USEFULNESS OF BOOTSTRAPPING IN PORTFOLIO MANAGEMENT

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1 USEFULNESS OF BOOTSTRAPPING IN PORTFOLIO MANAGEMENT Radovaov Bors Faculty of Ecoomcs Subotca Segedsk put 9-11 Subotca E-mal: Marckć Aleksadra Faculty of Ecoomcs Subotca Segedsk put 9-11 Subotca E-mal: Abstract Ths paper cotas a comparso of -sample ad out-of-sample performaces betwee the resampled effcecy techque, pateted by Rchard Mchaud ad Robert Mchaud (1999), ad tradtoal Mea-Varace portfolo selecto, preseted by Harry Markowtz (1952). Based o the Mote Carlo smulato, data (samples) geerato process determes the algorthms by usg both, parametrc ad oparametrc bootstrap techques. Resampled effcecy provdes the soluto to use ucerta formato wthout the eed for costras portfolo optmzato. Parametrc bootstrap process starts wth a parametrc model specfcato, where we apply Captal Asset Prcg Model. After the estmato of specfed model, the seres of resduals are used for resamplg process. O the other had, oparametrc bootstrap dvdes seres of prce returs to the ew seres of blocks cotag prevous determed umber of cosecutve prce returs. Ths procedure eables smooth resamplg process ad preserves the orgal structure of data seres. Key words: Resampled effcecy, Mea-Varace portfolo, parametrc bootstrap, oparametrc bootstrap, Captal asset prcg model 1. INTRODUCTION The purpose of ths paper s to develop a more robust methodology for asset allocato for the stock vestmet markets whch takes more serously to accout heret valuato ad data ssues. Ths cludes the tegrato of mea-varace optmzer usg resampled data puts, passve vestmet 68

2 maagemet, the selecto of approprate asset classes ad tme rebalacg techque to esure that the portfolo remas alged wth the dyamc ature of stock markets. The proposed methodology wll prove to be useful for makg asset allocato decsos, especally hghly volatle facal markets. The chose bootstrap procedure selectvely resamples the retur tme seres by matag the ecoomc cycle. After costructg resampled effcet portfolos, the research process resumes wth comparso made o the tradtoal Mea-Varace portfolo optmzato problem. 2. MARKOWITZ EFFICIENCY Sxty years ago Harry Markowtz (1952) developed the portfolo selecto theory that became a foudato of facal ecoomcs for asset maagemet ad revolutozed vestmet practce. Markowtz otced well a basc premse for hs theory that all ecoomc decsos are made upo trade-offs. I stuato of vestmet selecto, a trade-off, rsk versus expected retur, s observed. The theory exteds the techques of lear programmg to develop the crtcal le algorthm. Metoed algorthm detfes all feasble portfolos that mmze rsk for a gve level of expected retur ad maxmze expected retur for a gve level of rsk order to form a set of portfolos graphcally preseted as the effcet froter. Showg the level of dversfcato portfolo selecto, the effcet froter dcates the mportace of achevg rsk reducto. However, the portfolo selecto s based o assumpto that vestmet decso oly depeds o 2 expected value E(R p ) ad varace p of the total portfolo retur. Fallowg ths backgroud, the portfolo optmzato procedure requres the kowledge of E(R ) as the expected retur of the asset, σ as the stadard devato of the retur of the asset, ρ j as the correlato betwee the returs of the assets ad j for, j 1,...,, ad σ j as the covarace betwee two asset or securty returs. Cosequetly, the classcal Mea-Varace optmzato model s preseted followg form: M st 1 j 1 1 E( R ) x M 1 x x x x j 0 1 j 1,..., (1) 69

