Investment Portfolo Evaluaton by the Fuzzy Approach Lambovska Maya, Marchev Angel Abstract Ths paper presents a new fuzzy approach for the evaluaton of nvestment portfolo, where the approach s vewed by the authors as a sub-phase of the management process of these portfolos. The approach defnes the mutual and delayed effects among the sgnfcant varables of the nvestment portfolo. The evaluaton of the effects s descrbed as fuzzy trapezodal numbers and they are aggregated by mathematcal operatons wth ncdence matrces and fuzzy functons experton. Key words: management process of nvestment portfolo, fuzz y evaluaton; fuzz y expertons and ncdence matrces; delayed effects 1. INTRODUCTION Portfolo management s a well-researched nterdscplnary feld. At the same tme, there are many new possbltes for nnovaton through applcaton of varous new methods for solvng the problem. Fuzzy logc and fuzzy sets are ncreasngly popular n portfolo management. The man focus of ths paper s on proposng a new fuzzy approach for evaluatng nvestment portfolos. Ths am s acheved by consequently fulfllng several research tasks. Frst s to revew the general concept of nvestment portfolo and the process of nvestment portfolo management. Next s to pont out possble fuzzy approaches for portfolo management. Then are defned the termnology of the used methods and stages of the portfolo evaluaton. Fnally an emprcal approbaton s conducted. In methodcal terms the approach suggested uses tools of the theory of confdence ntervals, theory of fuzzy subsets and method of expertse. A range of fuzzy nstruments are used fuzzy trapezodal numbers, fuzzy functons experton and fuzzy random ncdence matrces. The paper s dvded nto sx chapters, correspondng to the structure of a scentfc artcle. Some mportant terms of the portfolo theory are defned n sub-chapter 2.1. Process of the portfolo management s descrbes and dssects n sub-chapter 2.2. Known and possble fuzzy approaches to portfolo management are revewed n sub-chapter 2.3. Sub-chapter 3.1 outlnes the concept of the current proposal. Tools necessary for the proposal fulfllment are descrbed n sub-chapter 3.2. Stages of the approach suggested are presents n sub-chapter 3.3. Results of the approach approbaton are presented n chapter 4, followed by short dscusson (chapter 5) and concluson (chapter 6). 13
2. THEORETICAL SOLUTIONS 2.1 General outlne of the nvestment portfolo The nvestment portfolo s a combnaton of securtes owned by an nvestor. Securtes are nvestment opportuntes (nvestment assets), traded freely on a transparent market. Transparent market s one publcly transmts enough relevant nformaton. The most mportant feature of the securtes s that they are nterdependent ther prces covarate reflectng the real-world nterdependence of the ssung companes. The purpose of usng a portfolo approach s to mprove the condtons of the nvestment process by obtanng such nvestment propertes of the combnaton of securtes, whch are not obtanable by any sngle securty. The most rottenly consdered sgnfcant varables for any nvestment are rsk and return. A certan confguraton of rsk and return s only possble wthn a gven confguraton of securtes. Dversfcaton s the effect of combnng multple securtes such that t mproves the rsk condtons of nvestment. A portfolo conssts of k+1 postons each wth respectve weghts, where k s the total number of postons traded on the market (formula 1). The nvested sum of unwanted postons s set to 0. The non-nvested amount s assumed a cash poston C. If the cash poston s less than 0, then there are borrowed funds. Short postons are also possble, n whch case the sum nvested n securty s negatve. (1) Pt s t Ct k 1 - seral number of poston; k - total number of possble non-cash postons; t - tme of observaton; P - value of the portfolo at the tme t ; t s t - allocated nvestment of poston at the tme t ; C t - value of cash poston at the tme t. Return of an nvestment portfolo (see formula 2) s calculated as a weghted average of the returns of all ncluded securtes. The weghts correspond to the confguraton of the portfolo the allocated nvestment n each poston. The sum of all weghts (ncludng cash poston) s always equal to 1. The return of a cash poston s normally assumed 0. (2) R t w t.r t p k 1 R p t - return of the portfolo p at the tme t ; R t - return of the securty at the tme t ; w t - relatve weght of poston at the tme t. 14 Journal of Compettveness Issue 3/2011
