A two-stage stochastic mixed-integer program modelling and hybrid solution approach to portfolio selection problems

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1 A two-stage stochastc mxed-teger program modellg ad hybrd soluto approach to portfolo selecto problems Fag He, Rog Qu The Automated Schedulg, Optmsato ad Plag (ASAP) Group, School of Computer Scece The Uversty of Nottgham, Nottgham, NG8 1BB, UK } Abstract I ths paper, we vestgate a mult-perod portfolo selecto problem wth a comprehesve set of real-world tradg costrats as well as market radom ucertaty terms of asset prces. We formulate the problem to a two-stage stochastc mxed-teger program (SMIP) wth recourse. The set of costrats s modelled as mxed-teger program, whle a set of decso varables to rebalace the portfolo multple perods s explctly troduced as the recourse varables the secod stage of stochastc program. Although the combato of stochastc program ad mxed-teger program leads to computatoal challeges fdg solutos to the problem, the proposed SMIP model provdes a sghtful ad flexble descrpto of the problem. The model also eables the vestors the make decsos subject to real-world tradg costrats ad market ucertaty. To deal wth the computatoal dffculty of the proposed model, a smplfcato ad hybrd soluto method s appled the paper. The smplfcato method ams to elmate the dffcult costrats the model, resultg to easer sub-problems compared to the orgal oe. The hybrd method s developed to tegrate local search wth Brach-ad-Boud (B&B) to solve the problem heurstcally. We preset computatoal results of the hybrd approach to aalyse the performace of the proposed method. The results llustrate that the hybrd method ca geerate good solutos a reasoable amout of computatoal tme. We also compare the obtaed portfolo values agast a dex value to llustrate the performace ad stregths of the proposed SMIP model. Implcatos of the model ad future work are also dscussed. Key words: Stochastc programmg; hybrd algorthm; Brach-ad-Boud; local search; portfolo selecto problems 1. Itroducto The essece of portfolo selecto problem (PSP) ca be descrbed as fdg a combato of assets that best satsfes a vestor s eeds. The theory of PSP was developed by Harry Markowtz frstly the 1950 s. I hs work, the PSP was formulated as the mea varace (MV) model (Markowtz 1952), a quadratc optmsato model. The basc MV model selects the composto of assets whch ether acheves a predetermed level of expected retur whle mmzg the rsk, or acheves the maxmum expected retur wth a predefed level of rsk. From a practcal pot of vew, however, the MV model s ofte cosdered to be too basc, as t gores may costrats faced by real-world vestors. Preseces of real-world tradg costrats dramatcally crease the complexty of PSP. These costrats clude the cardalty costrat (a lmt o the total umber of assets held a portfolo), the mmum ad/or maxmum posto sze costrat (bouds o the amout of each asset), the mmum trade sze costrat (bouds o the amout of trasacto occurred o each asset) ad trasacto costs, etc. Whe such costrats are 1

2 cosdered to the basc MV model, the problem s recogzed to be NP-complete (Bestock 1996, Mas ad Speraza 1999). Alog wth the tradg costrats, aother mportat factor faced by the vestors to make a proper vestmet decso s the market ucertaty. However, the classc MV model ad other models of PSP (Chag, Meade et al. 2000, Kellerer, Mas et al. 2000, Crama ad Schys 2003, Koo ad Yamamoto 2005, Mas ad Speraza 2005), the expected retur ad covarace betwee assets are usually based drectly o hstorcal data. These models do ot accout for the ucertates of the market. Facal markets are upredctable ad decso should be made wth the cosderato of market codtos (J, Zhu et al. 2005, Baldacc, Boschett et al. 2009). Moreover, vestors apply flexble portfolo maagemet strateges by rebalacg portfolos perodcally respose to ew ucerta market codtos (.e. chagg perceptos regardg the radom asset prce). Radom ucertates of market,.e., terms of asset prces ad currecy exchage rates etc. are ma factors of market. Several o-probablstc ucertaty factors such as vagueess ad ambguty are vestgated maly by fuzzy techques (Gupta, Mehlawat et al. 2010, Gupta, Iuguch et al. 2013, L ad Xu 2013). I ths paper we focus o radom ucertaty of the market,.e., term of asset prces. Stochastc programmg becomes a creasgly popular techque to model decso makg uder radom ucertaty. It s able to model ucertates a flexble way ad mpose real-world costrats relatvely easly (Kg ad Wallace 2012). It uses formato the future to make curret decsos. Hgle ad Wallace (Hgle ad Wallace 2003) vestgate the ucertaty factors geeral stochastc problems. Geerally speakg, takg to accout of ucertaty helps to mprove decso makg. Stochastc programmg has bee appled to descrbe a varety of portfolo optmzato problems. Models for asset-lablty ad rsk maagemet have bee proposed (Mulvey ad Vladmrou 1992), (Topaloglou, Vladmrou et al. 2008, Stoya ad Kwo 2011). New approaches based o stochastc programmg for portfolo maagemet have also bee proposed (Mulvey ad Vladmrou 1992, Flete, Høylad et al. 2002, Hgle ad Wallace 2003, Barro ad Caestrell 2005, Gavorosk, Krylov et al. 2005, J, Zhu et al. 2005, Escudero, Garí et al. 2007, Topaloglou, Vladmrou et al. 2008, Baldacc, Boschett et al. 2009, Stoya ad Kwo 2010, Stoya ad Kwo 2011). I portfolo optmzato problem doma, Gavorosk, Krylov et al. (Gavorosk, Krylov et al. 2005) vestgate the ssue of optmal portfolo rebalacg ad try to determe whether to rebalace a gve portfolo based o the trasacto costs ad ew formato of market. Flete, Hoylad et al. (Flete, Høylad et al. 2002) demostrate how to evaluate stochastc programmg models by comparg two dfferet approaches to asset lablty maagemet. The frst uses multstage stochastc programmg, whle the other s a statc approach. They show that the stochastc programmg method domates the statc method to the asset lablty maagemet problem due to ts ablty that adapts the market formato. To solve the stochastc model of PSP, the lterature a wde rage of decomposto techques have bee developed (Brge 1985, Mulvey ad Ruszczyńsk 1995, Barro ad Caestrell 2005, Escudero, Garí et al. 2007, Stoya ad Kwo 2010, Stoya ad Kwo 2011). (Brge 1985) proposed the ested varat of the Beders decomposto. Mulvey ad Ruszczyńsk proposed the augmeted Lagraga decomposto (Mulvey ad Ruszczyńsk 1995). (Stoya ad Kwo 2011) proposed a ovel decomposto method based o the partcular structure of the problem cocered. It decomposes the problem geographcally to securty ad bod sub-problems, whch are the further broke to smaller sub-problems. I ths paper, we model the mult-perod PSP wth a comprehesve set of real-world tradg costrats cludg the cardalty costrat, the mmum posto sze, the mmum trade sze costrat, ad trasacto costs, etc. I order to model these costrats, we apply a mxed-teger optmzato approach. I addto, to accout for the ucertaty the market term of asset prces to ehace the decso makg of vestors, we apply stochastc programmg to capture future formato the market,.e. asset prces, represeted by a set of radom varables. A set of decso 2

