A Stochastic Programming Approach For Multi-Period Portfolio Optimization With Transaction Costs

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1 SOUTHEAST EUROPE JOURNAL OF SOFT COMPUTNG Avalable ole at A Stochastc Programmg Approach For Mult-Perod Portfolo Optmzato Wth Trasacto Costs Mehmet Ca a, Narela Bajram b a Faculty of Egeerg, teratoal Uversty of Sarajevo, Hrasčka Cesta 15, Sarajevo Bosa ad Herzegova, mca@us.edu.ba b Faculty of Maagemet, teratoal Uversty of Sarajevo, Hrasčka Cesta 15, Sarajevo Bosa ad Herzegova, arela@us.edu.ba Abstract Ths paper uses stochastc programmg to solve mult-perod vestmet problems. We combe the feature of asset retur predctablty wth practcally relevat costrats arsg a mult-perod vestmet cotext. The objectve s to maxmze the expected utlty of the returs the perods to balace the labltes. Asset returs ad state varables follow a frstorder vector auto-regresso ad the assocated ucertaty s descrbed by dscrete scearo trees. To deal wth the log tme tervals volved mult-perod problems, we cosder short-term decsos, ad corporate a soluto for the log, subsequet steady-state perod to accout for ed effects. Keywords Portfolo optmzato, mult- perod asset allocato, stochastc programmg, Scearo trees, trasacto costs 25

2 NTRODUCTON The classcal treatmets of strategc asset allocato ca be traced back to Samuelso (1969) ad Merto (1969, 1971). the lght of Markowtz semal papers o sgle-perod portfolo selecto, the early lterature focused o codtos leadg to the optmalty of myopc polces,.e., codtos uder whch portfolo decsos for mult-perod problems cocde wth those for sgle perod problems. addto, the lack of computg power lead to formulate models drve by the quest for closed form solutos. To acheve these objectves, rather restrctve assumptos were made, ad may of these models results tured out to be cosstet wth covetoal wsdom as expressed by the so-called Samuelso puzzle: whereas oe of the ma results from early mult-perod portfolo models s that the fractos of rsky assets are costat over tme, ths cotradcts the advce obtaed from may professoals practce that vestors should hold a share of rsky assets whch decles steadly as they approach retremet (ofte called the age effect). Sce the, may researchers have tred to resolve ths puzzle whch s maly rooted some of the (smplfyg) assumptos used early models (fxed plag horzo, tme-costat vestmet opportutes, o termedate cosumpto, etc.). Research the area of lfe-cycle asset allocato models regaed mometum the early 1990s for two ma reasos: frst, a umber of ecoomc factors creased the umber of people wth szeable wealth to vest (the geerato of hers ), coupled wth creased ucertaty about the securty of publc peso systems. Secod, the eormous crease computer power eabled the soluto of models wth more realstc assumptos. A umber of addtoal features have bee added to the classcal models, may cases wth the goal of resolvg the Samuelso puzzle: stochastc labor come, tme-varyg vestmet opportutes, parameter ucertaty (wth ad wthout learg), specal treatmet of certa asset classes (real estate), ad habt formato, to ame just the most mportat developmets. cotrast to other approaches the lterature usg olear optmzato (see, e.g., Blomvall ad Ldberg 2002; Godzo ad Grothey 2007), we use mult-perod stochastc lear programmg (SLP) to solve the problem of optmal lfe-cycle asset allocato ad cosumpto. Ths method has bee explctly chose wth the practcal applcato of the approach md. May features whch are cosdered mportat for vestmet decsos practce ca be easly corporated whe usg SLP. For example, persoal characterstcs of the vestor ca be take to accout (e.g., mortalty rsk, rsk atttude, retremet, future cash flows for major purchases or assocated wth other lfe evets). Combed wth the avalablty of effcet solvers, ths explas why the SLP approach has bee successfully appled to a wde rage of problems (see, e.g., Wallace ad Zemba 2005). To est classcal aalytcal results from ths area wth our