An asymptotic Markovian approach to the portfolio selection problem

Transcription

1 An asymptotc Marovan approach to the portfolo selecton problem Enrco Angelell, Sergo Ortobell Lozza, an Gaetano Iaqunta. Abstract In ths paper we propose a portfolo choce problem uner the hypothess of Marovan returns. In partcular, we assume stable Paretan strbute returns whch mports a more flexble envronment rather than the tratonal Gaussan moelng. herefore uner these assumptons we perform an ex-post analyss to nvestgate the real beneft of our approach an raw some remarable conclusons. Keywors ynamc portfolo selecton, stable Paretan strbutons, Marov chan, maret stochastc bouns. I I. INRODUCION N ths paper we propose a methoology to optmze portfolo value n a choce problem framewor usng a marovan structure to moel the asset portfolo returns. A comparson between fferent portfolo selecton strateges s prove. he propose methoology s teste n an ex-post analyss an the last crss pero ata are use to assess the gooness of the metho. A normal strbuton of asset returns s a tratonal an basc assumpton n many theoretcal fnancal stues. However, many emprcal stues reject the hypothess that asset returns are normally strbute see the funamental wors of Manelbrot 963 an Fama 965 an, among others, Rachev an Mttn, Rachev et al. 7 an the references theren. Moreover, many fnancal events are consere as real wtnesses of falure of normal strbuton hypothess n the fnancal returns.e. stoc maret crash n October 987, Asan fnancal crss n 997, hghly volatle pero after September,, an the most recent subprme mortgage crss an cret rs crss 8-. herefore a flexblty an statstcal relablty n fnancal moel are requre to cope wth that unrealstc hypothess. Researchers have spent many efforts to mprove methos an propose better moels for fnancal marets. Among the numerous moels propose a frutful research fel appears to be the stable Paretan framewor e.g., Samorontsy an aqqu, 994 whch assumes a fnancal return strbuton Enrco Angelell s wth Unversty of Bresca Department Economcs an Management, Bresca Italy e-mal: angele@eco.unbs.t. Sergo Ortobell Lozza., s wth Unversty of Bergamo Department SAEMQ 47 Bergamo Italy, an VSB-U of Ostrava Department of Fnance 7 Ostrava, Czech Republc. corresponng author to prove e- mal: sol@unbg.t. Gaetano Iaqunta s a prvate research analyst, 7 Rho Mlan Italy emal: gaetano.aqunta@unbg.t. more flexble than the tratonal one. Any portfolo ynamc moel has to tae nto account for: a Heavy tals an asymmetrc shape n returns strbuton. b A multvarate strbuton of unerlyng asset returns an correlaton among asset returns more flexble than the smple Pearson lnear correlaton. c A ynamc portfolo strategy has to be base on the entre sample paths. In ths paper we scuss a portfolo selecton moel for fnancal marets base on these three themes wth a partcular attenton to theme c. In orer to evaluate an estmate the path epenent portfolo strateges we approxmate the return tme evoluton by usng Marovan trees. hs approach, orgnally evelope n the opton theory see Cox et al.979, can be effcently use for portfolo selecton problems as shown by Angelell an Ortobell 9, Iaqunta et al., Angelell et al. 3 an n sever other fnancal fels as scusse by D Amco et al.. In ths framewor the evoluton of the wealth s erve as a non parametrc Marov process. he Marovan approach allows to compute: the statstcal strbuton of any contngent clam, the strbuton of stoppng tmes or frst passage tme see Angelell an Ortobell 9 an Angelell et al., an the jont Marov strbutons of rsy varables. he portfolo selecton strateges base on Marovan trees mport several results obtane n opton theory: path epenent portfolo selecton strateges, arbtrage strateges for hege funs, an strateges base on stoppng tmes of the ranom wealth process. In orer to account of the epenence structure we use the methoology scusse n Ortobell et al. an Angelell et al.. he epenence structure allows us to solve two stnct problems n portfolo choce: Account for the common behavor of the returns n the portfolo choces: funamental n any portfolo choce consstent wth nvestor's preferences, snce every nvestor sorts amssble portfolos wth respect to hs/her preferences an, ong so, he/she shoul account for the common behavor of the returns. Reuce the mensonalty of the large scale portfolo problem: t s well nown that the number of observatons necessary n optmzaton problems ncreases proportonally wth the number of the ranom varables see, among others, Papp et al. 5, Konor et al. 7. herefore, we eal wth the curse of mensonalty. o reuce the mensonalty we use the Issue, Volume 7, 3 936