3 varables to rebalace the portfolo s explctly troduced as the recourse varables the secod stage of stochastc program. These recourse varables are used to amed the frst stage decso varables based o the realzato of the radom varables of asset prces. Thus, the problem s formulated to a two-stage stochastc mxed-teger program. We set the model the framework of stochastc programmg ad formulate the problem a twostage settg. Ths s for smplcty but ot prmarly.what s more, we see ths as a approprate framework to fulfl the purpose of our vestmet. The tegral ad stochastc ature of the problem makes the two-stage model complex eough. The purpose of the model s makg curret decso uder a set of real-world tradg costrat whle takg accout of future asset prce chages. We do ot tet to provde tactcal rebalace strategy for each future tradg but to make curret more flexble decso wth recourse for the future. We vestgate the effect of asset prce ucertaty upo the soluto to the portfolo optmzato problem. Although stochastc programmg models have bee used for other types of portfolo costructo (Mulvey ad Vladmrou 1992, Gavorosk, Krylov et al. 2005, J, Zhu et al. 2005, Topaloglou, Vladmrou et al. 2008, Stoya ad Kwo 2010, Stoya ad Kwo 2011), to our kowledge, ths s the frst work to develop a mult-perod PSP model wth a comprehesve set of real-world costrats as well as market ucertaty. I our prevous work (He ad Qu 2013), tal tests of a hybrd local search method have bee coducted o a relatvely easer model for determstc PSPs wthout ucertates. I ths preset work, to address the complex two-stage portfolo selecto problem wth ucertaty, the model combes tegral ad stochastc varables. Ths leads to a more computatoal challegg SMIP model. Therefore, we adapt a smplfcato ad hybrd method based o our prevous method whch hybrds local search wth B&B (amed LS-B&B) to solve the SMIP heurstcally. I the LS-B&B, varable fxg together wth a local search are appled to geerate a sequece of smplfed subproblems. The default B&B search the solves these restrcted ad smplfed sub-problems more easly due to ther reduced sze comparg to the orgal oe. The dea s to perform computatoally expesve local search o the surface of certa varables, ad the explore the sub-problems by B&B to completo. I summary, ths paper presets two major cotrbutos: (1) a two-stage stochastc mxed-teger program model that formulates a comprehesve set of real-world tradg costrats, as well as radom ucertaly the market employg the cocepts of scearos ad stages. (2) the applcato ad aalyss of a effcet smplfcato ad hybrd soluto method to the ew two-stage SMIP model, whch s proposed ad examed the lterature for the frst tme ths work. The remader of ths paper s orgazed as follows. I Secto 2, we descrbe ad formulate the PSP wth ucertaty to a two-stage stochastc mxed-teger program model. I Secto 3, we preset the smplfcato ad hybrd soluto method. I Secto 4, we aalyse the performace of the hybrd method. Fally, we draw our coclusos Secto Problem Modellg We cosder the problem of a decso maker who s cocered wth the actve maagemet of a set of facal assets, to acheve certa goal whle satsfyg a set of market ad tradg costrats. These costrats clude the cardalty costrat, the mmum posto sze costrat, the mmum trade sze costrat ad trasacto costs. I order to model these costrats, mxedteger formulato s ecessary. May of the decsos,.e., purchase or sell a asset, are based o the predcted prce of assets. Dfferet future prces ca lead to dfferet vestmet choces. Ths s the ssue of market radomess (Kg ad Wallace 2012). To accout for the ucertaty of the stock market term of asset prce, we use stochastc program techque to capture ay chages that may occur wth the future asset prce. That s, by usg stochastc program we ca use future formato to make curret decsos. 3

4 2.1 Scearo We formulate the problem as a two-stage stochastc mxed-teger program wth recourse. The frststage decsos represet curret portfolo costructo decsos whle the secod-stage recourse varables represet the amedmet of the portfolo based o the future market formato. The key ucertaty of the stock market we vestgated,.e. the assets prces, or, equvaletly, ther returs s represeted by radom varables. We ca capture may possble future market realzatos by expressg a fte umber of possble outcomes of these radom varables as scearos. Hece, the secod-stage, f oe of may market realzatos occurs, the model has accout for ths stuato ad the decsos are made based o these future market realzatos captured the model. Thus we ca say that the model respod to the ew formato o market codtos. Two mportat cocepts, stage ad scearo, should be clarfed before we preset the model. A formato stage, ormally smply called stage, s a pot tme where decsos are made based o the ew formato. It oly makes sese to dstgush two pots tme as dfferet stages f we observe somethg relevat (ew formato) betwee (Kg ad Wallace 2010). The cocept of stage s dfferet from perod. Perod s the tme pot where decso ca be made regardless the cosderato of ew formato realzato. Ths meas we ca have more tha oe perod wth oe stage f o ew formato s added betwee the perods. I Fg. 1, (a) s a mult-stage multperod scearo tree; (b) s a two-stage mult-perod scearo tree; whle (c) s a two-stage two-perod scearo tree. I ths work, we study a two-stage mult-perod PSP problem as show (b) Fg.1. Sce there s o ew formato comg,.e. o ew radom varable realzato betwee each perod wth each stage, we ca aggregate the perods to a stage as show Fg.1 (c). Fg. 1 Examples of scearo trees The secod mportat cocept s scearo. The possble outcomes of the radom varables whch represet the ucertaty are defed as scearos. The progressve evoluto of the radom varables our two-stage model ca be descrbed by a scearo tree as (c) Fg.1. The root ode of the scearo tree represets the tal state at the curret tme. The dscrete evolutos of the radom varables durg the plag horzo are specfed by odes ad arcs. I Fg.1 (c), ode = 1-9 represets each realzato of radom varables the secod stage,.e. the state of the market. The arcs represet the possble trastos betwee two adjacet states. Each path, startg from the root ode to leaf odes, represets oe scearo. The model based o the scearo tree Fg.1 (c) ca be modelled as lear stochastc program for certa PSPs f the objectve fucto ad costrats of the problems ca be represeted by lear fucto; or t ca be formulated as a mxed-teger lear program f teger varables are volved. I ether case, the scearo tree represets the evoluto of the radom varables over the plag horzo. For mult-stage stochastc program model, oe logcal requremet must be coformed. Ths logcal requremet s usually termed as o-atcpatve costrat. Sce we study a two-stage 4

