APPLICATION OF Q-MEASURE IN A REAL TIME FUZZY SYSTEM FOR MANAGING FINANCIAL ASSETS

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 APPLICATION OF Q-MEASURE IN A REAL TIME FUZZY SYSTEM FOR MANAGING FINANCIAL ASSETS Penka Georgieva and Ivan Pochev 2 Burgas Free Universiy, Faculy of Engineering and Comuer Science, Burgas, Bulgaria enka.v.georgieva@homail.com 2 Bulgarian Academy of Science, Sofia, Bulgaria iochev@ii.bas.bg ABSTRACT One of he maor roblems ha a financial manager faces is he enormous amoun of financial daa. There is a variey of sofware sysems used o suor he rocess of invesmen decision making. In his aer, a sofware sysem for financial asse managemen is resened. The sysem is based on fuzzy logic, oeraes in real ime and differs from exising sysems for orfolio managemen in five key asecs. The sysem is esed on real daa from Bulgarian Sock Exchange. KEYWORDS Fuzzy Sysem, Financial Daa, Asse Managemen, Porfolio Managemen, Q-measure. INTRODUCTION Financial invesors use mainly wo mehodologies for analysing financial daa in order o increase heir wealh: fundamenal analysis and echnical analysis. While he former is based on exloring he counless macroeconomic facors of economic environmen, he laer is abou keeing rack of rice changes and finding aerns in heir ime series. Boh aim a redicing he fuure behaviour of asse rices and making invesmen decisions deending on hese redicions. However, he argumen beween fundamenal and echnical analysers is no an issue in his work. In his aer, a sofware sysem for managing daa and invesmens (boh individual and orfolio) is resened. The sysem is based on ools ha fuzzy logic rovides and is an alicaion of he Q-measure of an asse. [] The resened sysem differs from he exising sysems for orfolio managemen in five key asecs: ) There are no requiremens for robabiliy disribuions of reurns. 2) Geomeric mean of daily reurns is used as a measure of changes in rices. 3) A soluion of he cardinaliy roblem in orfolio heory is roosed. 4) Financial daa is rocessed in real ime. 5) The sysem is alicable o individual asse managemen as well as o orfolio consrucion. DOI: 0.52/isc.202.3403 2

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 The sysem consiss of hree modules (Fig.): Module Daa Managing Module (DMM) Module 2 Q-Measure Fuzzy Logic Module (QFLM) Module 3 Porfolio Consrucion Module (PCM). RAW DATA DATA MANAGING MODULE Q-MEASURE FUZZY LOGIC MODULE PORTFOLIO CONSTRUCTION MODULE Figure. A real ime sofware sysem for consrucing financial orfolios DMM is an alicaion for collecing, soring and managing financial daa in real ime. In addiion, in his module calculaions of recise measuremens of imoran asse characerisics (reurn, risk and q-raio) are imlemened. QFLM consiss of alicaions ions based on fuzzy logic. Inu daa for his module are he cris numerical values of asse characerisics from DMM. These cris values are fuzzified and afer alying he aggregaion rules a fuzzy variable Q-measure for each of he asses is obained. The ouu of his module is a defuzzified cris value of Q-measure. In PCM several orfolios are consruced. The invesors uiliy references are he key facor for choosing he oimal orfolio. 2. DATA MANAGING MODULE (DMM) 2.. Reurn, Risk and q-raio Le P, P2,..., PT be he sequence of daily close rices P of an asse, where =, 2,..., T. The use of geomeric mean of reurns aims a analysing he changes in he invesed caial in is dynamics and is calculaed as follows: T = T g r = 2 r, where P r = is he daily reurn for day and P = 2,..., T. [], [3] 22