3 The formulato (1) dcates a covex quadratc programmg problem where s deote by M the requred level of retur for the portfolo ad by x the fracto of a gve captal to be vested each asset. 3. RESAMPLED EFFICIENCY Wth ofte use of Markowtz portfolo selecto procedure t became obvous that problems creatg adequate composto are occurrg extreme portfolo weghts, a ubalaced asset allocato ad a lack of dversfcato. I fact, the composto of optmal portfolos s very sestve to chages expected returs, varaces ad covaraces. Tedg to pck those assets wth most attractve features ad to short or deselect those wth worst features are exactly the cases whch estmato error s lkely to be hghest. Hece, the process maxmzes the mpact of estmato error o portfolo weghts ad decreasg out-of-sample performaces. There are several attempts made to reduce estmato errors ad mprove portfolo performace. Ths paper embraces the resampled effcecy techque pateted by Rchard Mchaud ad Robert Mchaud (1999), whch s based o resamplg of portfolo returs to reflect the ucertaty retur process. I order to aalyze the performace of the resampled effcecy, some studes made the comparso betwee mea-varace optmzato ad metoed resampled effcecy. I may of those studes Mchaud s procedure outperforms the approach of Markowtz. I smulato studes of Mchaud ad Mchaud (2008) ad also Markowtz ad Usme (2003) was foud a strog evdece for better performace of resampled effcecy. However, there are also completely dfferet results. For stace, study of Harvey ad other authors (2008), wth more sophstcated pror dstrbuto ad more approprate algorthm, obta rather balaced results betwee the resampled effcecy ad the optmzato of Markowtz or eve better results usg ther Bayesa estmator. Nevertheless, eve f there are some studes comparg these two techques, each of them cocetrates o a specfc settg, whch rarely leads to geeral recommedatos. Based o the aalyss of the metoed papers the results of Markowtz versus Mchaud are rather balaced ad very sestve to the legth of the estmato horzo wth capablty to gve advce for dfferet tal stuatos of vestmet. Why resampled effcecy? The startg pot to expla the purpose of the resamplg effcecy s a well kow set of rgd assumptos used Markowtz optmzato framework. Thus the utlty fucto ofte became more complex volvg prefereces besde mea ad varace. Istead of fallowg dyamc ature of market Markowtz selecto model maly offers statc optmzato (oeperod optmzato). Wth such a rgd set of costras small chages put assumptos could 70

4 mply large chages the optmzed portfolo. All prevously observed facts magfy estmato errors ad the ed lead to decrease of utlty value of the portfolo selecto model. Therefore, the resamplg effcecy techque has bee preseted to overcome shortcomgs the portfolo selecto procedure. Mchaud (1999) pateted the resampled effcecy 3FTM, but keepg some uderlyg assumptos from the portfolo selecto procedure. Scherer (2002) summarzed ths procedure as fallows: 1) Estmate varace-covarace matrx ad mea vector of hstorcal puts. 2) Resample from puts by takg B draws from put dstrbuto θ (ths paper cludes both, parametrc ad oparametrc bootstrap procedure). The umber of draws reflects the degree of ucertaty the puts. Calculate ew varace-covarace matrx from sampled seres. 3) Calculate the effcet froter from puts derved secod terato ad save optmal portfolo weghts for m equally dstrbuted returs alog the froter. 4) After repeatg step 2 ad 3 B tmes, calculate average portfolo weghts for each retur pot. Recreatg the hstory of tme seres bootstrap procedures mply dfferet output results the stadard portfolo selecto procedure. O the other had, wth cosderato of the ablty to preset a varety of dfferet vestmet solutos, resampled portfolos have desrable characterstcs for vestors. Delcourt ad Pettjea (2011) were elaborated the opo that low degree of dversfcato ad the sudde shfts allocato alog portfolos are udesrable characterstcs of mea-varace portfolo. 4. BOOTSTRAPPING TIME SERIES Bootstrappg s related wth smulato, but wth oe crucal dfferece. Wth smulato, the data are costructed completely artfcally, whle bootstrappg obtas a descrpto of the propertes of estmators by usg the sample data pots themselves, ad volves samplg repeatedly wth replacemet from the actual data. There are two obvous advatages of bootstrap procedure over aalytcal results of tradtoal statstcal methods. Frst, bootstrappg allows the researcher to make fereces wthout makg strog dstrbutoal assumptos. The bootstrap volves emprcally estmatg the samplg dstrbuto by lookg at the varato of the statstc wth sample. Hece, ths procedure treatg the sample as a populato from whch samples ca be draw. Secod, the bootstrap are more robust the the classcal statstcal methods. Therefore, t could be used effectvely TM U.S. Patet #6,003,018 by Mchaud et al., December 19,