2 same preselecton approach evelope from Ortobell et al. b whch preselects an aequate number of assets conserng ther forecaste future performance. hen we use a non-gaussan factor analyss that accounts the jont Marov evoluton of returns an ther asymptotc behavor. In the emprcal comparson we analyzes the mpact of some propose portfolo selecton strateges apple to US maret stoc returns ata. he ex-post analyss prove s base on two fferent atasets: the last ten years an the last sx months. he use of the two fferent atasets allows to value the mpact of the most recent frms on portfolo selecton problems. On these assets the Ortobell et al. technques of mensonalty reucton are apple. hen, the optmal portfolos of fferent rewar-rs strateges are etermne. Fnally, t s evaluate the mpact of conserng heavy tals comparng the sample paths of the ex-post wealth obtane from the fferent portfolo strateges. he paper s organze as follows. In Secton II we scuss how moelng return seres an ntrouce a set of performance ratos. Secton III eals wth the ex-post comparson among fferent portfolo strateges. Fnally Secton IV raws some remarable conclusons. II. PORFOLIO SELECION PROBLEM AND SRAEGIES A. A non parametrc Marovan framewor In ths secton we eal wth the returns moelzaton by Marov process wth heavy tale strbutons. e show how to etermne the future wealth strbuton. Let us ntrouce some notaton. e conser a screte sequence of nvestor wealth equally space n tme,,, e.g. ays. he ntal wealth.e. s nveste at tme n n rsy assets. he gross returns on ate t of the n assets are enote as z z,, z ]'. Generally, we assume the t [, t n, t stanar efnton of gross return between tme t an tme s, t,[ t, t] t of asset, as z, t, where s, t s the s prce of the -th asset at tme t an, [ t, t] s the total amount of cash vens pa by the asset between t an t. e stngush the efnton of gross return from the efnton of return,.e., z or the alternatve efnton of log return r, t log z, t, t. he vector x x,, x ]' ncates the, t [ n postons taen n the n assets. Assumng that no short sales are allowe, the vector x of portfolo weghts belongs to the n n n -mensonal smplex S { x x ; x }. he portfolo weght x represents the percentage of wealth nveste n the -th asset. In a ynamc framewor the percentage of wealth nveste n each asset coul change over tme. However, for sae of smplcty, n ths paper we stuy x t x an escrbe all amssble wealth processes t epenng on an ntal portfolo of weghts x S that s assume constant over tme. Moreover, we assume that these wealth processes are aapte processes efne on a fltere,,pr. hus, the gross return of probablty space, t t a portfolo x urng a pero [ t, t ] s gven by n z x, t x' zt x z, t. From a fnancal moel pont of vew we assume that the gross returns have a Marovan behavor an can be moele wth an homogeneous Marov chan. hus, we have to scretze the support of any portfolo. Gven a set x { zx, h h,, H } of H past observatons of the portfolo gross returns, we efne N states enote as [,, N Z x z x zx ]' n the nterval s mn x; max where w.l.o.g. we assume z s x z x for s,, N. In general, the wealth obtane wth the portfolo x S at tme,, s a ranom varable x wth a number of possble values ncreasng as a polynomal of orer N n varable. In orer to eep the complexty of the computaton reasonable, we frst ve the portfolo support mn x; max n N ntervals a x,, ax, where a x, ecreasng wth nex s gven x x / N mn by: ax, max x,,, N max ; then, we compute the return assocate to each state as the geometrc average of the extremes of the nterval a, a that s x, x, N s max x zx ax, sax, s max Yx mn x, s,,, N As a consequence, an the wealth s s s max x zx zxu where u N mnx x obtane along a path after steps.e. at tme can only assume N stnct values nstea of O N. e enote such property as the recombnng effect. hans to the recombnng effect of the wealth x, the possble values of x up to tme,, can be store n a matrx wth columns an N rows resultng n O N memory space requrement. he transton matrx P x value at tme [ p, j; x], jn p, j ; x measures the probabltes value at tme of the transton process from state j z x at tme to state z x at tme. In ths paper we only conser homogeneous Marov chans, so transton matrx oes not epen on tme an t can be smply enote by P x. In orer to smplfy the notaton, when the choce of the portfolo can be tactly unerstoo, we omt the reference to the portfolo x. hus, the transton matrx wll be enote smply as P an smlarly we get the probablty, the p, j Issue, Volume 7, 3 937

