FUZZINESS AND PROBABILITY FOR PORTFOLIO MANAGEMENT

Transcription

1 portfolo of assets, fuzzy numbers, optmzaton Anna WALASZEK-BABISZEWSKA, Wojcech MENDECKI ** FUZZINESS AND POBABILITY FO POTFOLIO MANAGEMENT Abstract In the paper the portfolo of fnancal assets has been formulated usng the fuzzy numbers dea and the statstcal nformaton concernng assets. The optmzaton task of the portfolo structure has been formulated for two types of fuzzy numbers. The data from Polsh market have been used for exemplary calculatons. 1. INTODUCTION POBABILISTIC TOOLS IN FINANCIAL MAKET ANALYSIS Fnancal markets have been nvestgated n probablstc and stochastc categores by economsts, mathematcans and researchers of natural scences for over a hundred years. It s justfed to menton French mathematcan, Bacheler, who presented hs doctoral thess n 1900 at the Academy of Pars. Hs poneerng work dealt wth the prcng of optons and he formulated, the random walk process as a model of prce changes. After years, ths process has been more mathematcally formalzed and developed by Ensten and then by Wener. Problems of the prce changng have been modellng usng many classes of models. The Black and Scholes opton prcng model (1973) provded a very mportant nstrument for buldng a strategy for market nvestors [4]. Let X(t) be the prce of a chosen fnancal asset at tme t. There are varables derved from prces of fnancal goods whch are commonly consdered by market researchers: prce changes as dfferences dscounted prce changes Y ( t) X ( t + t) X ( t), (1) [ X ( t + t) X ( t) ] D( ) Y D ( t) t, (2) where D(t) means a dscountng factor, Ph.D. Anna Walaszek-Babszewska, prof. PO, Instytut Automatyk Informatyk, Poltechnka Opolska, ul. Mkołajczyka 5, Opole, walaszek-babszewska@po.opole.pl ** M.Sc. Wojcech Mendeck, Energonstal S.A., Al. ozdzeńskego 188d, Katowce, mendeck@o2.pl 93

2 returns X ( t + t) X ( t) ( t), (3) X ( t) dfferences of the natural logarthm of prces X ( t + t) Z( t) ln X ( t + t) ln X ( t) ln. (4) X ( t) In ths paper, returns of assets are expressed by (3). Changes of assets depend on: macroeconomc and poltcal stuaton of the country, fluctuatons of the global market, stuaton of the company. Stochastc modellng has been developed by many authors, for the prce process X(t) and also for ts ndcators, under dfferent assumptons, market hypotheses. The problem of the dstrbuton of prce changes was connected wth the stochastc models of stock prces. Authors proposed Gaussan dstrbuton, log-normal dstrbuton, power-low dstrbuton or Levy dstrbuton for understandng or to predct the behavour of real prce changes and for the rsk predcton. The portfolo theory proposed models whch tred to fnd a rskless portfolo, or an optmum portfolo, characterzed by the hgh portfolo return at a relatvely low rsk [1]. The portfolo return s a weghted sum of returns of component assets. In many approaches of portfolo creatng, rsk as a probablstc category, s measured e.g. by a varance of the portfolo return. In such approach, the mean value of the portfolo return s a weghted sum of the expected values of component returns. The varance of the portfolo return s a varance of the random varable whch s the weghted sum of component random varables (assets). In ths paper we dscuss the methods of portfolo management whch are based on the fuzzy set theory and the statstcal analyss. The formal descrpton uses also percepton-based experts nformaton. 2. FUZZY POTFOLIO MANAGEMENT The dea of a fuzzy model of portfolo management has been formulated n [7]. The future values of returns, accordng to the market uncertanty, are estmated by fnancal nvestors rather as certan ntervals than numbers. Ths s the justfcaton that fuzzy numbers could represent an uncertanty of asset returns. We defne a fuzzy return n a space of real numbers, by means of a membershp functon µ (r) that assgns the level of afflaton of any value r to the fuzzy set, as follows: µ ( r) : [0,1] (5) r 0 : 0 µ ( r ) 1 (6) 94

