The efficient frontier for a portfolio that includes one risk-free asset

Transcription

1 Recent Researches n led Mathematcs, Smulaton and Modellng The effcent fronter for a ortfolo that ncludes one rsk-free asset Florentn Serban, Mara Vorca Stefanescu, Slva Dedu bstract Ths aer resents a descrton of the effcent fronter of a ortfolo comosed by three assets, ncludng a rsk-free asset We use a data analyss method to obtan two classes of assets and then we estmate the rsk of each asset corresondng to each class Thus, we get the best two assets among those consdered rsky for whch we buld the effcent fronter f we consder that the ortfolo conssts of these two rsky assets and a rsk-free asset The orgnalty of our study les n the combnaton of classfcaton theory wth rsk estmaton theory to determne the best assets To llustrate the effectveness of the method used, we resent a case study that refers to the domestc fnancal market We construct the effcent fronter, based on the correlaton of the best stocks that we have obtaned through data analyss (for classfcaton), and by assessng the loss dstrbuton (for rsk assessment), takng nto account that the ortfolo contans an asset wthout rsk Keywords effcent fronter, otmzaton, ortfolo selecton, rncal comonent analyss, rsk estmaton INTRODUCTION Fnancal assets ortfolo otmzaton s an mortant area, whch develoed from Markowtz's mean-varance theory We roose to solve the roblem n two stages: assets selecton and rsk estmaton The selecton of assets s realzed by alyng rncal comonents analyss n order to reduce the number of characterstcs of the assets to be taken nto account Methodology based on clusterng technques s a useful tool for understandng and detectng the structure and the herarchy n the fnancal data The dea to obtan clusters that characterze a set of assets can be found also n Kask et al[], Stefanescu et al [7], Mantegna [5], Brda and Rsso [] who aled clusterng technques to classfy the stocks of Mlan and Frankfurt Stock Exchanges usng the Pearson correlaton coeffcent In the second stage, we wll resent an aroach to estmatng rsk usng hstorcal smulaton method Then we wll construct the effcent fronter of the roftablty of a ortfolo consstng of the best two rsky assets that we obtan and a rsk-free asset t the end we solve a case study for the stocks lsted on the Bucharest Stock Exchange We wll buld a ortfolo and we construct the effcent fronter THE SELECTION OF SSETS SSETS CLUSTERING In the context of nowadays fnancal markets t s a huge amount of avalable fnancal data It s therefore very dffcult to make use of such an amount of nformaton and to fnd basc atterns, relatonshs or trends n the data We aly data analyss technques n order to dscover nformaton relevant to fnancal data, whch wll be useful durng the selecton of assets and decson makng Consder that we have collected nformaton on a number N of assets, each wth P characterstcs, whch reresent varous fnancal ndcators, stll called varables Denote by x the -th varable corresondng to the asset Prncal comonent analyss (PC) nvolves a mathematcal rocedure that transforms P varables, usually correlated, nto a number of P uncorrelated varables, called rncal comonents We use clusterng technques n order to fnd smlartes and dfferences between the assets under consderaton The dea of clusterng s an assgnment of the vectors X, X,, X N n T classes C, C,, CT PHSE ESTIMTION RISK Denote by P (t) the closng rce corresondng to the asset at tme t We defne the loss random varable corresondng to the asset, for [ t, t+ ] as: L ( t) R ( t) ln P ( t) ln P ( t+ ),, N Usng Rockafellar et al [6], we wll defne the VaR rsk measure corresondng to the loss random varable L Let we denote the robablty of L not to exceed a threshold z R by G ( z) P( L z) L The Value-at-Rsk of loss random varable L assocated wth the value of asset ncome and corresondng to the robablty level α (0,) s defned by: ( L ) mn{ z G ( z) } VaRα R L α One of the most frequently used mehods for estmatng the rsk s the hstorcal smulaton method The great advantage of ths method s that t makes no assumton of robablty dstrbuton, usng the emrcal dstrbuton obtaned from analyss of ast data Dsadvantage of ths method s that t redcts the future develoment based on hstorcal data, whch could lead to naccurate estmates f the trend of the ast no longer corresonds If L s the loss random varable we can rove that n VaR( L ) mn z R I α { L z}, where I n reresents the ndcator functon of the set L ISBN:

2 Recent Researches n led Mathematcs, Smulaton and Modellng THE EFFICIENT FRONTIER OF THE RENTBILITY OF PORTFOLIO CONSISTING OF TWO RISKY SSETS ND RISK-FREE SSET ssume there are assets on the market The asset has ercentage of rentablty r, where the average M ( r ) µ and standard devaton of r For,, ;, assume that we knows the covarance Total correlaton matrx s (,) V We denote by,, x - the weght of the actve wthn the ortfolo The random varable whch gves the roftablty of the ortfolo: r x r Defnton : ortofolo s called roftable f among all the ortofolo wth the same standard devaton of the rentablty, has the best average Defnton : The set of roftable ortofolos s called the fronter of rentablty (Markowtz, 990) Consder that the weghts x,, of the assets wthn the ortofolo are ostve and that the rsks r,, of the actves have the averages µ,, and devatons,, Therefore, the weghts{ x }, satsfy the relatons : x > 0 ;, x ; x n n m µ ; x x x, In order to determne the otmal{ } n, we use the ( mn) nonlnear model: x x µ m, x x x > 0;, - The effcent fronter of the ortfolo n the ( ; ) lane s a arabola: m Remarks: Only the segment (C) s a fronter of rentablty: a rudent nvesttor cannot accet the segment (BC) because any ortofolo on the segment (BC) s strctly domnated by a ortofolo on the segment (C) whch has the same rsk but a better roftablty 4 PPLICTION OF OPTIMIZTION OF PORTFOLIO OF STOCKS ON BSE 4 FINNCIL RTIOS USED IN THE STOCKS EVLUTION We wll resent the fnancal ndcators that we wll use n our study: - PER ndcator (net ncome er stock) s calculated by dvdng the current market rce to the value of net roft er stock for the ast four consecutve quarters - The P / BV (book value of stock) s calculated by dvdng the current tradng rce to book value er stock determned accordng to the latest fnancal reortng, - The rato of value traded n last 5 weeks and market catalzaton; the reort shows the lqudty of the stock - Evoluton of rce: to observe the rce level at a gven tme we take nto acount the maxmum rce and mnmum rce acheved n the last 9 months - Dvy ndex measures the erformance of dvdend and s calculated as the rato between the amount of the dvdend and book value or market value of the stock We used nformaton on a total of 60 stocks reresentng stocks of Class I and II, traded on the Bucharest Stock Exchange on 0 Then we consdered only the stocks for whch t s ossble to calculate most the ndcators mentoned resultng n 40 stocks Snce the Bucharest Stock Exchange s not mature enough, we can not afford to use a sngle fnancal ndex, such as, for examle, the closng rce So we take nto account several characterstcs for each stock, we use data analyss technques n order to rocess ths vast amount of nformaton We consder for each asset the values of fve characterstcs descrbed above on 0 µ C B ISBN:

3 Recent Researches n led Mathematcs, Smulaton and Modellng TBLE THE VLUE OF THE 5 CHRCTERISTICS No Smbol PER P/BV DIVY Mn/ P Max/P FP SIF SIF BRD TLV BVB BCM SIF SIF SIF DFR TEL COMI TGN BRK SNP TB ZO BIO PREH LR SNO SCD VESY OIL MO COTR RMH SPCU CEON SIF PTR RPH LT MPN ELJ CMP ROCE LU CGC SOURCE: WWWBVBRO 4 PRINCIPL COMPONENT NLYSIS We aly data analyss technques to dscover the smlartes and dfferences between the stocks of the Bucharest Stock Exchange, usng the ackage XL STT 8 Fgure contans the tree resulted from PC Stocks belongng to the same cluster are smlar n terms of characterstcss taken nto account In order to buld a dversfed ortfolo, we frst choose the number of clusters (for our study, we chose ), whch wll be taken nto account FIGURE GROUP OF STOCKS SOURCE: PCKGE OF PROGRMMES XL STT Remarks: We observe the classes n whch the stocks were groued Those classes are resented stll 4 RISK ESTIMTION We used the closng rce values daly for each stock, corresondng to a tme horzon of 50 days to measure VaR for each stock We used the data avalable on the Bucharest Stock Exchange from January - February The followng tables contan values of VaR for each stock and three levels of robablty values TBLE VR FOR ECH STOCK Class RS MO BCM BIO BRK BVB COTR CGC DFR ELJ EFO FP MPN PTR ROCE RPH SIF SIF SIF SIF SIF SNO TGN VESY I ISBN:

4 Recent Researches n led Mathematcs, Smulaton and Modellng Class LT LU LR ZO LR TB BCM BRD BRM CEON CMP COMI OIL PREH RMH SCD SOCP SRT SNP 0, SPCU TEL TLV TUFE TRP VNC SOURCE: PCKGE OF PROGRMMES XL STT 44 CONSTRUCTION OF N OPTIML PORTFOLIO MDE OF STOCK We start from the classes we formed above and we choose from each of the stock whch has mnmal VaR for the robablty 099; we get the two stocks (TLV, BIO) whch, together wth the rsk-free asset, wll form the ortfolo to whch we wll buld the effcent fronter 45 EFFICIENT FRONTIER FOR PORTFOLIO OF TWO RISKY SSETS ND RISK-FREE SSET We selected the closng rces for the erod for those selected stocks: TLV and BIO We wll transform the rces n anual rentablty by the formula (fnal rce-ntal rce)x60/ ntal rce and we get : : X TLV X BIO : µ and devaton 6 ; µ 7 and devaton 4 7 ; cov We consder µ 8 and devaton 0 We have cov 0 ; cov 0 Let xyz be the weghts of the stocks Followng the relatons from secton we obtan: m x + 7 y; 56 x + y + xy We therefore have the followng roblem of nonlnear otmzaton: Mn 56 x + y + 4 xy x + 7 y + 8 z m x + y + z x, y,z > 0 () () () We relace x and y n () and () n the goal functon and we get : 960z - (64-7m)z + m -86 m +48 Mnmum of ths functon s acheved for 64 7m z m 960 We relace z n the goal functon and we get 50 m - 70 m + 47 In the roftablty-rsk lane the revous relaton s the fronter of roftablty for the ortfolo made of the BIO, TLV and one rsk-free asset Grahcly, we get the reresentaton below, wth the remark that the efcent frronter s ust the curve C µ C B CKNOWLEDGMENT Ths work was suorted from the Euroean Socal Fund through Sectoral Oeratonal Programme Human Resources Develoment 07-, roect number POSDRU/89/5/S/5984 Performance and excellence n ostdoctoral research n Romanan economcs scence doman We aly the formulae secfc to statstcs and we obtan: ISBN:

5 Recent Researches n led Mathematcs, Smulaton and Modellng REFERENCES [] Brda, J G, Rsso, W, Herarchcal Structure of the german stock market avalable on-lne at htt://ssrncom/,07 [] Fulga, C, Dedu, S, Şerban, F, Portfolo Otmzaton wth Pror Stock Selecton, Economc Comutaton and Economc Cybernetcs Studes and Research: 4, 09, 57-7 [] Kask, K, Onnela, JP, Chakrabort, Dynamcs of Market Correlatons: Taxonomy and Portfolo nalyss, Physcal Revew E, 68, 0 [4]Larsen, N, Mauser, H, Uryasev, S, lgorthms for otmzaton of value-at-rsk, Fnancal Engneerng E- Commerce and Suly Chan, 0, 9-57 [5] Mantegna, RN, Herarchcal Structure n Fnancal Markets The Euroean Physcal Journal B,, 999, 9-97 [6] Rockafellar, T, Uryasev, S, Otmzaton of Condtonal Value-at-Rsk, Journal of Rsk,, 00 [7] Ştefănescu, V, Şerban F, Buşu, M, Ferrara, M, Portfolo Otmzaton usng Classfcaton and Functonal Data nalyss Technques, Economc Comutaton and Economc Cybernetc Studes and Research, 44, 0, 9-08 [8] htt://bvbro/ ISBN: