44
1 1,2 1 1 2
n. 586 January 2017
ISSN: 0870-8541
The Method of
Market Multiples on the Valuation of Companies:
A Multivariate Approach
José Couto1
Paula Brito1,2
António Cerqueira1
1 FEP-UP, School of Economics and Management, University of Porto2 LIAAD/INESC TEC
1
The Method of Market Multiples
on the Valuation of Companies: A Multivariate Approach
José Couto
Faculdade de Economia, Universidade do Porto, Portugal
Paula Brito
Faculdade de Economia & LIAAD INESC TEC, Universidade do Porto, Portugal
António Cerqueira
Faculdade de Economia, Universidade do Porto, Portugal
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
2
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).
3
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 cash-
flow 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,
4
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 three-
digit 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
5
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.
6
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 forward-
looking 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 industry-
preferred 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.
7
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 = |�̂�𝑦,𝑖𝑡 − 𝑚𝑦,𝑖𝑡
𝑚𝑦,𝑖𝑡| = |
�̂�𝑦,𝑖𝑡
𝑚𝑦,𝑖𝑡− 1| (3.1)
where m̂y,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
8
variable but absolute valuation errors of entity multiples compares the deviation on the entity
variable. This may be proven by noticing that
|(𝐸𝑉
𝑆⁄ )̂
(𝐸𝑉𝑆⁄ )
− 1| = |𝐸�̂�
𝐸𝑉− 1| (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⁄ )̂
(𝐸𝑉𝐸𝐵𝐼𝑇𝐷𝐴⁄ )
− 1| = |𝐸�̂�
𝐸𝑉− 1| (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.
9
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.
Figure 4.1: Evolution of the median of the entity multiples during the period 2000-2011
Source: Own elaboration
0,0
2,0
4,0
6,0
8,0
10,0
12,0
14,0
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
EV/S EV/GI EV/EBITDA EV/EBIT EV/TA
10
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.
0
2
4
6
8
10
12
14
16
18
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
P/S P/GI P/EBITDA P/EBIT
P/EBT PER P/B P/TA
11
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 0,0 0,6 1,3 2,6 9,4 2,0 1,0 6.088 1,7 2,4
EV/GI 4,8 0,0 2,2 3,8 6,3 17,4 3,7 0,8 5.319 1,3 1,4
EV/EBITDA 8,7 0,1 5,2 7,6 10,9 25,5 4,8 0,6 5.700 1,1 1,0
EV/EBIT 12,0 0,1 7,2 10,6 14,9 35,8 6,9 0,6 5.476 1,1 1,2
EV/TA 1,5 0,0 0,9 1,2 1,8 4,9 0,9 0,6 6.224 1,5 1,9
EV/OCF 11,4 0,1 6,4 9,6 14,6 36,0 7,1 0,6 5.795 1,2 1,2
EV/FCFF 15,3 0,0 6,1 11,5 20,4 63,4 13,1 0,9 3.855 1,5 2,0
Equity market multiples:
P/S 1,5 0,0 0,5 1,0 2,1 6,9 1,5 1,0 6.334 1,6 2,0
P/GI 3,8 0,0 1,7 3,0 5,1 14,0 2,8 0,7 5.392 1,2 1,1
P/EBITDA 6,8 0,0 4,0 5,9 8,7 20,6 3,9 0,6 5.847 1,1 1,0
P/EBIT 9,3 0,0 5,7 8,2 11,8 27,7 5,3 0,6 5.619 1,1 1,1
P/EBT 10,8 0,2 6,7 9,5 13,5 32,0 5,9 0,5 5.490 1,1 1,2
P/E 14,9 0,2 9,2 13,2 18,6 43,6 8,0 0,5 5.475 1,1 1,2
P/B 1,7 0,0 0,9 1,3 2,2 5,9 1,2 0,7 6.566 1,3 1,4
P/TA 1,4 0,0 0,6 1,0 1,8 5,6 1,2 0,8 6.320 1,5 2,0
P/OCF 8,9 0,1 4,9 7,7 11,8 28,7 5,5 0,6 5.984 1,1 1,0
P/FCFF 12,1 0,0 4,1 9,0 16,9 54,0 10,9 0,9 4.048 1,4 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
12
Mean Min. χ25 χ50 χ75 Max. S.D. n Skew. Kurt.
Growth rates (in %):
grSales(1y) 9,9 -61,6 0,6 8,6 18,9 63,2 17,4 6.471 0,0 1,6
grSales(CAGR4y) 6,3 -45,7 -1,2 4,9 12,9 44,9 12,8 6.369 0,2 1,3
grNI(1y) 11,8 -99,8 -14,5 10,0 34,0 159,0 47,1 5.134 0,4 0,7
grNI(CAGR4y) 3,6 -84,5 -8,9 3,8 16,2 70,0 22,3 5.201 -0,1 0,8
Income Statement margins (as % of Sales):
GI margin 40,1 -71,0 22,3 36,0 56,7 100,0 24,3 5.874 0,4 0,1
EBITDA margin 19,1 -61,4 7,8 15,6 27,7 76,4 18,3 6.382 0,4 1,8
EBIT margin 13,4 -53,5 4,4 10,5 20,7 61,9 15,4 6.314 0,3 2,0
EBT margin 11,0 -45,9 3,6 9,0 17,8 53,1 13,7 6.315 0,1 1,8
NI margin 7,8 -35,6 2,4 6,4 12,9 39,7 10,5 6.236 0,0 2,0
Balance Sheet Structure (items written as % of Sales)
FxdAssts_%Sales 62,1 0,0 20,7 41,6 83,6 291,7 60,1 6.322 1,6 2,2
NWC_%Sales -3,1 -141,8 -13,7 1,2 13,6 91,2 31,7 5.971 -1,1 3,1
Invtmts_%Sales 13,3 -3,1 0,1 1,9 9,4 247,6 33,0 5.993 4,2 19,0
TA_%Sales 111,0 -422,2 38,2 71,6 138,2 633,9 120,7 6.207 1,8 4,0
Debt_%Sales 16,3 -300,0 -6,6 6,7 31,0 299,7 61,8 6.446 0,8 5,5
Eqty_%Sales 80,1 -261,9 33,0 60,1 106,1 349,5 70,0 6.231 1,4 2,5
PrefStock_%Sales 6,6 0,0 0,0 0,0 0,0 35.976,7 431,8 7.020 82,5 6.862,6
MinInter_%Sales 5,1 -5,6 0,0 0,1 2,0 460,3 21,3 6.998 10,3 142,8
D/E 0,6 -93,6 -0,1 0,2 0,8 89,7 3,7 7.195 -0,9 249,8
ROA 8,6 -454,4 2,0 6,9 14,5 493,2 41,7 6.949 -0,1 47,1
ROE 7,8 -492,0 3,5 10,0 17,2 450,3 35,9 7.068 -3,6 54,6
Cash-flow Structure (items written as % of Sales)
OCF_%Sales 16,0 -196,7 6,4 13,0 24,7 190,1 23,8 6.863 -1,3 16,8
varNWCch_%Sales -4,2 -199,9 -4,9 0,1 4,1 200,0 35,1 6.762 -1,3 10,9
CapexCh_%Sales 12,9 -195,8 1,6 5,0 14,3 198,5 29,7 6.780 1,9 12,7
varInvtmts_%Sales 5,5 -197,0 -0,3 0,0 1,0 199,4 33,3 6.659 2,1 14,6
FCFFCh_%Sales 1,2 -199,0 -5,0 3,7 13,3 199,3 42,4 6.556 -0,7 6,2
Div_%Sales 5,5 0,0 0,0 1,6 4,8 191,4 12,3 6.983 5,6 47,4
Payout_%Sales 35,7 0,0 8,8 27,2 51,0 199,7 35,6 5.780 1,5 2,6
FreeFloat 66 0 44 71 92 100 28 7.261 -0,5 -1,0
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.