The frst assumpton to calculate return of each securty at the tme t(r (t)) s that the tme doman s dscrete. The return Also should account for market frctons, such as taxes, brokerages, nflaton rate, etc. The most general case s shown n formula 3, wth some further elaboraton n formulas 4, 5. (3) RP ( t). 1 Dp RC ( t). 1 Dc R t I b s D D P b, ;, 0,1 p c RP ( t ) - return from captal gans of securty at the tme t ; RC ( t ) - return from complmentary benefts of securty at the tme t ; - functon for tax rate on captal gans; D p D c - functon for tax rate on complmentary benefts; P b - buy prce at the tme b of securty ; I b, s - nflaton rate between tme b and s ; 1, b s (4) RP ( t) P s. h P b K s K b; h 1, b s P s - sell prce at the tme s of securty ; P b - buy prce at the tme b of securty ; h - stock splt correcton coeffcent of securty ; K - brokerage at the tme s ; s K b - brokerage at the tme b. (5) B ( t ) max( b, s) RC ( t) B ( t) tmn( b, s) - quantfed complmentary benefts of securty at tme t Formula 4 accounts for the possblty of short sellng: t s not known whch of the tmes s and b precedes the other. It also accounts for brokerages. To take account of possble stock splt operatons durng the perod of nvestng n the securty, a correcton coeffcent h s ntroduced. Dependng on the type of poston long or short the value of h could be: h >1 for long postons.e. x/1, where x s the stock splt rato or h <1 for short postons.e. 1/x, where x s the stock splt rato Formula 5 accounts for the fact that besdes the return derved from prce change, there are other forms of return of a securty emergng durng the tme of nvestng. These nclude dvdends, nterest, non fnancal benefts and etc. All these must be estmated as fnancal nflow or outflow per one share (e.g. f whle holdng a short poston there s a dvdend of z amount per share, then ths s a negatve return of z). There are several approaches to calculatng portfolo rsk. The domnant concept s to use hstorcal varance and/or standard devaton. Other rottenly used measures are hstorcal vola- 15
tlty and value at rsk. A good case could be buld around usng nformaton entropy as a rsk measure of a portfolo. So measurng the rsk of an ndvdual securty may be formulated as functon of hstorcal data of the return of securty (see formula 6): V t F t d, t (6) V t - rsk of the securty at the tme t ; F - functon for measurng the rsk of securty ; d - number (depth) of hstorcal data consdered for calculaton of rsk. No matter what measure s used the rsk of a portfolo depends not only on the rsks of every ncluded securty, but also on the mutual dependence (nterdependence) among the securtes, except for cash poston, whch s assumed to have a rsk of 0. A typcal approach to measure portfolo rsk s a sum of two addends one for weghted average of the rsks of ncluded securtes and the other for calculatng the nterdependence of the securtes (Formula 7). k k k 2 (7) Vp t w t V. t w t.w j t. V t,v j t 1 1 j1 V p t - rsk of the portfolo p at the tme t ; V t - rsk of the securty at the tme t ; V t,v t - measure for nterdependence of the securtes and j. j 2.2 Phases n the process of portfolo management The process of portfolo management could be analyzed n several phases that are arranged wthn a control cycle. At the same tme portfolo management s an nformaton transformng process. As such t may be analyzed as consstng of three general phases whch could be dssected further nto functonal sub-phases, as follows: 1. Informaton nput In ths phase the ngong nformatonal flow s encoded n an understandable form. 1.1. Settng goals A goal s a desred state (confguraton) of the sgnfcant varables. After the frst controllng cycle, an addtonal task s ncluded n goal settng comparng the current state wth the desred one. Crtera for evaluatng portfolo performance may be used. Very sutable for the task s the Sortno rato or ts modfcaton. The rato s naturally goal orented as t compares the acheved return wth a desred return. 1.2. Recevng, collectng, systemzng nformaton on the behavor and the structure of the portfolo. - Ths sub-phase closes the feedback loop of the controlled process. 1.3. Recevng, collectng, systemzng nformaton about market (envronment) Ths subphase works wth nformaton from the known, observed external factors (market condtons and constrants, obtanable nvestment opportuntes), nfluencng the portfolo management process. 2. Informaton processng Ths phase s assocated wth makng the best possble use of the nformaton obtaned accordng to the needed functon of portfolo management. 16 Journal of Compettveness Issue 3/2011
