206 M. Marozzi and L. Santamaria
- possibly transform the original data into comparable data through a proper func-
tionT(·)and obtain the partial indicators; - combine the partial indicators to obtain the composite indicator through a proper
link (combining) functionf(·).
IfX 1 ,...,XKare the measurable components of the complex variable, then the
composite indicator is defined as
M=f(T 1 (X 1 ),...,TK(XK)). (1)Fayers and Hand [3] report extensive literature on the practical application of com-
posite indicators (the authors call them multi-item measurement scales). In practice,
the simple weighted or unweighted summations are generally used as combining
functions. See Aiello and Attanasio [1] for a review on the most commonly used data
transformations to construct simple indicators.
The purpose of this paper is to reduce the dimensions of a composite indicator
for the easier practice of financial analysts. In the second section, we discuss how to
construct a composite indicator. A simple method to simplify a composite indicator
is presented in Section 3. A practical application to the listed company liquidity issue
is discussed in Section 4. Section 5 concludes with some remarks.
2 Composite indicator computation
LetXikdenote thekth financial ratio (partial component),k= 1 ,...,K,fortheith
company,i= 1 ,...,N. Let us suppose, without loss of generality, that the partial
components are positively correlated to the complex variable. To compute a composite
indicator, first of all one should transform the original data into comparable data in
order to obtain the partial indicators. Let us consider linear transformations. A linear
transformationLTchanges the origin and scale of the data, but does not change the
shape
LT(Xik)=a+bXik,a∈]−∞,+∞[,b> 0. (2)
Linear transformations allow us to maintain the same ratio between observations (they
are proportional transformations).
The four linear transformations most used in practice are briefly presented [4].
The first two linear transformations are defined as
LT 1 (Xik)=Xik
maxi(Xik)(3)
and
LT 2 (Xik)=Xik−mini(Xik)
maxi(Xik)−mini(Xik), (4)
which correspond to LTwherea = 0andb = maxi^1 (Xik),andwherea =
−mini(Xik)
maxi(Xik)−mini(Xik)andb=
1
maxi(Xik)−mini(Xik)respectively.LT^1 andLT^2 cancel