0
0,1
0,0
0,5
1,7
4,0
5,7
6,7
9,2
0,9
0,6
4,9
4,1
1,0
3,0
5,9
8,2
9,5
13,2
1,3
1,0
7,7
9,0
2,1
5,1
8,7
11,8
13,5
18,6
2,2
1,8
11,8
16,9
6,9
14,0
20,6
27,7
32,0
43,6
5,9
5,6
28,7
54,0
1,5
2,8
3,9
5,3
5,9
8,0
1,2
1,2
5,5
10,9
1,0
0,7
0,6
0,6
0,5
0,5
0,7
0,8
0,6
0,9
6.334
5.392
5.847
5.619
5.490
5.475
6.566
6.320
5.984
4.048
1,6
1,2
1,1
1,1
1,1
1,1
1,3
1,5
1,1
1,4
2,0
1,1
1,0
1,1
1,2
1,2
1,4
2,0
1,0
1,9
Source: Own elaboration
We can observe in Table 5.1 that the central tendency statistics of multiples of the income
statement increase when we move towards the net income, which is naturally a consequence of
the subtraction of costs. The dispersion, measured by the coefficient of variation, decreases
when we seek a similar pattern across multiples of the income statement. Cash-flow multiples
increases dispersion when we go from the top to bottom. The decrease on the number of valid
observations is due to the non-validity of negative multiples, which have no economic sense.
The exception goes to the Gross Income multiples whose number of observations decreases
further than that of the EBITDA multiples, this is because this item does not apply to banks and
insurance companies. Another finding of interest is that all multiples are positive biased, that is
to say, they have leptokurtic distributions (higher peak than a Normal distribution) indicated by
a positive kurtosis, and are right-tailed as the positive skewness values indicate.
Next we report the descriptive statistics of the ratios whose relation with multiples we study.
We did not exclude most severe outliers from these ratios because we did not want to add
another restriction to the relation between multiples and ratios, so these statistics may present
discrepant values to the experienced analyst. That won’t be a problem for our subsequent work.
Moreover, usually most analysts do not pay attention to these ratios when they value firms with
the multiples valuation method. The columns containing the maximum and minimum values in
Table 5.2 show how far we relaxed the outliers’ restrictions.
Table 5.2: Descriptive statistics of ratios in 2011
11
Mean
Min.
χ25
χ50
χ75
Max.
S.D.
n
Skew.
Kurt.
9,9
6,3
11,8
3,6
-61,6
-45,7
-99,8
-84,5
0,6
-1,2
-14,5
-8,9
8,6
4,9
10,0
3,8
18,9
12,9
34,0
16,2
63,2
44,9
159,0
70,0
17,4
12,8
47,1
22,3
6.471
6.369
5.134
5.201
0,0
0,2
0,4
-0,1
1,6
1,3
0,7
0,8
Income Statement margins (as % of Sales):
GI margin
40,1
-71,0
EBITDA margin
19,1
-61,4
EBIT margin
13,4
-53,5
EBT margin
11,0
-45,9
NI margin
7,8
-35,6
22,3
7,8
4,4
3,6
2,4
36,0
15,6
10,5
9,0
6,4
56,7
27,7
20,7
17,8
12,9
100,0
76,4
61,9
53,1
39,7
24,3
18,3
15,4
13,7
10,5
5.874
6.382
6.314
6.315
6.236
0,4
0,4
0,3
0,1
0,0
0,1
1,8
2,0
1,8
2,0
Balance Sheet Structure (items written as % of Sales)
FxdAssts_%Sales
62,1
0,0
20,7
NWC_%Sales
-3,1
-141,8
-13,7
Invtmts_%Sales
13,3
-3,1
0,1
TA_%Sales
111,0
-422,2
38,2
41,6
1,2
1,9
71,6
83,6
13,6
9,4
138,2
291,7
91,2
247,6
633,9
60,1
31,7
33,0
120,7
6.322
5.971
5.993
6.207
1,6
-1,1
4,2
1,8
2,2
3,1
19,0
4,0
Debt_%Sales
Eqty_%Sales
PrefStock_%Sales
MinInter_%Sales
Growth rates (in %):
grSales(1y)
grSales(CAGR4y)
grNI(1y)
grNI(CAGR4y)
16,3
80,1
6,6
5,1
-300,0
-261,9
0,0
-5,6
-6,6
33,0
0,0
0,0
6,7
60,1
0,0
0,1
31,0
106,1
0,0
2,0
299,7
349,5
35.976,7
460,3
61,8
70,0
431,8
21,3
6.446
6.231
7.020
6.998
0,8
1,4
82,5
10,3
5,5
2,5
6.862,6
142,8
0,6
8,6
7,8
-93,6
-454,4
-492,0
-0,1
2,0
3,5
0,2
6,9
10,0
0,8
14,5
17,2
89,7
493,2
450,3
3,7
41,7
35,9
7.195
6.949
7.068
-0,9
-0,1
-3,6
249,8
47,1
54,6
Cash-flow Structure (items written as % of Sales)
OCF_%Sales
16,0
-196,7
6,4
varNWCch_%Sales
-4,2
-199,9
-4,9
CapexCh_%Sales
12,9
-195,8
1,6
varInvtmts_%Sales
5,5
-197,0
-0,3
FCFFCh_%Sales
1,2
-199,0
-5,0
Div_%Sales
5,5
0,0
0,0
Payout_%Sales
35,7
0,0
8,8
13,0
0,1
5,0
0,0
3,7
1,6
27,2
24,7
4,1
14,3
1,0
13,3
4,8
51,0
190,1
200,0
198,5
199,4
199,3
191,4
199,7
23,8
35,1
29,7
33,3
42,4
12,3
35,6
6.863
6.762
6.780
6.659
6.556
6.983
5.780
-1,3
-1,3
1,9
2,1
-0,7
5,6
1,5
16,8
10,9
12,7
14,6
6,2
47,4
2,6
71
92
100
28
7.261
-0,5
-1,0
D/E
ROA
ROE
FreeFloat
66
0
44
Source: Own elaboration
5.2
Multivariate Analysis
It became clear above that the distributions of multiples are non-Normal distributions, having
skewness and kurtosis values larger than 1. For a distribution to be considered to follow a
Normal distribution it must have skewness and kurtosis values within the range ]-0,5;0,5[
(Maroco, 2007, p. 42). As a consequence we cannot perform the significance test for the
Pearson’s correlation coefficients, so that we report Spearman’s correlation coefficients in
Table 5.3 and Table 5.4. This non-parametric association measure allows for a non-parametric
test of significance. We omit some ratios indicated in Table 5.2 because their association with
any multiple was insignificant.
12
GI margin
0,55
0,00
0,05
0,00
0,16
0,00
0,11
0,00
0,20
0,00
0,14
0,00
0,10
0,00
0,59
0,00
0,13
0,00
0,19
0,00
0,15
0,00
0,14
0,00
0,11
0,00
0,21
0,00
0,18
0,00
0,21
0,00
0,15
0,00
EBITDA
margin
0,66
0,00
0,42
0,00
0,12
0,00
0,06
0,00
0,22
0,00
0,14
0,00
0,21
0,00
0,64
0,00
0,42
0,00
0,06
0,00
-0,01
0,62
0,02
0,26
0,00
0,94
0,24
0,00
0,13
0,00
0,12
0,00
0,20
0,00
EBIT
margin
0,63
0,00
0,41
0,00
0,12
0,00
-0,02
0,11
0,29
0,00
0,16
0,00
0,20
0,00
0,65
0,00
0,47
0,00
0,13
0,00
-0,05
0,00
-0,03
0,02
-0,05
0,00
0,30
0,00
0,22
0,00
0,19
0,00
0,22
0,00
EBT
margin
0,57
0,00
0,35
0,00
0,06
0,00
-0,10
0,00
0,31
0,00
0,10
0,00
0,20
0,00
0,67
0,00
0,47
0,00
0,17
0,00
-0,02
0,15
-0,06
0,00
-0,06
0,00
0,33
0,00
0,30
0,00
0,23
0,00
0,28
0,00
NI margin
0,57
0,00
0,36
0,00
0,09
0,00
-0,07
0,00
0,34
0,00
0,11
0,00
0,20
0,00
0,68
0,00
0,50
0,00
0,21
0,00
0,04
0,01
0,00
0,88
-0,09
0,00
0,35
0,00
0,33
0,00
0,25
0,00
0,28
0,00
TA
_%Sales
0,63
0,00
0,45
0,00
0,25
0,00
0,25
0,00
-0,13
0,00
0,25
0,00
0,08
0,00
0,42
0,00
0,25
0,00
-0,04
0,00
-0,03
0,02
0,05
0,00
0,03
0,03
-0,14
0,00
-0,29
0,00
-0,01
0,35
-0,05
0,01
Eqty
_%Sales
0,58
0,00
0,33
0,00
0,14
0,00
0,09
0,00
-0,10
0,00
0,11
0,00
-0,01
0,62
0,61
0,00
0,41
0,00
0,16
0,00
0,09
0,00
0,11
0,00
0,07
0,00
-0,10
0,00
-0,02
0,10
0,17
0,00
0,02
0,17
Table 5.3: Spearman’s correlation coefficients between multiples and ratios (Part 1)
EV/S
p-value
EV/GI
p-value
EV/EBITDA
p-value
EV/EBIT
p-value
EV/TA
p-value
EV/OCF
p-value
EV/FCFF
p-value
P/S
p-value
P/GI
p-value
P/EBITDA
p-value
P/EBIT
p-value
P/EBT
p-value
P/E
p-value
P/B
p-value
P/TA
p-value
P/OCF
p-value
P/FCFF
p-value
Source: Own elaboration
13
0,05
0,00
-0,02
0,24
-0,26
0,00
-0,44
0,00
0,46
0,00
-0,18
0,00
0,16
0,00
0,26
0,00
0,25
0,00
0,19
0,00
0,03
0,02
-0,13
0,00
-0,20
0,00
0,44
0,00
0,58
0,00
0,21
0,00
0,38
0,00
Roa
0,15
0,00
0,09
0,00
-0,12
0,00
-0,29
0,00
0,48
0,00
-0,05
0,00
0,24
0,00
0,24
0,00
0,22
0,00
0,05
0,00
-0,12
0,00
-0,21
0,00
-0,29
0,00
0,49
0,00
0,42
0,00
0,12
0,00
0,34
0,00
Roe
OCF
_%Sales
0,63
0,00
0,41
0,00
0,15
0,00
0,09
0,00
0,17
0,00
0,09
0,00
0,08
0,00
0,63
0,00
0,44
0,00
0,13
0,00
0,07
0,00
0,06
0,00
0,01
0,37
0,20
0,00
0,12
0,00
0,05
0,00
0,10
0,00
Capex
_%Sales
0,35
0,00
0,33
0,00
0,09
0,00
0,15
0,00
0,20
0,00
0,10
0,00
0,38
0,00
0,32
0,00
0,25
0,00
0,01
0,54
0,09
0,00
0,10
0,00
0,10
0,00
0,21
0,00
0,10
0,00
0,06
0,00
0,36
0,00
varNWC_ varInvtmts
%Sales
_%Sales
-0,05
0,11
0,00
0,00
0,00
0,12
0,76
0,00
-0,02
0,03
0,19
0,01
-0,01
-0,04
0,42
0,00
0,03
0,04
0,04
0,01
0,03
0,04
0,02
0,00
0,30
0,10
0,00
0,00
-0,04
0,11
0,00
0,00
0,00
0,11
0,77
0,00
0,02
-0,02
0,19
0,14
0,03
-0,11
0,02
0,00
0,01
-0,09
0,61
0,00
-0,02
-0,11
0,20
0,00
0,02
0,04
0,12
0,00
0,04
0,00
0,00
0,81
0,05
0,01
0,00
0,56
0,32
0,08
0,00
0,00
FCFF
_%Sales
0,09
0,00
-0,03
0,01
-0,06
0,00
-0,08
0,00
0,04
0,00
-0,17
0,00
-0,41
0,00
0,11
0,00
0,02
0,13
0,05
0,00
0,03
0,05
0,02
0,23
0,03
0,04
0,04
0,00
0,09
0,00
-0,09
0,00
-0,46
0,00
Divid
_%Sales
0,35
0,00
0,28
0,00
0,17
0,00
0,06
0,00
0,14
0,00
0,19
0,00
0,15
0,00
0,44
0,00
0,33
0,00
0,21
0,00
0,11
0,00
0,09
0,00
0,04
0,01
0,16
0,00
0,12
0,00
0,25
0,00
0,21
0,00
0,03
0,01
0,07
0,00
0,14
0,00
0,16
0,00
0,04
0,00
0,11
0,00
0,10
0,00
0,07
0,00
0,06
0,00
0,14
0,00
0,18
0,00
0,19
0,00
0,23
0,00
0,07
0,00
0,02
0,12
0,11
0,00
0,12
0,00
Payout
0,04
0,00
-0,02
0,12
-0,23
0,00
-0,43
0,00
0,55
0,00
-0,17
0,00
0,07
0,00
0,25
0,00
0,25
0,00
0,21
0,00
0,05
0,00
-0,13
0,00
-0,20
0,00
0,51
0,00
0,72
0,00
0,21
0,00
0,30
0,00
ln(Roa)
0,18
0,00
0,12
0,00
-0,08
0,00
-0,28
0,00
0,56
0,00
-0,01
0,58
0,17
0,00
0,23
0,00
0,21
0,00
0,05
0,00
-0,12
0,00
-0,21
0,00
-0,29
0,00
0,59
0,00
0,48
0,00
0,12
0,00
0,24
0,00
ln(Roe)
ln(OCF
_Sales)
0,72
0,00
0,45
0,00
0,16
0,00
0,09
0,00
0,16
0,00
0,08
0,00
0,06
0,00
0,72
0,00
0,48
0,00
0,13
0,00
0,06
0,00
0,06
0,00
0,00
0,89
0,18
0,00
0,11
0,00
0,05
0,00
0,05
0,01
ln(Capex
_%Sales)
0,45
0,00
0,40
0,00
0,14
0,00
0,19
0,00
0,14
0,00
0,16
0,00
0,31
0,00
0,38
0,00
0,30
0,00
0,02
0,08
0,09
0,00
0,11
0,00
0,09
0,00
0,15
0,00
0,03
0,03
0,07
0,00
0,24
0,00
ln(varNWC
_%Sales)
0,36
0,00
0,26
0,00
0,11
0,00
0,04
0,04
-0,08
0,00
0,14
0,00
0,06
0,02
0,28
0,00
0,22
0,00
0,03
0,11
-0,06
0,00
-0,02
0,35
-0,10
0,00
-0,08
0,00
-0,11
0,00
0,07
0,00
-0,01
0,73
Table 5.4: Spearman’s correlation coefficients between multiples and ratios (Part 2)
EV/S
p-value
EV/GI
p-value
EV/EBITDA
p-value
EV/EBIT
p-value
EV/TA
p-value
EV/OCF
p-value