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 If he sandard deviaion of he reurns is o be used as a measure of he risk, i is aroriae o ake he logarihms of reurns and hen he nex formulae are derived: ln ( r ) ln = ln( P ) ln( P ) = P P is he daily log reurn for = 2,..., T ; r g = ln( r g ) = ln T T r = = 2 T T = 2 ln r is he arihmeic mean of log reurns. [], [2] In ime series of asse rices, here are days wih no rading aciviy, e.g. here is missing daa. One way o deal wih his roblem is o coy he las rice corresonding number of imes, e.g. P, P, P, P 4 44 24 K 3. k This oeraion does no change he arihmeic mean of log reurns due o he roeries of logarihms. In case of daily observaions, he geomeric mean of reurns is calculaed as follows: k T = 2 r = k P T P, where k = T i= 2 and and decreased by. is he number of days beween he non-missing observaions a days If log reurns are used, he above consideraions should be made very carefully because of he differen number of days beween he observaions. Thus, if he reurn for he eriod is r r = +, hen he log reurn a he momen is ( r ) calculaed: r ln and so he arihmeic mean of log reurns is T T = ln T r = ln r. [] = 2 T = 2 23

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 Then he oal reurn is TR = e r r, he oal norm of reurn is TNR = e and he annual norm D of reurn is ANR = TNR., where D is he number of days in he financial year. k + Finally, he annual reurn is AR = ANR. In he roosed sysem, annual reurn is used as a measure of asse reurn as i is an adequae esimaor for he change of he invesmen. The commonly used idea for risk in invesmen heory is he variabiliy of reurns. The variabiliy is calculaed wih differen saisical ools, based on robabiliy disribuions bu mos ofen by he variance of he reurns. [], [6], [7], [8], [9] If log reurns are used hen he esimaor of variance as arihmeic mean of log reurns is calculaed as follows: T 2 s = ( ln( r ) r g ), T 2 = 2 and he q-raio equals he quoien of reurn and risk: AR q =. [] s Some invesors have secific execaions abou boh reurn and risk of heir invesmens. This is why here are wo addiional inus in he sysem- a lower hreshold of reurn r 0 and risk σ 0, and hese are redefined by he invesor. 2.2. Descriion of DMM DMM consiss of a hree-layer alicaion (Fig 2.) The funcion of his module is o collec raw daa (asse rices) in real ime, o sore and rocess his daa. Figure 2. Scheme of he Daa managing module 24

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 The ess of he sysem are conduced on daa from Bulgarian Sock Exchange. The changes in Bulgarian asse rices are ublished on he webage age of BSE (bse-sofia.bg). The informaion of ineres (dae, BSE code, oen, close, high and low rices) is in he hml code of his age. The alicaion is sared auomaically by Windows Task Scheduler. The daa access layer conains mehods for managing he daabase. The Requeser realizes he reques o he web age of BSE. The Parser selecs he daa ha is needed in a suiable form wih resec o he nex ses. For his reason, he hml code of he downloaded age by he Requeser is arsed wih regular exressions. The Filler is he ar of he alicaion ha deals wih he missing daa. There are wo yes of missing daa missing name and missing rice. The name of an asse is missing in he daabase if i is a new asse lised on he exchange or here has been no rade wih his asse for a eriod of ime being longer han he ime he sysem has been funcioning. The rice could be missing in he days wih no deals wih he given asse or on he holidays. The Calculaor uses he informaion obained by Filler and formulae from 2. o calculae he reurn, risk and q-raio for each asse. 3. Q-MEASURE FUZZY LOGIC MODULE (QFLM) Inu daa for QFLM is derived from he daabase afer he module for daa collecion and sorage has finished is work. The ouu is a cris value for he Q-measure of each asse. The scheme of his module is resened on Fig. 3. Figure 3. Scheme of QFLM The inu daa consiss of cris values for reurn, risk, q-raio, ID and name of each of he asses. These cris values are fuzzified according o he redefined linguisic variables. Each linguisic variable is a rile consising of a name, a domain and a se of fuzzy erms. The fuzzy erms are defined wih heir name and membershi funcion. Three yes of membershi funcions are used in his sysem: Gaussian, Bell and Sigmoid funcion. For each inu variable a degree of membershi o he corresonding erm is calculaed. Then he fuzzy rules are fired. The rules are of IF-THEN ye and model he decision making rocess. The fuzzy ouu variable Q-measure 25

