In the process of valuation of a company usually a number of experts work together in a team. The valuation formulas are usually based on several financial figures like the expected discounted cash flows and the expected cost of capital. But it is also based on an evaluation of the success factors, which are the causes of economic prosperity of the company. Especially in the new market or in the e-business these soft criteria determine the value of the companies. In this paper a Comparative Company Approach (CCA) based on the Analytic Hierarchy Process (AHP) (Saaty, 1990) is suggested. A company has to be compared to its competitors according to the predetermined criteria – especially the success factors. The results of this kind of bench-marking process are the relative values of the companies included in the investigation. The general method and the aggregation of individual benchmarking results will be described mathe-matically. The new idea is to use AHP as a basis for a Comparative Company Approach: Knowing the results of the AHP-benchmarking and the monetary value of the competitors (e.g. stock capitalization) it is possible to calculate a range of estimates for the monetary value of the company considered.Analytic Hierarchy Process, AHP, Multi-objective Decision Making, Group Decision Making, Valuation

Matthias Amen

English

Research Papers in Economics

 
   
 
ADVANCES 
in 
DECISION TECHNOLOGY 
and 
INTELLIGENT  INFORMATION  SYSTEMS 
Volume II 
 
 
 
 
 
 
 
Edited 
By 
 
 
KURT J. ENGEMANN 
Iona College 
U.S.A. 
GEORGE E. LASKER 
University of Windsor 
Canada 
 
 
 
 
Published by 
 
THE INTERNATIONAL INSTITUTE FOR ADVANCED STUDIES 
IN SYSTEMS RESEARCH AND CYBERNETICS  
   
Integrating Success Factors and Group Attitudes  
into the Valuation of a Company 
 
Dr. Matthias Amen 
Universität Bern - Institut für Unternehmensrechnung und Controlling  
CH-3012 Berne / Switzerland 
 
Abstract: In the process of valuation of a company usually a number of experts work together in 
a team. The valuation formulas are usually based on several financial figures like the expected 
discounted cash flows and the expected cost of capital. But it is also based on an evaluation of 
the success factors, which are the causes of economic prosperity of the company. Especially in 
the new market or in the e -business these soft criteria determine the value of the companies. In 
this paper a Comparative Company Approach (CCA) based o n the Analytic Hierarchy Process 
(AHP) (Saaty, 1990) is suggested. A company has to be compared to its competitors according to 
the predetermined criteria  – especially the success factors. The results of this kind of bench-
marking process are the relative  values of the companies included in the investigation. The gen-
eral method and the aggregation of individual benchmarking results will be described mathe-
matically. The new idea is to use AHP as a basis for a Comparative Company Approach: Know-
ing the results of the AHP-benchmarking and the monetary value of the competitors (e.g. stock 
capitalization) it is possible to calculate a range of estimates for the monetary value of the com-
pany considered. 
 
Keywords: Analytic Hierarchy Process, AHP, Multi-objective Decision Making, Group Deci-
sion Making, Valuation 
 
1 Hierarchical analysis of the problem 
 
Using the technique of the Analytic Hierarchy Process (AHP) (Saaty, 1990) we first have to de-
fine a hierarchy of soft and hard criteria (success factors and financial numbers) which are im-
portant for the valuation of a company. Within this Comparative Company Approach (CCA) 
(e.g. Damodaran, 1994, pp. 15-16) we assume that the criterion weights are the same for all 
companies within a certain industry. A conceptual example of such a hierarchy is shown in fig-
ure 1 (see next page).  
 
For a formal description we introduce the following symbols: 
 
I  number of elements (criteria, alternatives = companies) of the decision problem [-] 
i, j  elements of the decision problem (criteria, alternatives = companies); i, j ˛ {0, ..., I} 
i G   Set of elements (criteria, alternatives = companies) which have impact to element (criteria) i  
 
For simplification it is assumed that the nodes of the hierarchy are numbered topologically, i. e. 
j<i holds if  j i G ˛ . The top node is numbered as 0 and gives a description of the problem (in 
AHP-terminology: the goal). 2 
   
company competitor 1 competitor 2 competitor n
...
valuation of a company
success factors financial numbers
quality know-how
(human capital)
flexibility image
products processes technical entrepreneurial
total
equity
cash
flow
profit dynamic debt
ratio
 
Figure 1: Example of a hierarchy for the valuation of a company 
 
 
2 Determination of local priorities 
 
After the determination of the hierarchy the importance of each criterion has to be evaluated. 
Further the different companies are to be compared with respect to the different criteria. (Hafner 
(1988) uses AHP only for the determination of the criterion weights.) The relative importance/ 
value of an element with respect to a higher-level element of the hierarchy is defined as (local) 
priority: 
 
ij p   priority of element i with respect to element j [-],  j i G ˛ , j ˛ {0, ..., I} 
 
The priorities are normalized to the sum  1 p
j i
ij = ￿
G ˛
. The concrete determination depends on 
the nature of criteria/comparisons. 
 