5 model ths paper, we do ot eed to eforce ths costrat. We defe the decso varables o each ode, where ode 0 belogs to the frst stage whle odes 1, 2,.. belog to the secod stage. Scearo geerato methods for stochastc programme have bee extesvely studed the lterature. We apply a smple yet effectve method proposed by (Stoya ad Kwo 2010) to geerate scearos for the SMIP model by employg hstorcal market data. The reaso s that f market behavour s represetatve for a specfc hstorc tme terval, the we expect that ths patter wll cotue the ear future. (Jobst, Mtra et al. 2006) used the same ratoale for ther portfolo selecto model. We vestgate the past market asset prces ad produce scearos that ft a approxmato of that fucto. We refer to (Jobst, Mtra et al. 2006, Stoya ad Kwo 2010) for more formato about the scearo geerato methods. The formato about the hstorc data that are appled to geerate scearos ad umber of scearos geerated wll be preseted Secto SMIP model compoets The otatos appled ths paper are gve Table 1. Table1. Notatos Sets A The set of assets. Idex, A User-specfc parameters The target expected portfolo retur after the plag horzo Crtcal percetle for VaR ad CVaR Determstc put data h w 0 k w m t m F The tal avalable captal to vest the asset market The tal posto ( umber of uts ) of asset The fxed fee whe purchasg asset The fxed fee whe sellg asset The varable cost whe purchasg asset The varable cost whe sellg asset The fxed trasacto fee The varable trasacto cost The umber of assets held the portfolo The mmum hold posto The mmum tradg sze Feasble soluto set Stage (scearo) depedet data N The set of odes the secod-stage Fg.1, dex, N p Probablty of ode the secod stage 0 P P V The prce per ut of asset the frst stage The prce per ut of asset at ode the secod stage The fal wealth of the portfolo at ode R The retur of the portfolo at ode Auxlary varables 5

6 y z Decso varables b s w b s w c f g c f g Portfolo shortfall excess of VaR at ode The varable defto of CVaR whch equals to VaR the optmal soluto The umber of uts of asset purchased the frst stage The umber of uts of asset sold the frst stage The posto ( umber of uts ) of asset after trasactos the frst stage The umber of uts of asset purchased at ode the secod stage The umber of uts of asset sold at ode the secod stage The posto ( umber of uts ) of asset after trasactos the secod stage Bary varable the frst stage. Is 1 f we hold asset, 0 otherwse Bary varable the frst stage. Is 1 f we purchase asset, 0 otherwse Bary varable the frst stage. Is 1 f we sell asset, 0 otherwse Bary varable the secod stage. Is 1 f we hold asset at ode, 0 otherwse Bary varable the secod stage. Is 1 f we purchase asset at ode, 0 otherwse Bary varable the secod stage. Is 1 f we sell asset at ode, 0 otherwse The basc portfolo selecto problem (determstc) framework ca be expressed as: m rsk measure ( w ) (1) s.t reutr ( w ) ; (2) w F (3) where objectve (1) s to mmze the rsk of the portfolo. Costrat (2) esures the expected retur. F (3) represets the set of feasble portfolos subject to all the related costrats Costrat set I facal practce, the trasacto cost has sgfcat effects o portfolo selecto. It has bee show (Arott ad Wager 1990) that gorg the trasacto cost could result to effcet portfolos. Ths has also bee justfed by expermetal studes (Yoshmoto 1996). If the trasacto cost fucto s lear, whch leads to a covex optmsato problem, the the problem s geerally easy to solve. However, a fucto whch better reflects realstc trasacto costs s usually ocovex (Koo ad Wjayaayake 2001). The o-covex optmsato problem s more challegg. I ths paper, we cosder a model that cludes a fxed trasacto fee plus a lear cost, thus leads to a o-covex fucto show Fg. 2, as the cost decreases relatvely whe the tradg amout creases (Yoshmoto 1996, Koo ad Wjayaayake 2001, Koo ad Wjayaayake 2002). Ths fucto s also appled (Lobo, Fazel et al. 2007). The trasacto cost fucto s gve (4), ad show Fg. 2, otatos gve Table 1. I ths work we aggregate the costs occurred sellg ad buyg a asset, ad use a compact trasacto cost fucto b, b 0. 0, x 0; b, b 0; s, s 0; (4) 6

7 Fg.2 The trasacto cost fucto (Lobo, Fazel et al. 2007) Throughout the plag perod cash flows ad cash outflows occur due to the assets sellg, purchasg ad trasacto costs assocated wth asset tradg. I other words, ths costrat reflects the evoluto of the cash balace of the vestmet over tme. The cash balace costrat ca be explaed by Fg.3. Fg.3 Cash flow balace We state the cash balace costrat for the frst stage (5) ad secod stage (6) as the followg: h s P (1 ) b P (1 ) (5) 0 0 A A s P (1 ) b P (1 ) N (6) A A We have the asset balace codto for each asset at the frst stage (7) ad secod stage (8): w w b s A 0 (7) w w b s A, N (8) Next, we troduce a bary varable c ad c to cotrol the umber of assets to hold at the frst stage ad secod stage. c =1 f the vestor holds asset, c = 0 otherwse. The cardalty costrat states that the portfolo cossts of k assets as the followg: A c k (9) c k N A The mmum posto costrat prevets vestors from holdg very small postos after the rebalacg. We troduce a prescrbed postve percetage value w m. That s, holdg a posto strctly less tha w m s forbdde (Chag, Meade et al. 2000, Crama ad Schys 2003). The small (10) 7