EV/FCFF
p-value
P/S
p-value
P/GI
p-value
P/EBITDA
p-value
P/EBIT
p-value
P/EBT
p-value
P/E
p-value
P/B
p-value
P/TA
p-value
P/OCF
p-value
P/FCFF
p-value
Source: Own elaboration
14
ln(varInvtm
ts_%Sales)
0,36
0,00
0,31
0,00
0,16
0,00
0,03
0,09
-0,17
0,00
0,14
0,00
-0,12
0,00
0,28
0,00
0,23
0,00
0,01
0,77
-0,13
0,00
-0,07
0,00
-0,12
0,00
-0,17
0,00
-0,20
0,00
0,03
0,19
-0,22
0,00
ln(FCFF_
%Sales)
0,51
0,00
0,32
0,00
0,18
0,00
0,07
0,00
0,01
0,75
0,08
0,00
-0,42
0,00
0,44
0,00
0,30
0,00
0,12
0,00
0,02
0,24
0,06
0,00
0,02
0,18
0,01
0,73
-0,04
0,03
0,04
0,02
-0,46
0,00
ln(Divid_
%Sales)
0,64
0,00
0,46
0,00
0,28
0,00
0,14
0,00
0,18
0,00
0,31
0,00
0,14
0,00
0,72
0,00
0,54
0,00
0,31
0,00
0,18
0,00
0,16
0,00
0,06
0,00
0,21
0,00
0,16
0,00
0,37
0,00
0,17
0,00
ln(Payout)
0,09
0,00
0,11
0,00
0,23
0,00
0,28
0,00
0,05
0,00
0,17
0,00
0,08
0,00
0,11
0,00
0,09
0,00
0,21
0,00
0,28
0,00
0,32
0,00
0,34
0,00
0,08
0,00
0,01
0,37
0,15
0,00
0,07
0,00
grSales
(CAGR4y)
0,18
0,00
0,25
0,00
0,08
0,00
-0,01
0,37
0,29
0,00
0,10
0,00
0,23
0,00
0,20
0,00
0,26
0,00
0,10
0,00
0,02
0,11
0,01
0,62
0,00
0,79
0,31
0,00
0,23
0,00
0,13
0,00
0,26
0,00
grNI
(CAGR4y)
0,18
0,00
0,13
0,00
-0,03
0,04
-0,21
0,00
0,32
0,00
0,04
0,00
0,15
0,00
0,21
0,00
0,20
0,00
0,06
0,00
-0,08
0,00
-0,15
0,00
-0,19
0,00
0,34
0,00
0,29
0,00
0,12
0,00
0,21
0,00
In Table 5.3 and Table 5.4 we also report the p-values corresponding to the test: H0: s = 0 vs.
H1: s ≠ 0. The strongest correlation values are highlighted in bold and will be held in
consideration for the next step. Based on the strength of the Spearman’s rank correlation
coefficients we gathered together ratios that seemed to form natural sets due to their position in
the income statement, the cash-flow statement or the balance sheet. This criterion may look
somewhat arbitrary but it is strongly supported by the high correlation between all these ratios
to one or more multiples. We summarize these natural sets of ratios in Table 5.5 and add the
Industry criterion for future purposes.
Table 5.5: Sets of selected ratios
00
Industry
09
EBITDA TA RoE OCF Capex FCFF Divid
01
GI Ebitda Ebit Ebt NI
10
Ebitda TA OCF Capex
02
Ebitda Ebit Ebt NI
11
ln(RoA) ln(RoE)
03
TA Eqty
12
ln(OCF) ln(Capex) ln(varNWC) ln(varInvtmts)
04
RoA RoE
13
ln(FCFF)
05
OCF Capex varNWC varInvtmts
14
ln(Diviv)
06
OCF Capex
15
ln(FCFF) ln(Divid)
07
FCFF
16
ln(Payout)
08
Divid
17
grSales(CAGR) grNI(CAGR) ln(RoE)
Source: Own elaboration
These sets of selected ratios do not have all the same importance for every multiple as the
Spearman’s correlation coefficients show. We brief in Table 5.6 the relationships between
multiples and sets of ratios that will be carried further.
Table 5.6: Summary of held relationships between multiples and sets of ratios
00
01
02
03
EV/S
√
√
√
√
√
EV/GI
√
√
√
√
EV/EBITDA
√
EV/EBIT
√
EV/TA
√
EV/OCF
√
EV/FCFF
√
P/S
√
P/GI
√
P/EBITDA
√
P/EBIT
√
P/EBT
√
√
√
P/E
√
√
√
P/B
√
P/TA
√
P/OCF
√
P/FCFF
√
N
17
√
04
05
06
07
08
09
√
10
12
13
14
15
√
11
√
√
√
√
√
√
√
√
√
16
17
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
4
√
8
5
√
√
√
12
2
√
4
2
Source: Own elaboration
15
√
3
4
2
3
4
4
7
4
2
2
In the last line of Table 5.6 we may observe that the set of ratios 04 (RoA and RoE) is the one
that is more correlated to more multiples, retaining 12 ties. The second most “popular set of
ratios” among multiples is the set 02 (Ebitda margin, Ebit margin, Ebt margin and NI margin)
holding 8 ties, followed by the set 14 (logarithmic dividends) retaining 7 links. The multiple
P/EBIT does not hold any connection to any set of ratios due to its weak Spearman’s
coefficients with all ratios.
5.3
Cluster Analysis
Using the seventeen sets of ratios defined above we performed hierarchical and non-hierarchical
cluster analysis on the Training Group of the sample. The defined sets of ratios constitute, as
seen above, several attempts to identify the variables that better serve the purpose of dividing
the firms into different groups to perform valuations using multiples. It is here that we determine
the number of clusters for each set of ratios, or set of characteristics.
Hierarchical Cluster Analysis: To perform the hierarchical cluster analysis we selected the
Euclidean Distance to construct the dissimilarity matrix and the Complete Linkage (or FurthestNeighbour) method as clustering method. The use of the Euclidean Distance is related to its
popularity and simplicity. The use of the Complete Linkage method aims at avoiding chain
effects and favouring the appearance of compact clusters (Maroco, 2007, p. 428). We
standardized all variables, using Z scores, to eliminate the effect of different dispersions among
variables on the Euclidean distance.
The simple visual analysis of the dendrograms does not allow us to determine the number of
clusters due to the size of the Training Group (5.307 firms). So we analyse instead the
coefficients of the Agglomeration Schedule and the R2 calculated as follows (Maroco, 2007, p.
439):
2
∑𝑝𝑖=1 ∑𝑘𝑗=1 𝑛𝑖𝑗 (𝑋̅𝑖𝑗 − 𝑋̅𝑖 )
𝑆𝑄𝐶
2
𝑅 =
=
𝑆𝑄𝑇 ∑𝑝 ∑𝑘 ∑𝑛𝑖 (𝑋 − 𝑋̅)2
𝑖=1 𝑗=1 𝑙=1 𝑖𝑗𝑙
(5.1)
where SQC is the Sum of Squares Between Groups and SQT is the Total Sum of Squares.
Figure 5.1 shows the behaviour of the coefficients, measuring the distance between clusters,
and the R2 as we increase the number of clusters. This example (Figure 5.1) was made using
the set of ratios 01 (GI margin, Ebitda margin, Ebit margin, Ebt margin and NI margin).
16
250
1,0
200
0,8
150
0,6
100
0,4
50
0,2
0
0,0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Coefficients
R-squared
Figure 5.1: Visual representation of the coefficients and the R2 for the set of ratios 01
Source: Own elaboration
The main criterion to determine the number of clusters of each set of ratios was to achieve a R2
of at least 80%. Then, by the analysis of the slope of the coefficients, we intended to include
that number of clusters that capture a substantial sink of that distance. A third criterion was
implemented based on the relative increment of the R2 – that is, if after the 1st and the 2nd
criterion, there is another partition that increases the R2 considerably it should be included. The
analysis was performed for a maximum of 50 clusters for each set of ratios.
The “optimal” number of clusters for each set of ratios may be read on Table 5.7. All except
the set of ratios 09 exceed 80% of the R2. When a partition of 50 clusters is considered the set
of ratios 09 only reaches a value of 72%.
Non-hierarchical Cluster Analysis: Based on the partitions determined in the hierarchical
cluster analysis we performed a K-Means Cluster Analysis. This method consists in the
following: 1st) divide the elements in k clusters according to the researcher’s choice; 2nd)
compute/update the centre of each cluster; 3rd) assign each element to the cluster whose cluster
centre is closest; 4th) repeat all the process from the 2nd step until the minimum distance of all
elements to the respective cluster centre doesn’t change significantly (Maroco, 2007, p. 446).
This method allows an element to end up in a cluster different from the cluster it was assigned
at first. That does not occur in the hierarchical cluster analysis.
The R2 measures obtained for each set of ratios running the hierarchical clustering and K-Means
Cluster Analysis may be found at Table 5.7.
17
Table 5.7: Number of held clusters by set of ratios and their corresponding R2
Set of Ratios
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
Number of
Clusters
Classificatory Variables
GI; Ebitda; Ebit; Ebt; NI (all as % of Sales)
Ebitda; Ebit; Ebt; NI (all as % of Sales)
TA (%Sales); Eqty (%Sales)
RoA; RoE
OCF; Capex; varNWC; varInvtmts (all as % of Sales)
OCF; Capex (all as % of Sales)
FCFF (as % of Sales)
Divid (as % of Sales)
EBITDA; TA; RoE; OCF; Capex; FCFF; Divid (as % of Sales)
Ebitda TA OCF Capex (all as % of Sales)
ln(RoA) ln(RoE)
ln(OCF) ln(Capex) ln(varNWC) ln(varInvtmts)
ln(FCFF)
ln(Diviv)
ln(FCFF) ln(Divid)
ln(Payout)
grSales(CAGR) grNI(CAGR) ln(RoE)
20
8
10
19
43
19
5
8
30
43
8
39
7
9
16
6
24
R2
(Hierarchical
Analysis)
R2
(K-Means)
0,81
0,80
0,82
0,83
0,80
0,81
0,85
0,94
0,65
0,80
0,82
0,80
0,92
0,91
0,81
0,83
0,80
0,87
0,84
0,88
0,92
0,86
0,90
0,89
0,96
0,75
0,87
0,87
0,83
0,95
0,95
0,89
0,90
0,89
Source: Own elaboration
5.4
Conception and Analysis of the Prediction Errors
Implementation of the method of multiples: After the cluster analysis that divided our Training
Group into clusters according to the financial characteristics or sets of ratios, a broad
implementation of the method of multiples was carried out following the typical next steps: 1st)
computation of the mean, median, harmonic mean and geometric mean for all clusters obtained
in the cluster analysis as well as for all 4 ICB levels; 2nd) matching of each company of the Test
Group to its corresponding cluster of the Training Group using the Nearest Neighbour Analysis
built upon the consistent sets of ratios; 3rd) each Test Group company received the valuation
given by the mean, the median, the harmonic mean and the geometric mean of all relevant
multiples of its peers, defined by its corresponding cluster and its industry classification; 4th)
calculation of the estimation errors using formula 3.1.