Case 1: Hard criteria, which are proportional to the priority 
For h ard criteria, like total equity or a pre-calculated sum of discounted cash flows, which are 
proportional to the priority, the local priorities can easily be calculated: 
 
￿
=
G ˛ j k
kj
ij
ij w
w
: p  
 
ij w  “value” of company i for the hard criterion j [-],  j i G ˛ , j ˛ {0, ..., I} 3 
   
Case 2: Hard criteria, which are reciprocal to the priority 
For hard criteria, like the dynamic debt ratio (i.e. total debt / cash flow), which are reciprocal to 
the priority, the calculation of the local priorities is slightly more complicated: 
 
￿ ￿
￿
ł
￿
￿
￿
Ł
￿
￿
￿
ł
￿
￿
￿
Ł
￿
￿
￿
￿
ł
￿
￿
￿
Ł
￿
￿
=
￿
=
G ˛ G ˛
G ˛
G ˛
j k
kj
m
mj
ij
m
mj
j k kj
ij
ij
w w
w w
p
1
p
1
: p
j
j
 
ij p   priority of element i with respect to element j  assuming that “proportional importance” 
exists [-],  j i G ˛ , j ˛ {0, ..., I} 
 
For practical purpose the calculation can be done in two steps: First the local priorities  ij p  are 
calculated assuming that there is a proportional importance. Second the local priorities  ij p  are 
calculated on the basis of the reciprocals of the  ij p . 
 
Case 3: Soft criteria and criterion weights 
For soft criteria, like all different kinds of success factors, and for the determination of criterion 
weights the AHP technique derives the local priorities from comparison  matrices. The elements 
of a matrix are defined by pairwise comparisons of companies with respect to a criterion or by 
pairwise comparisons of criteria with respect to a higher-level criterion. The comparisons are 
made by the use of the scale suggested by Saaty (1990) or any other reasonable scale (table 1). 
 
1  equally good (large, important) 
3  moderately better (larger, more important) 
5  strongly better (larger, more important) 
7  very strongly better (larger, more important) 
9  extremely better (larger, more important) 
Saaty-
scale 
Intermediate values 2, 4, 6, 8 and reciprocals (e.g. 1/2; 1/7;) are also usable. 
Table 1: Saaty-scale 
 
For further discussion the following symbols were introduced:  
 
j A =( ) ikj a   comparison matrix for criterion j, j ˛ {0, ..., I} 
ikj a   result of comparisons of criterion/company i versus criterion/company k with respect 
to criterion j [-] 
j p =( ) ij p   priority vector for criterion j, j ˛ {0, ..., I} 
 4 
   
The entries in the comparison matrix are substitutes for the relations of the corresponding local 
priorities,  kj ij ikj p p a = . The quadratic matrix  j A  is reciprocal as  ikj ij kj kij a 1 p p a = = . For k 
= i,  1 aikj =  holds. Therefore the number of pairwise comparisons reduces to n(n–1)/2, where n is 
the number of columns/rows of matrix  j A . The matrix  j A  is consistent if  imj kmj ikj a a a = ￿  for 
any i and m ( transistency). For a consistent matrix the calculation of the local priority vector is 
easy. The local priorities are given by normalizing any column k of the matrix: 
￿ =
G ˛ j m
imj ikj ij a a p . 
 
As human individuals are not necessary consistent in their votings the transistency can be 
missed, especially for larger n. In the case of inconsistency we have to make the following con-
siderations: Multiplying a consistent matrix  j A  with the vector  j p  we get  j j p A ￿ = j p n￿ . In this 
formulation  j p  is known to be the eigenvector of  j A  with the eigenvalue n. For an inconsistent 
matrix  j A  we have to find a nontrivial solution for the eigenvalue problem  j j p A ￿  =  j max p ￿ l  
where  max l  is the greatest of n possible eigenvalues and the corresponding eigenvector  j p  rep-
resents the priority vector. Nevertheless it is important to achieve a consistent or an almost con-
sistent comparison matrix. Therefore the degree of consistency is measured in the following way: 
For a consistent matrix  max l =n  and for a inconsistent matrix  max l ‡n holds. Therefore the con-
sistency index CI:=( max l -n)/(n-1) is a possible measure for the degree of inconsistency. De-
fining the figure RI as the consistency index for a completely randomly generated matrix of size 
n, we c an measure the quality of consistency with the consistency ratio CR:= CI/RI. The figure 
CR describes the inconsistency, which occurred in the actual voting as a percentage of the incon-
sistency, which will occur by completely randomly voting. As a rule of thumb a value of CR  £ 
0.1 will be accepted. Otherwise a re-voting of the comparison-matrix has to be performed. 
 