2.1. Forecastng / estmatng the expected values of the sgnfcant varables of the obtanable nvestment opportuntes and the external factors. Statstcal analyss of the past portfolo structure s also necessary. 2.2. Soluton generaton Ths s the process of defnng and evaluatng feasble states of the portfolo as combnatons of multple securtes. There s a necessty of havng an external model to smulate possble solutons to the portfolo problem. It s not a compulsory component but usng an example model ( étalon ) s normal n nvestment portfolo management. It s a computerzed smulaton model for expermentng and evaluatng the generated solutons. In most cases, the computer smulaton would be programmed along a known (or new) theory (for nstance Markowtz Model). 2.3. Makng decson and selectng a portfolo structure. Only optmal (best possble) solutons out of all feasble are consdered. There s a need for usng mult-crtera optmzaton and enforcng the prncple of requste addton. An mportant varable to be consdered s the nvestor s ratonalty and hs preferences towards rsk (and towards other sgnfcant varables). 3. Informaton output Ths stage s assocated wth the transmsson (decodng) the nformaton necessary for the management effects of the portfolo At ths phase the controllng actons are emtted toward the portfolo, whch also means realzaton of the soluton. After comparson between the desred structure and the current structure of the portfolo, the dfferences are translated nto market orders. Several real lmtatons nterfere wth the realzaton of the soluton and thus the real mplementaton s always sub-optmal: Dscretzaton, dssectablty, avalablty of an ssue of a gven securty The numercal problem becomes a whole number optmzaton problem. Delay of the system reacton, ncludng the tme for executng an order, as well the tme for meetng the condtons of the order. The nertness of the controlled system also enforces delays. Market frcton s the cumulatve effect on the free trade from brokerages, the nflaton rate of the economy, taxes on captal gans and/or dvdends/nterests, etc. 2.3 Fuzzy approaches to the portfolo problem Snce the decson for a portfolo structure reles on ex-ante estmaton based on ex-post data, the process s carred out under uncertanty generated by the unknown future outcomes (Marcheva, 1995). Furthermore the huge complexty and abnormalty (Markowtz, Usmen, 1996, p. 22) of the fnancal markets makes the stochastc (let alone the determnstc) approach less and less applcable, because there s no base for assumng any gven probablty dstrbuton of the securty return. So other approaches to deal wth the uncertanty of the portfolo are beng sought by the researchng communty. There s an ongong dscusson on how to defne uncertanty n the context of fuzzy approach to portfolo management. A very systematc and tdy analyss s done by Zmeskal (2005). Uncertanty s a twofold meanng term. Frst uncertanty stands for measured uncertanty rsk. When dealng wth classcal defntons of rsk t s ether defned n determnstc or n stochastc terms, resultng n a crsp (as opposed to fuzzy) set of numbers and crsp values. The other way of defnng uncertanty s vagueness the unmeasured uncertanty. 17
When dscussng portfolo management both meanngs of uncertanty are consdered. Frst some sort of measurement s needed, but as well as a method to compensate the mprecseness of such method. A possble tool for the task s the fuzzy approach.e. usng fuzzy numbers and fuzzy sets to descrbe uncertan phenomena and/or usng fuzzy logc to process data from uncertan phenomena. A complete fuzzy approach for portfolo management would be a fuzzy control process entrely made of fuzzy sub-phases: Fuzzy nformaton nput fuzzfcaton of data from the portfolo and the envronment. As for the goal settng sub-phase, the goals orgnate as lngustc varables anyway. So t s just a matter to make them compatble wth the rest of the process n nformaton terms. Fuzzy nformaton processng would mostly use fuzzy logc and fuzzy mathematcs. There are already a lot of proposals of ths type to estmate the sgnfcant varables and generate solutons (see below). Some of them even suggest ways of fuzzy selecton and evaluaton of solutons by fuzzy functons. Fuzzy nformaton output would be the phase to conduct defuzzfcaton of the soluton and to carry out management actons on the portfolo. Once the fuzzy approach for solvng problems under condtons of uncertanty s becomng ncreasngly popular among researchers, t s qute expected that there s already a wde range of proposed