ANALYSIS OF THE MIB30 BASKET IN THE PERIOD BY FUNCTIONAL PC S

Transcription

1 COMPSTAT 2004 Symposium c Physica-Verlag/Springer 2004 ANALYSIS OF THE MIB30 BASKET IN THE PERIOD BY FUNCTIONAL PC S Damiana G. Costanzo and Salvatore Ingrassia Key words: Functional data, principal components, financial data. COMPSTAT 2004 section: Functional data analysis. Abstract: The MIB30 basket refers to the 30 most capitalised and traded companies on the Italian Stock Exchange. Related share prices and related quantities are updated during the phase of continuous trading at a frequency of one a minute on the basis of the prices of the latest contracts concluded on each share. The daily traded volumes of these 30 shares in the period from January 3rd, 2000 to ember 30th, 2002 are here investigated from an explorative point of view using functional principal component techniques. 1 Introduction Functional data are essentially curves and trajectories, the basic rationale is that we should think of observed data functions as single entities rather than merely a sequence of individual observations. Even though functional data analysis often deals with temporal data, its scope and objectives are quite different from time series analysis: while time series analysis mainly focuses on modeling data, or in predicting future observations, the techniques developed in FDA are essentially exploratory in nature: the emphasis is on trajectories and shapes; moreover unequally-spaced and/or different number of observations can be taken into account as well as series of observations with missing values, see Ramsay & Silverman [5], [6]. In this paper statistical properties of the daily series of the traded volumes of the shares composing the MIB30 basket in the period from January 3rd, 2000 to ember 30th, 2002 are investigated from a functional data analysis perspective. We remark that the MIB30 basket synthesizes the performance of the Italian Stock Exchange within the Telematic Share ket; at the beginning, in 1975, the term MIB was the acronym of Milano Indice Borsa ; subsequently it took on the new meaning of Mercato Italiano di Borsa since the market had become effectively national. The rest of the paper is organised as follows: in the next section we outline functional data modeling and give some details about functional principal component analysis; in Section 3 we introduce the MIB30 basket dataset and present the results of our analysis; finally in Section 4 we compare the dynamics of the first functional PC with the dynamics of the MIB30 index in order to explore the possibility of the construction of stock market indices based on functional indicators.

2 832 Damiana G. Costanzo and Salvatore Ingrassia 2 Functional PCA Let {ω 1,...,ω n } be a set of n units and let y i =(y i (t 1 ),...,y i (t p )) be a sample of measurements of a variable Y taken at p times t 1,...,t p T =[a, b] in the i-th unit ω i,(i =1,...,n). Such data y i (i =1,...,n) are regarded as functional because they are considered as single entities rather than merely sequences of individual observations, so they are called raw functional data; indeed the term functional refers to the intrinsic structure of the data rather than their explicit form. In order to convert raw functional data into a suitable functional form, a smooth function x i (t) is assumed to lie behind y i whichisreferredtoasthetrue functional form; this implies, in principle, that we can evaluate x at any point t T.ThesetX T = {x 1 (t),...,x n (t)} t T is the functional dataset. In functional data analysis the statistical techniques posit a vector space of real-valued functions defined on a closed interval for which the integral of their squares is finite. If attention is confined to functions having finite norms, then the resulting space is a Hilbert space; howeverwe often requirea stronger assumption so we assume H be a reproducing kernel Hilbert space (r.k.h.s.), which is a Hilbert space of real-valued functions on T with the property that, for each t T, the evaluation functional L t, which associates f with f(t), L t f f(t), is a bounded linear functional. In such spaces the objective in principal component analysis of functional data is the orthogonal decomposition of the empirical variance function: v(t, u) := 1 n 1 n {x i (t) x(t)}{x i (u) x(u)} (1) i=1 (which is the counterpart of the covariance matrix of a multidimensional dataset) in order to isolate the dominant components of functional variation. In analogy with the multivariate case, the functional PCA problem is characterized by the decomposition of the variance function: v(t, u) = j λ j ξ j (t)ξ j (u) (2) where λ j,ξ j (t) satisfy the eigenequation: v(s, ),ξ j h = λ j ξ j (u), where the eigenvalues: λ j := ξ j (t)v(t, u)ξ j (u)dt du T are positive and non decreasing while the eigenfunctions must satisfy the constraints: ξj 2 (t)dt =1 and ξ j ξ i (t)dt =0 (i<j). T The ξ j s are usually called principal component weight functions. Finally the T