model, we maxmze expected utlty of cosumpto over the vestor s lfetme ad expected utlty of bequest rather tha other objectves whch ca be mplemeted more easly (e.g., pecewse lear or quadratc pealty fuctos, or mmzg CVaR). The paper s orgazed as follows: Sect. 2 we provde a classfcato of the more recet lfe-cycle asset allocato models based o the type of avalable solutos. Secto 3 descrbes the stochastc programmg model, partcular the formulato of the objectve, the optmzato approach for ts learzato, ad the geerato of scearos. Sect. 4 results from the SLP are compared to those Campbell et al. (2003), ad results for a exteded settg are preseted. Secto 5 cocludes. OVERVEW OF SOLUTON METHODS May papers try to exted the classc Merto framework alog dfferet les whle matag aalytcal solutos (see, e.g., Bode et al. 1992; Balvers ad Mtchell 1997; Km ad Omberg 1996; Wachter 2002; Lu 2007). Aalytcal solutos are avalable for restrctve assumptos o the utlty structure ad the plag horzo. Aother set of models obta solutos whch are exact oly uder (geerally less strget) assumptos, ad approxmately correct f these assumptos are ot exactly met. some cases, these approxmate solutos are avalable closed form, whle others must be solved umercally. Approxmate aalytcal solutos are provded by, e.g., Campbell ad Vcera (1999, 2001, 2002), Campbell et al. (2004), ad Chacko ad Vcera (2005). Approxmate umercal solutos ca be foud, e.g., Schroder ad Skadas (1999) ad Campbell et al. (2003). To gve some examples for the restrctve assumptos metoed above, a umber of the models from ths category assume ether a determstc or a fte plag horzo. Some of the fte-horzo models defe utlty over termal wealth oly. These assumptos are clearly problematc for dvduals who face a ucerta lfetme ad derve ther utlty maly from what they cosume durg ther lves, ad ot oly from ther bequest. A mportat referece for the preset paper s Campbell et al. (2003). They model asset returs ad state varables as a frst-order vector autoregresso VAR(1) ad cosder Epste Z utlty wth a fte plag horzo. Addtoal assumptos clude the absece of borrowg ad short-sale costrats. Learzg the portfolo retur, the budget costrat, ad the Euler equato, they arrve at a system of lear-quadratc equatos for portfolo weghts ad cosumpto as fuctos of state varables. Ths system of equatos ca be solved aalytcally, yeldg solutos whch are exact oly for a specal case (very short tme tervals ad elastcty of tertemporal substtuto equal to oe), ad accurate approxmatos ts eghborhood. Sect. 4, we replcate ther results as far as possble ad subsequetly exemplfy the applcato of the SLP approach by vestgatg aspects beyod the scope of ther settg, such as costrats o asset weghts, trasacto costs, ad labor come. 26

3 Two ma types of umercal soluto methods ca be foud the lterature: Oe approach works va grd methods dscretzg the state space, the other s based o Mote Carlo smulato. Grd dscretzatos are used, amog others, Brea et al. (1997), Barbers (2000), Campbell et al. (2001), Cocco et al. (2005), ad Gomes ad Mchaeldes (2005). The ma drawback of ths approach s that the reducto the state-space dmesoalty, whch s crucal for the soluto terms of computato tme, requres to restrct the vestmet opportuty set (usually to oe rsky ad oe rskless asset). Ths may be approprate for may vestors. Detemple et al. (2003) ad Bradt et al. (2005) use smulato-based approaches.detemple et al. approxmate devatos from a closed-form soluto, whle Bradt et al. provde a approach that s spred by the opto prcg algorthm by Logstaff ad Schwartz (2001). The SLP used the preset paper has bee appled successfully to a umber of related problems. To cte oly a few examples, there are applcatos surace (Carño ad Zemba 1994, 1998; Carño et al. 1998), ad the peso fud dustry (e.g., Godzo ad Kouweberg 2001). Zeos (1999) surveys large-scale applcatos of SLP to fxed come portfolo maagemet. Geeral aspects of applyg such models a strategc asset allocato cotext are dscussed