solutons for dfferent phases and/or tasks of the process of portfolo management. The propostons are most often orented towards the two more techncal phases of portfolo management: Fuzz y approaches to estmatng sgnfcant varables of a portfolo Ths s the most common suggeston for usng fuzzy approach n portfolo management. The authors propose fuzzy measures of return and rsk of the portfolo. Typcally they are followed by a way to estmate the varance covarance matrx necessary for portfolo optmzaton. Good examples are the works of Katagr and Ish (1999), Mohamed et al. (2009), Petreska and Kolemsevska (2010) and Zhang et al. (2003). Fuzzy membershp functons are used to adjust the return and the rsk of the securtes n the study of Lan and L (2010). The portfolo rsk measure s a fuzzy estmated type of value at rsk n the studes of Lu et al. (2005) and Wang et al. (2009). An unorthodox measure of portfolo rsk s proposed by Huang (2008) the entropy of fuzzy returns of the securtes n the portfolo. An nterestng and somewhat related to the proposton n the current paper s the approach of Tastle and Werman (2009). The authors there use expert opnons to reach a degree of consensus on rsk estmaton. Also smlar to some extend s the study of Marcheva (1995). It s another research usng nterval numbers, where forecastng of shares prces s done by experts. Fuzz y approaches for generatng feasble solutons to a portfolo problem Authors focus on usng fuzzy reasonng.e. fuzzy subsets, fuzzy rules and lngustc varables for selectng portfolo structure or realzaton of nvestment strategy. In hs classcal book Bojadzev and Bojadzev (1997, pp. 157-164) propose such approach for one of the frst tmes. Later Chow and Inoue (2001) Ghandar et al. (2009) and Nakaoka et al. (2005) elaborate on fuzzy lngustc rules. 18 Journal of Compettveness Issue 3/2011
A fuzzy rankng strategy for portfolo selecton gvng best solutons for dfferent degrees of rsk-averson s proposed by Bermudez et al. (2007). And Tryak and Ahlatcoglu (2009) use a fuzzy analytcal herarchcal approach for mult-crtera selecton of securtes n a portfolo. 3. METHODOLOGY 3.1 The proposed fuzzy approach for portfolo evaluaton Current paper proposes a fuzzy approach for evaluatng a portfolo structure usng expertse. An mportant remark that has to be made upfront s that the term expertse s used n a broad sense. So an expert evaluaton may represent the computaton from a mathematcal algorthm, a statement form a person wth specal and extended knowledge on the subject or combnaton of both. The process of evaluaton of the portfolo begns after a portfolo structure has been already set. Second stage uses experts evaluatons or evaluatons from mathematcal algorthms (called method of expertse hereafter), presented n the form of fuzzy trapezodal numbers. The fuzzy trapezodal numbers have membershp functon whch specfcally dsplays a maxmum range (nstead of a pont) of values among the values of the estmated varable. The fuzzy numbers are then processed n a specfc method for dscoverng the nfluences of return on rsk among the securtes and wthn the portfolo. Analyss on delayed nfluences s later done. The am of the approach s to establsh a method for evaluatng nvestment portfolos by determnng the mutual nfluences among dfferent sgnfcant varables of the portfolo (n that case, return and rsk) and the hdden nfluences between them. The approach suggested could also be used as a base for comparson and/or rankng dfferent portfolos. Last but not least the used experts evaluatons may be aggregated results from other approaches for portfolo management. Thus, the approach could be descrbed as a unversal tool to combne several methods, whle averagng out ther extreme solutons. 3.2 Tools for portfolo evaluaton Portfolo evaluaton fnds expresson n two actvtes n ths approach. The frst actvty s evaluaton of the return nfluence on the rsk of shares n the portfolo takng nto account mutual nfluences between returns of shares and between ther rsks. The second actvty s evaluaton of delayed effects of returns on rsks of shares n the portfolo. Tools, suggested n the paper, for the portfolo evaluaton conssts of: method of expertse; mathematcal operatons wth confdence ntervals wth four evaluatons ( confdence fours ); and mathematcal operatons wth fuzzy trapezodal numbers (FTNs), fuzzy expertons, fuzzy random ncdence matrces. Method of expertse s used to evaluate returns and rsks of shares n the portfolo as well as the nfluence of returns on rsks of shares. The evaluatons are systematzed n fuzzy matrces of: nfluence of returns on rsks, mutual nfluences between returns of shares and mutual nflu- 19