Prediction errors analysis: The analysis of the error distributions, obtained implementing the
method of multiples through its several alternatives, is performed running paired t-student tests.
We may consider this parametric test as the size of the Test Group is far above 100 observations
to tested multiples. Thus, the hypotheses under analysis are as follows (Maroco, 2007, p. 271):
𝐻0 : 𝜇1 = 𝜇2
(6.2)
𝐻1 : 𝜇1 ≠ 𝜇2
where, μ represents the mean of populations 1 and 2 under comparison.
18
The t statistic is as follows:
𝑇=
̅
𝐷
𝑆𝐷′
⁄
√𝑛
(6.3)
̅ is the observed mean of Di = (X1i − X2i ), i=1,…, n, SD′ is the corrected standard
where, D
deviation of variable Di and n represents the number of observations of variable Di .
Due to the fact that we perform non-independent multiple comparisons of means, a Bonferroni
correction must be applied, so that the significance level shall be transformed into α′ = α/m,
where m represents the number of formal tests to perform (Dunn, 1961).
In order to compare such a great number of distributions we’ve encoded them according to the
keys in Table 5.8. For instance, a distribution coded as “03.3B” means that to predict the
companies’ value, we used the ICB system (1st cf. Table 5.8) considering as comparable
companies the ones belonging to the same Sector (2nd key cf. Table 5.8) and employed the
EV/EBITDA multiple (3rd key cf. Table 5.8) aggregating it applying the median (4th key cf.
Table 5.8) to the observed peer values. Alternatively, if a distribution is coded as “45:3B” it
means that to predict the companies’ value, we used the same EV/EBITDA multiple (3rd key
cf. Table 5.8) aggregating it using the median (4th key cf. Table 5.8) but recurring to a different
set of comparable companies: firms gathered using the set of rations 04 (1st cf. Table 5.8), i.e.
the Return on Assets (RoA) and the Return on Equity (RoE), running the Complete Linkage
procedure (2nd key cf. Table 5.8) for clustering peers.
Table 5.8: Reading diagram for the encoded distribution errors
First Code – Cluster Approach
0: ICB
1: Set of ratios 01
2: Set of ratios 02
3: Set of ratios 03
4: Set of ratios 04
5: Set of ratios 05
6: Set of ratios 06
7: Set of ratios 07
8: Set of ratios 08
9: Set of ratios 09
'0: Set of ratios 10
'1: Set of ratios 11
'2: Set of ratios 12
'3: Set of ratios 13
'4: Set of ratios 14
'5: Set of ratios 15
'6: Set of ratios 16
'7: Set of ratios 17
Second Code - Classification
1: Industry
2: Supersector
3: Sector
4: Subsector
5: Complete Linkage
6: K-Means
Source: Own elaboration
19
Third Code – Multiple
1: EV/S
2: EV/GI
3: EV/EBITDA
4: EV/EBIT
5: EV/TA
6: EV/OCF
7: EV/FCFF
8: P/S
9: P/GI
'0: P/EBITDA
'1: P/EBIT
'2: P/EBT
'3: P/E
'4: P/B
'5: P/TA
'6: P/OCF
'7: P/FCFF
Fourth Code – Selected Measure
A: Mean
B: Median
C: Harmonic Mean
D: Geometric Mean
5.5
Measure of Central Tendency
In this section we examine the question of which measure of central tendency (4th key cf. Table
5.8) provides the lowest prediction errors. We performed formal tests, for the four considered
measures of central tendency - mean (A), median (B), harmonic mean (C) and geometric mean
(D) - using several market multiples under different clustering procedures. In Table 5.9, we
may see two examples of how the tests were performed. The upper portion of the table presents
the paired t-statistics, the second presents the bilateral p-values for the tests in formula (6.2),
followed by the ascertained ranking of best measures and by three statistics of the distribution
errors indicated in the first line of the table.
EV/TA: Sector
t-Student Test
03.5A
03.5B
03.5C
03.5A
17,514
13,261
03.5B
7,878
03.5C
03.5A
0,000
0,000
03.5B
0,000
03.5C
Ranking
4th
2nd
1st
Mean
0,6699
0,5454
0,4825
Stand.-Dev. 1,7264
1,4858
1,2212
N
1.874
1.874
1.874
Descript.
Stat
p-value*
t-Stat.
EV/EBITDA: Industry t-Student Test
01.3A
01.3B
01.3C 01.3D
01.3A
11,629
6,256 9,517
01.3B
4,713 5,836
01.3C
- -4,291
01.3A
0,000
0,000 0,000
01.3B
0,000 0,000
01.3C
- 0,000
Ranking
4th
3rd
1st
2nd
Mean
0,7487
0,6685
0,5391 0,6306
Stand.-Dev.
2,7304
2,4759
1,4956 2,2601
N
1.712
1.712
1.712 1.712
Descript.
Stat
p-value*
t-Stat.
Table 5.9: Formal tests for the prediction errors associated with the use of different measures
03.5D
16,867
-4,194
-9,502
0,000
0,000
0,000
3rd
0,5543
1,4765
1.874
As we can notice in both cases the harmonic mean is the best measure of central tendency for
the multiple EV/EBITDA when the ICB Industry level is considered as well as for the EV/TA
multiple when an ICB Sector level is used. Table 5.10 and Table 5.11 summarize the results for
the best measure for all analysed multiples and clustering procedures.
Table 5.10: The best measure of central tendency (indicated by Distrib.) for each market
multiple and each ICB level characterized by the mean and the median of its distribution errors
Distrib.
EV/S
EV/GI
EV/EBITDA
EV/EBIT
EV/TA
EV/OCF
EV/FCFF
P/S
P/GI
P/EBITDA
P/EBIT
P/EBT
P/E
P/B
P/TA
P/OCF
P/FCFF
#01.1C
#01.2C
#01.3C
#01.4C
#01.5C
#01.6C
01.7C
#01.8C
#01.9C
#01.'0C
#01.'1C
#01.'2C
#01.'3C
#01.'4C
#01.'5C
#01.'6C
#01.'7C
Industry
Supersector
Sector
Mean Median Distrib. Mean Median Distrib. Mean Median
0,812
0,660
0,539
0,575
0,479
0,596
1,092
0,847
0,753
0,529
0,486
0,445
0,442
0,553
0,734
0,583
1,179
0,639
0,517
0,394
0,425
0,315
0,420
0,702
0,661
0,502
0,409
0,380
0,361
0,367
0,437
0,613
0,454
0,767
#02.1C
#02.2C
#02.3C
#02.4C
#02.5C
#02.6C
02.7C
#02.8C
#02.9C
#02.'0C
#02.'1C
#02.'2C
#02.'3C
#02.'4C
#02.'5C
#02.'6C
#02.'7C
0,810
0,655
0,542
0,580
0,485
0,575
1,170
0,838
0,647
0,518
0,483
0,445
0,440
0,551
0,715
0,566
1,180
0,623
0,514
0,382
0,412
0,324
0,408
0,660
0,647
0,515
0,398
0,375
0,360
0,365
0,421
0,564
0,417
0,742
Source: Own elaboration
20
#03.1C
#03.2C
#03.3C
#03.4C
#03.5C
#03.6C
03.7C
#03.8C
#03.9C
#03.'0C
#03.'1C
#03.'2C
#03.'3C
#03.'4C
#03.'5C
#03.'6C
#03.'7C
0,806
0,675
0,542
0,584
0,482
0,566
1,258
0,795
0,632
0,523
0,486
0,444
0,445
0,553
0,719
0,565
1,207
0,608
0,505
0,380
0,418
0,317
0,394
0,648
0,614
0,500
0,387
0,368
0,360
0,360
0,431
0,549
0,401
0,723
Distrib.
#04.1C
#04.2C
#04.3C
#04.4C
#04.5C
04.6C
#04.7C
#04.8C
#04.9C
#04.'0C
#04.'1C
#04.'2C
#04.'3C
#04.'4C
#04.'5C
#04.'6C
#04.'7C
Subsector
Mean Median
0,838
0,688
0,544
0,590
0,488
0,565
1,173
0,780
0,630
0,508
0,474
0,437
0,440
0,543
0,703
0,551
1,292
0,583
0,470
0,368
0,402
0,315
0,389
0,624
0,582
0,482
0,373
0,362
0,345
0,350
0,431
0,532
0,390
0,709
Table 5.11: The best measure of central tendency (indicated by Distrib.) for each market
multiple and each clustering process characterized by the mean and the median of its
distribution errors
Complete Linkage
K Means
Complete Linkage
K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV/S
EV/GI
EV/EBITDA
EV/EBIT
EV/TA
EV/OCF
EV/FCFF
P/S
P/GI
P/EBITDA
P/EBT
P/E
P/B
P/TA
P/OCF
P/FCFF
#15.1C
#35.1C
#85.1C
#'25.1C
#'45.1C
#25.2C
#65.2C
#'25.2C
#'45.2C
#35.3C
#45.4C
#25.5C
#'15.5C
#'45.6C
#25.7C
#55.7C
#95.7C
#15.8C
#35.8C
#65.8C
#95.8C
#'35.8C
#'55.8C
#25.9C
#45.9C
#85.9C
#'25.9C
#'45.9C
#'45.'0C
#45.'2C
#45.'3C
#15.'4C
#45.'4C
#'75.'4C
#45.'5C
#15.'6C
#'45.'6C
#25.'7C
#55.'7C
#95.'7C
0,574
0,700
0,796
0,530
0,660
0,599
0,601
0,582
0,599
0,485
0,519
0,507
0,398
0,559
0,920
0,787
0,846
0,553
0,742
0,671
0,747
0,762
0,566
0,553
0,648
0,633
0,572
0,546
0,465
0,444
0,448
0,524
0,551
0,448
0,756
0,511
0,495
0,935
0,894
0,767
0,427 #16.1C
0,502 #36.1C
0,640 #86.1C
0,431 #'26.1C
0,489 #'46.1C
0,467 #26.2C
0,466 #66.2C
0,419 #'26.2C
0,466 #'46.2C
0,364 #36.3C
0,379 #46.4C
0,330 #26.5C
0,288 #'16.5C
0,402 #'46.6C
0,641 #26.7C
0,596 #56.7C
0,605 #96.7C
0,439 #16.8C
0,569 #36.8C
0,526 #66.8C
0,536 #96.8C
0,615 #'36.8C
0,460 #'56.8C
0,442 #26.9C
0,526 #46.9C
0,507 #86.9C
0,480 #'26.9C
0,423 #'46.9C
0,349 #'46.'0C
0,354 #46.'2C
0,370 #46.'3C
0,411 #16.'4C
0,451 #46.'4C
0,372 #'76.'4C
0,627 #46.'5C
0,408 #16.'6C
0,392 #'46.'6C
0,705 #26.'7C
0,603 #56.'7C
0,581 #96.'7C
0,556
0,659
0,736
0,528
0,652
0,590
0,590
0,577
0,585
0,476
0,507
0,506
0,401
0,572
0,930
0,763
0,722
0,527
0,679
0,690
0,637
0,757
0,528
0,547
0,621
0,601
0,574
0,532
0,454
0,445
0,444
0,527
0,507
0,464
0,714
0,507
0,490
1,001
0,776
0,769
0,408
0,483
0,558
0,385
0,492
0,449
0,475
0,404
0,456
0,361
0,376
0,335
0,279
0,403
0,630
0,574
0,579
0,416
0,528
0,515
0,477
0,611
0,426
0,428
0,506
0,474
0,453
0,406
0,350
0,370
0,369
0,410
0,400
0,386
0,600
0,400
0,392
0,694
0,576
0,557
#25.1C
#65.1C
#'05.1C
#'35.1C
#'55.1C
#35.2C
#'05.2C
#'35.2C
#'55.2C
#45.3C
0,636
0,658
0,599
0,743
0,669
0,628
0,573
0,661
0,586
0,496
0,484
0,500
0,447
0,608
0,564
0,471
0,452
0,530
0,478
0,364
#26.1C
#66.1C
#'06.1C
#'36.1C
#'56.1C
#36.2C
#'06.2C
#'36.2C
#'56.2C
#46.3C
0,631
0,653
0,528
0,754
0,656
0,616
0,554
0,655
0,584
0,496
0,467
0,486
0,383
0,593
0,483
0,467
0,418
0,517
0,477
0,370
#45.5C
#'75.5C
0,462
0,455
0,318
0,302
#46.5C
#'76.5C
0,424
0,452
0,294
0,293
#45.7C
#75.7C
0,925
0,757
0,664
0,612
#46.7C
#76.7C
0,958
0,757
0,647
0,604
#25.8C
#45.8C
#85.8C
#'25.8C
#'45.8C
0,596
0,849
0,809
0,613
0,562
0,470
0,648
0,628
0,512
0,443
#26.8C
#46.8C
#86.8C
#'26.8C
#'46.8C
0,587
0,825
0,722
0,615
0,537
0,452
0,635
0,562
0,522
0,431
#35.9C
#65.9C
#95.9C
#'35.9C
#'55.9C
0,628
0,587
0,612
0,624
0,520
0,490
0,470
0,486
0,498
0,422
#36.9C
#66.9C
#96.9C
#'36.9C
#'56.9C
0,606
0,588
0,570
0,621
0,520
0,488
0,454
0,452
0,498
0,424
#'65.'2C
#'65.'3C
#25.'4C
#'15.'4C
0,420
0,412
0,555
0,446
0,331 #'66.'2C
0,318 #'66.'3C
0,440 #26.'4C
0,367 #'16.'4C
0,412
0,405
0,558
0,440
0,322
0,320
0,445
0,358
#'15.'5C
#45.'6C
0,530
0,576
0,427 #'16.'5C
0,436 #46.'6C
0,576
0,565
0,458
0,438
#45.'7C
#75.'7C
0,873
0,773
0,764
0,622
0,920
0,775
0,784
0,624
#46.'7C
#76.'7C
Source: Own elaboration
All our results show that the harmonic mean (marked with # before the cypher to denote the
rejection of the null hypothesis) is the measure that minimizes the prediction errors of valuations
using multiples. Only in four cases (EV/OCF Subsector, EV/FCFF Industry/ Supersector and
Sector), and just when the Bonferroni correction is considered, we may not reject the hypothesis
that the harmonic mean and the median produce similar results.