3 wealth, the state z s an so on. Moreover wth a lttle abuse of notaton we wll use the terms " s-th state" or "state s" of the Marov chan to pont both the return z s an the nex s tself; context wll mae clear the meanng of the term. Arcs connectng noes represent the transton from a state to a state j an are labelle wth the corresponng probablty p, j. Note that some states are not reachable n he entres B. he asymptotc behavour of the log returns Many emprcal fnngs show that log returns present a strbuton wth heaver tal than strbutons wth fnte varance. Several emprcal nvestgatons show that some noes of the tree. p, j of matrx P are estmate usng the maxmum lelhoo estmates pˆ, j j where j s the number of hstorcal observatons that transt from the state to the state j.e. from z to z j an s the number of hstorcal observatons n state. Pr lnz x u ~ u Lu as u where an Lu s a slowly varyng functon at he N values of the wealth [w l, ] l N Lcu for all c, see, among others, Lu Rachev an Mttn an the references theren. he tal behavor of returns mples that the vector of log-returns s n the oman of attracton of a n-mensonal stable law. Moreover, snce n all observe ata we get, then the relaton mples that log returns r x lnz x amt nfnty,.e., lm u after peros can be compute by the formula: w l, z u.l, l,, N thus, the l-th noe at tme of the wealth-tree correspons to wealth w l,. he proceure to compute the strbuton functon of the future gross returns s strctly connecte to the recombnng feature of the wealth-tree. fnte mean an not fnte varance. hs tal conton also mples that the portfolo log return r x s n the oman of Uner these assumptons Iaqunta an Ortobell 6, have shown how to compute the uncontonal an contonal contonal on the ntal state s,.e. z s probablty of each noe of the future wealth. attracton of an α-stable law. A smple way to trauce the asymptotc behavor of ata conssts n assumng the log wealth to be α stable strbute. hat s, for each portfolo x S the forecaste log wealth ln t lnz x,t at a gven future tme s n the oman of attracton of an x stable strbuton. Uner ths assumpton we mplctly assume that all optmal choces are entfe by four parameters an the forecaste log wealth of every portfolo can be well approxmate by a stable strbuton,.e.: ln S x x, x, where x,] s the Fg.: wealth-tree probabltes state representaton an nex of stablty, x s the scale parameter, x s the locaton parameter an x s the sewness parameter. he estmaton of the stable Paretan parameters can be one effcently n a neglgble computatonal tme by applyng the consstent quantle McCulloch's metho see McCulloch 986. In partcular, McCulloch's metho requres the nowlege of 5%, 5%, 5%, 75%, 95% quantles of the log wealth ln to obtan these estmates n an acceptable computatonal tme for any portfolo. hen, applyng some smple algorthms to compute rewar an rs measures wth stable strbutons we can easly get optmal portfolo strateges that account the Marovan an asymptotc behavor of the fnal wealth. transton In Fg. we prove a graphcal representaton of the wealthtree an the corresponng probabltes after steps, when we assume the return evolves startng from state followng a smple homogeneous 3-state process. Noes represent the possble values of wealth wl,. Namely, n column,, are represente the possble values of wealth after steps. he vector of the wealth after two steps s gven by [w,, w,, w 3,, w 4,, w 5, ]'. In each noe of the wealth-tree the 3 states of the Marov chan are emphasze. Issue, Volume 7, 3 C. Portfolo large scale strateges he classc statc portfolo selecton problem when no short sales are allowe, can be represente as the maxmzaton of a functonal f:ω,ℑ,p ℝ apple to the ranom portfolo of gross returns z x, subject to the portfolo weghts belongng to the n--mensonal smplex S,.e., 938