ences between rsks of shares. Possble nterval of change [0,1] s set for the evaluatons. The method of expertse s appled due to the authors belef n low utlty of statstcal methods for evaluatng under uncertanty. Confdence ntervals wth four evaluatons are a tool of the theory of ntervals. It s a branch of mathematcs appled to condtons of subjectvty and uncertanty (Kaufmann, Gl Aluja, 1990, p. 11). Accordng to the theory the evaluaton s descrbed by an nterval, whch s not characterzed by a possblty of occurrng and convexty (Kaufmann, Gl Aluja, 1990, p. 21). In ths context confdence fours are buldng elements of fuzzy random ncdence matrces and functons experton n the aggregaton procedure of the portfolo evaluatons. In ths approach confdence fours are presented n dscrete form (defuzzfcated) by so-called of the confdence four. It reflects the relatve lnear dstance of the nterval to the number zero on the explct condton of absent possblty of occurrng (Kaufmann, Gl Aluja, 1988, p. 74). Representatve numbers are used n the approach to defne delayed effects between returns and rsks as well as to present results of the portfolo evaluaton more clearly. Three types of tools of the theory of fuzzy subsets are used n the approach. The frst one s fuzzy subset/number. It s descrbed by confdence ntervals for any possblty of occurrng n the nterval [0,1] (Kaufmann, Gl Aluja, 1986, p. 37). Fuzzy trapezodal numbers are used to descrbe uncertan experts evaluatons of nfluences of: returns on rsks of the shares n the portfolo, returns between shares and rsks between them. The fuzzy trapezodal number s a fuzzy number/subset wth a lnear and contnuous characterstc functon, whch has two evaluatons of possblty of occurrng unty and two evaluatons of possblty of occurrng zero (Bojadzev, Bojadzev, 1997, pp. 24-25). Mathematcal operatons wth fuzzy random ncdence matrces (Kaufmann, Gl Aluja, 1988, p. 54) are used to aggregate evaluatons of nfluences and to study combned and delayed effects between returns and rsks. Three operatons wth fuzzy random matrces are used n the approach maxmn functon, calculaton of the mathematcal expectaton of matrces and dfference between matrces. The maxmn functon s appled for evaluaton of combned nfluences of I and II generatons of returns on rsks (formula 8). The mathematcal expectaton weghs the evaluatons of nfluences aganst the possbltes of ther occurrng. It s used as a bass for determnng the delayed effects of returns on rsks. Fuzzy functons experton are knd of fuzzy random matrces. They are used n the approach to aggregate the evaluatons. The experton functon s defned as a matrx descrbng the law on cumulatve (for all experts) complementary (n ths case to the number unty ) probable dstrbuton of evaluatons (Kaufmann, Gl Aluja, 1990, p. 54-55). 3.3 Stages of portfolo evaluaton Accordng to the authors dea the portfolo evaluaton could be mplemented n four stages: Stage I Determnng the portfolo ; Stage II Aggregaton of evaluatons of (mutual) nfluences between return and rsk of shares n the portfolo ; Stage III Evaluaton of combned nfluences (of I and II generatons) of returns on rsks of shares n the portfolo ; and 20 Journal of Compettveness Issue 3/2011
Stage IV Evaluaton of delayed effects of returns on rsks of shares n the portfolo. The frst stage conssts of procedures for portfolo generatng and evaluaton of (mutual) nfluences of returns and rsks of the shares n the portfolo. The frst procedure s not subject to ths publcaton. The second procedure covers actvtes of generatng matrces of (mutual) nfluence of returns and rsks n the portfolo, ncludng matrces of: nfluence of returns on rsks of the shares n the portfolo, mutual nfluence between returns of the shares and mutual nfluence between rsks of the shares. In mathematcal terms the evaluatons are represented by fuzzy trapezodal numbers. The evaluatons of nfluence of returns and rsks of the shares n the portfolo are aggregated at the second stage. Ths s acheved by formng experton functons, whch requre use of fuzzy trapezodal numbers as confdence fours. The second stage conssts of the followng procedures: calculaton of experton of mutual nfluences between returns of the shares; calculaton of experton of mutual nfluences between rsks of the shares; and calculaton of experton of nfluence of returns on rsks of the shares. Mutual nfluences between