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1、A simple dimension reduction procedure for corporatefinance composite indicatorsMarco Marozzi and Luigi SantamariaAbstract. Financial ratios provide useful quantitative financial information to both investorsandanalysts sothat they canrate a company.Many financialindicators from accountingbooks are
2、taken into account. Instead of sequentiallyexamining each ratio, one can analyse together different combinations of ratios in order to simultaneouslytake into accountdifferent aspects. Thismaybedonebycomputingacompositeindicator.Thefocusofthepaperisonreducingthe dimensionof a compositeindicator. A q
3、uick andcompactsolution is proposed,and a practicalapplicationto corporatefinanceis presented.In particular, the liquidity issueis addressed.The results suggest that analysts should take our method into consideration as it is much simpler than other dimension reduction methods suchas principal compo
4、nentor factor analysis andisthereforemucheasierto beusedin practicebynon-statisticians(asfinancialanalystsgenerally are). Moreover, the proposed method is always readily comprehended and requires milder assumptions.Key words: dimension reduction, composite indicator, financial ratios, liquidity1 Int
5、roductionFinancialratiosprovideusefulquantitativefinancialinformationtobothinvestorsandanalysts so that they can rate a company. Many financial indicators from accountingbooks are taken into account. In general, ratios measuring profitability, liquidity,solvency and efficiency are considered. Instea
6、d of sequentially examining each ratio, one can analyse different combina- tionsof ratiostogether inorder tosimultaneouslytake intoaccount different aspects. This can be done by computing a composite indicator. Complexvariables can be measured by means of compositeindicators.The basic idea is to bre
7、ak down a complex variable into components which are measurable by means of simple (partial) indicators. The partial indicators are then combined to obtain the composite indicator. To this end one shouldThe paper has been written by and the proposed methods are due to M. Marozzi. L. Santa- maria gav
8、e helpful comments to present the application results.M. Corazza et al. (eds.), Mathematical and Statistical Methods for Actuarial Sciences and Finance Springer-Verlag Italia 2010206M. Marozzi and L. Santamariapossiblytransform the original data into comparable data througha proper func- tion T() an
9、d obtain the partial indicators; combine the partial indicators to obtainthe composite indicator througha proper link (combining)function f ().If X1,., XKare the measurable components of the complex variable, then thecomposite indicatoris defined asM = f (T1(X1),.,TK(XK).(1)Fayers and Hand 3 report
10、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 Aielloand Attanasio1for a review onthe most commonlyused data tra
11、nsformationsto construct simple indicators. The purpose of this paper is to reduce the dimensions of a composite indicatorfor 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 isprese
12、nted inSection3. A practical applicationtothe listedcompany liquidityissue is discussed in Section 4. Section 5 concludes with some remarks.2 Composite indicator computationLet Xikdenote the kth financial ratio (partial component), k = 1,., K, for the ith company, i = 1,., N. Let us suppose, without
13、 loss of generality, that the partial componentsarepositivelycorrelatedtothecomplexvariable.Tocomputeacompositeindicator, 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 transformatio
14、n LT changes the origin and scale of the data, but does not change the shape LT(Xik) = a + bXik, a ,+, b 0.(2)Lineartransformationsallowustomaintainthesame ratiobetweenobservations(they are proportionaltransformations).The four linear transformations most used in practice are briefly presented 4.The
15、 first two linear transformationsare defined asLT1(Xik) =Xik maxi(Xik)(3)andLT2(Xik) =Xik mini(Xik) maxi(Xik) mini(Xik),(4)which correspond to LT where a = 0 and b =1 maxi(Xik), and where a = mini(Xik) maxi(Xik)mini(Xik)and b =1 maxi(Xik)mini(Xik)respectively. LT1and LT2cancelA simple dimension redu
16、ction procedurefor corporate financecomposite indicators207the measurement units and force the results into a short and well defined range: mini(Xik) maxi(Xik) LT1(Xik) 1 and 0 LT2(Xik) 1 respectively. LT1and LT2are readily comprehended.The third and fourth linear transformationsare defined asLT3(Xik) =Xik E(Xk) SD(Xk)(5)and LT4(Xik) =Xik MED(Xk) MAD(Xk),(6)which correspond to LT where a =E(Xk) SD(Xk)and b =1 SD(Xk), and where a = MED(Xk) MAD(