13
Ta
ble 5
.3: S
pearm
an’s co
rrelation
coefficien
ts betw
een m
ultip
les and
ratios (P
art 1)
G
I ma
rg
in
EB
ITD
A
ma
rg
in
EB
IT
ma
rg
in
EB
T
ma
rg
in
NI m
arg
in
TA
_%
Sa
les
Eq
ty
_%
Sa
les R
oa
Ro
e O
CF
_%
Sa
les
Ca
pex
_%
Sa
les
va
rN
WC
_
%S
ale
s
va
rIn
vtm
ts
_%
Sa
les
FC
FF
_%
Sa
les
EV
/S
0,5
5
0,6
6
0,6
3
0,5
7
0,5
7
0,6
3
0,5
8
0,0
5
0,1
5
0,6
3
0,3
5
-0,0
5
0,1
1
0,0
9
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
EV
/GI
0,0
5
0,4
2
0,4
1
0,3
5
0,3
6
0,4
5
0,3
3
-0,0
2
0,0
9
0,4
1
0,3
3
0,0
0
0,1
2
-0,0
3
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,2
4
0,0
0
0,0
0
0,0
0
0,7
6
0,0
0
0,0
1
EV
/EB
ITD
A
0,1
6
0,1
2
0,1
2
0,0
6
0,0
9
0,2
5
0,1
4
-0,2
6
-0,1
2
0,1
5
0,0
9
-0,0
2
0,0
3
-0,0
6
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,1
9
0,0
1
0,0
0
EV
/EB
IT
0,1
1
0,0
6
-0,0
2
-0,1
0
-0,0
7
0,2
5
0,0
9
-0,4
4
-0,2
9
0,0
9
0,1
5
-0,0
1
-0,0
4
-0,0
8
p-va
lue
0,0
0
0,0
0
0,1
1
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,4
2
0,0
0
0,0
0
EV
/TA
0
,20
0,2
2
0,2
9
0,3
1
0,3
4
-0,1
3
-0,1
0
0,4
6
0,4
8
0,1
7
0,2
0
0,0
3
0,0
4
0,0
4
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
4
0,0
1
0,0
0
EV
/OC
F
0,1
4
0,1
4
0,1
6
0,1
0
0,1
1
0,2
5
0,1
1
-0,1
8
-0,0
5
0,0
9
0,1
0
0,0
3
0,0
4
-0,1
7
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
2
0,0
0
0,0
0
EV
/FC
FF
0
,10
0,2
1
0,2
0
0,2
0
0,2
0
0,0
8
-0,0
1
0,1
6
0,2
4
0,0
8
0,3
8
0,3
0
0,1
0
-0,4
1
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,6
2
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
P/S
0
,59
0,6
4
0,6
5
0,6
7
0,6
8
0,4
2
0,6
1
0,2
6
0,2
4
0,6
3
0,3
2
-0,0
4
0,1
1
0,1
1
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
P/G
I 0
,13
0,4
2
0,4
7
0,4
7
0,5
0
0,2
5
0,4
1
0,2
5
0,2
2
0,4
4
0,2
5
0,0
0
0,1
1
0,0
2
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,7
7
0,0
0
0,1
3
P/E
BIT
DA
0
,19
0,0
6
0,1
3
0,1
7
0,2
1
-0,0
4
0,1
6
0,1
9
0,0
5
0,1
3
0,0
1
0,0
2
-0,0
2
0,0
5
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,5
4
0,1
9
0,1
4
0,0
0
P/E
BIT
0
,15
-0,0
1
-0,0
5
-0,0
2
0,0
4
-0,0
3
0,0
9
0,0
3
-0,1
2
0,0
7
0,0
9
0,0
3
-0,1
1
0,0
3
p-va
lue
0,0
0
0,6
2
0,0
0
0,1
5
0,0
1
0,0
2
0,0
0
0,0
2
0,0
0
0,0
0
0,0
0
0,0
2
0,0
0
0,0
5
P/E
BT
0
,14
0,0
2
-0,0
3
-0,0
6
0,0
0
0,0
5
0,1
1
-0,1
3
-0,2
1
0,0
6
0,1
0
0,0
1
-0,0
9
0,0
2
p-va
lue
0,0
0
0,2
6
0,0
2
0,0
0
0,8
8
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,6
1
0,0
0
0,2
3
P/E
0
,11
0,0
0
-0,0
5
-0,0
6
-0,0
9
0,0
3
0,0
7
-0,2
0
-0,2
9
0,0
1
0,1
0
-0,0
2
-0,1
1
0,0
3
p-va
lue
0,0
0
0,9
4
0,0
0
0,0
0
0,0
0
0,0
3
0,0
0
0,0
0
0,0
0
0,3
7
0,0
0
0,2
0
0,0
0
0,0
4
P/B
0
,21
0,2
4
0,3
0
0,3
3
0,3
5
-0,1
4
-0,1
0
0,4
4
0,4
9
0,2
0
0,2
1
0,0
2
0,0
4
0,0
4
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,1
2
0,0
0
0,0
0
P/T
A
0,1
8
0,1
3
0,2
2
0,3
0
0,3
3
-0,2
9
-0,0
2
0,5
8
0,4
2
0,1
2
0,1
0
0,0
4
0,0
0
0,0
9
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,1
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,8
1
0,0
0
P/O
CF
0
,21
0,1
2
0,1
9
0,2
3
0,2
5
-0,0
1
0,1
7
0,2
1
0,1
2
0,0
5
0,0
6
0,0
5
0,0
1
-0,0
9
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,3
5
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,5
6
0,0
0
P/F
CF
F
0,1
5
0,2
0
0,2
2
0,2
8
0,2
8
-0,0
5
0,0
2
0,3
8
0,3
4
0,1
0
0,3
6
0,3
2
0,0
8
-0,4
6
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
1
0,1
7
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
So
urce: O
wn elab
oratio
n
14
Ta
ble 5
.4: S
pearm
an’s co
rrelation
coefficien
ts betw
een m
ultip
les and
ratios (P
art 2)
D
ivid
_%
Sa
les P
ay
ou
t ln
(Ro
a)
ln(R
oe)
ln(O
CF
_S
ale
s)
ln(C
ap
ex
_%
Sa
les)
ln(v
arN
WC
_%
Sa
les)
ln(v
arIn
vtm
ts_%
Sa
les)
ln(F
CF
F_
%S
ale
s)
ln(D
ivid
_
%S
ale
s) ln
(Pay
ou
t) g
rS
ale
s
(CA
GR
4y
)
grN
I
(CA
GR
4y
)
EV
/S
0,3
5
0,0
3
0,0
4
0,1
8
0,7
2
0,4
5
0,3
6
0,3
6
0,5
1
0,6
4
0,0
9
0,1
8
0,1
8
p-va
lue
0,0
0
0,0
1
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
EV
/GI
0,2
8
0,0
7
-0,0
2
0,1
2
0,4
5
0,4
0
0,2
6
0,3
1
0,3
2
0,4
6
0,1
1
0,2
5
0,1
3
p-va
lue
0,0
0
0,0
0
0,1
2
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
EV
/EB
ITD
A
0,1
7
0,1
4
-0,2
3
-0,0
8
0,1
6
0,1
4
0,1
1
0,1
6
0,1
8
0,2
8
0,2
3
0,0
8
-0,0
3
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
4
EV
/EB
IT
0,0
6
0,1
6
-0,4
3
-0,2
8
0,0
9
0,1
9
0,0
4
0,0
3
0,0
7
0,1
4
0,2
8
-0,0
1
-0,2
1
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
4
0,0
9
0,0
0
0,0
0
0,0
0
0,3
7
0,0
0
EV
/TA
0
,14
0,0
4
0,5
5
0,5
6
0,1
6
0,1
4
-0,0
8
-0,1
7
0,0
1
0,1
8
0,0
5
0,2
9
0,3
2
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,7
5
0,0
0
0,0
0
0,0
0
0,0
0
EV
/OC
F
0,1
9
0,1
1
-0,1
7
-0,0
1
0,0
8
0,1
6
0,1
4
0,1
4
0,0
8
0,3
1
0,1
7
0,1
0
0,0
4
p-va
lue
0,0
0
0,0
0
0,0
0
0,5
8
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
EV
/FC
FF
0
,15
0,1
0
0,0
7
0,1
7
0,0
6
0,3
1
0,0
6
-0,1
2
-0,4
2
0,1
4
0,0
8
0,2
3
0,1
5
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
2
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
P/S
0
,44
0,0
7
0,2
5
0,2
3
0,7
2
0,3
8
0,2
8
0,2
8
0,4
4
0,7
2
0,1
1
0,2
0
0,2
1
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
P/G
I 0
,33
0,0
6
0,2
5
0,2
1
0,4
8
0,3
0
0,2
2
0,2
3
0,3
0
0,5
4
0,0
9
0,2
6
0,2
0
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
P/E
BIT
DA
0
,21
0,1
4
0,2
1
0,0
5
0,1
3
0,0
2
0,0
3
0,0
1
0,1
2
0,3
1
0,2
1
0,1
0
0,0
6
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
8
0,1
1
0,7
7
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
P/E
BIT
0
,11
0,1
8
0,0
5
-0,1
2
0,0
6
0,0
9
-0,0
6
-0,1
3
0,0
2
0,1
8
0,2
8
0,0
2
-0,0
8
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,2
4
0,0
0
0,0
0
0,1
1
0,0
0
P/E
BT
0
,09
0,1
9
-0,1
3
-0,2
1
0,0
6
0,1
1
-0,0
2
-0,0
7
0,0
6
0,1
6
0,3
2
0,0
1
-0,1
5
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,3
5
0,0
0
0,0
0
0,0
0
0,0
0
0,6
2
0,0
0
P/E
0
,04
0,2
3
-0,2
0
-0,2
9
0,0
0
0,0
9
-0,1
0
-0,1
2
0,0
2
0,0
6
0,3
4
0,0
0
-0,1
9
p-va
lue
0,0
1
0,0
0
0,0
0
0,0
0
0,8
9
0,0
0
0,0
0
0,0
0
0,1
8
0,0
0
0,0
0
0,7
9
0,0
0
P/B
0
,16
0,0
7
0,5
1
0,5
9
0,1
8
0,1
5
-0,0
8
-0,1
7
0,0
1
0,2
1
0,0
8
0,3
1
0,3
4
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,7
3
0,0
0
0,0
0
0,0
0
0,0
0
P/T
A
0,1
2
0,0
2
0,7
2
0,4
8
0,1
1
0,0
3
-0,1
1
-0,2
0
-0,0
4
0,1
6
0,0
1
0,2
3
0,2
9
p-va
lue
0,0
0
0,1
2
0,0
0
0,0
0
0,0
0
0,0
3
0,0
0
0,0
0
0,0
3
0,0
0
0,3
7
0,0
0
0,0
0
P/O
CF
0
,25
0,1
1
0,2
1
0,1
2
0,0
5
0,0
7
0,0
7
0,0
3
0,0
4
0,3
7
0,1
5
0,1
3
0,1
2
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,1
9
0,0
2
0,0
0
0,0
0
0,0
0
0,0
0
P/F
CF
F
0,2
1
0,1
2
0,3
0
0,2
4
0,0
5
0,2
4
-0,0
1
-0,2
2
-0,4
6
0,1
7
0,0
7
0,2
6
0,2
1
p-va
lue
0,0
0
0,0
0
0,0
0
0,0
0
0,0
1
0,0
0
0,7
3
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
0,0
0
So
urce: O
wn elab
oratio
n
15
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 04 05 06 07 08 09 10 11 12 13 14 15 16 17
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 4 8 5 12 2 4 2 3 4 2 3 4 4 7 4 2 2
Source: Own elaboration
16
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 Furthest-
Neighbour) 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):
𝑅2 =𝑆𝑄𝐶
𝑆𝑄𝑇=
∑ ∑ 𝑛𝑖𝑗(�̅�𝑖𝑗 − �̅�𝑖)2𝑘
𝑗=1𝑝𝑖=1
∑ ∑ ∑ (𝑋𝑖𝑗𝑙 − �̅�)2𝑛𝑖
𝑙=1𝑘𝑗=1
𝑝𝑖=1
(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).