4 max x S f zx ypcally, the functonal f. s a performance measure or an utlty functonal. In both cases the functonal f. shoul be sotonc wth a partcular orerng of preference, that s, f X s preferre to Y X Y, then fx fy. he choce of the functonal f. plays a crucal role n the portfolo strategy. Isotonc utlty functonals wth non satable an rs averse preferences have been use n many fnancal applcatons. In these cases we have fx=evx where v s an ncreasng an concave utlty functon. However, as suggeste n behavoural fnance, whle all nvestors prefer more to less they coul be nether rs averse nor rs lover. For ths reason t maes sense to conser functonals that are monotone, even though they are not consstent wth an uncertanty/aggressve orer see, among others, Rachev et al. 8. e call OA performance utlty functonal any functonal compute uner the assumpton that the gross return of each portfolo follows a Marov chan wth N states. In ths paper we wll use an escrbe only some OA functonals that conser the forecaste wealth at tme. hat s, nvestors have to perocally every peros compute the portfolo x S soluton of the problem: max f xs x Remar: he vector of weghts x soluton of the problem 3 represents the percentage of wealth that shoul be nveste n each asset urng the pero [,]. Snce the value of the assets change urng the pero [,], then even an OA portfolo strategy generally mples that the wealth coul be recalbrate more tmes urng the pero [,] n orer to mantan constant the percentages of the wealth nveste n each asset. If s very large an we o not recalbrate the portfolo perocally the pero shoul be the same use n the valuaton these percentages nveste n the assets coul be completely fferent at the en of nvestor's temporal horzon. hs pont has not been explctly aresse n Angelell an Ortobell's analyss 9 an coul have a very bg mpact n portfolo choces. In orer to etermne optmal soluton for OA functonals we have to choose fferent portfolo strateges for non satable nvestors whch account for asymptotc behavour of returns. In the followng subsecton we present some OA functonals use n the emprcal comparsons. In portfolo lterature more than one hunre statc rewar-rs performance measures have been propose see Cogneau an Hübner 9a, 9b. Here, we lst the Sharpe statc strategy an some OA performance functonals sotonc wth choces of non satable nvestors that wll be object of the followng emprcal analyss. For all the OA portfolo strateges we assume that nvestors have temporal horzon equal to. OA-Sharpe rato OA-SR. he classc verson of the Sharpe rato see Sharpe 994 values the expecte excess return for unty of rs stanar evaton. th the OA-Sharpe rato we value the expecte excess fnal wealth for unty of rs,.e., 3 OA SR E x x rb r where r s the fnal wealth at tme we obtan b nvestng n the benchmar r b. In the Marovan framewor we shoul conser the bvarate evoluton of the vector x r to value the stanar evaton b of x r. Yet, n the followng x rb b analyses we assume that the rsless asset s not allowe, thus, E x the OA-Sharpe Rato s smply gven by. hen b x the benchmar r b s the rs free rate, the Sharpe rato s sotonc wth non-satable rs averse preferences. However, usng Sharpe type measures we generally on't tae nto account the asymptotc behavor of the wealth except n the case the optmal portfolos are n the oman of attracton of the Gaussan law. OA-Asymptotc Sharpe rato OA-ASR hs performance functonal s efne as OA ASR E ln ln ln. f x. f x. where x s the mean of the stable strbuton ln that better approxmates the log fnal wealth: ln S x x, x, when x.. e assume OA ASR when x. snce low nexes of stablty mply so heavy tals that the. moment of the stable strbuton s nfnte. Observe that when the fnal wealth s log normal strbute. Moreover, f for all the portfolos, the maxmzaton of the OA asymptotc Sharpe rato s equvalent to the maxmzaton of the Sharpe rato of the log wealth. As for the Sharpe rato, ths rato s sotonc wth the preferences of non satable rs averse nvestors see Rachev et al. 8. In orer to maxmze the OA-ASR, we estmate the four stable parameters x, x, x, usng the McCulloch's quantle algorthm see McCulloch 986 an then we compute the. moment of the centere log wealth. OA-Stable stochastc bouns rato OA-SSBR hs performance functonal s efne as: 4 5 Issue, Volume 7, 3 939