21
For all multiples, except for the EV/TA and the P/B, we may rank the measures as follows: 1st)
harmonic mean, 2nd) geometric mean, 3rd) median, 4th) mean. For the multiples EV/TA and P/B
the rank generally changes to: 1st) harmonic mean, 2nd) median, 3rd) geometric mean, 4th) mean.
One may check in the appendix (Table A.1) an informal ranking to confirm this general rule.
Despite having performed all the formal tests, they are not shown in this document due to the
high amount of pages it would require. When variables are written in italic it indicates that the
null hypothesis may not be rejected.
5.6
Identifying the Best Clustering Method
Here we study which clustering procedure minimizes the estimation errors. The analysed
clustering procedures are: the four ICB levels – Industry (1); Supersector (2); Sector (3) and
Subsector (4), the hierarchical clustering with complete linkage (5) and the non-hierarchical kmeans (6). In our cypher system (see Table 5.8), these different proposals may be read in the
second key.
Table 5.12 and Table 5.13 summarize our conclusions regarding the best clustering procedure,
if any, to conduct a valuation using multiples. We marked the distributions with an asterisk
symbol (*) when the null hypothesis cannot be rejected, i.e., when there is no significant
difference between the clustering procedure, and we marked them with a hash symbol (#) when
the used clustering method minimizes the estimation errors.
Table 5.12: The best ICB level characterized by the mean and the median of its distribution
errors
Distrib.
*03.1C
*04.1C
*03.5C
*04.5C
ICB
#03.9C
#04.9C
*03.'3C
*04.'3C
*03.'7C
*04.'7C
Mean Median Distrib. Mean Median
EV/S
EV/GI
0,806
0,608 *03.2C
0,675
0,505
0,838
0,583 *04.2C
0,688
0,470
EV/TA
EV/OCF
0,482
0,317 *03.6C
0,566
0,394
0,488
0,315 *04.6C
0,565
0,389
P/GI
P/EBITDA
0,632
0,500 *04.'0C
0,508
0,373
0,630
0,482 *03.'0C
0,523
0,387
P/E
P/B
0,445
0,360 *03.'4C
0,553
0,431
0,440
0,350 *04.'4C
0,543
0,431
P/FCFF
1,207
0,723
1,292
0,709
Distrib. Mean Median
EV/EBITDA
*03.3C
0,542
0,380
*04.3C
0,544
0,368
EV/FCFF
*03.7C
1,258
0,648
*04.7C
1,173
0,624
P/EBIT
*03.'1C
0,486
0,368
*04.'1C
0,474
0,362
P/TA
*03.'5C
0,719
0,549
*04.'5C
0,703
0,532
Distrib.
*03.4C
*04.4C
#03.8C
#04.8C
*03.'2C
*04.'2C
*03.'6C
*04.'6C
Mean Median
EV/EBIT
0,584
0,418
0,590
0,402
P/S
0,795
0,614
0,780
0,582
P/EBT
0,444
0,360
0,437
0,345
P/OCF
0,565
0,401
0,551
0,390
Source: Own elaboration
The results shown in Table 5.12 are somewhat surprising because they reveal that there is no
significant difference on what ICB level to use when a valuation using multiples is conducted.
This conclusion conflicts with the results presented by Alford (1992, p. 106) and Schreiner
22
(2007a, p. 110). However, while Alford’s results are based in another classification system, the
SIC system, Schreiner’s ones are not supported by formal tests. This conclusion may reinforce
Schreiner’s idea that the use of a proprietary system should be encouraged because they are
regularly reviewed and adjusted (Schreiner, 2007a, p. 19&70) or may indicate that the number
of selected comparable firms also influences this issue, because we did not limit the number of
peers, or still that the broader sample that we considered can play an important role. Further
investigations on this subject should be carried out.
In fact just for the P/S and the P/GI multiples the results show that the first ICB level (i.e.
Industry) should be substituted by a narrow ICB level, for all other multiples it is irrelevant to
use a broader definition as the 1st ICB level or a narrow classification level.
23
Table 5.13: The best clustering procedure characterized by the mean and the median of its
distribution errors
Distrib.
Set of
ratios 01
Set of
ratios 02
Set of
ratios 04
Set of
ratios 05
Set of
ratios 06
Set of
ratios 07
Set of
ratios 08
Set of
ratios 09
Set of
ratios 10
Set of
ratios 11
Set of
ratios 12
Set of
ratios 13
Set of
ratios 14
Set of
ratios 15
Set of
ratios 16
Set of
ratios 17
Mean Median
EV/S
*16.1C
0,556
0,408
EV/S
*26.1C
0,6307
0,4665
P/S
*26.8C
0,587
0,452
EV/EBITDA
*46.3C
0,4961
0,370
P/S
#46.8C
0,825
0,635
P/B
#46.'4C
0,507
0,400
EV/FCFF
*56.7C
0,763
0,574
EV/S
*66.1C
0,653
0,486
EV/FCFF
*76.7C
0,757
0,604
EV/S
#86.1C
0,736
0,558
EV/FCFF
*96.7C
0,722
0,579
EV/S
#'06.1C
0,528
0,383
EV/TA
*'16.5C
0,401
0,279
EV/S
*'26.1C
0,528
0,385
EV/S
*'36.1C
0,754
0,593
EV/S
*'46.1C
0,652
0,492
P/GI
*'46.9C
0,532
0,406
EV/S
*'56.1C
0,656
0,483
P/EBT
#'66.'2C
0,412
0,322
EV/TA
*'76.5C
0,452
0,293
Distrib.
Mean Median
P/S
#16.8C
0,527
0,416
EV/GI
*26.2C
0,590
0,449
P/GI
*26.9C
0,547
0,428
EV/EBIT
*46.4C
0,507
0,376
P/GI
#46.9C
0,621
0,506
P/TA
#46.'5C
0,714
0,600
P/FCFF
*56.'7C
0,776
0,576
EV/GI
*66.2C
0,590
0,475
P/FCFF
*76.'7C
0,775
0,624
P/S
#86.8C
0,722
0,562
P/S
#96.8C
0,637
0,477
EV/GI
*'06.2C
0,554
0,418
P/B
*'16.'4C
0,440
0,358
EV/GI
*'26.2C
0,577
0,404
EV/GI
*'36.2C
0,655
0,517
EV/GI
*'46.2C
0,585
0,456
P/EBITDA
#'46.'0C
0,454
0,350
EV/GI
*'56.2C
0,584
0,477
P/E
#'66.'3C
0,405
0,320
P/B
#'75.'4C
0,448
0,372
Distrib.
*16.'4C
*26.5C
*26.'4C
#46.5C
*46.'2C
*46.'6C
*66.8C
#86.9C
#96.9C
#'15.'5C
*'26.8C
*'36.8C
*'46.6C
*'46.'6C
#''56.8C
Mean Median Distrib. Mean Median
P/B
P/OCF
0,527
0,410 *16.'6C
0,507
0,400
EV/TA
EV/FCFF
0,506
0,335 *26.7C
0,930
0,630
P/B
P/FCFF
0,558
0,445 *26.'7C
1,001
0,694
EV/TA
EV/FCFF
0,424
0,294 *46.7C
0,958
0,647
P/EBT
P/E
0,445
0,370 *46.'3C
0,444
0,369
P/OCF
P/FCFF
0,565
0,438 *46.'7C
0,920
0,784
P/S
0,690
P/GI
0,601
P/GI
0,570
P/TA
0,530
P/S
0,615
P/S
0,757
EV/OCF
0,572
P/OCF
0,490
P/S
0,528
0,515 *66.9C
P/GI
0,588
0,454
P/FCFF
0,769
0,557
0,474
0,452 *96.'7C
0,427
0,522 *'26.9C
0,611 *'36.9C
0,403 #'46.8C
P/GI
0,574
P/GI
0,621
P/S
0,537
0,453
0,498
0,431
0,392
0,426 *'56.9C
P/GI
0,520
0,424
Source: Own elaboration
Concerning the better clustering approach when a valuation is done upon a set of ratios, the
results show that, for most multiples, it is identical to use a hierarchical (5) or a k-means
clustering approach (6). In only 19 cases among 67 we concluded that the clustering method
has an impact on the estimation errors. When it was concluded for the preference of a clustering
procedure in almost all cases (17 cases), it is better to use the k-means approach. Further
considerations regarding these results and the establishment of regularities, are not likely to be
done.
24
In order to continue our study in the next sections, we will select the k-means approach when a
clustering approach may not be relegated, using sets of ratios. For the ICB approach we will
choose the 3-digit level, following Schreiner’s suggestion (2007a, p. 128), except for the
P/EBITDA multiple, for which we have chosen the 4-digit level due to an ad-hoc consideration.
5.7
The Best Performing Multiples
In this section we discuss which multiples are better suited to perform a valuation, having in
consideration the investigated clustering approaches. In Table 5.14 and Table 5.15 we
summarize the findings from our formal tests.
Table 5.14: The best market multiples characterized by the mean and the median of its
distribution errors – Part I
Clustering
method
ICB
Multiple
EV/TA
EV/EBITDA
EV/OCF
EV/EBIT
EV/GI
EV/S
EV/FCFF
Entity Multiples
Distrib.
Mean
Median
#03.5C
0,482
0,317
#03.3C
0,542
0,380
b03.6C
0,566
0,394
b03.4C
0,584
0,418
03.2C
0,675
0,505
03.1C
0,806
0,608
b03.7C
1,258
0,648
Multiple
P/EBT
P/E
P/EBIT
P/EBITDA
P/B
P/OCF
P/GI
P/TA
P/S
P/FCFF
Equity Multiples
Distrib.
Mean
Median
#03.'2C
0,444
0,360
#03.'3C
0,445
0,360
b03.'1C
0,486
0,368
b04.'0C
0,508
0,373
b03.'4C
0,553
0,431
b03.'6C
0,565
0,401
03.9C
0,632
0,500
03.'5C
0,719
0,549
03.8C
0,795
0,614
03.'7C
1,207
0,723
Source: Own elaboration
We marked the distributions’ code with a hash symbol (#) when the investigated multiple
provides similar results as the best performing multiple appearing in first place; we marked
them with a “b” when these multiples produce similar results as the best performing multiple
(ranked first) but only when the Bonferroni correction is considered. We have distinguished the
latter case from the first because the Bonferroni correction plays an important role when we
compare several multiples. For instance, for the ICB clustering method we compare 10 (𝑥)
different equity multiples which leads to a correction of 45 (𝑥 ∗ [𝑥 − 1]/2) times (α′ = α/45)
on the considered significance level. We also marked the distributions with an asterisk symbol
(*) when there is no statistical significant difference between the analysed multiples.
25
Table 5.15: The best market multiples characterized by the mean and the median of its
distribution errors – Part II
Clustering
method
Multiple
EV/S
Entity Multiples
Distrib.