Zemba ad Mulvey (1998)), Pflug ad Swetaowsk (2000), Godzo ad Kouweberg (2001), Wallace ad Zemba (2005), ad Geyer ad Zemba (2007). Partcular aspects that are relevat a lfe-cycle portfolo cotext are dscussed Geyer et al. (2007). A MULTSTAGE MODEL: ASSET- LABLTY MANAGEMENT The best way to troduce multstage stochastc models s a smple asset lablty maagemet (ALM) model (Brge, ad Louveaux 1967). We have a tal wealth W 0, that should be properly vested such a way to meet a lablty L at the ed of the plag horzo H. f possble, we would lke to ow a termal wealth W H larger tha L; however, we should accout properly for rsk averso, sce there could be some chace to ed up wth a termal wealth that s ot suffcet to pay for the lablty, whch case we wll have to borrow some moey. A olear, strctly cocave utlty fucto of the dfferece betwee the termal wealth W H, whch s a radom varable, ad the lablty L would do the job, but ths would lead to a olear programmg model. As a alteratve, we may buld a pecewse lear utlty fucto lke the oe llustrated Fg Fg. 1 Pecewse lear utlty fucto The utlty s zero whe the termal wealth W H matches the lablty exactly. f the slope r pealzg the shortfall s larger tha q, ths fucto s cocave (but ot strctly). The portfolo cossts of a set of assets. For smplcty, we assume that we may rebalace t oly at a dscrete set of tme stats t = 1,..., H-1, wth o trasacto cost; the tal portfolo s chose at tme t = 0, ad the lablty must be pad at tme H. Tme perod t s the perod betwee tme stats t - 1 ad t. order to represet ucertaty, we may buld a tree lke that Fg. 1 (Høylad K, Wallace SW 2001, Pflug GC 2001). Each ode k the tree correspods to a evet, where we should make some decso. We have a tal ode 0 correspodg to tme t = 0. Fg. 2 Scearo tree for a smple asset- lablty maagemet problem. The, for each evet ode, we have two braches; each brach s labeled by a codtoal probablty of occurrece, P( k ), where = a( k ) s the mmedate predecessor of ode k. Here, we have two odes at tme t = 1 ad four at tme t = 2, where we may rebalace our portfolo o the bass of the prevous asset returs. Fally, the eght odes correspodg to t = 3, the leaves of the tree, we just compare the termal wealth wth the lablty ad evaluate the utlty fucto. Each ode of the tree s assocated wth the set of asset returs durg the correspodg tme perod. A scearo cossts of a evet sequece,.e., a sequece of odes the tree, alog wth the assocated asset returs. We have 8 scearos Fg. 2. For stace, scearo 2 cossts of the ode sequece 3, 4, 5, 6. The probablty of each scearo depeds o the codtoal probablty of each ode o ts path (Samuelso PA 1969). f each brach at each ode s equprobable,.e., the codtoal probabltes are always, each scearo the fgure has probablty π s = 1/8, for s = 1,...,8. The brachg factor may be arbtrary prcple; the more braches we use, 27

4 the better our ablty to model ucertaty; ufortuately, the umber of odes grows expoetally wth the umber of stages, as well as the computatoal effort. At each ode the tree, we must make a set of decsos. practce, we are terested the decsos that must be mplemeted here ad ow,.e., those correspodg to the frst ode of the tree; the other (recourse) decso varables are strumetal to the am of devsg a robust pla, but they are ot mplemeted practce, as the multstage model s solved o a rollg-horzo bass. Ths suggests that, order to model the ucertaty as accurately as possble wth a. lmted computatoal effort, a possble dea s to brach may paths from the tal ode, ad less from the subsequet odes. Each decso at each stage may deped o the formato gathered so far, but ot o the future; ths requremet s called a. oatcpatvty codto. Essetally, ths meas that decsos made at tme t must be the same for scearos that caot be dstgushed at tme t. To buld a model esurg that the decso process makes sese, there are two choces: We ca troduce a set of decso varables x t s, represetg wealth allocated to asset at tme t o scearo s; we should force decso