5 OA SSBR. E - f an.. E - 6 otherwse where an are the nexes of stablty respectvely of ln ln mn z, ln max z ln x, whle mn z an max z are the forecaste wealths at tme obtane respectvely by the lower maret stochastc boun an the upper maret stochastc boun. Moreover, are the locaton parameters of the stable strbutons that better approxmate respectvely,, an S S,,. In orer to etermne the strbutons of ln ln Y z an ln z ln Y, where z mnn z an z max n z, we have to use the evoluton of the bvarate Marov processes t x, t z an t x, t z,. Recall that, when no short sales are allowe, the upper an the lower maret stochastc bouns among n assets wth gross returns z,, n are respectvely gven by z an z, snce z z x, z for any tme an for any vector of portfolo weghts x S for further generalzatons see Ortobell et al. an references theren. hs rato expresses the ea that nvestors want to maxmze the stance between the wealth an the lower maret stochastc boun, an to mnmze the stance between the wealth an the upper maret stochastc boun. OA-Stable loss rato OA-SLoss he OA stable loss rato values the expecte asymptotc log wealth for unty of loss. hs rato can be seen as a partcular case of the Starr rato apple to stable strbutons see, among others, Bglova et al. 4. hus, usng the asymptotc approxmaton of log wealth ln S x x, x, we can easly compute OA SLoss ln f x Eln ln 7 f x where x s the locaton parameter of the stable ln strbuton that better approxmates the fnal log wealth an Eln ln s obtane usng the Stoyanov et al.'s formula for stable strbutons see Stoyanov et al. 6. e assume OA SLoss when x snce low nexes of stablty mply so heavy tals that the frst moment of the stable strbuton s nfnte. he contonal expecte loss E X X of an stable ranom varable S a X,, s gven by: / cos E X X /, cos where arctan tan / see Stoyanov et al. 6. In orer to reuce the mensonalty of the problem we aopt the tecnques evelope by Ortobell et al. an Angelell et al. preselectng no more than 7 assets an then reucng the mensonalty of the preselecte assets entfyng some common factors to approxmate the asset returns. III. AN EMPIRICAL COMPARISON AMONG PORFOLIO SRAEGIES In ths secton, we evaluate the mpact of the propose moelzaton on the US stoc maret. In partcular, we conser the stocs trae on the NYSE an on the NASDAQ. Snce we want to propose as much as possble a realstc emprcal analyss, we have evelope a ynamc ataset escrbe here n the followng that uses all the useful fnancal ata from DataStream. A. Dynamc Dataset In ths paper we suggest a schema to solve large scale portfolo selecton. hs means that we expect to extract the lately tme seres of ajuste prces for a large number of assets from a atabase namely DataStream. Dealng wth tme seres wth mssng ata s not an easy tas an unfortunately the hunres of tme seres avalable from the atabase are often spole wth mssng ata. In ths paragraph we explan how we manage ths problem n orer to prouce a "clean", though large, set of tme seres to be submtte as nput to our portfolo selecton framewor. he objectve s to obtan a set of assets that are reasonably prce on a common set of ates. he tme seres are frst fltere so that the "ba" ones are rejecte an the corresponng assets wll not compete to enter the portfolo. hen, the promote seres are fxe, f neee, so as all assets are prce on the same set of ates. Once all assets are prce on a common set of ates, ata are reay to be passe to the portfolo selecton algorthm. he objectve s acheve by a number of steps: each prce seres avalable n the atabase for the chosen exchange s extracte from a fxe ate n the past up to the current ate n the analyss; a table s bult so that assets correspons to columns an each row correspon to a common ate for all prces each row wth more than 99% mssng ata s suppose to correspon to a ban holay for the exchange e.g. labour ay an s remove from the table; Issue, Volume 7, 3 94

6 3 each asset wth at least 3 consecutve mssng ata s remove from the table; 4 snce some pars of consecutve or sparse mssng ata can stll be present n some columns, we compute the number of mssng ata for each asset an tae the mnmum say m such quantty; afterwars, assets wth more than m mssng ata or at least one par of consecutve mssng ata are remove from the table; 5 now only sparse mssng ata can be present n the table; all mssng ata are pae forwar from the prevous ate. Naturally, f the mssng ata correspon to the frst ate, the pang s mae bacwar from the next ate; Dong so, we have a table wth no mssng prces an asset returns can be compute. B. An emprcal comparson In our emprcal analyss we use a ate set of about two years 5 aly observatons from 5-Sep-8 tll 3-Aug-, an assume the followng settngs: a that nvestors have a temporal horzon of worng ays thus, for each portfolo strategy we shoul optmze the portfolo every worng ays for a total of 5 optmzatons; b that nvestors cannot nvest more than ten percent n a