returns of shares n the portfolo are aggregated n the frst procedure. The procedure ncludes accumulaton of the evaluatons of mutual nfluence between returns of the shares by fuzzy random nfluence matrces and formaton of the experton of mutual nfluences between returns of the shares. Mutual nfluences between rsks of the shares n the portfolo are aggregated n the second procedure. Influences of returns of the shares on ther rsks are aggregated n the thrd procedure. The second and thrd procedures are realzed n analogy wth the frst procedure of the stage. Combned nfluences of I and II generatons of returns on rsks of the shares n the portfolo are evaluated at the thrd stage. It s mplemented by ntegratng mutual nfluences between returns of the shares, rsks of the shares and nfluence of returns on rsks nto so-called combned nfluences of I and II generatons. Combned nfluences are evaluated by applyng the functon maxmn to the expertons: return - return, rsk - rsk and return rsk (formula (8)). ~ (8) Y ~ E ~ R ~ Y ~ I E ~ R ~ I,II Y R Y R ~ - s fuzzy random matrx of combned nfluences of I and II generatons; I I,II, - are symbols to denote functons maxmn, max and mn respectvely; Y ~ - experton return - return ; R ~ - experton rsk - rsk ; E ~ Y R - experton return rsk. Delayed effects of returns on rsks of the shares n the portfolo are evaluated at the fourth stage. Ths stage ncludes de-accumulaton (to the number zero ) of the fuzzy matrces of nfluence of returns on rsks, calculaton of the mathematcal expectaton for fuzzy matrces of de-accumulated nfluences of returns on rsks and evaluaton of delayed effects of returns on 21
rsks. The frst actvty refers to the experton of nfluences of returns on rsks and to the fuzzy matrx of combned nfluences of I and II generatons of returns on rsks. The second actvty s amed at takng nto account possbltes of occurrng of de-accumulated evaluatons of the return nfluence on rsk. It s appled n respect to confdence fours of the experton of de-accumulated nfluences and of the fuzzy matrx of de-accumulated combned nfluences of I and II generatons as well as n respect to confdence fours of the portfolo n these experton and fuzzy matrx. Confdence fours of the mathematcal expectatons are substtuted by ther representatve numbers, whch are systematzed n so-called representatve matrces. Delayed effects are defned by: 1. formaton of the dfference between elements of the representatve matrces of mathematcal expectatons for returns nfluence on rsks (see formula (9) and for combned nfluences of I and II generatons; and 2. subsequent defnton as delayed effects of the dfferences, whch are equal to or hgher than gven constant (c), belongng to the nterval (0,1] (see formula (10). (9) D H D A Aj,H 2,A 1, A D A j,h Aj,H A Aj,H, A A A (10) Dde D for D c,c 0,1 A j Aj,H Aj,H 2,A 1, A, 0,1 A j,h D H - s the matrx of the dfference of mathematcal expectatons for returns nfluence on rsks, 1,A - of the mathematcal expectaton of de-accumulated return Aj,H nfluence of the share A on the rsk of the share Aj 2,A - of the mathematcal expectaton for de-accumulated A j,h combned nfluence of I and II generatons of the return nfluence of share A on the rsk of share Aj; Dde - delayed effect of the return nfluence of the share A on the rsk of the share Aj. A A j Aj,H 4. RESULTS Ths part of the artcle covers only an llustraton of the proposed fuzzy model for evaluaton of nvestment portfolos. The approbaton of the model wth real data suggests a separate survey. Accordng to the authors ts results can hardly be expressed n ths publcaton because of ts lmted volume. The approbaton of the suggested approach was accomplshed for three portfolos, each consstng of four shares (A1 to A4). Shares n all three portfolos are of the same knd, but partcpate n portfolos wth dfferent weghtngs. The results for the return nfluence on the rsk of the shares n portfolos 1, 2 and 3 are shown n Tables 1, 2 and 3 respectvely. Graphcal presentatons of the results for the return nfluence of the three portfolos on the rsk of the share A1 are done n fgure 1 (see tables 1, 2 and 3, column Share A1, row Portfolo ). 22 Journal of Compettveness Issue 3/2011