Mean
Median
16.1C
0,556
0,408
Set of ratios 01
Set of ratios 02
Set of ratios 03
Set of ratios 04
Set of ratios 05
Set of ratios 06
Set of ratios 07
Set of ratios 08
EV/TA
EV/GI
EV/S
EV/FCFF
EV/EBITDA
EV/GI
EV/S
EV/TA
EV/EBITDA
EV/EBIT
EV/FCFF
#26.5C
26.2C
26.1C
26.7C
#36.3C
36.2C
36.1C
#46.5C
46.3C
46.4C
46.7C
0,506
0,590
0,631
0,930
0,476
0,616
0,659
0,424
0,496
0,507
0,958
0,335
0,449
0,467
0,630
0,361
0,467
0,483
0,294
0,370
0,376
0,647
EV/FCFF
EV/GI
EV/S
EV/FCFF
EV/S
56.7C
*66.2C
*66.1C
76.7C
86.1C
0,763
0,590
0,653
0,757
0,736
0,574
0,475
0,486
0,604
0,558
EV/FCFF
96.7C
0,722
0,579
EV/S
EV/GI
EV/TA
#'06.1C
'06.2C
'16.5C
0,528
0,554
0,401
0,383
0,418
0,279
EV/S
EV/GI
EV/GI
EV/S
EV/OCF
EV/GI
EV/S
*'26.1C
*'26.2C
*'36.2C
*'36.1C
#'46.6C
'46.2C
#'46.1C
0,528
0,577
0,655
0,754
0,572
0,585
0,652
0,385
0,404
0,517
0,593
0,403
0,456
0,492
EV/GI
EV/S
*'56.2C
*'56.1C
0,584
0,656
0,477
0,483
EV/TA
'76.5C
0,452
0,293
Set of ratios 09
Set of ratios 10
Set of ratios 11
Set of ratios 12
Set of ratios 13
Set of ratios 14
Set of ratios 15
Set of ratios 16
Set of ratios 17
P/OCF
P/S
P/B
P/GI
P/B
P/S
P/FCFF
P/GI
P/S
Equity Multiples
Distrib.
Mean
Median
*16.'6C
0,507
0,400
*16.8C
0,527
0,416
*16.'4C
0,527
0,410
#26.9C
0,547
0,428
#26.'4C
0,558
0,445
#26.8C
0,587
0,452
26.'7C
1,001
0,694
#36.9C
0,606
0,488
36.8C
0,679
0,528
P/E
P/EBT
P/B
P/OCF
P/GI
P/TA
P/S
P/FCFF
P/FCFF
P/GI
P/S
P/FCFF
P/GI
P/S
P/GI
P/S
P/FCFF
#46.'3C
#46.'2C
#46.'4C
b46.'6C
46.9C
46.'5C
46.8C
46.'7C
56.'7C
#66.9C
66.8C
76.'7C
#86.9C
86.8C
#96.9C
b96.8C
96.'7C
0,444
0,445
0,507
0,565
0,621
0,714
0,825
0,920
0,776
0,588
0,690
0,775
0,601
0,722
0,570
0,637
0,769
0,369
0,370
0,400
0,438
0,506
0,600
0,635
0,784
0,576
0,454
0,515
0,624
0,474
0,562
0,452
0,477
0,557
P/B
P/TA
P/GI
P/S
P/GI
P/S
P/EBITDA
P/OCF
P/GI
P/S
P/GI
P/S
P/E
P/EBT
P/B
#'16.'4C
'15.'5C
*'26.9C
*'26.8C
#'36.9C
'36.8C
#'46.'0C
#'46.'6C
'46.9C
'46.8C
*'56.9C
*'56.8C
*'66.'3C
*'66.'2C
'75.'4C
0,440
0,530
0,574
0,615
0,621
0,757
0,454
0,490
0,532
0,537
0,520
0,528
0,405
0,412
0,448
0,358
0,427
0,453
0,522
0,498
0,611
0,350
0,392
0,406
0,431
0,424
0,426
0,320
0,322
0,372
Multiple
Source: Own elaboration
As we may notice on the above tables, the EV/TA and the EV/EBITDA multiples are the ones
amongst the better entity multiples for the considered clustering procedures, followed by the
EV/EBIT and the EV/OCF multiples. On the side of the equity multiples, the P/E, the P/EBT
and the P/B multiples rank always among the best market multiples, usually in this order.
26
However, as we referred on Section 3, we cannot directly compare entity to equity multiples
using the estimation errors since the underlying variables are not the same.
A more detailed ranking may not be given due to the impossibility to conduct a transitive
thinking when the comparison of multiples is performed running formal tests.
5.8
The Best Set of Ratios vs the ICB approach
At last, we investigate if a process of gathering firms in order to carry a valuation using
multiples may be better accomplished if we rely on the financial characteristics rather than the
same industry definition. We summarize our conclusions in Table 5.16 and Table 5.17, marking
the distributions’ codes with the same notations (#; “b” and *) as in Section 5.7.
Table 5.16: The best set of ratios vs the ICB approach characterized by the mean and the
median of its distribution errors – Part I
Multiple
EV/TA
EV/EBITDA
EV/EBIT
EV/OCF
Clustering
measures
Set of ratios 11
Set of ratios 04
Set of ratios 17
ICB
Set of ratios 02
Set of ratios 03
Set of ratios 04
ICB
Set of ratios 04
ICB
ICB
Set of ratios 14
Entity Multiples
Distrib.
Mean
Median
#'16.5C
0,401
0,279
b46.5C
0,424
0,294
'76.5C
0,452
0,293
03.5C
0,482
0,317
26.5C
0,506
0,335
#36.3C
0,476
0,361
46.3C
0,496
0,370
b03.3C
0,542
0,380
*46.4C
0,507
0,376
*03.4C
0,584
0,418
*03.6C
0,566
0,394
*'46.6C
0,572
0,403
Multiple
Clustering
measures
P/E
Set of ratios 16
Set of ratios 04
ICB
P/EBT
Set of ratios 16
ICB
Set of ratios 04
Set of ratios 11
Set of ratios 17
Set of ratios 04
Set of ratios 01
P/B
ICB
Set of ratios 02
Equity Multiples
Distrib.
Mean
Median
#'66.'3C
0,405
0,320
46.'3C
0,444
0,369
#03.'3C
0,445
0,360
#'66.'2C
#03.'2C
46.'2C
#'16.'4C
'75.'4C
46.'4C
16.'4C
03.'4C
26.'4C
0,412
0,444
0,445
0,440
0,448
0,507
0,527
0,553
0,558
0,322
0,360
0,370
0,358
0,372
0,400
0,410
0,431
0,445
Source: Own elaboration
The most promising multiples determined in Section 5.7, stated in Table 5.16, show how
effective the use of the financial characteristics to tie up comparable companies is. For the
EV/TA, the EV/EBITDA and the P/B multiples, the use of sets of ratios is highly compensated
by the decreasing of the estimation errors. In fact, even for the remaining multiples (EV/EBIT;
EV/OCF; P/E and P/EBT) the use of the financial characteristics to group the comparable firms
performs similarly well as the use of the industry criterion – some present average estimation
errors smaller but the difference is not statistically significant.
Here we may also relate how the performance of the used set of ratios is determined by the
correlation level analysed in Section 5.2. For instance, the EV/TA multiple is highly improved
when we use the set of ratios 11 - ln(RoA) and ln(RoE) – which present a Spearman’s
correlation with the EV/TA multiple of 0,55 and 0,56 respectively. Also, concerning the EV/TA
multiple, the set of ratios 04 – RoA and RoE (which is similar but does not force values to be
27
positive), has Spearman’s correlations of 0,46 and 0,48 respectively; and the set of ratios 17
with correlations of 0,28 with the growth rate of Sales (Compound Annual Growth Rate, or
CAGR, of the last 4 years), 0,32 with the growth rate of the Net Income (CAGR of the last 4
years) and 0,56 with the ln(RoE). The same applies to the other analysed multiples regarding
its associated set of ratios. This reinforces the idea that using sets of ratios is beneficial, but not
just any set, some customization is needed. Another positive fact is that the sets of ratios highly
ranked are relatively parsimonious as concerns the number of formed clusters: set of ratios 11
(8 clusters); set of ratios 04 (19 clusters); set of ratios 17 (24 clusters); set of ratios 16 (6
clusters); set of ratios 03 (10 clusters); and set of ratios 16 (6 clusters), to name a few.
Table 5.17: The best set of ratios vs the ICB approach characterized by the mean and the
median of its distribution errors - Part II
Multiple
EV/GI
EV/S
EV/FCFF
Clustering
measures
Set of ratios 10
Set of ratios 12
Set of ratios 15
Set of ratios 14
Set of ratios 06
Set of ratios 02
Set of ratios 03
Set of ratios 13
ICB
Set of ratios 10
Set of ratios 12
Set of ratios 01
Set of ratios 02
Set of ratios 14
Set of ratios 06
Set of ratios 15
Set of ratios 03
Set of ratios 08
Set of ratios 13
ICB
Set of ratios 09
Set of ratios 07
Set of ratios 05
Set of ratios 02
Set of ratios 04
ICB
Entity Multiples
Distrib.
Mean
Median
#'06.2C
0,554
0,418
#'26.2C
0,577
0,404
'56.2C
0,584
0,477
'46.2C
0,585
0,456
b66.2C
0,590
0,475
b26.2C
0,590
0,449
36.2C
0,616
0,467
'36.2C
0,655
0,517
b03.2C
0,675
0,505
#'06.1C
0,528
0,383
b'26.1C
0,528
0,385
16.1C
0,556
0,408
26.1C
0,631
0,467
'46.1C
0,652
0,492
66.1C
0,653
0,486
'56.1C
0,656
0,483
36.1C
0,659
0,483
86.1C
0,736
0,558
'36.1C
0,754
0,593
03.1C
0,806
0,608
*96.7C
0,722
0,579
*76.7C
0,757
0,604
*56.7C
0,763
0,574
*26.7C
0,930
0,630
*46.7C
0,958
0,647
*03.7C
1,258
0,648
Multiple
Clustering
measures
P/EBITDA
Set of ratios 14
P/OCF
P/GI
P/TA
P/S
P/FCFF
Source: Own elaboration
28
ICB
Set of ratios 14
Set of ratios 01
ICB
Set of ratios 04
Set of ratios 09
Set of ratios 15
Set of ratios 14
Set of ratios 02
Set of ratios 12
Set of ratios 06
Set of ratios 08
Set of ratios 03
Set of ratios 13
Set of ratios 04
ICB
Set of ratios 11
Set of ratios 04
ICB
Set of ratios 01
Set of ratios 15
Set of ratios 14
Set of ratios 02
Set of ratios 12
Set of ratios 09
Set of ratios 03
Set of ratios 06
Set of ratios 08
Set of ratios 13
ICB
Set of ratios 04
Set of ratios 09
Set of ratios 07
Set of ratios 05
Set of ratios 04
Set of ratios 02
ICB
Equity Multiples
Distrib.
Mean
Median
*'46.'0C
0,454
0,350
*04.'0C
0,508
0,373
*'46.'6C
0,490
0,392
*16.'6C
0,507
0,400
*03.'6C
0,565
0,401
*46.'6C
0,565
0,438
#96.9C
0,570
0,452
#'56.9C
0,520
0,424
#'46.9C
0,532
0,406
b26.9C
0,547
0,428
#'26.9C
0,574
0,453
#66.9C
0,588
0,454
b86.9C
0,601
0,474
b36.9C
0,606
0,488
'36.9C
0,621
0,498
46.9C
0,621
0,506
03.9C
0,632
0,500
#'15.'5C
0,530
0,427
46.'5C
0,714
0,600
03.'5C
0,719
0,549
#16.8C
0,527
0,416
'56.8C
0,528
0,426
b'46.8C
0,537
0,431
26.8C
0,587
0,452
b'26.8C
0,615
0,522
96.8C
0,637
0,477
36.8C
0,679
0,528
66.8C
0,690
0,515
86.8C
0,722
0,562
'36.8C
0,757
0,611
03.8C
0,795
0,614
46.8C
0,825
0,635
*96.'7C
0,769
0,557
*76.'7C
0,775
0,624
*56.'7C
0,776
0,576
*46.'7C
0,920
0,784
*26.'7C
1,001
0,694
*03.'7C
1,207
0,723
Table 5.17 presents identical results for the remaining multiples. The multiples for which the
estimation errors are lower using sets of ratios instead of the same industry criterion are the
EV/GI, the EV/S; the P/GI; the P/TA and the P/S. For the other multiples (EV/FCFF;
P/EBITDA; P/OCF and P/FCFF) it is similar to use an approach using the set of ratios ranked
first or the same industry criterion.