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 consiss of fuzzy erms ha describe he qualiy of an asse. The urose of he defuzzificaion is o obain a cris ouu value. 3.. Linguisic Variables The fuzzy sysem comrises of hree inu and one ouu variables. Inu variables are he characerisics of an asse: reurn K = { reurn}, risk K 2 = { Risk} and K3 = { q raio}. The ouu variable is he Q-measure of an asse. These four fuzzy variables have differen number of fuzzy erms wih Gaussian, Bell or Sigmoid membershi funcions which are shown on Fig. 4, 5, 6 Very low Low Neural High Very high -0.2-0. 0 0. 0.2 0.3 0.4 0.5 Figure 4. Inu linguisic variable K = { reurn} Very low Low Neural High Very high -0.2-0. 0 0. 0.2 0.3 0.4 0.5 Figure 5. Inu linguisic variable K = { Risk} 2 26

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 Small Neural Big -20 0 20 40 60 80 00 Figure 6. Inu linguisic variable K = { q raio} 3 Bad No good Neural Good Very good -0.5 0 0.5.5 3.2. Fuzzy Rules and Defuzzificaion Figure 7. Ouu linguisic variable Q-measure The goal of decision making is o find an oimal soluion for a siuaion where a number of ossible soluions exiss. Bellman and Zadeh roosed a fuzzy model for decision-making in which obecives and goals are described as fuzzy ses and he soluion is a suiable aggregaion of hese ses. [2] Le G i be a fuzzy se of obecives and le C be a fuzzy se of goals. Then, he fuzzy decision D under he given obecives G i and goals C, is he fuzzy se D( ) = min[ inf G ( ), inf C ( )], i N n i N for each P, where P is he universe of ossibiliies. [0], [2] If u i and v are non-negaive weighs such ha m n m + u i v i = = =, hen he fuzzy decision is obained as 27

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 n In he roosed sysem he rules have he form: D( ) = u G ( ) + v C ( ). m i i i= = IF (reurn K is high) AND (risk K 2 is low) AND (q-raio K 3 is big), THEN (Q-measure is good) There are 24 rules wih resecive weighs ha are imlemened in he sysem. [2] Alhough hese rules adequaely describe he mos imoran ossible siuaions ha migh arise in he rocess of invesmen decision-making, he lis of fuzzy rules can be exended wihou changing he sysem s archiecure. As a defuzzificaion mehod, he mehod of cenre of graviy has been chosen: d CoG ( D) = + + x. D D ( x) dx ( x)dx and hus a cris value for he asse qualiy is obained as an ouu of QFLM. Esimaed value for each asse is hen recorded in he daabase and used for managing individual asses or for consrucing invesmen orfolios. Thus a cris value for he asse qualiy is obained as an ouu of QFLM. Esimaed value for each asse is hen recorded in he daabase and used for managing individual asses or for consrucing invesmen orfolios. 4. INDIVIDUAL ASSET MANAGEMENT The cris values of Q-measure for each asse, obained in QFLM, can be used o suor invesors in he rocess of decision making. The Q-measure of an asse reflecs hree variables: reurn, risk and heir raio. In economics, i is believed ha reurn could always be raded for risk, i.e. higher reurns always bring more risk and vice versa. Emirical resuls confirm his belief. [4], [5] However, for he non-seculaive invesor i is essenial o wha exen hese wo characerisics (reurn and risk) are sable over ime. The roosed model for he assessmen of individual asses uses one addiional characerisic: he q-raio which is a quoien of reurn and risk and reflecs he exen o which he aken risk is usified by adequae reurns. The conduced emirical ess for all asses lised on BSE, for differen eriods of ime show ha Q-measure is a roer indicaor of he qualiy of he asse over ime. Some of he resuls are resened in 6. 28,