8 postos ca be forbdde by troducg the followg costrats for the frst stage (11) ad secod stage (12): wm c w A (11) wm c w A, N (12) I addto, we troduce a mmum tradg costrat to prevet tradg a very small amout of assets less tha a prescrbed percetage value t m. Two addtoal sets of bary varables, f ad g (ad f ad g ), are troduced to deote that the vestor buys or sells asset at the frst (ad secod) stage. Smlar to the modellg of the mmum posto costrat above, the mmum tradg costrat ca the be expressed as follows: t f w A m m t g w A m m (13) (14) t f w A, N (15) t g w A, N (16) We also troduce the exclusve costrat to prevet buyg ad sellg the same asset at the same tme: f g 1 A (17) f g 1 A, N (18) We deote the fal wealth of the portfolo at ode the scearo tree as V, calculated by: V P w N A (19) Based o the fal wealth we ca defe the retur of the portfolo as R : V R 1 N (20) 0 V Thus we have the expected retur costrat as follows: N pr The teger decso varables w,,,,, b s w b s allow detfyg the exact umber of asset uts to purchase or sell at a specfc tme. However, t dramatcally creases computatoal dffculty. We adapt the same method used the lterature (Gavorosk, Krylov et al. 2005, Woodsde-Orakh, Lucas et al. 2013), ad set w, b, s, w, b, s as real decso varables to represet the fracto vested asset the portfolo. However, the bary teger decso varables c, f, g, c, f, g stll preset great computatoal dffculty, for whch we apply the talored hybrd algorthm Objectve fucto (21) The covetoal MV model apples covarace of the assets the portfolo as a rsk measure, assumg ormal dstrbutos asset returs. However, practce returs of may facal securtes exhbt skewed ad leptokurtc dstrbutos (Kaut, Wallace et al. 2007). May other vestmets are exposed to multple rsk factors, thus jot effect o portfolo returs ofte caot be modelled by a ormal dstrbuto. Alteratve rsk measures have bee sought. Such measures, such as Value-at-rsk (VaR) (Joro 2001), are cocered wth addtoal characterstcs of the retur dstrbuto (e.g., the tals) besdes the varace. VaR s defed as the maxmal loss (or mmal retur) of a portfolo over a specfc tme horzo at a specfed cofdece level. However, VaR also suffers from a umber of theoretcal ad practcal lmtatos. The ma drawbacks clude that VaR s ot a coheret rsk measure the sese defed 8

9 by Artzer et al. (Artzer, Delbae et al. 1999). The VaR of a dversfed portfolo ca be larger tha the sum of the VaRs of ts costtuet asset compoets (Kaut, Wallace et al. 2007), thus fals to reward dversfcato. Moreover, whe the returs of assets are expressed terms of dscrete dstrbutos (.e., scearos), VaR s a o-smooth ad o-covex fucto of the portfolo postos ad exhbts multple local extreme (Rockafellar ad Uryasev 2002). Icorporatg such fuctos mathematcal programs s very dffcult, thus makg the use of VaR mpractcal portfolo optmzato models (Kaut, Wallace et al. 2007). To overcome the defceces of VaR, sutable alteratve rsk measures have bee sought. (Artzer, Delbae et al. 1999) dscussed the propertes that soud rsk measures should satsfy, ad specfed a famly of closely related coheret rsk measures termed as expected shortfall, mea excess loss, tal VaR, ad codtoal VaR to quatfy the mass the tal of the dstrbuto beyod VaR. (Rockafellar ad Uryasev 2002) troduced a defto of the codtoal value-at-rsk (CVaR) measure for geeral dstrbutos, cludg dscrete dstrbutos that exhbt dscotutes, ad showed that CVaR s a cotuous ad covex fucto of the portfolo postos. Most mportatly, they showed that a CVaR optmzato model ca be formulated as a lear program the case of dscrete dstrbutos of the stochastc put parameters. I ths work we choose CVaR as the rsk measure. The objectve of our SMIP model s to mmze the expected value of the CVaR the tal (beyod a specfc percetle, ) of the portfolo losses at the ed of the plag horzo. CVaR ca be trasformed to a lear programmg (Rockafellar ad Uryasev 2000) as the followg. A mmzato problem deoted as: m CVaR s. t. w F where w deotes decso varables ad F deotes ther feasble rego, ca be reduced to the followg lear programmg problem: 1 m z y (1 ) N s. t. y f ( w, y ) z; y 0; where s a user specfed percetle value (e.g. 95%), a parameter for VaR ad CVaR. f ( x, y ) represets portfolo loss over the plag horzo. z s the varable the defto of CVaR whch equals to the optmal VaR value. y are auxlary varables the lear programmg formulato whch represet the portfolo shortfall (.e. f ( x, y ) ) excess of VarR value (.e. z ) at ode. 2.3 SMIP model Based o the deftos of costrats ad objectve fucto, the two-stage stochastc mx-teger programmg model s defed as the followg: 9

10 1 m z (1 ) st.. w w b s, A 0 N w w b s A, N h s P (1 ) b P (1 ) 0 0 A A (SMIP) s P (1 ) b P (1 ) N A A A c c A m m m m m m k p y w c w A, N t f w A, N t g w A, N f g 1 A f g 1 A, N k N w c w A t f w A t g w A V P w N A R 1 N 0 y R z N y 0 N N w, b, s, w, b, s R c, f, g, c, f, g B z, y V V pr R 3. Smplfcato ad Hybrd Soluto Approach The basc MV model developed by Markowtz ca be formulated as a quadratc program model to mmse the rsk of the portfolo for a gve level of retur. However, the troducto of a sgle cardalty costrat whch restrcts the umber of assets the portfolo chages the classcal quadratc optmsato model to a quadratc mxed-teger problem that s NP-complete (Bestock 1996). Exact ad heurstc approaches for cardalty costraed PSP have bee wldly vestgated the lterature (Bestock 1996, Chag, Meade et al. 2000, Jobst, Horma et al. 2001, Crama ad Schys 2003). A brach-ad-boud algorthm based o a lfted polyhedral relaxato of coc quadratc costrats was proposed (Velma, Ahmed et al. 2008). (Shaw, Lu et al. 2008) preset a lagragea relaxato based procedure to search for exact solutos of the cardalty costraed PSP. For those large-scale costraed PSPs, heurstc (or meta-heurstc) methods have also bee wdely appled. (Chag, Meade et al. 2000) preset three heurstc algorthms based o a geetc algorthm, tabu search ad smulated aealg for fdg the cardalty costraed effcet froter. For more 10

