Mathematical and Statistical Methods for Actuarial Sciences and Finance

208 M. Marozzi and L. Santamaria

based for example on the rank transformation ofXiks or on Lago and Pesarin’s [6]
Nonparametric Combination of Dependent Rankings. For details on this matter see
[7] and [8].

3 To reduce the dimension of composite indicators

LetRK(Xk,k∈{ 1 ,...,K})=RKdenote the vector of ranks obtained following
the composite financial indicator

M 4 ,i=

∑K

k= 1

Xik−MED(Xk)

MAD(Xk)

, (8)

computed fori = 1 ,...,N, which combines all the partial financial indicators
X 1 ,...,XK. We consider the fourth method because the median absolute devia-
tion is the most useful ancillary estimate of scale [5, p. 107]. Suppose now thatXhis
excluded from the analysis. LethRK− 1 (Xk,k∈{ 1 ,...,K}−h)=hRK− 1 denote
the corresponding rank vector. IfRKandhRK− 1 are very similar, it follows that the
exclusion ofXhdoes not affect the ranking of the companies much. On the contrary,
if the two rank vectors are very different, by leaving outXhthe ranking process
is greatly influenced. To estimate the importance ofXhwe compute the Spearman
correlation coefficient betweenRKandhRK− 1

s(RK,hRK− 1 )= 1 −

6

∑N

i= 1 (RK[i]−hRK− 1 [i])

2

N(N^2 − 1 )

, (9)

whereRK[i]andhRK− 1 [i]aretheith element of the corresponding vector. The
closersis to 1, the less importantXhis. The idea is to leave out the partial indicator
Xhthat brings the greatests(RK,hRK− 1 ). The procedure may be repeated for the
K−2 rankings obtained by leaving out one more partial indicator. LetXlbe the next
indicator that is excluded from the ranking process. We computel,hRK− 2 (Xk,k∈
{ 1 ,...,K}−{l,h})=l,hRK− 2 ands(hRK− 1 ,l,hRK− 2 )forl= 1 ,...,K,l=h.
The partial indicator that brings the greatestsshould be excluded, and so on.
Even if the whole procedure naturally lasts until only one partial indicator is left to
be used by financial analysts, a natural question arises: when should the partial indi-
cator exclusion procedure be stopped? That is, how many partial financial indicators
should be excluded? Within this framework, it is assumed that the best ranking is the
one based on all the partial indicators. Of course, there is a trade-off between infor-
mation and variable number reduction. A natural stopping rule is: stop the procedure
as soon as the correlation coefficient is less than a fixed value.

4 A practical application

We present an application of the procedure for reducing the dimension of corporate
finance composite indicators. More precisely, the liquidity issue is considered. The
aim is to rate a set of companies on the basis of the following liquidity ratios.