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 If he Q-measure is less han 0.4 (whaever reurn and risk) a dramaic decrease of rice occurs in u o abou 3 monhs. A he same horizon and a Q-measure beween 0.4 and 0.6 he rice of he asse does no change significanly and even if i increases, he ransacion coss will exceed he oenial benefis. When he Q-measure is greaer han 0.6 he asse rice increases seadily and such an asse is considered suiable for urchase. 5. PORTFOLIO CONSTRUCTION MODULE (PCM) RESULTS FROM QFLM ASSET SELECTION COMBINATIONS GENERATOR PORTFOLIO CONSTRUCTOR DATA BASE PORTFOLIO ALLOCATION Figure 8. Scheme of PCM The scheme of PCM is shown on Fig. 8. Inu daa for PCM are received from he daabase afer QFLM has finished is work. Firs, in Asse Selecion all he asses are sored in descending order by heir Q-measure. Then several asses are seleced wih resec o a crierion se by he invesor, e.g. maximum number of asses or oher. Then all ossible combinaions of hese asses are generaed and hey form corresonding orfolios. If a subsanial amoun of caial is unused hen a rocedure for allocaion akes lace. This rocedure modifies he orfolio o he oimal reducion of unused caial. Consruced orfolios are sored in a daabase and he invesor can make a decision. Le A { A A ; A ; ; } = be a se of financial asses. The goal is o consruc invesmen ; 2 3 K A M orfolios (subses of A) according o he invesor s references. Here he case of max number of asses will be described. If he invesor wans no more han m asses in his orfolio, hen he firs m asses from he sor lis are exraced. Then hese asses are combined in all ossible ways o ses wih, 2,..., m 29

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 elemens. The number of hese ses is 2 m, because he emy se is no aken ino consideraion. Now for each of hese ses a orfolio is consruced. The share of asse A is denoed by x and equals x = n Q = Q, where n is he number of financial asses in he orfolio, n =,2, K, m, and Q is he Q-measure of A. The number of shares n, which are o be urchased from he asse A is calculaed as: n x. S =, P where x is he quoa he asse A, P he rice of one share of A and S is he caial (oal sum o be invesed). Due o he cardinaliy roblem, a his sage S is no fully used. Thus, he used caial is: u N S = n. P = and he remaining caial is S-S u. In case S-S u exceeds a given ercenage of he iniial caial an allocaion rocedure akes lace. In he firs se of his rocedure, he remaining caial is comared o he rice of he asse wih he highes Q-measure. If S-S u is bigger han ha rice hen addiional number of shares of his asse is added o he orfolio. Oherwise, he comarison is reeaed for he nex iem in he sor lis. This rocess is reeaed unil he remaining caial is small enough and no more shares can be bough. A each se, he number of addiional shares n of an asse is a x. n a = ( S S ) Finally, orfolio reurn R, risk σ and q-raio q are calculaed for all 2 m N N R R = x. r, σ = x. s, q =. σ = = P u. orfolios: Now each orfolio is u hrough QFLM in order o obain is Q-measure. The orfolios wih heir characerisics are sored in he daa base. 6. RESULTS In his secion some of he es resuls are demonsraed. 30

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 6.. Individual Asse Managemen As discussed in secion 4, he resuls of QFLM can be used for managing an individual asse. Four asses wih differen Q-measures are seleced. 6... 4EC On 5.02.202 he asse 4EC has Q-measure of 0.8 and as can be seen on Fig. 9 is rice seadily increases in he nex 3-4 monhs. 6..2. E4A Figure 9. One-year change in rice of 4EC On 5.02.202 he asse E4A has Q-measure of 0.8. Fig. 0 shows ha is rice decreases in he nex 3-4 monhs. Figure 0. One-year change in rice of E4A 3

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 6..3. 55B On 5.02.202 he asse 55B has Q-measure of 0.39 which indicaes no good qualiy. However, as i can be seen on Fig. in he nex 3-4 weeks is rice rises significanly and so does is Q- measure: 0.52, 0.7 and 0.82. Therefore, he sysem canno redic for a long eriod of ime, bu as far as i is sared daily i can quickly enough deec change in asse rice behavior. 6..4. 6F3 Figure. One-year change in rice of 55B On 5.02.202 he asse 6F3 has Q-measure of 0.24 which indicaes bad qualiy. As i can be seen on Fig. 2 in he nex 3-4 monhs is rice flucuaes, so he Q-measure is a rused indicaor in his case again. 6.2. Porfolios Figure 2. One-year change in rice of 6F3 The following resuls are obained afer ess conduced on 20.06.202 on Bulgarian Sock Exchange. The iniial caial is fixed. 32