11 formato wth regard to cardalty costraed PSP we refer to (Woodsde-Orakh, Lucas et al. 2011). Due to the teger varables ad the ature of the stochastc model, fdg solutos to the two-stage model s challegg. Therefore, a smplfcato method s appled to accommodate the complexty of our problem models. A large varety of decomposto methods have bee appled to SMIP models. A commo strategy stochastc programg s to decompose the problem based o tme-stage or scearos (Barro ad Caestrell 2005, Escudero, Garí et al. 2007, Shaw, Lu et al. 2008). Stoya ad Kwo (Stoya ad Kwo 2011) decomposed the problem geographcally to securty ad bod sub-problems. The subproblems are further broke dow by relaxg dffcult costrats. I our prevous work () Motvated by ths, ths paper, we adapt a smplfcato method our prevous work for solvg to elmate the dffcult costrats the SMIP model, leadg to easer sub-problems to tackle. Several researchers have poted out that the cardalty costrat presets the greatest computatoal challege to the problem (Bestock 1996, Jobst, Horma et al. 2001, Stoya ad Kwo 2010, Stoya ad Kwo 2011). Actually, the PSP wth cardalty costrat has bee recogzed to be NPcomplete (Bestock 1996, Mas ad Speraza 1999). To elmate the cardalty costrat, we detfy varables c whch defe the cardalty costrat c k as a set of core varables. Varable fxg (Bxby, Feelo et al. 2000) the s appled o ths A set of core varables c. There are two beefts after varable fxg. Frstly, the cardalty costrat whch s defed by these core varables s elmated (.e., each core varable has a value assgmet). Ths reduces the computatoal complexty. Secodly, sub-problems are geerated by varable fxg. Based o the varable fxg, we perform local search o these core varables, whch s computatoally expesve, ad the the default Brach-ad-Boud search s performed o the subproblems to geerate complete solutos to the orgal problems. That s, local search ad Brachad-Boud methods are tegrated to solve our SMIP models. I ths work, we apply varable fxg oly o the frst-stage varable c to smplfy the problem. How to apply varable fxg o more varables,.e. those the secod-stage wll be vestgated our future work. 3.1 Framework of Hybrd Local Search ad B&B Algorthm We preset the framework of hybrd Local Search ad B&B (LS-B&B), as show Fg.3. The framework of LS-B&B ca be summarsed as: we perform computatoally expesve local search (le 9) o the core varables, ad the perform default Brach-ad-Boud search (le 7) o the subproblems defed by varable fxg to geerate complete solutos to the orgal problems. These two search compoets work together a overall search procedure (whle loop) to obta solutos to the orgal problems. 11

12 LS- B&B /* LB: lower boud; UB: upper boud; (h, x,): a soluto (x) of the problem wth a correspodg objectve value h; solveb&b: a default B&B solver; C: set of c ; S ad S : two complemetary subsets of C,.e., S = C/ S P org : orgal problem defed by model (PSP); P, P : sub-problems defed by varable fxg; */ sub' y 1: Italzato phase 2: whle (the umber of teratos ot met) // overall search 3: If (LB ( P ) UB) 4: prue the sub-problem P ; 5: go to le 9; 6: Else 7: (h, x) = solveb&b( P ); 10: geerate sub-problems by varable fxg: P = Porg (c= 1), cs; P = P org (c = 0), c S ; sub' y 11: set (x*) as the best soluto amog all (x) ad h* be the correspodg objectve value; Fg. 3 Framework of LS-B&B LS-B&B cossts of four ma compoets. The frst compoet s the talzato phase (le 1). I ths phase, varable fxg s appled to geerate two braches of sub-problems. Lower boud ad upper boud of the problem are also talzed ths phase. The secod compoet s a default B&B search (le 7). It s called to solve the sub-problems to optmalty. From ths, the solutos ad the objectve value of the orgal problem ca be obtaed. The thrd compoet s a local search (le 9) whch s performed o set C of varable c to update sets S ad S. Wth the updated S ad S, the sub-problems are updated correspodgly. Therefore, we state that ths local search geerates a sequece of sub-problems. The fourth compoet s a overall search procedure (whle loop), whch we ame as Local Search B&B. I ths search, a local search (.e., the thrd compoet) s appled to geerate a sequece of subproblems. Ths Local Search B&B search cludes the prug of feror sub-problems ad solvg the promsg sub-problems to optmalty. We preset explaatos of key compoets ext. 3.2 Varable fxg 8: f h <UB, set UB =h ; 9: perform a Local search o the set C; (Hard) varable fxg has bee used MIP to dvde a problem to sub-problems (Bxby, Feelo et al. 2000). It assgs values to a subset of varables of the orgal problem (Bxby, Feelo et al. 2000, Lazc, Haaf et al. 2009). That s, certa varables are fxed to gve values. For a gve orgal MIP problem deoted as follows: P org T : m c x s. t. Ax b; x {0,1}, j B j x [0,1], j C j where x s the vector of varables. x are parttoed to two subsets: B correspods to the set of bary varables ad C correspods to the set of cotuous varables. 12