3 and ts -cut : { r : µ ( r) }, [0,1] (7) s a closed nterval n a space of real numbers (accordng to [2, 3, 8, 9]). In the classc portfolo analyss, the return P of the portfolo s a weghted sum of partcular returns: P I 1 c (8) where { }, 1,2,,I s a return set of component assets, and c s a real vector, such that T [ c c c ] c I, 0 c 1, 1,2,,I; c 1. (9) Then the portfolo return (8) fulflls the relatonshp Mn,...,... ) Max(,...,... ). (10) I 1 ( 1 I P 1 I The proposed n ths paper fuzzy portfolo analyss s based on fuzzy numbers. The shape of a membershp functon and a spread representng the rsk of the asset are very mportant parameters and should be accepted by experts of market studes. The parameters of the membershp functon one can choose based on the emprcal mean value m and the varance 2 s of the hstorcal data of returns, observed n a tme perod t T. Let now { }, 1,2,,I s a set of fuzzy numbers representng returns of component assets, and c s a real vector fulfllng (9). We consder an exemplary membershp functon, n a form of the trangular symmetrcal functon wth parameters r 0 and s: r r 0 µ when r s r r + s ( r) s (11) 0 otherwse where r 0 m s - parameter representng rsk of the asset, 1 s s or s max mn (12) 2 mn, max the observed lowest an hghest values of the return. 95

4 When membershp functons µ ( r), 1,2,..., I of the partcular returns of the portfolo components are trangular as (11), then the membershp functon of the portfolo return as a lnear functon of fuzzy numbers s also trangular and has a form: T r c r0 T T µ r when r c r c s and c p( ) 1 T 0 0 (13) c s 0 otherwse where and [ r... r r ] T, s [ s s... s ] T r0 0,1 0,... 0I T P r T 1... I (14) c r0, sp c s (15) are parameters of the membershp functons of the portfolo returns. Investors are lookng for such assets from the market data, whch are characterzed by hgh returns at a relatvely low rsk. It means that the prelmnary crteron of the choce s: s mn; r0, > 0. (16) r 0, The optmzaton task of the portfolo defned by (11) (16), can be formulated n a category of real numbers, as follows: - choose the set of real numbers c 0, 1,2,,I whch maxmzes the return r P or (and) mnmzes the rsk s P of the portfolo I r p cr0, max (17) 1 - under the condton I s P cs mn (18) 1 I 1 c 1. (19) More generally, for the any type of membershp functons µ ( r), 1,2,..., I of the partcular assets, the return of the portfolo can be calculated usng the extenson prncple, and the decomposton rule as follows: I ( P ) c ( ) 1 (20) 96

5 Υ ( ) (21) P [0,1] where Υ denotes the standard fuzzy unon [3] and means -cut of a fuzzy number. 3. EXEMPLAY OPTIMIZATION TASKS 3.1. Portfolo wth trangular fuzzy numbers Assume that the portfolo s composed of two assets A and B. eturn s a fuzzy number wth the membershp functon of a trangular type, for the asset A, as follows: P A mn, for ( A mn, A m) A m A mn µ A( ) A max (22), for <, ) A m A max A max A m 0 for (, A mn ) ( A max, ) where A mn, A max, Am are the lowest, hghest and mddle values of the return, observed n the hstorcal data set or estmated accordng to the expert opnon. Let c s a real number, 0 c 1, whch represents the share of the asset A n the portfolo. The portfolo return s a fuzzy number c + ( 1 c) (23) P where B s the return of the B asset, characterzed by the membershp functon (22) wth parameters B mn, B max, Bm, respectvely. Portfolo return (23) as a lnear combnaton of fuzzy components wth trangular membershp functons, has a form: A P mn, for ( P mn, P m) P m P mn µ P ( ) P max (24), for < P m, P max ) P max P m 0 for (, ) (, ) P mn P max B 97

6 where parameters of the membershp functon (24) of the portfolo return can be express by parameters of the partcular membershp functons of assets: c + ( 1 c) (25a) P m A m B m c c (25b) P mn A mn + ( 1 ) B mn c c (25c) P max A max + ( 1 ) B max It s very mportant for market nvestors to predct the reducton threat to expected value of returns. Denote the -cut of the portfolo return as a closed nterval of real numbers: ) [, ] (26) ( P P ( ) P ( + ) Denote also the crsp nterval as a measure of the reducton threat (rsk) to expected value of portfolo return wth the level of the truth equal to [0,1]. It can be calculated usng -cut of the membershp functon of the portfolo return, as follows: P m P( ) [ c( ) + (1 c)( ] ( 1 ) A m A mn B m B mn (27) Now, the task of the portfolo optmzaton has been formulated n the category of real numbers, not fuzzy numbers: choose the value of a share c, 0 c 1 whch fulflls: under the condton c + ( 1 c) max (28) P m A m B m f ( c, ) (29) cr where assets. s the crtcal value of rsk, accepted by nvestors, [0,1], for the set of tested cr Example 1. Let values of sx shares of Polsh stock exchange are gven, observed daly n the perod: May 2005 June The parameters of the shares have been calculated and presented n Table 1. The calculated parameters concernng these sx shares have been used for the smulaton test of potental portfolos, whch are composed of pars of assets taken from Table 1. The crsp nterval f ( c, ) 98