17
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.
0,0
0,2
0,4
0,6
0,8
1,0
0
50
100
150
200
250
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
18
Table 5.7: Number of held clusters by set of ratios and their corresponding R2
Set of Ratios Classificatory Variables Number of
Clusters
R2
(Hierarchical
Analysis)
R2
(K-Means)
01 GI; Ebitda; Ebit; Ebt; NI (all as % of Sales) 20 0,81 0,87
02 Ebitda; Ebit; Ebt; NI (all as % of Sales) 8 0,80 0,84
03 TA (%Sales); Eqty (%Sales) 10 0,82 0,88
04 RoA; RoE 19 0,83 0,92
05 OCF; Capex; varNWC; varInvtmts (all as % of Sales) 43 0,80 0,86
06 OCF; Capex (all as % of Sales) 19 0,81 0,90
07 FCFF (as % of Sales) 5 0,85 0,89
08 Divid (as % of Sales) 8 0,94 0,96
09 EBITDA; TA; RoE; OCF; Capex; FCFF; Divid (as % of Sales) 30 0,65 0,75
10 Ebitda TA OCF Capex (all as % of Sales) 43 0,80 0,87
11 ln(RoA) ln(RoE) 8 0,82 0,87
12 ln(OCF) ln(Capex) ln(varNWC) ln(varInvtmts) 39 0,80 0,83
13 ln(FCFF) 7 0,92 0,95
14 ln(Diviv) 9 0,91 0,95
15 ln(FCFF) ln(Divid) 16 0,81 0,89
16 ln(Payout) 6 0,83 0,90
17 grSales(CAGR) grNI(CAGR) ln(RoE) 24 0,80 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
𝐻1: 𝜇1 ≠ 𝜇2
(6.2)
where, μ represents the mean of populations 1 and 2 under comparison.
19
The t statistic is as follows:
𝑇 =
�̅�
𝑆𝐷′
√𝑛⁄
(6.3)
where, D̅ is the observed mean of Di = (X1i − X2i), i=1,…, n, SD′ is the corrected standard
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 Second Code - Classification Third Code – Multiple Fourth Code – Selected Measure
0: ICB 1: Industry 1: EV/S A: Mean
1: Set of ratios 01 2: Supersector 2: EV/GI B: Median
2: Set of ratios 02 3: Sector 3: EV/EBITDA C: Harmonic Mean
3: Set of ratios 03 4: Subsector 4: EV/EBIT D: Geometric Mean
4: Set of ratios 04 5: Complete Linkage 5: EV/TA
5: Set of ratios 05 6: K-Means 6: EV/OCF
6: Set of ratios 06 7: EV/FCFF
7: Set of ratios 07 8: P/S
8: Set of ratios 08 9: P/GI
9: Set of ratios 09 '0: P/EBITDA
'0: Set of ratios 10 '1: P/EBIT
'1: Set of ratios 11 '2: P/EBT
'2: Set of ratios 12 '3: P/E
'3: Set of ratios 13 '4: P/B
'4: Set of ratios 14 '5: P/TA
'5: Set of ratios 15 '6: P/OCF
'6: Set of ratios 16 '7: P/FCFF
'7: Set of ratios 17
Source: Own elaboration
20
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.
Table 5.9: Formal tests for the prediction errors associated with the use of different measures
EV/EBITDA: Industry t-Student Test
01.3A 01.3B 01.3C 01.3D
t-S
tat.
01.3A - 11,629 6,256 9,517
01.3B - 4,713 5,836
01.3C - -4,291
p-v
alu
e*
01.3A - 0,000 0,000 0,000
01.3B - 0,000 0,000
01.3C - 0,000
Ranking 4th 3rd 1st 2nd
Des
crip
t.
Sta
t
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
EV/TA: Sector t-Student Test
03.5A 03.5B 03.5C 03.5D
t-S
tat.
03.5A - 17,514 13,261 16,867
03.5B - 7,878 -4,194
03.5C - -9,502
p-v
alu
e*
03.5A - 0,000 0,000 0,000
03.5B - 0,000 0,000
03.5C - 0,000
Ranking 4th 2nd 1st 3rd
Des
crip
t.
Sta
t
Mean 0,6699 0,5454 0,4825 0,5543
Stand.-Dev. 1,7264 1,4858 1,2212 1,4765
N 1.874 1.874 1.874 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 Industry Supersector Sector Subsector
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV/S #01.1C 0,812 0,639 #02.1C 0,810 0,623 #03.1C 0,806 0,608 #04.1C 0,838 0,583
EV/GI #01.2C 0,660 0,517 #02.2C 0,655 0,514 #03.2C 0,675 0,505 #04.2C 0,688 0,470
EV/EBITDA #01.3C 0,539 0,394 #02.3C 0,542 0,382 #03.3C 0,542 0,380 #04.3C 0,544 0,368
EV/EBIT #01.4C 0,575 0,425 #02.4C 0,580 0,412 #03.4C 0,584 0,418 #04.4C 0,590 0,402
EV/TA #01.5C 0,479 0,315 #02.5C 0,485 0,324 #03.5C 0,482 0,317 #04.5C 0,488 0,315
EV/OCF #01.6C 0,596 0,420 #02.6C 0,575 0,408 #03.6C 0,566 0,394 04.6C 0,565 0,389
EV/FCFF 01.7C 1,092 0,702 02.7C 1,170 0,660 03.7C 1,258 0,648 #04.7C 1,173 0,624
P/S #01.8C 0,847 0,661 #02.8C 0,838 0,647 #03.8C 0,795 0,614 #04.8C 0,780 0,582
P/GI #01.9C 0,753 0,502 #02.9C 0,647 0,515 #03.9C 0,632 0,500 #04.9C 0,630 0,482
P/EBITDA #01.'0C 0,529 0,409 #02.'0C 0,518 0,398 #03.'0C 0,523 0,387 #04.'0C 0,508 0,373
P/EBIT #01.'1C 0,486 0,380 #02.'1C 0,483 0,375 #03.'1C 0,486 0,368 #04.'1C 0,474 0,362
P/EBT #01.'2C 0,445 0,361 #02.'2C 0,445 0,360 #03.'2C 0,444 0,360 #04.'2C 0,437 0,345
P/E #01.'3C 0,442 0,367 #02.'3C 0,440 0,365 #03.'3C 0,445 0,360 #04.'3C 0,440 0,350
P/B #01.'4C 0,553 0,437 #02.'4C 0,551 0,421 #03.'4C 0,553 0,431 #04.'4C 0,543 0,431
P/TA #01.'5C 0,734 0,613 #02.'5C 0,715 0,564 #03.'5C 0,719 0,549 #04.'5C 0,703 0,532
P/OCF #01.'6C 0,583 0,454 #02.'6C 0,566 0,417 #03.'6C 0,565 0,401 #04.'6C 0,551 0,390
P/FCFF #01.'7C 1,179 0,767 #02.'7C 1,180 0,742 #03.'7C 1,207 0,723 #04.'7C 1,292 0,709
Source: Own elaboration
21
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
#15.1C 0,574 0,427 #16.1C 0,556 0,408 #25.1C 0,636 0,484 #26.1C 0,631 0,467
#35.1C 0,700 0,502 #36.1C 0,659 0,483 #65.1C 0,658 0,500 #66.1C 0,653 0,486
#85.1C 0,796 0,640 #86.1C 0,736 0,558 #'05.1C 0,599 0,447 #'06.1C 0,528 0,383
#'25.1C 0,530 0,431 #'26.1C 0,528 0,385 #'35.1C 0,743 0,608 #'36.1C 0,754 0,593
#'45.1C 0,660 0,489 #'46.1C 0,652 0,492 #'55.1C 0,669 0,564 #'56.1C 0,656 0,483
EV/GI
#25.2C 0,599 0,467 #26.2C 0,590 0,449 #35.2C 0,628 0,471 #36.2C 0,616 0,467
#65.2C 0,601 0,466 #66.2C 0,590 0,475 #'05.2C 0,573 0,452 #'06.2C 0,554 0,418
#'25.2C 0,582 0,419 #'26.2C 0,577 0,404 #'35.2C 0,661 0,530 #'36.2C 0,655 0,517
#'45.2C 0,599 0,466 #'46.2C 0,585 0,456 #'55.2C 0,586 0,478 #'56.2C 0,584 0,477
EV/EBITDA #35.3C 0,485 0,364 #36.3C 0,476 0,361 #45.3C 0,496 0,364 #46.3C 0,496 0,370
EV/EBIT #45.4C 0,519 0,379 #46.4C 0,507 0,376
EV/TA #25.5C 0,507 0,330 #26.5C 0,506 0,335 #45.5C 0,462 0,318 #46.5C 0,424 0,294
#'15.5C 0,398 0,288 #'16.5C 0,401 0,279 #'75.5C 0,455 0,302 #'76.5C 0,452 0,293
EV/OCF #'45.6C 0,559 0,402 #'46.6C 0,572 0,403
EV/FCFF
#25.7C 0,920 0,641 #26.7C 0,930 0,630 #45.7C 0,925 0,664 #46.7C 0,958 0,647
#55.7C 0,787 0,596 #56.7C 0,763 0,574 #75.7C 0,757 0,612 #76.7C 0,757 0,604
#95.7C 0,846 0,605 #96.7C 0,722 0,579
P/S
#15.8C 0,553 0,439 #16.8C 0,527 0,416 #25.8C 0,596 0,470 #26.8C 0,587 0,452
#35.8C 0,742 0,569 #36.8C 0,679 0,528 #45.8C 0,849 0,648 #46.8C 0,825 0,635
#65.8C 0,671 0,526 #66.8C 0,690 0,515 #85.8C 0,809 0,628 #86.8C 0,722 0,562
#95.8C 0,747 0,536 #96.8C 0,637 0,477 #'25.8C 0,613 0,512 #'26.8C 0,615 0,522
#'35.8C 0,762 0,615 #'36.8C 0,757 0,611 #'45.8C 0,562 0,443 #'46.8C 0,537 0,431
#'55.8C 0,566 0,460 #'56.8C 0,528 0,426
P/GI
#25.9C 0,553 0,442 #26.9C 0,547 0,428 #35.9C 0,628 0,490 #36.9C 0,606 0,488
#45.9C 0,648 0,526 #46.9C 0,621 0,506 #65.9C 0,587 0,470 #66.9C 0,588 0,454
#85.9C 0,633 0,507 #86.9C 0,601 0,474 #95.9C 0,612 0,486 #96.9C 0,570 0,452
#'25.9C 0,572 0,480 #'26.9C 0,574 0,453 #'35.9C 0,624 0,498 #'36.9C 0,621 0,498
#'45.9C 0,546 0,423 #'46.9C 0,532 0,406 #'55.9C 0,520 0,422 #'56.9C 0,520 0,424
P/EBITDA #'45.'0C 0,465 0,349 #'46.'0C 0,454 0,350
P/EBT #45.'2C 0,444 0,354 #46.'2C 0,445 0,370 #'65.'2C 0,420 0,331 #'66.'2C 0,412 0,322
P/E #45.'3C 0,448 0,370 #46.'3C 0,444 0,369 #'65.'3C 0,412 0,318 #'66.'3C 0,405 0,320
P/B
#15.'4C 0,524 0,411 #16.'4C 0,527 0,410 #25.'4C 0,555 0,440 #26.'4C 0,558 0,445
#45.'4C 0,551 0,451 #46.'4C 0,507 0,400 #'15.'4C 0,446 0,367 #'16.'4C 0,440 0,358
#'75.'4C 0,448 0,372 #'76.'4C 0,464 0,386
P/TA #45.'5C 0,756 0,627 #46.'5C 0,714 0,600 #'15.'5C 0,530 0,427 #'16.'5C 0,576 0,458
P/OCF #15.'6C 0,511 0,408 #16.'6C 0,507 0,400 #45.'6C 0,576 0,436 #46.'6C 0,565 0,438
#'45.'6C 0,495 0,392 #'46.'6C 0,490 0,392
P/FCFF
#25.'7C 0,935 0,705 #26.'7C 1,001 0,694 #45.'7C 0,873 0,764 #46.'7C 0,920 0,784
#55.'7C 0,894 0,603 #56.'7C 0,776 0,576 #75.'7C 0,773 0,622 #76.'7C 0,775 0,624
#95.'7C 0,767 0,581 #96.'7C 0,769 0,557
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.