13 Varable fxg, as ts ame suggests, fx a restrcted subset S of the bary varable to certa values, e.g. 1. The we have a reduced problem P r as follows: T P : m c x r s. t. Ax b; x 1, j S B j x {0,1}, j B \ S j x [0,1], j C j We deote ths problem as a reduced problem because certa bary varables (.e., the varables set S) are already fxed. Note that the varables outsde the restrcted subset S,.e., varables B/S, are stll free bary varables. We apply varable fxg to smplfy the orgal problem to sub-problems by fxg varables the two complemetary subsets S ad S, S =B/S, to 1 ad 0, respectvely, to obta two sub-problems P ad Psub' y as the followg: P suby T : m c x s. t. Ax b; x 1, j S B j x {0,1}, j S ' B j x [0,1], j C j P sub' y T : m c x s. t. Ax b; x 0, j S ' B j x {0,1}, j S B j x [0,1], j C j 3.3. Italzato phase The ma task of the talzato phase, show Fg.4, s the geerato of two sub-problems ad that Psub' y by varable fxg o varables c o sets S ad S. From the defto of P P s the P org wth the talzato of varables S to 1. Ad Psub' y talzato of varables S to 0. Italzato phase R: cotuous relaxato of the frst-stage problem; solve: a quadratc program solver; C: set of bary varables c ; S, S : two complemet sets of C, S = C/S; k: the umber of assets allowed the portfolo, as defed the model (SMIP); P org : the orgal problem defed o model SMIP; 1: solve the cotuous relaxato problem R(P org ): solve (R(P org )); 2: sort the assets accordg to a sort rule; 3: cosder the sorted assets to geerate sub-problems by varable fxg: 4: select the frst k assets ad add them to set S; 5: geerate two braches of sub-problems by varable fxg: 6: P = P sub org (c = 1), c S; y 7: = P org (c = 0), c S ; 8: obta lower boud of sub-problem 9: set UB = ; P, dscard P : LB ( P sub' y )= solve( P ); P, we ca state s the P org wth the Fg.4. Italzato phase of Local Search B&B approach I the les 1 ad 2 of the talzato phase, a heurstc s appled to geerate a good frst subproblem. Ths heurstc s based o the frst-stage problem of SMIP model where the secod-stage 13

14 recourse varables ad costrats are dropped off. The teger decso varables c are relaxed as cotuous oes. I (Mas ad Speraza 2005), t has bee show that, the set of assets selected by the cotuous relaxato problem ofte cotas the assets whch are cluded the teger optmal soluto. Based o ths, we apply a heurstc to select the frst subset S to costruct the frst sub-problem P as the followg: 1. Relaxed problem soluto: solve the cotuous relaxed frst-stage problem. Save the soluto vector w; 2. Sort o the assets: sort the assets a o-decreasg order of the reduced cost of w of the cotuous relaxato to model (Mas ad Speraza 2005); 3. Select the assets: select the frst k assets of the soluto vector w. These assets form the frst subproblem. Ths heurstc s appled le 2 the algorthm preseted Fg. 4 to esure a proper selecto of the frst sub-problem. 3.4 Local search to geerate a sequece of sub-problems A local search s performed o the set C of varables c to geerate ew value assgmet for each varable c (see pseudo code Fg. 5). Each move of the local search updates the subsets S ad S thus a sequece of sub-problems s created. Theoretcally, ay local search techque ca be appled to search o c. I ths paper, we apply a varato of Varable Neghbourhood Search (VNS) (Hase, Mladeovc et al. 2001) to carry out the search o c. The evaluato value of soluto c ad ts eghbour are calculated by the cotuous relaxato soluto of the problems. Local search o Z c: curret assgmet of c e: evaluato value of a soluto: e = solve(r( )); 1: Select the set of eghbourhood structures N l, l = 1,, l max ; 2: Provde a tal soluto vector c (c represets the assgmet of z ) 3: Repeat the followg steps for t 1 teratos: 4: set l =1; 5: Repeat the followg steps for t 2 teratos: 6: Explorato of the eghbourhood N l of z wth the am to update the assgmet of c : Fd the frst mproved eghbour c of c; 7: Move or ot. If the ew soluto c s better tha z term of e, set c = c ; otherwse, set l = l+1; Fg.5 Steps of VNS local search Three eghbourhood structures N l are employed the algorthm. N 1 swaps oe par of elemets S ad S. N 2 ad N 3 swap two ad three pars of elemets, respectvely. For each curret eghbourhood structure N l, a gve umber of t 2 teratos are carred out before the search moves to the ext eghbourhood structure N l+1. Ths procedure termates after t 1 teratos. Therefore, Local Search B&B s a complete search. It ams to seek ear optmal solutos a lmted computatoal tme. 14

15 3.5 The overall search procedure o the sequece of sub-problems The overall search explores o ths sequece of sub-problems. Ths s show the whle loop Fg.3. I ths search, the lower boud of the sub-problem P s computed by a geeral LP solver, whch relaxes the sub-problem to a cotuous problem (le 3 Fg.3). The objectve value of the feasble soluto to the cocered sub-problem P serves as the upper boud of the orgal problem. If the lower boud of a sub-problem s above the curret upper boud foud so far, we ca prue ths sub-problem durg the search (le 4 Fg.3). Otherwse, these sub-problems are solved exactly by a default B&B a IP solver (le 7 Fg.3). The solutos to the sub-problems together wth the assgmets of core varables cosst of the feasble solutos to the complete orgal problem. These sub-problems are solved sequece ad the best soluto amog them approxmates the optmal soluto to the orgal problem. The whole procedure termates by termatg the geerato of sub-problems by the local search. Therefore, the search s a complete search. It caot guaratee optmalty of the soluto due to the ature of the local search o core varables c. 4. Expermetal results We mplemet our two-stage stochastc mxed teger program model C++ wth cocert techology CPLEX o top of CPLEX12.3 solver o a set of bechmark staces. The ams of our expermets are: frstly, to vestgate the effcecy of the proposed hybrd method for geeratg good solutos a reasoable computatoal tme; ad secodly, to evaluate the performace of the two-stage stochastc model. 4.1 Test problems I ths paper, we vestgate the mult-perod PSP wth a comprehesve set of real-world tradg costrats, together wth market ucertaty. We evaluate our algorthm o fve geeral bechmark staces (see Table 2) whch are publcly avalable the OR lbrary at We exted the problems wth addtoal costrats derved from a real-world problem at Socété Géérale Corporate & Ivestmet Bak. We use the 261 weekly hstorcal prce data for each asset to obta scearos for the SMIP model. We geerate market asset prce scearos for the SMIP model usg the framework preseted (Stoya ad Kwo 2010) based o the 261 weekly hstorcal prce data for each asset. A smlar approxmato techque s used (Jobst, Mtra et al. 2006). Table.2 Propertes of problem staces. m : the total umber of assets avalable; k : the umber of assets to be held. Istace m k Hag Seg DAX FTSE S&P Nkke We set the mmum proporto of wealth to be vested a asset, w m, to 0.01% ad the mmum trasacto amout, x m, to 0.01%. We also set the parameters the trasacto cost fucto ß to 0.01 for all the assets. Other values of k the cardalty costrat have bee tested, ragg from 10 to 150 for dfferet szes of portfolos (see Table 2). The tal portfolo oly volves cash the based currecy of the dex market, ad the model wll vest them to the avalable assets. We set the mmum expected retur, µ, as The crtcal percetle level of CVaR portfolo losses s set as 95%. 15