6 Conclusions and Future Research
The main purpose of this study was to investigate if relying on the financial characteristics in
order to conduct a valuation delivers better results than using the same industry criterion
commonly employed. We investigated further questions related to each step of the valuation
process such as the best aggregation measure; the best clustering procedure and the best
performing multiples. The main investigation questions were the following: 1) Which financial
ratios are highly correlated with the market multiples?; 2) What are the best central tendency
measures to perform a valuation (mean, median, harmonic mean or geometric mean)?; 3) The
use of economic and financial ratios to gather comparable companies performs better than the
use of same industry principle? Formal tests were run to answer these questions.
We conducted a broad investigation over 17 market multiples and several financial ratios, on a
sample of 7.590 companies from several countries of the world. The year to which the figures
are reported is the 2011.
We concluded that the harmonic mean performs better for all multiples and clustering
procedures. The second best measure is usually the geometric mean, followed by the median
and the mean. When it comes to the EV/TA and the P/B multiples, the median performs better
than the geometric mean but the other measures do not change their rank.
The best clustering approach examination, i.e. hierarchical vs. non-hierarchical clustering using
the selected sets of ratios or the ICB level when an industry classification is employed, allowed
concluding that, for almost all multiples, there aren’t significant dissimilarities subjacent to this
choice. When there is a difference statistically significant between the hierarchical and the nonhierarchical clustering we conclude that the k-means approach minimizes the estimation errors.
But even in these cases the hierarchical analysis was important because it allowed identifying
the number of subjacent clusters. The finding regarding the indifference of the classification
(ICB) level used conflicts with Alford (1992, p. 106) and Schreiner’s (2007a, p. 110), despite
the underlying methodology not being exactly comparable.
29
The market multiples that lead us to the smaller estimation errors were the EV/TA, the
EV/EBITDA, the EV/EBIT and the EV/OCF on the side of the entity multiples and the P/E, the
P/EBT and the P/B, when it comes to equity multiples. A broader ranking revealed hard to
establish due to the intransitivity of positions.
Finally, we found that employing sets of ratios, i.e. financial characteristics, to gather
comparable firms improves the estimation errors of almost all multiples. Even when we cannot
conclude that the use of financial parameters improves the estimation errors, they are equally
effective.
We believe that further investigations regarding the consistency of these results over time may
be of interest. Limiting the sample to more homogeneous countries or to countries alone may
also have an impact on the results. The exclusion from the sample of banks and insurance firms,
since they have different regulatory rules to fulfil and are usually treated separately on
valuations, could be of interest as well. The investigated procedures should also include
forecasted multiples (the forward P/E, for example) as they are quite well ranked in the related
literature.
In conclusion, we believe these results are important because they do not only indicate the more
reliable market multiples and procedures to conduct a valuation, but also indicate the different
financial factors with impact on each multiple.
References
Alford, A. W. (1992). The effect of the set of comparable firms on the accuracy of the priceearnings valuation method. Journal of Accounting Research, Vol. 30, 94-108.
Baker, M., & Ruback, R. S. (1999). Estimating Industry Multiples. Boston: Working Paper,
Harvard Business School.
Bhojraj, S., & Lee, C. M. (2002). Who Is My Peer? A Valuation-based approach to the selection
of comparable firms. Journal of Accounting Research Vol. 40, Issue 2, 407-439.
Bodie, Z., Kane, A., & Marcus, A. J. (2008). Investments (7th ed.). McGraw-Hill International
Edition.
Brandão, E. (2002). Finanças (2ª ed.). Porto Editora.
Brealey, R. A., Myers, S. C., & Allen, F. (2008). Principles of Corporate Finance (9th ed.).
McGraw-Hill.
30
Brigham, E. F., & Gapenski, L. C. (1994). Financial Management - Theory and Practice (7ª
ed.). The Dryden Press.
Cheng, C., & McNamara , R. (2000). The Valuation Accuracy of the Price-Earnings and PriceBook Benchmark Valuation Methods. Review of Quantitative Finance and Accounting Vol. 15,
349-370.
Cohen, R. D. (2005). The Relative Valuation of an Equity Price Index. Obtido de
http://rdcohen.50megs.com/RVEPIabstract.htm
Cooper, I., & Cordeiro, L. (2008). Optimal Equity Valuation Using Multiples: The Number of
Comparable Firms. SSRN Working Paper n. 1272349.
Damodaran, A. (1997). Corporate Finance - Theory and Practice. New York: John Wiley &
Sons, Inc.
Damodaran, A. (2002). Investment Valuation: Tools and Techniques for Determining the Value
of Any Asset (2nd ed.). New York: John Wiley and Sons Inc.
Dittmann, I., & Weiner, C. (2005). Selecting Comparables for the Valuation of European Firms.
SFB 649 Discussion Paper n. SFB649DP2005-002, Humboldt-Universität zu Berlin, School of
Business and Economics, Germany.
Dunn, O. (1961). Multiple Comparisons Among Means. Journal of the American Statistical
Association, 56, pp. 52-64.
Fernández, P. (2007). Company Valuation Methods: The Most Common Errors in Valuations.
SSRN Working Paper n. 1019977.
Guimarães, J. F. (1995). A obrigatoriedade da nomeação de um ROC pelas empresas. Boletim
da CROC, n.º3, Abril/Junho, 23-4.
Herrmann, V., & Richter, F. (Julho de 2003). Pricing with Performance-Controlled Multiples.
Schmalenbach Business Review,Vol. 55, 194–219.
Kim, M., & Ritter, J. R. (Setembro de 1999). Valuing IPOs. Journal of Financial Economics,
Vol. 53, No. 3, 409-437.
Liu, J., Nissim, D., & Thomas, J. (2002). Equity Valuation Using Multiples. Journal of
Accounting Research Vol. 40, Issue 1: 135-172.
Maroco, J. (2007). Análise Estatística - Com utilização do SPSS (3ª ed.). Lisboa: Edições
Sílabo, Lda.
Rodrigues, J. (2009). Sistema de Normalização Contabilística Explicado. Porto: Porto Editora.
Schreiner, A. (2007a). Equity Valuation Using Multiples: An Empirical Investigation¸
Dissertation, University of St.Gallen, N.º 3313. Austria.
31
Sloan, R. G. (2002). Discussion of Who Is My Peer? A Valuation-Based Approach to the
Selection of Comparable Firms. Journal of Accounting Research, Vol. 40, Issue 2: 441-444.
Thomson Reuters. (29 de Setembro de 2010). Worldscope Database: Data Definitions Guide.
Yoo, Y. K. (2006). The Valuation Accuracy of Equity Valuation Using a Combination of
Multiples. Review of Accounting and Finance, Vol. 5: 108-123.
32
Appendix
Table A.1: Estimation errors by central tendency measure, market multiple and ICB level,
characterized by the mean and the median of its distribution errors - Part I
P/E
P/EBT
P/EBIT
P/EBITDA
P/GI
P/S
EV/FCFF
EV/OCF
EV/TA
EV/EBIT
EV/EBITDA
EV/GI
EV/S
Distrib.
#01.1C
01.1D
01.1B
01.1A
#01.2C
01.2D
01.2B
01.2A
#01.3C
01.3D
01.3B
01.3A
#01.4C
01.4D
01.4B
01.4A
#01.5C
01.5B
01.5D
01.5A
#01.6C
01.6D
01.6B
01.6A
01.7C
01.7D
01.7B
01.7A
#01.8C
01.8D
01.8B
01.8A
#01.9C
01.9D
01.9B
01.9A
#01.'0C
01.'0D
01.'0B
01.'0A
#01.'1C
01.'1D
01.'1B
01.'1A
#01.'2C
01.'2D
01.'2B
01.'2A
#01.'3C
01.'3D
01.'3B
01.'3A
Industry
Mean Median
0,812
1,238
1,348
1,934
0,660
0,850
0,916
1,129
0,539
0,631
0,668
0,749
0,575
0,678
0,717
0,822
0,479
0,542
0,561
0,685
0,596
0,740
0,813
0,917
1,092
2,094
2,486
3,411
0,847
1,299
1,372
2,051
0,753
0,835
0,895
1,116
0,529
0,611
0,629
0,734
0,486
0,549
0,562
0,662
0,445
0,491
0,496
0,574
0,442
0,481
0,490
0,562
0,639
0,590
0,579
0,641
0,517
0,458
0,440
0,465
0,394
0,342
0,338
0,363
0,425
0,346
0,351
0,367
0,315
0,328
0,336
0,380
0,420
0,374
0,360
0,388
0,702
0,514
0,520
0,567
0,661
0,603
0,603
0,635
0,502
0,479
0,480
0,498
0,409
0,365
0,365
0,386
0,380
0,340
0,343
0,358
0,361
0,336
0,339
0,355
0,367
0,332
0,334
0,338
Distrib.
#02.1C
02.1D
02.1B
02.1A
#02.2C
02.2D
02.2B
02.2A
#02.3C
02.3D
02.3B
02.3A
#02.4C
02.4D
02.4B
02.4A
#02.5C
02.5B
02.5D
02.5A
#02.6C
02.6D
02.6B
02.6A
02.7C
02.7D
02.7B
02.7A
#02.8C
02.8D
02.8B
02.8A
#02.9C
02.9D
02.9B
02.9A
#02.'0C
02.'0D
02.'0B
02.'0A
#02.'1C
02.'1D
02.'1B
02.'1A
#02.'2C
02.'2D
02.'2B
02.'2A
#02.'3C
02.'3D
02.'3B
02.'3A
Supersector
Mean Median
0,810
1,209
1,345
1,832
0,655
0,831
0,884
1,092
0,542
0,619
0,661
0,734
0,580
0,669
0,712
0,812
0,485
0,543
0,557
0,680
0,575
0,697
0,741
0,857
1,170
2,152
2,576
3,428
0,838
1,270
1,357
1,956
0,647
0,817
0,866
1,087
0,518
0,602
0,624
0,723
0,483
0,550
0,569
0,658
0,445
0,490
0,501
0,573
0,440
0,481
0,490
0,564
0,623
0,557
0,559
0,600
0,514
0,427
0,430
0,458
0,382
0,334
0,329
0,349
0,412
0,339
0,350
0,363
0,324
0,323
0,329
0,376
0,408
0,357
0,348
0,368
0,660
0,527
0,520
0,555
0,647
0,576
0,557
0,585
0,515
0,480
0,476
0,490
0,398
0,360
0,362
0,371
0,375
0,334
0,331
0,354
0,360
0,340
0,335
0,352
0,365
0,336
0,334
0,333
33
Distrib.
Sector
Mean
Median
Distrib.
#03.1C
03.1D
03.1B
03.1A
#03.2C
03.2D
03.2B
03.2A
#03.3C
03.3D
03.3B
03.3A
#03.4C
03.4D
03.4B
03.4A
#03.5C
03.5B
03.5D
03.5A
#03.6C
03.6D
03.6B
03.6A
03.7C
03.7D
03.7B
03.7A
#03.8C
03.8D
03.8B
03.8A
#03.9C
03.9D
03.9B
03.9A
#03.'0C
03.'0D
03.'0B
03.'0A
#03.'1C
03.'1D
03.'1B
03.'1A
#03.'2C
03.'2D
03.'2B
03.'2A
#03.'3C
03.'3D
03.'3B
03.'3A
0,806
1,163
1,290
1,721
0,675
0,837
0,880
1,082
0,542
0,616
0,656
0,729
0,584
0,667
0,711
0,806
0,482
0,545
0,554
0,670
0,566
0,684
0,729
0,837
1,258
2,145
2,500
3,359
0,795
1,190
1,288
1,827
0,632
0,788
0,829
1,049
0,523
0,596
0,621
0,712
0,486
0,546
0,565
0,650
0,444
0,488
0,495
0,569
0,445
0,480
0,489
0,560
0,608
0,540
0,531
0,578
0,505
0,437
0,417
0,444
0,380
0,325
0,322
0,343
0,418
0,346
0,348
0,355
0,317
0,322
0,325
0,368
0,394
0,353
0,347
0,348
0,648
0,516
0,521
0,560
0,614
0,558
0,543
0,574
0,500
0,466
0,470
0,482
0,387
0,355
0,353
0,364
0,368
0,334
0,334
0,347
0,360
0,335
0,335
0,349
0,360
0,332
0,328
0,329
#04.1C
04.1D
04.1B
04.1A
#04.2C
04.2D
04.2B
04.2A
#04.3C
04.3D
04.3B
04.3A
#04.4C
04.4D
04.4B
04.4A
#04.5C
04.5B
04.5D
04.5A
04.6C
04.6D
04.6B
04.6A
#04.7C
04.7D
04.7B
04.7A
#04.8C
04.8D
04.8B
04.8A
#04.9C
04.9D
04.9B
04.9A
#04.'0C
04.'0D
04.'0B
04.'0A
#04.'1C
04.'1D
04.'1B
04.'1A
#04.'2C
04.'2D
04.'2B
04.'2A
#04.'3C
04.'3D
04.'3B
04.'3A
Subsector
Mean Median
0,838
1,140
1,248
1,623
0,688
0,828
0,876
1,051
0,544
0,611
0,648
0,721
0,590
0,666
0,705
0,797
0,488
0,551
0,553
0,660
0,565
0,677
0,720
0,828
1,173
1,876
2,079
2,985
0,780
1,125
1,223
1,670
0,630
0,770
0,806
1,015
0,508
0,580
0,604
0,693
0,474
0,529
0,541
0,624
0,437
0,479
0,486
0,555
0,440
0,470
0,478
0,545
0,583
0,513
0,488
0,544
0,470
0,419
0,408
0,451
0,368
0,330
0,324
0,336
0,402
0,350
0,344
0,358
0,315
0,311
0,317
0,358
0,389
0,347
0,348
0,342
0,624
0,523
0,528
0,582
0,582
0,537
0,504
0,548
0,482
0,453
0,455
0,470
0,373
0,355
0,357
0,370
0,362
0,337
0,339
0,356
0,345
0,331
0,331
0,355
0,350
0,324
0,316
0,340
Table A.1: Estimation errors by central tendency measure, market multiple and ICB level,
characterized by the mean and the median of its distribution errors - Part II
P/FCFF
P/OCF
P/TA
P/B
Distrib.