Tab. 1 Mathematcal Expectatons for Portfolo 1. Source: own creaton Mathematcal expectaton for the evaluatons of return nfluence on rsk of the shares of portfolo 1 Shares Share A1 Share A2 Share A1 evaluatons 0,465 0,534 0,900 1,000 0,399 0,467 0,900 0,967 0,465 0,534 0,833 1,000 0,399 0,467 0,800 0,967 Share 2 0,72 0,68 0,70 0,65 evaluatons 0,367 0,534 0,833 0,900 0,367 0,499 0,800 0,899 0,367 0,534 0,833 0,900 0,367 0,499 0,800 0,934 Share 3 0,67 0,64 0,67 0,65 evaluatons 0,567 0,600 0,833 0,967 0,467 0,500 0,767 0,899 0,466 0,600 0,833 0,967 0,567 0,600 0,800 0,933 Share 4 0,73 0,65 0,72 0,72 evaluatons 0,400 0,533 0,833 0,934 0,400 0,533 0,833 0,899 0,400 0,533 0,767 0,934 0,400 0,533 0,800 0,900 Portfolo 1 (shares A1 tll A4) 0,68 0,67 0,66 0,66 evaluatons 0,450 0,550 0,850 0,950 0,408 0,500 0,825 0,916 0,425 0,550 0,817 0,950 0,433 0,525 0,800 0,934 0,70 0,66 0,68 0,67 Share A3 Share A4 Tab. 2 Mathematcal Expectatons for Portfolo 2. Source: own creaton Mathematcal expectaton for the evaluatons of return nfluence on rsk of the shares of portfolo 2 Shares Share A1 Share A2 Share A1 evaluatons 0,367 0,500 0,833 0,900 0,567 0,600 0,767 0,899 0,567 0,600 0,733 0,733 0,466 0,600 0,833 0,899 Share 2 0,66 0,70 0,66 0,71 evaluatons 0,499 0,567 0,800 0,967 0,533 0,600 0,800 0,900 0,533 0,600 0,767 0,866 0,499 0,600 0,800 0,900 0,70 0,71 0,69 0,70 Share 3 evaluatons 0,433 0,533 0,800 0,933 0,501 0,567 0,834 1,000 0,501 0,567 0,834 1,000 0,501 0,567 0,800 0,899 0,67 0,72 0,72 0,69 Share 4 evaluatons 0,499 0,567 0,799 0,967 0,567 0,600 0,767 0,866 0,567 0,600 0,700 0,767 0,500 0,600 0,799 0,867 Portfolo 2 (shares A1 tll A4) 0,70 0,69 0,66 0,69 evaluatons 0,450 0,542 0,808 0,942 0,542 0,592 0,792 0,916 0,542 0,592 0,759 0,842 0,492 0,592 0,808 0,891 0,68 0,70 0,68 0,70 Share A3 Tab. 3 Mathematcal Expectatons for Portfolo 3. Source: own creaton Shares Share A1 Share A2 Share A4 Share A1 evaluatons 0,567 0,600 0,867 0,967 0,600 0,600 0,799 0,833 0,600 0,600 0,833 0,967 0,500 0,600 0,867 0,934 Share 2 0,74 0,71 0,74 0,73 evaluatons 0,567 0,700 0,867 0,967 0,533 0,700 0,799 0,867 0,567 0,700 0,767 0,867 0,567 0,600 0,867 0,967 0,78 0,73 0,73 0,74 Share 3 evaluatons 0,567 0,600 0,833 0,900 0,533 0,600 0,834 0,899 0,567 0,600 0,833 0,900 0,567 0,600 0,767 0,833 Share 4 0,72 0,72 0,72 0,69 evaluatons 0,567 0,700 0,833 0,967 0,533 0,700 0,767 0,867 0,567 0,700 0,767 0,867 0,567 0,600 0,833 0,967 Portfolo 3 (shares A1 tll A4) 0,77 0,72 0,73 0,73 evaluatons 0,567 0,650 0,850 0,950 0,550 0,650 0,800 0,867 0,575 0,650 0,800 0,900 0,550 0,600 0,834 0,925 0,75 0,72 0,73 0,72 Mathematcal expectaton for the evaluatons of return nfluence on rsk of the shares of portfolo 3 Share A3 Share A4 23
Tab. 4 Mathematcal Expectatons for Delayed Effects of Portfolo 3. Source: own creaton Shares Mathematcal expectaton for the evaluatons of delayed effects of return on rsk of the shares of portfolo 3 Share A1 Share A2 Share A3 Share A4 A1 0,045 0,055 0,079 0,148 A2 0,208 0,173 0,148 0,115 A3 0,102 0,107 0,072 0,129 A4 0,137 0,052 0,048 0,063 Portfolo 3 0,123 0,089 0,079 0,114 Fg. 1 Fuzz y evaluatons of return nfluence of portfolo 1, 2 and 3 on the rsk of share A1. Source: Own creaton. 5. DISCUSSION It s obvous from tables 1 to 3 that the three portfolos are characterzed by hgh degree of the return nfluence on the rsk belongng to the range [0,66;0,75]. The hghest result s that of portfolo 3 (table 3). Therefore other thngs beng equal the choce s defntely for portfolo 3. The results of the approach approbaton show that the delayed effects of returns on rsks n the evaluaton of combned nfluences of I and II generaton for the three portfolos are lower than 0,21. These delayed effects are defned as very low or neglgble. Table 4 presents evaluatons of delayed effects of portfolo 3. That s the portfolo wth the hghest evaluatons of delayed effects among the three portfolos (see table 4, row Share A2 and column Share A1 ). Ths result s logcal gven that portfolo 3 s the portfolo wth the hghest degree of return nfluence on the rsk of the shares. 6. CONCLUSION Ths paper presents a new approach for evaluatng nvestment portfolos through fuzzy tools of the theory of confdence ntervals and theory of fuzzy subsets. The approach conssts n determnng mutual and hdden nfluences between the sgnfcant varables of the nvestment portfolo n whch evaluatons of the nfluences are descrbed by fuzzy trapezodal numbers and are aggregated by mathematcal operatons on fuzzy ncdence matrces and fuzzy functons experton. 24 Journal of Compettveness Issue 3/2011