Extension to Case 3: Aggregation of single votings to a group voting 
Usually a number of experts is engaged in the process of benchmarking and determining the cri-
terion weights. Therefore we have to aggregate the individual votings to a group comparison 
matrix. With the arithmetic mean as well as with the median it is possible to get an inconsistent 
group matrix from consistent individual votings. Furthermore with the arithmetic mean it is also 
possible to get a non-reciprocal group matrix. Therefore we use the geometric mean: 
 
Z
Z
1 z
ikjz ikj a : a ￿ =
=
 
Z  number of individuals 
z  index of individual, z ˛ {1, ..., Z} 
ikjz a   result of comparisons of criterion/company i versus criterion/company k with respect 
to criterion j made by individual z [-], z ˛ {1, ..., Z} 5 
   
To measure the diversity within the group we can calculate a “geometric variance” which is also 
used in software package “Team Expert Choice™ Version 9.5”. It is defined as follows: 
 
U log
a
a
,
a
a
max log : var Z
Z
1 z ikjz
ikj
ikj
ikjz
ikj ￿
￿
￿
￿
￿
￿
￿
=
=
 
U  Upper bound of the scale used for pairwise comparisons [-]. 
For the Saaty-scale U=9 holds. 
 
This measure of diversity is defined on [0, 1]. During the voting process it is important to show 
the diversity and to discuss the meaning of the criterion if the geometric variance exceeds a pre-
determined value.  
 
3 Relative values and estimation of the monetary value of a company 
 
The relative values of the companies can be calculated by synthesizing the (local) priorities of 
the total hierarchy. Defining 
 
i p   overall priority of element i [-], i ˛ {0, ..., I} 
 
we can use the recursive calculation formula for the overall priority of element i: 
 
1 p0 =    
￿ ￿ =
G ˛ j i j
j ij i p p : p  for i ˛{1, …, I} 
 
The overall priorities for the companies, i.e. the nodes i for which  i G =˘ holds, represent the 
total relative values of the companies included in the comparison. Defining  
 
i m   monetary value of company i [monetary units], i ˛ {0, ..., I}, 
 
e.g. capitalized stock, we can calculate different estimates (optimistic, pessimistic, average) for 
the value of a certain company i ˆ  if we know the monetary values of its competitors j: 
 
{ } i ˆ j j
j
opt
i ˆ p p m max : m ￿ =  
{ } i ˆ j j
j
pes
i ˆ p p m min : m ￿ =  
i ˆ
j
j
j
j
ave
i ˆ p p m : m ￿ ￿
￿
ł
￿
￿
￿
Ł
￿
￿ ￿ =  6 
   
The calculation for company A is demonstrated in table 2 (MU monetary units; for clarification 
the index i is substituted by the company name A, B, C or D): 
 
company  total rela-
tive value 
known value of 
the competitor  multiplier 
estimation for 
the value of 
company A 
 
  [-]  [MU]  [MU]  [MU]   
a  b  c  d=c/b  e=d*0.275   
A  0.275  ?       
B  0.242  35,000  144,628  39,773   
C  0.247  40,000  161,943  44,534  optimistic 
D  0.236  30,000  127,119  34,958  pessimistic 
Total B, C, D  0.725  105,000  144,828  39,828  average 
Table 2: Example for calculating estimates for the monetary value of a company 
 
Of course, it is recommended to perform a sensitivity analysis to see how the range of estimates 
will change if the criterion weights would be chosen differently. With the sensitivity analysis it is 
also possible to identify the critical success factors. Companies are encouraged to invest in these 
critical success factors to improve their competitive position. 
 
4 Conclusion 
 
In this paper a method has been suggested which enables to combine success factors and finan-
cial data within the process of the valuation of a company. The method is based on the Analytic 
Hierarchy Process (AHP). The results of the AHP approach can be interpreted as relative values 
of the companies included in the investigation. The new idea is to take the results of the AHP as 
input data for a Comparative Company Approach (CCA). With this further calculation it is pos-
sible to obtain a range of estimates for the value of a company. 
 
References 
 
Damodaran, A. (1994); Damodaran on Valuation  – Security Analysis for Investment and Cor-
porate Finance; New York et al.: John Wiley & Sons 
Expert Choice Incorporated (1999); Team Expert Choice™ Version 9.5; Pittsburgh, PA 
Hafner, R. (1988); Unternehmensbewertung bei mehrfacher Zielsetzung; Betriebswirtschaftli-
che Forschung und Praxis, Vol. 40, pp. 485-504 
Saaty, T. L. (1990); Multicriteria Decision Making  - The Analytic Hierarchy Process  - Plan-
ning, Priority Setting, Resource Allocation; Pittsburgh: RWS Publications 
 

Damodaran on Valuation – Security Analysis for Investment and Corporate Finance; New York et al.:

Integrating success factors and group attitudes into the valuation of a company

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