3 Analysis of the MIB30 basket in period by functional PC s 833 principal component scores (of ξ(t)) of the units in the dataset are the values w i given by: w (j) i := x i,ξ j = T ξ(t)x i (t)dt. (3) The decomposition (2) defined by the eigenequation (2) permits a reduced rank least squares approximation to the empirical covariance function v. Thus, the leading eigenfunctions ξ define the principal components of variation among the sample functions x i. 3 A functional PC analysis of the MIB30 basket dataset Raw data considered here consist of the total value of the traded volumes of the shares composing the MIB30 basket in the period January 3rd, ember 30th, They have been collected in a matrix. We remark that an important characteristic of this basket is that it is open in that the composition of the index is normally updated twice a year, in the months of ch and tember (ordinary revisions). Moreover, in response to extraordinary events, or for technical reasons ordinary revisions may be brought forward or postponed with respect to the scheduled date. In particular in our data set there are 21 companies which have remained in the basket for the three years while the other 9 places in the basket have been shared by a set of other companies which have been remaining in the basket for shorter periods. Due to the connection among the international financial markets, data concerning the closing days (as week-ends and holidays) are regarded here as missing data. In literature functional PCA is usually performed from original data (x ij ); here we prefer to work on the daily standardized raw functional data: z ij := x ij x j s j (i =1,...,30, j =1,...,758), (4) where x j and s j are respectively the daily mean and standard deviation of the e.e.v of the shares in the basket. We shall exhibit later how such transformation can gain an insight into the PC trajectories understanding. The first PC alone accounts for the 89.4% and the second PC accounts for the 6.9% of the whole variability. In Figure 1 we give the trajectories of the first two functional principal components which show the way in which such set of functional data varies from its mean, and, in terms of these modes of variability, quantifies the discrepancy from the mean of each individual functional datum. The meaning of functional principal component analysis is a more complicated task than the usual multidimensional analysis, however here it emerges the following interpretation:

4 834 Damiana G. Costanzo and Salvatore Ingrassia Figure 1: Plot of the first 2 functional principal components. i. The first functional PC is always positive, then shares with large scores of this component during the considered period have a large traded volume as compared to the mean value on the basket; it can be interpreted as a long term trend component. ii. The second functional PC changes sign at t = 431 which corresponds to tember 11th, 2001 and the final values, in absolute value, are greater than the initial values: this means that shares having good (bad) performances before tember 11th, 2001 have been going down(rising) after this date; it can be interpreted as a shock component. This interpretation is confirmed by the following analysis of the raw data. As it concerns the first PC, for each company we considered its minimum standardized value over the three years z (min) i =min j=1,...,758 z ij (i =1,...,30). In particular z (min) i is positive (negative) when the traded volumes of the i-th share are always greater (less) than the mean value of the MIB30 basket during the three years. As for the second PC, let x Bi be the average of the traded volumes of the ith company over the days: 1,...,431 (i.e. before tember 11th, 2001) and x Ai the corresponding mean value after tember 11th, Let us consider the variation per cent: δ i := x Ai x Bi x Bi 100% i =1,...,30. If δ i is positive (negative) then the ith company increased (decreased) its mean e.e.v. after the tember 11, Consider the scores on the two first PCs given in (3), respectively w (1) i and w (2) i. We observed that the correlation coefficient between the z (min) i and w (1) i w (2) i is equal to 0.96 and the correlation coefficient between the δ i and is equal to 0.84, see Ingrassia and Costanzo [3] for details.