22
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 k-
means (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. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
ICB
EV/S EV/GI EV/EBITDA EV/EBIT
*03.1C 0,806 0,608 *03.2C 0,675 0,505 *03.3C 0,542 0,380 *03.4C 0,584 0,418
*04.1C 0,838 0,583 *04.2C 0,688 0,470 *04.3C 0,544 0,368 *04.4C 0,590 0,402
EV/TA EV/OCF EV/FCFF P/S
*03.5C 0,482 0,317 *03.6C 0,566 0,394 *03.7C 1,258 0,648 #03.8C 0,795 0,614
*04.5C 0,488 0,315 *04.6C 0,565 0,389 *04.7C 1,173 0,624 #04.8C 0,780 0,582
P/GI P/EBITDA P/EBIT P/EBT
#03.9C 0,632 0,500 *04.'0C 0,508 0,373 *03.'1C 0,486 0,368 *03.'2C 0,444 0,360
#04.9C 0,630 0,482 *03.'0C 0,523 0,387 *04.'1C 0,474 0,362 *04.'2C 0,437 0,345
P/E P/B P/TA P/OCF
*03.'3C 0,445 0,360 *03.'4C 0,553 0,431 *03.'5C 0,719 0,549 *03.'6C 0,565 0,401
*04.'3C 0,440 0,350 *04.'4C 0,543 0,431 *04.'5C 0,703 0,532 *04.'6C 0,551 0,390
P/FCFF
*03.'7C 1,207 0,723
*04.'7C 1,292 0,709
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
23
(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.
24
Table 5.13: The best clustering procedure characterized by the mean and the median of its
distribution errors Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
Set of
ratios 01
EV/S P/S P/B P/OCF
*16.1C 0,556 0,408 #16.8C 0,527 0,416 *16.'4C 0,527 0,410 *16.'6C 0,507 0,400
Set of
ratios 02
EV/S EV/GI EV/TA EV/FCFF
*26.1C 0,6307 0,4665 *26.2C 0,590 0,449 *26.5C 0,506 0,335 *26.7C 0,930 0,630
P/S P/GI P/B P/FCFF
*26.8C 0,587 0,452 *26.9C 0,547 0,428 *26.'4C 0,558 0,445 *26.'7C 1,001 0,694
Set of
ratios 04
EV/EBITDA EV/EBIT EV/TA EV/FCFF
*46.3C 0,4961 0,370 *46.4C 0,507 0,376 #46.5C 0,424 0,294 *46.7C 0,958 0,647
P/S P/GI P/EBT P/E
#46.8C 0,825 0,635 #46.9C 0,621 0,506 *46.'2C 0,445 0,370 *46.'3C 0,444 0,369
P/B P/TA P/OCF P/FCFF
#46.'4C 0,507 0,400 #46.'5C 0,714 0,600 *46.'6C 0,565 0,438 *46.'7C 0,920 0,784
Set of
ratios 05
EV/FCFF P/FCFF
*56.7C 0,763 0,574 *56.'7C 0,776 0,576
Set of
ratios 06
EV/S EV/GI P/S P/GI
*66.1C 0,653 0,486 *66.2C 0,590 0,475 *66.8C 0,690 0,515 *66.9C 0,588 0,454
Set of
ratios 07
EV/FCFF P/FCFF
*76.7C 0,757 0,604 *76.'7C 0,775 0,624
Set of
ratios 08
EV/S P/S P/GI
#86.1C 0,736 0,558 #86.8C 0,722 0,562 #86.9C 0,601 0,474
Set of
ratios 09
EV/FCFF P/S P/GI P/FCFF
*96.7C 0,722 0,579 #96.8C 0,637 0,477 #96.9C 0,570 0,452 *96.'7C 0,769 0,557
Set of
ratios 10
EV/S EV/GI
#'06.1C 0,528 0,383 *'06.2C 0,554 0,418
Set of
ratios 11
EV/TA P/B P/TA
*'16.5C 0,401 0,279 *'16.'4C 0,440 0,358 #'15.'5C 0,530 0,427
Set of
ratios 12
EV/S EV/GI P/S P/GI
*'26.1C 0,528 0,385 *'26.2C 0,577 0,404 *'26.8C 0,615 0,522 *'26.9C 0,574 0,453
Set of
ratios 13
EV/S EV/GI P/S P/GI
*'36.1C 0,754 0,593 *'36.2C 0,655 0,517 *'36.8C 0,757 0,611 *'36.9C 0,621 0,498
Set of
ratios 14
EV/S EV/GI EV/OCF P/S
*'46.1C 0,652 0,492 *'46.2C 0,585 0,456 *'46.6C 0,572 0,403 #'46.8C 0,537 0,431
P/GI P/EBITDA P/OCF
*'46.9C 0,532 0,406 #'46.'0C 0,454 0,350 *'46.'6C 0,490 0,392
Set of
ratios 15
EV/S EV/GI P/S P/GI
*'56.1C 0,656 0,483 *'56.2C 0,584 0,477 #''56.8C 0,528 0,426 *'56.9C 0,520 0,424
Set of
ratios 16
P/EBT P/E
#'66.'2C 0,412 0,322 #'66.'3C 0,405 0,320
Set of
ratios 17
EV/TA P/B
*'76.5C 0,452 0,293 #'75.'4C 0,448 0,372
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.
25
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 Multiple
Entity Multiples Multiple
Equity Multiples
Distrib. Mean Median Distrib. Mean Median
ICB
EV/TA #03.5C 0,482 0,317 P/EBT #03.'2C 0,444 0,360
EV/EBITDA #03.3C 0,542 0,380 P/E #03.'3C 0,445 0,360
EV/OCF b03.6C 0,566 0,394 P/EBIT b03.'1C 0,486 0,368
EV/EBIT b03.4C 0,584 0,418 P/EBITDA b04.'0C 0,508 0,373
EV/GI 03.2C 0,675 0,505 P/B b03.'4C 0,553 0,431
EV/S 03.1C 0,806 0,608 P/OCF b03.'6C 0,565 0,401
EV/FCFF b03.7C 1,258 0,648 P/GI 03.9C 0,632 0,500 P/TA 03.'5C 0,719 0,549 P/S 03.8C 0,795 0,614 P/FCFF 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.
