Functional Principal Component Analysis of Financial Time Series G. Damiana Costanzo Dipartimento di Economia e Statistica, Universita` della Calabria 87036 Arcavacata di Rende (CS), Italy [email protected]

Cnam - Paris, November 23rd 2005 Summary 1. Introduction 2. Functional data vs. Multidimensional data modeling 3. Functional PCA 4. The e.e.v. MIB30 dataset 5. The statistical analysis 6. Conclusions and perspectives

Ouverture The problem (methodological perspective): • Dimensional reduction of a functional data set with homogeneous piecewise components The datasets: • Daily quantities (prices and e.e.v.) of the shares constituting the MIB30 basket in the period: January 3rd, 2000 - December 30th, 2002 (courtesy of the Research & Development DBMS (Borsa Italiana)). The statistical method: • Functional principal component analysis

Why Functional Data F dA is a generalization of classical M vA when data are functions, curves or trajectories. Such data arise quite naturally in different fields. For example in phenomena where measurements come from an automated on-line collection process (on-line sensing and monitoring equipments): in economic analysis, statistical quality control of manufacturing process, shape analysis and natural science: seismology, meteorology, physiology and medicine (see recent paper on gait data by Preda & Saporta, 2005)

MD vs. FD modeling Given a set of n units ω1, . . . , ωn. Multidimensional Data

Data set:

# variables

Functional Data

X: set of points in Rp

XT : set of functions on T

x1 x2 .. .. xn

x1(t) x2(t) .. .. xn(t)

p 40 w1 > 40 w1 > 40 10 < w1 10 < w1 10 < w1 10 < w1 near 0 near 0 near 0

The correlation r (w1i, zij ) = 0.96

< 40 < 40 < 40 < 40

Analysis of the 2nd PC Let x ¯Bi the mean value of the e.e.v. of the ith company over the days: 1,...,431 (i.e. before September 11th, 2001) and x ¯Ai the corresponding mean value after September 11th, 2001. Let us consider the variation per cent: x ¯ −x ¯Bi 100% δi = Ai x ¯Bi δi

Company

Score

-80.20% -58.50% -47.08%

Seat Pagine Gialle w2 > 10 Olivetti w2 > 10 Enel 0 < w2 < 10

63.21% 83.10% 133.79%

Unicredito Autostrade T9

−20 < w2 < −10 −10 < w2 < 0 w2 < −20

The companies with large positive (negative) scores on the 2nd PC present the largest decrements (increments) after September 11th, 2001. The correlation r (w2i, zij ) = 0.84

Conclusions and perspectives for future researches 1. Functional PCA looks an interesting tools in order to gain insight in functional dataset.

2. Does it open methodological perspectives for the construction of new financial indeces ? Some existing stock market indices have been criticized (e.g. Elton and Gruber, 1973, 1995): the famous U.S. Dow Jones presents some statistical flaws. In Italy MIB30 basket is summarized by the MIB30 index =⇒ analysis of the MIB30 index within the FDA framework: ′how′ and how much is it statistical representative of the basket?

35 30 25 15

20

MIB30 Index

2000.0

2000.5

2001.0

2001.5

2002.0

2002.5

2003.0

Year

in fact MIB30 is calculated according to the formula: M IB30 = 10000



30 X pit 

i=1 pi0



wiT  rT

(3)

where the weight of the i-th share in the basket (i.e. the weight of each company in the index) is: p q wiT = P30i0 i0 i=1 pi0 qi0

REFERENCES - Costanzo, G. D. (2003), A graphical analysis of the dynamics of the MIB30 index in the period 2000-2002 by a functional data approach. SIS 2003, Napoli. - Elton, E.J. and Gruber, M. J. (1973), Estimating the dependence structures of share prices. Implications for Portfolio. Journal of Finance, 1203-1232. - Elton, E.J. and Gruber, M. J. (1995), Modern Portfolio Theory and Investment Analysis. John Wiley and Sons, New York. - Ingrassia, S. and Costanzo, G. D. (2004), Functional principal component analysis of financial time series, New Developments in Classification and Data Analysis, Spriger-Verlag, Berlin.

- Preda C. and Saporta G. (2005), PLS discriminant analysis for functional data. XI ASMDA Symposium, Brest, May 2005. - Ramsay, J. O.(1982), When the data are functions, Psychometrika, 47. - Ramsay, J. O. and Silverman, B. W. (1997), Functional Data Analysis, Springer-Verlag, New York. - Ramsay, J. O and Silverman, B. W. (2002), Applied Functional Data Analysis, Springer-Verlag, New York. - Saporta, G. (1985), Data Analysis For Numerical and Categorical Individual Time Series, Applied Stochastic Models and Data Analysis, 1. - Simonoff, J. S. (1996), Smoothing Methods in Statistics, SpringerVerlag, New York.