Mathematical and Statistical Methods for Actuarial Sciences and Finance

A simple dimension reduction procedure for corporate finance composite indicators 207

the measurement units and force the results into a short and well defined range:
mini(Xik)
maxi(Xik)≤LT^1 (Xik)≤1and0≤LT^2 (Xik)≤1 respectively.LT^1 andLT^2 are
readily comprehended.
The third and fourth linear transformations are defined as

LT 3 (Xik)=

Xik−E(Xk)

SD(Xk)

(5)

and

LT 4 (Xik)=

Xik−MED(Xk)

MAD(Xk)

, (6)

which correspond toLTwherea = −SDE((XXkk))andb = SD(^1 Xk), and wherea =
−MED(Xk)
MAD(Xk) andb=

1
MAD(Xk), respectively.LT^3 (Xik)indicates how farXiklies from
the meanE(Xk)in terms of the standard deviationSD(Xk).LT 4 is similar toLT 3 and
uses the medianMED(Xk)instead of the mean as location measure and the median
absolute deviationMAD(Xk)instead of the standard deviation as scale measure.
By means ofLTh(where the subscripth=1, 2, 3 or 4 denotes the various methods)
the original data are transformed into comparable data. The composite indicator is
then defined using the sum as the combining function, in accordance with general
practice (see [1])

Mh,i=

∑K

k= 1

LTh(Xik),h= 1 , 2 , 3 , 4. (7)

Mh,is are used to rank the units. Note that the first and second method may be ap-
plied also to ordered categorical variables, or to mixed variables, partly quantitative
and partly ordered categorical, with the unique concern of how to score the ordered
categories.
In Section 4 we analyse a data set about listed companies, in particular we con-
sider four different liquidity ratios. For theith company we denote these ratios by
Xi 1 ,Xi 2 ,Xi 3 ,Xi 4. Note thatT(Xi 1 ),T(Xi 2 ),T(Xi 3 ),T(Xi 4 )are partial financial
indicators since they correspond to a unique financial ratio:T(Xik)>T(Xjk)lets
the analyst conclude that companyiis better than companyjfor what concerns fi-
nancial ratioXk(since of courseT(Xik)>T(Xjk)⇔ Xik>Xjk), whereasMi
is a composite financial indicator since it simultaneously considers every financial
ratio.M 1 ,...,MNallow the analyst to rank the companies sinceMi>Mjmeans
that companyiis better than companyjregarding all the financial ratios together.
There is reason to believe that financial ratios are correlated. This central question
is addressed in the next section: a simple method for reducing the number of partial
indicators underlying a composite indicator is proposed.
It is important to emphasise that in this paper we do not consider composite
indicators based on non-linear transformations since Arboretti and Marozzi (2005)
showed that such composite indicators perform better than those based on linear
transformations only when distributions ofX 1 ,...,XKparent populations are very
heavy-tailed. Preliminary analyses on our data show that parent distributions are
not heavy-tailed. Composite indicators based on non-linear transformations may be