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 6.2.. Two mehods of orfolio consrucion The firs orfolio consiss of 25 asses wih quoas deending on heir Q-measure. The BSE code, rice, number of shares, and characerisics are resened on Table. Iniial caial is BGN 00 000. This orfolio has reurn R =. 4652573, risk σ = 0.085322 and Q-measure=0.7583479. The unused caial is BGN 232.27. Table. Invesmen orfolio of 25 asses wih quoas deending on heir Q-measure BSE code Price Number of shares Quoa Risk Annual norm of reurn Q-measure 3JU 45 40 0.0633698 0.02493.99408284 0.8299283 5ORG 90 68 0.0620385 0.00247399 0.8245 6A6.74 3565 0.06203792 0.0237877.85524372 0.8244236 BLKC 0.47 37 0.06203554 0.0222776 2.5288465 0.8249 4EC.9 3247 0.0620326 0.02754392 2.86830926 0.8237285 SO5.846 3360 0.0620397 0.00874433.8275862 0.8236439 5BD 0.754 8225 0.0620702 0.0087467.0485752 0.82686 5BN 3.95 566 0.0686838 0.06723.2027027 0.8022204 57B 50 23 0.0679298 0.03305052 2.6875 0.80923458 55B 42.5 45 0.067658 0.027792.9634966 0.8088787 4L4 40 43 0.05733093 0.0049 0.8574286 0.75080002 VX 2.73 209 0.0330793 0.0839974.992875 0.4323995 4BJ.88 2767 0.0328755 0.02609853.22826087 0.43053437 5BU 0.5 6447 0.03223697 0.044866 0.9880478 0.4227205 6F3 0.244 456 0.02795274 0.08326.0373444 0.3660662 5H4 4 77 0.0247837 0.095233 0.8965572 0.32455794 6C4P.707 243 0.0223304 0.075938 0.7095238 0.27806573 3NB 27 78 0.027488 0.039949 0.5965385 0.277304 E4A 3.03 67 0.020235 0.0936448 0.6554054 0.26494962 5V2 5.55 356 0.097827 0.0385523 0.9205263 0.25906524 6AB 47 4 0.096254 0.0235358 0.748632 0.2569669 3JR 2.25 856 0.092648 0.0357075 0.4333093 0.2522899 4I8.46 246 0.08253 0.024523 0.6856676 0.2384963 AO0.493 065 0.0590433 0.03634636 0.73777506 0.2082844 3MZ 0.70 2202 0.054427 0.0332062 0.59309494 0.2022736 33

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 If an invesmen orfolio consiss of he same 25 asses wih equal quoas of 0.04 hen he reurn is R =.223736 (lower), he risk isσ = 0.099578 (slighly higher) and he Q-measure is 0.4444046 (lower). The unused caial is BGN 20.65. The only advanage of he orfolio wih equal quoas is he smaller amoun of unused caial. Now, le us ake only he firs en asses and consruc a orfolio (Table 2) wih quoas deending on heir Q-measure. Now he reurn is R =.832328, he risk is σ = 0.07806 and he unused caial is BGN 37.89. Table 2. Invesmen orfolio of firs 0 asses wih quoas deending on heir Q-measure BSE code Price 3JU 45 226 5ORG 90 6A6.74 574 BLKC 0.47 22 4EC.9 5230 SO5.846 54 5BD 0.754 3246 5BN 3.95 2522 57B 50 99 55B 42.5 234 Number of shares The resuls show ha an invesmen orfolio wih quoas, deending on he Q-measure, shows higher Q- measure desie he fac ha some of he asses in he orfolio have low Q-measure. For a orfolio wih equal quoas he Q-measure is smaller as i consiss of significanly more shares of asses wih low Q-measure. However, he bes choice is o ick he bes asses. 6.2.2. Allocaion rocedure To sudy he relaionshi beween he size of he invesmen caial and he need o imlemen addiional allocaion rocedure, orfolios of seven asses have been consruced. These orfolios consis of he same asses, ye wih differen amouns of invesmen caial. I is imoran o noe ha an addiional allocaion rocedure does no aly if unused caial is less han 0.5% of he iniial caial. This ercenage is used o avoid ransacion coss, whenever unused caial is relaively small. Iniial caial BGN,000. Iniial orfolio (Table 3): reurn R =.7838324, risk σ = 0.059687, Q-measure 0.8230537 and unused caial BGN 66.58. 34