sngle asset.e.: x [,.] ; c Marov chans have N 9 states; the ntal wealth s equal to at the ate 5-Sep- 8. e perform a comparsons to evaluate the mpact of the Stable Paretan approxmaton by comparng the ex-post performance of fferent portfolo strateges base on: the OA-Sharpe rato 4, the OA-Asymptotc Sharpe rato 5, the OA-Stable loss rato 7, the OA-Stable stochastc bouns rato 6. Even n ths analyss we preselect assets among all those actve ether n the last ten years or n the last sx months. hen we approxmate the returns to reuce the ranomness of the problem. For each strategy, we have to compute the optmal portfolo composton 6 tmes an at the -th optmzaton,,,,5, two man steps are performe to compute the ex-post fnal wealth: Step Determne the maret portfolo x M that maxmzes the performance rato assocate to the strategy,.e. the "eal" soluton of the followng optmzaton problem: max x x s. t. x x ' e,.; x,, n Angelell an Ortobell 9 have observe that the complexty of the portfolo problem s much hgher n vew of a Marovan evoluton of the wealth process. In orer to overcome ths lmt we use the Angelell an Ortobell's heurstc algorthm that coul be apple to any complex portfolo selecton problem that amt more local optma. Step Durng the pero [ t, t ] where t t we have to recalbrate aly the portfolo mantanng the percentages nveste n each asset equal to those of the maret portfolo x. hus, the ex-post fnal wealth s gven by: where ex post t t t x M ' z ex post t 8 z s the vector of observe aly gross returns between t an t. Steps an are repeate for all performance ratos untl some observatons are avalable. Fg.: Ex-post comparson of OA Sharpe rato, OA Asymptotc Sharpe rato, OA stable loss rato, OA stochastc bouns rato apple to preselecte assets among all the actve n the last years. Fg.3: Ex-post comparson of OA Sharpe rato, OA Asymptotc Sharpe rato, OA stable loss rato, OA stochastc bouns rato apple to preselecte assets among all the actve n the last 6 months. he output of ths analyss s gven n Fg. an 3 an able. Fg. an 3 report the results of all strateges apple to the preselecte assets among all the actve assets respectvely n the last ten years an n the last sx months. he comparson between these fgures confrms that the recent entres n the maret have an mportant mpact n the portfolo choces. As a matter of fact, the results obtane from the stable Paretan Issue, Volume 7, 3 94

7 strateges.e., OA-Asymptotc Sharpe rato 5, the OA-Stable loss rato 7, the OA-Stable stochastc bouns rato 6 apple to preselecte assets among all the actve n the last sx months present outstanng results conserng that we apply the moel urng a pero of global crss. he OA Stable loss rato best strategy gves more than the 3% for year. Frst of all we observe that among the assets selecte there are several ones that have Moreover from ths comparson t s stll clear that the OA asymptotc Sharpe strategy, as all the other stable Paretan strateges, presents hgher fnal wealth than the OA Sharpe strategy apple to the preselecte assets among all actve ones ether n the last ten years or n the last sx months. However, urng some peros of the ex-post comparson the OA Sharpe strategy presents hgher wealth than the analogous Stable type strateges. hus n orer to account more precsely these results we have to conser some emprcal statstcs on the expost returns of the portfolo strateges. able reports, for all the strateges, the values of: two rewar measures of the ex-post returns the emprcal mean, the AVaR of the opposte ranom varable.e., EL.5 -X where EL Y AVaR Y E Y Y FY ; two rs measures of the ex-post returns the stanar evaton σx an the AVaR of the centre ranom varable EL.5 X-EX; 3 all the possble rewar rs ratos ervng from these two measures. Actve last 6 months OA stable loss OA stable stochastc bouns OA asymptotc Sharpe OA Sharpe meanx St.evX EL-X ELX-EX Sharpe rato Mean/ ELX-EX EL-X/ ELX-EX EL-X/St.evX Actve last years OA stable loss OA stable stochastc bouns OA asymptotc Sharpe OA Sharpe meanx St.evX EL-X ELX-EX Sharpe rato Mean/ ELX-EX EL-X/ ELX-EX EL-X/St.evX able Emprcal mean, stanar evaton, EL.5-X an EL.5X- EX on the ex-post returns. able suggests that the OA Sharpe strategy s omnate from the stable Paretan type strateges snce the ex-post returns obtane by the OA Sharpe strategy presents lower rewar/rs performance than almost all the stable Paretan strateges except for the 6 months stable stochastc bouns