5 Analysis of the MIB30 basket in period by functional PC s A comparison between the dynamics of the first FPC and the MIB30 index In our opinion, the obtained results open methodological perspectives for the construction of new financial indices having some suitable statistical properties. As a matter of fact, the construction of some existing stock market indices has been criticized by several authors, see e.g. Elton and Gruber [2]. In Costanzo [1] the dynamics of the MIB30 index in the three years 2000, 2001 and 2002 have been investigated using the phase-plane plot; herewe compare such dynamics with the ones coming from the first functional. This technique may provide information about pure dynamics of the event and compare it within time, since it focuses as on the rate of change of the index rather than its actual size. The functional index is regarded as a harmonic process in which energy is exchanged between the potential and the kinetic states; thus, as mentioned above, the phase-plane plot draws the acceleration against the velocity. As a matter of fact, the kinetic energy is proportional to the square of the velocity (i.e. the first derivative of the function) and the potential energy is proportional to the acceleration (i.e. the second derivative). In economics, potential energy corresponds to available capital, human resources, raw material and other resources that are at hand to be used in economic activity; kinetic energy corresponds to the manufacturing process in full swing, when these resources are moving along the assembly line (see [6]). For a general dynamic process, velocity represents its rate of change, while acceleration indicates the input or whatever resources or forces produce this change. Thus, in financial markets, the potential energy may correspond to strength of the economic situation, both the realistic one and the one perceived by financial operators. The kinetic energy corresponds to the confidence of the financial operators in economy in such strength. In fact, a financial index reflects the confidence of the financial operators in the economic situation: when the markets are confident, the value of the index tends to increase and exhibits stable patterns over time; shocks, as wars or dramatic events, produce both short transitory effects and longer-lasting readjustments in market behaviour. In the phase-plane we plot acceleration on the vertical axis, versus the velocity on the horizontal axis. The interpretation of this graphical representation follows Ramsay and Silverman [6], Ramsay and Ramsey [4]. In Figure 2 (left column) we give the phase plane plots of the MIB30 index in the three years 2000, 2001 and Concerning the year 2000, we note three large cycles surrounding zero, plus two small cycles that are much closer to the origin. The largest one starts in the beginning of ruary (), with positive velocity but near zero acceleration. In this period the velocity of the index reaches its absolute maximum value in the year; the other maximum value of the velocity is roughly at the beginning of ember. As remarked before, the size of the radius of the cycles is an important aspect of the plot. When the markets consider the economic situation good, they react with

6 836 Damiana G. Costanzo and Salvatore Ingrassia high level of confidence: the level of trading increases and the rate of change of the index is high. This large cycle ends to the middle of April (Apr). The cusp corresponding to the smallest cycle starts at the middle of and lasts for about two months, until the middle of July (Jul). During this period both velocity and acceleration are very low, near zero. Note that the location of the center of this smallest cycle indicates a positive and decreasing velocity. The second largest cycle starts at the middle of October (Oct) and it continues until the end of the year. As for the year 2001 there is just one evident large cycle. It is represented by the intermediate cycle starting in January till about the end of April. Note that its horizontal center is located to the left of the plot. The subsequent smaller cycle moves from the middle of till the end of July. The largest cycle starts at the beginning of August with negative velocity and acceleration. The cycle moves clockwise through August and passes the horizontal zero acceleration line around the end of the month. We see now in the plot a bulge to the left starting from the middle of tember, roughly when the catastrophic event took place: acceleration reached its maximum value but it was not followed by a positive velocity. stays negative until the beginning of October. This cycle closes at the end of October with very low velocity of the index. Note that in this plot the cusp now corresponds to the cycle spanning the last two months of the year and it is very small in size, with very low velocity and acceleration. To summarize, the year 2001 started with a not very good performance of the index in terms of its rate of change, this is shown by the first two cycles corresponding to the first six/seven months of the year in the plot. After that it started to perform better; note, in fact, how the largest cycle is characterized by net negative velocity for its evident horizontal location of the center to the left, but at the same time net velocity increases for its vertical location of the center is above zero. The shock event in tember interrupted some way this better performance: the year closed with a very small cycle. Finally for the year 2002, we remark the rate of change of the index lasts in the following year. Note indeed the impressive change in the cycles in the year 2002 from the year Almost all cycles shift to the left; that is, they have now their horizontal centers on the left, which denotes a net negative velocity almost during the whole year. Further, the size of their radius with respect to the horizontal axes of the plot, denotes lower velocity of the index compared to the year Note that the cusp occurs again in the same period mid - mid July, as in year 2000, but it has dramatically shifted to the left. Strictly speaking in year 2002, as a consequence of tember 11th in the previous year, the overall slope of the index becomes more negative and the amplitude of the short term cycles shrinks. The phase plane plot of the first PC concerning the years 2000, 2001 and 2002 is given in Figure 2 (right column). The interpretation of the plot