16 4.2 Evaluatos o hybrd LS-B&B algorthm I ths secto, we aalyse our proposed hybrd local search ad B&B algorthm for SMIP from dfferet aspects, cludg sub-problem solvg ad overall problem solvg. The hybrd local search ad B&B algorthm s mplemeted C++ wth cocert techology CPLEX o top of CPLEX12.3 solver. All expermets have bee carred out o a Itel Core Duo mache wth 3.16GHz 3.17GHz ad 2.97GB memory The sze of the orgal problem ad sub-problems Frst, we evaluate the effectveess of the problem smplfcato of the varable fxg o the SMIP model. The orgal problem s SMIP ad sub-problems are stll MIP due to the bary varables f, g. We compare the sze of the orgal problem ad sub-problems after the varable fxg Table 3. Due to the varables eeded to defe each scearo at each ode, the sze of the orgal problem s very large. After fxg the values for all varables c by varable fxg, the resultg sub-problems are much smaller tha the orgal oes. We ca also observe that the sze of the sub-problems are smlar (Table 3 presets the approxmate average sze of sub-problems). Note that ths work we am to reduce the computatoal tme of solvg the problem by heurstcally smplfyg the orgal problem. As the problem sze s reduced by usg varable fxg o c, the sub-problems ca be solved effcetly by the default B&B CPLEX 12.3 our expermets. Table 2. Sze of the orgal SMIP problem ad the approxmate average sze of SMIP sub-problems Istace Orgal problem Sub-problem No. of rows No. of colums No. of ozeros No. of rows No. of colums No. of ozeros Hag Seg DAX FTSE S&P Nkke Computatoal alyss of the sub-problem solvg I ths secto, we aalyse the deferet performace (.e., CPU tme sped) of the sub-problem solvg. I the LS-B&B algorthm Fg.3, after the smplfcato, the sub-problems are solved by default B&B. Wth the calculato of the lower boud ad upper boud the search, some subproblem ca be prued. Actually, whe these sub-problems are processed, four possble stuatos could emerge: (1) a sub-problem could be solved by B&B to optmalty; (2) the reparg heurstc mbedded CPLEX could be evoked ad appled to a sub-problem to obta a feasble soluto heurstcally; (3) a sub-problem could be prued; ths wll happe f the optmal soluto uder LP relaxato s larger tha the curret upper boud; ad (4) the soluto of a sub-problem could be feasble. Table 4. Computatoal aalyss o sub-problem processg. Istace Avg CPU tme per sub-p for sub-problem solved Avg CPU tme per sub-p for sub-problem repared Avg CPU tme per sub-p for sub-problem prued Avg CPU tme per sub-p for sub-problem feasble Hag Seg DAX FTSE S&P Nkke

17 Table 4 clearly dcates that the CPU tme for detfyg feasblty s eglgble. The more suproblems prued, the more effcet the search s. The CPU tme for prug the feror sub-problem (by calculatg ts optmal soluto of the LP relaxato) s qute effcet. 4.3 Evaluatos o SMIP model solutos I Secto 4.2, we aalyse the proposed LS-B&B from the aspects of solvg the sub-problems ad the overall problem. I ths secto, we exame the performace of the SMIP model ad solutos geerated by the proposed LS-B&B statc as well as dyamc tests Statc tests o SMIP model solutos It s worth otg that LS-B&B s a heurstc approach. It caot prove optmalty of the soluto to the overall problem due to the ature of the local search o core varables z, although the subproblems ca be measured by a optmalty gap. I order to evaluate the qualty of the solutos we obtaed from LS-B&B, we compare t agast the approxmate optmal soluto to the problem. It s very dffcult, f ot mpossble, to obta ad prove the optmal solutos to the problems cocered. We therefore calculate the approxmate optmal soluto to the problem cocered by rug the default B&B algorthm CPLEX12.3 for a extesve amout of tme. We ru our hybrd algorthm ad default B&B CPLEX o the same staces wth the parameters settg stated above (w m, x m,µ etc.), for a gve same expected retur µ. We calculate the CVaR at level =95% of portfolo loss. We preset the results of hybrd LS-B&B comparso wth the default B&B CPLEX 12.3 ad cosder the soluto qualty ad CPU tme over the test staces. The two methods solve the same model, that s, employg the same rsk measure (CVaR of the portfolo loss), startg wth the same tal portfolo. We plot the resultg rsk-retur effcet froters of the two methods Fg. 6 for the Hag Sed stace. From Fg. 6, we ca see that the soluto obtaed by the hybrd method domates the soluto of the default CPLEX method. For ay expected retur target, the hybrd method ca obta better soluto where rsk s lower tha the soluto from CPLEX. Fg. 6 Effcet froters from the hybrd LS-B&B ad pure B&B CPLEX I Fg. 6, we have computed the froter by settg the expected retur from 0.01 to 0.06 wth step of 0.01 (sx portfolos are thus plotted o the approxmate effcet froter). Lear terpolato s used to plot the termedate values. For each pot o the froter, the CPU tme eeded to obta the 17