strategy. hus we essentally confrm the results observe n Fgs an 3. he portfolo composton generally changes a lot urng the ex-post pero. hs s confrme from Fg. 4 that escrbes the portfolo turnover an ts versfcaton. In partcular, t examnes how the portfolo composton of the Stable loss strategy changes urng the ex-post pero. In the frst subfgure Fg. 4a we have the percentages nveste n each assets at each computaton of the optmal portfolo. Fg. 4: Portfolo composton an portfolo varatons of the OA stable loss strategy apple to preselecte assets among all the actve n the last 6 months. he secon sub-fgure Fg. 4b ponts out the percentages =,...,5 of the portfolo change every ays obtane by the formula: n x M, x M, In partcular shoul belong to the nterval [,], where the value means that the portfolo composton s not change urng the pero [t -,t ], whle the value correspons to the case we sell the portfolo an we buy a completely fferent portfolo. he last sub-fgure Fg. 4c ponts out the number of: the quantty of assets use.e. those assets whose percentages are greater than zero x >, =,...,n; M, the quantty of enterng assets; 3 the quantty of extng assets. As we observe the portfolo s well versfe among all preselecte assets even f there are always some assets n whch the strategy suggests to nvests the maxmum possble Issue, Volume 7, 3 94

8 .e. %. Moreover we also observe that the portfolo change a lot every ays an even on these changes we shoul pay the transacton costs. he transacton costs are also pa ay by ay when we recalbrate the portfolo to mantan constant the percentages of the portfolos. hus f we assume.5% as aly average of transacton costs these transacton costs are hgh enough for nsttutonal nvestors we shoul get more than.5⁵⁰⁰.84 n the 5 ays of the ex-post analyss. However, most of the stable Paretan strateges prouce some profts snce they present a fnal wealth greater than.84 urng the last two years of ex post analyss. However, these results o not conser the transacton costs whch must be pa aly n orer to mantan constant the percentages nveste n each asset. Moreover from ths comparson t s stll clear that the OA asymptotc Sharpe strategy, as all the other stable Paretan strateges, presents hgher fnal wealth than the OA Sharpe strategy apple to the preselecte assets among all actve ones ether n the last ten years or n the last sx months. IV. CONCLUSION hs paper escrbes a Marovan approach apple to portfolo problems where nnovatons belong to the stable strbutons oman. In partcular, we frst examne how to approxmate the Marovan an asymptotc behavor of wealth. hen, we examne several portfolo strateges uner the propose envronment. Fnally, we propose an emprcal comparson among several strateges that account heavy tals of log return portfolos. he emprcal analyss shows that the asymptotc behavors of the wealth an the recent entres n the maret have an mportant mpact n the portfolo choces apple to the US stoc maret. Moreover, several new questons rse from the propose methoology an emprcal analyss. As a matter of fact, the experment on the US maret suggests that further nvestgatons neee to value: the maret effcency, lquty constrants, an the mpact of portfolo strateges base on the use of proper stoppng tmes. ACKNOLEDGMEN he paper has been supporte by the Italan funs ex MURS 6% 3 an MIUR PRIN MISURA Project, he research was also supporte through the Czech Scence Founaton GAČR uner project 3-34S an SP3/3, an SGS research project of VSB-U Ostrava, an furthermore by the European Regonal Develop-ment Fun n the I4Innovatons Centre of Excellence project CZ..5/../.7, nclung the access to the supercomputng capacty, an the European Socal Fun n the framewor of CZ..7/.3./.96 secon author. References [] Angelell, E. an S. Ortobell 9, "Amercan an European portfolo selecton strateges: the Marovan approach", n P. N. Catlere. E., Fnancal Hegng, Nova Scence, New Yor, pp [] Angelell E., Ortobell L.S. an Iaqunta G. 3 "Portfolo choce: a non parametrc Marovan framewor" ISBN: n "Mathematcal Applcatons n Scence & Mechancs" es. rsovc N., Rastero D., enoro A., Sman D., Mnea A., Roushy M., Salem A. pag seas Press [3] Angelell, E., A. Banch an S. Ortobell, "Fnancal Applcatons of bvarate Marov processes", Mathematcal Problems n Engneerng Volume Artcle ID 34764, 5 pages,o:.55// [4] Bglova, A., S. Ortobell, S. Rachev an S. Stoyanov, 4, "Dfferent approaches to rs estmaton n portfolo theory", Journal