BSE code Price Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 Table 3. Iniial orfolio wih 7 asses, S u =BGN 66.58 Number of shares Quoa Risk Annual norm of reurn Q-measure 3JU 45 3 0.4538565 0.02493.99408284 0.8299283 5ORG 90 0.4245006 0.00247399 0.8245 6A6.74 8 0.4244872 0.0237877.85524372 0.8244236 BLKC 0.47 302 0.4244326 0.0222776 2.5288465 0.8249 4EC.9 74 0.4243653 0.02754392 2.86830926 0.8237285 SO5.846 77 0.4243505 0.00874433.8275862 0.8236439 5BD 0.754 88 0.4240073 0.0087467.0485752 0.82686 Originally, he orfolio is consruced wih a high ercenage of unused funds (over 0.5%). To reduce his rae, addiional allocaion rocedure is alied. Porfolio afer he firs allocaion rocedure (Table 4): reurn R =.79868443, risk σ = 0.055607, Q-measure 0.822824 and unused caial BGN 5.97. BSE code Table 4. Porfolio wih 7 asses afer he firs allocaion, S u =BGN 5.97 Price Number of shares Quoa Risk Annual norm of reurn Q-measure 3JU 45 3 0.35 0.02493.99408284 0.8299283 5ORG 90 0.09 0.00247399 0.8245 6A6.74 87 0.538 0.0237877.85524372 0.8244236 BLKC 0.47 325 0.53075 0.0222776 2.5288465 0.8249 4EC.9 79 0.5089 0.02754392 2.86830926 0.8237285 SO5.846 82 0.5372 0.00874433.8275862 0.8236439 5BD 0.754 202 0.52308 0.0087467.0485752 0.82686 I can be noed ha he risk of his orfolio is less han he risk of he iniial orfolio, he reurn is higher and he Q-measure is slighly smaller. The unused caial is reduced o BGN 5.97, bu an addiional allocaion rocedure is sill ossible. Porfolio afer he second alicaion of he addiional allocaion (Table 5): reurn R =.8988325, risk σ = 0.0574007, Q-measure 0.822887 and unused caial BGN 4.64. 35

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 Table 5. Porfolio wih 7 asses afer he second allocaion, S u =BGN 4.64 BSE Number Annual norm Price Quoa Risk code of shares of reurn Q-measure 3JU 45 3 0.35 0.02493.99408284 0.8299283 5ORG 90 0.09 0.00247399 0.8245 6A6.74 88 0.532 0.0237877.85524372 0.8244236 BLKC 0.47 33 0.5590 0.0222776 2.5288465 0.8249 4EC.9 80 0.528 0.02754392 2.86830926 0.8237285 SO5.846 83 0.5328 0.00874433.8275862 0.8236439 5BD 0.754 206 0.55324 0.0087467.0485752 0.82686 Afer consrucing orfolios wih various invesmen caial a conclusion can be derived. I can be observed ha unused caial is significanly larger if he raio of caial/maximum rice is small. The roorion of unused caial decreases wih he increase of invesmen caial or, in oher words, he more iniial caial one has, he less ar of i remains unused (Table 6). Table 6. Correlaion beween he amoun of invesmen caial and he unused caial Caial/max rice er share Unused caial 000/90 =.() 6.658 % 0000/90 =.() 0.926 % 00000/90 =.() 0.597 % 6.2.3. Consrucion of all ossible orfolios wih redefined maximum number of asses The roosed sysem rovides he invesor wih he ooruniy o selec orfolios from all ossible orfolios wih redefined maximum number financial asses. These orfolios are consruced by he combinaion consrucor. If he bes five asses are seleced hen 3 invesmen orfolios are consruced. Each of hem is reresened as a oin on he grah shown in Fig. 3, invesmen risk being loed on he x axis, and he reurn being loed on he y axis. reurn 2.5 2.5 asse 2 asses 3 asses 4 asses 5 asses 0.5 0.005 0.0 0.05 0.02 0.025 risk Figure 3. Porfolios wih differen numbers of asses 36