5 wth relatvely small samples ad preserved the estmator stablty durg the perods of uexpected volatlty shfts. The bootstrap, orgally created by Efro (1979), begs wth a set of depedet ad detcally dstrbuted (d) observatos wth dstrbuto fucto F ad ukow parameter θ as a fucto of F. The bootstrap methodology allows a approxmato of the dstrbuto of θ uder very geeral codtos ad t s based o obtag a bootstrap replcate of the avalable data set by drawg wth replacemet radom samples from F. A descrbed method s the smplest verso ad s oly vald the case of depedet ad detcally dstrbuted observatos. If the d bootstrap s appled drectly to depedet observatos, the resampled data wll ot preserve the propertes of the orgal data set, provdg cosstet statstcal results. Icludg dyamc correlato ad codtoal heteroscedastcty, Ruz ad Pasqual (2002) offered two, parametrc ad oparametrc, bootstrap procedures recetly developed for tme seres data. There are several versos of parametrc ad oparametrc bootstrap method, but ths paper cotas two most popular, the resdual bootstrap ad the movg block bootstrap method Resdual bootstrap The parametrc bootstrap procedure s based o assumpto that there s always a specfc model sutable eough for tme seres data. I ths case, t s usually ot recommeded to bootstrap from the row data but from the resduals of a gve model. However, t s ecessary to decde whch form of model to be used ad whch resduals to be bootstrapped. Ths paper uses the Captal Asset Prcg Model (CAPM) for estmatg returs of observed tme seres as the fallowg regresso equato: E( R, t ) R f ( E( RM, t ) R f ) ut (2) where, E(R,t ) securty expected retur based o a cocept of the radom varable shows the weghtedaverage retur of -th securty observed tme t R f rsk free rate E(R M,t ) market expected retur, calculated from tme seres of the BELEX15 stock exchage dex returs α slope coeffcet β measure of sestvty to a movemet the overall market 72

6 After defg the form of estmato model, the resdual bootstrap procedure cotas followg four steps: 1) Estmate the model o the actual data, obta the ftted values of depedet varable ad calculate the resduals 2) Take the sample of sze wth replacemet from these resduals ad geerate a bootstrapped depedet varable by addg the ftted values to the bootstrapped resduals ( E( R ) R f ) ( E( R ) R f ) u 3) Regress ths depedet varable o the orgal data E( R M ) R ) to get a bootstrapped coeffcet 4) Go back to stage 2, ad repeat a total of B tmes. ( f Nevertheless, t s mportat to emphasze the other forms of regresso models, partcularly stuatos where some adjustmets are eeded. The parametrc model has to preset a good approxmato of true model. Thus the utlty value of the resdual bootstrap procedure predomatly depeds o approprate model selecto process Movg block bootstrap If the seral depedece of the date s msspecfed, the parametrc bootstrap could be cosstet. Cosequetly, alteratve approaches that ot requre fttg a parametrc model have bee developed to deal wth depedet tme seres data. Kusch (1989) proposed the movg block bootstrap method that dvdes the data to overlappg blocks of fxed legth ad resample wth replacemet from these blocks. Metoed method preserves the orgal structure of tme seres by dog the resample process wth defed blocks. However, the accuracy of the movg block bootstrap procedure maly depeds o optmal block legth selecto. Otherwse, the optmal block legth selecto depeds o sample sze, appled data geeratg process ad chose statstcs of terest. Whe sample sze creases, the block legth must follow the chages order to secure the bootstrap cosstecy ad emprcal dstrbuto fucto. By choosg the optmal block legth t s possble to mmze the mea squared error. The movg block bootstrap method cotas four evtable teratos order to assemble a effcet resample algorthm: 73

7 1) Dvde tme seres data to the equal sze blocks wth overlappg, where frst block cotas the set of X 1,,X l elemets, secod X 2,,X l+1 etc. 2) Do the resamplg process wth overlappg wth defed blocks ad alg resampled block oe bootstrap sample X 1,..., X 3) Estmate the statstcs of terest by usg the costructed bootstrap sample T T ( X,..., X ) 4) Repeat steps 2 ad 3 B tmes to get bootstrap dstrbuto ad probablty of obtag a test statstc ˆ 1 G ( t, F ) P ( T t) I( TN, b t) B B b EMPIRICAL EVIDENCE ON THE BELGRADE STOCK EXCHANGE Cosderg the effects of the facal crses ths paper volves 45 mothly stock prces data from the begg of the year 2009 order to preserve relatve vestmet stablty volated durg the year Wth relatvely small umber of avalable data, the resamplg algorthms fd ther place makg a approprate data set. Sx stocks from the Belgrade stock exchage wth a hgh turover rate comparg wth other tradg stocks are cluded portfolo aalyss. The sx compay stocks deoted by the stock symbols are: IMLK Imlek, BMBI Bamb Baat, MTLC Metalac, AIKB AIK baka, FITO Galeka Ftofarmacja ad GMON Goša motaža I-sample portfolo aalyss At the begg of the metoed aalyss we compute mea-varace effcet froter from the orgal set of puts ad emphasze that oly the weghts computed wth the Markowtz equatos are optmal regardg to orgal set of puts. Followg the frst terato, the aalyss resumes wth the resampled effcecy procedure usg two separate bootstrap algorthms. I combato wth the orgal set of puts, all resampled portfolo weghts wll form froters below the mea-varace effcet froter, dcated Fgure 1. Fgure 1. gves us a startg pot fulfllg the dea o how samplg errors ca effect the determato of a effcet froter. Ths fgure demostrates that eve small chages the sample data ca cause sgfcat chages mea-varace effcet portfolo decso. We otced that the movg block bootstrap effcet froter has the hgher slope coeffcet, whle the mea-varace effcet froter ad ts resdual bootstrap aalogue are approxmately parallel. 74