26
Table 5.15: The best market multiples characterized by the mean and the median of its
distribution errors – Part II Clustering
method Multiple
Entity Multiples Multiple
Equity Multiples
Distrib. Mean Median Distrib. Mean Median
Set of ratios 01
EV/S 16.1C 0,556 0,408 P/OCF *16.'6C 0,507 0,400
P/S *16.8C 0,527 0,416
P/B *16.'4C 0,527 0,410
Set of ratios 02
EV/TA #26.5C 0,506 0,335 P/GI #26.9C 0,547 0,428
EV/GI 26.2C 0,590 0,449 P/B #26.'4C 0,558 0,445
EV/S 26.1C 0,631 0,467 P/S #26.8C 0,587 0,452
EV/FCFF 26.7C 0,930 0,630 P/FCFF 26.'7C 1,001 0,694
Set of ratios 03
EV/EBITDA #36.3C 0,476 0,361 P/GI #36.9C 0,606 0,488
EV/GI 36.2C 0,616 0,467 P/S 36.8C 0,679 0,528
EV/S 36.1C 0,659 0,483
Set of ratios 04
EV/TA #46.5C 0,424 0,294 P/E #46.'3C 0,444 0,369
EV/EBITDA 46.3C 0,496 0,370 P/EBT #46.'2C 0,445 0,370
EV/EBIT 46.4C 0,507 0,376 P/B #46.'4C 0,507 0,400
EV/FCFF 46.7C 0,958 0,647 P/OCF b46.'6C 0,565 0,438
P/GI 46.9C 0,621 0,506 P/TA 46.'5C 0,714 0,600 P/S 46.8C 0,825 0,635 P/FCFF 46.'7C 0,920 0,784
Set of ratios 05 EV/FCFF 56.7C 0,763 0,574 P/FCFF 56.'7C 0,776 0,576
Set of ratios 06 EV/GI *66.2C 0,590 0,475 P/GI #66.9C 0,588 0,454
EV/S *66.1C 0,653 0,486 P/S 66.8C 0,690 0,515
Set of ratios 07 EV/FCFF 76.7C 0,757 0,604 P/FCFF 76.'7C 0,775 0,624
Set of ratios 08 EV/S 86.1C 0,736 0,558 P/GI #86.9C 0,601 0,474
P/S 86.8C 0,722 0,562
Set of ratios 09
EV/FCFF 96.7C 0,722 0,579 P/GI #96.9C 0,570 0,452
P/S b96.8C 0,637 0,477
P/FCFF 96.'7C 0,769 0,557
Set of ratios 10 EV/S #'06.1C 0,528 0,383
EV/GI '06.2C 0,554 0,418
Set of ratios 11 EV/TA '16.5C 0,401 0,279 P/B #'16.'4C 0,440 0,358
P/TA '15.'5C 0,530 0,427
Set of ratios 12 EV/S *'26.1C 0,528 0,385 P/GI *'26.9C 0,574 0,453
EV/GI *'26.2C 0,577 0,404 P/S *'26.8C 0,615 0,522
Set of ratios 13 EV/GI *'36.2C 0,655 0,517 P/GI #'36.9C 0,621 0,498
EV/S *'36.1C 0,754 0,593 P/S '36.8C 0,757 0,611
Set of ratios 14
EV/OCF #'46.6C 0,572 0,403 P/EBITDA #'46.'0C 0,454 0,350
EV/GI '46.2C 0,585 0,456 P/OCF #'46.'6C 0,490 0,392
EV/S #'46.1C 0,652 0,492 P/GI '46.9C 0,532 0,406
P/S '46.8C 0,537 0,431
Set of ratios 15 EV/GI *'56.2C 0,584 0,477 P/GI *'56.9C 0,520 0,424
EV/S *'56.1C 0,656 0,483 P/S *'56.8C 0,528 0,426
Set of ratios 16 P/E *'66.'3C 0,405 0,320 P/EBT *'66.'2C 0,412 0,322
Set of ratios 17 EV/TA '76.5C 0,452 0,293 P/B '75.'4C 0,448 0,372
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.
27
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 Clustering
measures
Entity Multiples Multiple
Clustering
measures
Equity Multiples
Distrib. Mean Median Distrib. Mean Median
EV/TA Set of ratios 11 #'16.5C 0,401 0,279 P/E Set of ratios 16 #'66.'3C 0,405 0,320
Set of ratios 04 b46.5C 0,424 0,294 Set of ratios 04 46.'3C 0,444 0,369
Set of ratios 17 '76.5C 0,452 0,293 ICB #03.'3C 0,445 0,360
ICB 03.5C 0,482 0,317
Set of ratios 02 26.5C 0,506 0,335
EV/EBITDA Set of ratios 03 #36.3C 0,476 0,361 P/EBT Set of ratios 16 #'66.'2C 0,412 0,322
Set of ratios 04 46.3C 0,496 0,370 ICB #03.'2C 0,444 0,360
ICB b03.3C 0,542 0,380 Set of ratios 04 46.'2C 0,445 0,370
EV/EBIT Set of ratios 04 *46.4C 0,507 0,376 P/B Set of ratios 11 #'16.'4C 0,440 0,358
ICB *03.4C 0,584 0,418 Set of ratios 17 '75.'4C 0,448 0,372
EV/OCF ICB *03.6C 0,566 0,394 Set of ratios 04 46.'4C 0,507 0,400
Set of ratios 14 *'46.6C 0,572 0,403 Set of ratios 01 16.'4C 0,527 0,410
ICB 03.'4C 0,553 0,431
Set of ratios 02 26.'4C 0,558 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
28
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 Clustering
measures
Entity Multiples Multiple
Clustering
measures
Equity Multiples
Distrib. Mean Median Distrib. Mean Median
EV/GI Set of ratios 10 #'06.2C 0,554 0,418 P/EBITDA Set of ratios 14 *'46.'0C 0,454 0,350
Set of ratios 12 #'26.2C 0,577 0,404 ICB *04.'0C 0,508 0,373
Set of ratios 15 '56.2C 0,584 0,477 P/OCF Set of ratios 14 *'46.'6C 0,490 0,392
Set of ratios 14 '46.2C 0,585 0,456 Set of ratios 01 *16.'6C 0,507 0,400
Set of ratios 06 b66.2C 0,590 0,475 ICB *03.'6C 0,565 0,401
Set of ratios 02 b26.2C 0,590 0,449 Set of ratios 04 *46.'6C 0,565 0,438
Set of ratios 03 36.2C 0,616 0,467 P/GI Set of ratios 09 #96.9C 0,570 0,452
Set of ratios 13 '36.2C 0,655 0,517 Set of ratios 15 #'56.9C 0,520 0,424
ICB b03.2C 0,675 0,505 Set of ratios 14 #'46.9C 0,532 0,406
EV/S Set of ratios 10 #'06.1C 0,528 0,383 Set of ratios 02 b26.9C 0,547 0,428
Set of ratios 12 b'26.1C 0,528 0,385 Set of ratios 12 #'26.9C 0,574 0,453
Set of ratios 01 16.1C 0,556 0,408 Set of ratios 06 #66.9C 0,588 0,454
Set of ratios 02 26.1C 0,631 0,467 Set of ratios 08 b86.9C 0,601 0,474
Set of ratios 14 '46.1C 0,652 0,492 Set of ratios 03 b36.9C 0,606 0,488
Set of ratios 06 66.1C 0,653 0,486 Set of ratios 13 '36.9C 0,621 0,498
Set of ratios 15 '56.1C 0,656 0,483 Set of ratios 04 46.9C 0,621 0,506
Set of ratios 03 36.1C 0,659 0,483 ICB 03.9C 0,632 0,500
Set of ratios 08 86.1C 0,736 0,558 P/TA Set of ratios 11 #'15.'5C 0,530 0,427
Set of ratios 13 '36.1C 0,754 0,593 Set of ratios 04 46.'5C 0,714 0,600
ICB 03.1C 0,806 0,608 ICB 03.'5C 0,719 0,549
EV/FCFF Set of ratios 09 *96.7C 0,722 0,579 P/S Set of ratios 01 #16.8C 0,527 0,416
Set of ratios 07 *76.7C 0,757 0,604 Set of ratios 15 '56.8C 0,528 0,426
Set of ratios 05 *56.7C 0,763 0,574 Set of ratios 14 b'46.8C 0,537 0,431
Set of ratios 02 *26.7C 0,930 0,630 Set of ratios 02 26.8C 0,587 0,452
Set of ratios 04 *46.7C 0,958 0,647 Set of ratios 12 b'26.8C 0,615 0,522
ICB *03.7C 1,258 0,648 Set of ratios 09 96.8C 0,637 0,477
Set of ratios 03 36.8C 0,679 0,528
Set of ratios 06 66.8C 0,690 0,515
Set of ratios 08 86.8C 0,722 0,562
Set of ratios 13 '36.8C 0,757 0,611
ICB 03.8C 0,795 0,614
Set of ratios 04 46.8C 0,825 0,635
P/FCFF Set of ratios 09 *96.'7C 0,769 0,557
Set of ratios 07 *76.'7C 0,775 0,624
Set of ratios 05 *56.'7C 0,776 0,576
Set of ratios 04 *46.'7C 0,920 0,784
Set of ratios 02 *26.'7C 1,001 0,694
ICB *03.'7C 1,207 0,723
Source: Own elaboration
29
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 non-
hierarchical 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.
30
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.