7 representng rsk of the reducton of the mean values of portfolo returns can be calculated accordng to (27), n a form of the equaton depended on parameters: c - share of the frst asset n the portfolo, - assumed level of the truth (possblty) of the occurrence the nterval. Tab. 1. The parameters of the returns for sx chosen shares of Polsh stock exchange ( [5]) AMC BPH DUD PKN TPS WWL mn max m m - mn Tab. 2. The equatons of the rsk of portfolo returns for pars of assets of Polsh stock exchange (accordng to [5]) PAIS OF ASSETS f ( c, ) AMC/BPH 0.71c AMC/ DUD 0.49c AMC/ PKN 0.67c (1 ) AMC/ TPS 0.46c AMC/ WWL 0.06c BPH/ DUD 0.22c BPH/ PKN 0.04c BPH/ TPS 1.18c BPH/ WWL 0.77c DUD/ PKN 0.18c DUD/ TPS 0.95c DUD/ WWL 0.55c PKN/ TPS 1.13c

8 PKN/ WWL 0.73c TPS / WWL 0.41c Assumng 0. 75, cr 0. 7 and usng data and equatons from Table 2. we derve only fve values of c, fulfllng 0 c 1. Table 3 shows the calculated shares c and mean values of portfolos Pm. For the new value cr 0. 8 and we derve as results of calculatons nne portfolos, fulfllng condtons of c, 0 c 1 (Table 4). Tab. 3. The values of shares c and mean values of portfolos Pm calculated on the base of data from Table 2. and assumed values 0. 75, cr 0. 7 (accordng to [5]) PAIS OF ASSETS f ( c, ) 0.7 c, P m AMC/ TPS c 0.50, BPH/ TPS c 0.20, DUD/ TPS c 0.24, PKN/ TPS c 0.20, TPS / WWL c 0.43, The optmal portfolo s composed of 20% BPH assets and 80% TPS assets. The mean return s Tab. 4. The values of shares c and mean values of portfolos Pm calculated on the base of data from Table 2. and assumed values 0. 75, cr 0. 8 (accordng to [5]) PAIS OF ASSETS f ( c, ) 0.8 c, P m AMC/BPH c 0.77, AMC/ DUD c 0.66, AMC/ PKN c 0.75, BPH/ TPS c 0.54,

9 BPH/ WWL c 0.29, 0. 3 DUD/ TPS c 0.66, DUD/ WWL c 0.41, PKN/ TPS c 0.56, PKN/ WWL c 0.31, When the crtcal rsk nterval ncreases, then the set of portfolos ncreases too. We have the set of nne portfolos, the optmal portfolo s characterzed by the mean value of return equal to 0.39, 54% BPH assets and 46% TPS assets. Smlarly we can calculate the nterval of the chance of ncreasng the mean value of portfolo returns + P ( + ) ( 1 ) for the assumed level of possblty. P m [ c( ) + (1 c)( ] A max 3.2. Portfolo optmzaton task wth exponental fuzzy numbers A m B max Assume, lke n paragraph 3.1, that the portfolo s composed of two assets A and B. We defne returns of the partcular components, usng fuzzy number of the L- type, n the exponental form: B m (30) m L, m sl µ ( ) (31) m P, m s 2 m exp, m sl µ ( ) (32) 2 m exp, m s 101