varables to take the same value whe approprate, by wrtg explct oatcpatvty costrats for scearos that caot be dstgushed at tme t. We ca assocate decso varables wth odes the scearo trees ad wrte the model a. way that relates each ode to ts predecessors. A TWO-STAGE, THREE ASSETS MODEL: ASSET- LABLTY MANAGEMENT We wll llustrate the secod alteratve detal, usg the followg umercal data: The tal wealth s W 0 = 50. The target lablty s L s = 100. There are three assets, say, stocks A ad B, ad bods; hece, = 3. the scearo tree of Fg. 2 we have up - ad dowbraches; the (lucky) upbraches, total retur s 1.28 for stocks A, 1.40 for stocks B ad 1.20 for bods; the (bad) dowbraches, total retur s 1.08 for stocks A, 0.99 for stocks B ad 1.12 for bods (Barbers NC 2000). We see that bods play the role of safer assets, ad stocks B are very rsky assets here. We also see that returs are a sequece of..d. radom varables, but more realstc scearos ca be defed. The reward rate q for excess wealth above the target lablty s 1. The pealty rate r for the shortfall below the target lablty s 4. Let us troduce the followg otato: N s the set of evet odes; our case N = { 0, 1, 2,, 14 } Each ode N, 0, apart from the root ode 0, has a uque drect predecessor ode, deoted by a(): for stace, a( 3 ) = 1. There s a set S N of leaf (termal) odes; our case S = { 7, 8, 9,, 14 }, For each ode s S we have surplus ad shortfall varables w + s ad w s, related to the dfferece betwee termal wealth ad lablty. There s a set T N of termedate odes, where portfolo rebalacg may occur after the tal allocato ode o; our case T = { 1, 2,, 6 } For each ode T 0 there s a decso varable x, expressg the moey vested asset at ode. Wth ths otato, the model may be wrtte as follows: max π s (qw s s s S + rw ) (1) Such that x 0 =1 = W 0 (2) R a() =1 x = =1 x, T (3) R s a(s) =1 x = L s + w s + w s, s S (4) x, w + s, w s 0 (5) where R s the total retur for asset durg the perod that leads to ode, ad π s s the probablty of reachg the termal ode s S ; ths probablty s the product of all the codtoal probabltes o the path that leads from root ode o to leaf ode s. Let us choose a olear utlty fucto, such that the objectve of the optmzato problem becomes: π s ((w + s ) 2 (w s ) 2 ) s S (7) Fg. 3 The olear utlty fucto ths case we get a better dversfcato: Table 1. vestmet strategy for a smple ALM problem wth olear utlty fuctos. Node Stocks A Stocks B Bods

5 ASSET-LABLTY MANAGEMENT WTH TRANSACTON COSTS To gve the reader a dea of how to buld otrval facal plag models, we geeralze a bt the model formulato of the prevous secto, order to accout for proportoal trasacto costs. The assumptos ad the lmtatos behd ths exteded model are the followg: We are gve a set of tal holdgs for each asset; ths s a more realstc assumpto, sce we should use the model to rebalace the portfolo perodcally, accordg to a rollg-horzo strategy. We take proportoal (lear) trasacto costs to accout; the trasacto cost s a percetage c of the traded value, for both buyg ad sellg a asset. We wat to maxmze the expected utlty of the termal wealth. There s a stream of ucerta labltes that we have to meet. We do ot cosder the possblty of borrowg moey; we assume that all of the avalable wealth at each rebalacg perod s vested the avalable assets; actually, the possblty of vestg a rskfree asset s mplct the model. We do ot cosder the possblty of vestg ew cash at each rebalacg date (as would be the case, e.g. for a peso fud). Some of the lmtatos of the model may easly be relaxed. The mportat pot we make s that whe trasacto costs are volved, we have to troduce ew decso varables to express the amout of assets (umber of shares, ot the moetary value) held, sold, ad bought at each rebalacg date. We use a otato whch s smlar to that used the prevous ALM formulato: N s the set of odes the tree; 0 s the root ode. The (uque) predecessor of ode N, 0, s deoted by a(); the set of termal odes s deoted by S; as the prevous formulato, each of