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33
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 Industry Supersector Sector Subsector
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/S
#01.1C 0,812 0,639 #02.1C 0,810 0,623 #03.1C 0,806 0,608 #04.1C 0,838 0,583
01.1D 1,238 0,590 02.1D 1,209 0,557 03.1D 1,163 0,540 04.1D 1,140 0,513
01.1B 1,348 0,579 02.1B 1,345 0,559 03.1B 1,290 0,531 04.1B 1,248 0,488
01.1A 1,934 0,641 02.1A 1,832 0,600 03.1A 1,721 0,578 04.1A 1,623 0,544
EV
/GI
#01.2C 0,660 0,517 #02.2C 0,655 0,514 #03.2C 0,675 0,505 #04.2C 0,688 0,470
01.2D 0,850 0,458 02.2D 0,831 0,427 03.2D 0,837 0,437 04.2D 0,828 0,419
01.2B 0,916 0,440 02.2B 0,884 0,430 03.2B 0,880 0,417 04.2B 0,876 0,408
01.2A 1,129 0,465 02.2A 1,092 0,458 03.2A 1,082 0,444 04.2A 1,051 0,451
EV
/EB
ITD
A
#01.3C 0,539 0,394 #02.3C 0,542 0,382 #03.3C 0,542 0,380 #04.3C 0,544 0,368
01.3D 0,631 0,342 02.3D 0,619 0,334 03.3D 0,616 0,325 04.3D 0,611 0,330
01.3B 0,668 0,338 02.3B 0,661 0,329 03.3B 0,656 0,322 04.3B 0,648 0,324
01.3A 0,749 0,363 02.3A 0,734 0,349 03.3A 0,729 0,343 04.3A 0,721 0,336
EV
/EB
IT #01.4C 0,575 0,425 #02.4C 0,580 0,412 #03.4C 0,584 0,418 #04.4C 0,590 0,402
01.4D 0,678 0,346 02.4D 0,669 0,339 03.4D 0,667 0,346 04.4D 0,666 0,350
01.4B 0,717 0,351 02.4B 0,712 0,350 03.4B 0,711 0,348 04.4B 0,705 0,344
01.4A 0,822 0,367 02.4A 0,812 0,363 03.4A 0,806 0,355 04.4A 0,797 0,358
EV
/TA
#01.5C 0,479 0,315 #02.5C 0,485 0,324 #03.5C 0,482 0,317 #04.5C 0,488 0,315
01.5B 0,542 0,328 02.5B 0,543 0,323 03.5B 0,545 0,322 04.5B 0,551 0,311
01.5D 0,561 0,336 02.5D 0,557 0,329 03.5D 0,554 0,325 04.5D 0,553 0,317
01.5A 0,685 0,380 02.5A 0,680 0,376 03.5A 0,670 0,368 04.5A 0,660 0,358
EV
/OC
F #01.6C 0,596 0,420 #02.6C 0,575 0,408 #03.6C 0,566 0,394 04.6C 0,565 0,389
01.6D 0,740 0,374 02.6D 0,697 0,357 03.6D 0,684 0,353 04.6D 0,677 0,347
01.6B 0,813 0,360 02.6B 0,741 0,348 03.6B 0,729 0,347 04.6B 0,720 0,348
01.6A 0,917 0,388 02.6A 0,857 0,368 03.6A 0,837 0,348 04.6A 0,828 0,342
EV
/FC
FF
01.7C 1,092 0,702 02.7C 1,170 0,660 03.7C 1,258 0,648 #04.7C 1,173 0,624
01.7D 2,094 0,514 02.7D 2,152 0,527 03.7D 2,145 0,516 04.7D 1,876 0,523
01.7B 2,486 0,520 02.7B 2,576 0,520 03.7B 2,500 0,521 04.7B 2,079 0,528
01.7A 3,411 0,567 02.7A 3,428 0,555 03.7A 3,359 0,560 04.7A 2,985 0,582
P/S
#01.8C 0,847 0,661 #02.8C 0,838 0,647 #03.8C 0,795 0,614 #04.8C 0,780 0,582
01.8D 1,299 0,603 02.8D 1,270 0,576 03.8D 1,190 0,558 04.8D 1,125 0,537
01.8B 1,372 0,603 02.8B 1,357 0,557 03.8B 1,288 0,543 04.8B 1,223 0,504
01.8A 2,051 0,635 02.8A 1,956 0,585 03.8A 1,827 0,574 04.8A 1,670 0,548
P/G
I
#01.9C 0,753 0,502 #02.9C 0,647 0,515 #03.9C 0,632 0,500 #04.9C 0,630 0,482
01.9D 0,835 0,479 02.9D 0,817 0,480 03.9D 0,788 0,466 04.9D 0,770 0,453
01.9B 0,895 0,480 02.9B 0,866 0,476 03.9B 0,829 0,470 04.9B 0,806 0,455
01.9A 1,116 0,498 02.9A 1,087 0,490 03.9A 1,049 0,482 04.9A 1,015 0,470
P/E
BIT
DA
#01.'0C 0,529 0,409 #02.'0C 0,518 0,398 #03.'0C 0,523 0,387 #04.'0C 0,508 0,373
01.'0D 0,611 0,365 02.'0D 0,602 0,360 03.'0D 0,596 0,355 04.'0D 0,580 0,355
01.'0B 0,629 0,365 02.'0B 0,624 0,362 03.'0B 0,621 0,353 04.'0B 0,604 0,357
01.'0A 0,734 0,386 02.'0A 0,723 0,371 03.'0A 0,712 0,364 04.'0A 0,693 0,370
P/E
BIT
#01.'1C 0,486 0,380 #02.'1C 0,483 0,375 #03.'1C 0,486 0,368 #04.'1C 0,474 0,362
01.'1D 0,549 0,340 02.'1D 0,550 0,334 03.'1D 0,546 0,334 04.'1D 0,529 0,337
01.'1B 0,562 0,343 02.'1B 0,569 0,331 03.'1B 0,565 0,334 04.'1B 0,541 0,339
01.'1A 0,662 0,358 02.'1A 0,658 0,354 03.'1A 0,650 0,347 04.'1A 0,624 0,356
P/E
BT
#01.'2C 0,445 0,361 #02.'2C 0,445 0,360 #03.'2C 0,444 0,360 #04.'2C 0,437 0,345
01.'2D 0,491 0,336 02.'2D 0,490 0,340 03.'2D 0,488 0,335 04.'2D 0,479 0,331
01.'2B 0,496 0,339 02.'2B 0,501 0,335 03.'2B 0,495 0,335 04.'2B 0,486 0,331
01.'2A 0,574 0,355 02.'2A 0,573 0,352 03.'2A 0,569 0,349 04.'2A 0,555 0,355
P/E
#01.'3C 0,442 0,367 #02.'3C 0,440 0,365 #03.'3C 0,445 0,360 #04.'3C 0,440 0,350
01.'3D 0,481 0,332 02.'3D 0,481 0,336 03.'3D 0,480 0,332 04.'3D 0,470 0,324
01.'3B 0,490 0,334 02.'3B 0,490 0,334 03.'3B 0,489 0,328 04.'3B 0,478 0,316
01.'3A 0,562 0,338 02.'3A 0,564 0,333 03.'3A 0,560 0,329 04.'3A 0,545 0,340
34
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 Industry Supersector Sector Subsector
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/B
#01.'4C 0,553 0,437 #02.'4C 0,551 0,421 #03.'4C 0,553 0,431 #04.'4C 0,543 0,431
01.'4B 0,668 0,433 02.'4B 0,660 0,431 03.'4B 0,661 0,436 04.'4D 0,643 0,417
01.'4D 0,672 0,437 02.'4D 0,666 0,434 03.'4D 0,659 0,429 04.'4B 0,659 0,422
01.'4A 0,862 0,477 02.'4A 0,854 0,481 03.'4A 0,836 0,468 04.'4A 0,808 0,454
P/T
A
#01.'5C 0,734 0,613 #02.'5C 0,715 0,564 #03.'5C 0,719 0,549 #04.'5C 0,703 0,532
01.'5D 1,089 0,497 02.'5D 1,041 0,499 03.'5D 1,022 0,499 04.'5D 0,975 0,491
01.'5B 1,190 0,503 02.'5B 1,128 0,499 03.'5B 1,094 0,500 04.'5B 1,045 0,485
01.'5A 1,603 0,551 02.'5A 1,533 0,530 03.'5A 1,523 0,522 04.'5A 1,413 0,511
P/O
CF
#01.'6C 0,583 0,454 #02.'6C 0,566 0,417 #03.'6C 0,565 0,401 #04.'6C 0,551 0,390
01.'6D 0,713 0,415 02.'6D 0,686 0,394 03.'6D 0,679 0,378 04.'6D 0,659 0,378
01.'6B 0,756 0,411 02.'6B 0,728 0,387 03.'6B 0,725 0,382 04.'6B 0,685 0,371
01.'6A 0,885 0,415 02.'6A 0,848 0,400 03.'6A 0,835 0,392 04.'6A 0,814 0,388
P/F
CF
F #01.'7C 1,179 0,767 #02.'7C 1,180 0,742 #03.'7C 1,207 0,723 #04.'7C 1,292 0,709
01.'7D 2,269 0,572 02.'7D 2,234 0,557 03.'7D 2,212 0,553 04.'7D 2,132 0,565
01.'7B 2,653 0,573 02.'7B 2,610 0,553 03.'7B 2,533 0,564 04.'7B 2,373 0,565
01.'7A 3,832 0,602 02.'7A 3,740 0,595 03.'7A 3,684 0,593 04.'7A 3,527 0,608
35
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
Set of ratios 01 Set of ratios 02
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/S
#15.1C 0,574 0,427 #16.1C 0,556 0,408 #25.1C 0,636 0,484 #26.1C 0,631 0,467
15.1D 0,724 0,402 16.1D 0,672 0,369 25.1D 0,830 0,416 26.1D 0,818 0,421
15.1B 0,742 0,406 16.1B 0,694 0,373 25.1B 0,848 0,421 26.1B 0,833 0,412
15.1A 0,998 0,449 16.1A 0,905 0,416 25.1A 1,196 0,480 26.1A 1,174 0,472
Set of ratios 03 Set of ratios 06
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/S
#35.1C 0,700 0,502 #36.1C 0,659 0,483 #65.1C 0,658 0,500 #66.1C 0,653 0,486
35.1D 0,945 0,451 36.1D 0,859 0,438 65.1D 0,865 0,480 66.1D 0,845 0,435
35.1B 0,961 0,455 36.1B 0,878 0,435 65.1B 0,868 0,475 66.1B 0,888 0,431
35.1A 1,382 0,503 36.1A 1,215 0,492 65.1A 1,311 0,578 66.1A 1,210 0,501
Set of ratios 08 Set of ratios 10
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/S
#85.1C 0,796 0,640 #86.1C 0,736 0,558 #'05.1C 0,599 0,447 #'06.1C 0,528 0,383
85.1D 1,236 0,587 86.1B 1,080 0,533 05.1D 0,744 0,403 06.1D 0,610 0,344
85.1B 1,278 0,593 86.1D 1,089 0,531 05.1B 0,777 0,393 06.1B 0,630 0,346
85.1A 2,011 0,684 86.1A 1,792 0,648 05.1A 0,978 0,427 06.1A 0,765 0,373
Set of ratios 12 Set of ratios 13
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/S
#'25.1C 0,530 0,431 #'26.1C 0,528 0,385 #'35.1C 0,743 0,608 #'36.1C 0,754 0,593
25.1D 0,585 0,439 26.1D 0,586 0,382 35.1D 1,051 0,523 36.1D 1,051 0,526
25.1B 0,609 0,442 26.1B 0,613 0,396 35.1B 1,121 0,524 36.1B 1,117 0,517
25.1A 0,701 0,434 26.1A 0,695 0,431 35.1A 1,543 0,589 36.1A 1,534 0,579
Set of ratios 14 Set of ratios 15
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/S
#'45.1C 0,660 0,489 #'46.1C 0,652 0,492 #'55.1C 0,669 0,564 #'56.1C 0,656 0,483
45.1D 0,873 0,458 46.1D 0,857 0,458 55.1D 0,847 0,458 56.1D 0,797 0,434
45.1B 0,870 0,454 46.1B 0,863 0,453 55.1B 0,885 0,455 56.1B 0,834 0,434
45.1A 1,337 0,568 46.1A 1,282 0,548 55.1A 1,139 0,475 56.1A 1,049 0,453
Set of ratios 02 Set of ratios 03
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/GI
#25.2C 0,599 0,467 #26.2C 0,590 0,449 #35.2C 0,628 0,471 #36.2C 0,616 0,467
25.2D 0,750 0,429 26.2D 0,745 0,416 35.2D 0,775 0,414 36.2D 0,763 0,413
25.2B 0,763 0,429 26.2B 0,769 0,418 35.2B 0,814 0,408 36.2B 0,790 0,413
25.2A 1,000 0,462 26.2A 0,994 0,459 35.2A 1,011 0,427 36.2A 0,994 0,432
Set of ratios 06 Set of ratios 10
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/GI
#65.2C 0,601 0,466 #66.2C 0,590 0,475 #'05.2C 0,573 0,452 #'06.2C 0,554 0,418
65.2D 0,750 0,437 66.2D 0,740 0,433 05.2D 0,712 0,401 06.2D 0,674 0,396
65.2B 0,776 0,439 66.2B 0,770 0,435 05.2B 0,740 0,405 06.2B 0,690 0,398
65.2A 0,997 0,482 66.2A 0,970 0,451 05.2A 0,933 0,420 06.2A 0,891 0,419
Set of ratios 12 Set of ratios 13
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/GI
#'25.2C 0,582 0,419 #'26.2C 0,577 0,404 #'35.2C 0,661 0,530 #'36.2C 0,655 0,517
25.2D 0,641 0,364 26.2D 0,659 0,377 35.2D 0,822 0,448 36.2D 0,812 0,455
25.2B 0,677 0,370 26.2B 0,679 0,387 35.2B 0,858 0,447 36.2B 0,852 0,454
25.2A 0,767 0,393 26.2A 0,806 0,407 35.2A 1,069 0,455 36.2A 1,069 0,478
36
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 Set of ratios 15
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/GI
#'45.2C 0,599 0,466 #'46.2C 0,585 0,456 #'55.2C 0,586 0,478 #'56.2C 0,584 0,477
45.2D 0,732 0,406 46.2D 0,716 0,414 55.2D 0,690 0,397 56.2D 0,671 0,425
45.2B 0,748 0,413 46.2B 0,730 0,411 55.2B 0,723 0,403 56.2B 0,692 0,417
45.2A 0,961 0,460 46.2A 0,929 0,440 55.2CA 0,866 0,422 56.2A 0,844 0,423
Set of ratios 03 Set of ratios 04
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/EB
ITD
A
#35.3C 0,485 0,364 #36.3C 0,476 0,361 #45.3C 0,496 0,364 #46.3C 0,496 0,370
35.3D 0,548 0,321 36.3D 0,542 0,322 45.3D 0,564 0,346 46.3D 0,570 0,359
35.3B 0,566 0,322 36.3B 0,558 0,325 45.3B 0,584 0,351 46.3B 0,587 0,360
35.3A 0,638 0,338 36.3A 0,638 0,343 45.3A 0,667 0,373 46.3A 0,675 0,369
Set of ratios 04
Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median
EV
/EB
IT #45.4C 0,519 0,379 #46.4C 0,507 0,376
45.4D 0,585 0,350 46.4D 0,575 0,349
45.4B 0,608 0,353 46.4B 0,594 0,346
45.4A 0,698 0,369 46.4A 0,679 0,363
Set of ratios 02 Set of ratios 04
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/TA
#25.5C 0,507 0,330 #26.5C 0,506 0,335 #45.5C 0,462 0,318 #46.5C 0,424 0,294
25.5B 0,575 0,332 26.5B 0,576 0,334 45.5B 0,515 0,336 46.5B 0,469 0,270
25.5D 0,595 0,341 26.5D 0,591 0,346 45.5D 0,550 0,355 46.5D 0,473 0,278
25.5A 0,716 0,400 26.5A 0,710 0,399 45.5A 0,676 0,429 46.5A 0,549 0,316
Set of ratios 11 Set of ratios 17
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/TA
#'15.5C 0,398 0,288 #'16.5C 0,401 0,279 #'75.5C 0,455 0,302 #'76.5C 0,452 0,293
15.5D 0,431 0,261 16.5D 0,428 0,257 75.5B 0,488 0,293 76.5B 0,486 0,303
15.5B 0,432 0,256 16.5B 0,433 0,254 75.5D 0,500 0,299 76.5D 0,508 0,317
15.5A 0,489 0,280 16.5A 0,479 0,262 75.5A 0,575 0,325 76.5A 0,601 0,358
Set of ratios 14
Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median
EV
/OC
F #'45.6C 0,559 0,402 #'46.6C 0,572 0,403
45.6D 0,639 0,339 46.6D 0,650 0,351
45.6B 0,664 0,329 46.6B 0,673 0,343
45.6A 0,761 0,358 46.6A 0,768 0,366
Set of ratios 02 Set of ratios 04
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/FC
FF
#25.7C 0,920 0,641 #26.7C 0,930 0,630 #45.7C 0,925 0,664 #46.7C 0,958 0,647
25.7D 1,436 0,506 26.7D 1,408 0,507 45.7D 1,491 0,509 46.7D 1,484 0,511
25.7B 1,566 0,511 26.7B 1,536 0,497 45.7B 1,606 0,519 46.7B 1,593 0,507
25.7A 2,113 0,562 26.7A 2,046 0,552 45.7A 2,216 0,583 46.7A 2,208 0,587
Set of ratios 05 Set of ratios 07
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
EV
/FC
FF
#55.7C 0,787 0,596 #56.7C 0,763 0,574 #75.7C 0,757 0,612 #76.7C 0,757 0,604
55.7D 1,164 0,517 56.7D 1,074 0,491 75.7D 1,110 0,521 76.7D 1,107 0,514
55.7B 1,323 0,523 56.7B 1,124 0,494 75.7B 1,200 0,507 76.7B 1,199 0,508
55.7A 1,781 0,562 56.7A 1,606 0,549 75.7A 1,619 0,549 76.7A 1,603 0,551
37
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
Set of ratios 09
Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median
EV
/FC
FF
#95.7C 0,846 0,605 #96.7C 0,722 0,579
95.7D 1,251 0,501 96.7D 0,975 0,495
95.7B 1,332 0,510 96.7B 1,014 0,501
95.7A 1,807 0,567 96.7A 1,401 0,563
Set of ratios 01 Set of ratios 02
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/S
#15.8C 0,553 0,439 #16.8C 0,527 0,416 #25.8C 0,596 0,470 #26.8C 0,587 0,452
15.8D 0,689 0,422 16.8D 0,641 0,385 25.8D 0,780 0,433 26.8D 0,765 0,427
15.8B 0,711 0,426 16.8B 0,637 0,380 25.8B 0,789 0,432 26.8B 0,778 0,424
15.8A 0,977 0,453 16.8A 0,900 0,419 25.8A 1,124 0,467 26.8A 1,106 0,464
Set of ratios 03 Set of ratios 04
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/S
#35.8C 0,742 0,569 #36.8C 0,679 0,528 #45.8C 0,849 0,648 #46.8C 0,825 0,635
35.8D 1,058 0,545 36.8D 0,929 0,532 45.8D 1,303 0,622 46.8D 1,245 0,604
35.8B 1,074 0,547 36.8B 0,928 0,531 45.8B 1,337 0,622 46.8B 1,266 0,610
35.8A 1,637 0,587 36.8A 1,385 0,590 45.8A 2,159 0,697 46.8A 2,043 0,699
Set of ratios 06 Set of ratios 08
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/S
#65.8C 0,671 0,526 #66.8C 0,690 0,515 #85.8C 0,809 0,628 #86.8C 0,722 0,562
65.8D 0,898 0,497 66.8D 0,901 0,478 85.8D 1,218 0,579 86.8D 1,045 0,510
65.8B 0,921 0,503 66.8B 0,938 0,478 85.8B 1,277 0,579 86.8B 1,044 0,506
65.8A 1,312 0,565 66.8A 1,257 0,507 85.8A 1,921 0,635 86.8A 1,658 0,590
Set of ratios 09 Set of ratios 12
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/S
#95.8C 0,747 0,536 #96.8C 0,637 0,477 #'25.8C 0,613 0,512 #'26.8C 0,615 0,522
95.8D 0,985 0,497 96.8D 0,789 0,443 25.8D 0,729 0,517 26.8D 0,749 0,505
95.8B 1,010 0,493 96.8B 0,818 0,442 25.8B 0,760 0,528 26.8B 0,820 0,512
95.8A 1,450 0,561 96.8A 1,053 0,468 25.8A 0,927 0,527 26.8A 0,933 0,486
Set of ratios 13 Set of ratios 14
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/S
#'35.8C 0,762 0,615 #'36.8C 0,757 0,611 #'45.8C 0,562 0,443 #'46.8C 0,537 0,431