#01.'4C
01.'4B
01.'4D
01.'4A
#01.'5C
01.'5D
01.'5B
01.'5A
#01.'6C
01.'6D
01.'6B
01.'6A
#01.'7C
01.'7D
01.'7B
01.'7A
Industry
Mean Median
0,553
0,668
0,672
0,862
0,734
1,089
1,190
1,603
0,583
0,713
0,756
0,885
1,179
2,269
2,653
3,832
0,437
0,433
0,437
0,477
0,613
0,497
0,503
0,551
0,454
0,415
0,411
0,415
0,767
0,572
0,573
0,602
Distrib.
#02.'4C
02.'4B
02.'4D
02.'4A
#02.'5C
02.'5D
02.'5B
02.'5A
#02.'6C
02.'6D
02.'6B
02.'6A
#02.'7C
02.'7D
02.'7B
02.'7A
Supersector
Mean Median
0,551
0,660
0,666
0,854
0,715
1,041
1,128
1,533
0,566
0,686
0,728
0,848
1,180
2,234
2,610
3,740
0,421
0,431
0,434
0,481
0,564
0,499
0,499
0,530
0,417
0,394
0,387
0,400
0,742
0,557
0,553
0,595
34
Distrib.
Sector
Mean
Median
Distrib.
#03.'4C
03.'4B
03.'4D
03.'4A
#03.'5C
03.'5D
03.'5B
03.'5A
#03.'6C
03.'6D
03.'6B
03.'6A
#03.'7C
03.'7D
03.'7B
03.'7A
0,553
0,661
0,659
0,836
0,719
1,022
1,094
1,523
0,565
0,679
0,725
0,835
1,207
2,212
2,533
3,684
0,431
0,436
0,429
0,468
0,549
0,499
0,500
0,522
0,401
0,378
0,382
0,392
0,723
0,553
0,564
0,593
#04.'4C
04.'4D
04.'4B
04.'4A
#04.'5C
04.'5D
04.'5B
04.'5A
#04.'6C
04.'6D
04.'6B
04.'6A
#04.'7C
04.'7D
04.'7B
04.'7A
Subsector
Mean Median
0,543
0,643
0,659
0,808
0,703
0,975
1,045
1,413
0,551
0,659
0,685
0,814
1,292
2,132
2,373
3,527
0,431
0,417
0,422
0,454
0,532
0,491
0,485
0,511
0,390
0,378
0,371
0,388
0,709
0,565
0,565
0,608
Table A.2: Estimation errors by central tendency measure, market multiple and clustering
method, characterized by the mean and the median of its distribution errors - Part I
EV/S
Set of ratios 01
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#15.1C
15.1D
15.1B
15.1A
0,574
0,724
0,742
0,998
0,427
0,402
0,406
0,449
#16.1C
16.1D
16.1B
16.1A
0,556
0,672
0,694
0,905
0,408
0,369
0,373
0,416
EV/S
Set of ratios 03
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#35.1C
35.1D
35.1B
35.1A
0,700
0,945
0,961
1,382
0,502
0,451
0,455
0,503
#36.1C
36.1D
36.1B
36.1A
0,659
0,859
0,878
1,215
0,483
0,438
0,435
0,492
EV/S
EV/S
EV/S
Set of ratios 08
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#85.1C
85.1D
85.1B
85.1A
0,796
1,236
1,278
2,011
0,640
0,587
0,593
0,684
#86.1C
86.1B
86.1D
86.1A
0,736
1,080
1,089
1,792
0,558
0,533
0,531
0,648
EV/GI
0,636
0,830
0,848
1,196
0,484
0,416
0,421
0,480
#26.1C
26.1D
26.1B
26.1A
0,631
0,818
0,833
1,174
0,467
0,421
0,412
0,472
Set of ratios 06
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#65.1C
65.1D
65.1B
65.1A
0,658
0,865
0,868
1,311
0,500
0,480
0,475
0,578
#66.1C
66.1D
66.1B
66.1A
0,653
0,845
0,888
1,210
0,486
0,435
0,431
0,501
Set of ratios 10
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'05.1C
05.1D
05.1B
05.1A
0,599
0,744
0,777
0,978
0,447
0,403
0,393
0,427
#'06.1C
06.1D
06.1B
06.1A
0,528
0,610
0,630
0,765
0,383
0,344
0,346
0,373
Set of ratios 13
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'25.1C
25.1D
25.1B
25.1A
#'35.1C
35.1D
35.1B
35.1A
0,530
0,585
0,609
0,701
0,431
0,439
0,442
0,434
#'26.1C
26.1D
26.1B
26.1A
0,528
0,586
0,613
0,695
0,385
0,382
0,396
0,431
0,743
1,051
1,121
1,543
0,608
0,523
0,524
0,589
#'36.1C
36.1D
36.1B
36.1A
0,754
1,051
1,117
1,534
0,593
0,526
0,517
0,579
Set of ratios 14
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 15
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'45.1C
45.1D
45.1B
45.1A
#'55.1C
55.1D
55.1B
55.1A
0,660
0,873
0,870
1,337
0,489
0,458
0,454
0,568
#'46.1C
46.1D
46.1B
46.1A
0,652
0,857
0,863
1,282
0,492
0,458
0,453
0,548
#25.2C
25.2D
25.2B
25.2A
0,599
0,750
0,763
1,000
0,467
0,429
0,429
0,462
#26.2C
26.2D
26.2B
26.2A
0,590
0,745
0,769
0,994
0,449
0,416
0,418
0,459
Set of ratios 06
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
EV/GI
#25.1C
25.1D
25.1B
25.1A
Set of ratios 12
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 02
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
EV/GI
Set of ratios 02
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#65.2C
65.2D
65.2B
65.2A
0,601
0,750
0,776
0,997
0,466
0,437
0,439
0,482
#66.2C
66.2D
66.2B
66.2A
0,590
0,740
0,770
0,970
0,475
0,433
0,435
0,451
0,669
0,847
0,885
1,139
0,564
0,458
0,455
0,475
#'56.1C
56.1D
56.1B
56.1A
0,656
0,797
0,834
1,049
0,483
0,434
0,434
0,453
Set of ratios 03
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#35.2C
35.2D
35.2B
35.2A
0,628
0,775
0,814
1,011
0,471
0,414
0,408
0,427
#36.2C
36.2D
36.2B
36.2A
0,616
0,763
0,790
0,994
0,467
0,413
0,413
0,432
Set of ratios 10
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'05.2C
05.2D
05.2B
05.2A
0,573
0,712
0,740
0,933
0,452
0,401
0,405
0,420
#'06.2C
06.2D
06.2B
06.2A
0,554
0,674
0,690
0,891
0,418
0,396
0,398
0,419
Set of ratios 12
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 13
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'25.2C
25.2D
25.2B
25.2A
#'35.2C
35.2D
35.2B
35.2A
0,582
0,641
0,677
0,767
0,419
0,364
0,370
0,393
#'26.2C
26.2D
26.2B
26.2A
0,577
0,659
0,679
0,806
0,404
0,377
0,387
0,407
35
0,661
0,822
0,858
1,069
0,530
0,448
0,447
0,455
#'36.2C
36.2D
36.2B
36.2A
0,655
0,812
0,852
1,069
0,517
0,455
0,454
0,478
EV/GI
Table A.2: Estimation errors by central tendency measure, market multiple and clustering
method, characterized by the mean and the median of its distribution errors - Part II
Set of ratios 14
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 15
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'45.2C
45.2D
45.2B
45.2A
#'55.2C
55.2D
55.2B
55.2CA
0,599
0,732
0,748
0,961
0,466
0,406
0,413
0,460
#'46.2C
46.2D
46.2B
46.2A
0,585
0,716
0,730
0,929
0,456
0,414
0,411
0,440
EV/EBITDA
Set of ratios 03
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#35.3C
35.3D
35.3B
35.3A
0,485
0,548
0,566
0,638
0,364
0,321
0,322
0,338
#36.3C
36.3D
36.3B
36.3A
0,476
0,542
0,558
0,638
0,361
0,322
0,325
0,343
0,586
0,690
0,723
0,866
0,478
0,397
0,403
0,422
#'56.2C
56.2D
56.2B
56.2A
0,584
0,671
0,692
0,844
0,477
0,425
0,417
0,423
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#45.3C
45.3D
45.3B
45.3A
0,496
0,564
0,584
0,667
0,364
0,346
0,351
0,373
#46.3C
46.3D
46.3B
46.3A
0,496
0,570
0,587
0,675
0,370
0,359
0,360
0,369
EV/EBIT
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#45.4C
45.4D
45.4B
45.4A
0,519
0,585
0,608
0,698
0,379
0,350
0,353
0,369
#46.4C
46.4D
46.4B
46.4A
0,507
0,575
0,594
0,679
0,376
0,349
0,346
0,363
EV/TA
EV/TA
Set of ratios 02
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#25.5C
25.5B
25.5D
25.5A
0,507
0,575
0,595
0,716
0,330
0,332
0,341
0,400
#26.5C
26.5B
26.5D
26.5A
0,506
0,576
0,591
0,710
0,335
0,334
0,346
0,399
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#45.5C
45.5B
45.5D
45.5A
0,462
0,515
0,550
0,676
0,318
0,336
0,355
0,429
#46.5C
46.5B
46.5D
46.5A
0,424
0,469
0,473
0,549
0,294
0,270
0,278
0,316
Set of ratios 11
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 17
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'15.5C
15.5D
15.5B
15.5A
#'75.5C
75.5B
75.5D
75.5A
0,398
0,431
0,432
0,489
0,288
0,261
0,256
0,280
#'16.5C
16.5D
16.5B
16.5A
0,401
0,428
0,433
0,479
0,279
0,257
0,254
0,262
0,455
0,488
0,500
0,575
0,302
0,293
0,299
0,325
#'76.5C
76.5B
76.5D
76.5A
0,452
0,486
0,508
0,601
0,293
0,303
0,317
0,358
EV/OCF
Set of ratios 14
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'45.6C
45.6D
45.6B
45.6A
0,559
0,639
0,664
0,761
0,402
0,339
0,329
0,358
#'46.6C
46.6D
46.6B
46.6A
0,572
0,650
0,673
0,768
0,403
0,351
0,343
0,366
EV/FCFF
Set of ratios 02
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#25.7C
25.7D
25.7B
25.7A
0,920
1,436
1,566
2,113
0,641
0,506
0,511
0,562
#26.7C
26.7D
26.7B
26.7A
0,930
1,408
1,536
2,046
0,630
0,507
0,497
0,552
EV/FCFF
Set of ratios 05
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#55.7C
55.7D
55.7B
55.7A
0,787
1,164
1,323
1,781
0,596
0,517
0,523
0,562
#56.7C
56.7D
56.7B
56.7A
0,763
1,074
1,124
1,606
0,574
0,491
0,494
0,549
36
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#45.7C
45.7D
45.7B
45.7A
0,925
1,491
1,606
2,216
0,664
0,509
0,519
0,583
#46.7C
46.7D
46.7B
46.7A
0,958
1,484
1,593
2,208
0,647
0,511
0,507
0,587
Set of ratios 07
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#75.7C
75.7D
75.7B
75.7A
0,757
1,110
1,200
1,619
0,612
0,521
0,507
0,549
#76.7C
76.7D
76.7B
76.7A
0,757
1,107
1,199
1,603
0,604
0,514
0,508
0,551
Table A.2: Estimation errors by central tendency measure, market multiple and clustering
method, characterized by the mean and the median of its distribution errors - Part III
EV/FCFF
Set of ratios 09
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#95.7C
95.7D
95.7B
95.7A
0,846
1,251
1,332
1,807
0,605
0,501
0,510
0,567
#96.7C
96.7D
96.7B
96.7A
0,722
0,975
1,014
1,401
0,579
0,495
0,501
0,563
P/S
Set of ratios 01
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#15.8C
15.8D
15.8B
15.8A
0,553
0,689
0,711
0,977
0,439
0,422
0,426
0,453
#16.8C
16.8D
16.8B
16.8A
0,527
0,641
0,637
0,900
0,416
0,385
0,380
0,419
P/S
Set of ratios 03
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#35.8C
35.8D
35.8B
35.8A
0,742
1,058
1,074
1,637
0,569
0,545
0,547
0,587
#36.8C
36.8D
36.8B
36.8A
0,679
0,929