of Portfolo Management 3, 3-. [5] Cogneau, P. an Hübner, G. 9a, he more than ays to Measure Portfolo Performance - Part : Stanarze Rs-Ajuste Measures, Journal of Performance Measurement, vol. 3, n 4, pp [6] Cogneau, P. an Hübner, G. 9b, he more than ays to Measure Portfolo Performance - Part : Specal Measures an Comparson, Journal of Performance Measurement, vol. 4, n, pp [7] Cox, J. C., S. A. Ross an M. Rubnsten, 979, "Opton prcng: a smplfe approach", Journal of Fnancal Economcs 7, [8] D Amco, G., G. D Base, J. Janssen an R. Manca, "Sem- Marov Bacwar Cret Rs Mgraton. Moels: a Case Stuy", Internatonal Journal of Mathematcal Moels an Methos n Apple Scences, 4, 8-9. [9] Fama, E., 965, "he behavor of stoc maret prces", Journal of Busness 38, [] Iaqunta, G., an S., Ortobell 6, "Dstrbutonal approxmaton of asset returns wth nonparametrc marovan trees" n Internatonal Journal of Computer Scence & Networ Securty 6, [] Iaqunta G., Ortobell L.S., Angelell E. "GARCH type portfolo selecton moels wth the Marovan approach", Internatonal Journal of Mathematcal Moels an Methos n Apple Scences, Volume 5, pp [] Iaqunta G., Ortobell L.S., Angelell E. "he Marovan portfolo selecton moel wth GARCH volatlty ynamcs" n M. Prtea, M. Mazlu, J. Strouhal es. Selecte opcs n Economy & Management ransformaton Vol. I, ISSN: , ISBN: pp seas Press [3] Manelbrot, B., 963, "he varaton of certan speculatve prces, " Journal of Busness 6, [4] McCulloch, J.H., 986, "Smple consstent estmators of stable strbuton parameters", Communcatons n Statstcs - Smulaton an Computaton 54, [5] Ortobell, S., E. Angelell an D. onnell,, "Set-portfolo selecton wth the use of maret stochastc bouns", Emergng Marets Fnance an rae 475, 5-4. [6] Ortobell, S., A. Bglova, S. Rachev an S. Stoyanov,, "Portfolo selecton base on a smulate copula", Journal of Apple Functonal Analyss 5/, [7] Rachev, S. an S. Mttn,, "Stable Paretan Moels n Fnance", ley & Sons, New Yor. [8] Rachev, S., S. Mttn, F. Fabozz, S. Focar an. Jasc, 7, "Fnancal econometrcs: from bascs to avance moelng technques", ley & Sons, Hoboen. [9] Rachev, S., S. Ortobell, S. Stoyanov, F. Fabozz an A. Bglova, 8, "Desrable propertes of an eal rs measure n portfolo theory", Internatonal Journal of heoretcal an Apple Fnance, [] Sae, A. an N. Lmnos,, "Asymptotc propertes for maxmum lelhoo estmators for relablty an falure rates of Marov chans", Communcaton n Statstcs-heory an Methos. [] Samorontsy, G. an M. aqqu, 994, "Stable non-gaussan ranom processes: Stochastc moels wth nfnte varance", Chapman an Hall, New Yor. [] Sharpe,.F., 994, "he Sharpe rato", Journal of Portfolo Management, [3] Stoyanov, S., G. Samorontsy, S. Rachev an S. Ortobell, 6, "Computng the portfolo contonal value-at-rs n the α-stable case", Probablty an Mathematcal Statstcs 6, -. Enrco Angelell s an assocate professor n Mathematcal Fnance at the Unversty of Bresca. He hols a Ph.D. n Computatonal Methos for Fnancal an Economc Forecastng an Decsons from the Unversty of Bergamo. Issue, Volume 7, 3 943

9 He taught numerous courses at the Unversty of Bresca, nclung Computer scence for fnance, Algorthms an computer programmng, Informaton systems. Hs research focuses on the applcaton of operatonal research technques to optmzaton problems rangng from portfolo selecton to scheulng an transportaton. Sergo Ortobell Lozza s an assocate professor n Mathematcal Fnance at the Unversty of Bergamo. He s also vstng Professor at VSB U Ostrava Department of Fnance, Czech Republc. He hols a Ph.D. n Computatonal Methos for Fnancal an Economc Forecastng an Decsons from the Unversty of Bergamo. He taught numerous courses at the Unverstes of Bergamo, Calabra an Mlan, nclung basc an avance calculus, measure theory, stochastc processes, portfolo theory, an avance mathematcal fnance. Hs research, publshe n varous acaemc journals n mathematcs an fnance, focuses on the applcaton of probablty theory an operatonal research to portfolo theory, rs management, an opton theory. Gaetano Iaqunta s a prvate research analyst. He cooperates wth fnancal nsttutons n Mlan Italy. He hols a Ph.D. n Computatonal Methos for Fnancal an Economc Forecastng an Decsons from the Unversty of Bergamo an vstng Ph.D stuent n Fnance at Unversty of Lugano Swtzerlan. Hs worng areas an research nterests nclue computatonal fnance, opton prcng theory, rs management, multstage stochastc optmzaton an scenaro generaton. Issue, Volume 7, 3 944