35.8D 1,062 0,550 36.8D 1,050 0,553 45.8D 0,681 0,427 46.8D 0,647 0,415
35.8B 1,148 0,559 36.8B 1,114 0,556 45.8B 0,690 0,429 46.8B 0,649 0,411
35.8A 1,578 0,604 36.8A 1,549 0,602 45.8A 0,947 0,492 46.8A 0,903 0,466
Set of ratios 15
Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median
P/S
#'55.8C 0,566 0,460 #'56.8C 0,528 0,426
55.8D 0,656 0,432 56.8D 0,603 0,389
55.8B 0,666 0,435 56.8B 0,617 0,389
55.8A 0,845 0,467 56.8A 0,773 0,429
Set of ratios 02 Set of ratios 03
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/G
I
#25.9C 0,553 0,442 #26.9C 0,547 0,428 #35.9C 0,628 0,490 #36.9C 0,606 0,488
25.9D 0,683 0,428 26.9D 0,677 0,421 35.9D 0,781 0,448 36.9D 0,751 0,453
25.9B 0,692 0,434 26.9B 0,688 0,423 35.9B 0,821 0,449 36.9B 0,767 0,452
25.9A 0,914 0,473 26.9A 0,910 0,461 35.9A 1,036 0,489 36.9A 0,992 0,485
38
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
Set of ratios 04 Set of ratios 06
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/G
I
#45.9C 0,648 0,526 #46.9C 0,621 0,506 #65.9C 0,587 0,470 #66.9C 0,588 0,454
45.9D 0,836 0,503 46.9D 0,796 0,479 65.9D 0,724 0,441 66.9D 0,721 0,441
45.9B 0,867 0,504 46.9B 0,812 0,476 65.9B 0,736 0,447 66.9B 0,747 0,438
45.9A 1,154 0,528 46.9A 1,103 0,535 65.9A 0,961 0,482 66.9A 0,945 0,463
Set of ratios 08 Set of ratios 09
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/G
I
#85.9C 0,633 0,507 #86.9C 0,601 0,474 #95.9C 0,612 0,486 #96.9C 0,570 0,452
85.9D 0,801 0,469 86.9D 0,757 0,450 95.9D 0,747 0,452 96.9D 0,690 0,431
85.9B 0,853 0,473 86.9B 0,791 0,455 95.9B 0,773 0,455 96.9B 0,709 0,435
85.9A 1,073 0,493 86.9A 1,018 0,492 95.9A 0,991 0,486 96.9A 0,900 0,474
Set of ratios 12 Set of ratios 13
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/G
I
#'25.9C 0,572 0,480 #'26.9C 0,574 0,453 #'35.9C 0,624 0,498 #'36.9C 0,621 0,498
25.9D 0,664 0,427 26.9D 0,676 0,398 35.9D 0,762 0,464 36.9D 0,756 0,463
25.9B 0,727 0,457 26.9B 0,722 0,435 35.9B 0,805 0,467 36.9B 0,798 0,455
25.9A 0,822 0,428 26.9A 0,854 0,443 35.9A 0,988 0,464 36.9A 0,986 0,470
Set of ratios 14 Set of ratios 15
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/G
I
#'45.9C 0,546 0,423 #'46.9C 0,532 0,406 #'55.9C 0,520 0,422 #'56.9C 0,520 0,424
45.9D 0,637 0,398 46.9D 0,617 0,388 55.9D 0,585 0,385 56.9D 0,586 0,390
45.9B 0,652 0,394 46.9B 0,634 0,391 55.9B 0,601 0,384 56.9B 0,603 0,393
45.9A 0,808 0,423 46.9A 0,779 0,411 55.9A 0,715 0,408 56.9A 0,714 0,405
Set of ratios 14
Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median
P/E
BIT
DA
#'45.'0C 0,465 0,349 #'46.'0C 0,454 0,350
45.'0D 0,511 0,328 46.'0D 0,503 0,323
45.'0B 0,522 0,329 46.'0B 0,516 0,329
45.'0A 0,602 0,335 46.'0A 0,597 0,335
Set of ratios 04 Set of ratios 16
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/E
BT
#45.'2C 0,444 0,354 #46.'2C 0,445 0,370 #'65.'2C 0,420 0,331 #'66.'2C 0,412 0,322
45.'2D 0,492 0,348 46.'2D 0,492 0,344 65.'2D 0,454 0,302 66.'2D 0,446 0,295
45.'2B 0,504 0,348 46.'2B 0,499 0,345 65.'2B 0,458 0,306 66.'2B 0,452 0,296
45.'2A 0,576 0,362 46.'2A 0,574 0,361 65.'2A 0,518 0,322 66.'2A 0,509 0,321
Set of ratios 04 Set of ratios 16
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/E
#45.'3C 0,448 0,370 #46.'3C 0,444 0,369 #'65.'3C 0,412 0,318 #'66.'3C 0,405 0,320
45.'3D 0,491 0,342 46.'3D 0,486 0,345 65.'3D 0,443 0,302 66.'3D 0,435 0,300
45.'3B 0,503 0,339 46.'3B 0,497 0,338 65.'3B 0,448 0,302 66.'3B 0,440 0,299
45.'3A 0,571 0,347 46.'3A 0,563 0,346 65.'3A 0,505 0,311 66.'3A 0,494 0,305
Set of ratios 01 Set of ratios 02
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/B
#15.'4C 0,524 0,411 #16.'4C 0,527 0,410 #25.'4C 0,555 0,440 #26.'4C 0,558 0,445
15.'4B 0,633 0,419 16.'4B 0,613 0,401 25.'4D 0,664 0,429 26.'4D 0,668 0,428
15.'4D 0,627 0,422 16.'4D 0,621 0,409 25.'4B 0,666 0,428 26.'4B 0,675 0,430
15.'4A 0,791 0,447 16.'4A 0,779 0,445 25.'4A 0,847 0,462 26.'4A 0,850 0,464
39
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 Set of ratios 11
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/B
#45.'4C 0,551 0,451 #46.'4C 0,507 0,400 #'15.'4C 0,446 0,367 #'16.'4C 0,440 0,358
45.'4B 0,669 0,458 46.'4B 0,596 0,377 15.'4D 0,499 0,352 16.'4D 0,489 0,357
45.'4D 0,684 0,461 46.'4D 0,594 0,387 15.'4B 0,497 0,344 16.'4B 0,492 0,348
45.'4A 0,889 0,516 46.'4A 0,735 0,408 15.'4A 0,598 0,383 16.'4A 0,578 0,367
Set of ratios 17
Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median
P/B
#'75.'4C 0,448 0,372 #'76.'4C 0,464 0,386
75.'4B 0,503 0,373 76.'4B 0,518 0,382
75.'4D 0,505 0,374 76.'4D 0,529 0,387
75.'4A 0,606 0,385 76.'4A 0,651 0,418
Set of ratios 04 Set of ratios 11
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/T
A
#45.'5C 0,756 0,627 #46.'5C 0,714 0,600 #'15.'5C 0,530 0,427 #'16.'5C 0,576 0,458
45.'5D 1,146 0,512 46.'5D 0,976 0,439 15.'5D 0,623 0,391 16.'5D 0,703 0,395
45.'5B 1,188 0,517 46.'5B 1,049 0,437 15.'5B 0,643 0,394 16.'5B 0,750 0,385
45.'5A 1,720 0,612 46.'5A 1,362 0,470 15.'5A 0,805 0,431 16.'5A 0,915 0,395
Set of ratios 01 Set of ratios 04
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/O
CF
#15.'6C 0,511 0,408 #16.'6C 0,507 0,400 #45.'6C 0,576 0,436 #46.'6C 0,565 0,438
15.'6D 0,585 0,376 16.'6D 0,579 0,364 45.'6D 0,689 0,419 46.'6D 0,671 0,405
15.'6B 0,603 0,378 16.'6B 0,600 0,368 45.'6B 0,721 0,421 46.'6B 0,698 0,403
15.'6A 0,701 0,390 16.'6A 0,699 0,399 45.'6A 0,853 0,431 46.'6A 0,827 0,411
Set of ratios 14
Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median
P/O
CF
#'45.'6C 0,495 0,392 #'46.'6C 0,490 0,392
45.'6D 0,554 0,358 46.'6D 0,547 0,337
45.'6B 0,575 0,346 46.'6B 0,567 0,343
45.'6A 0,657 0,364 46.'6A 0,652 0,362
Set of ratios 02 Set of ratios 04
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/F
CF
F #25.'7C 0,935 0,705 #26.'7C 1,001 0,694 #45.'7C 0,873 0,764 #46.'7C 0,920 0,784
25.'7D 1,552 0,532 26.'7D 1,622 0,530 45.'7D 1,488 0,533 46.'7D 1,450 0,513
25.'7B 1,795 0,532 26.'7B 1,847 0,523 45.'7B 1,665 0,543 46.'7B 1,617 0,514
25.'7A 2,510 0,577 26.'7A 2,522 0,577 45.'7A 2,342 0,578 46.'7A 2,260 0,587
Set of ratios 05 Set of ratios 07
Complete Linkage K Means Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median Distrib. Mean Median
P/F
CF
F #55.'7C 0,894 0,603 #56.'7C 0,776 0,576 #75.'7C 0,773 0,622 #76.'7C 0,775 0,624
55.'7D 1,288 0,525 56.'7D 1,091 0,499 75.'7D 1,129 0,548 76.'7D 1,133 0,544
55.'7B 1,387 0,538 56.'7B 1,161 0,514 75.'7B 1,212 0,553 76.'7B 1,224 0,541
55.'7A 1,884 0,566 56.'7A 1,616 0,555 75.'7A 1,712 0,558 76.'7A 1,698 0,555
Set of ratios 09
Complete Linkage K Means
Distrib. Mean Median Distrib. Mean Median
P/F
CF
F #95.'7C 0,767 0,581 #96.'7C 0,769 0,557
95.'7D 1,070 0,512 96.'7D 1,068 0,501
95.'7B 1,143 0,526 96.'7B 1,128 0,507
95.'7A 1,603 0,571 96.'7A 1,554 0,557
40
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