these odes correspods to a scearo, whch s the sequece of evet odes alog the uque path leadg from 0 to s S, wth probablty π s. T = N\({ 0 } S) s the set of termedate tradg odes. L s the lablty we have to meet ode N; labltes are ode depedet ad stochastc. c s the percetage trasacto cost. h 0 s the tal holdg for asset = 1,..., at the root ode. P s the prce for asset at ode. z s the amout of asset purchased at ode. y s the amout of asset sold at ode. x s the amout of asset we hold at ode, after rebalacg. W s s the wealth at termal ode s S. u(w) s the utlty for wealth W; ths fucto s used to express utlty of termal wealth. O the bass ths otato, we may wrte the followg model: max s S π s u(w s ) (8) Such that x 0 = h 0 + z 0 y 0, = 1,..., (9) x = x a() + z y (10) (1 c) P y (1 + c) P z = L, =1 T 0 (11) W s s = =1 P x a(s) L s, s S (12) x, y, z, W s 0 (13) =1 The objectve (8) s the expected utlty of the termal wealth; f we approxmate ths olear cocave fucto by a pecewse lear cocave fucto, we get a LP problem. Equato (9) expresses the tal asset balace, takg the curret holdgs to accout; the asset balace at termedate tradg dates s take to accout by Eq. (10). Eq. (11) esures that eough cash s geerated by sellg assets order to meet the labltes; we may also revest the proceeds of what we sell ew asset holdgs; ote how the trasacto costs are expressed for sellg ad purchasg. Eq. (12) s used to evaluate termal wealth at leaf odes; ote here that we have ot take to accout the eed to sell assets order to geerate the cash requred by the last lablty; but ths would make oly sese f the whole fud s lqudated at the ed of the plag horzo. f so, we could rewrte Eq. (12) as P s W s = (1 c) =1 x a(s) L s, s S (12 ) practce, we would repeatedly solve the model o a rollghorzo bass, so the exact expresso of the objectve fucto s a bt debatable. The role of termal utlty s just to esure that we are left a good posto at the ed of the plag horzo. Let us choose a olear utlty fucto. The the objectve of the optmzato problem s: s S π s (W s ) 2/3 (15) Fg. 4 The olear utlty fucto ths case we get a better dversfcato: 29

6 Table 2. vestmet strategy for a smple ALM problem wth trasacto costs, ad wth olear utlty fuctos. Node Stocks Bods Wallace SW, Zemba WT (eds) (2005) Applcatos of stochastc programmg. MPS-SAM book seres o optmzato CONCLUSON The most mportat pot s that we have assumed that the labltes must be met. Ths may be a very hard costrat; f extreme scearos are cluded the formulato, t may well be the case that the model above s feasble. Therefore, the formulato should be relaxed a sesble way; we could cosder the possblty of borrowg cash; we could also troduce sutable pealtes for ot meetg the labltes. prcple, we could also requre that the probablty of ot meetg the labltes s small eough; ths leads to chacecostraed formulatos, for whch we refer the reader to the lterature (Campbell JY, Vcera LM 2002, Hetsch H, Römsch W 2003, Hochreter R, Pflug GC 2007, Klaasse P 2002, Lu J 2007, Wallace SW, Zemba WT (eds) 2005). REFERENCES Barbers NC (2000) vestg for the log ru whe returs are predctable. J Face 55: pp Brge J R, Louveaux F (1967) troducto to stochastc programmg, Sprger Verlag, New York: pp Campbell JY, Vcera LM (2002) Strategc asset allocato. Oxford Uversty Press, USA Hetsch H, Römsch W (2003) Scearo reducto algorthms stochastc programmg. Comput Optm Appl 24: pp Hochreter R, Pflug GC (2007) Facal scearo geerato for stochastc mult-stage decso processes as faclty locato problems. A Oper Res 152: pp Høylad K, Wallace SW (2001) Geeratg scearo trees for multstage decso problems. Maage Sc 47(2): pp Klaasse P (2002) Commet o geeratg scearo trees for multstage decso problems. Maage Sc 48: pp Lu J (2007) Portfolo selecto stochastc evromets. Rev Fac Stud 20: pp 1 39 Pflug GC (2001) Optmal scearo tree geerato for multperod facal plag. Math Program 89(2): pp Samuelso PA (1969) Lfetme portfolo selecto by dyamc stochastc programmg. Rev Eco Stat 51: pp