7 Analysis of the MIB30 basket in period by functional PC s 837 is analogous as in the MIB30 case. However, the first PC index summarizes in a clearer way than the MIB30 index the dynamics of the events in the three years here considered. In fact, concerning the year 2000 the plot shows a very largecycle with a positive velocity during all the year but with an acceleration which is positive until the end of ruary only. Following the economic interpretation previously outlined, this kind of circle denotes a positive trend which is characterized however by a kinetic energy (the confidence of the financial operators in the strength of the economic situation) which starts to decrease already in the first part of the year. Concerning the year 2001, it is strongly evident in the plot the shock event of tember 11th, since velocity even though positive during all the year, it is strongly increasing before that date then it is rapidly decreasing. For the year 2002 we remark the consequence of the last event in the rate of change of the first PC index. In fact the phase plane plot shows negative and decreasing both velocity and acceleration. The results here presented suggest that the shares scores on this harmonic could be a good ingredient for a new family of financial indices trying to capture as most as possible of the variability of the prices in the share basket. This provides ideas for further developments of functional principal component techniques in the financial field. References [1] Costanzo G.D. (2003). A graphical analysis of the dynamics of the MIB30 index in the period by a functional data approach. In: Atti Riunione Scientifica SIS 2003, Rocco Curto Editore, Napoli, [2] Elton E.J. and Gruber M.J. (1995). Modern Portfolio Theory and Investment Analysis. John Wiley & Sons, New York. [3] Ingrassia S. and Costanzo G.D. (2003). Functional principal component analysis of financial time series. Book of Short papers of CLADAG 2003, [4] Ramsay J.O. and Ramsey J.B. (2002). Functional data analysis of the dynamics of the monthly index on nondurable good production. Journal of Econometrics, 107, [5] Ramsay J.O. and Silverman B.W. (1997). Functional Data Analysis. Springer-Verlag, New York. [6] Ramsay J.O. and Silverman B.W. (2002). Applied Functional Data Analysis. Springer-Verlag, New York. Acknowledgement: Dataset used in this paper have been collected by the Italian Stock Exchange. The authors thank Research & Development DBMS (Borsa Italiana). Address: G.D. Costanzo, S. Ingrassia, Dipartimento di Economia e Statistica, Università della Calabria [dm.costanzo, s.ingrassia]@unical.it

8 838 Damiana G. Costanzo and Salvatore Ingrassia Year 2000 Year Apr Oct Jan Aug Jun Jul Nov -8*10^-8-4*10^-8 0 2*10^-8 Nov Oct Aug Jul Jan Apr Jun ^-5 1.4*10^-5 1.8*10^-5 2.2*10^-5 Year 2001 Year Jun Jul Nov Jan Apr Oct Aug -8*10^-8-6*10^-8-4*10^-8-2*10^-8 Nov Aug Oct Jul Jun Apr Jan *10^-6 6*10^-6 10^-5 Year 2002 Year JulJun Oct Apr Nov Jan Aug -6*10^-7-4*10^-7-2*10^-7 Nov Oct Aug Jan Apr Jun Jul *10^-5-6*10^-5-4*10^-5-2*10^-5 0 Figure 2: Phase plane plot of the Mib30 index daily series (left column) and of the first functional PC of the MIB3 basket (right column) for the years 2000,2001 and 2002 (note the different scale of the acceleration in the plot concerning the year 2001).