General concept of the nvestment portfolo s revewed n the paper. Phases of the process of managng the nvestment portfolo are determned. Important remarks about realzaton of a proposed optmal soluton to a portfolo problem are ponted out. Need for fuzzy approaches to solve ths task n the context of complexty and abnormalty of the fnancal markets s substantated. Concept of the fuzzy approach suggested by the authors of the artcle s presented. Tools and stages of the methods for the mplementaton of the approach are characterzed. Results of the approach approbaton are systematzed and analyzed. The approbaton s realzed through the case data and s amed only to demonstrate the approach and ts applcablty. Accordng to the authors the approach suggested could also be used as a base for comparson and/or rankng dfferent portfolos. Last but not least the used experts evaluatons may be aggregated results from other approaches for portfolo management. Thus, the approach could be descrbed as a unversal tool to combne several methods. References 1. Bermudez, J., Segura, J. & Vercher, E. (2007). A fuzzy rankng strategy for portfolo selecton appled to the Spansh stock market. IEEE Internatonal Fuzz y Systems Conference. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Bojadzev, G. & Bojadzev, M. (1997). Fuzz y logc for busness, fnance, and management. Sngapore: World Scentfc publshng. Chow, Y. & Inoue, A. (2001). On-lne Investment usng Soft Computng. IFSA World Congress and 20th NAFIPS Internatonal Conference. Ghandar, A., Mchalewcz, Z., Schmdt, M., To, T. & Zurbrugg, R. (2009). Computatonal Intellgence for Evolvng Tradng Rules. IEEE Transactons On Evolutonary Computaton, Vol. 13, No. 1. Huang, X. (2008). Mean-Entropy Models for Fuzzy Portfolo Selecton. IEEE Transactons On Fuzz y Systems Vol. 16, No. 4 Jones, C. (1994). Investments: Analyss and Management. New York: John Wley & Sons. Katagr, H. & Ish, H. (1999). Fuzzy Portfolo Selecton Problem. Systems, Man and Cybernetcs. Kaufmann, A. & Gl Aluja, J. (1986). Introduccon de la teora de los subconjuntos borrosos a la geston de la empresas. Santago de Compostela: Ed. Mlladoro. Kaufmann, A. & Gl Aluja, J. (1988). Modelos para la nvestgacon de efectos olvdados. Vgo: Pugalsa S. A. Kaufmann, A. & Gl Aluja, J. (1990). Laz matematcas del azar y de la ncertdumbre. Madrd: Edcones Grafcas. Lan, K. & L, C. (2010). A Fuzzy Decson Maker for Portfolo Problems. IEEE Internatonal Conference On Systems, Man And Cybernetcs. Lu, Y., Wang, T., Gao, L., Ren, P. & Lu, B. (2005) Fuzzy portfolo optmzaton model based on worst-case VaR. Fourth Internatonal Conference on Machne Learnng and Cybernetcs, Guangzhou. Marcheva, A. (1995) Makng a decson for portfolo management n the condtons of rsk and uncertanty. Sofa: Techncal Unversty. 14. Markowtz, H. & Usmen, N. (1996). The Lkelhood of Varous Stock Market Return Dstrbutons, Part 2: Emprcal Results. Journal of Rsk and Uncertanty, vol. 13(3), 221-247. 25
15. 16. 17. 18. 19. 20. 21. 22. Mohamed, Z., Mohamad, D. & Samat, O. (2009). A Fuzzy Approach to Portfolo Selecton. Sans Malaysana, 38(6), 895 899. Nakaoka, I., Tan, K., Hoshno, Y. & Kame, K. (2005). A Portfolo Selecton by SOM and An Asset Allocaton of Rsk / Nonrsk Assets by Fuzzy Reasonng Usng the Selected Brands. IEEE Internatonal Conference on Systems, Man and Cybernetcs. Petreska, B. & Kolemsevska-Gugulovska, T. (2010). A Fuzzy Rate-of-Return Based Model for Portfolo Selecton and Rsk Estmaton. IEEE Internatonal Conference On Systems, Man And Cybernetcs. Tastle, W. & Werman, M. (2009). Vsualzaton of Mutual Fund Rsk Usng the Consensus Theory Measure of Agreement 28th North Amercan Fuzz y Informaton Processng Socety Annual Conference. Tryak, F. & Ahlatcoglu, B. (2009). Fuzzy portfolo selecton usng fuzzy analytc herarchy process. Informaton Scences. Wang, B., Wang, S. & Watada, J. (2009). Fuzzy Portfolo Selecton based on Value-at-Rsk. IEEE Internatonal Conference on Systems, Man and Cybernetcs. Zhang, W., Zhang, Q. & Ne, Z. (2003). A class of fuzzy portfolo selecton problems. Proceedngs of the Second Internatonal Conference on Machne Learnng and Cybernetcs. Zmeskal, Z. (2005). Value at rsk methodology of nternatonal ndex portfolo under soft condtons (fuzzy-stochastc approach). Internatonal Revew of Fnancal Analyss, 14, 263 275. Contact nformaton Assoc. Prof. eng. Maya Lambovska, PhD Unversty of Natonal and World Economy - Sofa, Bulgara, Department of Management Sofa, 1700, Studentsk Grad Hrsto Botev Emal: mlambovska@abv.bg Assst. Prof. Angel Marchev, Jr. Unversty of Natonal and World Economy - Sofa, Bulgara, Department of Management Sofa, 1700, Studentsk Grad Hrsto Botev Emal: angel.marchev@yahoo.com JEL Classfcaton: G11, M20, C69 26 Journal of Compettveness Issue 3/2011