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 All orfolios lie along an almos sraigh line and his is no incidenally. A basic idea in economics is ha reurn is roorional o risk. Invesors can choose any orfolio deending on heir reference. For examle, if hey refer an asse orfolio wih only one asse, hey have five differen choices. Or if hey refer a orfolio wih fixed reurn hey have several choices again. 7. CONCLUSIONS AND FUTURE WORK In his aer a real ime sofware sysem for managing individual asses and orfolio invesmens is roosed. This sysem is based on he Q-measure of an asse. The Q-measure incororaes reurn, risk and heir raio, and being modelled wih fuzzy logic ools i inuiively reflecs he rocess of invesmen decisions in economic environmen wih enormous amoun of daa, which is ofen uncomleed and imrecise. The sysem is esed wih real daa from BSE. Maor difference from exising similar sysems is ha here are no requiremens for robabiliy disribuions of reurns. In addiion, he sysem rovides a rocedure for orfolio allocaion ha aims a maximal ossible use of invesmen caial. Alhough i is no based on an oimizaion algorihm i solves he cardinaliy roblem in orfolio managemen. Finally, one advanage of he sysem is ha he invesor can chose beween several orfolios. The sysem does no deermine he bes orfolio bu allows he invesors o base heir decision based on financial marke informaion rovided by he model and on heir ersonal references. The auhors lan o develo he sysem in wo direcions: o incororae a neural nework for auonomous adusing he arameers of he fuzzy erms, and second, o include a module for arsing oher sock exchanges. REFERENCES [] Cambell J., Lo A., MacKinlay C., (997) The Economerics of Financial Markes, Princeon, [2] Cover Th., (984) An Algorihm for Maximizing Execed Log Invesmen Reurn, IEEE Transacions on Informaion Theory, Vol. 30, No 2 [3] Judge, G. G., Hill, R. C., Griffihs, W. E., ec., (988) Inroducion o he Theory and Pracice of Economerics, New York, Wiley [4] Georgieva P., (2006) Analysis of Probabiliy Disribuion Characerisics of Finance Asses, Modern Managemen Pracices IV, BFU [5] Georgieva P., (2009) Disribuion of Financial Asses Characerisics, SIELA, vol., ISBN 978-954- 323-530-8 [6] Hanke J., Reisch A., (990) Undersanding Business Saisics, Homewood [7] Holon Gl., (2004) Defining Risk, Financial Analys Journal, Vol. 60, CFA Insiue [8] Leland H., (998) Beyond Mean-Variance: Risk and Performance Measures for Porfolios wih Nonsymmeric Reurn Disribuions, Berkley [9] Nawrocki D., (992) Managemen Using Porfolio Theory Techniques and he PMSP, Financial Markes and Porfolio Managemen, No. 2 [0] Peneva, V., Pochev I., (200) Fuzzy muli-crieria decision making algorihms. - Com. Rend. Acad. Bulg. Sci., Vol. 63 [] Pochev I., Georgieva P., (2008) Fuzzy Aroach for Solving Mulicrieria Invesmen Problems, Innovaive Techniques in Insrucion Technology, E-learning, E-assessmen and Educaion, Sringer [2] Zadeh, L.; Bellman, R., (970) Decision-making in A Fuzzy Environmen. Managemen Science 37

Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 Auhors PENKA VALKOVA GEORGIEVA P. Georgieva is an assisan rofessor and lecor in Mahemaics in he Faculy of Comuer Science and Engineering in Burgas Free Universiy, Bulgaria. P. Georgieva is working on her PHD hesis and has resened more han 6 aers a Bulgarian and inernaional conferences and ournals in he area of Fuzzy logic and sysems. IVAN PETKOV POPCHEV I. Pochev is a Full Member of he Bulgarian Academy of Sciences. I. Pochev has ublished 30 books. I. Pochev has resened and ublished more han 350 aers a world congresses, inernaional symosia and in inernaional ournals (eriodicals) in English, German, Bulgarian and Russian in he areas of Conrol Theory, Informaics and Decision Theory. 38