8 weghts E(R) Croata Operatoal Research Revew (CRORR), Vol. 3, Mea-Varace Effcet Froter Resdual Bootstrap Effcet Froter Movg Block Bootstrap Effcet Froter s Fgure 1: Mea-varace ad resampled effcet froters 100% 90% FITO 80% 70% 60% 50% MTLC 40% 30% BMBI 20% 10% 0% rak Fgure 2: Mea-varace portfolo allocato Durg the optmzato process the allocato problem has become more mportat feature vestmet strategy. Fgure 2. shows that mea-varace optmzato applyg quadratc programmg wth o addtoal costras emphasze weak dversfcato amogst sx selected stocks. Accordg to the mea-varace procedure, oly few termedate raks (raks defe the expected value of stock retur rate) clude three commo stocks, whereas the dversfcato of oly two commo stocks smaller or bgger raks (lowest or hghest expected returs) are show. At the other had, two resampled versos, preseted Fgures 3. ad 4., volve all sx aalyzed stocks. Two resampled portfolo solutos show smoothed trasto allocato alog the resampled froter. There are o sudde shfts weghts accordg to chages expected retur, partcularly the movg block bootstrap example. Show Fgure 4., the resdual bootstrap smoothes the orgal set of data, but keepg the level of average retur per stock t reduces weghts of the small retur 75

9 weghts weghts Croata Operatoal Research Revew (CRORR), Vol. 3, 2012 stocks the hghest raks of portfolo choce. The portfolo wth characterstcs of mea-varace portfolo model s lkely to maxmze samplg errors ad exhbt poor out-of-sample performace. I cotrast to mea-varace allocato, dversfcato s preserved the resamplg procedures where resampled portfolos show a tedecy towards the BMBI commo stock. 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% GMON FITO AIKB MTLC BMBI IMLK rak Fgure 3: Movg block bootstrap portfolo allocato 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% GMON FITO AIKB MTLC BMBI IMLK rak Fgure 4: Resdual bootstrap portfolo allocato Preseted dffereces betwee mea-varace portfolo selecto ad resampled allocato are mplcato of hstorcal varace. They are more lkely to be far from ther hstorcal value amog all dfferet scearos the smulato process. Hece, the rsker s the portfolo, the hgher s the estmato rsk. 76

10 5.2. Out-of-sample portfolo aalyss Out-of sample smulato study s performg to compare the performace betwee mea-varace ad two resampled portfolo allocato strateges. Aalogous wth smulato study elaborated by Delcourt ad Pettjea (2011) we cosder 12 mothly perods ahead for three dfferet expected retur (mmum, termedate ad maxmum retur). Optmal portfolos are computed for each perod, the the sample perod s moved foreword a moth ad optmzato process s repeated. At the begg, we computed the realzed returs geerated by the optmal portfolos. Afterwards, we estmate the average realzed returs ad the average rsk of the portfolos durg the out of sample perod. Therefore, we are capable to compute a average Sharpe rato as the approprate out of sample performace measure. Table 1: Out-of-sample performace measured by Sharpe rato method rak retur rsk Sharpe Meavarace Movg block bootstrap Resdual bootstrap Source: Authors Followg the out of sample methodology we could summarze measured performaces Table 1. Selected strategy vares due to the vestor s requred retur. The movg block bootstrap process shows domace towards hgher expected returs, whereas the resdual bootstrap procedure advatage towards lower expected returs are otced. I ths case, the estmato perod s qute short ad t s therefore reasoable to expect that the effect estmato rsk wll be more sgfcat, favorg the resampled methods. Hece, the larger the sample sze, the better the performace of the meavarace portfolos wth respect to the resamplg procedure. 77