10 where s L and s mean the standard devaton, calculated for left ( m - mn ) and rght ( max - m ) ntervals of observed fuzzy data. Generally, f fuzzy returns of assets A and B are characterzed by membershp functons wth parameters (33), noted accordng to [2]: A, s, s ), B, s, s ) (33) ( Am AL A ( Bm BL B then the portfolo return has parameters of the membershp functon c + ( 1 c) (34a) Pm PL Am AL Bm s cs + ( 1 c) s (34b) P A BL s cs + ( 1 c) s (34c) We are nterested n calculatng a measure of the reducton threat (rsk) to expected value of portfolo return, for the assumed level of possblty [0,1]. Takng nto account relatons (32) - (34c), we receved the real nterval as a functon of and c: 1/ 2 B ( ln) [ cs + (1 c) ] (35) AL s BL The task of the portfolo optmzaton can be agan formulated, n the meanng wrtten by (28), (29) and usng (34a) and (35). Example2. The set of assets presented n Table 1. has been consdered as a set of potental components of portfolos. Parameters s L of partcular returns have been calculated accordng to daly observaton n the perod May 2005 June Assumng the level of truth 0. 75, the crtcal value of rsk cr 0. 7 and usng equaton (35) we can calculate share values c of partcular pars of assets. Table 5. shows the calculated shares c, fulfllng 0 c 1 and mean values of portfolo returns Pm. Tab. 5. The values of shares c and mean values of portfolos Pm calculated on the base of exponental membershp functons; 0. 75, cr 0. 7 (accordng to [5]) PAIS OF ASSETS f ( c, ) 0.7 c, P m AMC/ TPS c 0.79, AMC/ WWL c 0.54, BPH/ TPS c 0.28,

11 BPH/ WWL c 0.11, DUD/ TPS c 0.62, DUD/ WWL c 0.33, PKN/ TPS c 0.34, PKN/ WWL c 0.14, The structure of the optmal portfolo calculated on the base of the exponental fuzzy numbers (28% BPH assets, 72% TPS assets, the mean return 0.35) s smlar to the structure of the optmal portfolo calculated for the trangular fuzzy numbers, shown n Table 3. (20% BPH, 80% TPS, the mean return 0.34). The set of portfolos from Table 5. conssts of eght pars of assets but n the case of trangular membershp functons the set of portfolos - only fve pars. Both portfolos have been calculated for the crtcal rsk nterval cr 0. 7 and the level of possblty CONCLUSION Ths paper dscusses possbltes of utlsng fuzzy set theory for makng decson n market nvestgatons. Fuzzy numbers represent the uncertanty of future values of returns. The type of fuzzy numbers and the shape of membershp functons are very mportant elements of the optmzaton task of the portfolo. Experences and knowledge of market experts are very helpful n such tasks. eferences [1] JAJUGA K. T.: Investments, fnancal nstruments, fnancal rsk, fnancal engneerng, (n Polsh), PWN, Warszawa, 1998 [2] KACPZYK J.: Fuzzy sets n system analyss, (n Polsh), PWN, Warszawa, 1986 [3] KLI G. J. YUAN BO: Fuzzy sets and Fuzzy Logc. Theory and Applcatons, Prentce Hall PT, New Jersey, 1995 [4] MANTEGNA.N., STANLEY H.E.: An Introducton to Econophyscs. Correlatons and Complexty n Fnance, Cambrdge Unversty Press, Cambrdge, 2000 [5] MENDECKI W.: Fuzzy models of the creaton and analyss of an assets portfolo, (Master Degree Work unpublshed, n Polsh), Slesan Unversty of Technology, Glwce, 2005 [6] WALASZEK-BABISZEWSKA A.: Fuzzy Probablty for Modellng of Partcle Preparaton Processes, The 4 th Int. Conf. Intellgent Processng and Manufacturng of Materals, Senda, Japan 2003; Intellgence n a Small Materal World, Selected papers from IPMM-2003, Meech J. A., 103

12 Kawazoe Y., Kumar V., Magure J.F. (Eds.), DEStech Publcatons, Inc., Lancaster, Pensylwana, 2005, pp [7] WALASZEK-BABISZEWSKA A.: Applcatons of Fuzzy and Probablstc Methods n Fnancal Market Analyses, Frst Warsaw Internatonal Semnar on Intellgent Systems (WISIS 2004), Polsh Academy of Scences; Issues n Intellgent Systems. Models and Technques, Dramńsk M. Grzegorzewsk P., Trojanowsk K., Zadrożny S. (Eds.), book seres: Problemy Współczesnej Nauk, Informatyka, Akademcka Ofcyna Wydawncza EXIT, Warszawa, 2005, pp [8] YAGE.., FILEV D. F.: Essental of Fuzzy Modellng and Control, (n Polsh), WNT, Warszawa, 1995 [9] ZADEH L. A.: Probablty Measures of Fuzzy Events, Journal of Math. Anal. Appl., 23/1968, pp