0,928
1,385
0,528
0,532
0,531
0,590
P/S
Set of ratios 06
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#65.8C
65.8D
65.8B
65.8A
0,671
0,898
0,921
1,312
0,526
0,497
0,503
0,565
#66.8C
66.8D
66.8B
66.8A
0,690
0,901
0,938
1,257
0,515
0,478
0,478
0,507
P/S
P/S
Set of ratios 09
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#95.8C
95.8D
95.8B
95.8A
0,747
0,985
1,010
1,450
0,536
0,497
0,493
0,561
#96.8C
96.8D
96.8B
96.8A
0,637
0,789
0,818
1,053
0,477
0,443
0,442
0,468
Set of ratios 02
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#25.8C
25.8D
25.8B
25.8A
0,596
0,780
0,789
1,124
0,470
0,433
0,432
0,467
#26.8C
26.8D
26.8B
26.8A
0,587
0,765
0,778
1,106
0,452
0,427
0,424
0,464
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#45.8C
45.8D
45.8B
45.8A
0,849
1,303
1,337
2,159
0,648
0,622
0,622
0,697
#46.8C
46.8D
46.8B
46.8A
0,825
1,245
1,266
2,043
0,635
0,604
0,610
0,699
Set of ratios 08
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#85.8C
85.8D
85.8B
85.8A
0,809
1,218
1,277
1,921
0,628
0,579
0,579
0,635
#86.8C
86.8D
86.8B
86.8A
0,722
1,045
1,044
1,658
0,562
0,510
0,506
0,590
Set of ratios 12
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'25.8C
25.8D
25.8B
25.8A
0,613
0,729
0,760
0,927
0,512
0,517
0,528
0,527
#'26.8C
26.8D
26.8B
26.8A
0,615
0,749
0,820
0,933
0,522
0,505
0,512
0,486
Set of ratios 13
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 14
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'35.8C
35.8D
35.8B
35.8A
#'45.8C
45.8D
45.8B
45.8A
0,762
1,062
1,148
1,578
0,615
0,550
0,559
0,604
#'36.8C
36.8D
36.8B
36.8A
0,757
1,050
1,114
1,549
0,611
0,553
0,556
0,602
0,562
0,681
0,690
0,947
0,443
0,427
0,429
0,492
#'46.8C
46.8D
46.8B
46.8A
0,537
0,647
0,649
0,903
0,431
0,415
0,411
0,466
P/S
Set of ratios 15
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'55.8C
55.8D
55.8B
55.8A
0,566
0,656
0,666
0,845
0,460
0,432
0,435
0,467
#'56.8C
56.8D
56.8B
56.8A
0,528
0,603
0,617
0,773
0,426
0,389
0,389
0,429
P/GI
Set of ratios 02
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#25.9C
25.9D
25.9B
25.9A
0,553
0,683
0,692
0,914
0,442
0,428
0,434
0,473
#26.9C
26.9D
26.9B
26.9A
0,547
0,677
0,688
0,910
0,428
0,421
0,423
0,461
37
Set of ratios 03
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#35.9C
35.9D
35.9B
35.9A
0,628
0,781
0,821
1,036
0,490
0,448
0,449
0,489
#36.9C
36.9D
36.9B
36.9A
0,606
0,751
0,767
0,992
0,488
0,453
0,452
0,485
Table A.2: Estimation errors by central tendency measure, market multiple and clustering
method, characterized by the mean and the median of its distribution errors - Part IV
P/GI
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#45.9C
45.9D
45.9B
45.9A
0,648
0,836
0,867
1,154
0,526
0,503
0,504
0,528
#46.9C
46.9D
46.9B
46.9A
0,621
0,796
0,812
1,103
0,506
0,479
0,476
0,535
P/GI
P/GI
P/GI
Set of ratios 08
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#85.9C
85.9D
85.9B
85.9A
0,633
0,801
0,853
1,073
0,507
0,469
0,473
0,493
#86.9C
86.9D
86.9B
86.9A
0,601
0,757
0,791
1,018
0,474
0,450
0,455
0,492
Set of ratios 06
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#65.9C
65.9D
65.9B
65.9A
0,587
0,724
0,736
0,961
0,470
0,441
0,447
0,482
#66.9C
66.9D
66.9B
66.9A
0,588
0,721
0,747
0,945
0,454
0,441
0,438
0,463
Set of ratios 09
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#95.9C
95.9D
95.9B
95.9A
0,612
0,747
0,773
0,991
0,486
0,452
0,455
0,486
#96.9C
96.9D
96.9B
96.9A
0,570
0,690
0,709
0,900
0,452
0,431
0,435
0,474
Set of ratios 12
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 13
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'25.9C
25.9D
25.9B
25.9A
#'35.9C
35.9D
35.9B
35.9A
0,572
0,664
0,727
0,822
0,480
0,427
0,457
0,428
#'26.9C
26.9D
26.9B
26.9A
0,574
0,676
0,722
0,854
0,453
0,398
0,435
0,443
0,624
0,762
0,805
0,988
0,498
0,464
0,467
0,464
#'36.9C
36.9D
36.9B
36.9A
0,621
0,756
0,798
0,986
0,498
0,463
0,455
0,470
Set of ratios 14
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 15
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'45.9C
45.9D
45.9B
45.9A
#'55.9C
55.9D
55.9B
55.9A
0,546
0,637
0,652
0,808
0,423
0,398
0,394
0,423
#'46.9C
46.9D
46.9B
46.9A
0,532
0,617
0,634
0,779
0,406
0,388
0,391
0,411
0,520
0,585
0,601
0,715
0,422
0,385
0,384
0,408
#'56.9C
56.9D
56.9B
56.9A
0,520
0,586
0,603
0,714
0,424
0,390
0,393
0,405
P/B
P/E
P/EBT
P/EBITDA
Set of ratios 14
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'45.'0C
45.'0D
45.'0B
45.'0A
0,465
0,511
0,522
0,602
0,349
0,328
0,329
0,335
#'46.'0C
46.'0D
46.'0B
46.'0A
0,454
0,503
0,516
0,597
0,350
0,323
0,329
0,335
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 16
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#45.'2C
45.'2D
45.'2B
45.'2A
#'65.'2C
65.'2D
65.'2B
65.'2A
0,444
0,492
0,504
0,576
0,354
0,348
0,348
0,362
#46.'2C
46.'2D
46.'2B
46.'2A
0,445
0,492
0,499
0,574
0,370
0,344
0,345
0,361
0,420
0,454
0,458
0,518
0,331
0,302
0,306
0,322
#'66.'2C
66.'2D
66.'2B
66.'2A
0,412
0,446
0,452
0,509
0,322
0,295
0,296
0,321
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 16
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#45.'3C
45.'3D
45.'3B
45.'3A
#'65.'3C
65.'3D
65.'3B
65.'3A
0,448
0,491
0,503
0,571
0,370
0,342
0,339
0,347
#46.'3C
46.'3D
46.'3B
46.'3A
0,444
0,486
0,497
0,563
0,369
0,345
0,338
0,346
0,412
0,443
0,448
0,505
0,318
0,302
0,302
0,311
#'66.'3C
66.'3D
66.'3B
66.'3A
0,405
0,435
0,440
0,494
0,320
0,300
0,299
0,305
Set of ratios 01
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 02
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#15.'4C
15.'4B
15.'4D
15.'4A
#25.'4C
25.'4D
25.'4B
25.'4A
0,524
0,633
0,627
0,791
0,411
0,419
0,422
0,447
#16.'4C
16.'4B
16.'4D
16.'4A
0,527
0,613
0,621
0,779
0,410
0,401
0,409
0,445
38
0,555
0,664
0,666
0,847
0,440
0,429
0,428
0,462
#26.'4C
26.'4D
26.'4B
26.'4A
0,558
0,668
0,675
0,850
0,445
0,428
0,430
0,464
P/B
Table A.2: Estimation errors by central tendency measure, market multiple and clustering
method, characterized by the mean and the median of its distribution errors - Part V
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 11
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#45.'4C
45.'4B
45.'4D
45.'4A
#'15.'4C
15.'4D
15.'4B
15.'4A
0,551
0,669
0,684
0,889
0,451
0,458
0,461
0,516
#46.'4C
46.'4B
46.'4D
46.'4A
0,507
0,596
0,594
0,735
0,400
0,377
0,387
0,408
0,446
0,499
0,497
0,598
0,367
0,352
0,344
0,383
#'16.'4C
16.'4D
16.'4B
16.'4A
0,440
0,489
0,492
0,578
0,358
0,357
0,348
0,367
P/OCF
P/TA
P/B
Set of ratios 17
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'75.'4C
75.'4B
75.'4D
75.'4A
0,448
0,503
0,505
0,606
0,372
0,373
0,374
0,385
#'76.'4C
76.'4B
76.'4D
76.'4A
0,464
0,518
0,529
0,651
0,386
0,382
0,387
0,418
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 11
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#45.'5C
45.'5D
45.'5B
45.'5A
#'15.'5C
15.'5D
15.'5B
15.'5A
0,756
1,146
1,188
1,720
0,627
0,512
0,517
0,612
#46.'5C
46.'5D
46.'5B
46.'5A
0,714
0,976
1,049
1,362
0,600
0,439
0,437
0,470
0,530
0,623
0,643
0,805
0,427
0,391
0,394
0,431
#'16.'5C
16.'5D
16.'5B
16.'5A
0,576
0,703
0,750
0,915
0,458
0,395
0,385
0,395
Set of ratios 01
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#15.'6C
15.'6D
15.'6B
15.'6A
#45.'6C
45.'6D
45.'6B
45.'6A
0,511
0,585
0,603
0,701
0,408
0,376
0,378
0,390
#16.'6C
16.'6D
16.'6B
16.'6A
0,507
0,579
0,600
0,699
0,400
0,364
0,368
0,399
0,576
0,689
0,721
0,853
0,436
0,419
0,421
0,431
#46.'6C
46.'6D
46.'6B
46.'6A
0,565
0,671
0,698
0,827
0,438
0,405
0,403
0,411
P/FCFF
P/FCFF
P/OCF
Set of ratios 14
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#'45.'6C
45.'6D
45.'6B
45.'6A
0,495
0,554
0,575
0,657
0,392
0,358
0,346
0,364
#'46.'6C
46.'6D
46.'6B
46.'6A
0,490
0,547
0,567
0,652
0,392
0,337
0,343
0,362
Set of ratios 02
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 04
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#25.'7C
25.'7D
25.'7B
25.'7A
#45.'7C
45.'7D
45.'7B
45.'7A
0,935
1,552
1,795
2,510
0,705
0,532
0,532
0,577
#26.'7C
26.'7D
26.'7B
26.'7A
1,001
1,622
1,847
2,522
0,694
0,530
0,523
0,577
0,764
0,533
0,543
0,578
#46.'7C
46.'7D
46.'7B
46.'7A
0,920
1,450
1,617
2,260
0,784
0,513
0,514
0,587
Set of ratios 05
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
Set of ratios 07
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
#55.'7C
55.'7D
55.'7B
55.'7A
#75.'7C
75.'7D
75.'7B
75.'7A
0,894
1,288
1,387
1,884
0,603
0,525
0,538
0,566
#56.'7C
56.'7D
56.'7B
56.'7A
0,776
1,091
1,161
1,616
0,576
0,499
0,514
0,555
Set of ratios 09
Complete Linkage
K Means
Distrib.
Mean Median Distrib.
Mean Median
P/FCFF
0,873
1,488
1,665
2,342
#95.'7C
95.'7D
95.'7B
95.'7A
0,767
1,070
1,143
1,603
0,581
0,512
0,526
0,571
#96.'7C
96.'7D
96.'7B
96.'7A
0,769
1,068
1,128
1,554
0,557
0,501
0,507
0,557
39
0,773
1,129
1,212
1,712
0,622
0,548
0,553
0,558
#76.'7C
76.'7D
76.'7B
76.'7A
0,775
1,133
1,224
1,698
0,624
0,544
0,541
0,555
40
Editorial Board ([email protected])
Download available at:
also in
http://wps.fep.up.pt/wplist.php
http://ideas.repec.org/PaperSeries.html