18 solutos Fg. 6 s preseted Table 5. For larger staces t s very tme cosumg to plot eve a sgle pot o the froters. For example, CPLEX eeds 7258 secods to plot oe portfolo. The hybrd LS-B&B sgfcatly outperforms the default CPLEX all of the staces. I the worst case, CPLEX obtas a feasble soluto secods whle the hybrd method oly eeds secods. Table 5. Comparso o CPU tme betwee the hybrd LS-B&B method ad CPLEX ( secods) Dyamc tests o portfolo retur values Istace Hybrd LS-B&B CPLEX Hag Seg DAX FTSE S&P Nkke The statc test results Fg. 6 preset formato o the soluto qualty at certa pots of tme the plag horzo. However, practce, vestors seldom make decsos ad evaluate the costructed portfolos based o a sgle statc result. We ru dyamc tests to evaluate the costructed portfolos. Ths dyamc test reflects formato o the soluto qualty a more practcal maer. I (Topaloglou, Vladmrou et al. 2008), authors ra dyamc tests to assess the performace of the models backtrackg smulatos. The key dea of the test method s to compare the results (.e. retur values) that have bee geerated by the model wth the actual retur values. Ths s a wdelyused method, wth several varats, to evaluate algorthm performace the Stochastc Programmg lterature (Flete, Høylad et al. 2002, Stoya ad Kwo 2010, Stoya ad Kwo 2011). I ths secto, we adapt the same dea to evaluate solutos geerated by the proposed LS-B&B for the twostage SMIP model. Oe of the methods a vestor evaluates the performace of a portfolo s comparg the portfolo values agast a well-kow bechmark dex. Ths s called dex trackg, a well-kow passve vestmet strategy (Caakgoz ad Beasley 2009). I ths secto, we llustrate the performace ad stregths of the proposed stochastc model by presetg the obtaed portfolo values agast the actual market dex value. I Fg 7, for each dex of the fve world-wde dex markets, we have 291 weekly (5 years) dex values (sold les). Our SMIP model was ru, o a rollg horzo bass, at tervals of every sx moths. Startg from week oe, the model s ru to decde the assets ad ther postos the portfolo, as well as the value of the portfolo. The tme moves forward 6 moths ad the model s ru aga to obta ew portfolos. Therefore, we ca plot 10 pots of the costructed portfolo values (wth sx-moth tervals over 5 years), obtaed by the SMIP model. From Fg. 7 we ca see that some cases, the costructed portfolos by the SMIP model outperform the market dex value. I some cases t has dffculty to match the dex values. However, the portfolo cotas a low umber of assets (k<m) ad t s ot very susceptble to market fluctuatos. I order to see more clearly the fluctuatos of the costructed portfolos, we use a movg smooth average tred le over the 10 values of the costructed portfolos. We average every two cosecutve pots to draw the tred le (dotted les), whch clearly shows that the costructed portfolo values match the market dex wth less holdg of assets (k<m). 18

19 Fg. 7 Portfolo values from the SMIP model agast the market dex values I order to observe closer detals Fg. 7, we preset the detaled data Table. 6. The left colum V presets the absoluto dfferece betwee the dex value dex Vportfolo ad portfolo value. The rght colum presets the percetage dfferece betwee the dex value ad portfolo value, from whch we ca see that for three staces (Hag Seg, DAX ad S&P), our method obtaed slghtly better solutos o average. For stace FTSE, the portfolo values are worse tha the dex values, however at a eglgble dfferece. Table 6.Comparsos betwee portfolo value from the SMIP model ad dex value Istace Absolute Value(= V portfolo V dex ) Percetage Value(= ( Vportfolo Vdex ) / V ) dex Best Worst Average S.D Best Worst Average S.D. Hag Seg DAX

20 FTSE S&P Nkke The evaluato method appled above,.e. comparg the retur values obtaed from the model agast actual market values, s a reasoable ad well-justfed method for the followg reasos: (1) Comparg the results obtaed from the model wth the actual market values s covcg sce facts speak for themselves. (2) The portfolo selecto problem s oe of the most studed topcs face. A wde rage of models have bee proposed to tackle the problems. Dfferet varable deftos, objectve fuctos, costrats, ad data sets have bee proposed. It s therefore dffcult to coduct far ad exhaustve comparsos of all the publshed work. What s more, to the best of our kowledge, our two-stage SMIP model wth the comprehesve set of costrats, as well as ucertates by employg scearos, s preseted the lterature for the frst tme. There s o exstg work we could coduct comparsos agast our hybrd LS-B&B approach o the same data. (3) The ature of SP models,.e. the radomess, make t dffcult or eve mpossble to make parallel comparg,.e., comparg wth other methods the lterature. 5. Coclusos I ths paper, we vestgate a mult-perod portfolo selecto problem uder radom ucertaty term of asset prce ad wth a comprehesve set of real-world costrats. We formulate the problem to a two-stage stochastc mxed-teger program, ad apply a hybrd method, where a smplfcato method ad a local search wth B&B are employed. The hybrd local search ad B&B method ca obta good solutos for the hghly computatoal expesve problem less computatoal tme comparg wth default B&B. The stochastc mxed-teger programmg model maages the rsk by mmzg the codtoal value at rsk (CVaR) of portfolo loss. We ru statc ad dyamc expermets to vestgate the performace of the model. We demostrate that the costructed portfolo value ca match the market dex wth holdg of less umber of assets. The stochastc model provdes a flexble framework to mult-perod portfolo optmzato. I ths work, we test ad aalyse the performace of the two-stage model. I our future work, we wll aalyse qualtatvely what ways the optmal soluto chages as the ucertaty s troduced. We wll also vestgate alteratve objectve fuctos to reflect vestors prefereces ad corporate addtoal practcal costrats. Refereces Arott, R. D. ad W. H. Wager (1990). The measuremet ad cotrol of tradg costs. Facal aalusts joural 4(6): Artzer, P., F. Delbae, J. M. Eber ad D. Heath (1999). Coheret measures of rsk. Mathematcal Face 9(3): Baldacc, R., M. A. Boschett, N. Chrstofdes ad S. Chrstofdes (2009). Exact methods for large-scale mult-perod facal plag problems. Computatoal Maagemet Scece 6(3): Barro, D. ad E. Caestrell (2005). Dyamc portfolo optmzato: Tme decomposto usg the Maxmum Prcple wth a scearo approach. Europea Joural of Operatoal Research 163(1): Bestock, D. (1996). Computatoal study of a famly of mxed-teger quadratc programmg problems. Mathematcal Programmg 74(2): Brge, J. R. (1985). Decomposto ad parttog methods for multstage stochastc lear programs. Operatos Research 33(5):

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