11 6. CONCLUSION Keep md that the dfferece betwee the resampled ad the tradtoal effcet froter arses because resamplg provdes portfolos that are too dversfed. Cocerg Scherer (2002), staces ca occur resamplg whch dversfcato becomes smaller as the maxmum-retur soluto s approached. However, all resamplgs are derved from the same vector ad covarace matrx, where true dstrbuto s ukow. Hece, all resampled portfolos wll suffer from the devato of the parameters. Averagg wll ot help greatly ths case because the averaged weghts are the results of the put vector, whch s tself very ucerta. Portfolo resamplg offers a tutve way to develop tests for statstcal dfferece betwee portfolos. Smulated retur ad rsk help to quatfy the effect o the optmzato process of ucertaty heret the vestmet decso. The comparso betwee mea-varace optmzato ad resampled optmzato shows that resampled strateges lead to more stable ad more dversfed portfolos regardless to trasacto costs. Moreover, t s mportat to otce that there s sgfcat dfferece betwee two resamplg optmzato procedures, but ther commo feature s greater portfolo stablty ad dversfcato over the mea-varace optmzato. The ed result may be useful for cotrollg rsk ad structurg the allocato so that s cosstet wth vestor objectves. REFERENCES Aloso, A., Peña, D., Romo, J. (2004) Itroducg Model Ucertaty Tme Seres Bootstrap, Statstca Sca, Vol. 14, No. 1, pp Bauwes, L., Pohlmeer, W., Veredas, D. (2006) Hgh Frequecy Facal Ecoometrcs: Recet Developmets, Physca-Verlag, Hedelberg Berkowtz, J., Kla, L. (2000) Recet Developmets Bootstrappg Tme Seres, Ecoometrc Revew, Taylor ad Fracs Jourals, Vol. 19, No. 1, pp Brooks, C. (2004) Itroductory Ecoometrcs for Face, Cambrdge Uversty Press Bühlma, P. (2002) Bootstraps for Tme Seres, Statstcal Scece, Vol. 17, No. 1, pp Chou, P., Zhou, G. (2006) Usg Bootstrap to Test Portfolo Effcecy, Aals of Ecoomcs ad Face, Vol. 7, No. 2, pp Cogeau, P., Zakomoule, V. (2010) Bootstrap Methods for Face: Revew ad Aalyss, Delcourt, F., Pettjea, M. (2011) To What Extet s Resamplg Useful Portfolo Maagemet, Appled Ecoomcs Letters, Vol. 18, No. 3, pp Frake, J., Härdle, W., Hafer, C. (2004) Statstcs of Facal Markets: A Itroducto, Sprger- Verlag, Berl 78

12 Harvey, C., Lechty, J., Lechty, M. (2008) Bayes vs. Resamplg: A Rematch, Joural of Ivestmet Maagemet, Vol. 6, No. 1, pp Km, J. (2011) Captal Allocato Usg the Bootstrap, North Amerca Actuaral Joural, Vol. 15, No. 4, pp Küsch, H. (1989) The Jackkfe ad the Bootstrap for Geeral Statoary Observatos, Vol. 17, No. 3, pp Markowtz, H. (1952) Portfolo Selecto, The Joural of Face, Vol. 7, No. 1, pp Markowtz, H., Usme, N. (2003) Resampled Froter Versus Dffuse Bayes: A Expermet, Joural of Ivestmet Maagemet, Vol. 1, No. 4, pp Mchaud, R., Mchaud, R. (2008) Effcet Asset Maagemet: A Practcal Gude to Stock Portfolo Optmzato ad Asset Alocato, 2 d Edto, Oxford Uversty Press Ruz, E., Pascual, L. (2002) Bootstrappg Facal Tme Seres, Joural of Ecoomc Surveys, Vol. 16, No. 3, pp Scherer, B. (2002) Portfolo Resamplg: Revew ad Crtque, Facal Aalsts Joural, Vol. 58, No. 6, pp Srvatsa, R., Smth, A., Lekader, J. (2010) Portfolo Optmsato ad Bootstrappg, Joural of Property Ivestmet ad Face, Vol. 28, No. 1, pp Zobrowsk, A., Cheg, P., Zobrowsk, B. (1997) Usg a Bootstrap to Measure Optmum Mxed- Asset Portfolo Composto: A Commet, Real Estate Ecoomcs, Vol. 25, pp